]> https://gitweb.dealii.org/ - dealii.git/commitdiff
Add headers for tests common to all supported AD frameworks
authorJean-Paul Pelteret <jppelteret@gmail.com>
Fri, 22 Mar 2019 09:09:11 +0000 (10:09 +0100)
committerJean-Paul Pelteret <jppelteret@gmail.com>
Sat, 23 Mar 2019 17:40:30 +0000 (18:40 +0100)
24 files changed:
tests/ad_common_tests/helper_scalar_coupled_2_components_01.h [new file with mode: 0644]
tests/ad_common_tests/helper_scalar_coupled_2_components_02.h [new file with mode: 0644]
tests/ad_common_tests/helper_scalar_coupled_2_components_03.h [new file with mode: 0644]
tests/ad_common_tests/helper_scalar_coupled_2_components_04.h [new file with mode: 0644]
tests/ad_common_tests/helper_scalar_coupled_2_components_05.h [new file with mode: 0644]
tests/ad_common_tests/helper_scalar_coupled_2_components_06.h [new file with mode: 0644]
tests/ad_common_tests/helper_scalar_coupled_2_components_07.h [new file with mode: 0644]
tests/ad_common_tests/helper_scalar_coupled_2_components_08.h [new file with mode: 0644]
tests/ad_common_tests/helper_scalar_coupled_3_components_01.h [new file with mode: 0644]
tests/ad_common_tests/helper_scalar_coupled_3_components_01_tapeless_only.h [new file with mode: 0644]
tests/ad_common_tests/helper_scalar_single_component_01.h [new file with mode: 0644]
tests/ad_common_tests/helper_scalar_single_component_02.h [new file with mode: 0644]
tests/ad_common_tests/helper_scalar_single_component_03.h [new file with mode: 0644]
tests/ad_common_tests/helper_scalar_single_component_04.h [new file with mode: 0644]
tests/ad_common_tests/helper_vector_1_indep_1_dep_vars_01.h [new file with mode: 0644]
tests/ad_common_tests/helper_vector_1_indep_2_dep_vars_01.h [new file with mode: 0644]
tests/ad_common_tests/helper_vector_2_indep_1_dep_vars_01.h [new file with mode: 0644]
tests/ad_common_tests/helper_vector_2_indep_2_dep_vars_01.h [new file with mode: 0644]
tests/ad_common_tests/step-44-helper_scalar_01.h [new file with mode: 0644]
tests/ad_common_tests/step-44-helper_vector_01.h [new file with mode: 0644]
tests/ad_common_tests/symmetric_tensor_functions_02.h [new file with mode: 0644]
tests/ad_common_tests/symmetric_tensor_functions_03.h [new file with mode: 0644]
tests/ad_common_tests/symmetric_tensor_functions_04.h [new file with mode: 0644]
tests/ad_common_tests/tensor_functions_02.h [new file with mode: 0644]

diff --git a/tests/ad_common_tests/helper_scalar_coupled_2_components_01.h b/tests/ad_common_tests/helper_scalar_coupled_2_components_01.h
new file mode 100644 (file)
index 0000000..2383e54
--- /dev/null
@@ -0,0 +1,228 @@
+// ---------------------------------------------------------------------
+//
+// Copyright (C) 2016 - 2018 by the deal.II authors
+//
+// This file is part of the deal.II library.
+//
+// The deal.II library is free software; you can use it, redistribute
+// it, and/or modify it under the terms of the GNU Lesser General
+// Public License as published by the Free Software Foundation; either
+// version 2.1 of the License, or (at your option) any later version.
+// The full text of the license can be found in the file LICENSE at
+// the top level of the deal.II distribution.
+//
+// ---------------------------------------------------------------------
+
+
+// Header file:
+// Evaluation of a coupled system (scalar + scalar components)
+// using a helper class
+
+#include <deal.II/base/logstream.h>
+#include <deal.II/base/symmetric_tensor.h>
+#include <deal.II/base/tensor.h>
+
+#include <deal.II/differentiation/ad.h>
+
+#include <deal.II/fe/fe_values_extractors.h>
+
+#include <deal.II/lac/full_matrix.h>
+#include <deal.II/lac/vector.h>
+
+#include <iostream>
+
+#include "../tests.h"
+
+using namespace dealii;
+namespace AD = dealii::Differentiation::AD;
+
+// Function and its derivatives
+template <int dim, typename NumberType>
+struct FunctionsTestScalarScalarCoupled
+{
+  static NumberType
+  psi(const NumberType &s1, const NumberType &s2)
+  {
+    return 2.0 * std::pow(s1, 4) * std::pow(s2, 3);
+  };
+
+  static NumberType
+  dpsi_ds1(const NumberType &s1, const NumberType &s2)
+  {
+    return 8.0 * std::pow(s1, 3) * std::pow(s2, 3);
+  };
+
+  static NumberType
+  dpsi_ds2(const NumberType &s1, const NumberType &s2)
+  {
+    return 6.0 * std::pow(s1, 4) * std::pow(s2, 2);
+  };
+
+  static NumberType
+  d2psi_ds1_ds1(const NumberType &s1, const NumberType &s2)
+  {
+    return 24.0 * std::pow(s1, 2) * std::pow(s2, 3);
+  };
+
+  static NumberType
+  d2psi_ds2_ds1(const NumberType &s1, const NumberType &s2)
+  {
+    return 24.0 * std::pow(s1, 3) * std::pow(s2, 2);
+  };
+
+  static NumberType
+  d2psi_ds1_ds2(const NumberType &s1, const NumberType &s2)
+  {
+    return d2psi_ds2_ds1(s1, s2);
+  };
+
+  static NumberType
+  d2psi_ds2_ds2(const NumberType &s1, const NumberType &s2)
+  {
+    return 12.0 * std::pow(s1, 4) * std::pow(s2, 1);
+  };
+};
+
+template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+void
+test_scalar_scalar_coupled()
+{
+  typedef AD::ScalarFunction<dim, ad_type_code, number_t> ADHelper;
+  typedef typename ADHelper::ad_type                      ADNumberType;
+  typedef typename ADHelper::scalar_type                  ScalarNumberType;
+
+  std::cout << "*** Test variables: Scalar + Scalar (coupled), "
+            << "dim = " << Utilities::to_string(dim) << ", "
+            << "Type code: " << static_cast<int>(ad_type_code) << std::endl;
+
+  // Values computed from the AD energy function
+  ScalarNumberType             psi;
+  Vector<ScalarNumberType>     Dpsi;
+  FullMatrix<ScalarNumberType> D2psi;
+
+  // Function and its derivatives
+  typedef FunctionsTestScalarScalarCoupled<dim, ADNumberType> func_ad;
+
+  // Setup the variable components and choose a value at which to
+  // evaluate the tape
+  const FEValuesExtractors::Scalar s1_dof(0);
+  const FEValuesExtractors::Scalar s2_dof(1);
+  const unsigned int               n_AD_components = 2;
+  ADHelper                         ad_helper(n_AD_components);
+  ad_helper.set_tape_buffer_sizes(); // Increase the buffer size from the
+                                     // default values
+
+  ScalarNumberType s1 = 3.1;
+  ScalarNumberType s2 = 5.9;
+
+  // Configure tape
+  const int  tape_no = 1;
+  const bool is_recording =
+    ad_helper.start_recording_operations(tape_no /*material_id*/,
+                                         true /*overwrite_tape*/,
+                                         true /*keep*/);
+  if (is_recording == true)
+    {
+      ad_helper.register_independent_variable(s1, s1_dof);
+      ad_helper.register_independent_variable(s2, s2_dof);
+
+      const ADNumberType s1_ad = ad_helper.get_sensitive_variables(s1_dof);
+      const ADNumberType s2_ad = ad_helper.get_sensitive_variables(s2_dof);
+
+      const ADNumberType psi(func_ad::psi(s1_ad, s2_ad));
+
+      ad_helper.register_dependent_variable(psi);
+      ad_helper.stop_recording_operations(false /*write_tapes_to_file*/);
+
+      std::cout << "Taped data..." << std::endl;
+      std::cout << "independent variable values: " << std::flush;
+      ad_helper.print_values(std::cout);
+      std::cout << "s1_ad: " << s1_ad << std::endl;
+      std::cout << "s2_ad: " << s2_ad << std::endl;
+      std::cout << "psi: " << psi << std::endl;
+      std::cout << std::endl;
+    }
+  else
+    {
+      Assert(is_recording == true, ExcInternalError());
+    }
+
+  // Do some work :-)
+  // Set a new evaluation point
+  if (AD::ADNumberTraits<ADNumberType>::is_taped == true)
+    {
+      std::cout
+        << "Using tape with different values for independent variables..."
+        << std::endl;
+      s1 = 4.9;
+      s2 = 0.87;
+      ad_helper.activate_recorded_tape(tape_no);
+      ad_helper.set_independent_variable(s1, s1_dof);
+      ad_helper.set_independent_variable(s2, s2_dof);
+    }
+
+  std::cout << "independent variable values: " << std::flush;
+  ad_helper.print_values(std::cout);
+
+  // Compute the function value, gradient and hessian for the new evaluation
+  // point
+  psi = ad_helper.compute_value();
+  ad_helper.compute_gradient(Dpsi);
+  if (AD::ADNumberTraits<ADNumberType>::n_supported_derivative_levels >= 2)
+    {
+      ad_helper.compute_hessian(D2psi);
+    }
+
+  // Output the full stored function, gradient vector and hessian matrix
+  std::cout << "psi: " << psi << std::endl;
+  std::cout << "Dpsi: \n";
+  Dpsi.print(std::cout);
+  if (AD::ADNumberTraits<ADNumberType>::n_supported_derivative_levels >= 2)
+    {
+      std::cout << "D2psi: \n";
+      D2psi.print_formatted(std::cout, 3, true, 0, "0.0");
+    }
+
+  // Extract components of the solution
+  const ScalarNumberType dpsi_ds1 =
+    ad_helper.extract_gradient_component(Dpsi, s1_dof);
+  const ScalarNumberType dpsi_ds2 =
+    ad_helper.extract_gradient_component(Dpsi, s2_dof);
+  std::cout << "extracted Dpsi (s1): " << dpsi_ds1 << "\n"
+            << "extracted Dpsi (s2): " << dpsi_ds2 << "\n";
+
+  // Verify the result
+  typedef FunctionsTestScalarScalarCoupled<dim, ScalarNumberType> func;
+  static const ScalarNumberType                                   tol =
+    1e5 * std::numeric_limits<ScalarNumberType>::epsilon();
+  Assert(std::abs(psi - func::psi(s1, s2)) < tol,
+         ExcMessage("No match for function value."));
+  Assert(std::abs(dpsi_ds1 - func::dpsi_ds1(s1, s2)) < tol,
+         ExcMessage("No match for first derivative."));
+  Assert(std::abs(dpsi_ds2 - func::dpsi_ds2(s1, s2)) < tol,
+         ExcMessage("No match for first derivative."));
+  if (AD::ADNumberTraits<ADNumberType>::n_supported_derivative_levels >= 2)
+    {
+      const ScalarNumberType d2psi_ds1_ds1 =
+        ad_helper.extract_hessian_component(D2psi, s1_dof, s1_dof);
+      const ScalarNumberType d2psi_ds2_ds1 =
+        ad_helper.extract_hessian_component(D2psi, s1_dof, s2_dof);
+      const ScalarNumberType d2psi_ds1_ds2 =
+        ad_helper.extract_hessian_component(D2psi, s2_dof, s1_dof);
+      const ScalarNumberType d2psi_ds2_ds2 =
+        ad_helper.extract_hessian_component(D2psi, s2_dof, s2_dof);
+      std::cout << "extracted D2psi (s1,s1): " << d2psi_ds1_ds1 << "\n"
+                << "extracted D2psi (s1,s2): " << d2psi_ds2_ds1 << "\n"
+                << "extracted D2psi (s2,s1): " << d2psi_ds1_ds2 << "\n"
+                << "extracted D2psi (s2,s2): " << d2psi_ds2_ds2 << "\n"
+                << std::endl;
+      Assert(std::abs(d2psi_ds1_ds1 - func::d2psi_ds1_ds1(s1, s2)) < tol,
+             ExcMessage("No match for second derivative."));
+      Assert(std::abs(d2psi_ds2_ds1 - func::d2psi_ds2_ds1(s1, s2)) < tol,
+             ExcMessage("No match for second derivative."));
+      Assert(std::abs(d2psi_ds1_ds2 - func::d2psi_ds1_ds2(s1, s2)) < tol,
+             ExcMessage("No match for second derivative."));
+      Assert(std::abs(d2psi_ds2_ds2 - func::d2psi_ds2_ds2(s1, s2)) < tol,
+             ExcMessage("No match for second derivative."));
+    }
+}
diff --git a/tests/ad_common_tests/helper_scalar_coupled_2_components_02.h b/tests/ad_common_tests/helper_scalar_coupled_2_components_02.h
new file mode 100644 (file)
index 0000000..530e3c9
--- /dev/null
@@ -0,0 +1,233 @@
+// ---------------------------------------------------------------------
+//
+// Copyright (C) 2016 - 2018 by the deal.II authors
+//
+// This file is part of the deal.II library.
+//
+// The deal.II library is free software; you can use it, redistribute
+// it, and/or modify it under the terms of the GNU Lesser General
+// Public License as published by the Free Software Foundation; either
+// version 2.1 of the License, or (at your option) any later version.
+// The full text of the license can be found in the file LICENSE at
+// the top level of the deal.II distribution.
+//
+// ---------------------------------------------------------------------
+
+
+// Header file:
+// Evaluation of a coupled system (vector + scalar components)
+// using a helper class
+
+#include <deal.II/base/logstream.h>
+#include <deal.II/base/symmetric_tensor.h>
+#include <deal.II/base/tensor.h>
+
+#include <deal.II/differentiation/ad.h>
+
+#include <deal.II/fe/fe_values_extractors.h>
+
+#include <deal.II/lac/full_matrix.h>
+#include <deal.II/lac/vector.h>
+
+#include <iostream>
+
+#include "../tests.h"
+
+using namespace dealii;
+namespace AD = dealii::Differentiation::AD;
+
+// Function and its derivatives
+template <int dim, typename NumberType>
+struct FunctionsTestVectorScalarCoupled
+{
+  static NumberType
+  psi(const Tensor<1, dim, NumberType> &v, const NumberType &s)
+  {
+    return (v * v) * std::pow(s, 3);
+  };
+
+  static Tensor<1, dim, NumberType>
+  dpsi_dv(const Tensor<1, dim, NumberType> &v, const NumberType &s)
+  {
+    return 2.0 * v * std::pow(s, 3);
+  };
+
+  static NumberType
+  dpsi_ds(const Tensor<1, dim, NumberType> &v, const NumberType &s)
+  {
+    return 3.0 * (v * v) * std::pow(s, 2);
+  };
+
+  static Tensor<2, dim, NumberType>
+  d2psi_dv_dv(const Tensor<1, dim, NumberType> &v, const NumberType &s)
+  {
+    static const Tensor<2, dim, NumberType> I(
+      unit_symmetric_tensor<dim, NumberType>());
+    return 2.0 * I * std::pow(s, 3);
+  };
+
+  static Tensor<1, dim, NumberType>
+  d2psi_ds_dv(const Tensor<1, dim, NumberType> &v, const NumberType &s)
+  {
+    return 6.0 * v * std::pow(s, 2);
+  };
+
+  static Tensor<1, dim, NumberType>
+  d2psi_dv_ds(const Tensor<1, dim, NumberType> &v, const NumberType &s)
+  {
+    return d2psi_ds_dv(v, s);
+  };
+
+  static NumberType
+  d2psi_ds_ds(const Tensor<1, dim, NumberType> &v, const NumberType &s)
+  {
+    return 6.0 * (v * v) * std::pow(s, 1);
+  };
+};
+
+template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+void
+test_vector_scalar_coupled()
+{
+  typedef AD::ScalarFunction<dim, ad_type_code, number_t> ADHelper;
+  typedef typename ADHelper::ad_type                      ADNumberType;
+  typedef typename ADHelper::scalar_type                  ScalarNumberType;
+
+  std::cout << "*** Test variables: Vector + Scalar (coupled), "
+            << "dim = " << Utilities::to_string(dim) << ", "
+            << "Type code: " << static_cast<int>(ad_type_code) << std::endl;
+
+  // Values computed from the AD energy function
+  ScalarNumberType             psi;
+  Vector<ScalarNumberType>     Dpsi;
+  FullMatrix<ScalarNumberType> D2psi;
+
+  // Function and its derivatives
+  typedef FunctionsTestVectorScalarCoupled<dim, ADNumberType> func_ad;
+
+  // Setup the variable components and choose a value at which to
+  // evaluate the tape
+  const FEValuesExtractors::Vector v_dof(0);
+  const FEValuesExtractors::Scalar s_dof(
+    Tensor<1, dim>::n_independent_components);
+  const unsigned int n_AD_components = dim + 1;
+  ADHelper           ad_helper(n_AD_components);
+  ad_helper.set_tape_buffer_sizes(); // Increase the buffer size from the
+                                     // default values
+
+  ScalarNumberType                 s = 3.0;
+  Tensor<1, dim, ScalarNumberType> v;
+  for (unsigned int i = 0; i < dim; ++i)
+    v[i] = 1.0 + i;
+
+  const int  tape_no = 1;
+  const bool is_recording =
+    ad_helper.start_recording_operations(tape_no /*material_id*/,
+                                         true /*overwrite_tape*/,
+                                         true /*keep*/);
+  if (is_recording == true)
+    {
+      ad_helper.register_independent_variable(v, v_dof);
+      ad_helper.register_independent_variable(s, s_dof);
+
+      const Tensor<1, dim, ADNumberType> v_ad =
+        ad_helper.get_sensitive_variables(v_dof);
+      const ADNumberType s_ad = ad_helper.get_sensitive_variables(s_dof);
+
+      const ADNumberType psi(func_ad::psi(v_ad, s_ad));
+
+      ad_helper.register_dependent_variable(psi);
+      ad_helper.stop_recording_operations(false /*write_tapes_to_file*/);
+
+      std::cout << "Taped data..." << std::endl;
+      std::cout << "independent variable values: " << std::flush;
+      ad_helper.print_values(std::cout);
+      std::cout << "v_ad: " << v_ad << std::endl;
+      std::cout << "s_ad: " << s_ad << std::endl;
+      std::cout << "psi: " << psi << std::endl;
+      std::cout << std::endl;
+    }
+  else
+    {
+      Assert(is_recording == true, ExcInternalError());
+    }
+
+  // Do some work :-)
+  // Set a new evaluation point
+  if (AD::ADNumberTraits<ADNumberType>::is_taped == true)
+    {
+      std::cout
+        << "Using tape with different values for independent variables..."
+        << std::endl;
+      ad_helper.activate_recorded_tape(tape_no);
+      s = 2.75;
+      v *= 1.3;
+      ad_helper.set_independent_variable(v, v_dof);
+      ad_helper.set_independent_variable(s, s_dof);
+    }
+
+  std::cout << "independent variable values: " << std::flush;
+  ad_helper.print_values(std::cout);
+
+  // Compute the function value, gradient and hessian for the new evaluation
+  // point
+  psi = ad_helper.compute_value();
+  ad_helper.compute_gradient(Dpsi);
+  if (AD::ADNumberTraits<ADNumberType>::n_supported_derivative_levels >= 2)
+    {
+      ad_helper.compute_hessian(D2psi);
+    }
+
+  // Output the full stored function, gradient vector and hessian matrix
+  std::cout << "psi: " << psi << std::endl;
+  std::cout << "Dpsi: \n";
+  Dpsi.print(std::cout);
+  if (AD::ADNumberTraits<ADNumberType>::n_supported_derivative_levels >= 2)
+    {
+      std::cout << "D2psi: \n";
+      D2psi.print_formatted(std::cout, 3, true, 0, "0.0");
+    }
+
+  // Extract components of the solution
+  const Tensor<1, dim, ScalarNumberType> dpsi_dv =
+    ad_helper.extract_gradient_component(Dpsi, v_dof);
+  const ScalarNumberType dpsi_ds =
+    ad_helper.extract_gradient_component(Dpsi, s_dof);
+  std::cout << "extracted Dpsi (v): " << dpsi_dv << "\n"
+            << "extracted Dpsi (s): " << dpsi_ds << "\n";
+
+  // Verify the result
+  typedef FunctionsTestVectorScalarCoupled<dim, ScalarNumberType> func;
+  static const ScalarNumberType                                   tol =
+    1e5 * std::numeric_limits<ScalarNumberType>::epsilon();
+  Assert(std::abs(psi - func::psi(v, s)) < tol,
+         ExcMessage("No match for function value."));
+  Assert(std::abs((dpsi_dv - func::dpsi_dv(v, s)).norm()) < tol,
+         ExcMessage("No match for first derivative."));
+  Assert(std::abs(dpsi_ds - func::dpsi_ds(v, s)) < tol,
+         ExcMessage("No match for first derivative."));
+  if (AD::ADNumberTraits<ADNumberType>::n_supported_derivative_levels >= 2)
+    {
+      const Tensor<2, dim, ScalarNumberType> d2psi_dv_dv =
+        ad_helper.extract_hessian_component(D2psi, v_dof, v_dof);
+      const Tensor<1, dim, ScalarNumberType> d2psi_ds_dv =
+        ad_helper.extract_hessian_component(D2psi, s_dof, v_dof);
+      const Tensor<1, dim, ScalarNumberType> d2psi_dv_ds =
+        ad_helper.extract_hessian_component(D2psi, v_dof, s_dof);
+      const ScalarNumberType d2psi_ds_ds =
+        ad_helper.extract_hessian_component(D2psi, s_dof, s_dof);
+      std::cout << "extracted D2psi (v,v): " << d2psi_dv_dv << "\n"
+                << "extracted D2psi (v,s): " << d2psi_ds_dv << "\n"
+                << "extracted D2psi (s,v): " << d2psi_dv_ds << "\n"
+                << "extracted D2psi (s,s): " << d2psi_ds_ds << "\n"
+                << std::endl;
+      Assert(std::abs((d2psi_dv_dv - func::d2psi_dv_dv(v, s)).norm()) < tol,
+             ExcMessage("No match for second derivative."));
+      Assert(std::abs((d2psi_ds_dv - func::d2psi_ds_dv(v, s)).norm()) < tol,
+             ExcMessage("No match for second derivative."));
+      Assert(std::abs((d2psi_dv_ds - func::d2psi_dv_ds(v, s)).norm()) < tol,
+             ExcMessage("No match for second derivative."));
+      Assert(std::abs(d2psi_ds_ds - func::d2psi_ds_ds(v, s)) < tol,
+             ExcMessage("No match for second derivative."));
+    }
+}
diff --git a/tests/ad_common_tests/helper_scalar_coupled_2_components_03.h b/tests/ad_common_tests/helper_scalar_coupled_2_components_03.h
new file mode 100644 (file)
index 0000000..a563b78
--- /dev/null
@@ -0,0 +1,244 @@
+// ---------------------------------------------------------------------
+//
+// Copyright (C) 2016 - 2018 by the deal.II authors
+//
+// This file is part of the deal.II library.
+//
+// The deal.II library is free software; you can use it, redistribute
+// it, and/or modify it under the terms of the GNU Lesser General
+// Public License as published by the Free Software Foundation; either
+// version 2.1 of the License, or (at your option) any later version.
+// The full text of the license can be found in the file LICENSE at
+// the top level of the deal.II distribution.
+//
+// ---------------------------------------------------------------------
+
+
+// Header file:
+// Evaluation of a coupled system (tensor + scalar components)
+// using a helper class
+
+#include <deal.II/base/logstream.h>
+#include <deal.II/base/symmetric_tensor.h>
+#include <deal.II/base/tensor.h>
+
+#include <deal.II/differentiation/ad.h>
+
+#include <deal.II/fe/fe_values_extractors.h>
+
+#include <deal.II/lac/full_matrix.h>
+#include <deal.II/lac/vector.h>
+
+#include <iostream>
+
+#include "../tests.h"
+
+using namespace dealii;
+namespace AD = dealii::Differentiation::AD;
+
+// Function and its derivatives
+template <int dim, typename NumberType>
+struct FunctionsTestTensorScalarCoupled
+{
+  static NumberType
+  psi(const Tensor<2, dim, NumberType> &t, const NumberType &s)
+  {
+    return double_contract(t, t) * std::pow(s, 3);
+  };
+
+  static Tensor<2, dim, NumberType>
+  dpsi_dt(const Tensor<2, dim, NumberType> &t, const NumberType &s)
+  {
+    return 2.0 * t * std::pow(s, 3);
+  };
+
+  static NumberType
+  dpsi_ds(const Tensor<2, dim, NumberType> &t, const NumberType &s)
+  {
+    return 3.0 * double_contract(t, t) * std::pow(s, 2);
+  };
+
+  static Tensor<4, dim, NumberType>
+  d2psi_dt_dt(const Tensor<2, dim, NumberType> &t, const NumberType &s)
+  {
+    // Non-symmetric fourth order identity tensor
+    static const SymmetricTensor<2, dim, NumberType> I(
+      unit_symmetric_tensor<dim, NumberType>());
+    Tensor<4, dim, NumberType> II;
+    for (unsigned int i = 0; i < dim; ++i)
+      for (unsigned int j = 0; j < dim; ++j)
+        for (unsigned int k = 0; k < dim; ++k)
+          for (unsigned int l = 0; l < dim; ++l)
+            II[i][j][k][l] = I[i][k] * I[j][l];
+
+    return 2.0 * II * std::pow(s, 3);
+  };
+
+  static Tensor<2, dim, NumberType>
+  d2psi_ds_dt(const Tensor<2, dim, NumberType> &t, const NumberType &s)
+  {
+    return 6.0 * t * std::pow(s, 2);
+  };
+
+  static Tensor<2, dim, NumberType>
+  d2psi_dt_ds(const Tensor<2, dim, NumberType> &t, const NumberType &s)
+  {
+    return d2psi_ds_dt(t, s);
+  };
+
+  static NumberType
+  d2psi_ds_ds(const Tensor<2, dim, NumberType> &t, const NumberType &s)
+  {
+    return 6.0 * double_contract(t, t) * std::pow(s, 1);
+  };
+};
+
+template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+void
+test_tensor_scalar_coupled()
+{
+  typedef AD::ScalarFunction<dim, ad_type_code, number_t> ADHelper;
+  typedef typename ADHelper::ad_type                      ADNumberType;
+  typedef typename ADHelper::scalar_type                  ScalarNumberType;
+
+  std::cout << "*** Test variables: Tensor + Scalar (coupled), "
+            << "dim = " << Utilities::to_string(dim) << ", "
+            << "Type code: " << static_cast<int>(ad_type_code) << std::endl;
+
+  // Values computed from the AD energy function
+  ScalarNumberType             psi;
+  Vector<ScalarNumberType>     Dpsi;
+  FullMatrix<ScalarNumberType> D2psi;
+
+  // Function and its derivatives
+  typedef FunctionsTestTensorScalarCoupled<dim, ADNumberType> func_ad;
+
+  const FEValuesExtractors::Tensor<2> t_dof(0);
+  const FEValuesExtractors::Scalar    s_dof(
+    Tensor<2, dim>::n_independent_components);
+  const unsigned int n_AD_components =
+    Tensor<2, dim>::n_independent_components + 1;
+  ADHelper ad_helper(n_AD_components);
+  ad_helper.set_tape_buffer_sizes(); // Increase the buffer size from the
+                                     // default values
+
+  ScalarNumberType                 s = 7.5;
+  Tensor<2, dim, ScalarNumberType> t =
+    unit_symmetric_tensor<dim, ScalarNumberType>();
+  for (unsigned int i = 0; i < t.n_independent_components; ++i)
+    t[t.unrolled_to_component_indices(i)] += 0.18 * (i + 0.12);
+
+  const int  tape_no = 1;
+  const bool is_recording =
+    ad_helper.start_recording_operations(tape_no /*material_id*/,
+                                         true /*overwrite_tape*/,
+                                         true /*keep*/);
+  if (is_recording == true)
+    {
+      ad_helper.register_independent_variable(t, t_dof);
+      ad_helper.register_independent_variable(s, s_dof);
+
+      const Tensor<2, dim, ADNumberType> t_ad =
+        ad_helper.get_sensitive_variables(t_dof);
+      const ADNumberType s_ad = ad_helper.get_sensitive_variables(s_dof);
+
+      const ADNumberType psi(func_ad::psi(t_ad, s_ad));
+
+      ad_helper.register_dependent_variable(psi);
+      ad_helper.stop_recording_operations(false /*write_tapes_to_file*/);
+
+
+      std::cout << "Recorded data..." << std::endl;
+      std::cout << "independent variable values: " << std::flush;
+      ad_helper.print_values(std::cout);
+      std::cout << "t_ad: " << t_ad << std::endl;
+      std::cout << "s_ad: " << s_ad << std::endl;
+      std::cout << "psi: " << psi << std::endl;
+      std::cout << std::endl;
+    }
+  else
+    {
+      Assert(is_recording == true, ExcInternalError());
+    }
+
+  // Do some work :-)
+  // Set a new evaluation point
+  if (AD::ADNumberTraits<ADNumberType>::is_taped == true)
+    {
+      std::cout
+        << "Using tape with different values for independent variables..."
+        << std::endl;
+      ad_helper.activate_recorded_tape(tape_no);
+      s = 1.2;
+      t *= 1.75;
+      ad_helper.set_independent_variable(t, t_dof);
+      ad_helper.set_independent_variable(s, s_dof);
+    }
+
+  std::cout << "independent variable values: " << std::flush;
+  ad_helper.print_values(std::cout);
+
+  // Compute the function value, gradient and hessian for the new evaluation
+  // point
+  psi = ad_helper.compute_value();
+  ad_helper.compute_gradient(Dpsi);
+  if (AD::ADNumberTraits<ADNumberType>::n_supported_derivative_levels >= 2)
+    {
+      ad_helper.compute_hessian(D2psi);
+    }
+
+  // Output the full stored function, gradient vector and hessian matrix
+  std::cout << "psi: " << psi << std::endl;
+  std::cout << "Dpsi: \n";
+  Dpsi.print(std::cout);
+  if (AD::ADNumberTraits<ADNumberType>::n_supported_derivative_levels >= 2)
+    {
+      std::cout << "D2psi: \n";
+      D2psi.print_formatted(std::cout, 3, true, 0, "0.0");
+    }
+
+  // Extract components of the solution
+  const Tensor<2, dim, ScalarNumberType> dpsi_dt =
+    ad_helper.extract_gradient_component(Dpsi, t_dof);
+  const ScalarNumberType dpsi_ds =
+    ad_helper.extract_gradient_component(Dpsi, s_dof);
+  std::cout << "extracted Dpsi (t): " << dpsi_dt << "\n"
+            << "extracted Dpsi (s): " << dpsi_ds << "\n";
+  ;
+
+  // Verify the result
+  typedef FunctionsTestTensorScalarCoupled<dim, ScalarNumberType> func;
+  static const ScalarNumberType                                   tol =
+    1e5 * std::numeric_limits<ScalarNumberType>::epsilon();
+
+  Assert(std::abs(psi - func::psi(t, s)) < tol,
+         ExcMessage("No match for function value."));
+  Assert(std::abs((dpsi_dt - func::dpsi_dt(t, s)).norm()) < tol,
+         ExcMessage("No match for first derivative."));
+  Assert(std::abs(dpsi_ds - func::dpsi_ds(t, s)) < tol,
+         ExcMessage("No match for first derivative."));
+  if (AD::ADNumberTraits<ADNumberType>::n_supported_derivative_levels >= 2)
+    {
+      const Tensor<4, dim, ScalarNumberType> d2psi_dt_dt =
+        ad_helper.extract_hessian_component(D2psi, t_dof, t_dof);
+      const Tensor<2, dim, ScalarNumberType> d2psi_ds_dt =
+        ad_helper.extract_hessian_component(D2psi, t_dof, s_dof);
+      const Tensor<2, dim, ScalarNumberType> d2psi_dt_ds =
+        ad_helper.extract_hessian_component(D2psi, t_dof, s_dof);
+      const ScalarNumberType d2psi_ds_ds =
+        ad_helper.extract_hessian_component(D2psi, s_dof, s_dof);
+      std::cout << "extracted D2psi (t,t): " << d2psi_dt_dt << "\n"
+                << "extracted D2psi (t,s): " << d2psi_ds_dt << "\n"
+                << "extracted D2psi (s,t): " << d2psi_dt_ds << "\n"
+                << "extracted D2psi (s,s): " << d2psi_ds_ds << "\n"
+                << std::endl;
+      Assert(std::abs((d2psi_dt_dt - func::d2psi_dt_dt(t, s)).norm()) < tol,
+             ExcMessage("No match for second derivative."));
+      Assert(std::abs((d2psi_ds_dt - func::d2psi_ds_dt(t, s)).norm()) < tol,
+             ExcMessage("No match for second derivative."));
+      Assert(std::abs((d2psi_dt_ds - func::d2psi_dt_ds(t, s)).norm()) < tol,
+             ExcMessage("No match for second derivative."));
+      Assert(std::abs(d2psi_ds_ds - func::d2psi_ds_ds(t, s)) < tol,
+             ExcMessage("No match for second derivative."));
+    }
+}
diff --git a/tests/ad_common_tests/helper_scalar_coupled_2_components_04.h b/tests/ad_common_tests/helper_scalar_coupled_2_components_04.h
new file mode 100644 (file)
index 0000000..f69b51f
--- /dev/null
@@ -0,0 +1,260 @@
+// ---------------------------------------------------------------------
+//
+// Copyright (C) 2016 - 2018 by the deal.II authors
+//
+// This file is part of the deal.II library.
+//
+// The deal.II library is free software; you can use it, redistribute
+// it, and/or modify it under the terms of the GNU Lesser General
+// Public License as published by the Free Software Foundation; either
+// version 2.1 of the License, or (at your option) any later version.
+// The full text of the license can be found in the file LICENSE at
+// the top level of the deal.II distribution.
+//
+// ---------------------------------------------------------------------
+
+
+// Header file:
+// Evaluation of a coupled system (vector + vector components)
+// using a helper class
+
+#include <deal.II/base/logstream.h>
+#include <deal.II/base/symmetric_tensor.h>
+#include <deal.II/base/tensor.h>
+
+#include <deal.II/differentiation/ad.h>
+
+#include <deal.II/fe/fe_values_extractors.h>
+
+#include <deal.II/lac/full_matrix.h>
+#include <deal.II/lac/vector.h>
+
+#include <iostream>
+
+#include "../tests.h"
+
+using namespace dealii;
+namespace AD = dealii::Differentiation::AD;
+
+// Function and its derivatives
+template <int dim, typename NumberType>
+struct FunctionsTestVectorVectorCoupled
+{
+  static NumberType
+  psi(const Tensor<1, dim, NumberType> &v1,
+      const Tensor<1, dim, NumberType> &v2)
+  {
+    return std::pow(v1 * v1, 2) * std::pow(v2 * v2, 3);
+  };
+
+  static Tensor<1, dim, NumberType>
+  dpsi_dv1(const Tensor<1, dim, NumberType> &v1,
+           const Tensor<1, dim, NumberType> &v2)
+  {
+    return 2.0 * std::pow(v1 * v1, 1) * 2.0 * v1 * std::pow(v2 * v2, 3);
+  };
+
+  static Tensor<1, dim, NumberType>
+  dpsi_dv2(const Tensor<1, dim, NumberType> &v1,
+           const Tensor<1, dim, NumberType> &v2)
+  {
+    return std::pow(v1 * v1, 2) * 3.0 * std::pow(v2 * v2, 2) * 2.0 * v2;
+  };
+
+  static Tensor<2, dim, NumberType>
+  d2psi_dv1_dv1(const Tensor<1, dim, NumberType> &v1,
+                const Tensor<1, dim, NumberType> &v2)
+  {
+    const Tensor<2, dim, NumberType> I(
+      unit_symmetric_tensor<dim, NumberType>());
+    return 2.0 * 2.0 * std::pow(v2 * v2, 3) *
+           (pow(v1 * v1, 0) * 2.0 * outer_product(v1, v1) +
+            std::pow(v1 * v1, 1) * I);
+  };
+
+  static Tensor<2, dim, NumberType>
+  d2psi_dv2_dv1(const Tensor<1, dim, NumberType> &v1,
+                const Tensor<1, dim, NumberType> &v2)
+  {
+    // Note: This is not symmetric, and this is why we
+    // don't set the return type for hessian extractor (vector,vector)
+    // operations as SymmetricTensor.
+    return (2.0 * std::pow(v1 * v1, 1) * 2.0) *
+           (3.0 * std::pow(v2 * v2, 2) * 2.0) * outer_product(v1, v2);
+  };
+
+  static Tensor<2, dim, NumberType>
+  d2psi_dv1_dv2(const Tensor<1, dim, NumberType> &v1,
+                const Tensor<1, dim, NumberType> &v2)
+  {
+    return (2.0 * std::pow(v1 * v1, 1) * 2.0) *
+           (3.0 * std::pow(v2 * v2, 2) * 2.0) * outer_product(v2, v1);
+  };
+
+  static Tensor<2, dim, NumberType>
+  d2psi_dv2_dv2(const Tensor<1, dim, NumberType> &v1,
+                const Tensor<1, dim, NumberType> &v2)
+  {
+    const Tensor<2, dim, NumberType> I(
+      unit_symmetric_tensor<dim, NumberType>());
+    // return std::pow(v1*v1,2)*3.0*
+    //        ( 2.0*std::pow(v2*v2,1)*2.0*outer_product(v2,v2) +
+    //        std::pow(v2*v2,2)*2.0*I);
+    return std::pow(v1 * v1, 2) * 3.0 * 2.0 *
+           (2.0 * std::pow(v2 * v2, 1) * 2.0 * outer_product(v2, v2) +
+            std::pow(v2 * v2, 2) * I);
+  };
+};
+
+template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+void
+test_vector_vector_coupled()
+{
+  typedef AD::ScalarFunction<dim, ad_type_code, number_t> ADHelper;
+  typedef typename ADHelper::ad_type                      ADNumberType;
+  typedef typename ADHelper::scalar_type                  ScalarNumberType;
+
+  std::cout << "*** Test variables: Vector + Vector (coupled), "
+            << "dim = " << Utilities::to_string(dim) << ", "
+            << "Type code: " << static_cast<int>(ad_type_code) << std::endl;
+
+  // Values computed from the AD energy function
+  ScalarNumberType             psi;
+  Vector<ScalarNumberType>     Dpsi;
+  FullMatrix<ScalarNumberType> D2psi;
+
+  // Function and its derivatives
+  typedef FunctionsTestVectorVectorCoupled<dim, ADNumberType> func_ad;
+
+  const FEValuesExtractors::Vector v1_dof(0);
+  const FEValuesExtractors::Vector v2_dof(
+    Tensor<1, dim>::n_independent_components);
+  const unsigned int n_AD_components = dim + dim;
+  ADHelper           ad_helper(n_AD_components);
+  ad_helper.set_tape_buffer_sizes(); // Increase the buffer size from the
+                                     // default values
+
+  Tensor<1, dim, ScalarNumberType> v1, v2;
+  for (unsigned int i = 0; i < dim; ++i)
+    {
+      v1[i] = 0.2 * (1.0 + i);
+      v2[i] = 0.1 * (2.0 + i * 1.7);
+    }
+
+  const int  tape_no = 1;
+  const bool is_recording =
+    ad_helper.start_recording_operations(tape_no /*material_id*/,
+                                         true /*overwrite_tape*/,
+                                         true /*keep*/);
+  if (is_recording == true)
+    {
+      ad_helper.register_independent_variable(v1, v1_dof);
+      ad_helper.register_independent_variable(v2, v2_dof);
+
+      const Tensor<1, dim, ADNumberType> v1_ad =
+        ad_helper.get_sensitive_variables(v1_dof);
+      const Tensor<1, dim, ADNumberType> v2_ad =
+        ad_helper.get_sensitive_variables(v2_dof);
+
+      const ADNumberType psi(func_ad::psi(v1_ad, v2_ad));
+
+      ad_helper.register_dependent_variable(psi);
+      ad_helper.stop_recording_operations(false /*write_tapes_to_file*/);
+
+      std::cout << "Recorded data..." << std::endl;
+      std::cout << "independent variable values: " << std::flush;
+      ad_helper.print_values(std::cout);
+      std::cout << "v1_ad: " << v1_ad << std::endl;
+      std::cout << "v2_ad: " << v2_ad << std::endl;
+      std::cout << "psi: " << psi << std::endl;
+      std::cout << std::endl;
+    }
+  else
+    {
+      Assert(is_recording == true, ExcInternalError());
+    }
+
+  // Do some work :-)
+  // Set a new evaluation point
+  if (AD::ADNumberTraits<ADNumberType>::is_taped == true)
+    {
+      std::cout
+        << "Using tape with different values for independent variables..."
+        << std::endl;
+      ad_helper.activate_recorded_tape(tape_no);
+      v1 *= 1.7;
+      v2 *= 0.25;
+      ad_helper.set_independent_variable(v1, v1_dof);
+      ad_helper.set_independent_variable(v2, v2_dof);
+    }
+
+  std::cout << "independent variable values: " << std::flush;
+  ad_helper.print_values(std::cout);
+
+  // Compute the function value, gradient and hessian for the new evaluation
+  // point
+  psi = ad_helper.compute_value();
+  ad_helper.compute_gradient(Dpsi);
+  if (AD::ADNumberTraits<ADNumberType>::n_supported_derivative_levels >= 2)
+    {
+      ad_helper.compute_hessian(D2psi);
+    }
+
+  // Output the full stored function, gradient vector and hessian matrix
+  std::cout << "psi: " << psi << std::endl;
+  std::cout << "Dpsi: \n";
+  Dpsi.print(std::cout);
+  if (AD::ADNumberTraits<ADNumberType>::n_supported_derivative_levels >= 2)
+    {
+      std::cout << "D2psi: \n";
+      D2psi.print_formatted(std::cout, 3, true, 0, "0.0");
+    }
+
+  // Extract components of the solution
+  const Tensor<1, dim, ScalarNumberType> dpsi_dv1 =
+    ad_helper.extract_gradient_component(Dpsi, v1_dof);
+  const Tensor<1, dim, ScalarNumberType> dpsi_dv2 =
+    ad_helper.extract_gradient_component(Dpsi, v2_dof);
+  std::cout << "extracted Dpsi (v1): " << dpsi_dv1 << "\n"
+            << "extracted Dpsi (v2): " << dpsi_dv2 << "\n";
+
+  // Verify the result
+  typedef FunctionsTestVectorVectorCoupled<dim, ScalarNumberType> func;
+  static const ScalarNumberType                                   tol =
+    1e5 * std::numeric_limits<ScalarNumberType>::epsilon();
+
+  Assert(std::abs(psi - func::psi(v1, v2)) < tol,
+         ExcMessage("No match for function value."));
+  Assert(std::abs((dpsi_dv1 - func::dpsi_dv1(v1, v2)).norm()) < tol,
+         ExcMessage("No match for first derivative."));
+  Assert(std::abs((dpsi_dv2 - func::dpsi_dv2(v1, v2)).norm()) < tol,
+         ExcMessage("No match for first derivative."));
+  if (AD::ADNumberTraits<ADNumberType>::n_supported_derivative_levels >= 2)
+    {
+      const Tensor<2, dim, ScalarNumberType> d2psi_dv1_dv1 =
+        ad_helper.extract_hessian_component(D2psi, v1_dof, v1_dof);
+      const Tensor<2, dim, ScalarNumberType> d2psi_dv2_dv1 =
+        ad_helper.extract_hessian_component(D2psi, v1_dof, v2_dof);
+      const Tensor<2, dim, ScalarNumberType> d2psi_dv1_dv2 =
+        ad_helper.extract_hessian_component(D2psi, v2_dof, v1_dof);
+      const Tensor<2, dim, ScalarNumberType> d2psi_dv2_dv2 =
+        ad_helper.extract_hessian_component(D2psi, v2_dof, v2_dof);
+      std::cout << "extracted D2psi (v1,v1): " << d2psi_dv1_dv1 << "\n"
+                << "extracted D2psi (v1,v2): " << d2psi_dv2_dv1 << "\n"
+                << "extracted D2psi (v2,v1): " << d2psi_dv1_dv2 << "\n"
+                << "extracted D2psi (v2,v2): " << d2psi_dv2_dv2 << "\n"
+                << std::endl;
+      Assert(std::abs((d2psi_dv1_dv1 - func::d2psi_dv1_dv1(v1, v2)).norm()) <
+               tol,
+             ExcMessage("No match for second derivative."));
+      Assert(std::abs((d2psi_dv2_dv1 - func::d2psi_dv2_dv1(v1, v2)).norm()) <
+               tol,
+             ExcMessage("No match for second derivative."));
+      Assert(std::abs((d2psi_dv1_dv2 - func::d2psi_dv1_dv2(v1, v2)).norm()) <
+               tol,
+             ExcMessage("No match for second derivative."));
+      Assert(std::abs((d2psi_dv2_dv2 - func::d2psi_dv2_dv2(v1, v2)).norm()) <
+               tol,
+             ExcMessage("No match for second derivative."));
+    }
+}
diff --git a/tests/ad_common_tests/helper_scalar_coupled_2_components_05.h b/tests/ad_common_tests/helper_scalar_coupled_2_components_05.h
new file mode 100644 (file)
index 0000000..bb9fdf2
--- /dev/null
@@ -0,0 +1,332 @@
+// ---------------------------------------------------------------------
+//
+// Copyright (C) 2016 - 2018 by the deal.II authors
+//
+// This file is part of the deal.II library.
+//
+// The deal.II library is free software; you can use it, redistribute
+// it, and/or modify it under the terms of the GNU Lesser General
+// Public License as published by the Free Software Foundation; either
+// version 2.1 of the License, or (at your option) any later version.
+// The full text of the license can be found in the file LICENSE at
+// the top level of the deal.II distribution.
+//
+// ---------------------------------------------------------------------
+
+
+// Header file:
+// Evaluation of a coupled system (tensor + vector components)
+// using a helper class
+
+#include <deal.II/base/logstream.h>
+#include <deal.II/base/symmetric_tensor.h>
+#include <deal.II/base/tensor.h>
+
+#include <deal.II/differentiation/ad.h>
+
+#include <deal.II/fe/fe_values_extractors.h>
+
+#include <deal.II/lac/full_matrix.h>
+#include <deal.II/lac/vector.h>
+
+#include <iostream>
+
+#include "../tests.h"
+
+using namespace dealii;
+namespace AD = dealii::Differentiation::AD;
+
+// Function and its derivatives
+template <int dim, typename NumberType>
+struct FunctionsTestTensorVectorCoupled
+{
+  static NumberType
+  det_t(const Tensor<2, dim, NumberType> &t)
+  {
+    return determinant(t);
+  }
+
+  static Tensor<2, dim, NumberType>
+  ddet_t_dt(const Tensor<2, dim, NumberType> &t)
+  {
+    return det_t(t) * transpose(invert(t));
+  }
+
+  static Tensor<4, dim, NumberType>
+  d2det_t_dt_dt(const Tensor<2, dim, NumberType> &t)
+  {
+    const Tensor<2, dim, NumberType> t_inv = invert(t);
+    Tensor<4, dim, NumberType>       dt_inv_trans_dt;
+    // https://en.wikiversity.org/wiki/Introduction_to_Elasticity/Tensors#Derivative_of_the_determinant_of_a_tensor
+    for (unsigned int i = 0; i < dim; ++i)
+      for (unsigned int j = 0; j < dim; ++j)
+        for (unsigned int k = 0; k < dim; ++k)
+          for (unsigned int l = 0; l < dim; ++l)
+            dt_inv_trans_dt[i][j][k][l] = -t_inv[l][i] * t_inv[j][k];
+
+    return det_t(t) * outer_product(transpose(t_inv), transpose(t_inv)) +
+           det_t(t) * dt_inv_trans_dt;
+  }
+
+  static NumberType
+  v_squ(const Tensor<1, dim, NumberType> &v)
+  {
+    return v * v;
+  }
+
+  static Tensor<1, dim, NumberType>
+  dv_squ_dv(const Tensor<1, dim, NumberType> &v)
+  {
+    return 2.0 * v;
+  }
+
+  static Tensor<2, dim, NumberType>
+  d2v_squ_dv_dv(const Tensor<1, dim, NumberType> &v)
+  {
+    static const Tensor<2, dim, NumberType> I(
+      unit_symmetric_tensor<dim, NumberType>());
+    return 2.0 * I;
+  }
+
+  // --------
+
+  static NumberType
+  psi(const Tensor<2, dim, NumberType> &t, const Tensor<1, dim, NumberType> &v)
+  {
+    return std::pow(det_t(t), 2) * std::pow(v_squ(v), 3);
+  };
+
+  static Tensor<2, dim, NumberType>
+  dpsi_dt(const Tensor<2, dim, NumberType> &t,
+          const Tensor<1, dim, NumberType> &v)
+  {
+    return 2.0 * std::pow(det_t(t), 1) * ddet_t_dt(t) * std::pow(v_squ(v), 3);
+  };
+
+  static Tensor<1, dim, NumberType>
+  dpsi_dv(const Tensor<2, dim, NumberType> &t,
+          const Tensor<1, dim, NumberType> &v)
+  {
+    return std::pow(det_t(t), 2) * 3.0 * std::pow(v_squ(v), 2) * dv_squ_dv(v);
+  };
+
+  static Tensor<4, dim, NumberType>
+  d2psi_dt_dt(const Tensor<2, dim, NumberType> &t,
+              const Tensor<1, dim, NumberType> &v)
+  {
+    return 2.0 * std::pow(v_squ(v), 3) *
+           (pow(det_t(t), 0) * outer_product(ddet_t_dt(t), ddet_t_dt(t)) +
+            std::pow(det_t(t), 1) * d2det_t_dt_dt(t));
+  };
+
+  static Tensor<3, dim, NumberType>
+  d2psi_dv_dt(const Tensor<2, dim, NumberType> &t,
+              const Tensor<1, dim, NumberType> &v)
+  {
+    return 2.0 * std::pow(det_t(t), 1) * 3.0 * std::pow(v_squ(v), 2) *
+           outer_product(ddet_t_dt(t), dv_squ_dv(v));
+  };
+
+  static Tensor<3, dim, NumberType>
+  d2psi_dt_dv(const Tensor<2, dim, NumberType> &t,
+              const Tensor<1, dim, NumberType> &v)
+  {
+    return 2.0 * std::pow(det_t(t), 1) * 3.0 * std::pow(v_squ(v), 2) *
+           outer_product(dv_squ_dv(v), ddet_t_dt(t));
+  };
+
+  static Tensor<2, dim, NumberType>
+  d2psi_dv_dv(const Tensor<2, dim, NumberType> &t,
+              const Tensor<1, dim, NumberType> &v)
+  {
+    return std::pow(det_t(t), 2) * 3.0 *
+           (2.0 * std::pow(v_squ(v), 1) *
+              outer_product(dv_squ_dv(v), dv_squ_dv(v)) +
+            std::pow(v_squ(v), 2) * d2v_squ_dv_dv(v));
+  };
+};
+
+template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+void
+test_tensor_vector_coupled()
+{
+  typedef AD::ScalarFunction<dim, ad_type_code, number_t> ADHelper;
+  typedef typename ADHelper::ad_type                      ADNumberType;
+  typedef typename ADHelper::scalar_type                  ScalarNumberType;
+
+  std::cout << "*** Test variables: Tensor + Vector (coupled), "
+            << "dim = " << Utilities::to_string(dim) << ", "
+            << "Type code: " << static_cast<int>(ad_type_code) << std::endl;
+
+  // Values computed from the AD energy function
+  ScalarNumberType             psi;
+  Vector<ScalarNumberType>     Dpsi;
+  FullMatrix<ScalarNumberType> D2psi;
+
+  // Function and its derivatives
+  typedef FunctionsTestTensorVectorCoupled<dim, ADNumberType> func_ad;
+
+  // Setup the variable components and choose a value at which to
+  // evaluate the tape
+  const FEValuesExtractors::Tensor<2> t_dof(0);
+  const FEValuesExtractors::Vector    v_dof(
+    Tensor<2, dim>::n_independent_components);
+  const unsigned int n_AD_components =
+    Tensor<2, dim>::n_independent_components +
+    Tensor<1, dim>::n_independent_components;
+  ADHelper ad_helper(n_AD_components);
+  ad_helper.set_tape_buffer_sizes(); // Increase the buffer size from the
+                                     // default values
+
+  Tensor<2, dim, ScalarNumberType> t =
+    unit_symmetric_tensor<dim, ScalarNumberType>();
+  for (unsigned int i = 0; i < t.n_independent_components; ++i)
+    t[t.unrolled_to_component_indices(i)] += 0.21 * (i + 0.045);
+  Tensor<1, dim, ScalarNumberType> v;
+  for (unsigned int i = 0; i < dim; ++i)
+    v[i] = 0.275 * (1.0 + i);
+
+  const int  tape_no = 1;
+  const bool is_recording =
+    ad_helper.start_recording_operations(tape_no /*material_id*/,
+                                         true /*overwrite_tape*/,
+                                         true /*keep*/);
+  if (is_recording == true)
+    {
+      ad_helper.register_independent_variable(t, t_dof);
+      ad_helper.register_independent_variable(v, v_dof);
+
+      const Tensor<2, dim, ADNumberType> t_ad =
+        ad_helper.get_sensitive_variables(t_dof);
+      const Tensor<1, dim, ADNumberType> v_ad =
+        ad_helper.get_sensitive_variables(v_dof);
+
+      const ADNumberType psi(func_ad::psi(t_ad, v_ad));
+
+      ad_helper.register_dependent_variable(psi);
+      ad_helper.stop_recording_operations(false /*write_tapes_to_file*/);
+
+      std::cout << "Recorded data..." << std::endl;
+      std::cout << "independent variable values: " << std::flush;
+      ad_helper.print_values(std::cout);
+      std::cout << "t_ad: " << t_ad << std::endl;
+      std::cout << "v_ad: " << v_ad << std::endl;
+      std::cout << "psi: " << psi << std::endl;
+      std::cout << std::endl;
+    }
+  else
+    {
+      Assert(is_recording == true, ExcInternalError());
+    }
+
+  // Do some work :-)
+  // Set a new evaluation point
+  if (AD::ADNumberTraits<ADNumberType>::is_taped == true)
+    {
+      std::cout
+        << "Using tape with different values for independent variables..."
+        << std::endl;
+      ad_helper.activate_recorded_tape(tape_no);
+      t *= 0.9;
+      v *= 0.63;
+      ad_helper.set_independent_variable(t, t_dof);
+      ad_helper.set_independent_variable(v, v_dof);
+    }
+
+  std::cout << "independent variable values: " << std::flush;
+  ad_helper.print_values(std::cout);
+
+  // Compute the function value, gradient and hessian for the new evaluation
+  // point
+  psi = ad_helper.compute_value();
+  ad_helper.compute_gradient(Dpsi);
+  if (AD::ADNumberTraits<ADNumberType>::n_supported_derivative_levels >= 2)
+    {
+      ad_helper.compute_hessian(D2psi);
+    }
+
+  // Output the full stored function, gradient vector and hessian matrix
+  std::cout << "psi: " << psi << std::endl;
+  std::cout << "Dpsi: \n";
+  Dpsi.print(std::cout);
+  if (AD::ADNumberTraits<ADNumberType>::n_supported_derivative_levels >= 2)
+    {
+      std::cout << "D2psi: \n";
+      D2psi.print_formatted(std::cout, 3, true, 0, "0.0");
+    }
+
+  // Extract components of the solution
+  const Tensor<2, dim, ScalarNumberType> dpsi_dt =
+    ad_helper.extract_gradient_component(Dpsi, t_dof);
+  const Tensor<1, dim, ScalarNumberType> dpsi_dv =
+    ad_helper.extract_gradient_component(Dpsi, v_dof);
+  std::cout << "extracted Dpsi (t): " << dpsi_dt << "\n"
+            << "extracted Dpsi (v): " << dpsi_dv << "\n";
+
+  // Verify the result
+  typedef FunctionsTestTensorVectorCoupled<dim, ScalarNumberType> func;
+  static const ScalarNumberType                                   tol =
+    1e5 * std::numeric_limits<ScalarNumberType>::epsilon();
+  std::cout << "dpsi_dt:            " << dpsi_dt << std::endl;
+  std::cout << "func::dpsi_dt(t,v): " << func::dpsi_dt(t, v) << std::endl;
+  std::cout << "diff: " << std::abs((dpsi_dt - func::dpsi_dt(t, v)).norm())
+            << std::endl;
+  std::cout << "dpsi_dv:            " << dpsi_dv << std::endl;
+  std::cout << "func::dpsi_dv(t,v): " << func::dpsi_dv(t, v) << std::endl;
+  std::cout << "diff: " << std::abs((dpsi_dv - func::dpsi_dv(t, v)).norm())
+            << std::endl;
+  Assert(std::abs(psi - func::psi(t, v)) < tol,
+         ExcMessage("No match for function value."));
+  Assert(std::abs((dpsi_dt - func::dpsi_dt(t, v)).norm()) < tol,
+         ExcMessage("No match for first derivative."));
+  Assert(std::abs((dpsi_dv - func::dpsi_dv(t, v)).norm()) < tol,
+         ExcMessage("No match for first derivative."));
+  if (AD::ADNumberTraits<ADNumberType>::n_supported_derivative_levels >= 2)
+    {
+      const Tensor<4, dim, ScalarNumberType> d2psi_dt_dt =
+        ad_helper.extract_hessian_component(D2psi, t_dof, t_dof);
+      const Tensor<3, dim, ScalarNumberType> d2psi_dv_dt =
+        ad_helper.extract_hessian_component(D2psi, t_dof, v_dof);
+      const Tensor<3, dim, ScalarNumberType> d2psi_dt_dv =
+        ad_helper.extract_hessian_component(D2psi, v_dof, t_dof);
+      const Tensor<2, dim, ScalarNumberType> d2psi_dv_dv =
+        ad_helper.extract_hessian_component(D2psi, v_dof, v_dof);
+      std::cout << "extracted D2psi (t,t): " << d2psi_dt_dt << "\n"
+                << "extracted D2psi (t,v): " << d2psi_dv_dt << "\n"
+                << "extracted D2psi (v,t): " << d2psi_dt_dv << "\n"
+                << "extracted D2psi (v,v): " << d2psi_dv_dv << "\n"
+                << std::endl;
+      std::cout << "d2psi_dt_dt:            " << d2psi_dt_dt << std::endl;
+      std::cout << "func::d2psi_dt_dt(t,v): " << func::d2psi_dt_dt(t, v)
+                << std::endl;
+      std::cout << "diff: "
+                << std::abs((d2psi_dt_dt - func::d2psi_dt_dt(t, v)).norm())
+                << std::endl;
+      std::cout << "d2psi_dv_dt:            " << d2psi_dv_dt << std::endl;
+      std::cout << "func::d2psi_dv_dt(t,v): " << func::d2psi_dv_dt(t, v)
+                << std::endl;
+      std::cout << "diff: "
+                << std::abs((d2psi_dv_dt - func::d2psi_dv_dt(t, v)).norm())
+                << std::endl;
+      std::cout << "d2psi_dt_dv:            " << d2psi_dt_dv << std::endl;
+      std::cout << "func::d2psi_dt_dv(t,v): " << func::d2psi_dt_dv(t, v)
+                << std::endl;
+      std::cout << "diff: "
+                << std::abs((d2psi_dt_dv - func::d2psi_dt_dv(t, v)).norm())
+                << std::endl;
+      std::cout << "d2psi_dv_dv:            " << d2psi_dv_dv << std::endl;
+      std::cout << "func::d2psi_dv_dv(t,v): " << func::d2psi_dv_dv(t, v)
+                << std::endl;
+      std::cout << "diff: "
+                << std::abs((d2psi_dv_dv - func::d2psi_dv_dv(t, v)).norm())
+                << std::endl;
+      Assert(std::abs((d2psi_dt_dt - func::d2psi_dt_dt(t, v)).norm()) < tol,
+             ExcMessage("No match for second derivative."));
+      Assert(std::abs((d2psi_dv_dt - func::d2psi_dv_dt(t, v)).norm()) < tol,
+             ExcMessage("No match for second derivative."));
+      Assert(std::abs((d2psi_dt_dv - func::d2psi_dt_dv(t, v)).norm()) < tol,
+             ExcMessage("No match for second derivative."));
+      Assert(std::abs((d2psi_dv_dv - func::d2psi_dv_dv(t, v)).norm()) < tol,
+             ExcMessage("No match for second derivative."));
+    }
+}
diff --git a/tests/ad_common_tests/helper_scalar_coupled_2_components_06.h b/tests/ad_common_tests/helper_scalar_coupled_2_components_06.h
new file mode 100644 (file)
index 0000000..f255145
--- /dev/null
@@ -0,0 +1,342 @@
+// ---------------------------------------------------------------------
+//
+// Copyright (C) 2016 - 2018 by the deal.II authors
+//
+// This file is part of the deal.II library.
+//
+// The deal.II library is free software; you can use it, redistribute
+// it, and/or modify it under the terms of the GNU Lesser General
+// Public License as published by the Free Software Foundation; either
+// version 2.1 of the License, or (at your option) any later version.
+// The full text of the license can be found in the file LICENSE at
+// the top level of the deal.II distribution.
+//
+// ---------------------------------------------------------------------
+
+
+// Header file:
+// Evaluation of a coupled system (symmetric tensor + vector components)
+// using a helper class
+
+#include <deal.II/base/logstream.h>
+#include <deal.II/base/symmetric_tensor.h>
+#include <deal.II/base/tensor.h>
+
+#include <deal.II/differentiation/ad.h>
+
+#include <deal.II/fe/fe_values_extractors.h>
+
+#include <deal.II/lac/full_matrix.h>
+#include <deal.II/lac/vector.h>
+
+#include <iostream>
+
+#include "../tests.h"
+
+using namespace dealii;
+namespace AD = dealii::Differentiation::AD;
+
+// Function and its derivatives
+template <int dim, typename NumberType>
+struct FunctionsTestSymmetricTensorVectorCoupled
+{
+  static NumberType
+  det_t(const SymmetricTensor<2, dim, NumberType> &t)
+  {
+    return determinant(t);
+  }
+
+  static SymmetricTensor<2, dim, NumberType>
+  ddet_t_dt(const SymmetricTensor<2, dim, NumberType> &t)
+  {
+    const SymmetricTensor<2, dim, NumberType> t_inv =
+      symmetrize(invert(static_cast<Tensor<2, dim, NumberType>>(t)));
+    return det_t(t) * t_inv;
+  }
+
+  static SymmetricTensor<4, dim, NumberType>
+  d2det_t_dt_dt(const SymmetricTensor<2, dim, NumberType> &t)
+  {
+    const SymmetricTensor<2, dim, NumberType> t_inv =
+      symmetrize(invert(static_cast<Tensor<2, dim, NumberType>>(t)));
+    SymmetricTensor<4, dim, NumberType> dt_inv_dt;
+    // https://en.wikiversity.org/wiki/Introduction_to_Elasticity/Tensors#Derivative_of_the_determinant_of_a_tensor
+    for (unsigned int i = 0; i < dim; ++i)
+      for (unsigned int j = i; j < dim; ++j)
+        for (unsigned int k = 0; k < dim; ++k)
+          for (unsigned int l = k; l < dim; ++l)
+            dt_inv_dt[i][j][k][l] =
+              -0.5 * (t_inv[i][k] * t_inv[j][l] + t_inv[i][l] * t_inv[j][k]);
+
+    return det_t(t) * outer_product(t_inv, t_inv) + det_t(t) * dt_inv_dt;
+  }
+
+  static NumberType
+  v_squ(const Tensor<1, dim, NumberType> &v)
+  {
+    return v * v;
+  }
+
+  static Tensor<1, dim, NumberType>
+  dv_squ_dv(const Tensor<1, dim, NumberType> &v)
+  {
+    return 2.0 * v;
+  }
+
+  static SymmetricTensor<2, dim, NumberType>
+  d2v_squ_dv_dv(const Tensor<1, dim, NumberType> &v)
+  {
+    static const SymmetricTensor<2, dim, NumberType> I(
+      unit_symmetric_tensor<dim, NumberType>());
+    return SymmetricTensor<2, dim, NumberType>(2.0 * I);
+  }
+
+  // -------
+
+  static NumberType
+  psi(const SymmetricTensor<2, dim, NumberType> &t,
+      const Tensor<1, dim, NumberType> &         v)
+  {
+    return std::pow(det_t(t), 2) * std::pow(v_squ(v), 3);
+  };
+
+  static SymmetricTensor<2, dim, NumberType>
+  dpsi_dt(const SymmetricTensor<2, dim, NumberType> &t,
+          const Tensor<1, dim, NumberType> &         v)
+  {
+    return SymmetricTensor<2, dim, NumberType>(
+      2.0 * std::pow(det_t(t), 1) * ddet_t_dt(t) * std::pow(v_squ(v), 3));
+  };
+
+  static Tensor<1, dim, NumberType>
+  dpsi_dv(const SymmetricTensor<2, dim, NumberType> &t,
+          const Tensor<1, dim, NumberType> &         v)
+  {
+    return std::pow(det_t(t), 2) * 3.0 * std::pow(v_squ(v), 2) * dv_squ_dv(v);
+  };
+
+  static SymmetricTensor<4, dim, NumberType>
+  d2psi_dt_dt(const SymmetricTensor<2, dim, NumberType> &t,
+              const Tensor<1, dim, NumberType> &         v)
+  {
+    return SymmetricTensor<4, dim, NumberType>(
+      2.0 * std::pow(v_squ(v), 3) *
+      (pow(det_t(t), 0) * outer_product(ddet_t_dt(t), ddet_t_dt(t)) +
+       std::pow(det_t(t), 1) * d2det_t_dt_dt(t)));
+  };
+
+  static Tensor<3, dim, NumberType>
+  d2psi_dv_dt(const SymmetricTensor<2, dim, NumberType> &t,
+              const Tensor<1, dim, NumberType> &         v)
+  {
+    return 2.0 * std::pow(det_t(t), 1) * 3.0 * std::pow(v_squ(v), 2) *
+           outer_product(static_cast<Tensor<2, dim, NumberType>>(ddet_t_dt(t)),
+                         dv_squ_dv(v));
+  };
+
+  static Tensor<3, dim, NumberType>
+  d2psi_dt_dv(const SymmetricTensor<2, dim, NumberType> &t,
+              const Tensor<1, dim, NumberType> &         v)
+  {
+    return 2.0 * std::pow(det_t(t), 1) * 3.0 * std::pow(v_squ(v), 2) *
+           outer_product(dv_squ_dv(v),
+                         static_cast<Tensor<2, dim, NumberType>>(ddet_t_dt(t)));
+  };
+
+  static Tensor<2, dim, NumberType>
+  d2psi_dv_dv(const SymmetricTensor<2, dim, NumberType> &t,
+              const Tensor<1, dim, NumberType> &         v)
+  {
+    return std::pow(det_t(t), 2) * 3.0 *
+           (2.0 * std::pow(v_squ(v), 1) *
+              outer_product(dv_squ_dv(v), dv_squ_dv(v)) +
+            std::pow(v_squ(v), 2) * d2v_squ_dv_dv(v));
+  };
+};
+
+template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+void
+test_symmetric_tensor_vector_coupled()
+{
+  typedef AD::ScalarFunction<dim, ad_type_code, number_t> ADHelper;
+  typedef typename ADHelper::ad_type                      ADNumberType;
+  typedef typename ADHelper::scalar_type                  ScalarNumberType;
+
+  std::cout << "*** Test variables: SymmetricTensor + Vector (coupled), "
+            << "dim = " << Utilities::to_string(dim) << ", "
+            << "Type code: " << static_cast<int>(ad_type_code) << std::endl;
+
+  // Values computed from the AD energy function
+  ScalarNumberType             psi;
+  Vector<ScalarNumberType>     Dpsi;
+  FullMatrix<ScalarNumberType> D2psi;
+
+  // Function and its derivatives
+  typedef FunctionsTestSymmetricTensorVectorCoupled<dim, ADNumberType> func_ad;
+
+  // Setup the variable components and choose a value at which to
+  // evaluate the tape
+  const FEValuesExtractors::SymmetricTensor<2> t_dof(0);
+  const FEValuesExtractors::Vector             v_dof(
+    SymmetricTensor<2, dim>::n_independent_components);
+  const unsigned int n_AD_components =
+    SymmetricTensor<2, dim>::n_independent_components +
+    Tensor<1, dim>::n_independent_components;
+  ADHelper ad_helper(n_AD_components);
+  ad_helper.set_tape_buffer_sizes(); // Increase the buffer size from the
+                                     // default values
+
+  SymmetricTensor<2, dim, ScalarNumberType> t =
+    unit_symmetric_tensor<dim, ScalarNumberType>();
+  for (unsigned int i = 0; i < t.n_independent_components; ++i)
+    t[t.unrolled_to_component_indices(i)] += 0.165 * (i + 0.1);
+  Tensor<1, dim, ScalarNumberType> v;
+  for (unsigned int i = 0; i < dim; ++i)
+    v[i] = 1.3 + 0.5 * i;
+
+  const int  tape_no = 1;
+  const bool is_recording =
+    ad_helper.start_recording_operations(tape_no /*material_id*/,
+                                         true /*overwrite_tape*/,
+                                         true /*keep*/);
+  if (is_recording == true)
+    {
+      ad_helper.register_independent_variable(t, t_dof);
+      ad_helper.register_independent_variable(v, v_dof);
+
+      const SymmetricTensor<2, dim, ADNumberType> t_ad =
+        ad_helper.get_sensitive_variables(t_dof);
+      const Tensor<1, dim, ADNumberType> v_ad =
+        ad_helper.get_sensitive_variables(v_dof);
+
+      const ADNumberType psi(func_ad::psi(t_ad, v_ad));
+
+      ad_helper.register_dependent_variable(psi);
+      ad_helper.stop_recording_operations(false /*write_tapes_to_file*/);
+
+      std::cout << "Recorded data..." << std::endl;
+      std::cout << "independent variable values: " << std::flush;
+      ad_helper.print_values(std::cout);
+      std::cout << "t_ad: " << t_ad << std::endl;
+      std::cout << "v_ad: " << v_ad << std::endl;
+      std::cout << "psi: " << psi << std::endl;
+      std::cout << std::endl;
+    }
+  else
+    {
+      Assert(is_recording == true, ExcInternalError());
+    }
+
+  // Do some work :-)
+  // Set a new evaluation point
+  if (AD::ADNumberTraits<ADNumberType>::is_taped == true)
+    {
+      std::cout
+        << "Using tape with different values for independent variables..."
+        << std::endl;
+      ad_helper.activate_recorded_tape(tape_no);
+      t *= 0.9;
+      v *= 0.63;
+      ad_helper.set_independent_variable(t, t_dof);
+      ad_helper.set_independent_variable(v, v_dof);
+    }
+
+  std::cout << "independent variable values: " << std::flush;
+  ad_helper.print_values(std::cout);
+
+  // Compute the function value, gradient and hessian for the new evaluation
+  // point
+  psi = ad_helper.compute_value();
+  ad_helper.compute_gradient(Dpsi);
+  if (AD::ADNumberTraits<ADNumberType>::n_supported_derivative_levels >= 2)
+    {
+      ad_helper.compute_hessian(D2psi);
+    }
+
+  // Output the full stored function, gradient vector and hessian matrix
+  std::cout << "psi: " << psi << std::endl;
+  std::cout << "Dpsi: \n";
+  Dpsi.print(std::cout);
+  if (AD::ADNumberTraits<ADNumberType>::n_supported_derivative_levels >= 2)
+    {
+      std::cout << "D2psi: \n";
+      D2psi.print_formatted(std::cout, 3, true, 0, "0.0");
+    }
+
+  // Extract components of the solution
+  const SymmetricTensor<2, dim, ScalarNumberType> dpsi_dt =
+    ad_helper.extract_gradient_component(Dpsi, t_dof);
+  const Tensor<1, dim, ScalarNumberType> dpsi_dv =
+    ad_helper.extract_gradient_component(Dpsi, v_dof);
+  std::cout << "extracted Dpsi (t): " << dpsi_dt << "\n"
+            << "extracted Dpsi (v): " << dpsi_dv << "\n";
+
+  // Verify the result
+  typedef FunctionsTestSymmetricTensorVectorCoupled<dim, ScalarNumberType> func;
+  static const ScalarNumberType                                            tol =
+    1e5 * std::numeric_limits<ScalarNumberType>::epsilon();
+  std::cout << "dpsi_dt:            " << dpsi_dt << std::endl;
+  std::cout << "func::dpsi_dt(t,v): " << func::dpsi_dt(t, v) << std::endl;
+  std::cout << "diff: " << std::abs((dpsi_dt - func::dpsi_dt(t, v)).norm())
+            << std::endl;
+  std::cout << "dpsi_dv:            " << dpsi_dv << std::endl;
+  std::cout << "func::dpsi_dv(t,v): " << func::dpsi_dv(t, v) << std::endl;
+  std::cout << "diff: " << std::abs((dpsi_dv - func::dpsi_dv(t, v)).norm())
+            << std::endl;
+  Assert(std::abs(psi - func::psi(t, v)) < tol,
+         ExcMessage("No match for function value."));
+  Assert(std::abs((dpsi_dt - func::dpsi_dt(t, v)).norm()) < tol,
+         ExcMessage("No match for first derivative."));
+  Assert(std::abs((dpsi_dv - func::dpsi_dv(t, v)).norm()) < tol,
+         ExcMessage("No match for first derivative."));
+  if (AD::ADNumberTraits<ADNumberType>::n_supported_derivative_levels >= 2)
+    {
+      const SymmetricTensor<4, dim, ScalarNumberType> d2psi_dt_dt =
+        ad_helper.extract_hessian_component(D2psi, t_dof, t_dof);
+      const Tensor<3, dim, ScalarNumberType> d2psi_dv_dt =
+        ad_helper.extract_hessian_component(D2psi, t_dof, v_dof);
+      const Tensor<3, dim, ScalarNumberType> d2psi_dt_dv =
+        ad_helper.extract_hessian_component(D2psi, v_dof, t_dof);
+      const Tensor<2, dim, ScalarNumberType> d2psi_dv_dv =
+        ad_helper.extract_hessian_component(D2psi, v_dof, v_dof);
+      std::cout << "extracted Dpsi (t): " << dpsi_dt << "\n"
+                << "extracted Dpsi (v): " << dpsi_dv << "\n"
+                << "extracted D2psi (t,t): " << d2psi_dt_dt << "\n"
+                << "extracted D2psi (t,v): " << d2psi_dv_dt << "\n"
+                << "extracted D2psi (v,t): " << d2psi_dt_dv << "\n"
+                << "extracted D2psi (v,v): " << d2psi_dv_dv << "\n"
+                << std::endl;
+      std::cout << "d2psi_dt_dt:            " << d2psi_dt_dt << std::endl;
+      std::cout << "func::d2psi_dt_dt(t,v): " << func::d2psi_dt_dt(t, v)
+                << std::endl;
+      std::cout << "diff: "
+                << std::abs((d2psi_dt_dt - func::d2psi_dt_dt(t, v)).norm())
+                << std::endl;
+      std::cout << "d2psi_dv_dt:            " << d2psi_dv_dt << std::endl;
+      std::cout << "func::d2psi_dv_dt(t,v): " << func::d2psi_dv_dt(t, v)
+                << std::endl;
+      std::cout << "diff: "
+                << std::abs((d2psi_dv_dt - func::d2psi_dv_dt(t, v)).norm())
+                << std::endl;
+      std::cout << "d2psi_dt_dv:            " << d2psi_dt_dv << std::endl;
+      std::cout << "func::d2psi_dt_dv(t,v): " << func::d2psi_dt_dv(t, v)
+                << std::endl;
+      std::cout << "diff: "
+                << std::abs((d2psi_dt_dv - func::d2psi_dt_dv(t, v)).norm())
+                << std::endl;
+      std::cout << "d2psi_dv_dv:            " << d2psi_dv_dv << std::endl;
+      std::cout << "func::d2psi_dv_dv(t,v): " << func::d2psi_dv_dv(t, v)
+                << std::endl;
+      std::cout << "diff: "
+                << std::abs((d2psi_dv_dv - func::d2psi_dv_dv(t, v)).norm())
+                << std::endl;
+      Assert(std::abs((d2psi_dt_dt - func::d2psi_dt_dt(t, v)).norm()) < tol,
+             ExcMessage("No match for second derivative."));
+      Assert(std::abs((d2psi_dv_dt - func::d2psi_dv_dt(t, v)).norm()) < tol,
+             ExcMessage("No match for second derivative."));
+      Assert(std::abs((d2psi_dt_dv - func::d2psi_dt_dv(t, v)).norm()) < tol,
+             ExcMessage("No match for second derivative."));
+      Assert(std::abs((d2psi_dv_dv - func::d2psi_dv_dv(t, v)).norm()) < tol,
+             ExcMessage("No match for second derivative."));
+    }
+}
diff --git a/tests/ad_common_tests/helper_scalar_coupled_2_components_07.h b/tests/ad_common_tests/helper_scalar_coupled_2_components_07.h
new file mode 100644 (file)
index 0000000..8ce5d57
--- /dev/null
@@ -0,0 +1,361 @@
+// ---------------------------------------------------------------------
+//
+// Copyright (C) 2016 - 2018 by the deal.II authors
+//
+// This file is part of the deal.II library.
+//
+// The deal.II library is free software; you can use it, redistribute
+// it, and/or modify it under the terms of the GNU Lesser General
+// Public License as published by the Free Software Foundation; either
+// version 2.1 of the License, or (at your option) any later version.
+// The full text of the license can be found in the file LICENSE at
+// the top level of the deal.II distribution.
+//
+// ---------------------------------------------------------------------
+
+
+// Header file:
+// Evaluation of a coupled system (symmetric tensor + tensor components)
+// using a helper class
+
+#include <deal.II/base/logstream.h>
+#include <deal.II/base/symmetric_tensor.h>
+#include <deal.II/base/tensor.h>
+
+#include <deal.II/differentiation/ad.h>
+
+#include <deal.II/fe/fe_values_extractors.h>
+
+#include <deal.II/lac/full_matrix.h>
+#include <deal.II/lac/vector.h>
+
+#include <iostream>
+
+#include "../tests.h"
+
+using namespace dealii;
+namespace AD = dealii::Differentiation::AD;
+
+// Function and its derivatives
+template <int dim, typename NumberType>
+struct FunctionsTestSymmetricTensorTensorCoupled
+{
+  static NumberType
+  det_t(const SymmetricTensor<2, dim, NumberType> &t)
+  {
+    return determinant(t);
+  }
+
+  static SymmetricTensor<2, dim, NumberType>
+  ddet_t_dt(const SymmetricTensor<2, dim, NumberType> &t)
+  {
+    const SymmetricTensor<2, dim, NumberType> t_inv =
+      symmetrize(invert(static_cast<Tensor<2, dim, NumberType>>(t)));
+    return det_t(t) * t_inv;
+  }
+
+  static SymmetricTensor<4, dim, NumberType>
+  d2det_t_dt_dt(const SymmetricTensor<2, dim, NumberType> &t)
+  {
+    const SymmetricTensor<2, dim, NumberType> t_inv =
+      symmetrize(invert(static_cast<Tensor<2, dim, NumberType>>(t)));
+    SymmetricTensor<4, dim, NumberType> dt_inv_dt;
+    // https://en.wikiversity.org/wiki/Introduction_to_Elasticity/Tensors#Derivative_of_the_determinant_of_a_tensor
+    for (unsigned int i = 0; i < dim; ++i)
+      for (unsigned int j = i; j < dim; ++j)
+        for (unsigned int k = 0; k < dim; ++k)
+          for (unsigned int l = k; l < dim; ++l)
+            dt_inv_dt[i][j][k][l] =
+              -0.5 * (t_inv[i][k] * t_inv[j][l] + t_inv[i][l] * t_inv[j][k]);
+
+    return det_t(t) * outer_product(t_inv, t_inv) + det_t(t) * dt_inv_dt;
+  }
+
+  // -------
+
+  static NumberType
+  det_t(const Tensor<2, dim, NumberType> &t)
+  {
+    return determinant(t);
+  }
+
+  static Tensor<2, dim, NumberType>
+  ddet_t_dt(const Tensor<2, dim, NumberType> &t)
+  {
+    return det_t(t) * transpose(invert(t));
+  }
+
+  static Tensor<4, dim, NumberType>
+  d2det_t_dt_dt(const Tensor<2, dim, NumberType> &t)
+  {
+    const Tensor<2, dim, NumberType> t_inv = invert(t);
+    Tensor<4, dim, NumberType>       dt_inv_trans_dt;
+    // https://en.wikiversity.org/wiki/Introduction_to_Elasticity/Tensors#Derivative_of_the_determinant_of_a_tensor
+    for (unsigned int i = 0; i < dim; ++i)
+      for (unsigned int j = 0; j < dim; ++j)
+        for (unsigned int k = 0; k < dim; ++k)
+          for (unsigned int l = 0; l < dim; ++l)
+            dt_inv_trans_dt[i][j][k][l] = -t_inv[l][i] * t_inv[j][k];
+
+    return det_t(t) * outer_product(transpose(t_inv), transpose(t_inv)) +
+           det_t(t) * dt_inv_trans_dt;
+  }
+
+  // -------
+
+  static NumberType
+  psi(const SymmetricTensor<2, dim, NumberType> &t1,
+      const Tensor<2, dim, NumberType> &         t2)
+  {
+    return std::pow(det_t(t1), 2) * std::pow(det_t(t2), 3);
+  };
+
+  static SymmetricTensor<2, dim, NumberType>
+  dpsi_dt1(const SymmetricTensor<2, dim, NumberType> &t1,
+           const Tensor<2, dim, NumberType> &         t2)
+  {
+    return SymmetricTensor<2, dim, NumberType>(
+      2.0 * std::pow(det_t(t1), 1) * ddet_t_dt(t1) * std::pow(det_t(t2), 3));
+  };
+
+  static Tensor<2, dim, NumberType>
+  dpsi_dt2(const SymmetricTensor<2, dim, NumberType> &t1,
+           const Tensor<2, dim, NumberType> &         t2)
+  {
+    return std::pow(det_t(t1), 2) * 3.0 * std::pow(det_t(t2), 2) *
+           ddet_t_dt(t2);
+  };
+
+  static SymmetricTensor<4, dim, NumberType>
+  d2psi_dt1_dt1(const SymmetricTensor<2, dim, NumberType> &t1,
+                const Tensor<2, dim, NumberType> &         t2)
+  {
+    return SymmetricTensor<4, dim, NumberType>(
+      2.0 * std::pow(det_t(t2), 3) *
+      (pow(det_t(t1), 0) * outer_product(ddet_t_dt(t1), ddet_t_dt(t1)) +
+       std::pow(det_t(t1), 1) * d2det_t_dt_dt(t1)));
+  };
+
+  static Tensor<4, dim, NumberType>
+  d2psi_dt2_dt1(const SymmetricTensor<2, dim, NumberType> &t1,
+                const Tensor<2, dim, NumberType> &         t2)
+  {
+    return 2.0 * std::pow(det_t(t1), 1) * 3.0 * std::pow(det_t(t2), 2) *
+           outer_product(static_cast<Tensor<2, dim, NumberType>>(ddet_t_dt(t1)),
+                         ddet_t_dt(t2));
+  };
+
+  static Tensor<4, dim, NumberType>
+  d2psi_dt1_dt2(const SymmetricTensor<2, dim, NumberType> &t1,
+                const Tensor<2, dim, NumberType> &         t2)
+  {
+    return 2.0 * std::pow(det_t(t1), 1) * 3.0 * std::pow(det_t(t2), 2) *
+           outer_product(ddet_t_dt(t2),
+                         static_cast<Tensor<2, dim, NumberType>>(
+                           ddet_t_dt(t1)));
+  };
+
+  static Tensor<4, dim, NumberType>
+  d2psi_dt2_dt2(const SymmetricTensor<2, dim, NumberType> &t1,
+                const Tensor<2, dim, NumberType> &         t2)
+  {
+    return std::pow(det_t(t1), 2) * 3.0 *
+           (2.0 * std::pow(det_t(t2), 1) *
+              outer_product(ddet_t_dt(t2), ddet_t_dt(t2)) +
+            std::pow(det_t(t2), 2) * d2det_t_dt_dt(t2));
+  };
+};
+
+template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+void
+test_symmetric_tensor_tensor_coupled()
+{
+  typedef AD::ScalarFunction<dim, ad_type_code, number_t> ADHelper;
+  typedef typename ADHelper::ad_type                      ADNumberType;
+  typedef typename ADHelper::scalar_type                  ScalarNumberType;
+
+  std::cout << "*** Test variables: SymmetricTensor + Tensor (coupled), "
+            << "dim = " << Utilities::to_string(dim) << ", "
+            << "Type code: " << static_cast<int>(ad_type_code) << std::endl;
+
+  // Values computed from the AD energy function
+  ScalarNumberType             psi;
+  Vector<ScalarNumberType>     Dpsi;
+  FullMatrix<ScalarNumberType> D2psi;
+
+  // Function and its derivatives
+  typedef FunctionsTestSymmetricTensorTensorCoupled<dim, ADNumberType> func_ad;
+
+  // Setup the variable components and choose a value at which to
+  // evaluate the tape
+  const FEValuesExtractors::SymmetricTensor<2> t1_dof(0);
+  const FEValuesExtractors::Tensor<2>          t2_dof(
+    SymmetricTensor<2, dim>::n_independent_components);
+  const unsigned int n_AD_components =
+    SymmetricTensor<2, dim>::n_independent_components +
+    Tensor<2, dim>::n_independent_components;
+  ADHelper ad_helper(n_AD_components);
+  ad_helper.set_tape_buffer_sizes(); // Increase the buffer size from the
+                                     // default values
+
+  SymmetricTensor<2, dim, ScalarNumberType> t1 =
+    unit_symmetric_tensor<dim, ScalarNumberType>();
+  for (unsigned int i = 0; i < t1.n_independent_components; ++i)
+    t1[t1.unrolled_to_component_indices(i)] += 0.125 * (i + 0.15);
+  Tensor<2, dim, ScalarNumberType> t2 =
+    unit_symmetric_tensor<dim, ScalarNumberType>();
+  for (unsigned int i = 0; i < t2.n_independent_components; ++i)
+    t2[t2.unrolled_to_component_indices(i)] += 0.24 * (i + 0.05);
+
+  const int  tape_no = 1;
+  const bool is_recording =
+    ad_helper.start_recording_operations(tape_no /*material_id*/,
+                                         true /*overwrite_tape*/,
+                                         true /*keep*/);
+  if (is_recording == true)
+    {
+      ad_helper.register_independent_variable(t1, t1_dof);
+      ad_helper.register_independent_variable(t2, t2_dof);
+
+      const SymmetricTensor<2, dim, ADNumberType> t1_ad =
+        ad_helper.get_sensitive_variables(t1_dof);
+      const Tensor<2, dim, ADNumberType> t2_ad =
+        ad_helper.get_sensitive_variables(t2_dof);
+
+      const ADNumberType psi(func_ad::psi(t1_ad, t2_ad));
+
+      ad_helper.register_dependent_variable(psi);
+      ad_helper.stop_recording_operations(false /*write_tapes_to_file*/);
+
+      std::cout << "Recorded data..." << std::endl;
+      std::cout << "independent variable values: " << std::flush;
+      ad_helper.print_values(std::cout);
+      std::cout << "t1_ad: " << t1_ad << std::endl;
+      std::cout << "t2_ad: " << t2_ad << std::endl;
+      std::cout << "psi: " << psi << std::endl;
+      std::cout << std::endl;
+    }
+  else
+    {
+      Assert(is_recording == true, ExcInternalError());
+    }
+
+  // Do some work :-)
+  // Set a new evaluation point
+  if (AD::ADNumberTraits<ADNumberType>::is_taped == true)
+    {
+      std::cout
+        << "Using tape with different values for independent variables..."
+        << std::endl;
+      ad_helper.activate_recorded_tape(tape_no);
+      t1 *= 1.17;
+      t2 *= 0.92;
+      ad_helper.set_independent_variable(t1, t1_dof);
+      ad_helper.set_independent_variable(t2, t2_dof);
+    }
+
+  std::cout << "independent variable values: " << std::flush;
+  ad_helper.print_values(std::cout);
+
+  // Compute the function value, gradient and hessian for the new evaluation
+  // point
+  psi = ad_helper.compute_value();
+  ad_helper.compute_gradient(Dpsi);
+  if (AD::ADNumberTraits<ADNumberType>::n_supported_derivative_levels >= 2)
+    {
+      ad_helper.compute_hessian(D2psi);
+    }
+
+  // Output the full stored function, gradient vector and hessian matrix
+  std::cout << "psi: " << psi << std::endl;
+  std::cout << "Dpsi: \n";
+  Dpsi.print(std::cout);
+  if (AD::ADNumberTraits<ADNumberType>::n_supported_derivative_levels >= 2)
+    {
+      std::cout << "D2psi: \n";
+      D2psi.print_formatted(std::cout, 3, true, 0, "0.0");
+    }
+
+  // Extract components of the solution
+  const SymmetricTensor<2, dim, ScalarNumberType> dpsi_dt1 =
+    ad_helper.extract_gradient_component(Dpsi, t1_dof);
+  const Tensor<2, dim, ScalarNumberType> dpsi_dt2 =
+    ad_helper.extract_gradient_component(Dpsi, t2_dof);
+  std::cout << "extracted Dpsi (t1): " << dpsi_dt1 << "\n"
+            << "extracted Dpsi (t2): " << dpsi_dt2 << "\n";
+
+  // Verify the result
+  typedef FunctionsTestSymmetricTensorTensorCoupled<dim, ScalarNumberType> func;
+  static const ScalarNumberType                                            tol =
+    1e5 * std::numeric_limits<ScalarNumberType>::epsilon();
+  std::cout << "dpsi_dt1:            " << dpsi_dt1 << std::endl;
+  std::cout << "func::dpsi_dt1(t1,t2): " << func::dpsi_dt1(t1, t2) << std::endl;
+  std::cout << "diff: " << std::abs((dpsi_dt1 - func::dpsi_dt1(t1, t2)).norm())
+            << std::endl;
+  std::cout << "dpsi_dt2:            " << dpsi_dt2 << std::endl;
+  std::cout << "func::dpsi_dt2(t1,t2): " << func::dpsi_dt2(t1, t2) << std::endl;
+  std::cout << "diff: " << std::abs((dpsi_dt2 - func::dpsi_dt2(t1, t2)).norm())
+            << std::endl;
+  Assert(std::abs(psi - func::psi(t1, t2)) < tol,
+         ExcMessage("No match for function value."));
+  Assert(std::abs((dpsi_dt1 - func::dpsi_dt1(t1, t2)).norm()) < tol,
+         ExcMessage("No match for first derivative."));
+  Assert(std::abs((dpsi_dt2 - func::dpsi_dt2(t1, t2)).norm()) < tol,
+         ExcMessage("No match for first derivative."));
+  if (AD::ADNumberTraits<ADNumberType>::n_supported_derivative_levels >= 2)
+    {
+      const SymmetricTensor<4, dim, ScalarNumberType> d2psi_dt1_dt1 =
+        ad_helper.extract_hessian_component(D2psi, t1_dof, t1_dof);
+      const Tensor<4, dim, ScalarNumberType> d2psi_dt2_dt1 =
+        ad_helper.extract_hessian_component(D2psi, t1_dof, t2_dof);
+      const Tensor<4, dim, ScalarNumberType> d2psi_dt1_dt2 =
+        ad_helper.extract_hessian_component(D2psi, t2_dof, t1_dof);
+      const Tensor<4, dim, ScalarNumberType> d2psi_dt2_dt2 =
+        ad_helper.extract_hessian_component(D2psi, t2_dof, t2_dof);
+      std::cout << "extracted D2psi (t1,t1): " << d2psi_dt1_dt1 << "\n"
+                << "extracted D2psi (t1,t2): " << d2psi_dt2_dt1 << "\n"
+                << "extracted D2psi (t2,t1): " << d2psi_dt1_dt2 << "\n"
+                << "extracted D2psi (t2,t2): " << d2psi_dt2_dt2 << "\n"
+                << std::endl;
+      std::cout << "d2psi_dt1_dt1:            " << d2psi_dt1_dt1 << std::endl;
+      std::cout << "func::d2psi_dt1_dt1(t1,t2): " << func::d2psi_dt1_dt1(t1, t2)
+                << std::endl;
+      std::cout << "diff: "
+                << std::abs(
+                     (d2psi_dt1_dt1 - func::d2psi_dt1_dt1(t1, t2)).norm())
+                << std::endl;
+      std::cout << "d2psi_dt2_dt1:            " << d2psi_dt2_dt1 << std::endl;
+      std::cout << "func::d2psi_dt2_dt1(t1,t2): " << func::d2psi_dt2_dt1(t1, t2)
+                << std::endl;
+      std::cout << "diff: "
+                << std::abs(
+                     (d2psi_dt2_dt1 - func::d2psi_dt2_dt1(t1, t2)).norm())
+                << std::endl;
+      std::cout << "d2psi_dt1_dt2:            " << d2psi_dt1_dt2 << std::endl;
+      std::cout << "func::d2psi_dt1_dt2(t1,t2): " << func::d2psi_dt1_dt2(t1, t2)
+                << std::endl;
+      std::cout << "diff: "
+                << std::abs(
+                     (d2psi_dt1_dt2 - func::d2psi_dt1_dt2(t1, t2)).norm())
+                << std::endl;
+      std::cout << "d2psi_dt2_dt2:            " << d2psi_dt2_dt2 << std::endl;
+      std::cout << "func::d2psi_dt2_dt2(t1,t2): " << func::d2psi_dt2_dt2(t1, t2)
+                << std::endl;
+      std::cout << "diff: "
+                << std::abs(
+                     (d2psi_dt2_dt2 - func::d2psi_dt2_dt2(t1, t2)).norm())
+                << std::endl;
+      Assert(std::abs((d2psi_dt1_dt1 - func::d2psi_dt1_dt1(t1, t2)).norm()) <
+               tol,
+             ExcMessage("No match for second derivative."));
+      Assert(std::abs((d2psi_dt2_dt1 - func::d2psi_dt2_dt1(t1, t2)).norm()) <
+               tol,
+             ExcMessage("No match for second derivative."));
+      Assert(std::abs((d2psi_dt1_dt2 - func::d2psi_dt1_dt2(t1, t2)).norm()) <
+               tol,
+             ExcMessage("No match for second derivative."));
+      Assert(std::abs((d2psi_dt2_dt2 - func::d2psi_dt2_dt2(t1, t2)).norm()) <
+               tol,
+             ExcMessage("No match for second derivative."));
+    }
+}
diff --git a/tests/ad_common_tests/helper_scalar_coupled_2_components_08.h b/tests/ad_common_tests/helper_scalar_coupled_2_components_08.h
new file mode 100644 (file)
index 0000000..971885d
--- /dev/null
@@ -0,0 +1,337 @@
+// ---------------------------------------------------------------------
+//
+// Copyright (C) 2016 - 2018 by the deal.II authors
+//
+// This file is part of the deal.II library.
+//
+// The deal.II library is free software; you can use it, redistribute
+// it, and/or modify it under the terms of the GNU Lesser General
+// Public License as published by the Free Software Foundation; either
+// version 2.1 of the License, or (at your option) any later version.
+// The full text of the license can be found in the file LICENSE at
+// the top level of the deal.II distribution.
+//
+// ---------------------------------------------------------------------
+
+
+// Header file:
+// Evaluation of a coupled system (symmetric tensor + symmetric tensor
+// components) using a helper class
+
+#include <deal.II/base/logstream.h>
+#include <deal.II/base/symmetric_tensor.h>
+#include <deal.II/base/tensor.h>
+
+#include <deal.II/differentiation/ad.h>
+
+#include <deal.II/fe/fe_values_extractors.h>
+
+#include <deal.II/lac/full_matrix.h>
+#include <deal.II/lac/vector.h>
+
+#include <iostream>
+
+#include "../tests.h"
+
+using namespace dealii;
+namespace AD = dealii::Differentiation::AD;
+
+// Function and its derivatives
+template <int dim, typename NumberType>
+struct FunctionsTestSymmetricTensorSymmetricTensorCoupled
+{
+  static NumberType
+  det_t(const SymmetricTensor<2, dim, NumberType> &t)
+  {
+    return determinant(t);
+  }
+
+  static SymmetricTensor<2, dim, NumberType>
+  ddet_t_dt(const SymmetricTensor<2, dim, NumberType> &t)
+  {
+    const SymmetricTensor<2, dim, NumberType> t_inv =
+      symmetrize(invert(static_cast<Tensor<2, dim, NumberType>>(t)));
+    return det_t(t) * t_inv;
+  }
+
+  static SymmetricTensor<4, dim, NumberType>
+  d2det_t_dt_dt(const SymmetricTensor<2, dim, NumberType> &t)
+  {
+    const SymmetricTensor<2, dim, NumberType> t_inv =
+      symmetrize(invert(static_cast<Tensor<2, dim, NumberType>>(t)));
+    SymmetricTensor<4, dim, NumberType> dt_inv_dt;
+    // https://en.wikiversity.org/wiki/Introduction_to_Elasticity/Tensors#Derivative_of_the_determinant_of_a_tensor
+    for (unsigned int i = 0; i < dim; ++i)
+      for (unsigned int j = i; j < dim; ++j)
+        for (unsigned int k = 0; k < dim; ++k)
+          for (unsigned int l = k; l < dim; ++l)
+            dt_inv_dt[i][j][k][l] =
+              -0.5 * (t_inv[i][k] * t_inv[j][l] + t_inv[i][l] * t_inv[j][k]);
+
+    return det_t(t) * outer_product(t_inv, t_inv) + det_t(t) * dt_inv_dt;
+  }
+
+  // -------
+
+  static NumberType
+  psi(const SymmetricTensor<2, dim, NumberType> &t1,
+      const SymmetricTensor<2, dim, NumberType> &t2)
+  {
+    return std::pow(det_t(t1), 2) * std::pow(det_t(t2), 3);
+  };
+
+  static SymmetricTensor<2, dim, NumberType>
+  dpsi_dt1(const SymmetricTensor<2, dim, NumberType> &t1,
+           const SymmetricTensor<2, dim, NumberType> &t2)
+  {
+    return SymmetricTensor<2, dim, NumberType>(
+      2.0 * std::pow(det_t(t1), 1) * ddet_t_dt(t1) * std::pow(det_t(t2), 3));
+  };
+
+  static SymmetricTensor<2, dim, NumberType>
+  dpsi_dt2(const SymmetricTensor<2, dim, NumberType> &t1,
+           const SymmetricTensor<2, dim, NumberType> &t2)
+  {
+    return SymmetricTensor<2, dim, NumberType>(
+      pow(det_t(t1), 2) * 3.0 * std::pow(det_t(t2), 2) * ddet_t_dt(t2));
+  };
+
+  static SymmetricTensor<4, dim, NumberType>
+  d2psi_dt1_dt1(const SymmetricTensor<2, dim, NumberType> &t1,
+                const SymmetricTensor<2, dim, NumberType> &t2)
+  {
+    return SymmetricTensor<4, dim, NumberType>(
+      2.0 * std::pow(det_t(t2), 3) *
+      (pow(det_t(t1), 0) * outer_product(ddet_t_dt(t1), ddet_t_dt(t1)) +
+       std::pow(det_t(t1), 1) * d2det_t_dt_dt(t1)));
+  };
+
+  static SymmetricTensor<4, dim, NumberType>
+  d2psi_dt2_dt1(const SymmetricTensor<2, dim, NumberType> &t1,
+                const SymmetricTensor<2, dim, NumberType> &t2)
+  {
+    return SymmetricTensor<4, dim, NumberType>(
+      2.0 * std::pow(det_t(t1), 1) * 3.0 * std::pow(det_t(t2), 2) *
+      outer_product(ddet_t_dt(t1), ddet_t_dt(t2)));
+  };
+
+  static SymmetricTensor<4, dim, NumberType>
+  d2psi_dt1_dt2(const SymmetricTensor<2, dim, NumberType> &t1,
+                const SymmetricTensor<2, dim, NumberType> &t2)
+  {
+    return SymmetricTensor<4, dim, NumberType>(
+      2.0 * std::pow(det_t(t1), 1) * 3.0 * std::pow(det_t(t2), 2) *
+      outer_product(ddet_t_dt(t2), ddet_t_dt(t1)));
+  };
+
+  static SymmetricTensor<4, dim, NumberType>
+  d2psi_dt2_dt2(const SymmetricTensor<2, dim, NumberType> &t1,
+                const SymmetricTensor<2, dim, NumberType> &t2)
+  {
+    return SymmetricTensor<4, dim, NumberType>(
+      pow(det_t(t1), 2) * 3.0 *
+      (2.0 * std::pow(det_t(t2), 1) *
+         outer_product(ddet_t_dt(t2), ddet_t_dt(t2)) +
+       std::pow(det_t(t2), 2) * d2det_t_dt_dt(t2)));
+  };
+};
+
+template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+void
+test_symmetric_tensor_symmetric_tensor_coupled()
+{
+  typedef AD::ScalarFunction<dim, ad_type_code, number_t> ADHelper;
+  typedef typename ADHelper::ad_type                      ADNumberType;
+  typedef typename ADHelper::scalar_type                  ScalarNumberType;
+
+  std::cout
+    << "*** Test variables: SymmetricTensor + SymmetricTensor (coupled), "
+    << "dim = " << Utilities::to_string(dim) << ", "
+    << "Type code: " << static_cast<int>(ad_type_code) << std::endl;
+
+  // Values computed from the AD energy function
+  ScalarNumberType             psi;
+  Vector<ScalarNumberType>     Dpsi;
+  FullMatrix<ScalarNumberType> D2psi;
+
+  // Function and its derivatives
+  typedef FunctionsTestSymmetricTensorSymmetricTensorCoupled<dim, ADNumberType>
+    func_ad;
+
+  // Setup the variable components and choose a value at which to
+  // evaluate the tape
+  const FEValuesExtractors::SymmetricTensor<2> t1_dof(0);
+  const FEValuesExtractors::SymmetricTensor<2> t2_dof(
+    SymmetricTensor<2, dim>::n_independent_components);
+  const unsigned int n_AD_components =
+    SymmetricTensor<2, dim>::n_independent_components +
+    SymmetricTensor<2, dim>::n_independent_components;
+  ADHelper ad_helper(n_AD_components);
+  ad_helper.set_tape_buffer_sizes(); // Increase the buffer size from the
+                                     // default values
+
+  SymmetricTensor<2, dim, ScalarNumberType> t1 =
+    unit_symmetric_tensor<dim, ScalarNumberType>();
+  for (unsigned int i = 0; i < t1.n_independent_components; ++i)
+    t1[t1.unrolled_to_component_indices(i)] += 0.135 * (i + 0.195);
+  SymmetricTensor<2, dim, ScalarNumberType> t2 =
+    unit_symmetric_tensor<dim, ScalarNumberType>();
+  for (unsigned int i = 0; i < t2.n_independent_components; ++i)
+    t2[t2.unrolled_to_component_indices(i)] += 0.18 * (i + 0.09);
+
+  const int  tape_no = 1;
+  const bool is_recording =
+    ad_helper.start_recording_operations(tape_no /*material_id*/,
+                                         true /*overwrite_tape*/,
+                                         true /*keep*/);
+  if (is_recording == true)
+    {
+      ad_helper.register_independent_variable(t1, t1_dof);
+      ad_helper.register_independent_variable(t2, t2_dof);
+
+      const SymmetricTensor<2, dim, ADNumberType> t1_ad =
+        ad_helper.get_sensitive_variables(t1_dof);
+      const SymmetricTensor<2, dim, ADNumberType> t2_ad =
+        ad_helper.get_sensitive_variables(t2_dof);
+
+      const ADNumberType psi(func_ad::psi(t1_ad, t2_ad));
+
+      ad_helper.register_dependent_variable(psi);
+      ad_helper.stop_recording_operations(false /*write_tapes_to_file*/);
+
+      std::cout << "Recorded data..." << std::endl;
+      std::cout << "independent variable values: " << std::flush;
+      ad_helper.print_values(std::cout);
+      std::cout << "t1_ad: " << t1_ad << std::endl;
+      std::cout << "t2_ad: " << t2_ad << std::endl;
+      std::cout << "psi: " << psi << std::endl;
+      std::cout << std::endl;
+    }
+  else
+    {
+      Assert(is_recording == true, ExcInternalError());
+    }
+
+  // Do some work :-)
+  // Set a new evaluation point
+  if (AD::ADNumberTraits<ADNumberType>::is_taped == true)
+    {
+      std::cout
+        << "Using tape with different values for independent variables..."
+        << std::endl;
+      ad_helper.activate_recorded_tape(tape_no);
+      t1 *= 1.17;
+      t2 *= 0.92;
+      ad_helper.set_independent_variable(t1, t1_dof);
+      ad_helper.set_independent_variable(t2, t2_dof);
+    }
+
+  std::cout << "independent variable values: " << std::flush;
+  ad_helper.print_values(std::cout);
+
+  // Compute the function value, gradient and hessian for the new evaluation
+  // point
+  psi = ad_helper.compute_value();
+  ad_helper.compute_gradient(Dpsi);
+  if (AD::ADNumberTraits<ADNumberType>::n_supported_derivative_levels >= 2)
+    {
+      ad_helper.compute_hessian(D2psi);
+    }
+
+  // Output the full stored function, gradient vector and hessian matrix
+  std::cout << "psi: " << psi << std::endl;
+  std::cout << "Dpsi: \n";
+  Dpsi.print(std::cout);
+  if (AD::ADNumberTraits<ADNumberType>::n_supported_derivative_levels >= 2)
+    {
+      std::cout << "D2psi: \n";
+      D2psi.print_formatted(std::cout, 3, true, 0, "0.0");
+    }
+
+  // Extract components of the solution
+  const SymmetricTensor<2, dim, ScalarNumberType> dpsi_dt1 =
+    ad_helper.extract_gradient_component(Dpsi, t1_dof);
+  const SymmetricTensor<2, dim, ScalarNumberType> dpsi_dt2 =
+    ad_helper.extract_gradient_component(Dpsi, t2_dof);
+  std::cout << "extracted Dpsi (t1): " << dpsi_dt1 << "\n"
+            << "extracted Dpsi (t2): " << dpsi_dt2 << "\n";
+
+  // Verify the result
+  typedef FunctionsTestSymmetricTensorSymmetricTensorCoupled<dim,
+                                                             ScalarNumberType>
+                                func;
+  static const ScalarNumberType tol =
+    1e5 * std::numeric_limits<ScalarNumberType>::epsilon();
+  std::cout << "dpsi_dt1:            " << dpsi_dt1 << std::endl;
+  std::cout << "func::dpsi_dt1(t1,t2): " << func::dpsi_dt1(t1, t2) << std::endl;
+  std::cout << "diff: " << std::abs((dpsi_dt1 - func::dpsi_dt1(t1, t2)).norm())
+            << std::endl;
+  std::cout << "dpsi_dt2:            " << dpsi_dt2 << std::endl;
+  std::cout << "func::dpsi_dt2(t1,t2): " << func::dpsi_dt2(t1, t2) << std::endl;
+  std::cout << "diff: " << std::abs((dpsi_dt2 - func::dpsi_dt2(t1, t2)).norm())
+            << std::endl;
+  Assert(std::abs(psi - func::psi(t1, t2)) < tol,
+         ExcMessage("No match for function value."));
+  Assert(std::abs((dpsi_dt1 - func::dpsi_dt1(t1, t2)).norm()) < tol,
+         ExcMessage("No match for first derivative."));
+  Assert(std::abs((dpsi_dt2 - func::dpsi_dt2(t1, t2)).norm()) < tol,
+         ExcMessage("No match for first derivative."));
+  if (AD::ADNumberTraits<ADNumberType>::n_supported_derivative_levels >= 2)
+    {
+      const SymmetricTensor<4, dim, ScalarNumberType> d2psi_dt1_dt1 =
+        ad_helper.extract_hessian_component(D2psi, t1_dof, t1_dof);
+      const SymmetricTensor<4, dim, ScalarNumberType> d2psi_dt2_dt1 =
+        ad_helper.extract_hessian_component(D2psi, t1_dof, t2_dof);
+      const SymmetricTensor<4, dim, ScalarNumberType> d2psi_dt1_dt2 =
+        ad_helper.extract_hessian_component(D2psi, t2_dof, t1_dof);
+      const SymmetricTensor<4, dim, ScalarNumberType> d2psi_dt2_dt2 =
+        ad_helper.extract_hessian_component(D2psi, t2_dof, t2_dof);
+      std::cout << "extracted Dpsi (t1): " << dpsi_dt1 << "\n"
+                << "extracted Dpsi (t2): " << dpsi_dt2 << "\n"
+                << "extracted D2psi (t1,t1): " << d2psi_dt1_dt1 << "\n"
+                << "extracted D2psi (t1,t2): " << d2psi_dt2_dt1 << "\n"
+                << "extracted D2psi (t2,t1): " << d2psi_dt1_dt2 << "\n"
+                << "extracted D2psi (t2,t2): " << d2psi_dt2_dt2 << "\n"
+                << std::endl;
+      std::cout << "d2psi_dt1_dt1:            " << d2psi_dt1_dt1 << std::endl;
+      std::cout << "func::d2psi_dt1_dt1(t1,t2): " << func::d2psi_dt1_dt1(t1, t2)
+                << std::endl;
+      std::cout << "diff: "
+                << std::abs(
+                     (d2psi_dt1_dt1 - func::d2psi_dt1_dt1(t1, t2)).norm())
+                << std::endl;
+      std::cout << "d2psi_dt2_dt1:            " << d2psi_dt2_dt1 << std::endl;
+      std::cout << "func::d2psi_dt2_dt1(t1,t2): " << func::d2psi_dt2_dt1(t1, t2)
+                << std::endl;
+      std::cout << "diff: "
+                << std::abs(
+                     (d2psi_dt2_dt1 - func::d2psi_dt2_dt1(t1, t2)).norm())
+                << std::endl;
+      std::cout << "d2psi_dt1_dt2:            " << d2psi_dt1_dt2 << std::endl;
+      std::cout << "func::d2psi_dt1_dt2(t1,t2): " << func::d2psi_dt1_dt2(t1, t2)
+                << std::endl;
+      std::cout << "diff: "
+                << std::abs(
+                     (d2psi_dt1_dt2 - func::d2psi_dt1_dt2(t1, t2)).norm())
+                << std::endl;
+      std::cout << "d2psi_dt2_dt2:            " << d2psi_dt2_dt2 << std::endl;
+      std::cout << "func::d2psi_dt2_dt2(t1,t2): " << func::d2psi_dt2_dt2(t1, t2)
+                << std::endl;
+      std::cout << "diff: "
+                << std::abs(
+                     (d2psi_dt2_dt2 - func::d2psi_dt2_dt2(t1, t2)).norm())
+                << std::endl;
+      Assert(std::abs((d2psi_dt1_dt1 - func::d2psi_dt1_dt1(t1, t2)).norm()) <
+               tol,
+             ExcMessage("No match for second derivative."));
+      Assert(std::abs((d2psi_dt2_dt1 - func::d2psi_dt2_dt1(t1, t2)).norm()) <
+               tol,
+             ExcMessage("No match for second derivative."));
+      Assert(std::abs((d2psi_dt1_dt2 - func::d2psi_dt1_dt2(t1, t2)).norm()) <
+               tol,
+             ExcMessage("No match for second derivative."));
+      Assert(std::abs((d2psi_dt2_dt2 - func::d2psi_dt2_dt2(t1, t2)).norm()) <
+               tol,
+             ExcMessage("No match for second derivative."));
+    }
+}
diff --git a/tests/ad_common_tests/helper_scalar_coupled_3_components_01.h b/tests/ad_common_tests/helper_scalar_coupled_3_components_01.h
new file mode 100644 (file)
index 0000000..b12facf
--- /dev/null
@@ -0,0 +1,421 @@
+// ---------------------------------------------------------------------
+//
+// Copyright (C) 2016 - 2018 by the deal.II authors
+//
+// This file is part of the deal.II library.
+//
+// The deal.II library is free software; you can use it, redistribute
+// it, and/or modify it under the terms of the GNU Lesser General
+// Public License as published by the Free Software Foundation; either
+// version 2.1 of the License, or (at your option) any later version.
+// The full text of the license can be found in the file LICENSE at
+// the top level of the deal.II distribution.
+//
+// ---------------------------------------------------------------------
+
+
+// Header file:
+// Evaluation of a coupled system (tensor + vector + scalar components)
+// using a helper class
+
+#include <deal.II/base/conditional_ostream.h>
+#include <deal.II/base/logstream.h>
+#include <deal.II/base/symmetric_tensor.h>
+#include <deal.II/base/tensor.h>
+
+#include <deal.II/differentiation/ad.h>
+
+#include <deal.II/fe/fe_values_extractors.h>
+
+#include <deal.II/lac/full_matrix.h>
+#include <deal.II/lac/vector.h>
+
+#include <iostream>
+
+#include "../tests.h"
+
+using namespace dealii;
+namespace AD = dealii::Differentiation::AD;
+
+// Function and its derivatives
+template <int dim, typename NumberType>
+struct FunctionsTestTensorVectorScalarCoupled
+{
+  static NumberType
+  det_t(const Tensor<2, dim, NumberType> &t)
+  {
+    return determinant(t);
+  }
+
+  static Tensor<2, dim, NumberType>
+  ddet_t_dt(const Tensor<2, dim, NumberType> &t)
+  {
+    return det_t(t) * transpose(invert(t));
+  }
+
+  static Tensor<4, dim, NumberType>
+  d2det_t_dt_dt(const Tensor<2, dim, NumberType> &t)
+  {
+    const Tensor<2, dim, NumberType> t_inv = invert(t);
+    Tensor<4, dim, NumberType>       dt_inv_trans_dt;
+    // https://en.wikiversity.org/wiki/Introduction_to_Elasticity/Tensors#Derivative_of_the_determinant_of_a_tensor
+    for (unsigned int i = 0; i < dim; ++i)
+      for (unsigned int j = 0; j < dim; ++j)
+        for (unsigned int k = 0; k < dim; ++k)
+          for (unsigned int l = 0; l < dim; ++l)
+            dt_inv_trans_dt[i][j][k][l] = -t_inv[l][i] * t_inv[j][k];
+
+    return det_t(t) * outer_product(transpose(t_inv), transpose(t_inv)) +
+           det_t(t) * dt_inv_trans_dt;
+  }
+
+  static NumberType
+  v_squ(const Tensor<1, dim, NumberType> &v)
+  {
+    return v * v;
+  }
+
+  static Tensor<1, dim, NumberType>
+  dv_squ_dv(const Tensor<1, dim, NumberType> &v)
+  {
+    return 2.0 * v;
+  }
+
+  static Tensor<2, dim, NumberType>
+  d2v_squ_dv_dv(const Tensor<1, dim, NumberType> &v)
+  {
+    static const Tensor<2, dim, NumberType> I(
+      unit_symmetric_tensor<dim, NumberType>());
+    return 2.0 * I;
+  }
+
+  // --------
+
+  static const double sf;
+
+  static NumberType
+  psi(const Tensor<2, dim, NumberType> &t,
+      const Tensor<1, dim, NumberType> &v,
+      const NumberType &                s)
+  {
+    return std::pow(det_t(t), 2) * std::pow(v_squ(v), 3) * std::pow(s, sf);
+  };
+
+  static Tensor<2, dim, NumberType>
+  dpsi_dt(const Tensor<2, dim, NumberType> &t,
+          const Tensor<1, dim, NumberType> &v,
+          const NumberType &                s)
+  {
+    return 2.0 * std::pow(det_t(t), 1) * ddet_t_dt(t) * std::pow(v_squ(v), 3) *
+           std::pow(s, sf);
+  };
+
+  static Tensor<1, dim, NumberType>
+  dpsi_dv(const Tensor<2, dim, NumberType> &t,
+          const Tensor<1, dim, NumberType> &v,
+          const NumberType &                s)
+  {
+    return std::pow(det_t(t), 2) * 3.0 * std::pow(v_squ(v), 2) * dv_squ_dv(v) *
+           std::pow(s, sf);
+  };
+
+  static NumberType
+  dpsi_ds(const Tensor<2, dim, NumberType> &t,
+          const Tensor<1, dim, NumberType> &v,
+          const NumberType &                s)
+  {
+    return std::pow(det_t(t), 2) * std::pow(v_squ(v), 3) * sf *
+           std::pow(s, sf - 1.0);
+  };
+
+  static Tensor<4, dim, NumberType>
+  d2psi_dt_dt(const Tensor<2, dim, NumberType> &t,
+              const Tensor<1, dim, NumberType> &v,
+              const NumberType &                s)
+  {
+    return 2.0 * std::pow(v_squ(v), 3) *
+           (pow(det_t(t), 0) * outer_product(ddet_t_dt(t), ddet_t_dt(t)) +
+            std::pow(det_t(t), 1) * d2det_t_dt_dt(t)) *
+           std::pow(s, sf);
+  };
+
+  static Tensor<3, dim, NumberType>
+  d2psi_dv_dt(const Tensor<2, dim, NumberType> &t,
+              const Tensor<1, dim, NumberType> &v,
+              const NumberType &                s)
+  {
+    return 2.0 * std::pow(det_t(t), 1) * 3.0 * std::pow(v_squ(v), 2) *
+           outer_product(ddet_t_dt(t), dv_squ_dv(v)) * std::pow(s, sf);
+  };
+
+  static Tensor<2, dim, NumberType>
+  d2psi_ds_dt(const Tensor<2, dim, NumberType> &t,
+              const Tensor<1, dim, NumberType> &v,
+              const NumberType &                s)
+  {
+    return 2.0 * std::pow(det_t(t), 1) * ddet_t_dt(t) * std::pow(v_squ(v), 3) *
+           sf * std::pow(s, sf - 1.0);
+  };
+
+  static Tensor<3, dim, NumberType>
+  d2psi_dt_dv(const Tensor<2, dim, NumberType> &t,
+              const Tensor<1, dim, NumberType> &v,
+              const NumberType &                s)
+  {
+    return 2.0 * std::pow(det_t(t), 1) * 3.0 * std::pow(v_squ(v), 2) *
+           outer_product(dv_squ_dv(v), ddet_t_dt(t)) * std::pow(s, sf);
+  };
+
+  static Tensor<2, dim, NumberType>
+  d2psi_dv_dv(const Tensor<2, dim, NumberType> &t,
+              const Tensor<1, dim, NumberType> &v,
+              const NumberType &                s)
+  {
+    return std::pow(det_t(t), 2) * 3.0 *
+           (2.0 * std::pow(v_squ(v), 1) *
+              outer_product(dv_squ_dv(v), dv_squ_dv(v)) +
+            std::pow(v_squ(v), 2) * d2v_squ_dv_dv(v)) *
+           std::pow(s, sf);
+  };
+
+  static Tensor<1, dim, NumberType>
+  d2psi_ds_dv(const Tensor<2, dim, NumberType> &t,
+              const Tensor<1, dim, NumberType> &v,
+              const NumberType &                s)
+  {
+    return std::pow(det_t(t), 2) * 3.0 * std::pow(v_squ(v), 2) * dv_squ_dv(v) *
+           sf * std::pow(s, sf - 1.0);
+  };
+
+  static Tensor<2, dim, NumberType>
+  d2psi_dt_ds(const Tensor<2, dim, NumberType> &t,
+              const Tensor<1, dim, NumberType> &v,
+              const NumberType &                s)
+  {
+    return 2.0 * std::pow(det_t(t), 1) * ddet_t_dt(t) * std::pow(v_squ(v), 3) *
+           sf * std::pow(s, sf - 1.0);
+  };
+
+  static Tensor<1, dim, NumberType>
+  d2psi_dv_ds(const Tensor<2, dim, NumberType> &t,
+              const Tensor<1, dim, NumberType> &v,
+              const NumberType &                s)
+  {
+    return std::pow(det_t(t), 2) * 3.0 * std::pow(v_squ(v), 2) * dv_squ_dv(v) *
+           sf * std::pow(s, sf - 1.0);
+  };
+
+  static NumberType
+  d2psi_ds_ds(const Tensor<2, dim, NumberType> &t,
+              const Tensor<1, dim, NumberType> &v,
+              const NumberType &                s)
+  {
+    return std::pow(det_t(t), 2) * std::pow(v_squ(v), 3) * sf * (sf - 1.0) *
+           std::pow(s, sf - 2.0);
+  };
+};
+
+template <int dim, typename NumberType>
+const double FunctionsTestTensorVectorScalarCoupled<dim, NumberType>::sf = 2.2;
+
+template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+void
+test_tensor_vector_scalar_coupled()
+{
+  typedef AD::ScalarFunction<dim, ad_type_code, number_t> ADHelper;
+  typedef typename ADHelper::ad_type                      ADNumberType;
+  typedef typename ADHelper::scalar_type                  ScalarNumberType;
+
+  const unsigned int this_mpi_process =
+    Utilities::MPI::this_mpi_process(MPI_COMM_WORLD);
+  ConditionalOStream pcout(deallog.get_console(), this_mpi_process == 0);
+
+  pcout << "*** Test variables: Tensor + Vector + Scalar (coupled), "
+        << "dim = " << Utilities::to_string(dim) << ", "
+        << "Type code: " << static_cast<int>(ad_type_code) << std::endl;
+
+  // Values computed from the AD energy function
+  ScalarNumberType             psi;
+  Vector<ScalarNumberType>     Dpsi;
+  FullMatrix<ScalarNumberType> D2psi;
+
+  // Function and its derivatives
+  typedef FunctionsTestTensorVectorScalarCoupled<dim, ADNumberType> func_ad;
+
+  // Setup the variable components and choose a value at which to
+  // evaluate the tape
+  const FEValuesExtractors::Tensor<2> t_dof(0);
+  const FEValuesExtractors::Vector    v_dof(
+    Tensor<2, dim>::n_independent_components);
+  const FEValuesExtractors::Scalar s_dof(
+    Tensor<2, dim>::n_independent_components +
+    Tensor<1, dim>::n_independent_components);
+  const unsigned int n_AD_components =
+    Tensor<2, dim>::n_independent_components +
+    Tensor<1, dim>::n_independent_components + 1;
+  ADHelper ad_helper(n_AD_components);
+  ad_helper.set_tape_buffer_sizes(); // Increase the buffer size from the
+                                     // default values
+
+  Tensor<2, dim, ScalarNumberType> t =
+    unit_symmetric_tensor<dim, ScalarNumberType>();
+  for (unsigned int i = 0; i < t.n_independent_components; ++i)
+    t[t.unrolled_to_component_indices(i)] += 0.11 * (i + 0.125);
+  Tensor<1, dim, ScalarNumberType> v;
+  ScalarNumberType                 s = 0.57;
+  for (unsigned int i = 0; i < dim; ++i)
+    v[i] = 0.275 * (1.0 + i);
+
+  const int  tape_no = 1;
+  const bool is_recording =
+    ad_helper.start_recording_operations(tape_no /*material_id*/,
+                                         true /*overwrite_tape*/,
+                                         true /*keep*/);
+  if (is_recording == true)
+    {
+      ad_helper.register_independent_variable(t, t_dof);
+      ad_helper.register_independent_variable(v, v_dof);
+      ad_helper.register_independent_variable(s, s_dof);
+
+      const Tensor<2, dim, ADNumberType> t_ad =
+        ad_helper.get_sensitive_variables(t_dof);
+      const Tensor<1, dim, ADNumberType> v_ad =
+        ad_helper.get_sensitive_variables(v_dof);
+      ADNumberType s_ad = ad_helper.get_sensitive_variables(s_dof);
+
+      const ADNumberType psi(func_ad::psi(t_ad, v_ad, s_ad));
+
+      ad_helper.register_dependent_variable(psi);
+      ad_helper.stop_recording_operations(false /*write_tapes_to_file*/);
+
+      pcout << "Recorded data..." << std::endl;
+      pcout << "independent variable values: " << std::flush;
+      if (this_mpi_process == 0)
+        ad_helper.print_values(pcout.get_stream());
+      pcout << "t_ad: " << t_ad << std::endl;
+      pcout << "v_ad: " << v_ad << std::endl;
+      pcout << "s_ad: " << s_ad << std::endl;
+      pcout << "psi: " << psi << std::endl;
+      pcout << std::endl;
+    }
+  else
+    {
+      Assert(is_recording == true, ExcInternalError());
+    }
+
+  // Do some work :-)
+  // Set a new evaluation point
+  if (AD::ADNumberTraits<ADNumberType>::is_taped == true)
+    {
+      pcout << "Using tape with different values for independent variables..."
+            << std::endl;
+      ad_helper.activate_recorded_tape(tape_no);
+      t *= 0.9;
+      v *= 0.63;
+      s *= 1.21;
+      ad_helper.set_independent_variable(t, t_dof);
+      ad_helper.set_independent_variable(v, v_dof);
+      ad_helper.set_independent_variable(s, s_dof);
+    }
+
+  pcout << "independent variable values: " << std::flush;
+  if (this_mpi_process == 0)
+    ad_helper.print_values(pcout.get_stream());
+
+  // Compute the function value, gradient and hessian for the new evaluation
+  // point
+  psi = ad_helper.compute_value();
+  ad_helper.compute_gradient(Dpsi);
+  if (AD::ADNumberTraits<ADNumberType>::n_supported_derivative_levels >= 2)
+    {
+      ad_helper.compute_hessian(D2psi);
+    }
+
+  // Output the full stored function, gradient vector and hessian matrix
+  pcout << "psi: " << psi << std::endl;
+  pcout << "Dpsi: \n";
+  if (this_mpi_process == 0)
+    Dpsi.print(pcout.get_stream());
+  if (AD::ADNumberTraits<ADNumberType>::n_supported_derivative_levels >= 2)
+    {
+      pcout << "D2psi: \n";
+      if (this_mpi_process == 0)
+        D2psi.print_formatted(pcout.get_stream(), 3, true, 0, "0.0");
+    }
+
+  // Extract components of the solution
+  const Tensor<2, dim, ScalarNumberType> dpsi_dt =
+    ad_helper.extract_gradient_component(Dpsi, t_dof);
+  const Tensor<1, dim, ScalarNumberType> dpsi_dv =
+    ad_helper.extract_gradient_component(Dpsi, v_dof);
+  const Tensor<0, dim, ScalarNumberType> dpsi_ds =
+    ad_helper.extract_gradient_component(Dpsi, s_dof);
+  pcout << "extracted Dpsi (t): " << dpsi_dt << "\n"
+        << "extracted Dpsi (v): " << dpsi_dv << "\n"
+        << "extracted Dpsi (s): " << dpsi_ds << "\n";
+
+  // Verify the result
+  typedef FunctionsTestTensorVectorScalarCoupled<dim, ScalarNumberType> func;
+  static const ScalarNumberType                                         tol =
+    1e5 * std::numeric_limits<ScalarNumberType>::epsilon();
+  Assert(std::abs(psi - func::psi(t, v, s)) < tol,
+         ExcMessage("No match for function value."));
+  Assert(std::abs((dpsi_dt - func::dpsi_dt(t, v, s)).norm()) < tol,
+         ExcMessage("No match for first derivative."));
+  Assert(std::abs((dpsi_dv - func::dpsi_dv(t, v, s)).norm()) < tol,
+         ExcMessage("No match for first derivative."));
+  Assert(std::abs(dpsi_ds - func::dpsi_ds(t, v, s)) < tol,
+         ExcMessage("No match for first derivative."));
+  if (AD::ADNumberTraits<ADNumberType>::n_supported_derivative_levels >= 2)
+    {
+      const Tensor<4, dim, ScalarNumberType> d2psi_dt_dt =
+        ad_helper.extract_hessian_component(D2psi, t_dof, t_dof);
+      const Tensor<3, dim, ScalarNumberType> d2psi_dv_dt =
+        ad_helper.extract_hessian_component(D2psi, t_dof, v_dof);
+      const Tensor<2, dim, ScalarNumberType> d2psi_ds_dt =
+        ad_helper.extract_hessian_component(D2psi, t_dof, s_dof);
+      const Tensor<3, dim, ScalarNumberType> d2psi_dt_dv =
+        ad_helper.extract_hessian_component(D2psi, v_dof, t_dof);
+      const Tensor<2, dim, ScalarNumberType> d2psi_dv_dv =
+        ad_helper.extract_hessian_component(D2psi, v_dof, v_dof);
+      const Tensor<1, dim, ScalarNumberType> d2psi_ds_dv =
+        ad_helper.extract_hessian_component(D2psi, v_dof, s_dof);
+      const Tensor<2, dim, ScalarNumberType> d2psi_dt_ds =
+        ad_helper.extract_hessian_component(D2psi, s_dof, t_dof);
+      const Tensor<1, dim, ScalarNumberType> d2psi_dv_ds =
+        ad_helper.extract_hessian_component(D2psi, s_dof, v_dof);
+      const Tensor<0, dim, ScalarNumberType> d2psi_ds_ds =
+        ad_helper.extract_hessian_component(D2psi, s_dof, s_dof);
+      pcout << "extracted Dpsi (t): " << dpsi_dt << "\n"
+            << "extracted Dpsi (v): " << dpsi_dv << "\n"
+            << "extracted Dpsi (s): " << dpsi_ds << "\n"
+            << "extracted D2psi (t,t): " << d2psi_dt_dt << "\n"
+            << "extracted D2psi (t,v): " << d2psi_dv_dt << "\n"
+            << "extracted D2psi (t,s): " << d2psi_ds_dt << "\n"
+            << "extracted D2psi (v,t): " << d2psi_dt_dv << "\n"
+            << "extracted D2psi (v,v): " << d2psi_dv_dv << "\n"
+            << "extracted D2psi (v,s): " << d2psi_ds_dv << "\n"
+            << "extracted D2psi (s,t): " << d2psi_dt_ds << "\n"
+            << "extracted D2psi (s,v): " << d2psi_dv_ds << "\n"
+            << "extracted D2psi (s,s): " << d2psi_ds_ds << "\n"
+            << std::endl;
+      Assert(std::abs((d2psi_dt_dt - func::d2psi_dt_dt(t, v, s)).norm()) < tol,
+             ExcMessage("No match for second derivative."));
+      Assert(std::abs((d2psi_dv_dt - func::d2psi_dv_dt(t, v, s)).norm()) < tol,
+             ExcMessage("No match for second derivative."));
+      Assert(std::abs((d2psi_ds_dt - func::d2psi_ds_dt(t, v, s)).norm()) < tol,
+             ExcMessage("No match for second derivative."));
+      Assert(std::abs((d2psi_dt_dv - func::d2psi_dt_dv(t, v, s)).norm()) < tol,
+             ExcMessage("No match for second derivative."));
+      Assert(std::abs((d2psi_dv_dv - func::d2psi_dv_dv(t, v, s)).norm()) < tol,
+             ExcMessage("No match for second derivative."));
+      Assert(std::abs((d2psi_ds_dv - func::d2psi_ds_dv(t, v, s)).norm()) < tol,
+             ExcMessage("No match for second derivative."));
+      Assert(std::abs((d2psi_dt_ds - func::d2psi_dt_ds(t, v, s)).norm()) < tol,
+             ExcMessage("No match for second derivative."));
+      Assert(std::abs((d2psi_dv_ds - func::d2psi_dv_ds(t, v, s)).norm()) < tol,
+             ExcMessage("No match for second derivative."));
+      Assert(std::abs(d2psi_ds_ds - func::d2psi_ds_ds(t, v, s)) < tol,
+             ExcMessage("No match for second derivative."));
+    }
+}
diff --git a/tests/ad_common_tests/helper_scalar_coupled_3_components_01_tapeless_only.h b/tests/ad_common_tests/helper_scalar_coupled_3_components_01_tapeless_only.h
new file mode 100644 (file)
index 0000000..fe265ef
--- /dev/null
@@ -0,0 +1,400 @@
+// ---------------------------------------------------------------------
+//
+// Copyright (C) 2016 - 2018 by the deal.II authors
+//
+// This file is part of the deal.II library.
+//
+// The deal.II library is free software; you can use it, redistribute
+// it, and/or modify it under the terms of the GNU Lesser General
+// Public License as published by the Free Software Foundation; either
+// version 2.1 of the License, or (at your option) any later version.
+// The full text of the license can be found in the file LICENSE at
+// the top level of the deal.II distribution.
+//
+// ---------------------------------------------------------------------
+
+
+// Header file:
+// Evaluation of a coupled system (tensor + vector + scalar components)
+// using a helper class.
+// This test is based off of helper_scalar_coupled_3_components_01.h, and checks
+// that everything still works in tapeless only mode (i.e. when the
+// start_recording_operations and stop_recording_operations calls are
+// removed).
+
+#include <deal.II/base/conditional_ostream.h>
+#include <deal.II/base/logstream.h>
+#include <deal.II/base/symmetric_tensor.h>
+#include <deal.II/base/tensor.h>
+
+#include <deal.II/differentiation/ad.h>
+
+#include <deal.II/fe/fe_values_extractors.h>
+
+#include <deal.II/lac/full_matrix.h>
+#include <deal.II/lac/vector.h>
+
+#include <iostream>
+
+#include "../tests.h"
+
+using namespace dealii;
+namespace AD = dealii::Differentiation::AD;
+
+// Function and its derivatives
+template <int dim, typename NumberType>
+struct FunctionsTestTensorVectorScalarCoupled
+{
+  static NumberType
+  det_t(const Tensor<2, dim, NumberType> &t)
+  {
+    return determinant(t);
+  }
+
+  static Tensor<2, dim, NumberType>
+  ddet_t_dt(const Tensor<2, dim, NumberType> &t)
+  {
+    return det_t(t) * transpose(invert(t));
+  }
+
+  static Tensor<4, dim, NumberType>
+  d2det_t_dt_dt(const Tensor<2, dim, NumberType> &t)
+  {
+    const Tensor<2, dim, NumberType> t_inv = invert(t);
+    Tensor<4, dim, NumberType>       dt_inv_trans_dt;
+    // https://en.wikiversity.org/wiki/Introduction_to_Elasticity/Tensors#Derivative_of_the_determinant_of_a_tensor
+    for (unsigned int i = 0; i < dim; ++i)
+      for (unsigned int j = 0; j < dim; ++j)
+        for (unsigned int k = 0; k < dim; ++k)
+          for (unsigned int l = 0; l < dim; ++l)
+            dt_inv_trans_dt[i][j][k][l] = -t_inv[l][i] * t_inv[j][k];
+
+    return det_t(t) * outer_product(transpose(t_inv), transpose(t_inv)) +
+           det_t(t) * dt_inv_trans_dt;
+  }
+
+  static NumberType
+  v_squ(const Tensor<1, dim, NumberType> &v)
+  {
+    return v * v;
+  }
+
+  static Tensor<1, dim, NumberType>
+  dv_squ_dv(const Tensor<1, dim, NumberType> &v)
+  {
+    return 2.0 * v;
+  }
+
+  static Tensor<2, dim, NumberType>
+  d2v_squ_dv_dv(const Tensor<1, dim, NumberType> &v)
+  {
+    static const Tensor<2, dim, NumberType> I(
+      unit_symmetric_tensor<dim, NumberType>());
+    return 2.0 * I;
+  }
+
+  // --------
+
+  static const double sf;
+
+  static NumberType
+  psi(const Tensor<2, dim, NumberType> &t,
+      const Tensor<1, dim, NumberType> &v,
+      const NumberType &                s)
+  {
+    return std::pow(det_t(t), 2) * std::pow(v_squ(v), 3) * std::pow(s, sf);
+  };
+
+  static Tensor<2, dim, NumberType>
+  dpsi_dt(const Tensor<2, dim, NumberType> &t,
+          const Tensor<1, dim, NumberType> &v,
+          const NumberType &                s)
+  {
+    return 2.0 * std::pow(det_t(t), 1) * ddet_t_dt(t) * std::pow(v_squ(v), 3) *
+           std::pow(s, sf);
+  };
+
+  static Tensor<1, dim, NumberType>
+  dpsi_dv(const Tensor<2, dim, NumberType> &t,
+          const Tensor<1, dim, NumberType> &v,
+          const NumberType &                s)
+  {
+    return std::pow(det_t(t), 2) * 3.0 * std::pow(v_squ(v), 2) * dv_squ_dv(v) *
+           std::pow(s, sf);
+  };
+
+  static NumberType
+  dpsi_ds(const Tensor<2, dim, NumberType> &t,
+          const Tensor<1, dim, NumberType> &v,
+          const NumberType &                s)
+  {
+    return std::pow(det_t(t), 2) * std::pow(v_squ(v), 3) * sf *
+           std::pow(s, sf - 1.0);
+  };
+
+  static Tensor<4, dim, NumberType>
+  d2psi_dt_dt(const Tensor<2, dim, NumberType> &t,
+              const Tensor<1, dim, NumberType> &v,
+              const NumberType &                s)
+  {
+    return 2.0 * std::pow(v_squ(v), 3) *
+           (pow(det_t(t), 0) * outer_product(ddet_t_dt(t), ddet_t_dt(t)) +
+            std::pow(det_t(t), 1) * d2det_t_dt_dt(t)) *
+           std::pow(s, sf);
+  };
+
+  static Tensor<3, dim, NumberType>
+  d2psi_dv_dt(const Tensor<2, dim, NumberType> &t,
+              const Tensor<1, dim, NumberType> &v,
+              const NumberType &                s)
+  {
+    return 2.0 * std::pow(det_t(t), 1) * 3.0 * std::pow(v_squ(v), 2) *
+           outer_product(ddet_t_dt(t), dv_squ_dv(v)) * std::pow(s, sf);
+  };
+
+  static Tensor<2, dim, NumberType>
+  d2psi_ds_dt(const Tensor<2, dim, NumberType> &t,
+              const Tensor<1, dim, NumberType> &v,
+              const NumberType &                s)
+  {
+    return 2.0 * std::pow(det_t(t), 1) * ddet_t_dt(t) * std::pow(v_squ(v), 3) *
+           sf * std::pow(s, sf - 1.0);
+  };
+
+  static Tensor<3, dim, NumberType>
+  d2psi_dt_dv(const Tensor<2, dim, NumberType> &t,
+              const Tensor<1, dim, NumberType> &v,
+              const NumberType &                s)
+  {
+    return 2.0 * std::pow(det_t(t), 1) * 3.0 * std::pow(v_squ(v), 2) *
+           outer_product(dv_squ_dv(v), ddet_t_dt(t)) * std::pow(s, sf);
+  };
+
+  static Tensor<2, dim, NumberType>
+  d2psi_dv_dv(const Tensor<2, dim, NumberType> &t,
+              const Tensor<1, dim, NumberType> &v,
+              const NumberType &                s)
+  {
+    return std::pow(det_t(t), 2) * 3.0 *
+           (2.0 * std::pow(v_squ(v), 1) *
+              outer_product(dv_squ_dv(v), dv_squ_dv(v)) +
+            std::pow(v_squ(v), 2) * d2v_squ_dv_dv(v)) *
+           std::pow(s, sf);
+  };
+
+  static Tensor<1, dim, NumberType>
+  d2psi_ds_dv(const Tensor<2, dim, NumberType> &t,
+              const Tensor<1, dim, NumberType> &v,
+              const NumberType &                s)
+  {
+    return std::pow(det_t(t), 2) * 3.0 * std::pow(v_squ(v), 2) * dv_squ_dv(v) *
+           sf * std::pow(s, sf - 1.0);
+  };
+
+  static Tensor<2, dim, NumberType>
+  d2psi_dt_ds(const Tensor<2, dim, NumberType> &t,
+              const Tensor<1, dim, NumberType> &v,
+              const NumberType &                s)
+  {
+    return 2.0 * std::pow(det_t(t), 1) * ddet_t_dt(t) * std::pow(v_squ(v), 3) *
+           sf * std::pow(s, sf - 1.0);
+  };
+
+  static Tensor<1, dim, NumberType>
+  d2psi_dv_ds(const Tensor<2, dim, NumberType> &t,
+              const Tensor<1, dim, NumberType> &v,
+              const NumberType &                s)
+  {
+    return std::pow(det_t(t), 2) * 3.0 * std::pow(v_squ(v), 2) * dv_squ_dv(v) *
+           sf * std::pow(s, sf - 1.0);
+  };
+
+  static NumberType
+  d2psi_ds_ds(const Tensor<2, dim, NumberType> &t,
+              const Tensor<1, dim, NumberType> &v,
+              const NumberType &                s)
+  {
+    return std::pow(det_t(t), 2) * std::pow(v_squ(v), 3) * sf * (sf - 1.0) *
+           std::pow(s, sf - 2.0);
+  };
+};
+
+template <int dim, typename NumberType>
+const double FunctionsTestTensorVectorScalarCoupled<dim, NumberType>::sf = 2.2;
+
+template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+void
+test_tensor_vector_scalar_coupled()
+{
+  typedef AD::ScalarFunction<dim, ad_type_code, number_t> ADHelper;
+  typedef typename ADHelper::ad_type                      ADNumberType;
+  typedef typename ADHelper::scalar_type                  ScalarNumberType;
+
+  const unsigned int this_mpi_process =
+    Utilities::MPI::this_mpi_process(MPI_COMM_WORLD);
+  ConditionalOStream pcout(deallog.get_console(), this_mpi_process == 0);
+
+  pcout << "*** Test variables: Tensor + Vector + Scalar (coupled), "
+        << "dim = " << Utilities::to_string(dim) << ", "
+        << "Type code: " << static_cast<int>(ad_type_code) << std::endl;
+
+  // Values computed from the AD energy function
+  ScalarNumberType             psi;
+  Vector<ScalarNumberType>     Dpsi;
+  FullMatrix<ScalarNumberType> D2psi;
+
+  // Function and its derivatives
+  typedef FunctionsTestTensorVectorScalarCoupled<dim, ADNumberType> func_ad;
+
+  // Setup the variable components and choose a value at which to
+  // evaluate the tape
+  const FEValuesExtractors::Tensor<2> t_dof(0);
+  const FEValuesExtractors::Vector    v_dof(
+    Tensor<2, dim>::n_independent_components);
+  const FEValuesExtractors::Scalar s_dof(
+    Tensor<2, dim>::n_independent_components +
+    Tensor<1, dim>::n_independent_components);
+  const unsigned int n_AD_components =
+    Tensor<2, dim>::n_independent_components +
+    Tensor<1, dim>::n_independent_components + 1;
+  ADHelper ad_helper(n_AD_components);
+  ad_helper.set_tape_buffer_sizes(); // Increase the buffer size from the
+                                     // default values
+
+  Tensor<2, dim, ScalarNumberType> t =
+    unit_symmetric_tensor<dim, ScalarNumberType>();
+  for (unsigned int i = 0; i < t.n_independent_components; ++i)
+    t[t.unrolled_to_component_indices(i)] += 0.11 * (i + 0.125);
+  Tensor<1, dim, ScalarNumberType> v;
+  ScalarNumberType                 s = 0.57;
+  for (unsigned int i = 0; i < dim; ++i)
+    v[i] = 0.275 * (1.0 + i);
+
+  // Perform the differentiation operations
+  {
+    ad_helper.register_independent_variable(t, t_dof);
+    ad_helper.register_independent_variable(v, v_dof);
+    ad_helper.register_independent_variable(s, s_dof);
+
+    const Tensor<2, dim, ADNumberType> t_ad =
+      ad_helper.get_sensitive_variables(t_dof);
+    const Tensor<1, dim, ADNumberType> v_ad =
+      ad_helper.get_sensitive_variables(v_dof);
+    ADNumberType s_ad = ad_helper.get_sensitive_variables(s_dof);
+
+    const ADNumberType psi(func_ad::psi(t_ad, v_ad, s_ad));
+
+    ad_helper.register_dependent_variable(psi);
+
+    pcout << "Recorded data..." << std::endl;
+    pcout << "independent variable values: " << std::flush;
+    if (this_mpi_process == 0)
+      ad_helper.print_values(pcout.get_stream());
+    pcout << "t_ad: " << t_ad << std::endl;
+    pcout << "v_ad: " << v_ad << std::endl;
+    pcout << "s_ad: " << s_ad << std::endl;
+    pcout << "psi: " << psi << std::endl;
+    pcout << std::endl;
+  }
+
+  pcout << "independent variable values: " << std::flush;
+  if (this_mpi_process == 0)
+    ad_helper.print_values(pcout.get_stream());
+
+  // Compute the function value, gradient and hessian for the new evaluation
+  // point
+  psi = ad_helper.compute_value();
+  ad_helper.compute_gradient(Dpsi);
+  if (AD::ADNumberTraits<ADNumberType>::n_supported_derivative_levels >= 2)
+    {
+      ad_helper.compute_hessian(D2psi);
+    }
+
+  // Output the full stored function, gradient vector and hessian matrix
+  pcout << "psi: " << psi << std::endl;
+  pcout << "Dpsi: \n";
+  if (this_mpi_process == 0)
+    Dpsi.print(pcout.get_stream());
+  if (AD::ADNumberTraits<ADNumberType>::n_supported_derivative_levels >= 2)
+    {
+      pcout << "D2psi: \n";
+      if (this_mpi_process == 0)
+        D2psi.print_formatted(pcout.get_stream(), 3, true, 0, "0.0");
+    }
+
+  // Extract components of the solution
+  const Tensor<2, dim, ScalarNumberType> dpsi_dt =
+    ad_helper.extract_gradient_component(Dpsi, t_dof);
+  const Tensor<1, dim, ScalarNumberType> dpsi_dv =
+    ad_helper.extract_gradient_component(Dpsi, v_dof);
+  const Tensor<0, dim, ScalarNumberType> dpsi_ds =
+    ad_helper.extract_gradient_component(Dpsi, s_dof);
+  pcout << "extracted Dpsi (t): " << dpsi_dt << "\n"
+        << "extracted Dpsi (v): " << dpsi_dv << "\n"
+        << "extracted Dpsi (s): " << dpsi_ds << "\n";
+
+  // Verify the result
+  typedef FunctionsTestTensorVectorScalarCoupled<dim, ScalarNumberType> func;
+  static const ScalarNumberType                                         tol =
+    1e5 * std::numeric_limits<ScalarNumberType>::epsilon();
+  Assert(std::abs(psi - func::psi(t, v, s)) < tol,
+         ExcMessage("No match for function value."));
+  Assert(std::abs((dpsi_dt - func::dpsi_dt(t, v, s)).norm()) < tol,
+         ExcMessage("No match for first derivative."));
+  Assert(std::abs((dpsi_dv - func::dpsi_dv(t, v, s)).norm()) < tol,
+         ExcMessage("No match for first derivative."));
+  Assert(std::abs(dpsi_ds - func::dpsi_ds(t, v, s)) < tol,
+         ExcMessage("No match for first derivative."));
+  if (AD::ADNumberTraits<ADNumberType>::n_supported_derivative_levels >= 2)
+    {
+      const Tensor<4, dim, ScalarNumberType> d2psi_dt_dt =
+        ad_helper.extract_hessian_component(D2psi, t_dof, t_dof);
+      const Tensor<3, dim, ScalarNumberType> d2psi_dv_dt =
+        ad_helper.extract_hessian_component(D2psi, t_dof, v_dof);
+      const Tensor<2, dim, ScalarNumberType> d2psi_ds_dt =
+        ad_helper.extract_hessian_component(D2psi, t_dof, s_dof);
+      const Tensor<3, dim, ScalarNumberType> d2psi_dt_dv =
+        ad_helper.extract_hessian_component(D2psi, v_dof, t_dof);
+      const Tensor<2, dim, ScalarNumberType> d2psi_dv_dv =
+        ad_helper.extract_hessian_component(D2psi, v_dof, v_dof);
+      const Tensor<1, dim, ScalarNumberType> d2psi_ds_dv =
+        ad_helper.extract_hessian_component(D2psi, v_dof, s_dof);
+      const Tensor<2, dim, ScalarNumberType> d2psi_dt_ds =
+        ad_helper.extract_hessian_component(D2psi, s_dof, t_dof);
+      const Tensor<1, dim, ScalarNumberType> d2psi_dv_ds =
+        ad_helper.extract_hessian_component(D2psi, s_dof, v_dof);
+      const Tensor<0, dim, ScalarNumberType> d2psi_ds_ds =
+        ad_helper.extract_hessian_component(D2psi, s_dof, s_dof);
+      pcout << "extracted Dpsi (t): " << dpsi_dt << "\n"
+            << "extracted Dpsi (v): " << dpsi_dv << "\n"
+            << "extracted Dpsi (s): " << dpsi_ds << "\n"
+            << "extracted D2psi (t,t): " << d2psi_dt_dt << "\n"
+            << "extracted D2psi (t,v): " << d2psi_dv_dt << "\n"
+            << "extracted D2psi (t,s): " << d2psi_ds_dt << "\n"
+            << "extracted D2psi (v,t): " << d2psi_dt_dv << "\n"
+            << "extracted D2psi (v,v): " << d2psi_dv_dv << "\n"
+            << "extracted D2psi (v,s): " << d2psi_ds_dv << "\n"
+            << "extracted D2psi (s,t): " << d2psi_dt_ds << "\n"
+            << "extracted D2psi (s,v): " << d2psi_dv_ds << "\n"
+            << "extracted D2psi (s,s): " << d2psi_ds_ds << "\n"
+            << std::endl;
+      Assert(std::abs((d2psi_dt_dt - func::d2psi_dt_dt(t, v, s)).norm()) < tol,
+             ExcMessage("No match for second derivative."));
+      Assert(std::abs((d2psi_dv_dt - func::d2psi_dv_dt(t, v, s)).norm()) < tol,
+             ExcMessage("No match for second derivative."));
+      Assert(std::abs((d2psi_ds_dt - func::d2psi_ds_dt(t, v, s)).norm()) < tol,
+             ExcMessage("No match for second derivative."));
+      Assert(std::abs((d2psi_dt_dv - func::d2psi_dt_dv(t, v, s)).norm()) < tol,
+             ExcMessage("No match for second derivative."));
+      Assert(std::abs((d2psi_dv_dv - func::d2psi_dv_dv(t, v, s)).norm()) < tol,
+             ExcMessage("No match for second derivative."));
+      Assert(std::abs((d2psi_ds_dv - func::d2psi_ds_dv(t, v, s)).norm()) < tol,
+             ExcMessage("No match for second derivative."));
+      Assert(std::abs((d2psi_dt_ds - func::d2psi_dt_ds(t, v, s)).norm()) < tol,
+             ExcMessage("No match for second derivative."));
+      Assert(std::abs((d2psi_dv_ds - func::d2psi_dv_ds(t, v, s)).norm()) < tol,
+             ExcMessage("No match for second derivative."));
+      Assert(std::abs(d2psi_ds_ds - func::d2psi_ds_ds(t, v, s)) < tol,
+             ExcMessage("No match for second derivative."));
+    }
+}
diff --git a/tests/ad_common_tests/helper_scalar_single_component_01.h b/tests/ad_common_tests/helper_scalar_single_component_01.h
new file mode 100644 (file)
index 0000000..c6e0ab3
--- /dev/null
@@ -0,0 +1,179 @@
+// ---------------------------------------------------------------------
+//
+// Copyright (C) 2016 - 2018 by the deal.II authors
+//
+// This file is part of the deal.II library.
+//
+// The deal.II library is free software; you can use it, redistribute
+// it, and/or modify it under the terms of the GNU Lesser General
+// Public License as published by the Free Software Foundation; either
+// version 2.1 of the License, or (at your option) any later version.
+// The full text of the license can be found in the file LICENSE at
+// the top level of the deal.II distribution.
+//
+// ---------------------------------------------------------------------
+
+
+// Header file:
+// Evaluation of a single component (scalar) system using a helper class
+
+#include <deal.II/base/logstream.h>
+#include <deal.II/base/symmetric_tensor.h>
+#include <deal.II/base/tensor.h>
+
+#include <deal.II/differentiation/ad.h>
+
+#include <deal.II/fe/fe_values_extractors.h>
+
+#include <deal.II/lac/full_matrix.h>
+#include <deal.II/lac/vector.h>
+
+#include <iostream>
+
+#include "../tests.h"
+
+using namespace dealii;
+namespace AD = dealii::Differentiation::AD;
+
+template <int dim, typename NumberType>
+struct FunctionsTestScalar
+{
+  static NumberType
+  psi(const NumberType &s)
+  {
+    return 4.0 * std::pow(s, 4);
+  }
+
+  static NumberType
+  dpsi_ds(const NumberType &s)
+  {
+    return 16.0 * std::pow(s, 3);
+  }
+
+  static NumberType
+  d2psi_ds_ds(const NumberType &s)
+  {
+    return 48.0 * std::pow(s, 2);
+  }
+};
+
+template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+void
+test_scalar()
+{
+  typedef AD::ScalarFunction<dim, ad_type_code, number_t> ADHelper;
+  typedef typename ADHelper::ad_type                      ADNumberType;
+  typedef typename ADHelper::scalar_type                  ScalarNumberType;
+
+  std::cout << "*** Test variables: Scalar dof, "
+            << "dim = " << Utilities::to_string(dim) << ", "
+            << "Type code: " << static_cast<int>(ad_type_code) << std::endl;
+
+  // Values computed from the AD energy function
+  ScalarNumberType             psi;
+  Vector<ScalarNumberType>     Dpsi;
+  FullMatrix<ScalarNumberType> D2psi;
+
+  // Function and its derivatives
+  typedef FunctionsTestScalar<dim, ADNumberType> func_ad;
+
+  // Setup the variable components and choose a value at which to
+  // evaluate the tape
+  const FEValuesExtractors::Scalar s_dof(0);
+  const unsigned int               n_AD_components = 1;
+  ADHelper                         ad_helper(n_AD_components);
+  ad_helper.set_tape_buffer_sizes(); // Increase the buffer size from the
+                                     // default values
+
+  ScalarNumberType s = 1.2;
+
+  // Configure tape
+  const int  tape_no = 1;
+  const bool is_recording =
+    ad_helper.start_recording_operations(tape_no /*material_id*/,
+                                         true /*overwrite_tape*/,
+                                         true /*keep*/);
+  if (is_recording == true)
+    {
+      ad_helper.register_independent_variable(s, s_dof);
+
+      const ADNumberType s_ad = ad_helper.get_sensitive_variables(s_dof);
+
+      const ADNumberType psi(func_ad::psi(s_ad));
+
+      ad_helper.register_dependent_variable(psi);
+      ad_helper.stop_recording_operations(false /*write_tapes_to_file*/);
+
+      std::cout << "Recorded data..." << std::endl;
+      std::cout << "independent variable values: " << std::flush;
+      ad_helper.print_values(std::cout);
+      std::cout << "s_ad: " << s_ad << std::endl;
+      std::cout << "psi: " << psi << std::endl;
+      std::cout << std::endl;
+    }
+  else
+    {
+      Assert(is_recording == true, ExcInternalError());
+    }
+
+  // Do some work :-)
+  // Set a new evaluation point
+  if (AD::ADNumberTraits<ADNumberType>::is_taped == true)
+    {
+      std::cout
+        << "Using tape with different values for independent variables..."
+        << std::endl;
+      s = 1.5;
+      ad_helper.activate_recorded_tape(tape_no);
+      ad_helper.set_independent_variable(s, s_dof);
+
+      std::cout << "independent variable values: " << std::flush;
+      ad_helper.print_values(std::cout);
+    }
+
+  // Compute the function value, gradient and hessian for the new evaluation
+  // point
+  psi = ad_helper.compute_value();
+  ad_helper.compute_gradient(Dpsi);
+  if (AD::ADNumberTraits<ADNumberType>::n_supported_derivative_levels >= 2)
+    ad_helper.compute_hessian(D2psi);
+
+  // Output the full stored function, gradient vector and hessian matrix
+  std::cout << "psi: " << psi << std::endl;
+  std::cout << "Dpsi: \n";
+  Dpsi.print(std::cout);
+  if (AD::ADNumberTraits<ADNumberType>::n_supported_derivative_levels >= 2)
+    {
+      std::cout << "D2psi: \n";
+      D2psi.print_formatted(std::cout, 3, true, 0, "0.0");
+    }
+
+  // Extract components of the solution
+  const ScalarNumberType dpsi_ds =
+    ad_helper.extract_gradient_component(Dpsi, s_dof);
+
+  // Verify the result
+  typedef FunctionsTestScalar<dim, ScalarNumberType> func;
+  static const ScalarNumberType                      tol =
+    1e5 * std::numeric_limits<ScalarNumberType>::epsilon();
+  std::cout << "psi:              " << psi << std::endl;
+  std::cout << "func::psi(s):     " << func::psi(s) << std::endl;
+  Assert(std::abs(psi - func::psi(s)) < tol,
+         ExcMessage("No match for function value."));
+  std::cout << "dpsi_ds:              " << dpsi_ds << std::endl;
+  std::cout << "func::dpsi_ds(s):     " << func::dpsi_ds(s) << std::endl;
+  Assert(std::abs(dpsi_ds - func::dpsi_ds(s)) < tol,
+         ExcMessage("No match for first derivative."));
+  if (AD::ADNumberTraits<ADNumberType>::n_supported_derivative_levels >= 2)
+    {
+      const ScalarNumberType d2psi_ds_ds =
+        ad_helper.extract_hessian_component(D2psi, s_dof, s_dof);
+      std::cout << "d2psi_ds_ds:              " << d2psi_ds_ds << std::endl;
+      std::cout << "func::d2psi_ds_ds(s):     " << func::d2psi_ds_ds(s)
+                << std::endl;
+      Assert(std::abs(d2psi_ds_ds - func::d2psi_ds_ds(s)) < tol,
+             ExcMessage("No match for second derivative."));
+    }
+
+  std::cout << std::endl << std::endl;
+}
diff --git a/tests/ad_common_tests/helper_scalar_single_component_02.h b/tests/ad_common_tests/helper_scalar_single_component_02.h
new file mode 100644 (file)
index 0000000..e9b4e40
--- /dev/null
@@ -0,0 +1,190 @@
+// ---------------------------------------------------------------------
+//
+// Copyright (C) 2016 - 2018 by the deal.II authors
+//
+// This file is part of the deal.II library.
+//
+// The deal.II library is free software; you can use it, redistribute
+// it, and/or modify it under the terms of the GNU Lesser General
+// Public License as published by the Free Software Foundation; either
+// version 2.1 of the License, or (at your option) any later version.
+// The full text of the license can be found in the file LICENSE at
+// the top level of the deal.II distribution.
+//
+// ---------------------------------------------------------------------
+
+
+// Header file:
+// Evaluation of a single component (vector) system using a helper class
+
+#include <deal.II/base/logstream.h>
+#include <deal.II/base/symmetric_tensor.h>
+#include <deal.II/base/tensor.h>
+
+#include <deal.II/differentiation/ad.h>
+
+#include <deal.II/fe/fe_values_extractors.h>
+
+#include <deal.II/lac/full_matrix.h>
+#include <deal.II/lac/vector.h>
+
+#include <iostream>
+
+#include "../tests.h"
+
+using namespace dealii;
+namespace AD = dealii::Differentiation::AD;
+
+template <int dim, typename NumberType>
+struct FunctionsTestVector
+{
+  static NumberType
+  psi(const Tensor<1, dim, NumberType> &v)
+  {
+    // Potential memory corruption in adtl::adouble...
+    // Probably need to zero some temporary value
+    // in Tensor::operator* or Tensor::contract()
+    return 2.0 * (v * v);
+    // return 2.0*contract(v,v);
+    // return 2.0*scalar_product(v,v);
+  }
+
+  static Tensor<1, dim, NumberType>
+  dpsi_dv(const Tensor<1, dim, NumberType> &v)
+  {
+    return 4.0 * v;
+  }
+
+  static Tensor<2, dim, NumberType>
+  d2psi_dv_dv(const Tensor<1, dim, NumberType> &v)
+  {
+    const SymmetricTensor<2, dim, NumberType> I(unit_symmetric_tensor<dim>());
+    const Tensor<2, dim, NumberType>          I_ns(I);
+    return 4.0 * I_ns;
+  }
+};
+
+template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+void
+test_vector()
+{
+  typedef AD::ScalarFunction<dim, ad_type_code, number_t> ADHelper;
+  typedef typename ADHelper::ad_type                      ADNumberType;
+  typedef typename ADHelper::scalar_type                  ScalarNumberType;
+
+  std::cout << "*** Test variables: Vector dof, "
+            << "dim = " << Utilities::to_string(dim) << ", "
+            << "Type code: " << static_cast<int>(ad_type_code) << std::endl;
+
+  // Values computed from the AD energy function
+  ScalarNumberType             psi;
+  Vector<ScalarNumberType>     Dpsi;
+  FullMatrix<ScalarNumberType> D2psi;
+
+  // Function and its derivatives
+  typedef FunctionsTestVector<dim, ADNumberType> func_ad;
+
+  // Setup the variable components and choose a value at which to
+  // evaluate the tape
+  const FEValuesExtractors::Vector v_dof(0);
+  const unsigned int               n_AD_components = dim;
+  ADHelper                         ad_helper(n_AD_components);
+  ad_helper.set_tape_buffer_sizes(); // Increase the buffer size from the
+                                     // default values
+
+  Tensor<1, dim, ScalarNumberType> v;
+  for (unsigned int i = 0; i < dim; ++i)
+    v[i] = 1.0 + i;
+
+  const int  tape_no = 1;
+  const bool is_recording =
+    ad_helper.start_recording_operations(tape_no /*material_id*/,
+                                         true /*overwrite_tape*/,
+                                         true /*keep*/);
+  if (is_recording == true)
+    {
+      ad_helper.register_independent_variable(v, v_dof);
+
+      const Tensor<1, dim, ADNumberType> v_ad =
+        ad_helper.get_sensitive_variables(v_dof);
+
+      const ADNumberType psi = func_ad::psi(v_ad);
+
+      ad_helper.register_dependent_variable(psi);
+      ad_helper.stop_recording_operations(false /*write_tapes_to_file*/);
+
+      std::cout << "Recorded data..." << std::endl;
+      std::cout << "independent variable values: " << std::flush;
+      ad_helper.print_values(std::cout);
+      std::cout << "v_ad: " << v_ad << std::endl;
+      std::cout << "psi: " << psi << std::endl;
+      std::cout << std::endl;
+    }
+  else
+    {
+      Assert(is_recording == true, ExcInternalError());
+    }
+
+  // Do some work :-)
+  // Set a new evaluation point
+  if (AD::ADNumberTraits<ADNumberType>::is_taped == true)
+    {
+      std::cout
+        << "Using tape with different values for independent variables..."
+        << std::endl;
+      ad_helper.activate_recorded_tape(tape_no);
+      v *= 2.2;
+      ad_helper.set_independent_variable(v, v_dof);
+
+      std::cout << "independent variable values: " << std::flush;
+      ad_helper.print_values(std::cout);
+    }
+
+  // Compute the function value, gradient and hessian for the new evaluation
+  // point
+  psi = ad_helper.compute_value();
+  ad_helper.compute_gradient(Dpsi);
+  if (AD::ADNumberTraits<ADNumberType>::n_supported_derivative_levels >= 2)
+    {
+      ad_helper.compute_hessian(D2psi);
+    }
+
+  // Output the full stored function, gradient vector and hessian matrix
+  std::cout << "psi: " << psi << std::endl;
+  std::cout << "Dpsi: \n";
+  Dpsi.print(std::cout);
+  if (AD::ADNumberTraits<ADNumberType>::n_supported_derivative_levels >= 2)
+    {
+      std::cout << "D2psi: \n";
+      D2psi.print_formatted(std::cout, 3, true, 0, "0.0");
+    }
+
+  // Extract components of the solution
+  const Tensor<1, dim, ScalarNumberType> dpsi_dv =
+    ad_helper.extract_gradient_component(Dpsi, v_dof);
+
+  // Verify the result
+  typedef FunctionsTestVector<dim, ScalarNumberType> func;
+  static const ScalarNumberType                      tol =
+    1e5 * std::numeric_limits<ScalarNumberType>::epsilon();
+  std::cout << "psi:              " << psi << std::endl;
+  std::cout << "func::psi(v):     " << func::psi(v) << std::endl;
+  Assert(std::abs(psi - func::psi(v)) < tol,
+         ExcMessage("No match for function value."));
+  std::cout << "dpsi_dv:              " << dpsi_dv << std::endl;
+  std::cout << "func::dpsi_dv(v):     " << func::dpsi_dv(v) << std::endl;
+  Assert(std::abs((dpsi_dv - func::dpsi_dv(v)).norm()) < tol,
+         ExcMessage("No match for first derivative."));
+  if (AD::ADNumberTraits<ADNumberType>::n_supported_derivative_levels >= 2)
+    {
+      const Tensor<2, dim, ScalarNumberType> d2psi_dv_dv =
+        ad_helper.extract_hessian_component(D2psi, v_dof, v_dof);
+      std::cout << "d2psi_dv_dv:          " << d2psi_dv_dv << std::endl;
+      std::cout << "func::d2psi_dv_dv(v): " << func::d2psi_dv_dv(v)
+                << std::endl;
+      Assert(std::abs((d2psi_dv_dv - func::d2psi_dv_dv(v)).norm()) < tol,
+             ExcMessage("No match for second derivative."));
+    }
+
+  std::cout << std::endl << std::endl;
+}
diff --git a/tests/ad_common_tests/helper_scalar_single_component_03.h b/tests/ad_common_tests/helper_scalar_single_component_03.h
new file mode 100644 (file)
index 0000000..50309e7
--- /dev/null
@@ -0,0 +1,201 @@
+// ---------------------------------------------------------------------
+//
+// Copyright (C) 2016 - 2018 by the deal.II authors
+//
+// This file is part of the deal.II library.
+//
+// The deal.II library is free software; you can use it, redistribute
+// it, and/or modify it under the terms of the GNU Lesser General
+// Public License as published by the Free Software Foundation; either
+// version 2.1 of the License, or (at your option) any later version.
+// The full text of the license can be found in the file LICENSE at
+// the top level of the deal.II distribution.
+//
+// ---------------------------------------------------------------------
+
+
+// Header file:
+// Evaluation of a single component (tensor) system using a helper class
+
+#include <deal.II/base/logstream.h>
+#include <deal.II/base/symmetric_tensor.h>
+#include <deal.II/base/tensor.h>
+
+#include <deal.II/differentiation/ad.h>
+
+#include <deal.II/fe/fe_values_extractors.h>
+
+#include <deal.II/lac/full_matrix.h>
+#include <deal.II/lac/vector.h>
+
+#include <iostream>
+
+#include "../tests.h"
+
+using namespace dealii;
+namespace AD = dealii::Differentiation::AD;
+
+template <int dim, typename NumberType>
+struct FunctionsTestTensor
+{
+  static NumberType
+  psi(const Tensor<2, dim, NumberType> &t)
+  {
+    return 3.0 * determinant(t);
+  }
+
+  static Tensor<2, dim, NumberType>
+  dpsi_dt(const Tensor<2, dim, NumberType> &t)
+  {
+    // Wriggers2008 p521
+    const NumberType det_t = determinant(t);
+    // std::cout << "determinant(t): " << determinant(t) << std::endl;
+    // std::cout << "transpose(invert(t)): " << transpose(invert(t)) <<
+    // std::endl;
+    return 3.0 * (det_t * transpose(invert(t)));
+  }
+
+  static Tensor<4, dim, NumberType>
+  d2psi_dt_dt(const Tensor<2, dim, NumberType> &t)
+  {
+    const NumberType                 det_t = determinant(t);
+    const Tensor<2, dim, NumberType> t_inv = invert(t);
+    Tensor<4, dim, NumberType>       dt_inv_trans_dt;
+
+    // https://en.wikiversity.org/wiki/Introduction_to_Elasticity/Tensors#Derivative_of_the_determinant_of_a_tensor
+    for (unsigned int i = 0; i < dim; ++i)
+      for (unsigned int j = 0; j < dim; ++j)
+        for (unsigned int k = 0; k < dim; ++k)
+          for (unsigned int l = 0; l < dim; ++l)
+            dt_inv_trans_dt[i][j][k][l] = -t_inv[l][i] * t_inv[j][k];
+
+    return 3.0 * (det_t * outer_product(transpose(t_inv), transpose(t_inv)) +
+                  det_t * dt_inv_trans_dt);
+  }
+};
+
+template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+void
+test_tensor()
+{
+  typedef AD::ScalarFunction<dim, ad_type_code, number_t> ADHelper;
+  typedef typename ADHelper::ad_type                      ADNumberType;
+  typedef typename ADHelper::scalar_type                  ScalarNumberType;
+
+  std::cout << "*** Test variables: Tensor dof, "
+            << "dim = " << Utilities::to_string(dim) << ", "
+            << "Type code: " << static_cast<int>(ad_type_code) << std::endl;
+
+  // Values computed from the AD energy function
+  ScalarNumberType             psi;
+  Vector<ScalarNumberType>     Dpsi;
+  FullMatrix<ScalarNumberType> D2psi;
+
+  // Function and its derivatives
+  typedef FunctionsTestTensor<dim, ADNumberType> func_ad;
+
+  // Setup the variable components and choose a value at which to
+  // evaluate the tape
+  const FEValuesExtractors::Tensor<2> t_dof(0);
+  const unsigned int n_AD_components = Tensor<2, dim>::n_independent_components;
+  ADHelper           ad_helper(n_AD_components);
+  ad_helper.set_tape_buffer_sizes(); // Increase the buffer size from the
+                                     // default values
+
+  Tensor<2, dim, ScalarNumberType> t =
+    unit_symmetric_tensor<dim, ScalarNumberType>();
+  for (unsigned int i = 0; i < t.n_independent_components; ++i)
+    t[t.unrolled_to_component_indices(i)] += 0.14 * (i + 0.07);
+
+  const int  tape_no = 1;
+  const bool is_recording =
+    ad_helper.start_recording_operations(tape_no /*material_id*/,
+                                         true /*overwrite_tape*/,
+                                         true /*keep*/);
+  if (is_recording == true)
+    {
+      ad_helper.register_independent_variable(t, t_dof);
+
+      const Tensor<2, dim, ADNumberType> t_ad =
+        ad_helper.get_sensitive_variables(t_dof);
+
+      const ADNumberType psi(func_ad::psi(t_ad));
+
+      ad_helper.register_dependent_variable(psi);
+      ad_helper.stop_recording_operations(false /*write_tapes_to_file*/);
+
+      std::cout << "Recorded data..." << std::endl;
+      std::cout << "independent variable values: " << std::flush;
+      ad_helper.print_values(std::cout);
+      std::cout << "t_ad: " << t_ad << std::endl;
+      std::cout << "psi: " << psi << std::endl;
+      std::cout << std::endl;
+    }
+  else
+    {
+      Assert(is_recording == true, ExcInternalError());
+    }
+
+  // Do some work :-)
+  // Set a new evaluation point
+  if (AD::ADNumberTraits<ADNumberType>::is_taped == true)
+    {
+      std::cout
+        << "Using tape with different values for independent variables..."
+        << std::endl;
+      ad_helper.activate_recorded_tape(tape_no);
+      t *= 1.15;
+      ad_helper.set_independent_variable(t, t_dof);
+
+      std::cout << "independent variable values: " << std::flush;
+      ad_helper.print_values(std::cout);
+    }
+
+  // Compute the function value, gradient and hessian for the new evaluation
+  // point
+  psi = ad_helper.compute_value();
+  ad_helper.compute_gradient(Dpsi);
+  if (AD::ADNumberTraits<ADNumberType>::n_supported_derivative_levels >= 2)
+    {
+      ad_helper.compute_hessian(D2psi);
+    }
+
+  // Output the full stored function, gradient vector and hessian matrix
+  std::cout << "psi: " << psi << std::endl;
+  std::cout << "Dpsi: \n";
+  Dpsi.print(std::cout);
+  if (AD::ADNumberTraits<ADNumberType>::n_supported_derivative_levels >= 2)
+    {
+      std::cout << "D2psi: \n";
+      D2psi.print_formatted(std::cout, 3, true, 0, "0.0");
+    }
+
+  // Extract components of the solution
+  const Tensor<2, dim, ScalarNumberType> dpsi_dt =
+    ad_helper.extract_gradient_component(Dpsi, t_dof);
+
+  // Verify the result
+  typedef FunctionsTestTensor<dim, ScalarNumberType> func;
+  static const ScalarNumberType                      tol =
+    1e5 * std::numeric_limits<ScalarNumberType>::epsilon();
+  std::cout << "psi:              " << psi << std::endl;
+  std::cout << "func::psi(t):     " << func::psi(t) << std::endl;
+  Assert(std::abs(psi - func::psi(t)) < tol,
+         ExcMessage("No match for function value."));
+  std::cout << "dpsi_dt:              " << dpsi_dt << std::endl;
+  std::cout << "func::dpsi_dt(t):     " << func::dpsi_dt(t) << std::endl;
+  Assert(std::abs((dpsi_dt - func::dpsi_dt(t)).norm()) < tol,
+         ExcMessage("No match for first derivative."));
+  if (AD::ADNumberTraits<ADNumberType>::n_supported_derivative_levels >= 2)
+    {
+      const Tensor<4, dim, ScalarNumberType> d2psi_dt_dt =
+        ad_helper.extract_hessian_component(D2psi, t_dof, t_dof);
+      std::cout << "d2psi_dt_dt:          " << d2psi_dt_dt << std::endl;
+      std::cout << "func::d2psi_dt_dt(t): " << func::d2psi_dt_dt(t)
+                << std::endl;
+      Assert(std::abs((d2psi_dt_dt - func::d2psi_dt_dt(t)).norm()) < tol,
+             ExcMessage("No match for second derivative."));
+    }
+
+  std::cout << std::endl << std::endl;
+}
diff --git a/tests/ad_common_tests/helper_scalar_single_component_04.h b/tests/ad_common_tests/helper_scalar_single_component_04.h
new file mode 100644 (file)
index 0000000..75efb24
--- /dev/null
@@ -0,0 +1,205 @@
+// ---------------------------------------------------------------------
+//
+// Copyright (C) 2016 - 2018 by the deal.II authors
+//
+// This file is part of the deal.II library.
+//
+// The deal.II library is free software; you can use it, redistribute
+// it, and/or modify it under the terms of the GNU Lesser General
+// Public License as published by the Free Software Foundation; either
+// version 2.1 of the License, or (at your option) any later version.
+// The full text of the license can be found in the file LICENSE at
+// the top level of the deal.II distribution.
+//
+// ---------------------------------------------------------------------
+
+
+// Header file:
+// Evaluation of a single component (symmetric tensor) system using a helper
+// class
+
+#include <deal.II/base/logstream.h>
+#include <deal.II/base/symmetric_tensor.h>
+#include <deal.II/base/tensor.h>
+
+#include <deal.II/differentiation/ad.h>
+
+#include <deal.II/fe/fe_values_extractors.h>
+
+#include <deal.II/lac/full_matrix.h>
+#include <deal.II/lac/vector.h>
+
+#include <iostream>
+
+#include "../tests.h"
+
+using namespace dealii;
+namespace AD = dealii::Differentiation::AD;
+
+// Function and its derivatives
+template <int dim, typename NumberType>
+struct FunctionsTestSymmetricTensor
+{
+  static NumberType
+  psi(const SymmetricTensor<2, dim, NumberType> &t)
+  {
+    return 2.0 * determinant(t);
+  }
+
+  static SymmetricTensor<2, dim, NumberType>
+  dpsi_dt(const SymmetricTensor<2, dim, NumberType> &t)
+  {
+    // Wriggers2008 p521
+    const NumberType                          det_t = determinant(t);
+    const SymmetricTensor<2, dim, NumberType> t_inv =
+      symmetrize(invert(static_cast<Tensor<2, dim, NumberType>>(t)));
+    return SymmetricTensor<2, dim, NumberType>(2.0 * (det_t * t_inv));
+  }
+
+  static SymmetricTensor<4, dim, NumberType>
+  d2psi_dt_dt(const SymmetricTensor<2, dim, NumberType> &t)
+  {
+    const NumberType                          det_t = determinant(t);
+    const SymmetricTensor<2, dim, NumberType> t_inv =
+      symmetrize(invert(static_cast<Tensor<2, dim, NumberType>>(t)));
+    SymmetricTensor<4, dim, NumberType> dt_inv_dt;
+
+    // https://en.wikiversity.org/wiki/Introduction_to_Elasticity/Tensors#Derivative_of_the_determinant_of_a_tensor
+    for (unsigned int i = 0; i < dim; ++i)
+      for (unsigned int j = i; j < dim; ++j)
+        for (unsigned int k = 0; k < dim; ++k)
+          for (unsigned int l = k; l < dim; ++l)
+            dt_inv_dt[i][j][k][l] =
+              -0.5 * (t_inv[i][k] * t_inv[j][l] + t_inv[i][l] * t_inv[j][k]);
+
+    return SymmetricTensor<4, dim, NumberType>(
+      2.0 * (det_t * outer_product(t_inv, t_inv) + det_t * dt_inv_dt));
+  }
+};
+
+template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+void
+test_symmetric_tensor()
+{
+  typedef AD::ScalarFunction<dim, ad_type_code, number_t> ADHelper;
+  typedef typename ADHelper::ad_type                      ADNumberType;
+  typedef typename ADHelper::scalar_type                  ScalarNumberType;
+
+  std::cout << "*** Test variables: SymmetricTensor dof, "
+            << "dim = " << Utilities::to_string(dim) << ", "
+            << "Type code: " << static_cast<int>(ad_type_code) << std::endl;
+
+  // Values computed from the AD energy function
+  ScalarNumberType             psi;
+  Vector<ScalarNumberType>     Dpsi;
+  FullMatrix<ScalarNumberType> D2psi;
+
+  // Function and its derivatives
+  typedef FunctionsTestSymmetricTensor<dim, ADNumberType> func_ad;
+
+  // Setup the variable components and choose a value at which to
+  // evaluate the tape
+  const FEValuesExtractors::SymmetricTensor<2> t_dof(0);
+  const unsigned int                           n_AD_components =
+    SymmetricTensor<2, dim>::n_independent_components;
+  ADHelper ad_helper(n_AD_components);
+  ad_helper.set_tape_buffer_sizes(); // Increase the buffer size from the
+                                     // default values
+
+  SymmetricTensor<2, dim, ScalarNumberType> t =
+    unit_symmetric_tensor<dim, ScalarNumberType>();
+  for (unsigned int i = 0; i < t.n_independent_components; ++i)
+    t[t.unrolled_to_component_indices(i)] += 0.12 * (i + 0.02);
+
+  const int  tape_no = 1;
+  const bool is_recording =
+    ad_helper.start_recording_operations(tape_no /*material_id*/,
+                                         true /*overwrite_tape*/,
+                                         true /*keep*/);
+  if (is_recording == true)
+    {
+      ad_helper.register_independent_variable(t, t_dof);
+
+      const SymmetricTensor<2, dim, ADNumberType> t_ad =
+        ad_helper.get_sensitive_variables(t_dof);
+
+      const ADNumberType psi(func_ad::psi(t_ad));
+
+      ad_helper.register_dependent_variable(psi);
+      ad_helper.stop_recording_operations(false /*write_tapes_to_file*/);
+
+      std::cout << "Recorded data..." << std::endl;
+      std::cout << "independent variable values: " << std::flush;
+      ad_helper.print_values(std::cout);
+      std::cout << "t_ad: " << t_ad << std::endl;
+      std::cout << "psi: " << psi << std::endl;
+      std::cout << std::endl;
+    }
+  else
+    {
+      Assert(is_recording == true, ExcInternalError());
+    }
+
+  // Do some work :-)
+  // Set a new evaluation point
+  if (AD::ADNumberTraits<ADNumberType>::is_taped == true)
+    {
+      std::cout
+        << "Using tape with different values for independent variables..."
+        << std::endl;
+      ad_helper.activate_recorded_tape(tape_no);
+      t *= 4.0;
+      ad_helper.set_independent_variable(t, t_dof);
+
+      std::cout << "independent variable values: " << std::flush;
+      ad_helper.print_values(std::cout);
+    }
+
+  // Compute the function value, gradient and hessian for the new evaluation
+  // point
+  psi = ad_helper.compute_value();
+  ad_helper.compute_gradient(Dpsi);
+  if (AD::ADNumberTraits<ADNumberType>::n_supported_derivative_levels >= 2)
+    {
+      ad_helper.compute_hessian(D2psi);
+    }
+
+  // Output the full stored function, gradient vector and hessian matrix
+  std::cout << "psi: " << psi << std::endl;
+  std::cout << "Dpsi: \n";
+  Dpsi.print(std::cout);
+  if (AD::ADNumberTraits<ADNumberType>::n_supported_derivative_levels >= 2)
+    {
+      std::cout << "D2psi: \n";
+      D2psi.print_formatted(std::cout, 3, true, 0, "0.0");
+    }
+
+  // Extract components of the solution
+  const SymmetricTensor<2, dim, ScalarNumberType> dpsi_dt =
+    ad_helper.extract_gradient_component(Dpsi, t_dof);
+
+  // Verify the result
+  typedef FunctionsTestSymmetricTensor<dim, ScalarNumberType> func;
+  static const ScalarNumberType                               tol =
+    1e5 * std::numeric_limits<ScalarNumberType>::epsilon();
+  std::cout << "psi:              " << psi << std::endl;
+  std::cout << "func::psi(t):     " << func::psi(t) << std::endl;
+  Assert(std::abs(psi - func::psi(t)) < tol,
+         ExcMessage("No match for function value."));
+  std::cout << "dpsi_dt:              " << dpsi_dt << std::endl;
+  std::cout << "func::dpsi_dt(t):     " << func::dpsi_dt(t) << std::endl;
+  Assert(std::abs((dpsi_dt - func::dpsi_dt(t)).norm()) < tol,
+         ExcMessage("No match for first derivative."));
+  if (AD::ADNumberTraits<ADNumberType>::n_supported_derivative_levels >= 2)
+    {
+      const SymmetricTensor<4, dim, ScalarNumberType> d2psi_dt_dt =
+        ad_helper.extract_hessian_component(D2psi, t_dof, t_dof);
+      std::cout << "d2psi_dt_dt:          " << d2psi_dt_dt << std::endl;
+      std::cout << "func::d2psi_dt_dt(t): " << func::d2psi_dt_dt(t)
+                << std::endl;
+      Assert(std::abs((d2psi_dt_dt - func::d2psi_dt_dt(t)).norm()) < tol,
+             ExcMessage("No match for second derivative."));
+    }
+
+  std::cout << std::endl << std::endl;
+}
diff --git a/tests/ad_common_tests/helper_vector_1_indep_1_dep_vars_01.h b/tests/ad_common_tests/helper_vector_1_indep_1_dep_vars_01.h
new file mode 100644 (file)
index 0000000..8693976
--- /dev/null
@@ -0,0 +1,158 @@
+// ---------------------------------------------------------------------
+//
+// Copyright (C) 2016 - 2018 by the deal.II authors
+//
+// This file is part of the deal.II library.
+//
+// The deal.II library is free software; you can use it, redistribute
+// it, and/or modify it under the terms of the GNU Lesser General
+// Public License as published by the Free Software Foundation; either
+// version 2.1 of the License, or (at your option) any later version.
+// The full text of the license can be found in the file LICENSE at
+// the top level of the deal.II distribution.
+//
+// ---------------------------------------------------------------------
+
+
+// Header file:
+// Evaluation of a single component (scalar) system using a helper class
+// This is the equivalent of helper_scalar_single_component_01, but for
+// the other (vector) helper class
+
+#include <deal.II/base/logstream.h>
+#include <deal.II/base/symmetric_tensor.h>
+#include <deal.II/base/tensor.h>
+
+#include <deal.II/differentiation/ad.h>
+
+#include <deal.II/fe/fe_values_extractors.h>
+
+#include <deal.II/lac/full_matrix.h>
+#include <deal.II/lac/vector.h>
+
+#include <iostream>
+
+#include "../tests.h"
+
+using namespace dealii;
+namespace AD = dealii::Differentiation::AD;
+
+template <int dim, typename NumberType>
+struct FunctionsTestScalar
+{
+  static NumberType
+  f0(const NumberType &s0)
+  {
+    return 4.0 * std::pow(s0, 4);
+  }
+
+  static NumberType
+  df0_ds0(const NumberType &s0)
+  {
+    return 16.0 * std::pow(s0, 3);
+  }
+};
+
+template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+void
+test_AD_vector_jacobian()
+{
+  typedef AD::VectorFunction<dim, ad_type_code, number_t> ADHelper;
+  typedef typename ADHelper::ad_type                      ADNumberType;
+  typedef typename ADHelper::scalar_type                  ScalarNumberType;
+
+  std::cout << "*** Test variables: Variables: 1 independent, 1 dependent, "
+            << "dim = " << Utilities::to_string(dim) << ", "
+            << "Type code: " << static_cast<int>(ad_type_code) << std::endl;
+
+  // Function and its derivatives
+  typedef FunctionsTestScalar<dim, ADNumberType> func_ad;
+
+  // Setup the variable components and choose a value at which to
+  // evaluate the tape
+  const unsigned int n_vars_indep = 1;
+  const unsigned int n_vars_dep   = 1;
+  ADHelper           ad_helper(n_vars_indep, n_vars_dep);
+  ad_helper.set_tape_buffer_sizes(); // Increase the buffer size from the
+                                     // default values
+
+  std::vector<ScalarNumberType> s(n_vars_indep);
+  s[0] = 1.2;
+
+  // Configure tape
+  const int  tape_no = 1;
+  const bool is_recording =
+    ad_helper.start_recording_operations(tape_no /*material_id*/,
+                                         true /*overwrite_tape*/,
+                                         true /*keep*/);
+  if (is_recording == true)
+    {
+      ad_helper.register_independent_variables(s);
+
+      const std::vector<ADNumberType> s_ad =
+        ad_helper.get_sensitive_variables();
+
+      // ADNumberType f0 (s_ad*s_ad*s_ad);
+      std::vector<ADNumberType> f_ad(n_vars_dep, ADNumberType(0.0));
+      f_ad[0] = func_ad::f0(s_ad[0]);
+
+      ad_helper.register_dependent_variables(f_ad);
+      ad_helper.stop_recording_operations(false /*write_tapes_to_file*/);
+
+      std::cout << "Taped data..." << std::endl;
+      std::cout << "independent variable values: " << std::flush;
+      ad_helper.print_values(std::cout);
+      std::cout << "s_ad: ";
+      for (unsigned int i = 0; i < n_vars_indep; ++i)
+        std::cout << s_ad[i] << (i < (n_vars_indep - 1) ? "," : "");
+      std::cout << std::endl;
+      std::cout << "f_ad: ";
+      for (unsigned int i = 0; i < n_vars_dep; ++i)
+        std::cout << f_ad[i] << (i < (n_vars_dep - 1) ? "," : "");
+      std::cout << std::endl;
+    }
+  else
+    {
+      Assert(is_recording == true, ExcInternalError());
+    }
+
+  // Do some work :-)
+  // Set a new evaluation point
+  if (AD::ADNumberTraits<ADNumberType>::is_taped == true)
+    {
+      std::cout
+        << "Using tape with different values for independent variables..."
+        << std::endl;
+      s[0] = 1.5;
+      ad_helper.activate_recorded_tape(tape_no);
+      ad_helper.set_independent_variables(s);
+
+      std::cout << "independent variable values: " << std::flush;
+      ad_helper.print_values(std::cout);
+    }
+
+  // Compute the function values and their jacobian for the new evaluation point
+  Vector<ScalarNumberType>     funcs(n_vars_dep);
+  FullMatrix<ScalarNumberType> Dfuncs(n_vars_dep, n_vars_indep);
+  ad_helper.compute_values(funcs);
+  ad_helper.compute_jacobian(Dfuncs);
+
+  // Output the full stored function, gradient vector and hessian matrix
+  std::cout << "funcs: \n";
+  funcs.print(std::cout);
+  std::cout << "Dfuncs: \n";
+  Dfuncs.print_formatted(std::cout, 3, true, 0, "0.0");
+
+  // Verify the result
+  typedef FunctionsTestScalar<dim, ScalarNumberType> func;
+  static const ScalarNumberType                      tol =
+    1e5 * std::numeric_limits<ScalarNumberType>::epsilon();
+  std::cout << "funcs[0]: " << funcs[0]
+            << "\t func::f0(s[0])): " << func::f0(s[0]) << std::endl;
+  std::cout << "Dfuncs[0][0]: " << Dfuncs[0][0]
+            << "\t func::df0_ds0(s[0])): " << func::df0_ds0(s[0]) << std::endl;
+  Assert(std::abs(funcs[0] - func::f0(s[0])) < tol,
+         ExcMessage("No match for function value."));
+  Assert(std::abs(Dfuncs[0][0] - func::df0_ds0(s[0])) < tol,
+         ExcMessage("No match for first derivative."));
+}
diff --git a/tests/ad_common_tests/helper_vector_1_indep_2_dep_vars_01.h b/tests/ad_common_tests/helper_vector_1_indep_2_dep_vars_01.h
new file mode 100644 (file)
index 0000000..e7a2ee1
--- /dev/null
@@ -0,0 +1,178 @@
+// ---------------------------------------------------------------------
+//
+// Copyright (C) 2016 - 2018 by the deal.II authors
+//
+// This file is part of the deal.II library.
+//
+// The deal.II library is free software; you can use it, redistribute
+// it, and/or modify it under the terms of the GNU Lesser General
+// Public License as published by the Free Software Foundation; either
+// version 2.1 of the License, or (at your option) any later version.
+// The full text of the license can be found in the file LICENSE at
+// the top level of the deal.II distribution.
+//
+// ---------------------------------------------------------------------
+
+
+// Header file:
+// Evaluation of a vector of 2 dependent and 1 independent variables
+// using a helper class
+
+#include <deal.II/base/logstream.h>
+#include <deal.II/base/symmetric_tensor.h>
+#include <deal.II/base/tensor.h>
+
+#include <deal.II/differentiation/ad.h>
+
+#include <deal.II/fe/fe_values_extractors.h>
+
+#include <deal.II/lac/full_matrix.h>
+#include <deal.II/lac/vector.h>
+
+#include <iostream>
+
+#include "../tests.h"
+
+using namespace dealii;
+namespace AD = dealii::Differentiation::AD;
+
+// Function and its derivatives
+template <int dim, typename NumberType>
+struct FunctionsTestSquare
+{
+  static NumberType
+  f0(const NumberType &s0)
+  {
+    return 2.0 * std::pow(s0, 4);
+  };
+
+  static NumberType
+  df0_ds0(const NumberType &s0)
+  {
+    return 8.0 * std::pow(s0, 3);
+  };
+
+  static NumberType
+  f1(const NumberType &s0)
+  {
+    return 3.0 * std::pow(s0, 2);
+  };
+
+  static NumberType
+  df1_ds0(const NumberType &s0)
+  {
+    return 6.0 * std::pow(s0, 1);
+  };
+};
+
+template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+void
+test_AD_vector_jacobian()
+{
+  typedef AD::VectorFunction<dim, ad_type_code, number_t> ADHelper;
+  typedef typename ADHelper::ad_type                      ADNumberType;
+  typedef typename ADHelper::scalar_type                  ScalarNumberType;
+
+  std::cout << "*** Test variables: Variables: 1 independent, 2 dependent, "
+            << "dim = " << Utilities::to_string(dim) << ", "
+            << "Type code: " << static_cast<int>(ad_type_code) << std::endl;
+
+  // Function and its derivatives
+  typedef FunctionsTestSquare<dim, ADNumberType> func_ad;
+
+  // Setup the variable components and choose a value at which to
+  // evaluate the tape
+  const unsigned int n_vars_indep = 1;
+  const unsigned int n_vars_dep   = 2;
+  ADHelper           ad_helper(n_vars_indep, n_vars_dep);
+  ad_helper.set_tape_buffer_sizes(); // Increase the buffer size from the
+                                     // default values
+
+  std::vector<ScalarNumberType> s(n_vars_indep);
+  s[0] = 3.1;
+
+  // Configure tape
+  const int  tape_no = 1;
+  const bool is_recording =
+    ad_helper.start_recording_operations(tape_no /*material_id*/,
+                                         true /*overwrite_tape*/,
+                                         true /*keep*/);
+  if (is_recording == true)
+    {
+      ad_helper.register_independent_variables(s);
+
+      const std::vector<ADNumberType> s_ad =
+        ad_helper.get_sensitive_variables();
+
+      std::vector<ADNumberType> f_ad(n_vars_dep, ADNumberType(0.0));
+      f_ad[0] = func_ad::f0(s_ad[0]);
+      f_ad[1] = func_ad::f1(s_ad[0]);
+
+      ad_helper.register_dependent_variables(f_ad);
+      ad_helper.stop_recording_operations(false /*write_tapes_to_file*/);
+
+      std::cout << "Taped data..." << std::endl;
+      std::cout << "independent variable values: " << std::flush;
+      ad_helper.print_values(std::cout);
+      std::cout << "s_ad: ";
+      for (unsigned int i = 0; i < n_vars_indep; ++i)
+        std::cout << s_ad[i] << (i < (n_vars_indep - 1) ? "," : "");
+      std::cout << std::endl;
+      std::cout << "f_ad: ";
+      for (unsigned int i = 0; i < n_vars_dep; ++i)
+        std::cout << f_ad[i] << (i < (n_vars_dep - 1) ? "," : "");
+      std::cout << std::endl;
+    }
+  else
+    {
+      Assert(is_recording == true, ExcInternalError());
+    }
+
+  // Do some work :-)
+  // Set a new evaluation point
+  if (AD::ADNumberTraits<ADNumberType>::is_taped == true)
+    {
+      std::cout
+        << "Using tape with different values for independent variables..."
+        << std::endl;
+      s[0] = 4.9;
+      ad_helper.activate_recorded_tape(tape_no);
+      ad_helper.set_independent_variables(s);
+
+      std::cout << "independent variable values: " << std::flush;
+      ad_helper.print_values(std::cout);
+    }
+
+  // Compute the function values and their jacobian for the new evaluation point
+  Vector<ScalarNumberType>     funcs(n_vars_dep);
+  FullMatrix<ScalarNumberType> Dfuncs(n_vars_dep, n_vars_indep);
+  ad_helper.compute_values(funcs);
+  ad_helper.compute_jacobian(Dfuncs);
+
+  // Output the full stored function, gradient vector and hessian matrix
+  std::cout << "funcs: \n";
+  funcs.print(std::cout);
+  std::cout << "Dfuncs: \n";
+  Dfuncs.print_formatted(std::cout, 3, true, 0, "0.0");
+
+  // Verify the result
+  typedef FunctionsTestSquare<dim, ScalarNumberType> func;
+  static const ScalarNumberType                      tol =
+    1e5 * std::numeric_limits<ScalarNumberType>::epsilon();
+  std::cout << "funcs[0]: " << funcs[0]
+            << "\t func::f0(s[0])): " << func::f0(s[0]) << std::endl;
+  std::cout << "funcs[1]: " << funcs[1]
+            << "\t func::f1(s[0])): " << func::f1(s[0]) << std::endl;
+  std::cout << "Dfuncs[0][0]: " << Dfuncs[0][0]
+            << "\t func::df0_ds0(s[0])): " << func::df0_ds0(s[0]) << std::endl;
+  std::cout << "Dfuncs[1][0]: " << Dfuncs[1][0]
+            << "\t func::df1_ds0(s[0])): " << func::df1_ds0(s[0]) << std::endl;
+  Assert(std::abs(funcs[0] - func::f0(s[0])) < tol,
+         ExcMessage("No match for function 1 value."));
+  Assert(std::abs(funcs[1] - func::f1(s[0])) < tol,
+         ExcMessage("No match for function 2 value."));
+  Assert(std::abs(Dfuncs[0][0] - func::df0_ds0(s[0])) < tol,
+         ExcMessage("No match for function 1 first derivative.."));
+  Assert(std::abs(Dfuncs[1][0] - func::df1_ds0(s[0])) < tol,
+         ExcMessage("No match for function 2 first derivative.."));
+}
diff --git a/tests/ad_common_tests/helper_vector_2_indep_1_dep_vars_01.h b/tests/ad_common_tests/helper_vector_2_indep_1_dep_vars_01.h
new file mode 100644 (file)
index 0000000..e140a7e
--- /dev/null
@@ -0,0 +1,171 @@
+// ---------------------------------------------------------------------
+//
+// Copyright (C) 2016 - 2018 by the deal.II authors
+//
+// This file is part of the deal.II library.
+//
+// The deal.II library is free software; you can use it, redistribute
+// it, and/or modify it under the terms of the GNU Lesser General
+// Public License as published by the Free Software Foundation; either
+// version 2.1 of the License, or (at your option) any later version.
+// The full text of the license can be found in the file LICENSE at
+// the top level of the deal.II distribution.
+//
+// ---------------------------------------------------------------------
+
+
+// Header file:
+// Evaluation of a vector of 1 dependent and 2 independent variables
+// using a helper class
+
+#include <deal.II/base/logstream.h>
+#include <deal.II/base/symmetric_tensor.h>
+#include <deal.II/base/tensor.h>
+
+#include <deal.II/differentiation/ad.h>
+
+#include <deal.II/fe/fe_values_extractors.h>
+
+#include <deal.II/lac/full_matrix.h>
+#include <deal.II/lac/vector.h>
+
+#include <iostream>
+
+#include "../tests.h"
+
+using namespace dealii;
+namespace AD = dealii::Differentiation::AD;
+
+// Function and its derivatives
+template <int dim, typename NumberType>
+struct FunctionsTestSquare
+{
+  static NumberType
+  f0(const NumberType &s0, const NumberType &s1)
+  {
+    return 2.0 * std::pow(s0, 4) * std::pow(s1, 3);
+  };
+
+  static NumberType
+  df0_ds0(const NumberType &s0, const NumberType &s1)
+  {
+    return 8.0 * std::pow(s0, 3) * std::pow(s1, 3);
+  };
+
+  static NumberType
+  df0_ds1(const NumberType &s0, const NumberType &s1)
+  {
+    return 6.0 * std::pow(s0, 4) * std::pow(s1, 2);
+  };
+};
+
+template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+void
+test_AD_vector_jacobian()
+{
+  typedef AD::VectorFunction<dim, ad_type_code, number_t> ADHelper;
+  typedef typename ADHelper::ad_type                      ADNumberType;
+  typedef typename ADHelper::scalar_type                  ScalarNumberType;
+
+  std::cout << "*** Test variables: Variables: 2 independent, 1 dependent, "
+            << "dim = " << Utilities::to_string(dim) << ", "
+            << "Type code: " << static_cast<int>(ad_type_code) << std::endl;
+
+  // Function and its derivatives
+  typedef FunctionsTestSquare<dim, ADNumberType> func_ad;
+
+  // Setup the variable components and choose a value at which to
+  // evaluate the tape
+  const unsigned int n_vars_indep = 2;
+  const unsigned int n_vars_dep   = 1;
+  ADHelper           ad_helper(n_vars_indep, n_vars_dep);
+  ad_helper.set_tape_buffer_sizes(); // Increase the buffer size from the
+                                     // default values
+
+  std::vector<ScalarNumberType> s(n_vars_indep);
+  s[0] = 3.1;
+  s[1] = 5.9;
+
+  // Configure tape
+  const int  tape_no = 1;
+  const bool is_recording =
+    ad_helper.start_recording_operations(tape_no /*material_id*/,
+                                         true /*overwrite_tape*/,
+                                         true /*keep*/);
+  if (is_recording == true)
+    {
+      ad_helper.register_independent_variables(s);
+
+      const std::vector<ADNumberType> s_ad =
+        ad_helper.get_sensitive_variables();
+
+      std::vector<ADNumberType> f_ad(n_vars_dep, ADNumberType(0.0));
+      f_ad[0] = func_ad::f0(s_ad[0], s_ad[1]);
+
+      ad_helper.register_dependent_variables(f_ad);
+      ad_helper.stop_recording_operations(false /*write_tapes_to_file*/);
+
+      std::cout << "Taped data..." << std::endl;
+      std::cout << "independent variable values: " << std::flush;
+      ad_helper.print_values(std::cout);
+      std::cout << "s_ad: ";
+      for (unsigned int i = 0; i < n_vars_indep; ++i)
+        std::cout << s_ad[i] << (i < (n_vars_indep - 1) ? "," : "");
+      std::cout << std::endl;
+      std::cout << "f_ad: ";
+      for (unsigned int i = 0; i < n_vars_dep; ++i)
+        std::cout << f_ad[i] << (i < (n_vars_dep - 1) ? "," : "");
+      std::cout << std::endl;
+    }
+  else
+    {
+      Assert(is_recording == true, ExcInternalError());
+    }
+
+  // Do some work :-)
+  // Set a new evaluation point
+  if (AD::ADNumberTraits<ADNumberType>::is_taped == true)
+    {
+      std::cout
+        << "Using tape with different values for independent variables..."
+        << std::endl;
+      s[0] = 4.9;
+      s[1] = 0.87;
+      ad_helper.activate_recorded_tape(tape_no);
+      ad_helper.set_independent_variables(s);
+
+      std::cout << "independent variable values: " << std::flush;
+      ad_helper.print_values(std::cout);
+    }
+
+  // Compute the function values and their jacobian for the new evaluation point
+  Vector<ScalarNumberType>     funcs(n_vars_dep);
+  FullMatrix<ScalarNumberType> Dfuncs(n_vars_dep, n_vars_indep);
+  ad_helper.compute_values(funcs);
+  ad_helper.compute_jacobian(Dfuncs);
+
+  // Output the full stored function, gradient vector and hessian matrix
+  std::cout << "funcs: \n";
+  funcs.print(std::cout);
+  std::cout << "Dfuncs: \n";
+  Dfuncs.print_formatted(std::cout, 3, true, 0, "0.0");
+
+  // Verify the result
+  typedef FunctionsTestSquare<dim, ScalarNumberType> func;
+  static const ScalarNumberType                      tol =
+    1e5 * std::numeric_limits<ScalarNumberType>::epsilon();
+  std::cout << "funcs[0]: " << funcs[0]
+            << "\t func::f0(s[0],s[1])): " << func::f0(s[0], s[1]) << std::endl;
+  std::cout << "Dfuncs[0][1]: " << Dfuncs[0][0]
+            << "\t func::df0_ds0(s[0],s[1])): " << func::df0_ds0(s[0], s[1])
+            << std::endl;
+  std::cout << "Dfuncs[0][1]: " << Dfuncs[0][1]
+            << "\t func::df0_ds1(s[0],s[1])): " << func::df0_ds1(s[0], s[1])
+            << std::endl;
+  Assert(std::abs(funcs[0] - func::f0(s[0], s[1])) < tol,
+         ExcMessage("No match for function 1 value."));
+  Assert(std::abs(Dfuncs[0][0] - func::df0_ds0(s[0], s[1])) < tol,
+         ExcMessage("No match for function 1 first derivative.."));
+  Assert(std::abs(Dfuncs[0][1] - func::df0_ds1(s[0], s[1])) < tol,
+         ExcMessage("No match for function 1 first derivative.."));
+}
diff --git a/tests/ad_common_tests/helper_vector_2_indep_2_dep_vars_01.h b/tests/ad_common_tests/helper_vector_2_indep_2_dep_vars_01.h
new file mode 100644 (file)
index 0000000..79a4030
--- /dev/null
@@ -0,0 +1,204 @@
+// ---------------------------------------------------------------------
+//
+// Copyright (C) 2016 - 2018 by the deal.II authors
+//
+// This file is part of the deal.II library.
+//
+// The deal.II library is free software; you can use it, redistribute
+// it, and/or modify it under the terms of the GNU Lesser General
+// Public License as published by the Free Software Foundation; either
+// version 2.1 of the License, or (at your option) any later version.
+// The full text of the license can be found in the file LICENSE at
+// the top level of the deal.II distribution.
+//
+// ---------------------------------------------------------------------
+
+
+// Header file:
+// Evaluation of a vector of 2 dependent and 2 independent variables
+// using a helper class
+
+#include <deal.II/base/logstream.h>
+#include <deal.II/base/symmetric_tensor.h>
+#include <deal.II/base/tensor.h>
+
+#include <deal.II/differentiation/ad.h>
+
+#include <deal.II/fe/fe_values_extractors.h>
+
+#include <deal.II/lac/full_matrix.h>
+#include <deal.II/lac/vector.h>
+
+#include <iostream>
+
+#include "../tests.h"
+
+using namespace dealii;
+namespace AD = dealii::Differentiation::AD;
+
+// Function and its derivatives
+template <int dim, typename NumberType>
+struct FunctionsTestSquare
+{
+  static NumberType
+  f0(const NumberType &s0, const NumberType &s1)
+  {
+    return 2.0 * std::pow(s0, 4) * std::pow(s1, 3);
+  };
+
+  static NumberType
+  df0_ds0(const NumberType &s0, const NumberType &s1)
+  {
+    return 8.0 * std::pow(s0, 3) * std::pow(s1, 3);
+  };
+
+  static NumberType
+  df0_ds1(const NumberType &s0, const NumberType &s1)
+  {
+    return 6.0 * std::pow(s0, 4) * std::pow(s1, 2);
+  };
+
+  static NumberType
+  f1(const NumberType &s0, const NumberType &s1)
+  {
+    return 3.0 * std::pow(s0, 2) * std::pow(s1, 4);
+  };
+
+  static NumberType
+  df1_ds0(const NumberType &s0, const NumberType &s1)
+  {
+    return 6.0 * std::pow(s0, 1) * std::pow(s1, 4);
+  };
+
+  static NumberType
+  df1_ds1(const NumberType &s0, const NumberType &s1)
+  {
+    return 12.0 * std::pow(s0, 2) * std::pow(s1, 3);
+  };
+};
+
+template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+void
+test_AD_vector_jacobian()
+{
+  typedef AD::VectorFunction<dim, ad_type_code, number_t> ADHelper;
+  typedef typename ADHelper::ad_type                      ADNumberType;
+  typedef typename ADHelper::scalar_type                  ScalarNumberType;
+
+  std::cout << "*** Test variables: Variables: 2 independent, 2 dependent, "
+            << "dim = " << Utilities::to_string(dim) << ", "
+            << "Type code: " << static_cast<int>(ad_type_code) << std::endl;
+
+  // Function and its derivatives
+  typedef FunctionsTestSquare<dim, ADNumberType> func_ad;
+
+  // Setup the variable components and choose a value at which to
+  // evaluate the tape
+  const unsigned int n_vars_indep = 2;
+  const unsigned int n_vars_dep   = 2;
+  ADHelper           ad_helper(n_vars_indep, n_vars_dep);
+  ad_helper.set_tape_buffer_sizes(); // Increase the buffer size from the
+                                     // default values
+
+  std::vector<ScalarNumberType> s(n_vars_indep);
+  s[0] = 3.1;
+  s[1] = 5.9;
+
+  // Configure tape
+  const int  tape_no = 1;
+  const bool is_recording =
+    ad_helper.start_recording_operations(tape_no /*material_id*/,
+                                         true /*overwrite_tape*/,
+                                         true /*keep*/);
+  if (is_recording == true)
+    {
+      ad_helper.register_independent_variables(s);
+
+      const std::vector<ADNumberType> s_ad =
+        ad_helper.get_sensitive_variables();
+
+      std::vector<ADNumberType> f_ad(n_vars_dep, ADNumberType(0.0));
+      f_ad[0] = func_ad::f0(s_ad[0], s_ad[1]);
+      f_ad[1] = func_ad::f1(s_ad[0], s_ad[1]);
+
+      ad_helper.register_dependent_variables(f_ad);
+      ad_helper.stop_recording_operations(false /*write_tapes_to_file*/);
+
+      std::cout << "Taped data..." << std::endl;
+      std::cout << "independent variable values: " << std::flush;
+      ad_helper.print_values(std::cout);
+      std::cout << "s_ad: ";
+      for (unsigned int i = 0; i < n_vars_indep; ++i)
+        std::cout << s_ad[i] << (i < (n_vars_indep - 1) ? "," : "");
+      std::cout << std::endl;
+      std::cout << "f_ad: ";
+      for (unsigned int i = 0; i < n_vars_dep; ++i)
+        std::cout << f_ad[i] << (i < (n_vars_dep - 1) ? "," : "");
+      std::cout << std::endl;
+    }
+  else
+    {
+      Assert(is_recording == true, ExcInternalError());
+    }
+
+  // Do some work :-)
+  // Set a new evaluation point
+  if (AD::ADNumberTraits<ADNumberType>::is_taped == true)
+    {
+      std::cout
+        << "Using tape with different values for independent variables..."
+        << std::endl;
+      s[0] = 4.9;
+      s[1] = 0.87;
+      ad_helper.activate_recorded_tape(tape_no);
+      ad_helper.set_independent_variables(s);
+
+      std::cout << "independent variable values: " << std::flush;
+      ad_helper.print_values(std::cout);
+    }
+
+  // Compute the function values and their jacobian for the new evaluation point
+  Vector<ScalarNumberType>     funcs(n_vars_dep);
+  FullMatrix<ScalarNumberType> Dfuncs(n_vars_dep, n_vars_indep);
+  ad_helper.compute_values(funcs);
+  ad_helper.compute_jacobian(Dfuncs);
+
+  // Output the full stored function, gradient vector and hessian matrix
+  std::cout << "funcs: \n";
+  funcs.print(std::cout);
+  std::cout << "Dfuncs: \n";
+  Dfuncs.print_formatted(std::cout, 3, true, 0, "0.0");
+
+  // Verify the result
+  typedef FunctionsTestSquare<dim, ScalarNumberType> func;
+  static const ScalarNumberType                      tol =
+    1e5 * std::numeric_limits<ScalarNumberType>::epsilon();
+  std::cout << "funcs[0]: " << funcs[0]
+            << "\t func::f0(s[0],s[1])): " << func::f0(s[0], s[1]) << std::endl;
+  std::cout << "funcs[1]: " << funcs[1]
+            << "\t func::f1(s[0],s[1])): " << func::f1(s[0], s[1]) << std::endl;
+  std::cout << "Dfuncs[0][0]: " << Dfuncs[0][0]
+            << "\t func::df0_ds0(s[0],s[1])): " << func::df0_ds0(s[0], s[1])
+            << std::endl;
+  std::cout << "Dfuncs[0][1]: " << Dfuncs[0][1]
+            << "\t func::df0_ds1(s[0],s[1])): " << func::df0_ds1(s[0], s[1])
+            << std::endl;
+  std::cout << "Dfuncs[1][0]: " << Dfuncs[1][0]
+            << "\t func::df1_ds0(s[0],s[1])): " << func::df1_ds0(s[0], s[1])
+            << std::endl;
+  std::cout << "Dfuncs[1][1]: " << Dfuncs[1][1]
+            << "\t func::df1_ds1(s[0],s[1])): " << func::df1_ds1(s[0], s[1])
+            << std::endl;
+  Assert(std::abs(funcs[0] - func::f0(s[0], s[1])) < tol,
+         ExcMessage("No match for function 1 value."));
+  Assert(std::abs(funcs[1] - func::f1(s[0], s[1])) < tol,
+         ExcMessage("No match for function 2 value."));
+  Assert(std::abs(Dfuncs[0][0] - func::df0_ds0(s[0], s[1])) < tol,
+         ExcMessage("No match for function 1 first derivative.."));
+  Assert(std::abs(Dfuncs[0][1] - func::df0_ds1(s[0], s[1])) < tol,
+         ExcMessage("No match for function 1 first derivative.."));
+  Assert(std::abs(Dfuncs[1][0] - func::df1_ds0(s[0], s[1])) < tol,
+         ExcMessage("No match for function 2 first derivative.."));
+  Assert(std::abs(Dfuncs[1][1] - func::df1_ds1(s[0], s[1])) < tol,
+         ExcMessage("No match for function 2 first derivative.."));
+}
diff --git a/tests/ad_common_tests/step-44-helper_scalar_01.h b/tests/ad_common_tests/step-44-helper_scalar_01.h
new file mode 100644 (file)
index 0000000..4e80e43
--- /dev/null
@@ -0,0 +1,2278 @@
+// ---------------------------------------------------------------------
+//
+// Copyright (C) 2016 - 2018 by the deal.II authors
+//
+// This file is part of the deal.II library.
+//
+// The deal.II library is free software; you can use it, redistribute
+// it, and/or modify it under the terms of the GNU Lesser General
+// Public License as published by the Free Software Foundation; either
+// version 2.1 of the License, or (at your option) any later version.
+// The full text of the license can be found in the file LICENSE at
+// the top level of the deal.II distribution.
+//
+// ---------------------------------------------------------------------
+
+// Header file:
+// This is a modified version of step-44, which tests the implementation of
+// QP-level auto-differentiation.
+
+#include <deal.II/base/function.h>
+#include <deal.II/base/parameter_handler.h>
+#include <deal.II/base/point.h>
+#include <deal.II/base/quadrature_lib.h>
+#include <deal.II/base/quadrature_point_data.h>
+#include <deal.II/base/symmetric_tensor.h>
+#include <deal.II/base/tensor.h>
+#include <deal.II/base/timer.h>
+#include <deal.II/base/work_stream.h>
+
+#include <deal.II/differentiation/ad.h>
+
+#include <deal.II/dofs/dof_renumbering.h>
+#include <deal.II/dofs/dof_tools.h>
+
+#include <deal.II/fe/fe_dgp_monomial.h>
+#include <deal.II/fe/fe_q.h>
+#include <deal.II/fe/fe_system.h>
+#include <deal.II/fe/fe_tools.h>
+#include <deal.II/fe/fe_values.h>
+#include <deal.II/fe/mapping_q_eulerian.h>
+
+#include <deal.II/grid/grid_generator.h>
+#include <deal.II/grid/grid_in.h>
+#include <deal.II/grid/grid_tools.h>
+#include <deal.II/grid/tria.h>
+#include <deal.II/grid/tria_boundary_lib.h>
+
+#include <deal.II/lac/block_sparse_matrix.h>
+#include <deal.II/lac/block_vector.h>
+#include <deal.II/lac/constraint_matrix.h>
+#include <deal.II/lac/dynamic_sparsity_pattern.h>
+#include <deal.II/lac/full_matrix.h>
+#include <deal.II/lac/linear_operator.h>
+#include <deal.II/lac/packaged_operation.h>
+#include <deal.II/lac/precondition_selector.h>
+#include <deal.II/lac/solver_cg.h>
+#include <deal.II/lac/solver_selector.h>
+#include <deal.II/lac/sparse_direct.h>
+
+#include <deal.II/numerics/data_out.h>
+#include <deal.II/numerics/vector_tools.h>
+
+#include <fstream>
+#include <functional>
+#include <iostream>
+
+#include "../tests.h"
+namespace Step44
+{
+  using namespace dealii;
+  namespace AD = dealii::Differentiation::AD;
+  namespace Parameters
+  {
+    struct FESystem
+    {
+      unsigned int poly_degree;
+      unsigned int quad_order;
+      static void
+      declare_parameters(ParameterHandler &prm);
+      void
+      parse_parameters(ParameterHandler &prm);
+    };
+    void
+    FESystem::declare_parameters(ParameterHandler &prm)
+    {
+      prm.enter_subsection("Finite element system");
+      {
+        prm.declare_entry("Polynomial degree",
+                          "2",
+                          Patterns::Integer(0),
+                          "Displacement system polynomial order");
+        prm.declare_entry("Quadrature order",
+                          "3",
+                          Patterns::Integer(0),
+                          "Gauss quadrature order");
+      }
+      prm.leave_subsection();
+    }
+    void
+    FESystem::parse_parameters(ParameterHandler &prm)
+    {
+      prm.enter_subsection("Finite element system");
+      {
+        poly_degree = prm.get_integer("Polynomial degree");
+        quad_order  = prm.get_integer("Quadrature order");
+      }
+      prm.leave_subsection();
+    }
+    struct Geometry
+    {
+      unsigned int global_refinement;
+      double       scale;
+      double       p_p0;
+      static void
+      declare_parameters(ParameterHandler &prm);
+      void
+      parse_parameters(ParameterHandler &prm);
+    };
+    void
+    Geometry::declare_parameters(ParameterHandler &prm)
+    {
+      prm.enter_subsection("Geometry");
+      {
+        prm.declare_entry("Global refinement",
+                          "2",
+                          Patterns::Integer(0),
+                          "Global refinement level");
+        prm.declare_entry("Grid scale",
+                          "1e-3",
+                          Patterns::Double(0.0),
+                          "Global grid scaling factor");
+        prm.declare_entry("Pressure ratio p/p0",
+                          "100",
+                          Patterns::Selection("20|40|60|80|100"),
+                          "Ratio of applied pressure to reference pressure");
+      }
+      prm.leave_subsection();
+    }
+    void
+    Geometry::parse_parameters(ParameterHandler &prm)
+    {
+      prm.enter_subsection("Geometry");
+      {
+        global_refinement = prm.get_integer("Global refinement");
+        scale             = prm.get_double("Grid scale");
+        p_p0              = prm.get_double("Pressure ratio p/p0");
+      }
+      prm.leave_subsection();
+    }
+    struct Materials
+    {
+      double nu;
+      double mu;
+      static void
+      declare_parameters(ParameterHandler &prm);
+      void
+      parse_parameters(ParameterHandler &prm);
+    };
+    void
+    Materials::declare_parameters(ParameterHandler &prm)
+    {
+      prm.enter_subsection("Material properties");
+      {
+        prm.declare_entry("Poisson's ratio",
+                          "0.4999",
+                          Patterns::Double(-1.0, 0.5),
+                          "Poisson's ratio");
+        prm.declare_entry("Shear modulus",
+                          "80.194e6",
+                          Patterns::Double(),
+                          "Shear modulus");
+      }
+      prm.leave_subsection();
+    }
+    void
+    Materials::parse_parameters(ParameterHandler &prm)
+    {
+      prm.enter_subsection("Material properties");
+      {
+        nu = prm.get_double("Poisson's ratio");
+        mu = prm.get_double("Shear modulus");
+      }
+      prm.leave_subsection();
+    }
+    struct LinearSolver
+    {
+      std::string type_lin;
+      double      tol_lin;
+      double      max_iterations_lin;
+      bool        use_static_condensation;
+      std::string preconditioner_type;
+      double      preconditioner_relaxation;
+      static void
+      declare_parameters(ParameterHandler &prm);
+      void
+      parse_parameters(ParameterHandler &prm);
+    };
+    void
+    LinearSolver::declare_parameters(ParameterHandler &prm)
+    {
+      prm.enter_subsection("Linear solver");
+      {
+        prm.declare_entry("Solver type",
+                          "CG",
+                          Patterns::Selection("CG|Direct"),
+                          "Type of solver used to solve the linear system");
+        prm.declare_entry("Residual",
+                          "1e-6",
+                          Patterns::Double(0.0),
+                          "Linear solver residual (scaled by residual norm)");
+        prm.declare_entry(
+          "Max iteration multiplier",
+          "1",
+          Patterns::Double(0.0),
+          "Linear solver iterations (multiples of the system matrix size)");
+        prm.declare_entry("Use static condensation",
+                          "true",
+                          Patterns::Bool(),
+                          "Solve the full block system or a reduced problem");
+        prm.declare_entry("Preconditioner type",
+                          "ssor",
+                          Patterns::Selection("jacobi|ssor"),
+                          "Type of preconditioner");
+        prm.declare_entry("Preconditioner relaxation",
+                          "0.65",
+                          Patterns::Double(0.0),
+                          "Preconditioner relaxation value");
+      }
+      prm.leave_subsection();
+    }
+    void
+    LinearSolver::parse_parameters(ParameterHandler &prm)
+    {
+      prm.enter_subsection("Linear solver");
+      {
+        type_lin                  = prm.get("Solver type");
+        tol_lin                   = prm.get_double("Residual");
+        max_iterations_lin        = prm.get_double("Max iteration multiplier");
+        use_static_condensation   = prm.get_bool("Use static condensation");
+        preconditioner_type       = prm.get("Preconditioner type");
+        preconditioner_relaxation = prm.get_double("Preconditioner relaxation");
+      }
+      prm.leave_subsection();
+    }
+    struct NonlinearSolver
+    {
+      unsigned int max_iterations_NR;
+      double       tol_f;
+      double       tol_u;
+      static void
+      declare_parameters(ParameterHandler &prm);
+      void
+      parse_parameters(ParameterHandler &prm);
+    };
+    void
+    NonlinearSolver::declare_parameters(ParameterHandler &prm)
+    {
+      prm.enter_subsection("Nonlinear solver");
+      {
+        prm.declare_entry("Max iterations Newton-Raphson",
+                          "10",
+                          Patterns::Integer(0),
+                          "Number of Newton-Raphson iterations allowed");
+        prm.declare_entry("Tolerance force",
+                          "1.0e-9",
+                          Patterns::Double(0.0),
+                          "Force residual tolerance");
+        prm.declare_entry("Tolerance displacement",
+                          "1.0e-6",
+                          Patterns::Double(0.0),
+                          "Displacement error tolerance");
+      }
+      prm.leave_subsection();
+    }
+    void
+    NonlinearSolver::parse_parameters(ParameterHandler &prm)
+    {
+      prm.enter_subsection("Nonlinear solver");
+      {
+        max_iterations_NR = prm.get_integer("Max iterations Newton-Raphson");
+        tol_f             = prm.get_double("Tolerance force");
+        tol_u             = prm.get_double("Tolerance displacement");
+      }
+      prm.leave_subsection();
+    }
+    struct Time
+    {
+      double delta_t;
+      double end_time;
+      static void
+      declare_parameters(ParameterHandler &prm);
+      void
+      parse_parameters(ParameterHandler &prm);
+    };
+    void
+    Time::declare_parameters(ParameterHandler &prm)
+    {
+      prm.enter_subsection("Time");
+      {
+        prm.declare_entry("End time", "1", Patterns::Double(), "End time");
+        prm.declare_entry("Time step size",
+                          "0.1",
+                          Patterns::Double(),
+                          "Time step size");
+      }
+      prm.leave_subsection();
+    }
+    void
+    Time::parse_parameters(ParameterHandler &prm)
+    {
+      prm.enter_subsection("Time");
+      {
+        end_time = prm.get_double("End time");
+        delta_t  = prm.get_double("Time step size");
+      }
+      prm.leave_subsection();
+    }
+    struct AllParameters : public FESystem,
+                           public Geometry,
+                           public Materials,
+                           public LinearSolver,
+                           public NonlinearSolver,
+                           public Time
+    {
+      AllParameters(const std::string &input_file);
+      static void
+      declare_parameters(ParameterHandler &prm);
+      void
+      parse_parameters(ParameterHandler &prm);
+    };
+    AllParameters::AllParameters(const std::string &input_file)
+    {
+      ParameterHandler prm;
+      declare_parameters(prm);
+      prm.parse_input(input_file);
+      parse_parameters(prm);
+    }
+    void
+    AllParameters::declare_parameters(ParameterHandler &prm)
+    {
+      FESystem::declare_parameters(prm);
+      Geometry::declare_parameters(prm);
+      Materials::declare_parameters(prm);
+      LinearSolver::declare_parameters(prm);
+      NonlinearSolver::declare_parameters(prm);
+      Time::declare_parameters(prm);
+    }
+    void
+    AllParameters::parse_parameters(ParameterHandler &prm)
+    {
+      FESystem::parse_parameters(prm);
+      Geometry::parse_parameters(prm);
+      Materials::parse_parameters(prm);
+      LinearSolver::parse_parameters(prm);
+      NonlinearSolver::parse_parameters(prm);
+      Time::parse_parameters(prm);
+    }
+  } // namespace Parameters
+  template <int dim>
+  class StandardTensors
+  {
+  public:
+    static const SymmetricTensor<2, dim> I;
+    static const SymmetricTensor<4, dim> IxI;
+    static const SymmetricTensor<4, dim> II;
+    static const SymmetricTensor<4, dim> dev_P;
+  };
+  template <int dim>
+  const SymmetricTensor<2, dim>
+    StandardTensors<dim>::I = unit_symmetric_tensor<dim>();
+  template <int dim>
+  const SymmetricTensor<4, dim> StandardTensors<dim>::IxI = outer_product(I, I);
+  template <int dim>
+  const SymmetricTensor<4, dim>
+    StandardTensors<dim>::II = identity_tensor<dim>();
+  template <int dim>
+  const SymmetricTensor<4, dim>
+    StandardTensors<dim>::dev_P = deviator_tensor<dim>();
+  class Time
+  {
+  public:
+    Time(const double time_end, const double delta_t)
+      : timestep(0)
+      , time_current(0.0)
+      , time_end(time_end)
+      , delta_t(delta_t)
+    {}
+    virtual ~Time()
+    {}
+    double
+    current() const
+    {
+      return time_current;
+    }
+    double
+    end() const
+    {
+      return time_end;
+    }
+    double
+    get_delta_t() const
+    {
+      return delta_t;
+    }
+    unsigned int
+    get_timestep() const
+    {
+      return timestep;
+    }
+    void
+    increment()
+    {
+      time_current += delta_t;
+      ++timestep;
+    }
+
+  private:
+    unsigned int timestep;
+    double       time_current;
+    const double time_end;
+    const double delta_t;
+  };
+  template <int dim>
+  class Material_Compressible_Neo_Hook_Three_Field
+  {
+  public:
+    Material_Compressible_Neo_Hook_Three_Field(const double mu, const double nu)
+      : kappa((2.0 * mu * (1.0 + nu)) / (3.0 * (1.0 - 2.0 * nu)))
+      , c_1(mu / 2.0)
+      , det_F(1.0)
+      , p_tilde(0.0)
+      , J_tilde(1.0)
+      , b_bar(StandardTensors<dim>::I)
+    {
+      Assert(kappa > 0, ExcInternalError());
+    }
+    ~Material_Compressible_Neo_Hook_Three_Field()
+    {}
+    void
+    update_material_data(const Tensor<2, dim> &F,
+                         const double          p_tilde_in,
+                         const double          J_tilde_in)
+    {
+      det_F   = determinant(F);
+      b_bar   = std::pow(det_F, -2.0 / dim) * symmetrize(F * transpose(F));
+      p_tilde = p_tilde_in;
+      J_tilde = J_tilde_in;
+      Assert(det_F > 0, ExcInternalError());
+    }
+    template <typename NumberType>
+    NumberType
+    get_Psi_iso(const SymmetricTensor<2, dim, NumberType> &C_bar)
+    {
+      return c_1 * (trace(C_bar) - dim);
+    }
+    template <typename NumberType>
+    NumberType
+    get_Psi_vol(const NumberType &J_tilde)
+    {
+      return (kappa / 4.0) * (J_tilde * J_tilde - 1.0 - 2.0 * log(J_tilde));
+    }
+
+    // === OLD FUNCTIONS REMAIN FOR TESTING ===
+
+    SymmetricTensor<2, dim>
+    get_tau()
+    {
+      return get_tau_iso() + get_tau_vol();
+    }
+    SymmetricTensor<4, dim>
+    get_Jc() const
+    {
+      return get_Jc_vol() + get_Jc_iso();
+    }
+    double
+    get_dPsi_vol_dJ() const
+    {
+      return (kappa / 2.0) * (J_tilde - 1.0 / J_tilde);
+    }
+    double
+    get_d2Psi_vol_dJ2() const
+    {
+      return ((kappa / 2.0) * (1.0 + 1.0 / (J_tilde * J_tilde)));
+    }
+    double
+    get_det_F() const
+    {
+      return det_F;
+    }
+    double
+    get_p_tilde() const
+    {
+      return p_tilde;
+    }
+    double
+    get_J_tilde() const
+    {
+      return J_tilde;
+    }
+
+  protected:
+    const double            kappa;
+    const double            c_1;
+    double                  det_F;
+    double                  p_tilde;
+    double                  J_tilde;
+    SymmetricTensor<2, dim> b_bar;
+    SymmetricTensor<2, dim>
+    get_tau_vol() const
+    {
+      return p_tilde * det_F * StandardTensors<dim>::I;
+    }
+    SymmetricTensor<2, dim>
+    get_tau_iso() const
+    {
+      return StandardTensors<dim>::dev_P * get_tau_bar();
+    }
+    SymmetricTensor<2, dim>
+    get_tau_bar() const
+    {
+      return 2.0 * c_1 * b_bar;
+    }
+    SymmetricTensor<4, dim>
+    get_Jc_vol() const
+    {
+      return p_tilde * det_F *
+             (StandardTensors<dim>::IxI - (2.0 * StandardTensors<dim>::II));
+    }
+    SymmetricTensor<4, dim>
+    get_Jc_iso() const
+    {
+      const SymmetricTensor<2, dim> tau_bar = get_tau_bar();
+      const SymmetricTensor<2, dim> tau_iso = get_tau_iso();
+      const SymmetricTensor<4, dim> tau_iso_x_I =
+        outer_product(tau_iso, StandardTensors<dim>::I);
+      const SymmetricTensor<4, dim> I_x_tau_iso =
+        outer_product(StandardTensors<dim>::I, tau_iso);
+      const SymmetricTensor<4, dim> c_bar = get_c_bar();
+      return (2.0 / dim) * trace(tau_bar) * StandardTensors<dim>::dev_P -
+             (2.0 / dim) * (tau_iso_x_I + I_x_tau_iso) +
+             StandardTensors<dim>::dev_P * c_bar * StandardTensors<dim>::dev_P;
+    }
+    SymmetricTensor<4, dim>
+    get_c_bar() const
+    {
+      return SymmetricTensor<4, dim>();
+    }
+  };
+  template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+  class PointHistory
+  {
+  public:
+    PointHistory()
+      : F_inv(StandardTensors<dim>::I)
+      , tau(SymmetricTensor<2, dim>())
+      , d2Psi_vol_dJ2(0.0)
+      , dPsi_vol_dJ(0.0)
+      , Jc(SymmetricTensor<4, dim>())
+    {}
+    virtual ~PointHistory()
+    {}
+    void
+    setup_lqp(const Parameters::AllParameters &parameters)
+    {
+      material.reset(
+        new Material_Compressible_Neo_Hook_Three_Field<dim>(parameters.mu,
+                                                            parameters.nu));
+      update_values(Tensor<2, dim>(), 0.0, 1.0);
+    }
+    void
+    update_values(const Tensor<2, dim> &Grad_u_n,
+                  const double          p_tilde,
+                  const double          J_tilde)
+    {
+      const Tensor<2, dim> F =
+        (Tensor<2, dim>(StandardTensors<dim>::I) + Grad_u_n);
+      material->update_material_data(F, p_tilde, J_tilde);
+      F_inv = invert(F);
+
+      // Step 1: Update stress and material tangent
+      {
+        const FEValuesExtractors::SymmetricTensor<2> C_dofs(0);
+        const FEValuesExtractors::Scalar             p_dofs(
+          dealii::SymmetricTensor<2, dim>::n_independent_components);
+        const FEValuesExtractors::Scalar J_dofs(
+          dealii::SymmetricTensor<2, dim>::n_independent_components + 1);
+        const unsigned int n_independent_variables =
+          SymmetricTensor<2, dim>::n_independent_components + 1 + 1;
+
+        typedef AD::ScalarFunction<dim, ad_type_code, number_t> ADHelper;
+        typedef typename ADHelper::ad_type                      ADNumberType;
+        ADHelper ad_helper(n_independent_variables);
+        ad_helper.set_tape_buffer_sizes(); // Increase the buffer size from the
+                                           // default values
+
+        const int  tape_no = 1;
+        const bool is_recording =
+          ad_helper.start_recording_operations(tape_no, //  material_id
+                                               true,    // overwrite_tape
+                                               true);   // keep
+
+        const SymmetricTensor<2, dim> C = symmetrize(transpose(F) * F);
+
+        if (is_recording == true)
+          {
+            ad_helper.register_independent_variable(C, C_dofs);
+            ad_helper.register_independent_variable(p_tilde, p_dofs);
+            ad_helper.register_independent_variable(J_tilde, J_dofs);
+
+            const SymmetricTensor<2, dim, ADNumberType> C_AD =
+              ad_helper.get_sensitive_variables(C_dofs);
+            const ADNumberType p_tilde_AD =
+              ad_helper.get_sensitive_variables(p_dofs);
+            const ADNumberType J_tilde_AD =
+              ad_helper.get_sensitive_variables(J_dofs);
+
+            const ADNumberType det_F_AD = sqrt(determinant(C_AD));
+            SymmetricTensor<2, dim, ADNumberType> C_bar_AD(C_AD);
+            C_bar_AD *= std::pow(det_F_AD, -2.0 / dim);
+
+            ADNumberType psi_CpJ = material->get_Psi_iso(C_bar_AD);
+            psi_CpJ += p_tilde_AD * (det_F_AD - J_tilde_AD);
+
+            ad_helper.register_dependent_variable(psi_CpJ);
+            ad_helper.stop_recording_operations(false); // write_tapes_to_file
+          }
+        else
+          {
+            Assert(is_recording == true, ExcInternalError());
+          }
+
+        // ad_helper.activate_recorded_tape(tape_no);
+        // ad_helper.set_independent_variable(C, C_dofs);
+        // ad_helper.set_independent_variable(p_tilde, p_dofs);
+        // ad_helper.set_independent_variable(J_tilde, J_dofs);
+
+        const double   psi = ad_helper.compute_value();
+        Vector<double> Dpsi(n_independent_variables);
+        ad_helper.compute_gradient(Dpsi);
+        FullMatrix<double> D2psi(n_independent_variables,
+                                 n_independent_variables);
+        ad_helper.compute_hessian(D2psi);
+
+        const SymmetricTensor<2, dim> S =
+          2.0 * ad_helper.extract_gradient_component(Dpsi, C_dofs);
+        const SymmetricTensor<4, dim> H =
+          4.0 * ad_helper.extract_hessian_component(D2psi, C_dofs, C_dofs);
+
+        tau = 0.0;
+        Jc  = 0.0;
+        // Naive push forwards: Super slow!
+        for (unsigned int i = 0; i < dim; ++i)
+          for (unsigned int j = i; j < dim; ++j)
+            {
+              for (unsigned int I = 0; I < dim; ++I)
+                for (unsigned int J = 0; J < dim; ++J)
+                  tau[i][j] += F[i][I] * S[I][J] * F[j][J];
+
+              for (unsigned int k = 0; k < dim; ++k)
+                for (unsigned int l = k; l < dim; ++l)
+                  for (unsigned int I = 0; I < dim; ++I)
+                    for (unsigned int J = 0; J < dim; ++J)
+                      for (unsigned int K = 0; K < dim; ++K)
+                        for (unsigned int L = 0; L < dim; ++L)
+                          Jc[i][j][k][l] += F[i][I] * F[j][J] * H[I][J][K][L] *
+                                            F[k][K] * F[l][L];
+            }
+      }
+
+      // Step 2: Update volumetric penalty terms
+      {
+        const FEValuesExtractors::Scalar J_dofs(0);
+        const unsigned int               n_independent_variables = 1;
+
+        typedef typename AD::ScalarFunction<dim, ad_type_code, double>::ad_type
+                                                      ADNumberType;
+        AD::ScalarFunction<dim, ad_type_code, double> ad_helper(
+          n_independent_variables);
+        ad_helper.set_tape_buffer_sizes(); // Increase the buffer size from the
+                                           // default values
+
+        const int  tape_no = 1;
+        const bool is_recording =
+          ad_helper.start_recording_operations(tape_no, //  material_id
+                                               true,    // overwrite_tape
+                                               true);   // keep
+        if (is_recording == true)
+          {
+            ad_helper.register_independent_variable(J_tilde, J_dofs);
+
+            const ADNumberType J_tilde_AD =
+              ad_helper.get_sensitive_variables(J_dofs);
+
+            ADNumberType psi_vol = material->get_Psi_vol(J_tilde_AD);
+
+            ad_helper.register_dependent_variable(psi_vol);
+            ad_helper.stop_recording_operations(false); // write_tapes_to_file
+          }
+        else
+          {
+            Assert(is_recording == true, ExcInternalError());
+          }
+
+        // ad_helper.activate_recorded_tape(tape_no);
+        // ad_helper.set_independent_variable(J_tilde, J_dofs);
+
+        const double       psi = ad_helper.compute_value();
+        Vector<double>     Dpsi(n_independent_variables);
+        FullMatrix<double> D2psi(n_independent_variables,
+                                 n_independent_variables);
+        ad_helper.compute_gradient(Dpsi);
+        ad_helper.compute_hessian(D2psi);
+
+        dPsi_vol_dJ = ad_helper.extract_gradient_component(Dpsi, J_dofs);
+        d2Psi_vol_dJ2 =
+          ad_helper.extract_hessian_component(D2psi, J_dofs, J_dofs);
+      }
+
+      static const double tol =
+        1e-3; // Minor computation error due to order of operations
+      Assert((tau - material->get_tau()).norm() < tol,
+             ExcMessage("AD computed stress is incorrect."));
+      Assert((Jc - material->get_Jc()).norm() < tol,
+             ExcMessage("AD computed tangent is incorrect."));
+      Assert(std::abs(dPsi_vol_dJ - material->get_dPsi_vol_dJ()) < tol,
+             ExcMessage("AD computed dPsi_vol_dJ is incorrect."));
+      Assert(std::abs(d2Psi_vol_dJ2 - material->get_d2Psi_vol_dJ2()) < tol,
+             ExcMessage("AD computed d2Psi_vol_dJ2 is incorrect."));
+    }
+    double
+    get_J_tilde() const
+    {
+      return material->get_J_tilde();
+    }
+    double
+    get_det_F() const
+    {
+      return material->get_det_F();
+    }
+    const Tensor<2, dim> &
+    get_F_inv() const
+    {
+      return F_inv;
+    }
+    double
+    get_p_tilde() const
+    {
+      return material->get_p_tilde();
+    }
+    const SymmetricTensor<2, dim> &
+    get_tau() const
+    {
+      return tau;
+    }
+    double
+    get_dPsi_vol_dJ() const
+    {
+      return dPsi_vol_dJ;
+    }
+    double
+    get_d2Psi_vol_dJ2() const
+    {
+      return d2Psi_vol_dJ2;
+    }
+    const SymmetricTensor<4, dim> &
+    get_Jc() const
+    {
+      return Jc;
+    }
+
+  private:
+    std::shared_ptr<Material_Compressible_Neo_Hook_Three_Field<dim>> material;
+    Tensor<2, dim>                                                   F_inv;
+    SymmetricTensor<2, dim>                                          tau;
+    double                  d2Psi_vol_dJ2;
+    double                  dPsi_vol_dJ;
+    SymmetricTensor<4, dim> Jc;
+  };
+  template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+  class Solid
+  {
+  public:
+    Solid(const std::string &input_file);
+    virtual ~Solid();
+    void
+    run();
+
+  private:
+    struct PerTaskData_K;
+    struct ScratchData_K;
+    struct PerTaskData_RHS;
+    struct ScratchData_RHS;
+    struct PerTaskData_SC;
+    struct ScratchData_SC;
+    struct PerTaskData_UQPH;
+    struct ScratchData_UQPH;
+    void
+    make_grid();
+    void
+    system_setup();
+    void
+    determine_component_extractors();
+    void
+    assemble_system_tangent();
+    void
+    assemble_system_tangent_one_cell(
+      const typename DoFHandler<dim>::active_cell_iterator &cell,
+      ScratchData_K &                                       scratch,
+      PerTaskData_K &                                       data) const;
+    void
+    copy_local_to_global_K(const PerTaskData_K &data);
+    void
+    assemble_system_rhs();
+    void
+    assemble_system_rhs_one_cell(
+      const typename DoFHandler<dim>::active_cell_iterator &cell,
+      ScratchData_RHS &                                     scratch,
+      PerTaskData_RHS &                                     data) const;
+    void
+    copy_local_to_global_rhs(const PerTaskData_RHS &data);
+    void
+    assemble_sc();
+    void
+    assemble_sc_one_cell(
+      const typename DoFHandler<dim>::active_cell_iterator &cell,
+      ScratchData_SC &                                      scratch,
+      PerTaskData_SC &                                      data);
+    void
+    copy_local_to_global_sc(const PerTaskData_SC &data);
+    void
+    make_constraints(const int &it_nr);
+    void
+    setup_qph();
+    void
+    update_qph_incremental(const BlockVector<double> &solution_delta);
+    void
+    update_qph_incremental_one_cell(
+      const typename DoFHandler<dim>::active_cell_iterator &cell,
+      ScratchData_UQPH &                                    scratch,
+      PerTaskData_UQPH &                                    data);
+    void
+    copy_local_to_global_UQPH(const PerTaskData_UQPH & /*data*/)
+    {}
+    void
+    solve_nonlinear_timestep(BlockVector<double> &solution_delta);
+    std::pair<unsigned int, double>
+    solve_linear_system(BlockVector<double> &newton_update);
+    BlockVector<double>
+    get_total_solution(const BlockVector<double> &solution_delta) const;
+    void
+                              output_results() const;
+    Parameters::AllParameters parameters;
+    double                    vol_reference;
+    Triangulation<dim>        triangulation;
+    Time                      time;
+    mutable TimerOutput       timer;
+    CellDataStorage<typename Triangulation<dim>::cell_iterator,
+                    PointHistory<dim, number_t, ad_type_code>>
+                                     quadrature_point_history;
+    const unsigned int               degree;
+    const FESystem<dim>              fe;
+    DoFHandler<dim>                  dof_handler_ref;
+    const unsigned int               dofs_per_cell;
+    const FEValuesExtractors::Vector u_fe;
+    const FEValuesExtractors::Scalar p_fe;
+    const FEValuesExtractors::Scalar J_fe;
+    static const unsigned int        n_blocks          = 3;
+    static const unsigned int        n_components      = dim + 2;
+    static const unsigned int        first_u_component = 0;
+    static const unsigned int        p_component       = dim;
+    static const unsigned int        J_component       = dim + 1;
+    enum
+    {
+      u_dof = 0,
+      p_dof = 1,
+      J_dof = 2
+    };
+    std::vector<types::global_dof_index> dofs_per_block;
+    std::vector<types::global_dof_index> element_indices_u;
+    std::vector<types::global_dof_index> element_indices_p;
+    std::vector<types::global_dof_index> element_indices_J;
+    const QGauss<dim>                    qf_cell;
+    const QGauss<dim - 1>                qf_face;
+    const unsigned int                   n_q_points;
+    const unsigned int                   n_q_points_f;
+    ConstraintMatrix                     constraints;
+    BlockSparsityPattern                 sparsity_pattern;
+    BlockSparseMatrix<double>            tangent_matrix;
+    BlockVector<double>                  system_rhs;
+    BlockVector<double>                  solution_n;
+    struct Errors
+    {
+      Errors()
+        : norm(1.0)
+        , u(1.0)
+        , p(1.0)
+        , J(1.0)
+      {}
+      void
+      reset()
+      {
+        norm = 1.0;
+        u    = 1.0;
+        p    = 1.0;
+        J    = 1.0;
+      }
+      void
+      normalise(const Errors &rhs)
+      {
+        if (rhs.norm != 0.0)
+          norm /= rhs.norm;
+        if (rhs.u != 0.0)
+          u /= rhs.u;
+        if (rhs.p != 0.0)
+          p /= rhs.p;
+        if (rhs.J != 0.0)
+          J /= rhs.J;
+      }
+      double norm, u, p, J;
+    };
+    Errors error_residual, error_residual_0, error_residual_norm, error_update,
+      error_update_0, error_update_norm;
+    void
+    get_error_residual(Errors &error_residual);
+    void
+    get_error_update(const BlockVector<double> &newton_update,
+                     Errors &                   error_update);
+    std::pair<double, double>
+    get_error_dilation() const;
+    double
+    compute_vol_current() const;
+    static void
+    print_conv_header();
+    void
+    print_conv_footer();
+  };
+  template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+  Solid<dim, number_t, ad_type_code>::Solid(const std::string &input_file)
+    : parameters(input_file)
+    , triangulation(Triangulation<dim>::maximum_smoothing)
+    , time(parameters.end_time, parameters.delta_t)
+    , timer(std::cout, TimerOutput::never, TimerOutput::wall_times)
+    , degree(parameters.poly_degree)
+    , fe(FE_Q<dim>(parameters.poly_degree),
+         dim, // displacement
+         FE_DGPMonomial<dim>(parameters.poly_degree - 1),
+         1, // pressure
+         FE_DGPMonomial<dim>(parameters.poly_degree - 1),
+         1)
+    , // dilatation
+    dof_handler_ref(triangulation)
+    , dofs_per_cell(fe.dofs_per_cell)
+    , u_fe(first_u_component)
+    , p_fe(p_component)
+    , J_fe(J_component)
+    , dofs_per_block(n_blocks)
+    , qf_cell(parameters.quad_order)
+    , qf_face(parameters.quad_order)
+    , n_q_points(qf_cell.size())
+    , n_q_points_f(qf_face.size())
+  {
+    Assert(dim == 2 || dim == 3,
+           ExcMessage("This problem only works in 2 or 3 space dimensions."));
+    determine_component_extractors();
+  }
+  template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+  Solid<dim, number_t, ad_type_code>::~Solid()
+  {
+    dof_handler_ref.clear();
+  }
+  template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+  void
+  Solid<dim, number_t, ad_type_code>::run()
+  {
+    make_grid();
+    system_setup();
+    {
+      ConstraintMatrix constraints;
+      constraints.close();
+      const ComponentSelectFunction<dim> J_mask(J_component, n_components);
+      VectorTools::project(dof_handler_ref,
+                           constraints,
+                           QGauss<dim>(degree + 2),
+                           J_mask,
+                           solution_n);
+    }
+    // output_results();
+    time.increment();
+    BlockVector<double> solution_delta(dofs_per_block);
+    while (time.current() < time.end())
+      {
+        solution_delta = 0.0;
+        solve_nonlinear_timestep(solution_delta);
+        solution_n += solution_delta;
+        // output_results();
+        // Output displacement at centre of traction surface
+        {
+          const Point<dim> soln_pt(
+            dim == 3 ? Point<dim>(0.0, 1.0, 0.0) * parameters.scale :
+                       Point<dim>(0.0, 1.0) * parameters.scale);
+          typename DoFHandler<dim>::active_cell_iterator
+            cell = dof_handler_ref.begin_active(),
+            endc = dof_handler_ref.end();
+          for (; cell != endc; ++cell)
+            for (unsigned int v = 0; v < GeometryInfo<dim>::vertices_per_cell;
+                 ++v)
+              if (cell->vertex(v).distance(soln_pt) < 1e-6 * parameters.scale)
+                {
+                  Tensor<1, dim> soln;
+                  for (unsigned int d = 0; d < dim; ++d)
+                    soln[d] = solution_n(cell->vertex_dof_index(v, u_dof + d));
+                  deallog << "Timestep " << time.get_timestep() << ": " << soln
+                          << std::endl;
+                }
+        }
+        time.increment();
+      }
+  }
+  template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+  struct Solid<dim, number_t, ad_type_code>::PerTaskData_K
+  {
+    FullMatrix<double>                   cell_matrix;
+    std::vector<types::global_dof_index> local_dof_indices;
+    PerTaskData_K(const unsigned int dofs_per_cell)
+      : cell_matrix(dofs_per_cell, dofs_per_cell)
+      , local_dof_indices(dofs_per_cell)
+    {}
+    void
+    reset()
+    {
+      cell_matrix = 0.0;
+    }
+  };
+  template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+  struct Solid<dim, number_t, ad_type_code>::ScratchData_K
+  {
+    FEValues<dim>                                     fe_values_ref;
+    std::vector<std::vector<double>>                  Nx;
+    std::vector<std::vector<Tensor<2, dim>>>          grad_Nx;
+    std::vector<std::vector<SymmetricTensor<2, dim>>> symm_grad_Nx;
+    ScratchData_K(const FiniteElement<dim> &fe_cell,
+                  const QGauss<dim> &       qf_cell,
+                  const UpdateFlags         uf_cell)
+      : fe_values_ref(fe_cell, qf_cell, uf_cell)
+      , Nx(qf_cell.size(), std::vector<double>(fe_cell.dofs_per_cell))
+      , grad_Nx(qf_cell.size(),
+                std::vector<Tensor<2, dim>>(fe_cell.dofs_per_cell))
+      , symm_grad_Nx(qf_cell.size(),
+                     std::vector<SymmetricTensor<2, dim>>(
+                       fe_cell.dofs_per_cell))
+    {}
+    ScratchData_K(const ScratchData_K &rhs)
+      : fe_values_ref(rhs.fe_values_ref.get_fe(),
+                      rhs.fe_values_ref.get_quadrature(),
+                      rhs.fe_values_ref.get_update_flags())
+      , Nx(rhs.Nx)
+      , grad_Nx(rhs.grad_Nx)
+      , symm_grad_Nx(rhs.symm_grad_Nx)
+    {}
+    void
+    reset()
+    {
+      const unsigned int n_q_points      = Nx.size();
+      const unsigned int n_dofs_per_cell = Nx[0].size();
+      for (unsigned int q_point = 0; q_point < n_q_points; ++q_point)
+        {
+          Assert(Nx[q_point].size() == n_dofs_per_cell, ExcInternalError());
+          Assert(grad_Nx[q_point].size() == n_dofs_per_cell,
+                 ExcInternalError());
+          Assert(symm_grad_Nx[q_point].size() == n_dofs_per_cell,
+                 ExcInternalError());
+          for (unsigned int k = 0; k < n_dofs_per_cell; ++k)
+            {
+              Nx[q_point][k]           = 0.0;
+              grad_Nx[q_point][k]      = 0.0;
+              symm_grad_Nx[q_point][k] = 0.0;
+            }
+        }
+    }
+  };
+  template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+  struct Solid<dim, number_t, ad_type_code>::PerTaskData_RHS
+  {
+    Vector<double>                       cell_rhs;
+    std::vector<types::global_dof_index> local_dof_indices;
+    PerTaskData_RHS(const unsigned int dofs_per_cell)
+      : cell_rhs(dofs_per_cell)
+      , local_dof_indices(dofs_per_cell)
+    {}
+    void
+    reset()
+    {
+      cell_rhs = 0.0;
+    }
+  };
+  template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+  struct Solid<dim, number_t, ad_type_code>::ScratchData_RHS
+  {
+    FEValues<dim>                                     fe_values_ref;
+    FEFaceValues<dim>                                 fe_face_values_ref;
+    std::vector<std::vector<double>>                  Nx;
+    std::vector<std::vector<SymmetricTensor<2, dim>>> symm_grad_Nx;
+    ScratchData_RHS(const FiniteElement<dim> &fe_cell,
+                    const QGauss<dim> &       qf_cell,
+                    const UpdateFlags         uf_cell,
+                    const QGauss<dim - 1> &   qf_face,
+                    const UpdateFlags         uf_face)
+      : fe_values_ref(fe_cell, qf_cell, uf_cell)
+      , fe_face_values_ref(fe_cell, qf_face, uf_face)
+      , Nx(qf_cell.size(), std::vector<double>(fe_cell.dofs_per_cell))
+      , symm_grad_Nx(qf_cell.size(),
+                     std::vector<SymmetricTensor<2, dim>>(
+                       fe_cell.dofs_per_cell))
+    {}
+    ScratchData_RHS(const ScratchData_RHS &rhs)
+      : fe_values_ref(rhs.fe_values_ref.get_fe(),
+                      rhs.fe_values_ref.get_quadrature(),
+                      rhs.fe_values_ref.get_update_flags())
+      , fe_face_values_ref(rhs.fe_face_values_ref.get_fe(),
+                           rhs.fe_face_values_ref.get_quadrature(),
+                           rhs.fe_face_values_ref.get_update_flags())
+      , Nx(rhs.Nx)
+      , symm_grad_Nx(rhs.symm_grad_Nx)
+    {}
+    void
+    reset()
+    {
+      const unsigned int n_q_points      = Nx.size();
+      const unsigned int n_dofs_per_cell = Nx[0].size();
+      for (unsigned int q_point = 0; q_point < n_q_points; ++q_point)
+        {
+          Assert(Nx[q_point].size() == n_dofs_per_cell, ExcInternalError());
+          Assert(symm_grad_Nx[q_point].size() == n_dofs_per_cell,
+                 ExcInternalError());
+          for (unsigned int k = 0; k < n_dofs_per_cell; ++k)
+            {
+              Nx[q_point][k]           = 0.0;
+              symm_grad_Nx[q_point][k] = 0.0;
+            }
+        }
+    }
+  };
+  template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+  struct Solid<dim, number_t, ad_type_code>::PerTaskData_SC
+  {
+    FullMatrix<double>                   cell_matrix;
+    std::vector<types::global_dof_index> local_dof_indices;
+    FullMatrix<double>                   k_orig;
+    FullMatrix<double>                   k_pu;
+    FullMatrix<double>                   k_pJ;
+    FullMatrix<double>                   k_JJ;
+    FullMatrix<double>                   k_pJ_inv;
+    FullMatrix<double>                   k_bbar;
+    FullMatrix<double>                   A;
+    FullMatrix<double>                   B;
+    FullMatrix<double>                   C;
+    PerTaskData_SC(const unsigned int dofs_per_cell,
+                   const unsigned int n_u,
+                   const unsigned int n_p,
+                   const unsigned int n_J)
+      : cell_matrix(dofs_per_cell, dofs_per_cell)
+      , local_dof_indices(dofs_per_cell)
+      , k_orig(dofs_per_cell, dofs_per_cell)
+      , k_pu(n_p, n_u)
+      , k_pJ(n_p, n_J)
+      , k_JJ(n_J, n_J)
+      , k_pJ_inv(n_p, n_J)
+      , k_bbar(n_u, n_u)
+      , A(n_J, n_u)
+      , B(n_J, n_u)
+      , C(n_p, n_u)
+    {}
+    void
+    reset()
+    {}
+  };
+  template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+  struct Solid<dim, number_t, ad_type_code>::ScratchData_SC
+  {
+    void
+    reset()
+    {}
+  };
+  template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+  struct Solid<dim, number_t, ad_type_code>::PerTaskData_UQPH
+  {
+    void
+    reset()
+    {}
+  };
+  template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+  struct Solid<dim, number_t, ad_type_code>::ScratchData_UQPH
+  {
+    const BlockVector<double> & solution_total;
+    std::vector<Tensor<2, dim>> solution_grads_u_total;
+    std::vector<double>         solution_values_p_total;
+    std::vector<double>         solution_values_J_total;
+    FEValues<dim>               fe_values_ref;
+    ScratchData_UQPH(const FiniteElement<dim> & fe_cell,
+                     const QGauss<dim> &        qf_cell,
+                     const UpdateFlags          uf_cell,
+                     const BlockVector<double> &solution_total)
+      : solution_total(solution_total)
+      , solution_grads_u_total(qf_cell.size())
+      , solution_values_p_total(qf_cell.size())
+      , solution_values_J_total(qf_cell.size())
+      , fe_values_ref(fe_cell, qf_cell, uf_cell)
+    {}
+    ScratchData_UQPH(const ScratchData_UQPH &rhs)
+      : solution_total(rhs.solution_total)
+      , solution_grads_u_total(rhs.solution_grads_u_total)
+      , solution_values_p_total(rhs.solution_values_p_total)
+      , solution_values_J_total(rhs.solution_values_J_total)
+      , fe_values_ref(rhs.fe_values_ref.get_fe(),
+                      rhs.fe_values_ref.get_quadrature(),
+                      rhs.fe_values_ref.get_update_flags())
+    {}
+    void
+    reset()
+    {
+      const unsigned int n_q_points = solution_grads_u_total.size();
+      for (unsigned int q = 0; q < n_q_points; ++q)
+        {
+          solution_grads_u_total[q]  = 0.0;
+          solution_values_p_total[q] = 0.0;
+          solution_values_J_total[q] = 0.0;
+        }
+    }
+  };
+  template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+  void
+  Solid<dim, number_t, ad_type_code>::make_grid()
+  {
+    GridGenerator::hyper_rectangle(
+      triangulation,
+      (dim == 3 ? Point<dim>(0.0, 0.0, 0.0) : Point<dim>(0.0, 0.0)),
+      (dim == 3 ? Point<dim>(1.0, 1.0, 1.0) : Point<dim>(1.0, 1.0)),
+      true);
+    GridTools::scale(parameters.scale, triangulation);
+    triangulation.refine_global(std::max(1U, parameters.global_refinement));
+    vol_reference = GridTools::volume(triangulation);
+    std::cout << "Grid:\n\t Reference volume: " << vol_reference << std::endl;
+    typename Triangulation<dim>::active_cell_iterator cell = triangulation
+                                                               .begin_active(),
+                                                      endc =
+                                                        triangulation.end();
+    for (; cell != endc; ++cell)
+      for (unsigned int face = 0; face < GeometryInfo<dim>::faces_per_cell;
+           ++face)
+        {
+          if (cell->face(face)->at_boundary() == true &&
+              cell->face(face)->center()[1] == 1.0 * parameters.scale)
+            {
+              if (dim == 3)
+                {
+                  if (cell->face(face)->center()[0] < 0.5 * parameters.scale &&
+                      cell->face(face)->center()[2] < 0.5 * parameters.scale)
+                    cell->face(face)->set_boundary_id(6);
+                }
+              else
+                {
+                  if (cell->face(face)->center()[0] < 0.5 * parameters.scale)
+                    cell->face(face)->set_boundary_id(6);
+                }
+            }
+        }
+  }
+  template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+  void
+  Solid<dim, number_t, ad_type_code>::system_setup()
+  {
+    timer.enter_subsection("Setup system");
+    std::vector<unsigned int> block_component(n_components,
+                                              u_dof); // Displacement
+    block_component[p_component] = p_dof;             // Pressure
+    block_component[J_component] = J_dof;             // Dilatation
+    dof_handler_ref.distribute_dofs(fe);
+    DoFRenumbering::Cuthill_McKee(dof_handler_ref);
+    DoFRenumbering::component_wise(dof_handler_ref, block_component);
+    DoFTools::count_dofs_per_block(dof_handler_ref,
+                                   dofs_per_block,
+                                   block_component);
+    std::cout << "Triangulation:"
+              << "\n\t Number of active cells: "
+              << triangulation.n_active_cells()
+              << "\n\t Number of degrees of freedom: "
+              << dof_handler_ref.n_dofs() << std::endl;
+    tangent_matrix.clear();
+    {
+      const types::global_dof_index n_dofs_u = dofs_per_block[u_dof];
+      const types::global_dof_index n_dofs_p = dofs_per_block[p_dof];
+      const types::global_dof_index n_dofs_J = dofs_per_block[J_dof];
+      BlockDynamicSparsityPattern   dsp(n_blocks, n_blocks);
+      dsp.block(u_dof, u_dof).reinit(n_dofs_u, n_dofs_u);
+      dsp.block(u_dof, p_dof).reinit(n_dofs_u, n_dofs_p);
+      dsp.block(u_dof, J_dof).reinit(n_dofs_u, n_dofs_J);
+      dsp.block(p_dof, u_dof).reinit(n_dofs_p, n_dofs_u);
+      dsp.block(p_dof, p_dof).reinit(n_dofs_p, n_dofs_p);
+      dsp.block(p_dof, J_dof).reinit(n_dofs_p, n_dofs_J);
+      dsp.block(J_dof, u_dof).reinit(n_dofs_J, n_dofs_u);
+      dsp.block(J_dof, p_dof).reinit(n_dofs_J, n_dofs_p);
+      dsp.block(J_dof, J_dof).reinit(n_dofs_J, n_dofs_J);
+      dsp.collect_sizes();
+      Table<2, DoFTools::Coupling> coupling(n_components, n_components);
+      for (unsigned int ii = 0; ii < n_components; ++ii)
+        for (unsigned int jj = 0; jj < n_components; ++jj)
+          if (((ii < p_component) && (jj == J_component)) ||
+              ((ii == J_component) && (jj < p_component)) ||
+              ((ii == p_component) && (jj == p_component)))
+            coupling[ii][jj] = DoFTools::none;
+          else
+            coupling[ii][jj] = DoFTools::always;
+      DoFTools::make_sparsity_pattern(
+        dof_handler_ref, coupling, dsp, constraints, false);
+      sparsity_pattern.copy_from(dsp);
+    }
+    tangent_matrix.reinit(sparsity_pattern);
+    system_rhs.reinit(dofs_per_block);
+    system_rhs.collect_sizes();
+    solution_n.reinit(dofs_per_block);
+    solution_n.collect_sizes();
+    setup_qph();
+    timer.leave_subsection();
+  }
+  template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+  void
+  Solid<dim, number_t, ad_type_code>::determine_component_extractors()
+  {
+    element_indices_u.clear();
+    element_indices_p.clear();
+    element_indices_J.clear();
+    for (unsigned int k = 0; k < fe.dofs_per_cell; ++k)
+      {
+        const unsigned int k_group = fe.system_to_base_index(k).first.first;
+        if (k_group == u_dof)
+          element_indices_u.push_back(k);
+        else if (k_group == p_dof)
+          element_indices_p.push_back(k);
+        else if (k_group == J_dof)
+          element_indices_J.push_back(k);
+        else
+          {
+            Assert(k_group <= J_dof, ExcInternalError());
+          }
+      }
+  }
+  template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+  void
+  Solid<dim, number_t, ad_type_code>::setup_qph()
+  {
+    std::cout << "    Setting up quadrature point data..." << std::endl;
+    quadrature_point_history.initialize(triangulation.begin_active(),
+                                        triangulation.end(),
+                                        n_q_points);
+    for (typename Triangulation<dim>::active_cell_iterator cell =
+           triangulation.begin_active();
+         cell != triangulation.end();
+         ++cell)
+      {
+        const std::vector<
+          std::shared_ptr<PointHistory<dim, number_t, ad_type_code>>>
+          lqph = quadrature_point_history.get_data(cell);
+        Assert(lqph.size() == n_q_points, ExcInternalError());
+        for (unsigned int q_point = 0; q_point < n_q_points; ++q_point)
+          lqph[q_point]->setup_lqp(parameters);
+      }
+  }
+  template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+  void
+  Solid<dim, number_t, ad_type_code>::update_qph_incremental(
+    const BlockVector<double> &solution_delta)
+  {
+    timer.enter_subsection("Update QPH data");
+    std::cout << " UQPH " << std::flush;
+    const BlockVector<double> solution_total(
+      get_total_solution(solution_delta));
+    const UpdateFlags uf_UQPH(update_values | update_gradients);
+    PerTaskData_UQPH  per_task_data_UQPH;
+    ScratchData_UQPH  scratch_data_UQPH(fe, qf_cell, uf_UQPH, solution_total);
+    // ADOL-C is incompatible with TBB
+    // WorkStream::run(dof_handler_ref.begin_active(),
+    //                 dof_handler_ref.end(),
+    //                 *this,
+    //                 &Solid::update_qph_incremental_one_cell,
+    //                 &Solid::copy_local_to_global_UQPH,
+    //                 scratch_data_UQPH,
+    //                 per_task_data_UQPH);
+    for (typename DoFHandler<dim>::active_cell_iterator cell =
+           dof_handler_ref.begin_active();
+         cell != dof_handler_ref.end();
+         ++cell)
+      {
+        update_qph_incremental_one_cell(cell,
+                                        scratch_data_UQPH,
+                                        per_task_data_UQPH);
+        copy_local_to_global_UQPH(per_task_data_UQPH);
+      }
+    timer.leave_subsection();
+  }
+  template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+  void
+  Solid<dim, number_t, ad_type_code>::update_qph_incremental_one_cell(
+    const typename DoFHandler<dim>::active_cell_iterator &cell,
+    ScratchData_UQPH &                                    scratch,
+    PerTaskData_UQPH & /*data*/)
+  {
+    const std::vector<
+      std::shared_ptr<PointHistory<dim, number_t, ad_type_code>>>
+      lqph = quadrature_point_history.get_data(cell);
+    Assert(lqph.size() == n_q_points, ExcInternalError());
+    Assert(scratch.solution_grads_u_total.size() == n_q_points,
+           ExcInternalError());
+    Assert(scratch.solution_values_p_total.size() == n_q_points,
+           ExcInternalError());
+    Assert(scratch.solution_values_J_total.size() == n_q_points,
+           ExcInternalError());
+    scratch.reset();
+    scratch.fe_values_ref.reinit(cell);
+    scratch.fe_values_ref[u_fe].get_function_gradients(
+      scratch.solution_total, scratch.solution_grads_u_total);
+    scratch.fe_values_ref[p_fe].get_function_values(
+      scratch.solution_total, scratch.solution_values_p_total);
+    scratch.fe_values_ref[J_fe].get_function_values(
+      scratch.solution_total, scratch.solution_values_J_total);
+    for (unsigned int q_point = 0; q_point < n_q_points; ++q_point)
+      lqph[q_point]->update_values(scratch.solution_grads_u_total[q_point],
+                                   scratch.solution_values_p_total[q_point],
+                                   scratch.solution_values_J_total[q_point]);
+  }
+  template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+  void
+  Solid<dim, number_t, ad_type_code>::solve_nonlinear_timestep(
+    BlockVector<double> &solution_delta)
+  {
+    std::cout << std::endl
+              << "Timestep " << time.get_timestep() << " @ " << time.current()
+              << "s" << std::endl;
+    BlockVector<double> newton_update(dofs_per_block);
+    error_residual.reset();
+    error_residual_0.reset();
+    error_residual_norm.reset();
+    error_update.reset();
+    error_update_0.reset();
+    error_update_norm.reset();
+    print_conv_header();
+    unsigned int newton_iteration = 0;
+    for (; newton_iteration < parameters.max_iterations_NR; ++newton_iteration)
+      {
+        std::cout << " " << std::setw(2) << newton_iteration << " "
+                  << std::flush;
+        tangent_matrix = 0.0;
+        system_rhs     = 0.0;
+        assemble_system_rhs();
+        get_error_residual(error_residual);
+        if (newton_iteration == 0)
+          error_residual_0 = error_residual;
+        error_residual_norm = error_residual;
+        error_residual_norm.normalise(error_residual_0);
+        if (newton_iteration > 0 && error_update_norm.u <= parameters.tol_u &&
+            error_residual_norm.u <= parameters.tol_f)
+          {
+            std::cout << " CONVERGED! " << std::endl;
+            print_conv_footer();
+            break;
+          }
+        assemble_system_tangent();
+        make_constraints(newton_iteration);
+        constraints.condense(tangent_matrix, system_rhs);
+        const std::pair<unsigned int, double> lin_solver_output =
+          solve_linear_system(newton_update);
+        get_error_update(newton_update, error_update);
+        if (newton_iteration == 0)
+          error_update_0 = error_update;
+        error_update_norm = error_update;
+        error_update_norm.normalise(error_update_0);
+        solution_delta += newton_update;
+        update_qph_incremental(solution_delta);
+        std::cout << " | " << std::fixed << std::setprecision(3) << std::setw(7)
+                  << std::scientific << lin_solver_output.first << "  "
+                  << lin_solver_output.second << "  "
+                  << error_residual_norm.norm << "  " << error_residual_norm.u
+                  << "  " << error_residual_norm.p << "  "
+                  << error_residual_norm.J << "  " << error_update_norm.norm
+                  << "  " << error_update_norm.u << "  " << error_update_norm.p
+                  << "  " << error_update_norm.J << "  " << std::endl;
+      }
+    AssertThrow(newton_iteration <= parameters.max_iterations_NR,
+                ExcMessage("No convergence in nonlinear solver!"));
+  }
+  template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+  void
+  Solid<dim, number_t, ad_type_code>::print_conv_header()
+  {
+    static const unsigned int l_width = 155;
+    for (unsigned int i = 0; i < l_width; ++i)
+      std::cout << "_";
+    std::cout << std::endl;
+    std::cout << "                 SOLVER STEP                  "
+              << " |  LIN_IT   LIN_RES    RES_NORM    "
+              << " RES_U     RES_P      RES_J     NU_NORM     "
+              << " NU_U       NU_P       NU_J " << std::endl;
+    for (unsigned int i = 0; i < l_width; ++i)
+      std::cout << "_";
+    std::cout << std::endl;
+  }
+  template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+  void
+  Solid<dim, number_t, ad_type_code>::print_conv_footer()
+  {
+    static const unsigned int l_width = 155;
+    for (unsigned int i = 0; i < l_width; ++i)
+      std::cout << "_";
+    std::cout << std::endl;
+    const std::pair<double, double> error_dil = get_error_dilation();
+    std::cout << "Relative errors:" << std::endl
+              << "Displacement:\t" << error_update.u / error_update_0.u
+              << std::endl
+              << "Force: \t\t" << error_residual.u / error_residual_0.u
+              << std::endl
+              << "Dilatation:\t" << error_dil.first << std::endl
+              << "v / V_0:\t" << error_dil.second * vol_reference << " / "
+              << vol_reference << " = " << error_dil.second << std::endl;
+  }
+  template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+  double
+  Solid<dim, number_t, ad_type_code>::compute_vol_current() const
+  {
+    double        vol_current = 0.0;
+    FEValues<dim> fe_values_ref(fe, qf_cell, update_JxW_values);
+    for (typename Triangulation<dim>::active_cell_iterator cell =
+           triangulation.begin_active();
+         cell != triangulation.end();
+         ++cell)
+      {
+        fe_values_ref.reinit(cell);
+        const std::vector<
+          std::shared_ptr<const PointHistory<dim, number_t, ad_type_code>>>
+          lqph = quadrature_point_history.get_data(cell);
+        Assert(lqph.size() == n_q_points, ExcInternalError());
+        for (unsigned int q_point = 0; q_point < n_q_points; ++q_point)
+          {
+            const double det_F_qp = lqph[q_point]->get_det_F();
+            const double JxW      = fe_values_ref.JxW(q_point);
+            vol_current += det_F_qp * JxW;
+          }
+      }
+    Assert(vol_current > 0.0, ExcInternalError());
+    return vol_current;
+  }
+  template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+  std::pair<double, double>
+  Solid<dim, number_t, ad_type_code>::get_error_dilation() const
+  {
+    double        dil_L2_error = 0.0;
+    FEValues<dim> fe_values_ref(fe, qf_cell, update_JxW_values);
+    for (typename Triangulation<dim>::active_cell_iterator cell =
+           triangulation.begin_active();
+         cell != triangulation.end();
+         ++cell)
+      {
+        fe_values_ref.reinit(cell);
+        const std::vector<
+          std::shared_ptr<const PointHistory<dim, number_t, ad_type_code>>>
+          lqph = quadrature_point_history.get_data(cell);
+        Assert(lqph.size() == n_q_points, ExcInternalError());
+        for (unsigned int q_point = 0; q_point < n_q_points; ++q_point)
+          {
+            const double det_F_qp   = lqph[q_point]->get_det_F();
+            const double J_tilde_qp = lqph[q_point]->get_J_tilde();
+            const double the_error_qp_squared =
+              std::pow((det_F_qp - J_tilde_qp), 2);
+            const double JxW = fe_values_ref.JxW(q_point);
+            dil_L2_error += the_error_qp_squared * JxW;
+          }
+      }
+    return std::make_pair(std::sqrt(dil_L2_error),
+                          compute_vol_current() / vol_reference);
+  }
+  template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+  void
+  Solid<dim, number_t, ad_type_code>::get_error_residual(Errors &error_residual)
+  {
+    BlockVector<double> error_res(dofs_per_block);
+    for (unsigned int i = 0; i < dof_handler_ref.n_dofs(); ++i)
+      if (!constraints.is_constrained(i))
+        error_res(i) = system_rhs(i);
+    error_residual.norm = error_res.l2_norm();
+    error_residual.u    = error_res.block(u_dof).l2_norm();
+    error_residual.p    = error_res.block(p_dof).l2_norm();
+    error_residual.J    = error_res.block(J_dof).l2_norm();
+  }
+  template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+  void
+  Solid<dim, number_t, ad_type_code>::get_error_update(
+    const BlockVector<double> &newton_update,
+    Errors &                   error_update)
+  {
+    BlockVector<double> error_ud(dofs_per_block);
+    for (unsigned int i = 0; i < dof_handler_ref.n_dofs(); ++i)
+      if (!constraints.is_constrained(i))
+        error_ud(i) = newton_update(i);
+    error_update.norm = error_ud.l2_norm();
+    error_update.u    = error_ud.block(u_dof).l2_norm();
+    error_update.p    = error_ud.block(p_dof).l2_norm();
+    error_update.J    = error_ud.block(J_dof).l2_norm();
+  }
+  template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+  BlockVector<double>
+  Solid<dim, number_t, ad_type_code>::get_total_solution(
+    const BlockVector<double> &solution_delta) const
+  {
+    BlockVector<double> solution_total(solution_n);
+    solution_total += solution_delta;
+    return solution_total;
+  }
+  template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+  void
+  Solid<dim, number_t, ad_type_code>::assemble_system_tangent()
+  {
+    timer.enter_subsection("Assemble tangent matrix");
+    std::cout << " ASM_K " << std::flush;
+    tangent_matrix = 0.0;
+    const UpdateFlags uf_cell(update_values | update_gradients |
+                              update_JxW_values);
+    PerTaskData_K     per_task_data(dofs_per_cell);
+    ScratchData_K     scratch_data(fe, qf_cell, uf_cell);
+    WorkStream::run(
+      dof_handler_ref.begin_active(),
+      dof_handler_ref.end(),
+      std::bind(
+        &Solid<dim, number_t, ad_type_code>::assemble_system_tangent_one_cell,
+        this,
+        std::placeholders::_1,
+        std::placeholders::_2,
+        std::placeholders::_3),
+      std::bind(&Solid<dim, number_t, ad_type_code>::copy_local_to_global_K,
+                this,
+                std::placeholders::_1),
+      scratch_data,
+      per_task_data);
+    timer.leave_subsection();
+  }
+  template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+  void
+  Solid<dim, number_t, ad_type_code>::copy_local_to_global_K(
+    const PerTaskData_K &data)
+  {
+    for (unsigned int i = 0; i < dofs_per_cell; ++i)
+      for (unsigned int j = 0; j < dofs_per_cell; ++j)
+        tangent_matrix.add(data.local_dof_indices[i],
+                           data.local_dof_indices[j],
+                           data.cell_matrix(i, j));
+  }
+  template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+  void
+  Solid<dim, number_t, ad_type_code>::assemble_system_tangent_one_cell(
+    const typename DoFHandler<dim>::active_cell_iterator &cell,
+    ScratchData_K &                                       scratch,
+    PerTaskData_K &                                       data) const
+  {
+    data.reset();
+    scratch.reset();
+    scratch.fe_values_ref.reinit(cell);
+    cell->get_dof_indices(data.local_dof_indices);
+    const std::vector<
+      std::shared_ptr<const PointHistory<dim, number_t, ad_type_code>>>
+      lqph = quadrature_point_history.get_data(cell);
+    Assert(lqph.size() == n_q_points, ExcInternalError());
+    for (unsigned int q_point = 0; q_point < n_q_points; ++q_point)
+      {
+        const Tensor<2, dim> F_inv = lqph[q_point]->get_F_inv();
+        for (unsigned int k = 0; k < dofs_per_cell; ++k)
+          {
+            const unsigned int k_group = fe.system_to_base_index(k).first.first;
+            if (k_group == u_dof)
+              {
+                scratch.grad_Nx[q_point][k] =
+                  scratch.fe_values_ref[u_fe].gradient(k, q_point) * F_inv;
+                scratch.symm_grad_Nx[q_point][k] =
+                  symmetrize(scratch.grad_Nx[q_point][k]);
+              }
+            else if (k_group == p_dof)
+              scratch.Nx[q_point][k] =
+                scratch.fe_values_ref[p_fe].value(k, q_point);
+            else if (k_group == J_dof)
+              scratch.Nx[q_point][k] =
+                scratch.fe_values_ref[J_fe].value(k, q_point);
+            else
+              Assert(k_group <= J_dof, ExcInternalError());
+          }
+      }
+    for (unsigned int q_point = 0; q_point < n_q_points; ++q_point)
+      {
+        const Tensor<2, dim>          tau = lqph[q_point]->get_tau();
+        const SymmetricTensor<4, dim> Jc  = lqph[q_point]->get_Jc();
+        const double d2Psi_vol_dJ2        = lqph[q_point]->get_d2Psi_vol_dJ2();
+        const double det_F                = lqph[q_point]->get_det_F();
+        const std::vector<double> &                 N = scratch.Nx[q_point];
+        const std::vector<SymmetricTensor<2, dim>> &symm_grad_Nx =
+          scratch.symm_grad_Nx[q_point];
+        const std::vector<Tensor<2, dim>> &grad_Nx = scratch.grad_Nx[q_point];
+        const double JxW = scratch.fe_values_ref.JxW(q_point);
+        for (unsigned int i = 0; i < dofs_per_cell; ++i)
+          {
+            const unsigned int component_i =
+              fe.system_to_component_index(i).first;
+            const unsigned int i_group = fe.system_to_base_index(i).first.first;
+            for (unsigned int j = 0; j <= i; ++j)
+              {
+                const unsigned int component_j =
+                  fe.system_to_component_index(j).first;
+                const unsigned int j_group =
+                  fe.system_to_base_index(j).first.first;
+                if ((i_group == j_group) && (i_group == u_dof))
+                  {
+                    data.cell_matrix(i, j) += symm_grad_Nx[i] *
+                                              Jc // The material contribution:
+                                              * symm_grad_Nx[j] * JxW;
+                    if (component_i ==
+                        component_j) // geometrical stress contribution
+                      data.cell_matrix(i, j) += grad_Nx[i][component_i] * tau *
+                                                grad_Nx[j][component_j] * JxW;
+                  }
+                else if ((i_group == p_dof) && (j_group == u_dof))
+                  {
+                    data.cell_matrix(i, j) +=
+                      N[i] * det_F *
+                      (symm_grad_Nx[j] * StandardTensors<dim>::I) * JxW;
+                  }
+                else if ((i_group == J_dof) && (j_group == p_dof))
+                  data.cell_matrix(i, j) -= N[i] * N[j] * JxW;
+                else if ((i_group == j_group) && (i_group == J_dof))
+                  data.cell_matrix(i, j) += N[i] * d2Psi_vol_dJ2 * N[j] * JxW;
+                else
+                  Assert((i_group <= J_dof) && (j_group <= J_dof),
+                         ExcInternalError());
+              }
+          }
+      }
+    for (unsigned int i = 0; i < dofs_per_cell; ++i)
+      for (unsigned int j = i + 1; j < dofs_per_cell; ++j)
+        data.cell_matrix(i, j) = data.cell_matrix(j, i);
+  }
+  template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+  void
+  Solid<dim, number_t, ad_type_code>::assemble_system_rhs()
+  {
+    timer.enter_subsection("Assemble system right-hand side");
+    std::cout << " ASM_R " << std::flush;
+    system_rhs = 0.0;
+    const UpdateFlags uf_cell(update_values | update_gradients |
+                              update_JxW_values);
+    const UpdateFlags uf_face(update_values | update_normal_vectors |
+                              update_JxW_values);
+    PerTaskData_RHS   per_task_data(dofs_per_cell);
+    ScratchData_RHS   scratch_data(fe, qf_cell, uf_cell, qf_face, uf_face);
+    WorkStream::run(
+      dof_handler_ref.begin_active(),
+      dof_handler_ref.end(),
+      std::bind(
+        &Solid<dim, number_t, ad_type_code>::assemble_system_rhs_one_cell,
+        this,
+        std::placeholders::_1,
+        std::placeholders::_2,
+        std::placeholders::_3),
+      std::bind(&Solid<dim, number_t, ad_type_code>::copy_local_to_global_rhs,
+                this,
+                std::placeholders::_1),
+      scratch_data,
+      per_task_data);
+    timer.leave_subsection();
+  }
+  template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+  void
+  Solid<dim, number_t, ad_type_code>::copy_local_to_global_rhs(
+    const PerTaskData_RHS &data)
+  {
+    for (unsigned int i = 0; i < dofs_per_cell; ++i)
+      system_rhs(data.local_dof_indices[i]) += data.cell_rhs(i);
+  }
+  template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+  void
+  Solid<dim, number_t, ad_type_code>::assemble_system_rhs_one_cell(
+    const typename DoFHandler<dim>::active_cell_iterator &cell,
+    ScratchData_RHS &                                     scratch,
+    PerTaskData_RHS &                                     data) const
+  {
+    data.reset();
+    scratch.reset();
+    scratch.fe_values_ref.reinit(cell);
+    cell->get_dof_indices(data.local_dof_indices);
+    const std::vector<
+      std::shared_ptr<const PointHistory<dim, number_t, ad_type_code>>>
+      lqph = quadrature_point_history.get_data(cell);
+    Assert(lqph.size() == n_q_points, ExcInternalError());
+    for (unsigned int q_point = 0; q_point < n_q_points; ++q_point)
+      {
+        const Tensor<2, dim> F_inv = lqph[q_point]->get_F_inv();
+        for (unsigned int k = 0; k < dofs_per_cell; ++k)
+          {
+            const unsigned int k_group = fe.system_to_base_index(k).first.first;
+            if (k_group == u_dof)
+              scratch.symm_grad_Nx[q_point][k] = symmetrize(
+                scratch.fe_values_ref[u_fe].gradient(k, q_point) * F_inv);
+            else if (k_group == p_dof)
+              scratch.Nx[q_point][k] =
+                scratch.fe_values_ref[p_fe].value(k, q_point);
+            else if (k_group == J_dof)
+              scratch.Nx[q_point][k] =
+                scratch.fe_values_ref[J_fe].value(k, q_point);
+            else
+              Assert(k_group <= J_dof, ExcInternalError());
+          }
+      }
+    for (unsigned int q_point = 0; q_point < n_q_points; ++q_point)
+      {
+        const SymmetricTensor<2, dim> tau     = lqph[q_point]->get_tau();
+        const double                  det_F   = lqph[q_point]->get_det_F();
+        const double                  J_tilde = lqph[q_point]->get_J_tilde();
+        const double                  p_tilde = lqph[q_point]->get_p_tilde();
+        const double dPsi_vol_dJ = lqph[q_point]->get_dPsi_vol_dJ();
+        const std::vector<double> &                 N = scratch.Nx[q_point];
+        const std::vector<SymmetricTensor<2, dim>> &symm_grad_Nx =
+          scratch.symm_grad_Nx[q_point];
+        const double JxW = scratch.fe_values_ref.JxW(q_point);
+        for (unsigned int i = 0; i < dofs_per_cell; ++i)
+          {
+            const unsigned int i_group = fe.system_to_base_index(i).first.first;
+            if (i_group == u_dof)
+              data.cell_rhs(i) -= (symm_grad_Nx[i] * tau) * JxW;
+            else if (i_group == p_dof)
+              data.cell_rhs(i) -= N[i] * (det_F - J_tilde) * JxW;
+            else if (i_group == J_dof)
+              data.cell_rhs(i) -= N[i] * (dPsi_vol_dJ - p_tilde) * JxW;
+            else
+              Assert(i_group <= J_dof, ExcInternalError());
+          }
+      }
+    for (unsigned int face = 0; face < GeometryInfo<dim>::faces_per_cell;
+         ++face)
+      if (cell->face(face)->at_boundary() == true &&
+          cell->face(face)->boundary_id() == 6)
+        {
+          scratch.fe_face_values_ref.reinit(cell, face);
+          for (unsigned int f_q_point = 0; f_q_point < n_q_points_f;
+               ++f_q_point)
+            {
+              const Tensor<1, dim> &N =
+                scratch.fe_face_values_ref.normal_vector(f_q_point);
+              static const double p0 =
+                -4.0 / (parameters.scale * parameters.scale);
+              const double         time_ramp = (time.current() / time.end());
+              const double         pressure  = p0 * parameters.p_p0 * time_ramp;
+              const Tensor<1, dim> traction  = pressure * N;
+              for (unsigned int i = 0; i < dofs_per_cell; ++i)
+                {
+                  const unsigned int i_group =
+                    fe.system_to_base_index(i).first.first;
+                  if (i_group == u_dof)
+                    {
+                      const unsigned int component_i =
+                        fe.system_to_component_index(i).first;
+                      const double Ni =
+                        scratch.fe_face_values_ref.shape_value(i, f_q_point);
+                      const double JxW =
+                        scratch.fe_face_values_ref.JxW(f_q_point);
+                      data.cell_rhs(i) += (Ni * traction[component_i]) * JxW;
+                    }
+                }
+            }
+        }
+  }
+  template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+  void
+  Solid<dim, number_t, ad_type_code>::make_constraints(const int &it_nr)
+  {
+    std::cout << " CST " << std::flush;
+    if (it_nr > 1)
+      return;
+    constraints.clear();
+    const bool                       apply_dirichlet_bc = (it_nr == 0);
+    const FEValuesExtractors::Scalar x_displacement(0);
+    const FEValuesExtractors::Scalar y_displacement(1);
+    {
+      const int boundary_id = 0;
+      if (apply_dirichlet_bc == true)
+        VectorTools::interpolate_boundary_values(
+          dof_handler_ref,
+          boundary_id,
+          ZeroFunction<dim>(n_components),
+          constraints,
+          fe.component_mask(x_displacement));
+      else
+        VectorTools::interpolate_boundary_values(
+          dof_handler_ref,
+          boundary_id,
+          ZeroFunction<dim>(n_components),
+          constraints,
+          fe.component_mask(x_displacement));
+    }
+    {
+      const int boundary_id = 2;
+      if (apply_dirichlet_bc == true)
+        VectorTools::interpolate_boundary_values(
+          dof_handler_ref,
+          boundary_id,
+          ZeroFunction<dim>(n_components),
+          constraints,
+          fe.component_mask(y_displacement));
+      else
+        VectorTools::interpolate_boundary_values(
+          dof_handler_ref,
+          boundary_id,
+          ZeroFunction<dim>(n_components),
+          constraints,
+          fe.component_mask(y_displacement));
+    }
+    if (dim == 3)
+      {
+        const FEValuesExtractors::Scalar z_displacement(2);
+        {
+          const int boundary_id = 3;
+          if (apply_dirichlet_bc == true)
+            VectorTools::interpolate_boundary_values(
+              dof_handler_ref,
+              boundary_id,
+              ZeroFunction<dim>(n_components),
+              constraints,
+              (fe.component_mask(x_displacement) |
+               fe.component_mask(z_displacement)));
+          else
+            VectorTools::interpolate_boundary_values(
+              dof_handler_ref,
+              boundary_id,
+              ZeroFunction<dim>(n_components),
+              constraints,
+              (fe.component_mask(x_displacement) |
+               fe.component_mask(z_displacement)));
+        }
+        {
+          const int boundary_id = 4;
+          if (apply_dirichlet_bc == true)
+            VectorTools::interpolate_boundary_values(
+              dof_handler_ref,
+              boundary_id,
+              ZeroFunction<dim>(n_components),
+              constraints,
+              fe.component_mask(z_displacement));
+          else
+            VectorTools::interpolate_boundary_values(
+              dof_handler_ref,
+              boundary_id,
+              ZeroFunction<dim>(n_components),
+              constraints,
+              fe.component_mask(z_displacement));
+        }
+        {
+          const int boundary_id = 6;
+          if (apply_dirichlet_bc == true)
+            VectorTools::interpolate_boundary_values(
+              dof_handler_ref,
+              boundary_id,
+              ZeroFunction<dim>(n_components),
+              constraints,
+              (fe.component_mask(x_displacement) |
+               fe.component_mask(z_displacement)));
+          else
+            VectorTools::interpolate_boundary_values(
+              dof_handler_ref,
+              boundary_id,
+              ZeroFunction<dim>(n_components),
+              constraints,
+              (fe.component_mask(x_displacement) |
+               fe.component_mask(z_displacement)));
+        }
+      }
+    else
+      {
+        {
+          const int boundary_id = 3;
+          if (apply_dirichlet_bc == true)
+            VectorTools::interpolate_boundary_values(
+              dof_handler_ref,
+              boundary_id,
+              ZeroFunction<dim>(n_components),
+              constraints,
+              (fe.component_mask(x_displacement)));
+          else
+            VectorTools::interpolate_boundary_values(
+              dof_handler_ref,
+              boundary_id,
+              ZeroFunction<dim>(n_components),
+              constraints,
+              (fe.component_mask(x_displacement)));
+        }
+        {
+          const int boundary_id = 6;
+          if (apply_dirichlet_bc == true)
+            VectorTools::interpolate_boundary_values(
+              dof_handler_ref,
+              boundary_id,
+              ZeroFunction<dim>(n_components),
+              constraints,
+              (fe.component_mask(x_displacement)));
+          else
+            VectorTools::interpolate_boundary_values(
+              dof_handler_ref,
+              boundary_id,
+              ZeroFunction<dim>(n_components),
+              constraints,
+              (fe.component_mask(x_displacement)));
+        }
+      }
+    constraints.close();
+  }
+  template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+  void
+  Solid<dim, number_t, ad_type_code>::assemble_sc()
+  {
+    timer.enter_subsection("Perform static condensation");
+    std::cout << " ASM_SC " << std::flush;
+    PerTaskData_SC per_task_data(dofs_per_cell,
+                                 element_indices_u.size(),
+                                 element_indices_p.size(),
+                                 element_indices_J.size());
+    ScratchData_SC scratch_data;
+    WorkStream::run(dof_handler_ref.begin_active(),
+                    dof_handler_ref.end(),
+                    *this,
+                    &Solid::assemble_sc_one_cell,
+                    &Solid::copy_local_to_global_sc,
+                    scratch_data,
+                    per_task_data);
+    timer.leave_subsection();
+  }
+  template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+  void
+  Solid<dim, number_t, ad_type_code>::copy_local_to_global_sc(
+    const PerTaskData_SC &data)
+  {
+    for (unsigned int i = 0; i < dofs_per_cell; ++i)
+      for (unsigned int j = 0; j < dofs_per_cell; ++j)
+        tangent_matrix.add(data.local_dof_indices[i],
+                           data.local_dof_indices[j],
+                           data.cell_matrix(i, j));
+  }
+  template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+  void
+  Solid<dim, number_t, ad_type_code>::assemble_sc_one_cell(
+    const typename DoFHandler<dim>::active_cell_iterator &cell,
+    ScratchData_SC &                                      scratch,
+    PerTaskData_SC &                                      data)
+  {
+    data.reset();
+    scratch.reset();
+    cell->get_dof_indices(data.local_dof_indices);
+    data.k_orig.extract_submatrix_from(tangent_matrix,
+                                       data.local_dof_indices,
+                                       data.local_dof_indices);
+    data.k_pu.extract_submatrix_from(data.k_orig,
+                                     element_indices_p,
+                                     element_indices_u);
+    data.k_pJ.extract_submatrix_from(data.k_orig,
+                                     element_indices_p,
+                                     element_indices_J);
+    data.k_JJ.extract_submatrix_from(data.k_orig,
+                                     element_indices_J,
+                                     element_indices_J);
+    data.k_pJ_inv.invert(data.k_pJ);
+    data.k_pJ_inv.mmult(data.A, data.k_pu);
+    data.k_JJ.mmult(data.B, data.A);
+    data.k_pJ_inv.Tmmult(data.C, data.B);
+    data.k_pu.Tmmult(data.k_bbar, data.C);
+    data.k_bbar.scatter_matrix_to(element_indices_u,
+                                  element_indices_u,
+                                  data.cell_matrix);
+    data.k_pJ_inv.add(-1.0, data.k_pJ);
+    data.k_pJ_inv.scatter_matrix_to(element_indices_p,
+                                    element_indices_J,
+                                    data.cell_matrix);
+  }
+  template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+  std::pair<unsigned int, double>
+  Solid<dim, number_t, ad_type_code>::solve_linear_system(
+    BlockVector<double> &newton_update)
+  {
+    unsigned int lin_it  = 0;
+    double       lin_res = 0.0;
+    if (parameters.use_static_condensation == true)
+      {
+        BlockVector<double> A(dofs_per_block);
+        BlockVector<double> B(dofs_per_block);
+        {
+          assemble_sc();
+          tangent_matrix.block(p_dof, J_dof)
+            .vmult(A.block(J_dof), system_rhs.block(p_dof));
+          tangent_matrix.block(J_dof, J_dof)
+            .vmult(B.block(J_dof), A.block(J_dof));
+          A.block(J_dof) = system_rhs.block(J_dof);
+          A.block(J_dof) -= B.block(J_dof);
+          tangent_matrix.block(p_dof, J_dof)
+            .Tvmult(A.block(p_dof), A.block(J_dof));
+          tangent_matrix.block(u_dof, p_dof)
+            .vmult(A.block(u_dof), A.block(p_dof));
+          system_rhs.block(u_dof) -= A.block(u_dof);
+          timer.enter_subsection("Linear solver");
+          std::cout << " SLV " << std::flush;
+          if (parameters.type_lin == "CG")
+            {
+              const int solver_its = tangent_matrix.block(u_dof, u_dof).m() *
+                                     parameters.max_iterations_lin;
+              const double tol_sol =
+                parameters.tol_lin * system_rhs.block(u_dof).l2_norm();
+              SolverControl solver_control(solver_its, tol_sol, false, false);
+              GrowingVectorMemory<Vector<double>> GVM;
+              SolverCG<Vector<double>> solver_CG(solver_control, GVM);
+              PreconditionSelector<SparseMatrix<double>, Vector<double>>
+                preconditioner(parameters.preconditioner_type,
+                               parameters.preconditioner_relaxation);
+              preconditioner.use_matrix(tangent_matrix.block(u_dof, u_dof));
+              solver_CG.solve(tangent_matrix.block(u_dof, u_dof),
+                              newton_update.block(u_dof),
+                              system_rhs.block(u_dof),
+                              preconditioner);
+              lin_it  = solver_control.last_step();
+              lin_res = solver_control.last_value();
+            }
+          else if (parameters.type_lin == "Direct")
+            {
+              SparseDirectUMFPACK A_direct;
+              A_direct.initialize(tangent_matrix.block(u_dof, u_dof));
+              A_direct.vmult(newton_update.block(u_dof),
+                             system_rhs.block(u_dof));
+              lin_it  = 1;
+              lin_res = 0.0;
+            }
+          else
+            Assert(false, ExcMessage("Linear solver type not implemented"));
+          timer.leave_subsection();
+        }
+        constraints.distribute(newton_update);
+        timer.enter_subsection("Linear solver postprocessing");
+        std::cout << " PP " << std::flush;
+        {
+          tangent_matrix.block(p_dof, u_dof)
+            .vmult(A.block(p_dof), newton_update.block(u_dof));
+          A.block(p_dof) *= -1.0;
+          A.block(p_dof) += system_rhs.block(p_dof);
+          tangent_matrix.block(p_dof, J_dof)
+            .vmult(newton_update.block(J_dof), A.block(p_dof));
+        }
+        constraints.distribute(newton_update);
+        {
+          tangent_matrix.block(J_dof, J_dof)
+            .vmult(A.block(J_dof), newton_update.block(J_dof));
+          A.block(J_dof) *= -1.0;
+          A.block(J_dof) += system_rhs.block(J_dof);
+          tangent_matrix.block(p_dof, J_dof)
+            .Tvmult(newton_update.block(p_dof), A.block(J_dof));
+        }
+        constraints.distribute(newton_update);
+        timer.leave_subsection();
+      }
+    else
+      {
+        std::cout << " ------ " << std::flush;
+        timer.enter_subsection("Linear solver");
+        std::cout << " SLV " << std::flush;
+        if (parameters.type_lin == "CG")
+          {
+            const Vector<double> &f_u = system_rhs.block(u_dof);
+            const Vector<double> &f_p = system_rhs.block(p_dof);
+            const Vector<double> &f_J = system_rhs.block(J_dof);
+            Vector<double> &      d_u = newton_update.block(u_dof);
+            Vector<double> &      d_p = newton_update.block(p_dof);
+            Vector<double> &      d_J = newton_update.block(J_dof);
+            const auto            K_uu =
+              linear_operator(tangent_matrix.block(u_dof, u_dof));
+            const auto K_up =
+              linear_operator(tangent_matrix.block(u_dof, p_dof));
+            const auto K_pu =
+              linear_operator(tangent_matrix.block(p_dof, u_dof));
+            const auto K_Jp =
+              linear_operator(tangent_matrix.block(J_dof, p_dof));
+            const auto K_JJ =
+              linear_operator(tangent_matrix.block(J_dof, J_dof));
+            PreconditionSelector<SparseMatrix<double>, Vector<double>>
+              preconditioner_K_Jp_inv("jacobi");
+            preconditioner_K_Jp_inv.use_matrix(
+              tangent_matrix.block(J_dof, p_dof));
+            ReductionControl solver_control_K_Jp_inv(
+              tangent_matrix.block(J_dof, p_dof).m() *
+                parameters.max_iterations_lin,
+              1.0e-30,
+              parameters.tol_lin);
+            SolverSelector<Vector<double>> solver_K_Jp_inv;
+            solver_K_Jp_inv.select("cg");
+            solver_K_Jp_inv.set_control(solver_control_K_Jp_inv);
+            const auto K_Jp_inv =
+              inverse_operator(K_Jp, solver_K_Jp_inv, preconditioner_K_Jp_inv);
+            const auto K_pJ_inv     = transpose_operator(K_Jp_inv);
+            const auto K_pp_bar     = K_Jp_inv * K_JJ * K_pJ_inv;
+            const auto K_uu_bar_bar = K_up * K_pp_bar * K_pu;
+            const auto K_uu_con     = K_uu + K_uu_bar_bar;
+            PreconditionSelector<SparseMatrix<double>, Vector<double>>
+              preconditioner_K_con_inv(parameters.preconditioner_type,
+                                       parameters.preconditioner_relaxation);
+            preconditioner_K_con_inv.use_matrix(
+              tangent_matrix.block(u_dof, u_dof));
+            ReductionControl solver_control_K_con_inv(
+              tangent_matrix.block(u_dof, u_dof).m() *
+                parameters.max_iterations_lin,
+              1.0e-30,
+              parameters.tol_lin);
+            SolverSelector<Vector<double>> solver_K_con_inv;
+            solver_K_con_inv.select("cg");
+            solver_K_con_inv.set_control(solver_control_K_con_inv);
+            const auto K_uu_con_inv =
+              inverse_operator(K_uu_con,
+                               solver_K_con_inv,
+                               preconditioner_K_con_inv);
+            d_u =
+              K_uu_con_inv * (f_u - K_up * (K_Jp_inv * f_J - K_pp_bar * f_p));
+            timer.leave_subsection();
+            timer.enter_subsection("Linear solver postprocessing");
+            std::cout << " PP " << std::flush;
+            d_J     = K_pJ_inv * (f_p - K_pu * d_u);
+            d_p     = K_Jp_inv * (f_J - K_JJ * d_J);
+            lin_it  = solver_control_K_con_inv.last_step();
+            lin_res = solver_control_K_con_inv.last_value();
+          }
+        else if (parameters.type_lin == "Direct")
+          {
+            SparseDirectUMFPACK A_direct;
+            A_direct.initialize(tangent_matrix);
+            A_direct.vmult(newton_update, system_rhs);
+            lin_it  = 1;
+            lin_res = 0.0;
+            std::cout << " -- " << std::flush;
+          }
+        else
+          Assert(false, ExcMessage("Linear solver type not implemented"));
+        timer.leave_subsection();
+        constraints.distribute(newton_update);
+      }
+    return std::make_pair(lin_it, lin_res);
+  }
+  template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+  void
+  Solid<dim, number_t, ad_type_code>::output_results() const
+  {
+    DataOut<dim> data_out;
+    std::vector<DataComponentInterpretation::DataComponentInterpretation>
+      data_component_interpretation(
+        dim, DataComponentInterpretation::component_is_part_of_vector);
+    data_component_interpretation.push_back(
+      DataComponentInterpretation::component_is_scalar);
+    data_component_interpretation.push_back(
+      DataComponentInterpretation::component_is_scalar);
+    std::vector<std::string> solution_name(dim, "displacement");
+    solution_name.push_back("pressure");
+    solution_name.push_back("dilatation");
+    data_out.attach_dof_handler(dof_handler_ref);
+    data_out.add_data_vector(solution_n,
+                             solution_name,
+                             DataOut<dim>::type_dof_data,
+                             data_component_interpretation);
+    Vector<double> soln(solution_n.size());
+    for (unsigned int i = 0; i < soln.size(); ++i)
+      soln(i) = solution_n(i);
+    MappingQEulerian<dim> q_mapping(degree, dof_handler_ref, soln);
+    data_out.build_patches(q_mapping, degree);
+    std::ostringstream filename;
+    filename << "solution-" << dim << "d-" << time.get_timestep() << ".vtk";
+    std::ofstream output(filename.str().c_str());
+    data_out.write_vtk(output);
+  }
+} // namespace Step44
diff --git a/tests/ad_common_tests/step-44-helper_vector_01.h b/tests/ad_common_tests/step-44-helper_vector_01.h
new file mode 100644 (file)
index 0000000..a011e31
--- /dev/null
@@ -0,0 +1,1765 @@
+// ---------------------------------------------------------------------
+//
+// Copyright (C) 2016 - 2018 by the deal.II authors
+//
+// This file is part of the deal.II library.
+//
+// The deal.II library is free software; you can use it, redistribute
+// it, and/or modify it under the terms of the GNU Lesser General
+// Public License as published by the Free Software Foundation; either
+// version 2.1 of the License, or (at your option) any later version.
+// The full text of the license can be found in the file LICENSE at
+// the top level of the deal.II distribution.
+//
+// ---------------------------------------------------------------------
+
+// Header file:
+// This is a modified version of step-44, which tests the implementation of
+// cell-level auto-differentiation (linearisation of a residual vector).
+
+#include <deal.II/base/function.h>
+#include <deal.II/base/parameter_handler.h>
+#include <deal.II/base/point.h>
+#include <deal.II/base/quadrature_lib.h>
+#include <deal.II/base/quadrature_point_data.h>
+#include <deal.II/base/symmetric_tensor.h>
+#include <deal.II/base/tensor.h>
+#include <deal.II/base/timer.h>
+#include <deal.II/base/work_stream.h>
+
+#include <deal.II/differentiation/ad.h>
+
+#include <deal.II/dofs/dof_renumbering.h>
+#include <deal.II/dofs/dof_tools.h>
+
+#include <deal.II/fe/fe_dgp_monomial.h>
+#include <deal.II/fe/fe_q.h>
+#include <deal.II/fe/fe_system.h>
+#include <deal.II/fe/fe_tools.h>
+#include <deal.II/fe/fe_values.h>
+#include <deal.II/fe/mapping_q_eulerian.h>
+
+#include <deal.II/grid/grid_generator.h>
+#include <deal.II/grid/grid_in.h>
+#include <deal.II/grid/grid_tools.h>
+#include <deal.II/grid/tria.h>
+#include <deal.II/grid/tria_boundary_lib.h>
+
+#include <deal.II/lac/block_sparse_matrix.h>
+#include <deal.II/lac/block_vector.h>
+#include <deal.II/lac/constraint_matrix.h>
+#include <deal.II/lac/dynamic_sparsity_pattern.h>
+#include <deal.II/lac/full_matrix.h>
+#include <deal.II/lac/linear_operator.h>
+#include <deal.II/lac/packaged_operation.h>
+#include <deal.II/lac/precondition_selector.h>
+#include <deal.II/lac/solver_cg.h>
+#include <deal.II/lac/solver_selector.h>
+#include <deal.II/lac/sparse_direct.h>
+
+#include <deal.II/numerics/data_out.h>
+#include <deal.II/numerics/vector_tools.h>
+
+#include <fstream>
+#include <functional>
+#include <iostream>
+
+#include "../tests.h"
+namespace Step44
+{
+  using namespace dealii;
+  namespace AD = dealii::Differentiation::AD;
+  namespace Parameters
+  {
+    struct FESystem
+    {
+      unsigned int poly_degree;
+      unsigned int quad_order;
+      static void
+      declare_parameters(ParameterHandler &prm);
+      void
+      parse_parameters(ParameterHandler &prm);
+    };
+    void
+    FESystem::declare_parameters(ParameterHandler &prm)
+    {
+      prm.enter_subsection("Finite element system");
+      {
+        prm.declare_entry("Polynomial degree",
+                          "2",
+                          Patterns::Integer(0),
+                          "Displacement system polynomial order");
+        prm.declare_entry("Quadrature order",
+                          "3",
+                          Patterns::Integer(0),
+                          "Gauss quadrature order");
+      }
+      prm.leave_subsection();
+    }
+    void
+    FESystem::parse_parameters(ParameterHandler &prm)
+    {
+      prm.enter_subsection("Finite element system");
+      {
+        poly_degree = prm.get_integer("Polynomial degree");
+        quad_order  = prm.get_integer("Quadrature order");
+      }
+      prm.leave_subsection();
+    }
+    struct Geometry
+    {
+      unsigned int global_refinement;
+      double       scale;
+      double       p_p0;
+      static void
+      declare_parameters(ParameterHandler &prm);
+      void
+      parse_parameters(ParameterHandler &prm);
+    };
+    void
+    Geometry::declare_parameters(ParameterHandler &prm)
+    {
+      prm.enter_subsection("Geometry");
+      {
+        prm.declare_entry("Global refinement",
+                          "2",
+                          Patterns::Integer(0),
+                          "Global refinement level");
+        prm.declare_entry("Grid scale",
+                          "1e-3",
+                          Patterns::Double(0.0),
+                          "Global grid scaling factor");
+        prm.declare_entry("Pressure ratio p/p0",
+                          "100",
+                          Patterns::Selection("20|40|60|80|100"),
+                          "Ratio of applied pressure to reference pressure");
+      }
+      prm.leave_subsection();
+    }
+    void
+    Geometry::parse_parameters(ParameterHandler &prm)
+    {
+      prm.enter_subsection("Geometry");
+      {
+        global_refinement = prm.get_integer("Global refinement");
+        scale             = prm.get_double("Grid scale");
+        p_p0              = prm.get_double("Pressure ratio p/p0");
+      }
+      prm.leave_subsection();
+    }
+    struct Materials
+    {
+      double nu;
+      double mu;
+      static void
+      declare_parameters(ParameterHandler &prm);
+      void
+      parse_parameters(ParameterHandler &prm);
+    };
+    void
+    Materials::declare_parameters(ParameterHandler &prm)
+    {
+      prm.enter_subsection("Material properties");
+      {
+        prm.declare_entry("Poisson's ratio",
+                          "0.4999",
+                          Patterns::Double(-1.0, 0.5),
+                          "Poisson's ratio");
+        prm.declare_entry("Shear modulus",
+                          "80.194e6",
+                          Patterns::Double(),
+                          "Shear modulus");
+      }
+      prm.leave_subsection();
+    }
+    void
+    Materials::parse_parameters(ParameterHandler &prm)
+    {
+      prm.enter_subsection("Material properties");
+      {
+        nu = prm.get_double("Poisson's ratio");
+        mu = prm.get_double("Shear modulus");
+      }
+      prm.leave_subsection();
+    }
+    struct LinearSolver
+    {
+      std::string type_lin;
+      double      tol_lin;
+      double      max_iterations_lin;
+      bool        use_static_condensation;
+      std::string preconditioner_type;
+      double      preconditioner_relaxation;
+      static void
+      declare_parameters(ParameterHandler &prm);
+      void
+      parse_parameters(ParameterHandler &prm);
+    };
+    void
+    LinearSolver::declare_parameters(ParameterHandler &prm)
+    {
+      prm.enter_subsection("Linear solver");
+      {
+        prm.declare_entry("Solver type",
+                          "CG",
+                          Patterns::Selection("CG|Direct"),
+                          "Type of solver used to solve the linear system");
+        prm.declare_entry("Residual",
+                          "1e-6",
+                          Patterns::Double(0.0),
+                          "Linear solver residual (scaled by residual norm)");
+        prm.declare_entry(
+          "Max iteration multiplier",
+          "1",
+          Patterns::Double(0.0),
+          "Linear solver iterations (multiples of the system matrix size)");
+        prm.declare_entry("Use static condensation",
+                          "true",
+                          Patterns::Bool(),
+                          "Solve the full block system or a reduced problem");
+        prm.declare_entry("Preconditioner type",
+                          "ssor",
+                          Patterns::Selection("jacobi|ssor"),
+                          "Type of preconditioner");
+        prm.declare_entry("Preconditioner relaxation",
+                          "0.65",
+                          Patterns::Double(0.0),
+                          "Preconditioner relaxation value");
+      }
+      prm.leave_subsection();
+    }
+    void
+    LinearSolver::parse_parameters(ParameterHandler &prm)
+    {
+      prm.enter_subsection("Linear solver");
+      {
+        type_lin                  = prm.get("Solver type");
+        tol_lin                   = prm.get_double("Residual");
+        max_iterations_lin        = prm.get_double("Max iteration multiplier");
+        use_static_condensation   = prm.get_bool("Use static condensation");
+        preconditioner_type       = prm.get("Preconditioner type");
+        preconditioner_relaxation = prm.get_double("Preconditioner relaxation");
+      }
+      prm.leave_subsection();
+    }
+    struct NonlinearSolver
+    {
+      unsigned int max_iterations_NR;
+      double       tol_f;
+      double       tol_u;
+      static void
+      declare_parameters(ParameterHandler &prm);
+      void
+      parse_parameters(ParameterHandler &prm);
+    };
+    void
+    NonlinearSolver::declare_parameters(ParameterHandler &prm)
+    {
+      prm.enter_subsection("Nonlinear solver");
+      {
+        prm.declare_entry("Max iterations Newton-Raphson",
+                          "10",
+                          Patterns::Integer(0),
+                          "Number of Newton-Raphson iterations allowed");
+        prm.declare_entry("Tolerance force",
+                          "1.0e-9",
+                          Patterns::Double(0.0),
+                          "Force residual tolerance");
+        prm.declare_entry("Tolerance displacement",
+                          "1.0e-6",
+                          Patterns::Double(0.0),
+                          "Displacement error tolerance");
+      }
+      prm.leave_subsection();
+    }
+    void
+    NonlinearSolver::parse_parameters(ParameterHandler &prm)
+    {
+      prm.enter_subsection("Nonlinear solver");
+      {
+        max_iterations_NR = prm.get_integer("Max iterations Newton-Raphson");
+        tol_f             = prm.get_double("Tolerance force");
+        tol_u             = prm.get_double("Tolerance displacement");
+      }
+      prm.leave_subsection();
+    }
+    struct Time
+    {
+      double delta_t;
+      double end_time;
+      static void
+      declare_parameters(ParameterHandler &prm);
+      void
+      parse_parameters(ParameterHandler &prm);
+    };
+    void
+    Time::declare_parameters(ParameterHandler &prm)
+    {
+      prm.enter_subsection("Time");
+      {
+        prm.declare_entry("End time", "1", Patterns::Double(), "End time");
+        prm.declare_entry("Time step size",
+                          "0.1",
+                          Patterns::Double(),
+                          "Time step size");
+      }
+      prm.leave_subsection();
+    }
+    void
+    Time::parse_parameters(ParameterHandler &prm)
+    {
+      prm.enter_subsection("Time");
+      {
+        end_time = prm.get_double("End time");
+        delta_t  = prm.get_double("Time step size");
+      }
+      prm.leave_subsection();
+    }
+    struct AllParameters : public FESystem,
+                           public Geometry,
+                           public Materials,
+                           public LinearSolver,
+                           public NonlinearSolver,
+                           public Time
+    {
+      AllParameters(const std::string &input_file);
+      static void
+      declare_parameters(ParameterHandler &prm);
+      void
+      parse_parameters(ParameterHandler &prm);
+    };
+    AllParameters::AllParameters(const std::string &input_file)
+    {
+      ParameterHandler prm;
+      declare_parameters(prm);
+      prm.parse_input(input_file);
+      parse_parameters(prm);
+    }
+    void
+    AllParameters::declare_parameters(ParameterHandler &prm)
+    {
+      FESystem::declare_parameters(prm);
+      Geometry::declare_parameters(prm);
+      Materials::declare_parameters(prm);
+      LinearSolver::declare_parameters(prm);
+      NonlinearSolver::declare_parameters(prm);
+      Time::declare_parameters(prm);
+    }
+    void
+    AllParameters::parse_parameters(ParameterHandler &prm)
+    {
+      FESystem::parse_parameters(prm);
+      Geometry::parse_parameters(prm);
+      Materials::parse_parameters(prm);
+      LinearSolver::parse_parameters(prm);
+      NonlinearSolver::parse_parameters(prm);
+      Time::parse_parameters(prm);
+    }
+  } // namespace Parameters
+  template <int dim>
+  class StandardTensors
+  {
+  public:
+    static const SymmetricTensor<2, dim> I;
+    static const SymmetricTensor<4, dim> IxI;
+    static const SymmetricTensor<4, dim> II;
+    static const SymmetricTensor<4, dim> dev_P;
+  };
+  template <int dim>
+  const SymmetricTensor<2, dim>
+    StandardTensors<dim>::I = unit_symmetric_tensor<dim>();
+  template <int dim>
+  const SymmetricTensor<4, dim> StandardTensors<dim>::IxI = outer_product(I, I);
+  template <int dim>
+  const SymmetricTensor<4, dim>
+    StandardTensors<dim>::II = identity_tensor<dim>();
+  template <int dim>
+  const SymmetricTensor<4, dim>
+    StandardTensors<dim>::dev_P = deviator_tensor<dim>();
+  class Time
+  {
+  public:
+    Time(const double time_end, const double delta_t)
+      : timestep(0)
+      , time_current(0.0)
+      , time_end(time_end)
+      , delta_t(delta_t)
+    {}
+    virtual ~Time()
+    {}
+    double
+    current() const
+    {
+      return time_current;
+    }
+    double
+    end() const
+    {
+      return time_end;
+    }
+    double
+    get_delta_t() const
+    {
+      return delta_t;
+    }
+    unsigned int
+    get_timestep() const
+    {
+      return timestep;
+    }
+    void
+    increment()
+    {
+      time_current += delta_t;
+      ++timestep;
+    }
+
+  private:
+    unsigned int timestep;
+    double       time_current;
+    const double time_end;
+    const double delta_t;
+  };
+  template <int dim>
+  class Material_Compressible_Neo_Hook_Three_Field
+  {
+  public:
+    Material_Compressible_Neo_Hook_Three_Field(const double mu, const double nu)
+      : kappa((2.0 * mu * (1.0 + nu)) / (3.0 * (1.0 - 2.0 * nu)))
+      , c_1(mu / 2.0)
+    {
+      Assert(kappa > 0, ExcInternalError());
+    }
+    ~Material_Compressible_Neo_Hook_Three_Field()
+    {}
+
+    template <typename NumberType>
+    SymmetricTensor<2, dim, NumberType>
+    get_tau(const Tensor<2, dim, NumberType> &F,
+            const NumberType &                p_tilde) const
+    {
+      return get_tau_iso(F) + get_tau_vol(F, p_tilde);
+    }
+    template <typename NumberType>
+    NumberType
+    get_dPsi_vol_dJ(const NumberType &J_tilde) const
+    {
+      return (kappa / 2.0) * (J_tilde - 1.0 / J_tilde);
+    }
+
+  protected:
+    const double kappa;
+    const double c_1;
+
+    template <typename NumberType>
+    SymmetricTensor<2, dim, NumberType>
+    get_tau_vol(const Tensor<2, dim, NumberType> &F,
+                const NumberType &                p_tilde) const
+    {
+      const NumberType det_F = determinant(F);
+      // return p_tilde * det_F * StandardTensors<dim>::I;
+      SymmetricTensor<2, dim, NumberType> tau_vol(StandardTensors<dim>::I);
+      tau_vol *= p_tilde * det_F;
+      return tau_vol;
+    }
+    template <typename NumberType>
+    SymmetricTensor<2, dim, NumberType>
+    get_tau_iso(const Tensor<2, dim, NumberType> &F) const
+    {
+      // return StandardTensors<dim>::dev_P * get_tau_bar(F);
+
+      const SymmetricTensor<2, dim, NumberType> tau_bar = get_tau_bar(F);
+      SymmetricTensor<2, dim, NumberType>       tau_iso;
+      for (unsigned int i = 0; i < dim; ++i)
+        for (unsigned int j = i; j < dim; ++j)
+          for (unsigned int k = 0; k < dim; ++k)
+            for (unsigned int l = 0; l < dim; ++l)
+              tau_iso[i][j] +=
+                StandardTensors<dim>::dev_P[i][j][k][l] * tau_bar[k][l];
+
+      return tau_iso;
+    }
+    template <typename NumberType>
+    SymmetricTensor<2, dim, NumberType>
+    get_tau_bar(const Tensor<2, dim, NumberType> &F) const
+    {
+      const NumberType                    det_F = determinant(F);
+      SymmetricTensor<2, dim, NumberType> b_bar = symmetrize(F * transpose(F));
+      b_bar *= std::pow(det_F, -2.0 / dim);
+      return 2.0 * c_1 * b_bar;
+    }
+  };
+  template <int dim>
+  class PointHistory
+  {
+  public:
+    PointHistory()
+    {}
+    virtual ~PointHistory()
+    {}
+    void
+    setup_lqp(const Parameters::AllParameters &parameters)
+    {
+      material.reset(
+        new Material_Compressible_Neo_Hook_Three_Field<dim>(parameters.mu,
+                                                            parameters.nu));
+    }
+
+    template <typename NumberType>
+    SymmetricTensor<2, dim, NumberType>
+    get_tau(const Tensor<2, dim, NumberType> &F,
+            const NumberType &                p_tilde) const
+    {
+      return material->get_tau(F, p_tilde);
+    }
+    template <typename NumberType>
+    NumberType
+    get_dPsi_vol_dJ(const NumberType &J_tilde) const
+    {
+      return material->get_dPsi_vol_dJ(J_tilde);
+    }
+
+  private:
+    std::shared_ptr<Material_Compressible_Neo_Hook_Three_Field<dim>> material;
+  };
+  template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+  class Solid
+  {
+  public:
+    Solid(const std::string &input_file);
+    virtual ~Solid();
+    void
+    run();
+
+  private:
+    struct PerTaskData_ASM;
+    struct ScratchData_ASM;
+    struct PerTaskData_SC;
+    struct ScratchData_SC;
+    void
+    make_grid();
+    void
+    system_setup();
+    void
+    determine_component_extractors();
+    void
+    assemble_system(const BlockVector<double> &solution_delta);
+    void
+    assemble_system_one_cell(
+      const typename DoFHandler<dim>::active_cell_iterator &cell,
+      ScratchData_ASM &                                     scratch,
+      PerTaskData_ASM &                                     data) const;
+    void
+    copy_local_to_global_system(const PerTaskData_ASM &data);
+    void
+    assemble_sc();
+    void
+    assemble_sc_one_cell(
+      const typename DoFHandler<dim>::active_cell_iterator &cell,
+      ScratchData_SC &                                      scratch,
+      PerTaskData_SC &                                      data);
+    void
+    copy_local_to_global_sc(const PerTaskData_SC &data);
+    void
+    make_constraints(const int &it_nr);
+    void
+    setup_qph();
+    void
+    solve_nonlinear_timestep(BlockVector<double> &solution_delta);
+    std::pair<unsigned int, double>
+    solve_linear_system(BlockVector<double> &newton_update);
+    BlockVector<double>
+    get_total_solution(const BlockVector<double> &solution_delta) const;
+    void
+                              output_results() const;
+    Parameters::AllParameters parameters;
+    double                    vol_reference;
+    Triangulation<dim>        triangulation;
+    Time                      time;
+    mutable TimerOutput       timer;
+    CellDataStorage<typename Triangulation<dim>::cell_iterator,
+                    PointHistory<dim>>
+                                     quadrature_point_history;
+    const unsigned int               degree;
+    const FESystem<dim>              fe;
+    DoFHandler<dim>                  dof_handler_ref;
+    const unsigned int               dofs_per_cell;
+    const FEValuesExtractors::Vector u_fe;
+    const FEValuesExtractors::Scalar p_fe;
+    const FEValuesExtractors::Scalar J_fe;
+    static const unsigned int        n_blocks          = 3;
+    static const unsigned int        n_components      = dim + 2;
+    static const unsigned int        first_u_component = 0;
+    static const unsigned int        p_component       = dim;
+    static const unsigned int        J_component       = dim + 1;
+    enum
+    {
+      u_dof = 0,
+      p_dof = 1,
+      J_dof = 2
+    };
+    std::vector<types::global_dof_index> dofs_per_block;
+    std::vector<types::global_dof_index> element_indices_u;
+    std::vector<types::global_dof_index> element_indices_p;
+    std::vector<types::global_dof_index> element_indices_J;
+    const QGauss<dim>                    qf_cell;
+    const QGauss<dim - 1>                qf_face;
+    const unsigned int                   n_q_points;
+    const unsigned int                   n_q_points_f;
+    ConstraintMatrix                     constraints;
+    BlockSparsityPattern                 sparsity_pattern;
+    BlockSparseMatrix<double>            tangent_matrix;
+    BlockVector<double>                  system_rhs;
+    BlockVector<double>                  solution_n;
+    struct Errors
+    {
+      Errors()
+        : norm(1.0)
+        , u(1.0)
+        , p(1.0)
+        , J(1.0)
+      {}
+      void
+      reset()
+      {
+        norm = 1.0;
+        u    = 1.0;
+        p    = 1.0;
+        J    = 1.0;
+      }
+      void
+      normalise(const Errors &rhs)
+      {
+        if (rhs.norm != 0.0)
+          norm /= rhs.norm;
+        if (rhs.u != 0.0)
+          u /= rhs.u;
+        if (rhs.p != 0.0)
+          p /= rhs.p;
+        if (rhs.J != 0.0)
+          J /= rhs.J;
+      }
+      double norm, u, p, J;
+    };
+    Errors error_residual, error_residual_0, error_residual_norm, error_update,
+      error_update_0, error_update_norm;
+    void
+    get_error_residual(Errors &error_residual);
+    void
+    get_error_update(const BlockVector<double> &newton_update,
+                     Errors &                   error_update);
+    std::pair<double, double>
+    get_error_dilation(const BlockVector<double> &solution_total) const;
+    void
+    print_conv_header();
+    void
+    print_conv_footer(const BlockVector<double> &solution_delta);
+  };
+  template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+  Solid<dim, number_t, ad_type_code>::Solid(const std::string &input_file)
+    : parameters(input_file)
+    , triangulation(Triangulation<dim>::maximum_smoothing)
+    , time(parameters.end_time, parameters.delta_t)
+    , timer(std::cout, TimerOutput::never, TimerOutput::wall_times)
+    , degree(parameters.poly_degree)
+    , fe(FE_Q<dim>(parameters.poly_degree),
+         dim, // displacement
+         FE_DGPMonomial<dim>(parameters.poly_degree - 1),
+         1, // pressure
+         FE_DGPMonomial<dim>(parameters.poly_degree - 1),
+         1)
+    , // dilatation
+    dof_handler_ref(triangulation)
+    , dofs_per_cell(fe.dofs_per_cell)
+    , u_fe(first_u_component)
+    , p_fe(p_component)
+    , J_fe(J_component)
+    , dofs_per_block(n_blocks)
+    , qf_cell(parameters.quad_order)
+    , qf_face(parameters.quad_order)
+    , n_q_points(qf_cell.size())
+    , n_q_points_f(qf_face.size())
+  {
+    Assert(dim == 2 || dim == 3,
+           ExcMessage("This problem only works in 2 or 3 space dimensions."));
+    determine_component_extractors();
+  }
+  template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+  Solid<dim, number_t, ad_type_code>::~Solid()
+  {
+    dof_handler_ref.clear();
+  }
+  template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+  void
+  Solid<dim, number_t, ad_type_code>::run()
+  {
+    make_grid();
+    system_setup();
+    {
+      ConstraintMatrix constraints;
+      constraints.close();
+      const ComponentSelectFunction<dim> J_mask(J_component, n_components);
+      VectorTools::project(dof_handler_ref,
+                           constraints,
+                           QGauss<dim>(degree + 2),
+                           J_mask,
+                           solution_n);
+    }
+    // output_results();
+    time.increment();
+    BlockVector<double> solution_delta(dofs_per_block);
+    while (time.current() < time.end())
+      {
+        solution_delta = 0.0;
+        solve_nonlinear_timestep(solution_delta);
+        solution_n += solution_delta;
+        // output_results();
+        // Output displacement at centre of traction surface
+        {
+          const Point<dim> soln_pt(
+            dim == 3 ? Point<dim>(0.0, 1.0, 0.0) * parameters.scale :
+                       Point<dim>(0.0, 1.0) * parameters.scale);
+          typename DoFHandler<dim>::active_cell_iterator
+            cell = dof_handler_ref.begin_active(),
+            endc = dof_handler_ref.end();
+          for (; cell != endc; ++cell)
+            for (unsigned int v = 0; v < GeometryInfo<dim>::vertices_per_cell;
+                 ++v)
+              if (cell->vertex(v).distance(soln_pt) < 1e-6 * parameters.scale)
+                {
+                  Tensor<1, dim> soln;
+                  for (unsigned int d = 0; d < dim; ++d)
+                    soln[d] = solution_n(cell->vertex_dof_index(v, u_dof + d));
+                  deallog << "Timestep " << time.get_timestep() << ": " << soln
+                          << std::endl;
+                }
+        }
+        time.increment();
+      }
+  }
+  template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+  struct Solid<dim, number_t, ad_type_code>::PerTaskData_ASM
+  {
+    FullMatrix<double>                   cell_matrix;
+    Vector<double>                       cell_rhs;
+    std::vector<types::global_dof_index> local_dof_indices;
+    PerTaskData_ASM(const unsigned int dofs_per_cell)
+      : cell_matrix(dofs_per_cell, dofs_per_cell)
+      , cell_rhs(dofs_per_cell)
+      , local_dof_indices(dofs_per_cell)
+    {}
+    void
+    reset()
+    {
+      cell_matrix = 0.0;
+      cell_rhs    = 0.0;
+    }
+  };
+  template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+  struct Solid<dim, number_t, ad_type_code>::ScratchData_ASM
+  {
+    const BlockVector<double> &solution_total;
+    FEValues<dim>              fe_values_ref;
+    FEFaceValues<dim>          fe_face_values_ref;
+    ScratchData_ASM(const FiniteElement<dim> & fe_cell,
+                    const QGauss<dim> &        qf_cell,
+                    const UpdateFlags          uf_cell,
+                    const QGauss<dim - 1> &    qf_face,
+                    const UpdateFlags          uf_face,
+                    const BlockVector<double> &solution_total)
+      : solution_total(solution_total)
+      , fe_values_ref(fe_cell, qf_cell, uf_cell)
+      , fe_face_values_ref(fe_cell, qf_face, uf_face)
+    {}
+    ScratchData_ASM(const ScratchData_ASM &rhs)
+      : solution_total(rhs.solution_total)
+      , fe_values_ref(rhs.fe_values_ref.get_fe(),
+                      rhs.fe_values_ref.get_quadrature(),
+                      rhs.fe_values_ref.get_update_flags())
+      , fe_face_values_ref(rhs.fe_face_values_ref.get_fe(),
+                           rhs.fe_face_values_ref.get_quadrature(),
+                           rhs.fe_face_values_ref.get_update_flags())
+    {}
+    void
+    reset()
+    {}
+  };
+  template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+  struct Solid<dim, number_t, ad_type_code>::PerTaskData_SC
+  {
+    FullMatrix<double>                   cell_matrix;
+    std::vector<types::global_dof_index> local_dof_indices;
+    FullMatrix<double>                   k_orig;
+    FullMatrix<double>                   k_pu;
+    FullMatrix<double>                   k_pJ;
+    FullMatrix<double>                   k_JJ;
+    FullMatrix<double>                   k_pJ_inv;
+    FullMatrix<double>                   k_bbar;
+    FullMatrix<double>                   A;
+    FullMatrix<double>                   B;
+    FullMatrix<double>                   C;
+    PerTaskData_SC(const unsigned int dofs_per_cell,
+                   const unsigned int n_u,
+                   const unsigned int n_p,
+                   const unsigned int n_J)
+      : cell_matrix(dofs_per_cell, dofs_per_cell)
+      , local_dof_indices(dofs_per_cell)
+      , k_orig(dofs_per_cell, dofs_per_cell)
+      , k_pu(n_p, n_u)
+      , k_pJ(n_p, n_J)
+      , k_JJ(n_J, n_J)
+      , k_pJ_inv(n_p, n_J)
+      , k_bbar(n_u, n_u)
+      , A(n_J, n_u)
+      , B(n_J, n_u)
+      , C(n_p, n_u)
+    {}
+    void
+    reset()
+    {}
+  };
+  template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+  struct Solid<dim, number_t, ad_type_code>::ScratchData_SC
+  {
+    void
+    reset()
+    {}
+  };
+  template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+  void
+  Solid<dim, number_t, ad_type_code>::make_grid()
+  {
+    GridGenerator::hyper_rectangle(
+      triangulation,
+      (dim == 3 ? Point<dim>(0.0, 0.0, 0.0) : Point<dim>(0.0, 0.0)),
+      (dim == 3 ? Point<dim>(1.0, 1.0, 1.0) : Point<dim>(1.0, 1.0)),
+      true);
+    GridTools::scale(parameters.scale, triangulation);
+    triangulation.refine_global(std::max(1U, parameters.global_refinement));
+    vol_reference = GridTools::volume(triangulation);
+    std::cout << "Grid:\n\t Reference volume: " << vol_reference << std::endl;
+    typename Triangulation<dim>::active_cell_iterator cell = triangulation
+                                                               .begin_active(),
+                                                      endc =
+                                                        triangulation.end();
+    for (; cell != endc; ++cell)
+      for (unsigned int face = 0; face < GeometryInfo<dim>::faces_per_cell;
+           ++face)
+        {
+          if (cell->face(face)->at_boundary() == true &&
+              cell->face(face)->center()[1] == 1.0 * parameters.scale)
+            {
+              if (dim == 3)
+                {
+                  if (cell->face(face)->center()[0] < 0.5 * parameters.scale &&
+                      cell->face(face)->center()[2] < 0.5 * parameters.scale)
+                    cell->face(face)->set_boundary_id(6);
+                }
+              else
+                {
+                  if (cell->face(face)->center()[0] < 0.5 * parameters.scale)
+                    cell->face(face)->set_boundary_id(6);
+                }
+            }
+        }
+  }
+  template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+  void
+  Solid<dim, number_t, ad_type_code>::system_setup()
+  {
+    timer.enter_subsection("Setup system");
+    std::vector<unsigned int> block_component(n_components,
+                                              u_dof); // Displacement
+    block_component[p_component] = p_dof;             // Pressure
+    block_component[J_component] = J_dof;             // Dilatation
+    dof_handler_ref.distribute_dofs(fe);
+    DoFRenumbering::Cuthill_McKee(dof_handler_ref);
+    DoFRenumbering::component_wise(dof_handler_ref, block_component);
+    DoFTools::count_dofs_per_block(dof_handler_ref,
+                                   dofs_per_block,
+                                   block_component);
+    std::cout << "Triangulation:"
+              << "\n\t Number of active cells: "
+              << triangulation.n_active_cells()
+              << "\n\t Number of degrees of freedom: "
+              << dof_handler_ref.n_dofs() << std::endl;
+    tangent_matrix.clear();
+    {
+      const types::global_dof_index n_dofs_u = dofs_per_block[u_dof];
+      const types::global_dof_index n_dofs_p = dofs_per_block[p_dof];
+      const types::global_dof_index n_dofs_J = dofs_per_block[J_dof];
+      BlockDynamicSparsityPattern   dsp(n_blocks, n_blocks);
+      dsp.block(u_dof, u_dof).reinit(n_dofs_u, n_dofs_u);
+      dsp.block(u_dof, p_dof).reinit(n_dofs_u, n_dofs_p);
+      dsp.block(u_dof, J_dof).reinit(n_dofs_u, n_dofs_J);
+      dsp.block(p_dof, u_dof).reinit(n_dofs_p, n_dofs_u);
+      dsp.block(p_dof, p_dof).reinit(n_dofs_p, n_dofs_p);
+      dsp.block(p_dof, J_dof).reinit(n_dofs_p, n_dofs_J);
+      dsp.block(J_dof, u_dof).reinit(n_dofs_J, n_dofs_u);
+      dsp.block(J_dof, p_dof).reinit(n_dofs_J, n_dofs_p);
+      dsp.block(J_dof, J_dof).reinit(n_dofs_J, n_dofs_J);
+      dsp.collect_sizes();
+      Table<2, DoFTools::Coupling> coupling(n_components, n_components);
+      for (unsigned int ii = 0; ii < n_components; ++ii)
+        for (unsigned int jj = 0; jj < n_components; ++jj)
+          if (((ii < p_component) && (jj == J_component)) ||
+              ((ii == J_component) && (jj < p_component)) ||
+              ((ii == p_component) && (jj == p_component)))
+            coupling[ii][jj] = DoFTools::none;
+          else
+            coupling[ii][jj] = DoFTools::always;
+      DoFTools::make_sparsity_pattern(
+        dof_handler_ref, coupling, dsp, constraints, false);
+      sparsity_pattern.copy_from(dsp);
+    }
+    tangent_matrix.reinit(sparsity_pattern);
+    system_rhs.reinit(dofs_per_block);
+    system_rhs.collect_sizes();
+    solution_n.reinit(dofs_per_block);
+    solution_n.collect_sizes();
+    setup_qph();
+    timer.leave_subsection();
+  }
+  template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+  void
+  Solid<dim, number_t, ad_type_code>::determine_component_extractors()
+  {
+    element_indices_u.clear();
+    element_indices_p.clear();
+    element_indices_J.clear();
+    for (unsigned int k = 0; k < fe.dofs_per_cell; ++k)
+      {
+        const unsigned int k_group = fe.system_to_base_index(k).first.first;
+        if (k_group == u_dof)
+          element_indices_u.push_back(k);
+        else if (k_group == p_dof)
+          element_indices_p.push_back(k);
+        else if (k_group == J_dof)
+          element_indices_J.push_back(k);
+        else
+          {
+            Assert(k_group <= J_dof, ExcInternalError());
+          }
+      }
+  }
+  template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+  void
+  Solid<dim, number_t, ad_type_code>::setup_qph()
+  {
+    std::cout << "    Setting up quadrature point data..." << std::endl;
+    quadrature_point_history.initialize(triangulation.begin_active(),
+                                        triangulation.end(),
+                                        n_q_points);
+    for (typename Triangulation<dim>::active_cell_iterator cell =
+           triangulation.begin_active();
+         cell != triangulation.end();
+         ++cell)
+      {
+        const std::vector<std::shared_ptr<PointHistory<dim>>> lqph =
+          quadrature_point_history.get_data(cell);
+        Assert(lqph.size() == n_q_points, ExcInternalError());
+        for (unsigned int q_point = 0; q_point < n_q_points; ++q_point)
+          lqph[q_point]->setup_lqp(parameters);
+      }
+  }
+  template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+  void
+  Solid<dim, number_t, ad_type_code>::solve_nonlinear_timestep(
+    BlockVector<double> &solution_delta)
+  {
+    std::cout << std::endl
+              << "Timestep " << time.get_timestep() << " @ " << time.current()
+              << "s" << std::endl;
+    BlockVector<double> newton_update(dofs_per_block);
+    error_residual.reset();
+    error_residual_0.reset();
+    error_residual_norm.reset();
+    error_update.reset();
+    error_update_0.reset();
+    error_update_norm.reset();
+    print_conv_header();
+    unsigned int newton_iteration = 0;
+    for (; newton_iteration < parameters.max_iterations_NR; ++newton_iteration)
+      {
+        std::cout << " " << std::setw(2) << newton_iteration << " "
+                  << std::flush;
+        tangent_matrix = 0.0;
+        system_rhs     = 0.0;
+        make_constraints(newton_iteration);
+        assemble_system(solution_delta);
+        get_error_residual(error_residual);
+        if (newton_iteration == 0)
+          error_residual_0 = error_residual;
+        error_residual_norm = error_residual;
+        error_residual_norm.normalise(error_residual_0);
+        if (newton_iteration > 0 && error_update_norm.u <= parameters.tol_u &&
+            error_residual_norm.u <= parameters.tol_f)
+          {
+            std::cout << " CONVERGED! " << std::endl;
+            print_conv_footer(solution_delta);
+            break;
+          }
+        const std::pair<unsigned int, double> lin_solver_output =
+          solve_linear_system(newton_update);
+        get_error_update(newton_update, error_update);
+        if (newton_iteration == 0)
+          error_update_0 = error_update;
+        error_update_norm = error_update;
+        error_update_norm.normalise(error_update_0);
+        solution_delta += newton_update;
+        std::cout << " | " << std::fixed << std::setprecision(3) << std::setw(7)
+                  << std::scientific << lin_solver_output.first << "  "
+                  << lin_solver_output.second << "  "
+                  << error_residual_norm.norm << "  " << error_residual_norm.u
+                  << "  " << error_residual_norm.p << "  "
+                  << error_residual_norm.J << "  " << error_update_norm.norm
+                  << "  " << error_update_norm.u << "  " << error_update_norm.p
+                  << "  " << error_update_norm.J << "  " << std::endl;
+      }
+    AssertThrow(newton_iteration <= parameters.max_iterations_NR,
+                ExcMessage("No convergence in nonlinear solver!"));
+  }
+  template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+  void
+  Solid<dim, number_t, ad_type_code>::print_conv_header()
+  {
+    static const unsigned int l_width = 144;
+    for (unsigned int i = 0; i < l_width; ++i)
+      std::cout << "_";
+    std::cout << std::endl;
+    std::cout << "           SOLVER STEP             "
+              << " |  LIN_IT   LIN_RES    RES_NORM    "
+              << " RES_U     RES_P      RES_J     NU_NORM     "
+              << " NU_U       NU_P       NU_J " << std::endl;
+    for (unsigned int i = 0; i < l_width; ++i)
+      std::cout << "_";
+    std::cout << std::endl;
+  }
+  template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+  void
+  Solid<dim, number_t, ad_type_code>::print_conv_footer(
+    const BlockVector<double> &solution_delta)
+  {
+    static const unsigned int l_width = 144;
+    for (unsigned int i = 0; i < l_width; ++i)
+      std::cout << "_";
+    std::cout << std::endl;
+    const std::pair<double, double> error_dil =
+      get_error_dilation(get_total_solution(solution_delta));
+    std::cout << "Relative errors:" << std::endl
+              << "Displacement:\t" << error_update.u / error_update_0.u
+              << std::endl
+              << "Force: \t\t" << error_residual.u / error_residual_0.u
+              << std::endl
+              << "Dilatation:\t" << error_dil.first << std::endl
+              << "v / V_0:\t" << error_dil.second * vol_reference << " / "
+              << vol_reference << " = " << error_dil.second << std::endl;
+  }
+  template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+  std::pair<double, double>
+  Solid<dim, number_t, ad_type_code>::get_error_dilation(
+    const BlockVector<double> &solution_total) const
+  {
+    double        vol_current  = 0.0;
+    double        dil_L2_error = 0.0;
+    FEValues<dim> fe_values_ref(
+      fe, qf_cell, update_values | update_gradients | update_JxW_values);
+    std::vector<Tensor<2, dim>> solution_grads_u_total(qf_cell.size());
+    std::vector<double>         solution_values_J_total(qf_cell.size());
+    for (typename DoFHandler<dim>::active_cell_iterator cell =
+           dof_handler_ref.begin_active();
+         cell != dof_handler_ref.end();
+         ++cell)
+      {
+        fe_values_ref.reinit(cell);
+        fe_values_ref[u_fe].get_function_gradients(solution_total,
+                                                   solution_grads_u_total);
+        fe_values_ref[J_fe].get_function_values(solution_total,
+                                                solution_values_J_total);
+        const std::vector<std::shared_ptr<const PointHistory<dim>>> lqph =
+          quadrature_point_history.get_data(cell);
+        Assert(lqph.size() == n_q_points, ExcInternalError());
+        for (unsigned int q_point = 0; q_point < n_q_points; ++q_point)
+          {
+            const double det_F_qp = determinant(
+              StandardTensors<dim>::I + solution_grads_u_total[q_point]);
+            const double J_tilde_qp = solution_values_J_total[q_point];
+            const double the_error_qp_squared =
+              std::pow((det_F_qp - J_tilde_qp), 2);
+            const double JxW = fe_values_ref.JxW(q_point);
+            dil_L2_error += the_error_qp_squared * JxW;
+            vol_current += det_F_qp * JxW;
+          }
+      }
+    Assert(vol_current > 0.0, ExcInternalError());
+
+    return std::make_pair(std::sqrt(dil_L2_error), vol_current / vol_reference);
+  }
+  template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+  void
+  Solid<dim, number_t, ad_type_code>::get_error_residual(Errors &error_residual)
+  {
+    BlockVector<double> error_res(dofs_per_block);
+    for (unsigned int i = 0; i < dof_handler_ref.n_dofs(); ++i)
+      if (!constraints.is_constrained(i))
+        error_res(i) = system_rhs(i);
+    error_residual.norm = error_res.l2_norm();
+    error_residual.u    = error_res.block(u_dof).l2_norm();
+    error_residual.p    = error_res.block(p_dof).l2_norm();
+    error_residual.J    = error_res.block(J_dof).l2_norm();
+  }
+  template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+  void
+  Solid<dim, number_t, ad_type_code>::get_error_update(
+    const BlockVector<double> &newton_update,
+    Errors &                   error_update)
+  {
+    BlockVector<double> error_ud(dofs_per_block);
+    for (unsigned int i = 0; i < dof_handler_ref.n_dofs(); ++i)
+      if (!constraints.is_constrained(i))
+        error_ud(i) = newton_update(i);
+    error_update.norm = error_ud.l2_norm();
+    error_update.u    = error_ud.block(u_dof).l2_norm();
+    error_update.p    = error_ud.block(p_dof).l2_norm();
+    error_update.J    = error_ud.block(J_dof).l2_norm();
+  }
+  template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+  BlockVector<double>
+  Solid<dim, number_t, ad_type_code>::get_total_solution(
+    const BlockVector<double> &solution_delta) const
+  {
+    BlockVector<double> solution_total(solution_n);
+    solution_total += solution_delta;
+    return solution_total;
+  }
+  template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+  void
+  Solid<dim, number_t, ad_type_code>::assemble_system(
+    const BlockVector<double> &solution_delta)
+  {
+    timer.enter_subsection("Assemble system");
+    std::cout << " ASM_SYS " << std::flush;
+    system_rhs = 0.0;
+    const BlockVector<double> solution_total(
+      get_total_solution(solution_delta));
+    const UpdateFlags uf_cell(update_values | update_gradients |
+                              update_JxW_values);
+    const UpdateFlags uf_face(update_values | update_normal_vectors |
+                              update_JxW_values);
+    PerTaskData_ASM   per_task_data(dofs_per_cell);
+    ScratchData_ASM   scratch_data(
+      fe, qf_cell, uf_cell, qf_face, uf_face, solution_total);
+    // ADOL-C is incompatible with TBB
+    // WorkStream::run(dof_handler_ref.begin_active(),
+    //                 dof_handler_ref.end(),
+    //                 std::bind(&Solid<dim,number_t,ad_type_code>::assemble_system_one_cell,
+    //                                 this,
+    //                                 std::placeholders::_1,
+    //                                 std::placeholders::_2,
+    //                                 std::placeholders::_3),
+    //                 std::bind(&Solid<dim,number_t,ad_type_code>::copy_local_to_global_system,
+    //                                 this,
+    //                                 std::placeholders::_1),
+    //                 scratch_data,
+    //                 per_task_data);
+    for (typename DoFHandler<dim>::active_cell_iterator cell =
+           dof_handler_ref.begin_active();
+         cell != dof_handler_ref.end();
+         ++cell)
+      {
+        assemble_system_one_cell(cell, scratch_data, per_task_data);
+        copy_local_to_global_system(per_task_data);
+      }
+    timer.leave_subsection();
+  }
+  template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+  void
+  Solid<dim, number_t, ad_type_code>::copy_local_to_global_system(
+    const PerTaskData_ASM &data)
+  {
+    if (data.cell_matrix.frobenius_norm() > 1e-12)
+      constraints.distribute_local_to_global(data.cell_matrix,
+                                             data.cell_rhs,
+                                             data.local_dof_indices,
+                                             tangent_matrix,
+                                             system_rhs);
+  }
+  template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+  void
+  Solid<dim, number_t, ad_type_code>::assemble_system_one_cell(
+    const typename DoFHandler<dim>::active_cell_iterator &cell,
+    ScratchData_ASM &                                     scratch,
+    PerTaskData_ASM &                                     data) const
+  {
+    data.reset();
+    scratch.reset();
+    scratch.fe_values_ref.reinit(cell);
+    cell->get_dof_indices(data.local_dof_indices);
+
+    const std::vector<std::shared_ptr<const PointHistory<dim>>> lqph =
+      quadrature_point_history.get_data(cell);
+    Assert(lqph.size() == n_q_points, ExcInternalError());
+
+    const unsigned int n_independent_variables = data.local_dof_indices.size();
+    const unsigned int n_dependent_variables   = dofs_per_cell;
+    Assert(n_dependent_variables == n_independent_variables,
+           ExcMessage("Expect square system."));
+
+    typedef AD::VectorFunction<dim, ad_type_code, number_t> ADHelper;
+    typedef typename ADHelper::ad_type                      ADNumberType;
+    ADHelper ad_helper(n_independent_variables, n_dependent_variables);
+    ad_helper.set_tape_buffer_sizes(); // Increase the buffer size from the
+                                       // default values
+
+    // Values at all DoFs
+    std::vector<double> dof_values(n_independent_variables);
+    for (unsigned int i = 0; i < n_independent_variables; ++i)
+      dof_values[i] = scratch.solution_total(data.local_dof_indices[i]);
+
+    const int  tape_no = 1;
+    const bool is_recording =
+      ad_helper.start_recording_operations(tape_no, //  material_id
+                                           true,    // overwrite_tape
+                                           true);   // keep
+
+    if (is_recording == true)
+      {
+        ad_helper.register_independent_variables(dof_values);
+        const std::vector<ADNumberType> dof_values_ad =
+          ad_helper.get_sensitive_variables();
+        // Note: It is critical that this vector be initialised with zero'd
+        // values otherwise the results may be garbage!
+        std::vector<ADNumberType> residual_ad(n_dependent_variables,
+                                              ADNumberType(0.0));
+
+        // Compute all values, gradients etc. based on sensitive AD DoF values
+        std::vector<Tensor<2, dim, ADNumberType>> Grad_u(n_q_points);
+        std::vector<ADNumberType> p_tilde(n_q_points, ADNumberType(0.0));
+        std::vector<ADNumberType> J_tilde(n_q_points, ADNumberType(0.0));
+
+        for (unsigned int q_point = 0; q_point < n_q_points; ++q_point)
+          {
+            for (unsigned int k = 0; k < dofs_per_cell; ++k)
+              {
+                const unsigned int k_group =
+                  fe.system_to_base_index(k).first.first;
+                if (k_group == u_dof)
+                  Grad_u[q_point] +=
+                    dof_values_ad[k] *
+                    scratch.fe_values_ref[u_fe].gradient(k, q_point);
+                else if (k_group == p_dof)
+                  p_tilde[q_point] +=
+                    dof_values_ad[k] *
+                    scratch.fe_values_ref[p_fe].value(k, q_point);
+                else if (k_group == J_dof)
+                  J_tilde[q_point] +=
+                    dof_values_ad[k] *
+                    scratch.fe_values_ref[J_fe].value(k, q_point);
+                else
+                  Assert(k_group <= J_dof, ExcInternalError());
+              }
+          }
+
+        for (unsigned int q_point = 0; q_point < n_q_points; ++q_point)
+          {
+            const Tensor<2, dim, ADNumberType> F =
+              unit_symmetric_tensor<dim>() + Grad_u[q_point];
+            const Tensor<2, dim, ADNumberType> F_inv = invert(F);
+            const ADNumberType                 det_F = determinant(F);
+            Assert(numbers::value_is_greater_than(det_F, 0.0),
+                   ExcMessage("Negative jacobian detected!"));
+
+            const SymmetricTensor<2, dim, ADNumberType> tau =
+              lqph[q_point]->get_tau(F, p_tilde[q_point]);
+            const ADNumberType dPsi_vol_dJ =
+              lqph[q_point]->get_dPsi_vol_dJ(J_tilde[q_point]);
+
+            const double JxW = scratch.fe_values_ref.JxW(q_point);
+            for (unsigned int i = 0; i < dofs_per_cell; ++i)
+              {
+                const unsigned int i_group =
+                  fe.system_to_base_index(i).first.first;
+                if (i_group == u_dof)
+                  {
+                    const SymmetricTensor<2, dim, ADNumberType> symm_grad_Nx_i =
+                      symmetrize(
+                        scratch.fe_values_ref[u_fe].gradient(i, q_point) *
+                        F_inv);
+                    residual_ad[i] += (symm_grad_Nx_i * tau) * JxW;
+                  }
+                else if (i_group == p_dof)
+                  {
+                    const double N_i =
+                      scratch.fe_values_ref[p_fe].value(i, q_point);
+                    residual_ad[i] += N_i * (det_F - J_tilde[q_point]) * JxW;
+                  }
+                else if (i_group == J_dof)
+                  {
+                    const double N_i =
+                      scratch.fe_values_ref[J_fe].value(i, q_point);
+                    residual_ad[i] +=
+                      N_i * (dPsi_vol_dJ - p_tilde[q_point]) * JxW;
+                  }
+                else
+                  Assert(i_group <= J_dof, ExcInternalError());
+              }
+          }
+        for (unsigned int face = 0; face < GeometryInfo<dim>::faces_per_cell;
+             ++face)
+          if (cell->face(face)->at_boundary() == true &&
+              cell->face(face)->boundary_id() == 6)
+            {
+              scratch.fe_face_values_ref.reinit(cell, face);
+              for (unsigned int f_q_point = 0; f_q_point < n_q_points_f;
+                   ++f_q_point)
+                {
+                  const Tensor<1, dim> &N =
+                    scratch.fe_face_values_ref.normal_vector(f_q_point);
+                  static const double p0 =
+                    -4.0 / (parameters.scale * parameters.scale);
+                  const double time_ramp = (time.current() / time.end());
+                  const double pressure  = p0 * parameters.p_p0 * time_ramp;
+                  const Tensor<1, dim> traction = pressure * N;
+                  for (unsigned int i = 0; i < dofs_per_cell; ++i)
+                    {
+                      const unsigned int i_group =
+                        fe.system_to_base_index(i).first.first;
+                      if (i_group == u_dof)
+                        {
+                          const unsigned int component_i =
+                            fe.system_to_component_index(i).first;
+                          const double Ni =
+                            scratch.fe_face_values_ref.shape_value(i,
+                                                                   f_q_point);
+                          const double JxW =
+                            scratch.fe_face_values_ref.JxW(f_q_point);
+                          residual_ad[i] -= (Ni * traction[component_i]) * JxW;
+                        }
+                    }
+                }
+            }
+
+        ad_helper.register_dependent_variables(residual_ad);
+        ad_helper.stop_recording_operations(false /*write_tapes_to_file*/);
+      }
+    else
+      {
+        Assert(is_recording == true, ExcInternalError());
+      }
+
+    ad_helper.activate_recorded_tape(tape_no);
+    // ad_helper.set_independent_variables(dof_values); // Unnecessary when keep
+    // == true
+
+    // Compute the residual values and their jacobian for the new evaluation
+    // point
+    ad_helper.compute_values(data.cell_rhs);
+    data.cell_rhs *= -1.0; // RHS = - residual
+    ad_helper.compute_jacobian(data.cell_matrix);
+  }
+  template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+  void
+  Solid<dim, number_t, ad_type_code>::make_constraints(const int &it_nr)
+  {
+    std::cout << " CST " << std::flush;
+    if (it_nr > 1)
+      return;
+    constraints.clear();
+    const bool                       apply_dirichlet_bc = (it_nr == 0);
+    const FEValuesExtractors::Scalar x_displacement(0);
+    const FEValuesExtractors::Scalar y_displacement(1);
+    {
+      const int boundary_id = 0;
+      if (apply_dirichlet_bc == true)
+        VectorTools::interpolate_boundary_values(
+          dof_handler_ref,
+          boundary_id,
+          ZeroFunction<dim>(n_components),
+          constraints,
+          fe.component_mask(x_displacement));
+      else
+        VectorTools::interpolate_boundary_values(
+          dof_handler_ref,
+          boundary_id,
+          ZeroFunction<dim>(n_components),
+          constraints,
+          fe.component_mask(x_displacement));
+    }
+    {
+      const int boundary_id = 2;
+      if (apply_dirichlet_bc == true)
+        VectorTools::interpolate_boundary_values(
+          dof_handler_ref,
+          boundary_id,
+          ZeroFunction<dim>(n_components),
+          constraints,
+          fe.component_mask(y_displacement));
+      else
+        VectorTools::interpolate_boundary_values(
+          dof_handler_ref,
+          boundary_id,
+          ZeroFunction<dim>(n_components),
+          constraints,
+          fe.component_mask(y_displacement));
+    }
+    if (dim == 3)
+      {
+        const FEValuesExtractors::Scalar z_displacement(2);
+        {
+          const int boundary_id = 3;
+          if (apply_dirichlet_bc == true)
+            VectorTools::interpolate_boundary_values(
+              dof_handler_ref,
+              boundary_id,
+              ZeroFunction<dim>(n_components),
+              constraints,
+              (fe.component_mask(x_displacement) |
+               fe.component_mask(z_displacement)));
+          else
+            VectorTools::interpolate_boundary_values(
+              dof_handler_ref,
+              boundary_id,
+              ZeroFunction<dim>(n_components),
+              constraints,
+              (fe.component_mask(x_displacement) |
+               fe.component_mask(z_displacement)));
+        }
+        {
+          const int boundary_id = 4;
+          if (apply_dirichlet_bc == true)
+            VectorTools::interpolate_boundary_values(
+              dof_handler_ref,
+              boundary_id,
+              ZeroFunction<dim>(n_components),
+              constraints,
+              fe.component_mask(z_displacement));
+          else
+            VectorTools::interpolate_boundary_values(
+              dof_handler_ref,
+              boundary_id,
+              ZeroFunction<dim>(n_components),
+              constraints,
+              fe.component_mask(z_displacement));
+        }
+        {
+          const int boundary_id = 6;
+          if (apply_dirichlet_bc == true)
+            VectorTools::interpolate_boundary_values(
+              dof_handler_ref,
+              boundary_id,
+              ZeroFunction<dim>(n_components),
+              constraints,
+              (fe.component_mask(x_displacement) |
+               fe.component_mask(z_displacement)));
+          else
+            VectorTools::interpolate_boundary_values(
+              dof_handler_ref,
+              boundary_id,
+              ZeroFunction<dim>(n_components),
+              constraints,
+              (fe.component_mask(x_displacement) |
+               fe.component_mask(z_displacement)));
+        }
+      }
+    else
+      {
+        {
+          const int boundary_id = 3;
+          if (apply_dirichlet_bc == true)
+            VectorTools::interpolate_boundary_values(
+              dof_handler_ref,
+              boundary_id,
+              ZeroFunction<dim>(n_components),
+              constraints,
+              (fe.component_mask(x_displacement)));
+          else
+            VectorTools::interpolate_boundary_values(
+              dof_handler_ref,
+              boundary_id,
+              ZeroFunction<dim>(n_components),
+              constraints,
+              (fe.component_mask(x_displacement)));
+        }
+        {
+          const int boundary_id = 6;
+          if (apply_dirichlet_bc == true)
+            VectorTools::interpolate_boundary_values(
+              dof_handler_ref,
+              boundary_id,
+              ZeroFunction<dim>(n_components),
+              constraints,
+              (fe.component_mask(x_displacement)));
+          else
+            VectorTools::interpolate_boundary_values(
+              dof_handler_ref,
+              boundary_id,
+              ZeroFunction<dim>(n_components),
+              constraints,
+              (fe.component_mask(x_displacement)));
+        }
+      }
+    constraints.close();
+  }
+  template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+  void
+  Solid<dim, number_t, ad_type_code>::assemble_sc()
+  {
+    timer.enter_subsection("Perform static condensation");
+    std::cout << " ASM_SC " << std::flush;
+    PerTaskData_SC per_task_data(dofs_per_cell,
+                                 element_indices_u.size(),
+                                 element_indices_p.size(),
+                                 element_indices_J.size());
+    ScratchData_SC scratch_data;
+    WorkStream::run(dof_handler_ref.begin_active(),
+                    dof_handler_ref.end(),
+                    *this,
+                    &Solid::assemble_sc_one_cell,
+                    &Solid::copy_local_to_global_sc,
+                    scratch_data,
+                    per_task_data);
+    timer.leave_subsection();
+  }
+  template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+  void
+  Solid<dim, number_t, ad_type_code>::copy_local_to_global_sc(
+    const PerTaskData_SC &data)
+  {
+    for (unsigned int i = 0; i < dofs_per_cell; ++i)
+      for (unsigned int j = 0; j < dofs_per_cell; ++j)
+        tangent_matrix.add(data.local_dof_indices[i],
+                           data.local_dof_indices[j],
+                           data.cell_matrix(i, j));
+  }
+  template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+  void
+  Solid<dim, number_t, ad_type_code>::assemble_sc_one_cell(
+    const typename DoFHandler<dim>::active_cell_iterator &cell,
+    ScratchData_SC &                                      scratch,
+    PerTaskData_SC &                                      data)
+  {
+    data.reset();
+    scratch.reset();
+    cell->get_dof_indices(data.local_dof_indices);
+    data.k_orig.extract_submatrix_from(tangent_matrix,
+                                       data.local_dof_indices,
+                                       data.local_dof_indices);
+    data.k_pu.extract_submatrix_from(data.k_orig,
+                                     element_indices_p,
+                                     element_indices_u);
+    data.k_pJ.extract_submatrix_from(data.k_orig,
+                                     element_indices_p,
+                                     element_indices_J);
+    data.k_JJ.extract_submatrix_from(data.k_orig,
+                                     element_indices_J,
+                                     element_indices_J);
+    data.k_pJ_inv.invert(data.k_pJ);
+    data.k_pJ_inv.mmult(data.A, data.k_pu);
+    data.k_JJ.mmult(data.B, data.A);
+    data.k_pJ_inv.Tmmult(data.C, data.B);
+    data.k_pu.Tmmult(data.k_bbar, data.C);
+    data.k_bbar.scatter_matrix_to(element_indices_u,
+                                  element_indices_u,
+                                  data.cell_matrix);
+    data.k_pJ_inv.add(-1.0, data.k_pJ);
+    data.k_pJ_inv.scatter_matrix_to(element_indices_p,
+                                    element_indices_J,
+                                    data.cell_matrix);
+  }
+  template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+  std::pair<unsigned int, double>
+  Solid<dim, number_t, ad_type_code>::solve_linear_system(
+    BlockVector<double> &newton_update)
+  {
+    unsigned int lin_it  = 0;
+    double       lin_res = 0.0;
+    if (parameters.use_static_condensation == true)
+      {
+        BlockVector<double> A(dofs_per_block);
+        BlockVector<double> B(dofs_per_block);
+        {
+          assemble_sc();
+          tangent_matrix.block(p_dof, J_dof)
+            .vmult(A.block(J_dof), system_rhs.block(p_dof));
+          tangent_matrix.block(J_dof, J_dof)
+            .vmult(B.block(J_dof), A.block(J_dof));
+          A.block(J_dof) = system_rhs.block(J_dof);
+          A.block(J_dof) -= B.block(J_dof);
+          tangent_matrix.block(p_dof, J_dof)
+            .Tvmult(A.block(p_dof), A.block(J_dof));
+          tangent_matrix.block(u_dof, p_dof)
+            .vmult(A.block(u_dof), A.block(p_dof));
+          system_rhs.block(u_dof) -= A.block(u_dof);
+          timer.enter_subsection("Linear solver");
+          std::cout << " SLV " << std::flush;
+          if (parameters.type_lin == "CG")
+            {
+              const int solver_its = tangent_matrix.block(u_dof, u_dof).m() *
+                                     parameters.max_iterations_lin;
+              const double tol_sol =
+                parameters.tol_lin * system_rhs.block(u_dof).l2_norm();
+              SolverControl solver_control(solver_its, tol_sol, false, false);
+              GrowingVectorMemory<Vector<double>> GVM;
+              SolverCG<Vector<double>> solver_CG(solver_control, GVM);
+              PreconditionSelector<SparseMatrix<double>, Vector<double>>
+                preconditioner(parameters.preconditioner_type,
+                               parameters.preconditioner_relaxation);
+              preconditioner.use_matrix(tangent_matrix.block(u_dof, u_dof));
+              solver_CG.solve(tangent_matrix.block(u_dof, u_dof),
+                              newton_update.block(u_dof),
+                              system_rhs.block(u_dof),
+                              preconditioner);
+              lin_it  = solver_control.last_step();
+              lin_res = solver_control.last_value();
+            }
+          else if (parameters.type_lin == "Direct")
+            {
+              SparseDirectUMFPACK A_direct;
+              A_direct.initialize(tangent_matrix.block(u_dof, u_dof));
+              A_direct.vmult(newton_update.block(u_dof),
+                             system_rhs.block(u_dof));
+              lin_it  = 1;
+              lin_res = 0.0;
+            }
+          else
+            Assert(false, ExcMessage("Linear solver type not implemented"));
+          timer.leave_subsection();
+        }
+        constraints.distribute(newton_update);
+        timer.enter_subsection("Linear solver postprocessing");
+        std::cout << " PP " << std::flush;
+        {
+          tangent_matrix.block(p_dof, u_dof)
+            .vmult(A.block(p_dof), newton_update.block(u_dof));
+          A.block(p_dof) *= -1.0;
+          A.block(p_dof) += system_rhs.block(p_dof);
+          tangent_matrix.block(p_dof, J_dof)
+            .vmult(newton_update.block(J_dof), A.block(p_dof));
+        }
+        constraints.distribute(newton_update);
+        {
+          tangent_matrix.block(J_dof, J_dof)
+            .vmult(A.block(J_dof), newton_update.block(J_dof));
+          A.block(J_dof) *= -1.0;
+          A.block(J_dof) += system_rhs.block(J_dof);
+          tangent_matrix.block(p_dof, J_dof)
+            .Tvmult(newton_update.block(p_dof), A.block(J_dof));
+        }
+        constraints.distribute(newton_update);
+        timer.leave_subsection();
+      }
+    else
+      {
+        std::cout << " ------ " << std::flush;
+        timer.enter_subsection("Linear solver");
+        std::cout << " SLV " << std::flush;
+        if (parameters.type_lin == "CG")
+          {
+            const Vector<double> &f_u = system_rhs.block(u_dof);
+            const Vector<double> &f_p = system_rhs.block(p_dof);
+            const Vector<double> &f_J = system_rhs.block(J_dof);
+            Vector<double> &      d_u = newton_update.block(u_dof);
+            Vector<double> &      d_p = newton_update.block(p_dof);
+            Vector<double> &      d_J = newton_update.block(J_dof);
+            const auto            K_uu =
+              linear_operator(tangent_matrix.block(u_dof, u_dof));
+            const auto K_up =
+              linear_operator(tangent_matrix.block(u_dof, p_dof));
+            const auto K_pu =
+              linear_operator(tangent_matrix.block(p_dof, u_dof));
+            const auto K_Jp =
+              linear_operator(tangent_matrix.block(J_dof, p_dof));
+            const auto K_JJ =
+              linear_operator(tangent_matrix.block(J_dof, J_dof));
+            PreconditionSelector<SparseMatrix<double>, Vector<double>>
+              preconditioner_K_Jp_inv("jacobi");
+            preconditioner_K_Jp_inv.use_matrix(
+              tangent_matrix.block(J_dof, p_dof));
+            ReductionControl solver_control_K_Jp_inv(
+              tangent_matrix.block(J_dof, p_dof).m() *
+                parameters.max_iterations_lin,
+              1.0e-30,
+              parameters.tol_lin);
+            SolverSelector<Vector<double>> solver_K_Jp_inv;
+            solver_K_Jp_inv.select("cg");
+            solver_K_Jp_inv.set_control(solver_control_K_Jp_inv);
+            const auto K_Jp_inv =
+              inverse_operator(K_Jp, solver_K_Jp_inv, preconditioner_K_Jp_inv);
+            const auto K_pJ_inv     = transpose_operator(K_Jp_inv);
+            const auto K_pp_bar     = K_Jp_inv * K_JJ * K_pJ_inv;
+            const auto K_uu_bar_bar = K_up * K_pp_bar * K_pu;
+            const auto K_uu_con     = K_uu + K_uu_bar_bar;
+            PreconditionSelector<SparseMatrix<double>, Vector<double>>
+              preconditioner_K_con_inv(parameters.preconditioner_type,
+                                       parameters.preconditioner_relaxation);
+            preconditioner_K_con_inv.use_matrix(
+              tangent_matrix.block(u_dof, u_dof));
+            ReductionControl solver_control_K_con_inv(
+              tangent_matrix.block(u_dof, u_dof).m() *
+                parameters.max_iterations_lin,
+              1.0e-30,
+              parameters.tol_lin);
+            SolverSelector<Vector<double>> solver_K_con_inv;
+            solver_K_con_inv.select("cg");
+            solver_K_con_inv.set_control(solver_control_K_con_inv);
+            const auto K_uu_con_inv =
+              inverse_operator(K_uu_con,
+                               solver_K_con_inv,
+                               preconditioner_K_con_inv);
+            d_u =
+              K_uu_con_inv * (f_u - K_up * (K_Jp_inv * f_J - K_pp_bar * f_p));
+            timer.leave_subsection();
+            timer.enter_subsection("Linear solver postprocessing");
+            std::cout << " PP " << std::flush;
+            d_J     = K_pJ_inv * (f_p - K_pu * d_u);
+            d_p     = K_Jp_inv * (f_J - K_JJ * d_J);
+            lin_it  = solver_control_K_con_inv.last_step();
+            lin_res = solver_control_K_con_inv.last_value();
+          }
+        else if (parameters.type_lin == "Direct")
+          {
+            SparseDirectUMFPACK A_direct;
+            A_direct.initialize(tangent_matrix);
+            A_direct.vmult(newton_update, system_rhs);
+            lin_it  = 1;
+            lin_res = 0.0;
+            std::cout << " -- " << std::flush;
+          }
+        else
+          Assert(false, ExcMessage("Linear solver type not implemented"));
+        timer.leave_subsection();
+        constraints.distribute(newton_update);
+      }
+    return std::make_pair(lin_it, lin_res);
+  }
+  template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+  void
+  Solid<dim, number_t, ad_type_code>::output_results() const
+  {
+    DataOut<dim> data_out;
+    std::vector<DataComponentInterpretation::DataComponentInterpretation>
+      data_component_interpretation(
+        dim, DataComponentInterpretation::component_is_part_of_vector);
+    data_component_interpretation.push_back(
+      DataComponentInterpretation::component_is_scalar);
+    data_component_interpretation.push_back(
+      DataComponentInterpretation::component_is_scalar);
+    std::vector<std::string> solution_name(dim, "displacement");
+    solution_name.push_back("pressure");
+    solution_name.push_back("dilatation");
+    data_out.attach_dof_handler(dof_handler_ref);
+    data_out.add_data_vector(solution_n,
+                             solution_name,
+                             DataOut<dim>::type_dof_data,
+                             data_component_interpretation);
+    Vector<double> soln(solution_n.size());
+    for (unsigned int i = 0; i < soln.size(); ++i)
+      soln(i) = solution_n(i);
+    MappingQEulerian<dim> q_mapping(degree, dof_handler_ref, soln);
+    data_out.build_patches(q_mapping, degree);
+    std::ostringstream filename;
+    filename << "solution-" << dim << "d-" << time.get_timestep() << ".vtk";
+    std::ofstream output(filename.str().c_str());
+    data_out.write_vtk(output);
+  }
+} // namespace Step44
diff --git a/tests/ad_common_tests/symmetric_tensor_functions_02.h b/tests/ad_common_tests/symmetric_tensor_functions_02.h
new file mode 100644 (file)
index 0000000..202eca1
--- /dev/null
@@ -0,0 +1,226 @@
+// ---------------------------------------------------------------------
+//
+// Copyright (C) 2016 - 2018 by the deal.II authors
+//
+// This file is part of the deal.II library.
+//
+// The deal.II library is free software; you can use it, redistribute
+// it, and/or modify it under the terms of the GNU Lesser General
+// Public License as published by the Free Software Foundation; either
+// version 2.1 of the License, or (at your option) any later version.
+// The full text of the license can be found in the file LICENSE at
+// the top level of the deal.II distribution.
+//
+// ---------------------------------------------------------------------
+
+
+// Header file:
+// Test to check that tensor functions both compile and produce the right
+// result when differentiated using the various auto-differentiable number
+// types: Tensor inverse
+
+#include <deal.II/base/symmetric_tensor.h>
+
+#include <deal.II/differentiation/ad.h>
+
+#include <iostream>
+
+#include "../tests.h"
+
+using namespace dealii;
+namespace AD = dealii::Differentiation::AD;
+
+template <int dim, typename NumberType>
+struct FunctionsTestSymmetricTensor
+{
+  static SymmetricTensor<4, dim, NumberType>
+  dt_inv_dt(const SymmetricTensor<2, dim, NumberType> &t_inv)
+  {
+    // https://en.wikiversity.org/wiki/Introduction_to_Elasticity/Tensors#Derivative_of_the_inverse_of_a_tensor
+    SymmetricTensor<4, dim, NumberType> dt_inv_dt;
+    for (unsigned int i = 0; i < dim; ++i)
+      for (unsigned int j = i; j < dim; ++j)
+        for (unsigned int k = 0; k < dim; ++k)
+          for (unsigned int l = k; l < dim; ++l)
+            dt_inv_dt[i][j][k][l] =
+              -0.5 * (t_inv[i][k] * t_inv[j][l] + t_inv[i][l] * t_inv[j][k]);
+    return dt_inv_dt;
+  }
+
+  static NumberType
+  psi(const SymmetricTensor<2, dim, NumberType> &t)
+  {
+    // Previously, the invert function would hang for nested Sacado::Fad::DFad
+    const SymmetricTensor<2, dim, NumberType> t_inv = invert(t);
+    const SymmetricTensor<2, dim, NumberType> I =
+      unit_symmetric_tensor<dim, NumberType>();
+    return 3.0 * scalar_product(t_inv, I);
+  }
+
+  static SymmetricTensor<2, dim, NumberType>
+  dpsi_dt(const SymmetricTensor<2, dim, NumberType> &t)
+  {
+    const SymmetricTensor<2, dim, NumberType> t_inv = invert(t);
+    const SymmetricTensor<4, dim, NumberType> dt_inv_dt =
+      FunctionsTestSymmetricTensor::dt_inv_dt(t_inv);
+    const SymmetricTensor<2, dim, NumberType> I =
+      unit_symmetric_tensor<dim, NumberType>();
+    return 3.0 * (I * dt_inv_dt);
+  }
+
+  static Tensor<4, dim, NumberType>
+  d2psi_dt_dt(const SymmetricTensor<2, dim, NumberType> &t)
+  {
+    const SymmetricTensor<2, dim, NumberType> t_inv = invert(t);
+    const SymmetricTensor<4, dim, NumberType> dt_inv_dt =
+      FunctionsTestSymmetricTensor::dt_inv_dt(t_inv);
+    const SymmetricTensor<2, dim, NumberType> I =
+      unit_symmetric_tensor<dim, NumberType>();
+
+    SymmetricTensor<4, dim, NumberType> d2psi_dt_dt;
+    for (unsigned int i = 0; i < dim; ++i)
+      for (unsigned int j = i; j < dim; ++j)
+        for (unsigned int m = 0; m < dim; ++m)
+          for (unsigned int n = m; n < dim; ++n)
+            for (unsigned int k = 0; k < dim; ++k)
+              for (unsigned int l = k; l < dim; ++l)
+                d2psi_dt_dt[i][j][m][n] +=
+                  -3.0 * (0.5 * I[k][l] *
+                          (dt_inv_dt[i][k][m][n] * t_inv[j][l] +
+                           dt_inv_dt[i][l][m][n] * t_inv[j][k] +
+                           t_inv[i][k] * dt_inv_dt[j][l][m][n] +
+                           t_inv[i][l] * dt_inv_dt[j][k][m][n]));
+
+    return d2psi_dt_dt;
+  }
+};
+
+template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+void
+test_symmetric_tensor()
+{
+  typedef AD::ScalarFunction<dim, ad_type_code, number_t> ADHelper;
+  typedef typename ADHelper::ad_type                      ADNumberType;
+  typedef typename ADHelper::scalar_type                  ScalarNumberType;
+
+  std::cout << "*** dim = " << Utilities::to_string(dim) << ", "
+            << "Type code: " << static_cast<int>(ad_type_code) << std::endl;
+
+  // Tensor<2,dim,ADNumberType> grad_u;
+  // const Tensor<2,dim,ADNumberType> F =
+  // Physics::Elasticity::Kinematics::F(grad_u);
+
+  // Values computed from the AD energy function
+  ScalarNumberType             psi;
+  Vector<ScalarNumberType>     Dpsi;
+  FullMatrix<ScalarNumberType> D2psi;
+
+  // Function and its derivatives
+  typedef FunctionsTestSymmetricTensor<dim, ADNumberType> func_ad;
+
+  // Setup the variable components and choose a value at which to
+  // evaluate the tape
+  const FEValuesExtractors::SymmetricTensor<2> t_dof(0);
+  const unsigned int                           n_AD_components =
+    SymmetricTensor<2, dim>::n_independent_components;
+  ADHelper ad_helper(n_AD_components);
+  ad_helper.set_tape_buffer_sizes(); // Increase the buffer size from the
+                                     // default values
+
+  SymmetricTensor<2, dim, ScalarNumberType> t =
+    unit_symmetric_tensor<dim, ScalarNumberType>();
+  for (unsigned int i = 0; i < t.n_independent_components; ++i)
+    t[t.unrolled_to_component_indices(i)] += 0.12 * (i + 0.02);
+
+  const int  tape_no = 1;
+  const bool is_recording =
+    ad_helper.start_recording_operations(tape_no /*material_id*/,
+                                         true /*overwrite_tape*/,
+                                         true /*keep*/);
+  if (is_recording == true)
+    {
+      ad_helper.register_independent_variable(t, t_dof);
+
+      const SymmetricTensor<2, dim, ADNumberType> t_ad =
+        ad_helper.get_sensitive_variables(t_dof);
+
+      const ADNumberType psi(func_ad::psi(t_ad));
+
+      ad_helper.register_dependent_variable(psi);
+      ad_helper.stop_recording_operations(false /*write_tapes_to_file*/);
+
+      std::cout << "Recorded data..." << std::endl;
+      std::cout << "independent variable values: " << std::flush;
+      ad_helper.print_values(std::cout);
+      std::cout << "t_ad: " << t_ad << std::endl;
+      std::cout << "psi: " << psi << std::endl;
+      std::cout << std::endl;
+    }
+  else
+    {
+      Assert(is_recording == true, ExcInternalError());
+    }
+
+  // Do some work :-)
+  // Set a new evaluation point
+  if (AD::ADNumberTraits<ADNumberType>::is_taped == true)
+    {
+      std::cout
+        << "Using tape with different values for independent variables..."
+        << std::endl;
+      ad_helper.activate_recorded_tape(tape_no);
+      t *= 1.15;
+      ad_helper.set_independent_variable(t, t_dof);
+
+      std::cout << "independent variable values: " << std::flush;
+      ad_helper.print_values(std::cout);
+    }
+
+  // Compute the function value, gradient and hessian for the new evaluation
+  // point
+  psi = ad_helper.compute_value();
+  ad_helper.compute_gradient(Dpsi);
+  if (AD::ADNumberTraits<ADNumberType>::n_supported_derivative_levels >= 2)
+    {
+      ad_helper.compute_hessian(D2psi);
+    }
+
+  // Output the full stored function, gradient vector and hessian matrix
+  std::cout << "psi: " << psi << std::endl;
+  std::cout << "Dpsi: \n";
+  Dpsi.print(std::cout);
+  if (AD::ADNumberTraits<ADNumberType>::n_supported_derivative_levels >= 2)
+    {
+      std::cout << "D2psi: \n";
+      D2psi.print_formatted(std::cout, 3, true, 0, "0.0");
+    }
+
+  // Extract components of the solution
+  const SymmetricTensor<2, dim, ScalarNumberType> dpsi_dt =
+    ad_helper.extract_gradient_component(Dpsi, t_dof);
+
+  // Verify the result
+  typedef FunctionsTestSymmetricTensor<dim, ScalarNumberType> func;
+  static const ScalarNumberType                               tol =
+    1e5 * std::numeric_limits<ScalarNumberType>::epsilon();
+  std::cout << "psi:              " << psi << std::endl;
+  std::cout << "func::psi(t):     " << func::psi(t) << std::endl;
+  Assert(std::abs(psi - func::psi(t)) < tol,
+         ExcMessage("No match for function value."));
+  std::cout << "dpsi_dt:              " << dpsi_dt << std::endl;
+  std::cout << "func::dpsi_dt(t):     " << func::dpsi_dt(t) << std::endl;
+  Assert(std::abs((dpsi_dt - func::dpsi_dt(t)).norm()) < tol,
+         ExcMessage("No match for first derivative."));
+  if (AD::ADNumberTraits<ADNumberType>::n_supported_derivative_levels >= 2)
+    {
+      const Tensor<4, dim, ScalarNumberType> d2psi_dt_dt =
+        ad_helper.extract_hessian_component(D2psi, t_dof, t_dof);
+      std::cout << "d2psi_dt_dt:          " << d2psi_dt_dt << std::endl;
+      std::cout << "func::d2psi_dt_dt(t): " << func::d2psi_dt_dt(t)
+                << std::endl;
+      Assert(std::abs((d2psi_dt_dt - func::d2psi_dt_dt(t)).norm()) < tol,
+             ExcMessage("No match for second derivative."));
+    }
+
+  std::cout << std::endl << std::endl;
+}
diff --git a/tests/ad_common_tests/symmetric_tensor_functions_03.h b/tests/ad_common_tests/symmetric_tensor_functions_03.h
new file mode 100644 (file)
index 0000000..d85bef2
--- /dev/null
@@ -0,0 +1,445 @@
+// ---------------------------------------------------------------------
+//
+// Copyright (C) 2016 - 2018 by the deal.II authors
+//
+// This file is part of the deal.II library.
+//
+// The deal.II library is free software; you can use it, redistribute
+// it, and/or modify it under the terms of the GNU Lesser General
+// Public License as published by the Free Software Foundation; either
+// version 2.1 of the License, or (at your option) any later version.
+// The full text of the license can be found in the file LICENSE at
+// the top level of the deal.II distribution.
+//
+// ---------------------------------------------------------------------
+
+
+// Header file:
+// Test to check that tensor functions both compile and produce the right
+// result when differentiated using the various auto-differentiable number
+// types: Eigenvalues
+
+#include <deal.II/base/symmetric_tensor.h>
+
+#include <deal.II/differentiation/ad.h>
+
+#include "../tests.h"
+#ifdef DEAL_II_ADOLC_WITH_ADVANCED_BRANCHING
+#  include <deal.II/base/symmetric_tensor.templates.h>
+#endif
+
+#include <iostream>
+
+using namespace dealii;
+namespace AD = dealii::Differentiation::AD;
+
+double
+mu()
+{
+  return 100.0;
+}
+
+
+template <int dim, typename NumberType>
+NumberType
+J(const SymmetricTensor<2, dim, NumberType> &C)
+{
+  return std::sqrt(determinant(C));
+}
+
+template <int dim, typename NumberType>
+struct IncompressibleNeoHookean
+{
+  // Incompressible Neo-Hookean material
+  static NumberType
+  psi(const SymmetricTensor<2, dim, NumberType> &C)
+  {
+    return 0.5 * mu() * (trace(C) - dim) - mu() * std::log(J(C));
+  }
+
+  static SymmetricTensor<2, dim, NumberType>
+  dpsi_dC(const SymmetricTensor<2, dim, NumberType> &C)
+  {
+    const SymmetricTensor<2, dim, NumberType> I =
+      unit_symmetric_tensor<dim, NumberType>();
+    return 0.5 * mu() * (I - invert(C));
+  }
+
+  static SymmetricTensor<4, dim, NumberType>
+  d2psi_dC_dC(const SymmetricTensor<2, dim, NumberType> &C)
+  {
+    const SymmetricTensor<2, dim, NumberType> C_inv = invert(C);
+
+    SymmetricTensor<4, dim, NumberType> dC_inv_dC;
+    for (unsigned int A = 0; A < dim; ++A)
+      for (unsigned int B = A; B < dim; ++B)
+        for (unsigned int C = 0; C < dim; ++C)
+          for (unsigned int D = C; D < dim; ++D)
+            dC_inv_dC[A][B][C][D] -=
+              0.5 * (C_inv[A][C] * C_inv[B][D] + C_inv[A][D] * C_inv[B][C]);
+
+    return -0.5 * mu() * dC_inv_dC;
+  }
+};
+
+template <int dim, typename NumberType>
+struct IncompressibleNeoHookeanPrincipalStretches
+{
+  // Incompressible Neo-Hookean material
+  static NumberType
+  psi(const std::array<std::pair<NumberType, Tensor<1, dim, NumberType>>, dim>
+        eig_C)
+  {
+    NumberType psi = 0.0;
+    NumberType J   = 1.0;
+    for (unsigned int d = 0; d < dim; ++d)
+      {
+        const NumberType &lambda_squared = eig_C[d].first;
+        psi += 0.5 * mu() * (lambda_squared - 1.0);
+        J *= std::sqrt(lambda_squared);
+      }
+    psi -= mu() * std::log(J);
+    return psi;
+  }
+
+  //  static SymmetricTensor<2,dim,NumberType>
+  //  dpsi_dC (const std::array<std::pair<NumberType, Tensor<1,dim,NumberType>
+  //  >,dim> eig_C)
+  //  {
+  //    SymmetricTensor<2,dim,NumberType> C_inv;
+  //    for (unsigned int d=0; d<dim; ++d)
+  //      {
+  //        const NumberType &lambda_squared = eig_C[d].first;
+  //        const Tensor<1,dim,NumberType> &N = eig_C[d].second;
+  //        C_inv += (1.0/lambda_squared)*symmetrize(outer_product(N,N));
+  //      }
+  //
+  //    const SymmetricTensor<2,dim,NumberType> I =
+  //    unit_symmetric_tensor<dim,NumberType>(); return 0.5*mu()*(I - C_inv);
+  //  }
+};
+
+template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+void
+test_NH(const bool nontrivial_initial_values)
+{
+  typedef AD::ScalarFunction<dim, ad_type_code, number_t> ADHelper;
+  typedef typename ADHelper::ad_type                      ADNumberType;
+  typedef typename ADHelper::scalar_type                  ScalarNumberType;
+
+  std::cout << "*** Standard definition of incompressible NeoHookean material, "
+            << "dim = " << Utilities::to_string(dim) << ", "
+            << "Type code: " << static_cast<int>(ad_type_code) << ", "
+            << "Nontrivial initial values: " << std::boolalpha
+            << nontrivial_initial_values << std::endl;
+
+  // Values computed from the AD energy function
+  ScalarNumberType             psi;
+  Vector<ScalarNumberType>     Dpsi;
+  FullMatrix<ScalarNumberType> D2psi;
+
+  // Function and its derivatives
+  typedef IncompressibleNeoHookean<dim, ADNumberType> func_ad;
+
+  // Setup the variable components and choose a value at which to
+  // evaluate the tape
+  const FEValuesExtractors::SymmetricTensor<2> C_dof(0);
+  const unsigned int                           n_AD_components =
+    SymmetricTensor<2, dim>::n_independent_components;
+  ADHelper ad_helper(n_AD_components);
+  ad_helper.set_tape_buffer_sizes(); // Increase the buffer size from the
+                                     // default values
+
+  SymmetricTensor<2, dim, ScalarNumberType> C =
+    unit_symmetric_tensor<dim, ScalarNumberType>();
+  if (nontrivial_initial_values)
+    for (unsigned int i = 0; i < C.n_independent_components; ++i)
+      C[C.unrolled_to_component_indices(i)] += 0.12 * (i + 0.02);
+
+  const int  tape_no = 1;
+  const bool is_recording =
+    ad_helper.start_recording_operations(tape_no /*material_id*/,
+                                         true /*overwrite_tape*/,
+                                         true /*keep*/);
+  if (is_recording == true)
+    {
+      ad_helper.register_independent_variable(C, C_dof);
+
+      const SymmetricTensor<2, dim, ADNumberType> C_ad =
+        ad_helper.get_sensitive_variables(C_dof);
+
+      const ADNumberType psi(func_ad::psi(C_ad));
+
+      ad_helper.register_dependent_variable(psi);
+      ad_helper.stop_recording_operations(false /*write_tapes_to_file*/);
+
+      std::cout << "Recorded data..." << std::endl;
+      std::cout << "independent variable values: " << std::flush;
+      ad_helper.print_values(std::cout);
+      std::cout << "C_ad: " << C_ad << std::endl;
+      std::cout << "psi: " << psi << std::endl;
+      std::cout << std::endl;
+    }
+  else
+    {
+      Assert(is_recording == true, ExcInternalError());
+    }
+
+  // Do some work :-)
+  // Set a new evaluation point
+  if (AD::ADNumberTraits<ADNumberType>::is_taped == true)
+    {
+      std::cout
+        << "Using tape with different values for independent variables..."
+        << std::endl;
+      ad_helper.activate_recorded_tape(tape_no);
+      C *= 1.15;
+      ad_helper.set_independent_variable(C, C_dof);
+
+      std::cout << "independent variable values: " << std::flush;
+      ad_helper.print_values(std::cout);
+    }
+
+  // Compute the function value, gradient and hessian for the new evaluation
+  // point
+  psi = ad_helper.compute_value();
+  ad_helper.compute_gradient(Dpsi);
+  if (AD::ADNumberTraits<ADNumberType>::n_supported_derivative_levels >= 2)
+    {
+      ad_helper.compute_hessian(D2psi);
+    }
+
+  // Output the full stored function, gradient vector and hessian matrix
+  std::cout << "psi: " << psi << std::endl;
+  std::cout << "Dpsi: \n";
+  Dpsi.print(std::cout);
+  if (AD::ADNumberTraits<ADNumberType>::n_supported_derivative_levels >= 2)
+    {
+      std::cout << "D2psi: \n";
+      D2psi.print_formatted(std::cout, 3, true, 0, "0.0");
+    }
+
+  // Extract components of the solution
+  const SymmetricTensor<2, dim, ScalarNumberType> dpsi_dC =
+    ad_helper.extract_gradient_component(Dpsi, C_dof);
+
+  // Verify the result
+  typedef IncompressibleNeoHookean<dim, ScalarNumberType> func;
+  static const ScalarNumberType                           tol =
+    1e5 * std::numeric_limits<ScalarNumberType>::epsilon();
+  std::cout << "psi:              " << psi << std::endl;
+  std::cout << "func::psi(C):     " << func::psi(C) << std::endl;
+  Assert(std::abs(psi - func::psi(C)) < tol,
+         ExcMessage("No match for function value."));
+  std::cout << "dpsi_dC:              " << dpsi_dC << std::endl;
+  std::cout << "func::dpsi_dC(C):     " << func::dpsi_dC(C) << std::endl;
+  Assert(std::abs((dpsi_dC - func::dpsi_dC(C)).norm()) < tol,
+         ExcMessage("No match for first derivative."));
+  if (AD::ADNumberTraits<ADNumberType>::n_supported_derivative_levels >= 2)
+    {
+      const Tensor<4, dim, ScalarNumberType> d2psi_dC_dC =
+        ad_helper.extract_hessian_component(D2psi, C_dof, C_dof);
+      std::cout << "d2psi_dC_dC:          " << d2psi_dC_dC << std::endl;
+      std::cout << "func::d2psi_dC_dC(C): " << func::d2psi_dC_dC(C)
+                << std::endl;
+      Assert(std::abs((d2psi_dC_dC - func::d2psi_dC_dC(C)).norm()) < tol,
+             ExcMessage("No match for second derivative."));
+    }
+
+  std::cout << std::endl << std::endl;
+}
+
+
+
+template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+void
+test_NH_eigen_energy(const enum SymmetricTensorEigenvectorMethod method,
+                     const bool nontrivial_initial_values)
+{
+  typedef AD::ScalarFunction<dim, ad_type_code, number_t> ADHelper;
+  typedef typename ADHelper::ad_type                      ADNumberType;
+  typedef typename ADHelper::scalar_type                  ScalarNumberType;
+
+  std::cout
+    << "*** Principal stretch definition of incompressible NeoHookean material (from free energy function), "
+    << "dim = " << Utilities::to_string(dim) << ", "
+    << "Type code: " << static_cast<int>(ad_type_code) << ", "
+    << "Eig method: " << static_cast<int>(method) << ", "
+    << "Nontrivial initial values: " << std::boolalpha
+    << nontrivial_initial_values << std::endl;
+
+  // Values computed from the AD energy function
+  ScalarNumberType             psi;
+  Vector<ScalarNumberType>     Dpsi;
+  FullMatrix<ScalarNumberType> D2psi;
+
+  // Function and its derivatives
+  typedef IncompressibleNeoHookeanPrincipalStretches<dim, ADNumberType> func_ad;
+
+  // Setup the variable components and choose a value at which to
+  // evaluate the tape
+  const FEValuesExtractors::SymmetricTensor<2> C_dof(0);
+  const unsigned int                           n_AD_components =
+    SymmetricTensor<2, dim>::n_independent_components;
+  ADHelper ad_helper(n_AD_components);
+  ad_helper.set_tape_buffer_sizes(); // Increase the buffer size from the
+                                     // default values
+
+  SymmetricTensor<2, dim, ScalarNumberType> C =
+    unit_symmetric_tensor<dim, ScalarNumberType>();
+  if (nontrivial_initial_values)
+    for (unsigned int i = 0; i < C.n_independent_components; ++i)
+      C[C.unrolled_to_component_indices(i)] += 0.12 * (i + 0.02);
+
+  const int  tape_no = 1;
+  const bool is_recording =
+    ad_helper.start_recording_operations(tape_no /*material_id*/,
+                                         true /*overwrite_tape*/,
+                                         true /*keep*/);
+  if (is_recording == true)
+    {
+      ad_helper.register_independent_variable(C, C_dof);
+
+      const SymmetricTensor<2, dim, ADNumberType> C_ad =
+        ad_helper.get_sensitive_variables(C_dof);
+      const auto eig_C_ad = eigenvectors(C_ad, method);
+
+      const ADNumberType psi(func_ad::psi(eig_C_ad));
+
+      ad_helper.register_dependent_variable(psi);
+      ad_helper.stop_recording_operations(false /*write_tapes_to_file*/);
+
+      std::cout << "Recorded data..." << std::endl;
+      std::cout << "independent variable values: " << std::flush;
+      ad_helper.print_values(std::cout);
+      std::cout << "C_ad: " << C_ad << std::endl;
+      std::cout << "psi: " << psi << std::endl;
+      std::cout << std::endl;
+    }
+  else
+    {
+      Assert(is_recording == true, ExcInternalError());
+    }
+
+  // Do some work :-)
+  // Set a new evaluation point
+  if (AD::ADNumberTraits<ADNumberType>::is_taped == true)
+    {
+      std::cout
+        << "Using tape with different values for independent variables..."
+        << std::endl;
+      ad_helper.activate_recorded_tape(tape_no);
+      C *= 1.15;
+      ad_helper.set_independent_variable(C, C_dof);
+
+      std::cout << "independent variable values: " << std::flush;
+      ad_helper.print_values(std::cout);
+    }
+
+  // Compute the function value, gradient and hessian for the new evaluation
+  // point
+  psi = ad_helper.compute_value();
+  ad_helper.compute_gradient(Dpsi);
+  if (AD::ADNumberTraits<ADNumberType>::n_supported_derivative_levels >= 2)
+    {
+      ad_helper.compute_hessian(D2psi);
+    }
+
+  // Output the full stored function, gradient vector and hessian matrix
+  std::cout << "psi: " << psi << std::endl;
+  std::cout << "Dpsi: \n";
+  Dpsi.print(std::cout);
+  if (AD::ADNumberTraits<ADNumberType>::n_supported_derivative_levels >= 2)
+    {
+      std::cout << "D2psi: \n";
+      D2psi.print_formatted(std::cout, 3, true, 0, "0.0");
+    }
+
+  // Extract components of the solution
+  const SymmetricTensor<2, dim, ScalarNumberType> dpsi_dC =
+    ad_helper.extract_gradient_component(Dpsi, C_dof);
+
+  // Verify the result
+  typedef IncompressibleNeoHookean<dim, ScalarNumberType> func;
+  static const ScalarNumberType                           tol_val =
+    (nontrivial_initial_values ?
+       1e-6 :
+       1e5 * std::numeric_limits<ScalarNumberType>::epsilon());
+  static const ScalarNumberType tol_grad =
+    (nontrivial_initial_values ?
+       1e-4 :
+       1e5 * std::numeric_limits<ScalarNumberType>::epsilon());
+  static const ScalarNumberType tol_hess =
+    (nontrivial_initial_values ?
+       2e-2 /*1e-2,5e-3*/ :
+       1e5 * std::numeric_limits<ScalarNumberType>::epsilon());
+  std::cout << "psi:              " << psi << std::endl;
+  std::cout << "func::psi(C):     " << func::psi(C) << std::endl;
+  std::cout << "DIFF NORM: " << std::abs(psi - func::psi(C)) << std::endl;
+  Assert(std::abs(psi - func::psi(C)) < tol_val,
+         ExcMessage("No match for function value."));
+  std::cout << "dpsi_dC:              " << dpsi_dC << std::endl;
+  std::cout << "func::dpsi_dC(C):     " << func::dpsi_dC(C) << std::endl;
+  std::cout << "DIFF NORM: " << std::abs((dpsi_dC - func::dpsi_dC(C)).norm())
+            << std::endl;
+  Assert(std::abs((dpsi_dC - func::dpsi_dC(C)).norm()) < tol_grad,
+         ExcMessage("No match for first derivative."));
+  if (AD::ADNumberTraits<ADNumberType>::n_supported_derivative_levels >= 2)
+    {
+      const SymmetricTensor<4, dim, ScalarNumberType> d2psi_dC_dC =
+        ad_helper.extract_hessian_component(D2psi, C_dof, C_dof);
+      std::cout << "d2psi_dC_dC:          " << d2psi_dC_dC << std::endl;
+      std::cout << "func::d2psi_dC_dC(C): " << func::d2psi_dC_dC(C)
+                << std::endl;
+      //      std::cout << "DIFF: " << (d2psi_dC_dC - func::d2psi_dC_dC(C)) <<
+      //      std::endl;
+      std::cout << "DIFF NORM: "
+                << std::abs((d2psi_dC_dC - func::d2psi_dC_dC(C)).norm())
+                << std::endl;
+      Assert(std::abs((d2psi_dC_dC - func::d2psi_dC_dC(C)).norm()) < tol_hess,
+             ExcMessage("No match for second derivative."));
+    }
+
+  std::cout << std::endl << std::endl;
+}
+
+
+
+template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+void
+test_symmetric_tensor()
+{
+  // Then test the functions that compute the eigenvalues:
+  // Note: Derived from an energy function, only the eigenvalues are used
+
+  // Non-trivial initial values (unequal eigenvalues):
+  {
+    const bool nontrivial_initial_values = true;
+
+    // First verify that all manual calculations are correct
+    test_NH<dim, number_t, ad_type_code>(nontrivial_initial_values);
+
+    test_NH_eigen_energy<dim, number_t, ad_type_code>(
+      SymmetricTensorEigenvectorMethod::hybrid, nontrivial_initial_values);
+    test_NH_eigen_energy<dim, number_t, ad_type_code>(
+      SymmetricTensorEigenvectorMethod::ql_implicit_shifts,
+      nontrivial_initial_values);
+    test_NH_eigen_energy<dim, number_t, ad_type_code>(
+      SymmetricTensorEigenvectorMethod::jacobi, nontrivial_initial_values);
+  }
+  // Trivial initial values (equal eigenvalues):
+  {
+    const bool nontrivial_initial_values = false;
+
+    // First verify that all manual calculations are correct
+    test_NH<dim, number_t, ad_type_code>(nontrivial_initial_values);
+
+    //    test_NH_eigen_energy<dim,number_t,ad_type_code>(SymmetricTensorEigenvectorMethod::hybrid,nontrivial_initial_values);
+    //    // This will never work.
+    test_NH_eigen_energy<dim, number_t, ad_type_code>(
+      SymmetricTensorEigenvectorMethod::ql_implicit_shifts,
+      nontrivial_initial_values);
+    test_NH_eigen_energy<dim, number_t, ad_type_code>(
+      SymmetricTensorEigenvectorMethod::jacobi, nontrivial_initial_values);
+  }
+}
diff --git a/tests/ad_common_tests/symmetric_tensor_functions_04.h b/tests/ad_common_tests/symmetric_tensor_functions_04.h
new file mode 100644 (file)
index 0000000..52545da
--- /dev/null
@@ -0,0 +1,447 @@
+// ---------------------------------------------------------------------
+//
+// Copyright (C) 2016 - 2018 by the deal.II authors
+//
+// This file is part of the deal.II library.
+//
+// The deal.II library is free software; you can use it, redistribute
+// it, and/or modify it under the terms of the GNU Lesser General
+// Public License as published by the Free Software Foundation; either
+// version 2.1 of the License, or (at your option) any later version.
+// The full text of the license can be found in the file LICENSE at
+// the top level of the deal.II distribution.
+//
+// ---------------------------------------------------------------------
+
+
+// Header file:
+// Test to check that tensor functions both compile and produce the right
+// result when differentiated using the various auto-differentiable number
+// types: Eigenvalues and eignvectors
+
+#include <deal.II/base/symmetric_tensor.h>
+
+#include <deal.II/differentiation/ad.h>
+
+#include "../tests.h"
+#ifdef DEAL_II_ADOLC_WITH_ADVANCED_BRANCHING
+#  include <deal.II/base/symmetric_tensor.templates.h>
+#endif
+
+#include <iostream>
+
+using namespace dealii;
+namespace AD = dealii::Differentiation::AD;
+
+double
+mu()
+{
+  return 100.0;
+}
+
+
+template <int dim, typename NumberType>
+NumberType
+J(const SymmetricTensor<2, dim, NumberType> &C)
+{
+  return std::sqrt(determinant(C));
+}
+
+template <int dim, typename NumberType>
+struct IncompressibleNeoHookean
+{
+  // Incompressible Neo-Hookean material
+  static NumberType
+  psi(const SymmetricTensor<2, dim, NumberType> &C)
+  {
+    return 0.5 * mu() * (trace(C) - dim) - mu() * std::log(J(C));
+  }
+
+  static SymmetricTensor<2, dim, NumberType>
+  dpsi_dC(const SymmetricTensor<2, dim, NumberType> &C)
+  {
+    const SymmetricTensor<2, dim, NumberType> I =
+      unit_symmetric_tensor<dim, NumberType>();
+    return 0.5 * mu() * (I - invert(C));
+  }
+
+  static SymmetricTensor<4, dim, NumberType>
+  d2psi_dC_dC(const SymmetricTensor<2, dim, NumberType> &C)
+  {
+    const SymmetricTensor<2, dim, NumberType> C_inv = invert(C);
+
+    SymmetricTensor<4, dim, NumberType> dC_inv_dC;
+    for (unsigned int A = 0; A < dim; ++A)
+      for (unsigned int B = A; B < dim; ++B)
+        for (unsigned int C = 0; C < dim; ++C)
+          for (unsigned int D = C; D < dim; ++D)
+            dC_inv_dC[A][B][C][D] -=
+              0.5 * (C_inv[A][C] * C_inv[B][D] + C_inv[A][D] * C_inv[B][C]);
+
+    return -0.5 * mu() * dC_inv_dC;
+  }
+};
+
+template <int dim, typename NumberType>
+struct IncompressibleNeoHookeanPrincipalStretches
+{
+  //  // Incompressible Neo-Hookean material
+  //  static NumberType
+  //  psi (const std::array<std::pair<NumberType, Tensor<1,dim,NumberType>
+  //  >,dim> eig_C)
+  //  {
+  //    NumberType psi = 0.0;
+  //    NumberType J = 1.0;
+  //    for (unsigned int d=0; d<dim; ++d)
+  //      {
+  //        const NumberType &lambda_squared = eig_C[d].first;
+  //        psi += 0.5*mu()*(lambda_squared - 1.0);
+  //        J *= std::sqrt(lambda_squared);
+  //      }
+  //    psi -= mu()*std::log(J);
+  //    return psi;
+  //  }
+
+  static SymmetricTensor<2, dim, NumberType>
+  dpsi_dC(const std::array<std::pair<NumberType, Tensor<1, dim, NumberType>>,
+                           dim> eig_C)
+  {
+    SymmetricTensor<2, dim, NumberType> C_inv;
+    for (unsigned int d = 0; d < dim; ++d)
+      {
+        const NumberType &                lambda_squared = eig_C[d].first;
+        const Tensor<1, dim, NumberType> &N              = eig_C[d].second;
+        C_inv += (1.0 / lambda_squared) * symmetrize(outer_product(N, N));
+      }
+
+    const SymmetricTensor<2, dim, NumberType> I =
+      unit_symmetric_tensor<dim, NumberType>();
+    return 0.5 * mu() * (I - C_inv);
+  }
+};
+
+template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+void
+test_NH(const bool nontrivial_initial_values)
+{
+  typedef AD::ScalarFunction<dim, ad_type_code, number_t> ADHelper;
+  typedef typename ADHelper::ad_type                      ADNumberType;
+  typedef typename ADHelper::scalar_type                  ScalarNumberType;
+
+  std::cout << "*** Standard definition of incompressible NeoHookean material, "
+            << "dim = " << Utilities::to_string(dim) << ", "
+            << "Type code: " << static_cast<int>(ad_type_code) << ", "
+            << "Nontrivial initial values: " << std::boolalpha
+            << nontrivial_initial_values << std::endl;
+
+  // Values computed from the AD energy function
+  ScalarNumberType             psi;
+  Vector<ScalarNumberType>     Dpsi;
+  FullMatrix<ScalarNumberType> D2psi;
+
+  // Function and its derivatives
+  typedef IncompressibleNeoHookean<dim, ADNumberType> func_ad;
+
+  // Setup the variable components and choose a value at which to
+  // evaluate the tape
+  const FEValuesExtractors::SymmetricTensor<2> C_dof(0);
+  const unsigned int                           n_AD_components =
+    SymmetricTensor<2, dim>::n_independent_components;
+  ADHelper ad_helper(n_AD_components);
+  ad_helper.set_tape_buffer_sizes(); // Increase the buffer size from the
+                                     // default values
+
+  SymmetricTensor<2, dim, ScalarNumberType> C =
+    unit_symmetric_tensor<dim, ScalarNumberType>();
+  if (nontrivial_initial_values)
+    for (unsigned int i = 0; i < C.n_independent_components; ++i)
+      C[C.unrolled_to_component_indices(i)] += 0.12 * (i + 0.02);
+
+  const int  tape_no = 1;
+  const bool is_recording =
+    ad_helper.start_recording_operations(tape_no /*material_id*/,
+                                         true /*overwrite_tape*/,
+                                         true /*keep*/);
+  if (is_recording == true)
+    {
+      ad_helper.register_independent_variable(C, C_dof);
+
+      const SymmetricTensor<2, dim, ADNumberType> C_ad =
+        ad_helper.get_sensitive_variables(C_dof);
+
+      const ADNumberType psi(func_ad::psi(C_ad));
+
+      ad_helper.register_dependent_variable(psi);
+      ad_helper.stop_recording_operations(false /*write_tapes_to_file*/);
+
+      std::cout << "Recorded data..." << std::endl;
+      std::cout << "independent variable values: " << std::flush;
+      ad_helper.print_values(std::cout);
+      std::cout << "C_ad: " << C_ad << std::endl;
+      std::cout << "psi: " << psi << std::endl;
+      std::cout << std::endl;
+    }
+  else
+    {
+      Assert(is_recording == true, ExcInternalError());
+    }
+
+  // Do some work :-)
+  // Set a new evaluation point
+  if (AD::ADNumberTraits<ADNumberType>::is_taped == true)
+    {
+      std::cout
+        << "Using tape with different values for independent variables..."
+        << std::endl;
+      ad_helper.activate_recorded_tape(tape_no);
+      C *= 1.15;
+      ad_helper.set_independent_variable(C, C_dof);
+
+      std::cout << "independent variable values: " << std::flush;
+      ad_helper.print_values(std::cout);
+    }
+
+  // Compute the function value, gradient and hessian for the new evaluation
+  // point
+  psi = ad_helper.compute_value();
+  ad_helper.compute_gradient(Dpsi);
+  if (AD::ADNumberTraits<ADNumberType>::n_supported_derivative_levels >= 2)
+    {
+      ad_helper.compute_hessian(D2psi);
+    }
+
+  // Output the full stored function, gradient vector and hessian matrix
+  std::cout << "psi: " << psi << std::endl;
+  std::cout << "Dpsi: \n";
+  Dpsi.print(std::cout);
+  if (AD::ADNumberTraits<ADNumberType>::n_supported_derivative_levels >= 2)
+    {
+      std::cout << "D2psi: \n";
+      D2psi.print_formatted(std::cout, 3, true, 0, "0.0");
+    }
+
+  // Extract components of the solution
+  const SymmetricTensor<2, dim, ScalarNumberType> dpsi_dC =
+    ad_helper.extract_gradient_component(Dpsi, C_dof);
+
+  // Verify the result
+  typedef IncompressibleNeoHookean<dim, ScalarNumberType> func;
+  static const ScalarNumberType                           tol =
+    1e5 * std::numeric_limits<ScalarNumberType>::epsilon();
+  std::cout << "psi:              " << psi << std::endl;
+  std::cout << "func::psi(C):     " << func::psi(C) << std::endl;
+  Assert(std::abs(psi - func::psi(C)) < tol,
+         ExcMessage("No match for function value."));
+  std::cout << "dpsi_dC:              " << dpsi_dC << std::endl;
+  std::cout << "func::dpsi_dC(C):     " << func::dpsi_dC(C) << std::endl;
+  Assert(std::abs((dpsi_dC - func::dpsi_dC(C)).norm()) < tol,
+         ExcMessage("No match for first derivative."));
+  if (AD::ADNumberTraits<ADNumberType>::n_supported_derivative_levels >= 2)
+    {
+      const Tensor<4, dim, ScalarNumberType> d2psi_dC_dC =
+        ad_helper.extract_hessian_component(D2psi, C_dof, C_dof);
+      std::cout << "d2psi_dC_dC:          " << d2psi_dC_dC << std::endl;
+      std::cout << "func::d2psi_dC_dC(C): " << func::d2psi_dC_dC(C)
+                << std::endl;
+      Assert(std::abs((d2psi_dC_dC - func::d2psi_dC_dC(C)).norm()) < tol,
+             ExcMessage("No match for second derivative."));
+    }
+
+  std::cout << std::endl << std::endl;
+}
+
+
+
+template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+void
+test_NH_eigen_stress(const enum SymmetricTensorEigenvectorMethod method,
+                     const bool nontrivial_initial_values)
+{
+  typedef AD::VectorFunction<dim, ad_type_code, number_t> ADHelper;
+  typedef typename ADHelper::ad_type                      ADNumberType;
+  typedef typename ADHelper::scalar_type                  ScalarNumberType;
+
+  std::cout
+    << "*** Principal stretch definition of incompressible NeoHookean material (from stress), "
+    << "dim = " << Utilities::to_string(dim) << ", "
+    << "Type code: " << static_cast<int>(ad_type_code) << ", "
+    << "Eig method: " << static_cast<int>(method) << ", "
+    << "Nontrivial initial values: " << std::boolalpha
+    << nontrivial_initial_values << std::endl;
+
+  // Values computed from the AD energy function
+  // ScalarNumberType psi;
+  Vector<ScalarNumberType>     Dpsi;
+  FullMatrix<ScalarNumberType> D2psi;
+
+  // Function and its derivatives
+  typedef IncompressibleNeoHookeanPrincipalStretches<dim, ADNumberType> func_ad;
+
+  // Setup the variable components and choose a value at which to
+  // evaluate the tape
+  const FEValuesExtractors::SymmetricTensor<2> C_dof(0);
+  const unsigned int                           n_AD_independent_components =
+    SymmetricTensor<2, dim>::n_independent_components;
+  const unsigned int n_AD_dependent_components =
+    SymmetricTensor<2, dim>::n_independent_components;
+  ADHelper ad_helper(n_AD_independent_components, n_AD_dependent_components);
+  ad_helper.set_tape_buffer_sizes(); // Increase the buffer size from the
+                                     // default values
+
+  SymmetricTensor<2, dim, ScalarNumberType> C =
+    unit_symmetric_tensor<dim, ScalarNumberType>();
+  if (nontrivial_initial_values)
+    for (unsigned int i = 0; i < C.n_independent_components; ++i)
+      C[C.unrolled_to_component_indices(i)] += 0.12 * (i + 0.02);
+
+  const int  tape_no = 1;
+  const bool is_recording =
+    ad_helper.start_recording_operations(tape_no /*material_id*/,
+                                         true /*overwrite_tape*/,
+                                         true /*keep*/);
+  if (is_recording == true)
+    {
+      ad_helper.register_independent_variable(C, C_dof);
+
+      const SymmetricTensor<2, dim, ADNumberType> C_ad =
+        ad_helper.get_sensitive_variables(C_dof);
+      const auto eig_C_ad = eigenvectors(C_ad, method);
+
+      for (unsigned int d = 0; d < dim; ++d)
+        std::cout << "  Direction: " << d
+                  << "  Eigenvalue: " << eig_C_ad[d].first
+                  << "  Eigenvector: " << eig_C_ad[d].second << std::endl;
+
+      const SymmetricTensor<2, dim, ADNumberType> dpsi_dC(
+        func_ad::dpsi_dC(eig_C_ad));
+
+      ad_helper.register_dependent_variable(dpsi_dC, C_dof);
+      ad_helper.stop_recording_operations(false /*write_tapes_to_file*/);
+
+      std::cout << "Recorded data..." << std::endl;
+      std::cout << "independent variable values: " << std::flush;
+      ad_helper.print_values(std::cout);
+      std::cout << "C_ad: " << C_ad << std::endl;
+      std::cout << "dpsi_dC: " << dpsi_dC << std::endl;
+      std::cout << std::endl;
+    }
+  else
+    {
+      Assert(is_recording == true, ExcInternalError());
+    }
+
+  // Do some work :-)
+  // Set a new evaluation point
+  if (AD::ADNumberTraits<ADNumberType>::is_taped == true)
+    {
+      std::cout
+        << "Using tape with different values for independent variables..."
+        << std::endl;
+      ad_helper.activate_recorded_tape(tape_no);
+      C *= 1.15;
+      ad_helper.set_independent_variable(C, C_dof);
+
+      std::cout << "independent variable values: " << std::flush;
+      ad_helper.print_values(std::cout);
+    }
+
+  // Compute the function value, gradient and hessian for the new evaluation
+  // point
+  // psi = ad_helper.compute_value();
+  ad_helper.compute_values(Dpsi);
+  if (AD::ADNumberTraits<ADNumberType>::n_supported_derivative_levels >= 1)
+    {
+      ad_helper.compute_jacobian(D2psi);
+    }
+
+  // Output the full stored function, gradient vector and hessian matrix
+  // std::cout << "psi: " << psi << std::endl;
+  std::cout << "Dpsi: \n";
+  Dpsi.print(std::cout);
+  if (AD::ADNumberTraits<ADNumberType>::n_supported_derivative_levels >= 1)
+    {
+      std::cout << "D2psi: \n";
+      D2psi.print_formatted(std::cout, 3, true, 0, "0.0");
+    }
+
+  // Extract components of the solution
+  const SymmetricTensor<2, dim, ScalarNumberType> dpsi_dC =
+    ad_helper.extract_value_component(Dpsi, C_dof);
+
+  // Verify the result
+  typedef IncompressibleNeoHookean<dim, ScalarNumberType> func;
+  static const ScalarNumberType                           tol_val =
+    (nontrivial_initial_values ?
+       1e-4 :
+       1e6 * std::numeric_limits<ScalarNumberType>::epsilon());
+  static const ScalarNumberType tol_jac =
+    (nontrivial_initial_values ?
+       2.5e-3 :
+       1e6 * std::numeric_limits<ScalarNumberType>::epsilon());
+  //  std::cout << "psi:              " << psi << std::endl;
+  //  std::cout << "func::psi(C):     " << func::psi(C) << std::endl;
+  //  Assert(std::abs(psi - func::psi(C)) < tol, ExcMessage("No match for
+  //  function value."));
+  std::cout << "dpsi_dC:              " << dpsi_dC << std::endl;
+  std::cout << "func::dpsi_dC(C):     " << func::dpsi_dC(C) << std::endl;
+  //  std::cout << "DIFF NORM: " << std::abs((dpsi_dC -
+  //  func::dpsi_dC(C)).norm()) << std::endl;
+  Assert(std::abs((dpsi_dC - func::dpsi_dC(C)).norm()) < tol_val,
+         ExcMessage("No match for first derivative."));
+  if (AD::ADNumberTraits<ADNumberType>::n_supported_derivative_levels >= 1)
+    {
+      const SymmetricTensor<4, dim, ScalarNumberType> d2psi_dC_dC =
+        ad_helper.extract_jacobian_component(D2psi, C_dof, C_dof);
+      std::cout << "d2psi_dC_dC:          " << d2psi_dC_dC << std::endl;
+      std::cout << "func::d2psi_dC_dC(C): " << func::d2psi_dC_dC(C)
+                << std::endl;
+      //      std::cout << "DIFF: " << (d2psi_dC_dC - func::d2psi_dC_dC(C)) <<
+      //      std::endl; std::cout << "DIFF NORM: " << std::abs((d2psi_dC_dC -
+      //      func::d2psi_dC_dC(C)).norm()) << std::endl;
+      Assert(std::abs((d2psi_dC_dC - func::d2psi_dC_dC(C)).norm()) < tol_jac,
+             ExcMessage("No match for second derivative."));
+    }
+
+  std::cout << std::endl << std::endl;
+}
+
+
+
+template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+void
+test_symmetric_tensor()
+{
+  // Then test the functions that compute the eigenvalues/vectors:
+  // Note: Derived from a stress, both the eigenvalues and eigenvectors are used
+
+  // Non-trivial initial values (unequal eigenvalues):
+  {
+    const bool nontrivial_initial_values = true;
+
+    // First verify that all manual calculations are correct
+    test_NH<dim, number_t, ad_type_code>(nontrivial_initial_values);
+
+    test_NH_eigen_stress<dim, number_t, ad_type_code>(
+      SymmetricTensorEigenvectorMethod::hybrid, nontrivial_initial_values);
+    test_NH_eigen_stress<dim, number_t, ad_type_code>(
+      SymmetricTensorEigenvectorMethod::ql_implicit_shifts,
+      nontrivial_initial_values);
+    test_NH_eigen_stress<dim, number_t, ad_type_code>(
+      SymmetricTensorEigenvectorMethod::jacobi, nontrivial_initial_values);
+  }
+  // Trivial initial values (equal eigenvalues):
+  {
+    const bool nontrivial_initial_values = false;
+
+    // First verify that all manual calculations are correct
+    test_NH<dim, number_t, ad_type_code>(nontrivial_initial_values);
+
+    //    test_NH_eigen_stress<dim,number_t,ad_type_code>(SymmetricTensorEigenvectorMethod::hybrid,nontrivial_initial_values);
+    //    // This will never work.
+    test_NH_eigen_stress<dim, number_t, ad_type_code>(
+      SymmetricTensorEigenvectorMethod::ql_implicit_shifts,
+      nontrivial_initial_values);
+    test_NH_eigen_stress<dim, number_t, ad_type_code>(
+      SymmetricTensorEigenvectorMethod::jacobi, nontrivial_initial_values);
+  }
+}
diff --git a/tests/ad_common_tests/tensor_functions_02.h b/tests/ad_common_tests/tensor_functions_02.h
new file mode 100644 (file)
index 0000000..335060f
--- /dev/null
@@ -0,0 +1,220 @@
+// ---------------------------------------------------------------------
+//
+// Copyright (C) 2016 - 2018 by the deal.II authors
+//
+// This file is part of the deal.II library.
+//
+// The deal.II library is free software; you can use it, redistribute
+// it, and/or modify it under the terms of the GNU Lesser General
+// Public License as published by the Free Software Foundation; either
+// version 2.1 of the License, or (at your option) any later version.
+// The full text of the license can be found in the file LICENSE at
+// the top level of the deal.II distribution.
+//
+// ---------------------------------------------------------------------
+
+
+// Header file:
+// Test to check that tensor functions both compile and produce the right
+// result when differentiated using the various auto-differentiable number
+// types: Tensor inverse
+
+#include <deal.II/base/tensor.h>
+
+#include <deal.II/differentiation/ad.h>
+
+#include <iostream>
+
+#include "../tests.h"
+
+using namespace dealii;
+namespace AD = dealii::Differentiation::AD;
+
+template <int dim, typename NumberType>
+struct FunctionsTestTensor
+{
+  static Tensor<4, dim, NumberType>
+  dt_inv_dt(const Tensor<2, dim, NumberType> &t_inv)
+  {
+    // https://en.wikiversity.org/wiki/Introduction_to_Elasticity/Tensors#Derivative_of_the_inverse_of_a_tensor
+    Tensor<4, dim, NumberType> dt_inv_dt;
+    for (unsigned int i = 0; i < dim; ++i)
+      for (unsigned int J = 0; J < dim; ++J)
+        for (unsigned int k = 0; k < dim; ++k)
+          for (unsigned int L = 0; L < dim; ++L)
+            dt_inv_dt[J][i][k][L] = -t_inv[J][k] * t_inv[L][i];
+    return dt_inv_dt;
+  }
+
+  static NumberType
+  psi(const Tensor<2, dim, NumberType> &t)
+  {
+    // Previously, the invert function would hang for nested Sacado::Fad::DFad
+    const Tensor<2, dim, NumberType> t_inv = invert(t);
+    const Tensor<2, dim, NumberType> I =
+      unit_symmetric_tensor<dim, NumberType>();
+    return 3.0 * scalar_product(t_inv, I);
+  }
+
+  static Tensor<2, dim, NumberType>
+  dpsi_dt(const Tensor<2, dim, NumberType> &t)
+  {
+    const Tensor<2, dim, NumberType> t_inv = invert(t);
+    const Tensor<4, dim, NumberType> dt_inv_dt =
+      FunctionsTestTensor::dt_inv_dt(t_inv);
+    const Tensor<2, dim, NumberType> I =
+      unit_symmetric_tensor<dim, NumberType>();
+    return 3.0 * double_contract<0, 1, 1, 0>(I, dt_inv_dt);
+  }
+
+  static Tensor<4, dim, NumberType>
+  d2psi_dt_dt(const Tensor<2, dim, NumberType> &t)
+  {
+    const Tensor<2, dim, NumberType> t_inv = invert(t);
+    const Tensor<4, dim, NumberType> dt_inv_dt =
+      FunctionsTestTensor::dt_inv_dt(t_inv);
+    const Tensor<2, dim, NumberType> I =
+      unit_symmetric_tensor<dim, NumberType>();
+
+    Tensor<4, dim, NumberType> d2psi_dt_dt;
+    for (unsigned int i = 0; i < dim; ++i)
+      for (unsigned int J = 0; J < dim; ++J)
+        for (unsigned int k = 0; k < dim; ++k)
+          for (unsigned int L = 0; L < dim; ++L)
+            for (unsigned int alpha = 0; alpha < dim; ++alpha)
+              d2psi_dt_dt[i][J][k][L] +=
+                -3.0 * (dt_inv_dt[alpha][i][k][L] * t_inv[J][alpha] +
+                        t_inv[alpha][i] * dt_inv_dt[J][alpha][k][L]);
+
+    return d2psi_dt_dt;
+  }
+};
+
+template <int dim, typename number_t, enum AD::NumberTypes ad_type_code>
+void
+test_tensor()
+{
+  typedef AD::ScalarFunction<dim, ad_type_code, number_t> ADHelper;
+  typedef typename ADHelper::ad_type                      ADNumberType;
+  typedef typename ADHelper::scalar_type                  ScalarNumberType;
+
+  std::cout << "*** dim = " << Utilities::to_string(dim) << ", "
+            << "Type code: " << static_cast<int>(ad_type_code) << std::endl;
+
+  // Tensor<2,dim,ADNumberType> grad_u;
+  // const Tensor<2,dim,ADNumberType> F =
+  // Physics::Elasticity::Kinematics::F(grad_u);
+
+  // Values computed from the AD energy function
+  ScalarNumberType             psi;
+  Vector<ScalarNumberType>     Dpsi;
+  FullMatrix<ScalarNumberType> D2psi;
+
+  // Function and its derivatives
+  typedef FunctionsTestTensor<dim, ADNumberType> func_ad;
+
+  // Setup the variable components and choose a value at which to
+  // evaluate the tape
+  const FEValuesExtractors::Tensor<2> t_dof(0);
+  const unsigned int n_AD_components = Tensor<2, dim>::n_independent_components;
+  ADHelper           ad_helper(n_AD_components);
+  ad_helper.set_tape_buffer_sizes(); // Increase the buffer size from the
+                                     // default values
+
+  Tensor<2, dim, ScalarNumberType> t =
+    unit_symmetric_tensor<dim, ScalarNumberType>();
+  for (unsigned int i = 0; i < t.n_independent_components; ++i)
+    t[t.unrolled_to_component_indices(i)] += 0.14 * (i + 0.07);
+
+  const int  tape_no = 1;
+  const bool is_recording =
+    ad_helper.start_recording_operations(tape_no /*material_id*/,
+                                         true /*overwrite_tape*/,
+                                         true /*keep*/);
+  if (is_recording == true)
+    {
+      ad_helper.register_independent_variable(t, t_dof);
+
+      const Tensor<2, dim, ADNumberType> t_ad =
+        ad_helper.get_sensitive_variables(t_dof);
+
+      const ADNumberType psi(func_ad::psi(t_ad));
+
+      ad_helper.register_dependent_variable(psi);
+      ad_helper.stop_recording_operations(false /*write_tapes_to_file*/);
+
+      std::cout << "Recorded data..." << std::endl;
+      std::cout << "independent variable values: " << std::flush;
+      ad_helper.print_values(std::cout);
+      std::cout << "t_ad: " << t_ad << std::endl;
+      std::cout << "psi: " << psi << std::endl;
+      std::cout << std::endl;
+    }
+  else
+    {
+      Assert(is_recording == true, ExcInternalError());
+    }
+
+  // Do some work :-)
+  // Set a new evaluation point
+  if (AD::ADNumberTraits<ADNumberType>::is_taped == true)
+    {
+      std::cout
+        << "Using tape with different values for independent variables..."
+        << std::endl;
+      ad_helper.activate_recorded_tape(tape_no);
+      t *= 1.15;
+      ad_helper.set_independent_variable(t, t_dof);
+
+      std::cout << "independent variable values: " << std::flush;
+      ad_helper.print_values(std::cout);
+    }
+
+  // Compute the function value, gradient and hessian for the new evaluation
+  // point
+  psi = ad_helper.compute_value();
+  ad_helper.compute_gradient(Dpsi);
+  if (AD::ADNumberTraits<ADNumberType>::n_supported_derivative_levels >= 2)
+    {
+      ad_helper.compute_hessian(D2psi);
+    }
+
+  // Output the full stored function, gradient vector and hessian matrix
+  std::cout << "psi: " << psi << std::endl;
+  std::cout << "Dpsi: \n";
+  Dpsi.print(std::cout);
+  if (AD::ADNumberTraits<ADNumberType>::n_supported_derivative_levels >= 2)
+    {
+      std::cout << "D2psi: \n";
+      D2psi.print_formatted(std::cout, 3, true, 0, "0.0");
+    }
+
+  // Extract components of the solution
+  const Tensor<2, dim, ScalarNumberType> dpsi_dt =
+    ad_helper.extract_gradient_component(Dpsi, t_dof);
+
+  // Verify the result
+  typedef FunctionsTestTensor<dim, ScalarNumberType> func;
+  static const ScalarNumberType                      tol =
+    1e5 * std::numeric_limits<ScalarNumberType>::epsilon();
+  std::cout << "psi:              " << psi << std::endl;
+  std::cout << "func::psi(t):     " << func::psi(t) << std::endl;
+  Assert(std::abs(psi - func::psi(t)) < tol,
+         ExcMessage("No match for function value."));
+  std::cout << "dpsi_dt:              " << dpsi_dt << std::endl;
+  std::cout << "func::dpsi_dt(t):     " << func::dpsi_dt(t) << std::endl;
+  Assert(std::abs((dpsi_dt - func::dpsi_dt(t)).norm()) < tol,
+         ExcMessage("No match for first derivative."));
+  if (AD::ADNumberTraits<ADNumberType>::n_supported_derivative_levels >= 2)
+    {
+      const Tensor<4, dim, ScalarNumberType> d2psi_dt_dt =
+        ad_helper.extract_hessian_component(D2psi, t_dof, t_dof);
+      std::cout << "d2psi_dt_dt:          " << d2psi_dt_dt << std::endl;
+      std::cout << "func::d2psi_dt_dt(t): " << func::d2psi_dt_dt(t)
+                << std::endl;
+      Assert(std::abs((d2psi_dt_dt - func::d2psi_dt_dt(t)).norm()) < tol,
+             ExcMessage("No match for second derivative."));
+    }
+
+  std::cout << std::endl << std::endl;
+}

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