From 78958438c586bc6817abb7a70fd0fd133660c5d1 Mon Sep 17 00:00:00 2001 From: Maximilian Bergbauer Date: Thu, 5 Aug 2021 17:40:53 +0200 Subject: [PATCH] Integrate hessians on cells --- .../changes/minor/20210728MaximilianBergbauer | 4 +- .../deal.II/matrix_free/evaluation_kernels.h | 173 ++++- .../matrix_vector_hessians_cells.cc | 322 ++++++++ .../matrix_vector_hessians_cells.output | 733 ++++++++++++++++++ 4 files changed, 1210 insertions(+), 22 deletions(-) create mode 100644 tests/matrix_free/matrix_vector_hessians_cells.cc create mode 100644 tests/matrix_free/matrix_vector_hessians_cells.output diff --git a/doc/news/changes/minor/20210728MaximilianBergbauer b/doc/news/changes/minor/20210728MaximilianBergbauer index 91300d90cc..1ccf8c26f8 100644 --- a/doc/news/changes/minor/20210728MaximilianBergbauer +++ b/doc/news/changes/minor/20210728MaximilianBergbauer @@ -1,4 +1,4 @@ -Added: Evaluation and integration of hessians on faces is now available - in the matrix free framework. +Added: Evaluation and integration of hessians both on cells and on faces is now +available in the matrix-free framework.
(Maximilian Bergbauer, 2021/07/28) diff --git a/include/deal.II/matrix_free/evaluation_kernels.h b/include/deal.II/matrix_free/evaluation_kernels.h index 403539205e..ab10f44ba8 100644 --- a/include/deal.II/matrix_free/evaluation_kernels.h +++ b/include/deal.II/matrix_free/evaluation_kernels.h @@ -468,11 +468,6 @@ namespace internal Number * scratch_data, const bool add_into_values_array) { - // TODO: implement hessians - (void)hessians_quad; - AssertThrow(!(integration_flag & EvaluationFlags::hessians), - ExcNotImplemented()); - const EvaluatorVariant variant = EvaluatorSelector 4)>::variant; using Eval = EvaluatorTensorProduct(gradients_quad, values_dofs); } + if (integration_flag & EvaluationFlags::hessians) + { + if (integration_flag & EvaluationFlags::values || + integration_flag & EvaluationFlags::gradients || + add_into_values_array == true) + eval.template hessians<0, false, true>(hessians_quad, + values_dofs); + else + eval.template hessians<0, false, false>(hessians_quad, + values_dofs); + } // advance to the next component in 1D array values_dofs += dofs_per_comp; values_quad += n_q_points; gradients_quad += n_q_points; + hessians_quad += n_q_points; } break; @@ -583,11 +590,36 @@ namespace internal eval.template values<1, false, false>(gradients_quad, temp1); eval.template gradients<0, false, true>(temp1, values_dofs); } + if (integration_flag & EvaluationFlags::hessians) + { + // grad xx + eval.template values<1, false, false>(hessians_quad, temp1); + + if (integration_flag & EvaluationFlags::values || + integration_flag & EvaluationFlags::gradients || + add_into_values_array == true) + eval.template hessians<0, false, true>(temp1, values_dofs); + else + eval.template hessians<0, false, false>(temp1, values_dofs); + + // grad yy + eval.template hessians<1, false, false>(hessians_quad + + n_q_points, + temp1); + eval.template values<0, false, true>(temp1, values_dofs); + + // grad xy + eval.template gradients<1, false, false>(hessians_quad + + 2 * n_q_points, + temp1); + eval.template gradients<0, false, true>(temp1, values_dofs); + } // advance to the next component in 1D array values_dofs += dofs_per_comp; values_quad += n_q_points; gradients_quad += 2 * n_q_points; + hessians_quad += 3 * n_q_points; } break; @@ -624,11 +656,60 @@ namespace internal eval.template values<1, false, false>(temp1, temp2); eval.template gradients<0, false, true>(temp2, values_dofs); } + if (integration_flag & EvaluationFlags::hessians) + { + // grad xx + eval.template values<2, false, false>(hessians_quad, temp1); + eval.template values<1, false, false>(temp1, temp2); + + if (integration_flag & EvaluationFlags::values || + integration_flag & EvaluationFlags::gradients || + add_into_values_array == true) + eval.template hessians<0, false, true>(temp2, values_dofs); + else + eval.template hessians<0, false, false>(temp2, values_dofs); + + // grad yy + eval.template values<2, false, false>(hessians_quad + + n_q_points, + temp1); + eval.template hessians<1, false, false>(temp1, temp2); + eval.template values<0, false, true>(temp2, values_dofs); + + // grad zz + eval.template hessians<2, false, false>(hessians_quad + + 2 * n_q_points, + temp1); + eval.template values<1, false, false>(temp1, temp2); + eval.template values<0, false, true>(temp2, values_dofs); + + // grad xy + eval.template values<2, false, false>(hessians_quad + + 3 * n_q_points, + temp1); + eval.template gradients<1, false, false>(temp1, temp2); + eval.template gradients<0, false, true>(temp2, values_dofs); + + // grad xz + eval.template gradients<2, false, false>(hessians_quad + + 4 * n_q_points, + temp1); + eval.template values<1, false, false>(temp1, temp2); + eval.template gradients<0, false, true>(temp2, values_dofs); + + // grad yz + eval.template gradients<2, false, false>(hessians_quad + + 5 * n_q_points, + temp1); + eval.template gradients<1, false, false>(temp1, temp2); + eval.template values<0, false, true>(temp2, values_dofs); + } // advance to the next component in 1D array values_dofs += dofs_per_comp; values_quad += n_q_points; gradients_quad += 3 * n_q_points; + hessians_quad += 6 * n_q_points; } break; @@ -1374,15 +1455,9 @@ namespace internal Number * values_quad, Number * gradients_quad, Number * hessians_quad, - Number *, - const bool add_into_values_array) + Number * scratch_data, + const bool add_into_values_array) { - // TODO: implement hessians - (void)hessians_quad; - AssertThrow(!(integration_flag & EvaluationFlags::hessians), - ExcNotImplemented()); - - AssertDimension( shape_info.data.front().shape_gradients_collocation_eo.size(), (fe_degree + 2) / 2 * (fe_degree + 1)); @@ -1396,6 +1471,7 @@ namespace internal shape_info.data.front().shape_gradients_collocation_eo, shape_info.data.front().shape_hessians_collocation_eo); constexpr unsigned int n_q_points = Utilities::pow(fe_degree + 1, dim); + constexpr unsigned int hdim = (dim * (dim + 1)) / 2; for (unsigned int c = 0; c < n_components; c++) { @@ -1426,6 +1502,60 @@ namespace internal 2 * n_q_points, values_dofs); } + if (integration_flag & EvaluationFlags::hessians) + { + // diagonal + // grad xx + if (integration_flag & EvaluationFlags::values || + integration_flag & EvaluationFlags::gradients || + add_into_values_array == true) + eval.template hessians<0, false, true>(hessians_quad, + values_dofs); + else + eval.template hessians<0, false, false>(hessians_quad, + values_dofs); + // grad yy + if (dim > 1) + eval.template hessians<1, false, true>(hessians_quad + n_q_points, + values_dofs); + // grad zz + if (dim > 2) + eval.template hessians<2, false, true>(hessians_quad + + 2 * n_q_points, + values_dofs); + // off-diagonal + if (dim == 2) + { + // grad xy + eval.template gradients<0, false, false>(hessians_quad + + 2 * n_q_points, + scratch_data); + eval.template gradients<1, false, true>(scratch_data, + values_dofs); + } + if (dim == 3) + { + // grad xy + eval.template gradients<0, false, false>(hessians_quad + + 3 * n_q_points, + scratch_data); + eval.template gradients<1, false, true>(scratch_data, + values_dofs); + // grad xz + eval.template gradients<0, false, false>(hessians_quad + + 4 * n_q_points, + scratch_data); + eval.template gradients<2, false, true>(scratch_data, + values_dofs); + // grad yz + eval.template gradients<1, false, false>(hessians_quad + + 5 * n_q_points, + scratch_data); + eval.template gradients<2, false, true>(scratch_data, + values_dofs); + } + hessians_quad += hdim * n_q_points; + } gradients_quad += dim * n_q_points; values_quad += n_q_points; values_dofs += n_q_points; @@ -1539,11 +1669,11 @@ namespace internal Number>::integrate(const unsigned int n_components, const EvaluationFlags::EvaluationFlags integration_flag, const MatrixFreeFunctions::ShapeInfo &shape_info, - Number *values_dofs, - Number *values_quad, - Number *gradients_quad, - Number *hessians_quad, - Number *, + Number * values_dofs, + Number * values_quad, + Number * gradients_quad, + Number * hessians_quad, + Number * scratch_data, const bool add_into_values_array) { Assert(n_q_points_1d > fe_degree, @@ -1555,11 +1685,13 @@ namespace internal shape_info.data.front().shape_gradients_collocation_eo.size(), (n_q_points_1d + 1) / 2 * n_q_points_1d); constexpr unsigned int n_q_points = Utilities::pow(n_q_points_1d, dim); + constexpr unsigned int hdim = (dim * (dim + 1)) / 2; for (unsigned int c = 0; c < n_components; c++) { // apply derivatives in collocation space - if (integration_flag & EvaluationFlags::gradients) + if (integration_flag & + (EvaluationFlags::gradients | EvaluationFlags::hessians)) FEEvaluationImplCollocation:: integrate(1, integration_flag & (EvaluationFlags::gradients | @@ -1569,7 +1701,7 @@ namespace internal nullptr, gradients_quad, hessians_quad, - nullptr, + scratch_data, /*add_into_values_array=*/integration_flag & EvaluationFlags::values); @@ -1586,6 +1718,7 @@ namespace internal add_into_values_array, values_quad, values_dofs); + hessians_quad += hdim * n_q_points; gradients_quad += dim * n_q_points; values_quad += n_q_points; values_dofs += shape_info.dofs_per_component_on_cell; diff --git a/tests/matrix_free/matrix_vector_hessians_cells.cc b/tests/matrix_free/matrix_vector_hessians_cells.cc new file mode 100644 index 0000000000..11b803ecf8 --- /dev/null +++ b/tests/matrix_free/matrix_vector_hessians_cells.cc @@ -0,0 +1,322 @@ +// --------------------------------------------------------------------- +// +// Copyright (C) 2013 - 2021 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.md at +// the top level directory of deal.II. +// +// --------------------------------------------------------------------- + + +#include + +#include + +#include + +#include +#include +#include +#include + +#include + +#include +#include + +#include + +#include + +#include "../tests.h" + +// Tests if applying a matrix vector product with a matrix containing hessians +// on faces produces the same result with FEEvaluation and FEValues. +// This is checked for different combinations of EvaluationFlags, FE types and +// polynomial degrees. + +template +void +test_hessians(const dealii::FE_Poly & fe, + const dealii::Quadrature & quad, + const dealii::EvaluationFlags::EvaluationFlags evaluation_flags) +{ + using namespace dealii; + using VectorizedArrayType = VectorizedArray; + + Triangulation tria; + GridGenerator::hyper_cube(tria); + tria.refine_global(3); + + // TODO: make Hessians work for adaptive refinement + + // unsigned int counter = 0; + // for (auto cell = tria.begin_active(); cell != tria.end(); ++cell, + // ++counter) + // if (cell->is_locally_owned() && counter % 3 == 0) + // cell->set_refine_flag(); + // tria.execute_coarsening_and_refinement(); + + DoFHandler dof_handler(tria); + dof_handler.distribute_dofs(fe); + MappingQGeneric mapping(1); + + AffineConstraints constraints; + VectorTools::interpolate_boundary_values( + mapping, dof_handler, 0, Functions::ZeroFunction(), constraints); + constraints.close(); + + // FEEvaluation + typename MatrixFree::AdditionalData + additional_data; + additional_data.mapping_update_flags_inner_faces = + update_values | update_gradients | update_hessians; + + MatrixFree matrix_free; + matrix_free.reinit(mapping, dof_handler, constraints, quad, additional_data); + + if (Utilities::MPI::this_mpi_process(MPI_COMM_WORLD) == 0) + deallog << "Working with " << fe.get_name() << " and " + << dof_handler.n_dofs() << " dofs" << std::endl; + + LinearAlgebra::distributed::Vector src, dst, dst2; + matrix_free.initialize_dof_vector(src); + for (auto &v : src) + v = random_value(); + + matrix_free.initialize_dof_vector(dst); + matrix_free.initialize_dof_vector(dst2); + + // Setup FEFaceValues + FEValues fe_values(mapping, + fe, + quad, + update_values | update_gradients | update_hessians | + update_JxW_values); + Vector solution_values_local(fe.dofs_per_cell); + std::vector> solution_hessians(quad.size()); + std::vector> solution_gradients(quad.size()); + std::vector solution_values(quad.size()); + std::vector dof_indices(fe.dofs_per_cell); + src.update_ghost_values(); + dst2 = 0; + + + + matrix_free.template loop, + LinearAlgebra::distributed::Vector>( + [&]( + const auto &matrix_free, auto &dst, const auto &src, const auto &range) { + FEEvaluation fe_eval( + matrix_free); + for (unsigned int cell = range.first; cell < range.second; ++cell) + { + // FEEvaluation + fe_eval.reinit(cell); + fe_eval.gather_evaluate(src, evaluation_flags); + for (unsigned int q = 0; q < fe_eval.n_q_points; ++q) + { + if (evaluation_flags & EvaluationFlags::hessians) + fe_eval.submit_hessian(fe_eval.get_hessian(q), q); + if (evaluation_flags & EvaluationFlags::gradients) + fe_eval.submit_gradient(fe_eval.get_gradient(q), q); + if (evaluation_flags & EvaluationFlags::values) + fe_eval.submit_value(fe_eval.get_value(q), q); + } + fe_eval.integrate(evaluation_flags); + fe_eval.distribute_local_to_global(dst); + + // FEValues + for (unsigned int v = 0; + v < matrix_free.n_active_entries_per_cell_batch(cell); + ++v) + { + const auto cell_iterator = matrix_free.get_cell_iterator(cell, v); + + fe_values.reinit(cell_iterator); + + cell_iterator->get_dof_indices(dof_indices); + constraints.get_dof_values(src, + dof_indices.begin(), + solution_values_local.begin(), + solution_values_local.end()); + + for (unsigned int q = 0; q < quad.size(); ++q) + { + double values = 0.; + Tensor<1, dim> gradients; + Tensor<2, dim> hessians; + for (unsigned int i = 0; i < fe.dofs_per_cell; ++i) + { + if (evaluation_flags & EvaluationFlags::hessians) + hessians += solution_values_local(i) * + fe_values.shape_hessian(i, q); + if (evaluation_flags & EvaluationFlags::gradients) + gradients += + solution_values_local(i) * fe_values.shape_grad(i, q); + if (evaluation_flags & EvaluationFlags::values) + values += solution_values_local(i) * + fe_values.shape_value(i, q); + } + solution_hessians[q] = hessians * fe_values.JxW(q); + solution_gradients[q] = gradients * fe_values.JxW(q); + solution_values[q] = values * fe_values.JxW(q); + } + for (unsigned int i = 0; i < fe.dofs_per_cell; ++i) + { + double sum_hessians = 0.; + double sum_gradients = 0.; + double sum_values = 0.; + for (unsigned int q = 0; q < quad.size(); ++q) + { + if (evaluation_flags & EvaluationFlags::hessians) + sum_hessians += double_contract<0, 0, 1, 1>( + solution_hessians[q], fe_values.shape_hessian(i, q)); + if (evaluation_flags & EvaluationFlags::gradients) + sum_gradients += + solution_gradients[q] * fe_values.shape_grad(i, q); + if (evaluation_flags & EvaluationFlags::values) + sum_values += + solution_values[q] * fe_values.shape_value(i, q); + } + solution_values_local(i) = + sum_hessians + sum_gradients + sum_values; + } + constraints.distribute_local_to_global(solution_values_local, + dof_indices, + dst2); + } + } + }, + [&]( + const auto &matrix_free, auto &dst, const auto &src, const auto &range) { + (void)matrix_free; + (void)dst; + (void)src; + (void)range; + }, + [&]( + const auto &matrix_free, auto &dst, const auto &src, const auto &range) { + (void)matrix_free; + (void)dst; + (void)src; + (void)range; + }, + dst, + src, + true); + + const unsigned int end_of_print_dst = + dof_handler.n_dofs() > 9 ? 9 : dof_handler.n_dofs(); + + deallog << "dst FEE: "; + for (unsigned int i = 0; i < end_of_print_dst; ++i) + deallog << dst[i] << " "; + deallog << std::endl; + + deallog << "dst FEV: "; + for (unsigned int i = 0; i < end_of_print_dst; ++i) + deallog << dst2[i] << " "; + deallog << std::endl; + + // compare solutions of matrix vector product + { + dst2 -= dst; + + double error = 0.; + if (dst.l2_norm() > 0) + error = dst2.l2_norm() / dst.l2_norm(); + else + error = dst2.l2_norm(); + + if (Utilities::MPI::this_mpi_process(MPI_COMM_WORLD) == 0) + deallog << "FEValues verification: " << error << std::endl << std::endl; + } +} + +void +test_qiterated(dealii::EvaluationFlags::EvaluationFlags evaluation_flags) +{ + using namespace dealii; + deallog << "test_qiterated" << std::endl; + for (unsigned int i = 1; i < 4; ++i) + { + QIterated<1> quad_1(QGauss<1>(i + 1), 2); + QIterated<2> quad_2(QGauss<1>(i + 1), 2); + QIterated<3> quad_3(QGauss<1>(i + 1), 2); + test_hessians<1>(FE_Q<1>(i), quad_1, evaluation_flags); + test_hessians<2>(FE_Q<2>(i), quad_2, evaluation_flags); + test_hessians<3>(FE_Q<3>(i), quad_3, evaluation_flags); + test_hessians<1>(FE_DGQ<1>(i), quad_1, evaluation_flags); + test_hessians<2>(FE_DGQ<2>(i), quad_2, evaluation_flags); + test_hessians<3>(FE_DGQ<3>(i), quad_3, evaluation_flags); + } +} + +void +test_qgauss(dealii::EvaluationFlags::EvaluationFlags evaluation_flags) +{ + using namespace dealii; + deallog << "test_qgauss" << std::endl; + for (unsigned int i = 1; i < 4; ++i) + { + QGauss<1> quad_1(i + 1); + QGauss<2> quad_2(i + 1); + QGauss<3> quad_3(i + 1); + test_hessians<1>(FE_Q<1>(i), quad_1, evaluation_flags); + test_hessians<2>(FE_Q<2>(i), quad_2, evaluation_flags); + test_hessians<3>(FE_Q<3>(i), quad_3, evaluation_flags); + test_hessians<1>(FE_DGQ<1>(i), quad_1, evaluation_flags); + test_hessians<2>(FE_DGQ<2>(i), quad_2, evaluation_flags); + test_hessians<3>(FE_DGQ<3>(i), quad_3, evaluation_flags); + } +} + +int +main(int argc, char **argv) +{ + using namespace dealii; + Utilities::MPI::MPI_InitFinalize mpi(argc, argv, 1); + + initlog(); + deallog << std::setprecision(10); + + { + EvaluationFlags::EvaluationFlags evaluation_flags = + EvaluationFlags::hessians; + deallog << "test_hessians_only" << std::endl; + test_qgauss(evaluation_flags); + test_qiterated(evaluation_flags); + } + + { + EvaluationFlags::EvaluationFlags evaluation_flags = + EvaluationFlags::values | EvaluationFlags::hessians; + deallog << "test_hessians_with_values" << std::endl; + test_qgauss(evaluation_flags); + test_qiterated(evaluation_flags); + } + + { + EvaluationFlags::EvaluationFlags evaluation_flags = + EvaluationFlags::gradients | EvaluationFlags::hessians; + deallog << "test_hessians_with_gradients" << std::endl; + test_qgauss(evaluation_flags); + test_qiterated(evaluation_flags); + } + + { + EvaluationFlags::EvaluationFlags evaluation_flags = + EvaluationFlags::values | EvaluationFlags::gradients | + EvaluationFlags::hessians; + deallog << "test_hessians_with_gradients_and_values" << std::endl; + test_qgauss(evaluation_flags); + test_qiterated(evaluation_flags); + } +} diff --git a/tests/matrix_free/matrix_vector_hessians_cells.output b/tests/matrix_free/matrix_vector_hessians_cells.output new file mode 100644 index 0000000000..fe668b85a5 --- /dev/null +++ b/tests/matrix_free/matrix_vector_hessians_cells.output @@ -0,0 +1,733 @@ + +DEAL::test_hessians_only +DEAL::test_qgauss +DEAL::Working with FE_Q<1>(1) and 9 dofs +DEAL::dst FEE: 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000 +DEAL::dst FEV: 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000 +DEAL::FEValues verification: 0.000000000 +DEAL:: +DEAL::Working with FE_Q<2>(1) and 81 dofs +DEAL::dst FEE: 0.000000000 0.000000000 0.000000000 -127.9293550 0.000000000 154.0814910 0.000000000 84.32838283 133.7460355 +DEAL::dst FEV: 0.000000000 0.000000000 0.000000000 -127.9293550 0.000000000 154.0814910 0.000000000 84.32838283 133.7460355 +DEAL::FEValues verification: 0.000000000 +DEAL:: +DEAL::Working with FE_Q<3>(1) and 729 dofs +DEAL::dst FEE: 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000 63.80267270 0.000000000 +DEAL::dst FEV: 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000 63.80267270 0.000000000 +DEAL::FEValues verification: 0.000000000 +DEAL:: +DEAL::Working with FE_DGQ<1>(1) and 16 dofs +DEAL::dst FEE: 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000 +DEAL::dst FEV: 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000 +DEAL::FEValues verification: 0.000000000 +DEAL:: +DEAL::Working with FE_DGQ<2>(1) and 256 dofs +DEAL::dst FEE: 11.67296643 -11.67296643 -11.67296643 11.67296643 45.66021385 -45.66021385 -45.66021385 45.66021385 -73.78028658 +DEAL::dst FEV: 11.67296643 -11.67296643 -11.67296643 11.67296643 45.66021385 -45.66021385 -45.66021385 45.66021385 -73.78028658 +DEAL::FEValues verification: 0.000000000 +DEAL:: +DEAL::Working with FE_DGQ<3>(1) and 4096 dofs +DEAL::dst FEE: -2.614206237 5.261104676 -2.640250729 -0.006647709760 1.422705614 -4.069604052 3.831751352 -1.184852914 -5.754533413 +DEAL::dst FEV: -2.614206237 5.261104676 -2.640250729 -0.006647709760 1.422705614 -4.069604052 3.831751352 -1.184852914 -5.754533413 +DEAL::FEValues verification: 0.000000000 +DEAL:: +DEAL::Working with FE_Q<1>(2) and 17 dofs +DEAL::dst FEE: 0.000000000 3701.751955 2027.739533 10913.28312 -9431.243443 5858.851214 -12395.32281 6277.767337 677.6203769 +DEAL::dst FEV: 0.000000000 3701.751955 2027.739533 10913.28312 -9431.243443 5858.851214 -12395.32281 6277.767337 677.6203769 +DEAL::FEValues verification: 0.000000000 +DEAL:: +DEAL::Working with FE_Q<2>(2) and 289 dofs +DEAL::dst FEE: 0.000000000 0.000000000 0.000000000 -818.7652426 0.000000000 -1130.630042 0.000000000 2586.249308 804.8668216 +DEAL::dst FEV: 0.000000000 0.000000000 0.000000000 -818.7652426 0.000000000 -1130.630042 0.000000000 2586.249308 804.8668216 +DEAL::FEValues verification: 0.000000000 +DEAL:: +DEAL::Working with FE_Q<3>(2) and 4913 dofs +DEAL::dst FEE: 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000 194.3540325 0.000000000 +DEAL::dst FEV: 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000 194.3540325 0.000000000 +DEAL::FEValues verification: 0.000000000 +DEAL:: +DEAL::Working with FE_DGQ<1>(2) and 24 dofs +DEAL::dst FEE: -7171.305568 14342.61114 -7171.305568 -8136.233040 16272.46608 -8136.233040 -8257.416119 16514.83224 -8257.416119 +DEAL::dst FEV: -7171.305568 14342.61114 -7171.305568 -8136.233040 16272.46608 -8136.233040 -8257.416119 16514.83224 -8257.416119 +DEAL::FEValues verification: 0.000000000 +DEAL:: +DEAL::Working with FE_DGQ<2>(2) and 576 dofs +DEAL::dst FEE: -246.9400409 483.9173459 -420.5272225 -1221.020373 2364.748221 -776.6280129 606.3810345 -1125.506808 335.5758556 +DEAL::dst FEV: -246.9400409 483.9173459 -420.5272225 -1221.020373 2364.748221 -776.6280129 606.3810345 -1125.506808 335.5758556 +DEAL::FEValues verification: 0.000000000 +DEAL:: +DEAL::Working with 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