From 0e5db311fc1a577c610532055d731762e14a955a Mon Sep 17 00:00:00 2001
From: Denis Davydov <davydden@gmail.com>
Date: Fri, 17 Feb 2017 16:09:14 +0100
Subject: [PATCH] make GridTools::laplace_transform() internally solve for
 displacement by default

while there, adjust grid_transform_02 to produce a better
looking mesh.
---
 doc/news/changes/minor/20170217DenisDavydov |    6 +
 include/deal.II/grid/grid_tools.h           |   36 +-
 source/grid/grid_tools.cc                   |   17 +-
 source/grid/grid_tools.inst.in              |    3 +-
 tests/grid/grid_transform_02.cc             |    8 +-
 tests/grid/grid_transform_02.output         | 1282 ++++---------------
 tests/grid/grid_transform_03.cc             |  121 ++
 tests/grid/grid_transform_03.output         |  269 ++++
 tests/grid/grid_transform_3d.cc             |    2 +-
 tests/grid/grid_transform_coefficient.cc    |    2 +-
 10 files changed, 704 insertions(+), 1042 deletions(-)
 create mode 100644 doc/news/changes/minor/20170217DenisDavydov
 create mode 100644 tests/grid/grid_transform_03.cc
 create mode 100644 tests/grid/grid_transform_03.output

diff --git a/doc/news/changes/minor/20170217DenisDavydov b/doc/news/changes/minor/20170217DenisDavydov
new file mode 100644
index 0000000000..712b07cf23
--- /dev/null
+++ b/doc/news/changes/minor/20170217DenisDavydov
@@ -0,0 +1,6 @@
+Fixed: Fixed a bug in GridTools::laplace_transform() which was previously solving
+for the final position field instead of displacement. The two formulations are
+not equivalent when a non-constant coefficient is provided, which is commonly
+used with displacement formulation.
+<br>
+(Denis Davydov, 2017/02/17)
diff --git a/include/deal.II/grid/grid_tools.h b/include/deal.II/grid/grid_tools.h
index 6b4ed1ab68..a0feb09058 100644
--- a/include/deal.II/grid/grid_tools.h
+++ b/include/deal.II/grid/grid_tools.h
@@ -295,12 +295,27 @@ namespace GridTools
    * typically, each of the vertices at the boundary of the triangulation is
    * mapped to the corresponding points in the @p new_points map.
    *
-   * The way this function works is that it solves a Laplace equation for each
+   * The unknown displacement field $u_d(\mathbf x)$ in direction $d$ is obtained from
+   * the minimization problem
+   * \f[
+   *   \min\, \int
+   *   \frac{1}{2}
+   *   c(\mathbf x)
+   *   \mathbf \nabla u_d(\mathbf x) \cdot
+   *   \mathbf \nabla u_d(\mathbf x)
+   *   \,\rm d x
+   * \f]
+   * subject to prescribed constraints. The minimizer is obtained by solving the Laplace equation
    * of the dim components of a displacement field that maps the current
-   * domain into one described by @p new_points . The @p new_points array
-   * therefore represents the boundary values of this displacement field. The
-   * function then evaluates this displacement field at each vertex in the
-   * interior and uses it to place the mapped vertex where the displacement
+   * domain into one described by @p new_points . Linear finite elements with
+   * four Gaussian quadrature points in each direction are used. The difference
+   * between the vertex positions specified in @p new_points and their current
+   * value in @p tria therefore represents the prescribed values of this
+   * displacement field at the boundary of the domain, or more precisely at all
+   * of those locations for which @p new_points provides values (which may be
+   * at part of the boundary, or even in the interior of the domain). The
+   * function then evaluates this displacement field at each unconstrained
+   * vertex and uses it to place the mapped vertex where the displacement
    * field locates it. Because the solution of the Laplace equation is smooth,
    * this guarantees a smooth mapping from the old domain to the new one.
    *
@@ -323,12 +338,21 @@ namespace GridTools
    * Should this function be provided, sensible results can only be expected
    * if all coefficients are positive.
    *
+   * @param[in] solve_for_absolute_positions If set to <code>true</code>, the
+   * minimization problem is formulated with respect to the final vertex positions
+   * as opposed to their displacement. The two formulations are equivalent for
+   * the homogeneous problem (default value of @p coefficient), but they
+   * result in very different mesh motion otherwise. Since in most cases one will
+   * be using a non-constant coefficient in displacement formulation, the default
+   * value of this parameter is <code>false</code>.
+   *
    * @note This function is not currently implemented for the 1d case.
    */
   template <int dim>
   void laplace_transform (const std::map<unsigned int,Point<dim> > &new_points,
                           Triangulation<dim> &tria,
-                          const Function<dim,double> *coefficient = 0);
+                          const Function<dim,double> *coefficient = 0,
+                          const bool solve_for_absolute_positions = false);
 
   /**
    * Return a std::map with all vertices of faces located in the boundary
diff --git a/source/grid/grid_tools.cc b/source/grid/grid_tools.cc
index 249cc752f7..0720ae3360 100644
--- a/source/grid/grid_tools.cc
+++ b/source/grid/grid_tools.cc
@@ -693,7 +693,8 @@ namespace GridTools
   template <>
   void laplace_transform (const std::map<unsigned int,Point<1> > &,
                           Triangulation<1> &,
-                          const Function<1> *)
+                          const Function<1> *,
+                          const bool )
   {
     Assert(false, ExcNotImplemented());
   }
@@ -704,7 +705,8 @@ namespace GridTools
   void
   laplace_transform (const std::map<unsigned int,Point<dim> > &new_points,
                      Triangulation<dim> &triangulation,
-                     const Function<dim> *coefficient)
+                     const Function<dim> *coefficient,
+                     const bool solve_for_absolute_positions)
   {
     // first provide everything that is needed for solving a Laplace
     // equation.
@@ -743,6 +745,7 @@ namespace GridTools
              vertex_no<GeometryInfo<dim>::vertices_per_cell; ++vertex_no)
           {
             const unsigned int vertex_index=cell->vertex_index(vertex_no);
+            const Point<dim> &vertex_point=cell->vertex(vertex_no);
 
             const typename std::map<unsigned int,Point<dim> >::const_iterator map_iter
               = new_points.find(vertex_index);
@@ -751,7 +754,10 @@ namespace GridTools
               for (unsigned int i=0; i<dim; ++i)
                 fixed_dofs[i].insert(std::pair<types::global_dof_index,double>
                                      (cell->vertex_dof_index(vertex_no, 0),
-                                      map_iter->second(i)));
+                                      (solve_for_absolute_positions ?
+                                       map_iter->second(i) :
+                                       map_iter->second(i) - vertex_point[i])
+                                     ));
           }
       }
 
@@ -779,7 +785,10 @@ namespace GridTools
 
             const types::global_dof_index dof_index = cell->vertex_dof_index(vertex_no, 0);
             for (unsigned int i=0; i<dim; ++i)
-              v(i) = us[i](dof_index);
+              if (solve_for_absolute_positions)
+                v(i) = us[i](dof_index);
+              else
+                v(i) += us[i](dof_index);
 
             vertex_touched[cell->vertex_index(vertex_no)] = true;
           }
diff --git a/source/grid/grid_tools.inst.in b/source/grid/grid_tools.inst.in
index c01c63d55f..9d557cd056 100644
--- a/source/grid/grid_tools.inst.in
+++ b/source/grid/grid_tools.inst.in
@@ -213,7 +213,8 @@ for (deal_II_dimension : DIMENSIONS; deal_II_space_dimension :  SPACE_DIMENSIONS
     void
     laplace_transform (const std::map<unsigned int,Point<deal_II_dimension> > &new_points,
                        Triangulation<deal_II_dimension> &triangulation,
-                       const Function<deal_II_dimension> *coefficient);
+                       const Function<deal_II_dimension> *coefficient,
+                       const bool);
 
 #  endif
 
diff --git a/tests/grid/grid_transform_02.cc b/tests/grid/grid_transform_02.cc
index 585047ddb7..27237e0660 100644
--- a/tests/grid/grid_transform_02.cc
+++ b/tests/grid/grid_transform_02.cc
@@ -43,7 +43,7 @@ int main ()
 
   Triangulation< dim >  tria;
   std::map< unsigned int, Point< dim > > new_points;
-  const unsigned int N = 16;
+  const unsigned int N = 8;
   GridGenerator::subdivided_hyper_cube (tria, N, -5, 5);
 
   // find the vertex at the origin
@@ -64,9 +64,9 @@ int main ()
           best_dist   = dist;
         }
     }
-  // move the point at the origin by 2 units to the right
+  // move the point at the origin by 1 unit to the right
   new_points[best_vertex] = Point<dim>();
-  new_points[best_vertex][0] += 2;
+  new_points[best_vertex][0] += 1.;
 
   // now pin all of the points on the boundary
   cell = tria.begin_active();
@@ -83,7 +83,7 @@ int main ()
             }
 
   // then compute new point locations and output the result
-  GridTools::laplace_transform  (new_points,tria);
+  GridTools::laplace_transform<dim>  (new_points,tria, NULL, true);
   std::ofstream out ("output");
   GridOut grid_out;
   grid_out.write_eps (tria, out);
diff --git a/tests/grid/grid_transform_02.output b/tests/grid/grid_transform_02.output
index a792b88e24..b35bbeabbd 100644
--- a/tests/grid/grid_transform_02.output
+++ b/tests/grid/grid_transform_02.output
@@ -2,7 +2,7 @@
 %%Title: deal.II Output
 %%Creator: the deal.II library
 
-%%BoundingBox: 0 0 301 301
+%%BoundingBox: 0 0 301 300
 /m {moveto} bind def
 /x {lineto stroke} bind def
 /b {0 0 0 setrgbcolor} def
@@ -10,1028 +10,260 @@
 %%EndProlog
 
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diff --git a/tests/grid/grid_transform_03.cc b/tests/grid/grid_transform_03.cc
new file mode 100644
index 0000000000..1ec3480a31
--- /dev/null
+++ b/tests/grid/grid_transform_03.cc
@@ -0,0 +1,121 @@
+// ---------------------------------------------------------------------
+//
+// Copyright (C) 2017 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.
+//
+// ---------------------------------------------------------------------
+
+
+// Similar to grid_transform_02.cc but use coefficient function and solve
+// for displacement field.
+
+
+#include "../tests.h"
+#include <deal.II/base/logstream.h>
+#include <deal.II/grid/tria.h>
+#include <deal.II/grid/tria_accessor.h>
+#include <deal.II/grid/tria_boundary_lib.h>
+#include <deal.II/grid/grid_generator.h>
+#include <deal.II/grid/grid_out.h>
+#include <deal.II/grid/grid_tools.h>
+#include <deal.II/fe/mapping_q.h>
+
+#include <fstream>
+#include <iomanip>
+
+template <int dim>
+class L2_inverse : public Function<dim>
+{
+public:
+  L2_inverse (const std::vector<Point<dim> > &distance_source)
+    :
+    Function<dim> (),
+    distance_source (distance_source)
+  {
+    Assert (distance_source.size()>0,
+            ExcNotImplemented());
+  }
+
+  virtual double value (const dealii::Point<dim> &p,
+                        const unsigned int component = 0) const
+  {
+    double l2_inverse = std::numeric_limits<double>::max();
+
+    for (unsigned int d = 0; d<distance_source.size(); d++)
+      l2_inverse = std::min ((p - distance_source[d]).norm_square (),
+                             l2_inverse);
+
+    l2_inverse = std::max (l2_inverse, 1.e-5);
+
+    return 1.0/l2_inverse;
+  }
+
+private:
+  const std::vector<Point<dim> > distance_source;
+};
+
+
+int main ()
+{
+  const int dim = 2;
+
+  Triangulation< dim >  tria;
+  std::map< unsigned int, Point< dim > > new_points;
+  const unsigned int N = 8;
+  GridGenerator::subdivided_hyper_cube (tria, N, -5, 5);
+
+  // find the vertex at the origin
+  Triangulation<dim>::active_cell_iterator
+  cell = GridTools::find_active_cell_around_point (tria,Point<dim>());
+
+  unsigned int best_vertex   = cell->vertex_index(0); //vertex number on local triangulation
+  Point<dim>   best_pos      = cell->vertex(0);
+  double       best_dist     = Point<dim>().distance(best_pos);
+
+  for (unsigned int vertex_no = 1; vertex_no < GeometryInfo<dim>::vertices_per_cell; vertex_no++)
+    {
+      const double dist = Point<dim>().distance(cell->vertex(vertex_no));
+      if (dist < best_dist)
+        {
+          best_pos    = cell->vertex(vertex_no);
+          best_vertex = cell->vertex_index(vertex_no);
+          best_dist   = dist;
+        }
+    }
+  // move the point at the origin by 1 unit to the right
+  new_points[best_vertex] = Point<dim>();
+  new_points[best_vertex][0] += 1.;
+
+  // store the current location to be used in coefficient function
+  std::vector<Point<dim> > metric;
+  metric.push_back(best_pos);
+
+  // now pin all of the points on the boundary
+  cell = tria.begin_active();
+  Triangulation<dim>::active_cell_iterator endc = tria.end();
+
+  for ( ; cell != endc; ++cell)
+    if (cell->at_boundary() == true)
+      for (unsigned int face = 0; face < GeometryInfo<dim>::faces_per_cell; ++face)
+        if (cell->face(face)->at_boundary() == true)
+          for (unsigned int v=0; v < GeometryInfo<dim>::vertices_per_face; ++v)
+            {
+              unsigned int vertex_number = cell->face(face)->vertex_index(v);
+              new_points[vertex_number] =  cell->face(face)->vertex(v);
+            }
+
+  // then compute new point locations and output the result
+  L2_inverse<dim> coefficient(metric);
+  GridTools::laplace_transform  (new_points,tria, &coefficient);
+  std::ofstream out ("output");
+  GridOut grid_out;
+  grid_out.write_eps (tria, out);
+}
diff --git a/tests/grid/grid_transform_03.output b/tests/grid/grid_transform_03.output
new file mode 100644
index 0000000000..b2c40885b8
--- /dev/null
+++ b/tests/grid/grid_transform_03.output
@@ -0,0 +1,269 @@
+%!PS-Adobe-2.0 EPSF-1.2
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diff --git a/tests/grid/grid_transform_3d.cc b/tests/grid/grid_transform_3d.cc
index 6b2aee1910..a2d0e9937e 100644
--- a/tests/grid/grid_transform_3d.cc
+++ b/tests/grid/grid_transform_3d.cc
@@ -87,7 +87,7 @@ int main ()
             }
       }
 
-  GridTools::laplace_transform (new_points, tria);
+  GridTools::laplace_transform<dim> (new_points, tria, NULL, true);
 
 
   GridOut grid_out;
diff --git a/tests/grid/grid_transform_coefficient.cc b/tests/grid/grid_transform_coefficient.cc
index ac2e9134ec..eb4841820e 100644
--- a/tests/grid/grid_transform_coefficient.cc
+++ b/tests/grid/grid_transform_coefficient.cc
@@ -109,7 +109,7 @@ int main ()
     }
 
   Coefficient<dim> c;
-  GridTools::laplace_transform (new_points, tria, &c);
+  GridTools::laplace_transform (new_points, tria, &c, true);
   HyperBallBoundary<dim> inner_ball(n_center, n_radius);
   tria.set_boundary(1, inner_ball);
 
-- 
2.39.5