From: Luca Heltai Date: Thu, 18 May 2017 13:38:27 +0000 (+0200) Subject: Implemented regularize corner cells X-Git-Tag: v9.0.0-rc1~1581^2 X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=75c1053c33fa010c13f9e8b0ab23a48449eb5212;p=dealii.git Implemented regularize corner cells --- diff --git a/doc/doxygen/images/regularize_mesh_01.png b/doc/doxygen/images/regularize_mesh_01.png new file mode 100644 index 0000000000..3e681db891 Binary files /dev/null and b/doc/doxygen/images/regularize_mesh_01.png differ diff --git a/doc/doxygen/images/regularize_mesh_02.png b/doc/doxygen/images/regularize_mesh_02.png new file mode 100644 index 0000000000..9c5b6a543b Binary files /dev/null and b/doc/doxygen/images/regularize_mesh_02.png differ diff --git a/doc/doxygen/images/regularize_mesh_03.png b/doc/doxygen/images/regularize_mesh_03.png new file mode 100644 index 0000000000..08969ee873 Binary files /dev/null and b/doc/doxygen/images/regularize_mesh_03.png differ diff --git a/doc/news/changes/minor/20170518LucaHeltai b/doc/news/changes/minor/20170518LucaHeltai new file mode 100644 index 0000000000..30f83a2bca --- /dev/null +++ b/doc/news/changes/minor/20170518LucaHeltai @@ -0,0 +1,5 @@ +New: Added a GridTools::regularize_corner_cells function that +detects if the boundary cells of a mesh at corner positions +(with dim adjacent faces on the boundary) need to be split into +cells with smaller angles. +
(Luca Heltai, Martin Kronbichler, 2017/05/18) diff --git a/include/deal.II/grid/grid_tools.h b/include/deal.II/grid/grid_tools.h index 7b90b8672e..37198c9744 100644 --- a/include/deal.II/grid/grid_tools.h +++ b/include/deal.II/grid/grid_tools.h @@ -461,6 +461,99 @@ namespace GridTools const double max_ratio = 1.6180339887, const unsigned int max_iterations = 5); + /** + * Analyze the boundary cells of a mesh, and if one cell is found at + * a corner position (with dim adjacent faces on the boundary), and its + * dim-dimensional angle fraction exceeds @p limit_angle_fraction, + * refine globally once, and replace the children of such cell + * with children where the corner is no longer offending the given angle + * fraction. + * + * If no boundary cells exist with two adjacent faces on the boundary, then + * the triangulation is left untouched. If instead we do have cells with dim + * adjacent faces on the boundary, then the fraction between the dim-dimensional + * solid angle and dim*pi/2 is checked against the parameter @p limit_angle_fraction. + * If it is higher, the grid is refined once, and the children of the + * offending cell are replaced with some cells that instead respect the limit. After + * this process the triangulation is flattened, and all Manifold objects are restored + * as they were in the original triangulation. + * + * An example is given by the following mesh, obtained by attaching a SphericalManifold + * to a mesh generated using GridGenerator::hyper_cube: + * + * @code + * const SphericalManifold m0; + * Triangulation tria; + * GridGenerator::hyper_cube(tria,-1,1); + * tria.set_all_manifold_ids_on_boundary(0); + * tria.set_manifold(0, m0); + * tria.refine_global(4); + * @endcode + * + *

+ * @image html regularize_mesh_01.png + *

+ * + * The four cells that were originally the corners of a square will give you some troubles + * during computations, as the jacobian of the transformation from the reference cell to + * those cells will go to zero, affecting the error constants of the finite element estimates. + * + * Those cells have a corner with an angle that is very close to 180 degrees, i.e., an angle + * fraction very close to one. + * + * The same code, adding a call to regularize_corner_cells: + * @code + * const SphericalManifold m0; + * Triangulation tria; + * GridGenerator::hyper_cube(tria,-1,1); + * tria.set_all_manifold_ids_on_boundary(0); + * tria.set_manifold(0, m0); + * GridTools::regularize_corner_cells(tria); + * tria.refine_global(2); + * @endcode + * generates a mesh that has a much better behaviour w.r.t. the jacobian of the Mapping: + * + *

+ * @image html regularize_mesh_02.png + *

+ * + * This mesh is very similar to the one obtained by GridGenerator::hyper_ball. However, using + * GridTools::regularize_corner_cells one has the freedom to choose when to apply the + * regularization, i.e., one could in principle first refine a few times, and then call the + * regularize_corner_cells function: + * + * @code + * const SphericalManifold m0; + * Triangulation tria; + * GridGenerator::hyper_cube(tria,-1,1); + * tria.set_all_manifold_ids_on_boundary(0); + * tria.set_manifold(0, m0); + * tria.refine_global(2); + * GridTools::regularize_corner_cells(tria); + * tria.refine_global(1); + * @endcode + * + * This generates the following mesh: + * + *

+ * @image html regularize_mesh_03.png + *

+ * + * The function is currently implemented only for dim = 2 and + * will throw an exception if called with dim = 3. + * + * @param[in,out] tria Triangulation to regularize. + * + * @param[in] limit_angle_fraction Maximum ratio of angle or solid + * angle that is allowed for a corner element in the mesh. + * + * @author Luca Heltai, Martin Kronbichler, 2017 + */ + template + void + regularize_corner_cells(Triangulation &tria, + const double limit_angle_fraction=.75); + /*@}*/ /** * @name Finding cells and vertices of a triangulation diff --git a/source/grid/grid_tools.cc b/source/grid/grid_tools.cc index d4e54d2de6..b9616c22de 100644 --- a/source/grid/grid_tools.cc +++ b/source/grid/grid_tools.cc @@ -35,6 +35,7 @@ #include #include #include +#include #include #include #include @@ -4451,6 +4452,312 @@ next_cell: } } + + template + void regularize_corner_cells (Triangulation &tria, + const double limit_angle_fraction) + { + if (dim == 1) + return; // Nothing to do + + // Check that we don't have hanging nodes + AssertThrow(!tria.has_hanging_nodes(), ExcMessage("The input Triangulation cannot " + "have hanging nodes.")); + + + bool has_cells_with_more_than_dim_faces_on_boundary = true; + bool has_cells_with_dim_faces_on_boundary = false; + + unsigned int refinement_cycles = 0; + + while (has_cells_with_more_than_dim_faces_on_boundary) + { + has_cells_with_more_than_dim_faces_on_boundary = false; + + for (auto cell: tria.active_cell_iterators()) + { + unsigned int boundary_face_counter = 0; + for (unsigned int f=0; f::faces_per_cell; ++f) + if (cell->face(f)->at_boundary()) + boundary_face_counter++; + if (boundary_face_counter > dim) + { + has_cells_with_more_than_dim_faces_on_boundary = true; + break; + } + else if (boundary_face_counter == dim) + has_cells_with_dim_faces_on_boundary = true; + } + if (has_cells_with_more_than_dim_faces_on_boundary) + { + tria.refine_global(1); + refinement_cycles++; + } + } + + if (has_cells_with_dim_faces_on_boundary) + { + tria.refine_global(1); + refinement_cycles++; + } + else + { + while (refinement_cycles>0) + { + for (auto cell: tria.active_cell_iterators()) + cell->set_coarsen_flag(); + tria.execute_coarsening_and_refinement(); + refinement_cycles--; + } + return; + } + + std::vector cells_to_remove(tria.n_active_cells(), false); + std::vector > vertices = tria.get_vertices(); + + std::vector faces_to_remove(tria.n_raw_faces(),false); + + std::vector > cells_to_add; + SubCellData subcelldata_to_add; + + // Trick compiler for dimension independent things + const unsigned int + v0 = 0, v1 = 1, + v2 = (dim > 1 ? 2:0), v3 = (dim > 1 ? 3:0), + v4 = (dim > 2 ? 4:0), v5 = (dim > 2 ? 5:0), + v6 = (dim > 2 ? 6:0), v7 = (dim > 2 ? 7:0); + + for (auto cell : tria.active_cell_iterators()) + { + double angle_fraction = 0; + unsigned int vertex_at_corner = numbers::invalid_unsigned_int; + + if (dim == 2) + { + Tensor<1,spacedim> p0; + p0[spacedim > 1 ? 1 : 0] = 1; + Tensor<1,spacedim> p1; + p1[0] = 1; + + if (cell->face(v0)->at_boundary() && cell->face(v3)->at_boundary()) + { + p0 = cell->vertex(v0) - cell->vertex(v2); + p1 = cell->vertex(v3) - cell->vertex(v2); + vertex_at_corner = v2; + } + else if (cell->face(v3)->at_boundary() && cell->face(v1)->at_boundary()) + { + p0 = cell->vertex(v2) - cell->vertex(v3); + p1 = cell->vertex(v1) - cell->vertex(v3); + vertex_at_corner = v3; + } + else if (cell->face(1)->at_boundary() && cell->face(2)->at_boundary()) + { + p0 = cell->vertex(v0) - cell->vertex(v1); + p1 = cell->vertex(v3) - cell->vertex(v1); + vertex_at_corner = v1; + } + else if (cell->face(2)->at_boundary() && cell->face(0)->at_boundary()) + { + p0 = cell->vertex(v2) - cell->vertex(v0); + p1 = cell->vertex(v1) - cell->vertex(v0); + vertex_at_corner = v0; + } + p0 /= p0.norm(); + p1 /= p1.norm(); + angle_fraction = std::acos(p0*p1)/numbers::PI; + + } + else + { + Assert(false, ExcNotImplemented()); + } + + if (angle_fraction > limit_angle_fraction) + { + + auto flags_removal = [&](unsigned int f1, unsigned int f2, + unsigned int n1, unsigned int n2) -> void + { + cells_to_remove[cell->active_cell_index()] = true; + cells_to_remove[cell->neighbor(n1)->active_cell_index()] = true; + cells_to_remove[cell->neighbor(n2)->active_cell_index()] = true; + + faces_to_remove[cell->face(f1)->index()] = true; + faces_to_remove[cell->face(f2)->index()] = true; + + faces_to_remove[cell->neighbor(n1)->face(f1)->index()] = true; + faces_to_remove[cell->neighbor(n2)->face(f2)->index()] = true; + }; + + auto cell_creation = [&]( + const unsigned int vv0, + const unsigned int vv1, + const unsigned int f0, + const unsigned int f1, + + const unsigned int n0, + const unsigned int v0n0, + const unsigned int v1n0, + + const unsigned int n1, + const unsigned int v0n1, + const unsigned int v1n1) + { + CellData c1, c2; + CellData<1> l1, l2; + + c1.vertices[v0] = cell->vertex_index(vv0); + c1.vertices[v1] = cell->vertex_index(vv1); + c1.vertices[v2] = cell->neighbor(n0)->vertex_index(v0n0); + c1.vertices[v3] = cell->neighbor(n0)->vertex_index(v1n0); + + c1.manifold_id = cell->manifold_id(); + c1.material_id = cell->material_id(); + + c2.vertices[v0] = cell->vertex_index(vv0); + c2.vertices[v1] = cell->neighbor(n1)->vertex_index(v0n1); + c2.vertices[v2] = cell->vertex_index(vv1); + c2.vertices[v3] = cell->neighbor(n1)->vertex_index(v1n1); + + c2.manifold_id = cell->manifold_id(); + c2.material_id = cell->material_id(); + + l1.vertices[0] = cell->vertex_index(vv0); + l1.vertices[1] = cell->neighbor(n0)->vertex_index(v0n0); + + l1.boundary_id = cell->line(f0)->boundary_id(); + l1.manifold_id = cell->line(f0)->manifold_id(); + subcelldata_to_add.boundary_lines.push_back(l1); + + l2.vertices[0] = cell->vertex_index(vv0); + l2.vertices[1] = cell->neighbor(n1)->vertex_index(v0n1); + + l2.boundary_id = cell->line(f1)->boundary_id(); + l2.manifold_id = cell->line(f1)->manifold_id(); + subcelldata_to_add.boundary_lines.push_back(l2); + + cells_to_add.push_back(c1); + cells_to_add.push_back(c2); + }; + + if (dim == 2) + { + switch (vertex_at_corner) + { + case 0: + flags_removal(0,2,3,1); + cell_creation(0,3, 0,2, 3,2,3, 1,1,3); + break; + case 1: + flags_removal(1,2,3,0); + cell_creation(1,2, 2,1, 0,0,2, 3,3,2); + break; + case 2: + flags_removal(3,0,1,2); + cell_creation(2,1, 3,0, 1,3,1, 2,0,1); + break; + case 3: + flags_removal(3,1,0,2); + cell_creation(3,0, 1,3, 2,1,0, 0,2,0); + break; + } + } + else + { + Assert(false, ExcNotImplemented()); + } + } + } + + // if no cells need to be added, then no regularization is necessary. Restore things + // as they were before this function was called. + if (cells_to_add.size() == 0) + { + while (refinement_cycles>0) + { + for (auto cell: tria.active_cell_iterators()) + cell->set_coarsen_flag(); + tria.execute_coarsening_and_refinement(); + refinement_cycles--; + } + return; + } + + // add the cells that were not marked as skipped + for (auto cell : tria.active_cell_iterators()) + { + if (cells_to_remove[cell->active_cell_index()] == false) + { + CellData c; + for (unsigned int v=0; v::vertices_per_cell; ++v) + c.vertices[v] = cell->vertex_index(v); + c.manifold_id = cell->manifold_id(); + c.material_id = cell->material_id(); + cells_to_add.push_back(c); + } + } + + // Face counter for both dim == 2 and dim == 3 + typename Triangulation::active_face_iterator + face = tria.begin_active_face(), + endf = tria.end_face(); + for (; face != endf; ++face) + if ( (face->at_boundary() || face->manifold_id() != numbers::invalid_manifold_id) + && faces_to_remove[face->index()] == false) + { + for (unsigned int l=0; l::lines_per_face; ++l) + { + CellData<1> line; + if (dim == 2) + { + for (unsigned int v=0; v::vertices_per_cell; ++v) + line.vertices[v] = face->vertex_index(v); + line.boundary_id = face->boundary_id(); + line.manifold_id = face->manifold_id(); + } + else + { + for (unsigned int v=0; v::vertices_per_cell; ++v) + line.vertices[v] = face->line(l)->vertex_index(v); + line.boundary_id = face->line(l)->boundary_id(); + line.manifold_id = face->line(l)->manifold_id(); + } + subcelldata_to_add.boundary_lines.push_back(line); + } + if (dim == 3) + { + CellData<2> quad; + for (unsigned int v=0; v::vertices_per_cell; ++v) + quad.vertices[v] = face->vertex_index(v); + quad.boundary_id = face->boundary_id(); + quad.manifold_id = face->manifold_id(); + subcelldata_to_add.boundary_quads.push_back(quad); + } + } + GridTools::delete_unused_vertices(vertices, cells_to_add, subcelldata_to_add); + GridReordering::reorder_cells(cells_to_add, true); + + // Save manifolds + auto manifold_ids = tria.get_manifold_ids(); + std::map*> manifolds; + // Set manifolds in new Triangulation + for (auto manifold_id: manifold_ids) + if (manifold_id != numbers::invalid_manifold_id) + manifolds[manifold_id] = &tria.get_manifold(manifold_id); + + tria.clear(); + + tria.create_triangulation(vertices, cells_to_add, subcelldata_to_add); + + // Restore manifolds + for (auto manifold_id: manifold_ids) + if (manifold_id != numbers::invalid_manifold_id) + tria.set_manifold(manifold_id, *manifolds[manifold_id]); + } + + + } /* namespace GridTools */ diff --git a/source/grid/grid_tools.inst.in b/source/grid/grid_tools.inst.in index 1b593ec313..f148328ef1 100644 --- a/source/grid/grid_tools.inst.in +++ b/source/grid/grid_tools.inst.in @@ -411,6 +411,9 @@ for (deal_II_dimension : DIMENSIONS ; deal_II_space_dimension : SPACE_DIMENSIONS template void copy_material_to_manifold_id (Triangulation &, const bool); + template + void regularize_corner_cells + (Triangulation &, double); \} #endif } diff --git a/tests/grid/grid_tools_regularize_01.cc b/tests/grid/grid_tools_regularize_01.cc new file mode 100644 index 0000000000..743077afad --- /dev/null +++ b/tests/grid/grid_tools_regularize_01.cc @@ -0,0 +1,50 @@ +// --------------------------------------------------------------------- +// +// Copyright (C) 2001 - 2015 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. +// +// --------------------------------------------------------------------- + + +// GridTools::regularize_corner_cells + +#include "../tests.h" +#include +#include +#include +#include + +template +void test () +{ + const SphericalManifold m0; + Triangulation tria; + GridGenerator::hyper_cube(tria,-1,1); + tria.set_all_manifold_ids_on_boundary(0); + tria.set_manifold(0, m0); + + GridTools::regularize_corner_cells (tria); + + GridOut grid_out; + grid_out.write_msh (tria, deallog.get_file_stream()); +} + + + +int main () +{ + initlog(); + + test<2> (); + + return 0; +} + diff --git a/tests/grid/grid_tools_regularize_01.output b/tests/grid/grid_tools_regularize_01.output new file mode 100644 index 0000000000..214914d755 --- /dev/null +++ b/tests/grid/grid_tools_regularize_01.output @@ -0,0 +1,35 @@ +$NOD +17 +1 -1.00000 -1.00000 0 +2 1.00000 -1.00000 0 +3 -1.00000 1.00000 0 +4 1.00000 1.00000 0 +5 0.00000 -1.41421 0 +6 -1.41421 0.00000 0 +7 1.41421 0.00000 0 +8 0.00000 1.41421 0 +9 0.00000 0.00000 0 +10 0.00000 -0.707107 0 +11 0.00000 0.707107 0 +12 -0.707107 0.00000 0 +13 0.707107 0.00000 0 +14 -0.621135 -0.621135 0 +15 0.621135 -0.621135 0 +16 -0.621135 0.621135 0 +17 0.621135 0.621135 0 +$ENDNOD +$ELM +12 +1 3 0 0 4 12 6 1 14 +2 3 0 0 4 10 14 1 5 +3 3 0 0 4 10 5 2 15 +4 3 0 0 4 13 15 2 7 +5 3 0 0 4 11 8 3 16 +6 3 0 0 4 12 16 3 6 +7 3 0 0 4 13 7 4 17 +8 3 0 0 4 11 17 4 8 +9 3 0 0 4 9 12 14 10 +10 3 0 0 4 9 10 15 13 +11 3 0 0 4 9 11 16 12 +12 3 0 0 4 9 13 17 11 +$ENDELM diff --git a/tests/grid/grid_tools_regularize_02.cc b/tests/grid/grid_tools_regularize_02.cc new file mode 100644 index 0000000000..03981670f4 --- /dev/null +++ b/tests/grid/grid_tools_regularize_02.cc @@ -0,0 +1,55 @@ +// --------------------------------------------------------------------- +// +// Copyright (C) 2001 - 2015 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. +// +// --------------------------------------------------------------------- + + +// GridTools::regularize_corner_cells on more complicated mesh + +#include "../tests.h" +#include +#include +#include +#include + +int main() +{ + initlog(); + + Point<2> p0(0,0), p1(4,1); + Point<2> c0(.1, .5), c1(3.9,.5); + + SphericalManifold<2> m0(c0); + SphericalManifold<2> m1(c1); + + Triangulation<2> tria; + std::vector subdivisions(2); + subdivisions[0] = 4; + subdivisions[1] = 1; + GridGenerator::subdivided_hyper_rectangle(tria,subdivisions, p0,p1,true); + + GridTools::copy_boundary_to_manifold_id(tria); + + tria.set_manifold(0, m0); + tria.set_manifold(1, m1); + + GridTools::regularize_corner_cells (tria); + tria.refine_global(1); + + GridOut grid_out; + grid_out.write_msh (tria, deallog.get_file_stream()); + + return 0; +} + + diff --git a/tests/grid/grid_tools_regularize_02.output b/tests/grid/grid_tools_regularize_02.output new file mode 100644 index 0000000000..5d2bb06537 --- /dev/null +++ b/tests/grid/grid_tools_regularize_02.output @@ -0,0 +1,520 @@ + +$NOD +273 +1 0.00000 0.00000 0 +2 1.00000 0.00000 0 +3 2.00000 0.00000 0 +4 3.00000 0.00000 0 +5 4.00000 0.00000 0 +6 0.00000 1.00000 0 +7 1.00000 1.00000 0 +8 2.00000 1.00000 0 +9 3.00000 1.00000 0 +10 4.00000 1.00000 0 +11 0.500000 0.00000 0 +12 -0.409902 0.500000 0 +13 1.50000 0.00000 0 +14 1.00000 0.500000 0 +15 2.50000 0.00000 0 +16 2.00000 0.500000 0 +17 3.50000 0.00000 0 +18 3.00000 0.500000 0 +19 4.40990 0.500000 0 +20 0.500000 1.00000 0 +21 1.50000 1.00000 0 +22 2.50000 1.00000 0 +23 3.50000 1.00000 0 +24 0.448762 0.500000 0 +25 1.50000 0.500000 0 +26 2.50000 0.500000 0 +27 3.55124 0.500000 0 +28 0.750000 0.00000 0 +29 1.25000 0.00000 0 +30 1.75000 0.00000 0 +31 1.00000 0.250000 0 +32 1.00000 0.750000 0 +33 2.25000 0.00000 0 +34 2.75000 0.00000 0 +35 2.00000 0.250000 0 +36 2.00000 0.750000 0 +37 3.25000 0.00000 0 +38 3.00000 0.250000 0 +39 3.00000 0.750000 0 +40 0.750000 1.00000 0 +41 1.25000 1.00000 0 +42 1.75000 1.00000 0 +43 2.25000 1.00000 0 +44 2.75000 1.00000 0 +45 3.25000 1.00000 0 +46 0.474381 0.250000 0 +47 0.474381 0.750000 0 +48 0.0194302 0.500000 0 +49 0.724381 0.500000 0 +50 1.50000 0.250000 0 +51 1.50000 0.750000 0 +52 1.25000 0.500000 0 +53 1.75000 0.500000 0 +54 2.50000 0.250000 0 +55 2.50000 0.750000 0 +56 2.25000 0.500000 0 +57 2.75000 0.500000 0 +58 3.52562 0.250000 0 +59 3.52562 0.750000 0 +60 3.27562 0.500000 0 +61 3.98057 0.500000 0 +62 0.123543 0.240841 0 +63 0.737191 0.250000 0 +64 0.123543 0.759159 0 +65 0.737191 0.750000 0 +66 1.25000 0.250000 0 +67 1.75000 0.250000 0 +68 1.25000 0.750000 0 +69 1.75000 0.750000 0 +70 2.25000 0.250000 0 +71 2.75000 0.250000 0 +72 2.25000 0.750000 0 +73 2.75000 0.750000 0 +74 3.26281 0.250000 0 +75 3.87646 0.240841 0 +76 3.26281 0.750000 0 +77 3.87646 0.759159 0 +78 0.875000 0.00000 0 +79 1.87500 0.00000 0 +80 2.87500 0.00000 0 +81 0.875000 1.00000 0 +82 1.87500 1.00000 0 +83 2.87500 1.00000 0 +84 0.250000 0.00000 0 +85 -0.294329 0.176728 0 +86 -0.294329 0.823272 0 +87 1.37500 0.00000 0 +88 1.00000 0.375000 0 +89 1.00000 0.625000 0 +90 0.862191 0.500000 0 +91 2.37500 0.00000 0 +92 2.00000 0.375000 0 +93 2.00000 0.625000 0 +94 1.87500 0.500000 0 +95 3.75000 0.00000 0 +96 3.37500 0.00000 0 +97 3.00000 0.375000 0 +98 3.00000 0.625000 0 +99 2.87500 0.500000 0 +100 4.29433 0.176728 0 +101 4.29433 0.823272 0 +102 0.250000 1.00000 0 +103 1.37500 1.00000 0 +104 2.37500 1.00000 0 +105 3.75000 1.00000 0 +106 3.37500 1.00000 0 +107 0.461572 0.375000 0 +108 0.461572 0.625000 0 +109 0.234096 0.500000 0 +110 1.50000 0.375000 0 +111 1.50000 0.625000 0 +112 1.37500 0.500000 0 +113 2.50000 0.375000 0 +114 2.50000 0.625000 0 +115 2.37500 0.500000 0 +116 3.53843 0.375000 0 +117 3.53843 0.625000 0 +118 3.41343 0.500000 0 +119 3.76590 0.500000 0 +120 0.625000 0.00000 0 +121 1.12500 0.00000 0 +122 1.62500 0.00000 0 +123 1.00000 0.125000 0 +124 0.868595 0.250000 0 +125 1.00000 0.875000 0 +126 0.868595 0.750000 0 +127 2.12500 0.00000 0 +128 2.62500 0.00000 0 +129 2.00000 0.125000 0 +130 1.87500 0.250000 0 +131 2.00000 0.875000 0 +132 1.87500 0.750000 0 +133 3.12500 0.00000 0 +134 3.00000 0.125000 0 +135 2.87500 0.250000 0 +136 3.00000 0.875000 0 +137 2.87500 0.750000 0 +138 0.625000 1.00000 0 +139 1.12500 1.00000 0 +140 1.62500 1.00000 0 +141 2.12500 1.00000 0 +142 2.62500 1.00000 0 +143 3.12500 1.00000 0 +144 0.487191 0.125000 0 +145 0.298962 0.245420 0 +146 0.487191 0.875000 0 +147 0.298962 0.754580 0 +148 -0.195236 0.500000 0 +149 0.0714865 0.370420 0 +150 0.0714865 0.629580 0 +151 0.586572 0.500000 0 +152 0.730786 0.375000 0 +153 0.730786 0.625000 0 +154 1.50000 0.125000 0 +155 1.37500 0.250000 0 +156 1.50000 0.875000 0 +157 1.37500 0.750000 0 +158 1.12500 0.500000 0 +159 1.25000 0.375000 0 +160 1.25000 0.625000 0 +161 1.62500 0.500000 0 +162 1.75000 0.375000 0 +163 1.75000 0.625000 0 +164 2.50000 0.125000 0 +165 2.37500 0.250000 0 +166 2.50000 0.875000 0 +167 2.37500 0.750000 0 +168 2.12500 0.500000 0 +169 2.25000 0.375000 0 +170 2.25000 0.625000 0 +171 2.62500 0.500000 0 +172 2.75000 0.375000 0 +173 2.75000 0.625000 0 +174 3.51281 0.125000 0 +175 3.39421 0.250000 0 +176 3.70104 0.245420 0 +177 3.51281 0.875000 0 +178 3.39421 0.750000 0 +179 3.70104 0.754580 0 +180 3.13781 0.500000 0 +181 3.26921 0.375000 0 +182 3.26921 0.625000 0 +183 4.19524 0.500000 0 +184 3.92851 0.370420 0 +185 3.92851 0.629580 0 +186 0.0617714 0.120420 0 +187 0.743595 0.125000 0 +188 0.605786 0.250000 0 +189 0.0617714 0.879580 0 +190 0.743595 0.875000 0 +191 0.605786 0.750000 0 +192 1.25000 0.125000 0 +193 1.12500 0.250000 0 +194 1.75000 0.125000 0 +195 1.62500 0.250000 0 +196 1.12500 0.750000 0 +197 1.25000 0.875000 0 +198 1.75000 0.875000 0 +199 1.62500 0.750000 0 +200 2.25000 0.125000 0 +201 2.12500 0.250000 0 +202 2.75000 0.125000 0 +203 2.62500 0.250000 0 +204 2.12500 0.750000 0 +205 2.25000 0.875000 0 +206 2.75000 0.875000 0 +207 2.62500 0.750000 0 +208 3.25640 0.125000 0 +209 3.13140 0.250000 0 +210 3.93823 0.120420 0 +211 3.13140 0.750000 0 +212 3.25640 0.875000 0 +213 3.93823 0.879580 0 +214 -0.111421 0.273574 0 +215 0.274481 0.122710 0 +216 0.274481 0.877290 0 +217 -0.111421 0.726426 0 +218 3.72552 0.122710 0 +219 4.11142 0.273574 0 +220 4.11142 0.726426 0 +221 3.72552 0.877290 0 +222 0.266529 0.372710 0 +223 0.615393 0.125000 0 +224 0.871798 0.125000 0 +225 0.596179 0.375000 0 +226 0.865393 0.375000 0 +227 0.266529 0.627290 0 +228 0.596179 0.625000 0 +229 0.865393 0.625000 0 +230 0.615393 0.875000 0 +231 0.871798 0.875000 0 +232 1.12500 0.125000 0 +233 1.37500 0.125000 0 +234 1.12500 0.375000 0 +235 1.37500 0.375000 0 +236 1.62500 0.125000 0 +237 1.87500 0.125000 0 +238 1.62500 0.375000 0 +239 1.87500 0.375000 0 +240 1.12500 0.625000 0 +241 1.37500 0.625000 0 +242 1.12500 0.875000 0 +243 1.37500 0.875000 0 +244 1.62500 0.625000 0 +245 1.87500 0.625000 0 +246 1.62500 0.875000 0 +247 1.87500 0.875000 0 +248 2.12500 0.125000 0 +249 2.37500 0.125000 0 +250 2.12500 0.375000 0 +251 2.37500 0.375000 0 +252 2.62500 0.125000 0 +253 2.87500 0.125000 0 +254 2.62500 0.375000 0 +255 2.87500 0.375000 0 +256 2.12500 0.625000 0 +257 2.37500 0.625000 0 +258 2.12500 0.875000 0 +259 2.37500 0.875000 0 +260 2.62500 0.625000 0 +261 2.87500 0.625000 0 +262 2.62500 0.875000 0 +263 2.87500 0.875000 0 +264 3.12820 0.125000 0 +265 3.38461 0.125000 0 +266 3.13461 0.375000 0 +267 3.40382 0.375000 0 +268 3.73347 0.372710 0 +269 3.13461 0.625000 0 +270 3.40382 0.625000 0 +271 3.12820 0.875000 0 +272 3.38461 0.875000 0 +273 3.73347 0.627290 0 +$ENDNOD +$ELM +240 +1 3 3 0 4 48 148 214 149 +2 3 3 0 4 148 12 85 214 +3 3 3 0 4 149 214 186 62 +4 3 3 0 4 214 85 1 186 +5 3 3 0 4 46 145 215 144 +6 3 3 0 4 145 62 186 215 +7 3 3 0 4 144 215 84 11 +8 3 3 0 4 215 186 1 84 +9 3 3 0 4 47 146 216 147 +10 3 3 0 4 146 20 102 216 +11 3 3 0 4 147 216 189 64 +12 3 3 0 4 216 102 6 189 +13 3 3 0 4 48 150 217 148 +14 3 3 0 4 150 64 189 217 +15 3 3 0 4 148 217 86 12 +16 3 3 0 4 217 189 6 86 +17 3 3 0 4 58 174 218 176 +18 3 3 0 4 174 17 95 218 +19 3 3 0 4 176 218 210 75 +20 3 3 0 4 218 95 5 210 +21 3 3 0 4 61 184 219 183 +22 3 3 0 4 184 75 210 219 +23 3 3 0 4 183 219 100 19 +24 3 3 0 4 219 210 5 100 +25 3 3 0 4 61 183 220 185 +26 3 3 0 4 183 19 101 220 +27 3 3 0 4 185 220 213 77 +28 3 3 0 4 220 101 10 213 +29 3 3 0 4 59 179 221 177 +30 3 3 0 4 179 77 213 221 +31 3 3 0 4 177 221 105 23 +32 3 3 0 4 221 213 10 105 +33 3 3 0 4 24 109 222 107 +34 3 3 0 4 109 48 149 222 +35 3 3 0 4 107 222 145 46 +36 3 3 0 4 222 149 62 145 +37 3 3 0 4 63 188 223 187 +38 3 3 0 4 188 46 144 223 +39 3 3 0 4 187 223 120 28 +40 3 3 0 4 223 144 11 120 +41 3 3 0 4 31 124 224 123 +42 3 3 0 4 124 63 187 224 +43 3 3 0 4 123 224 78 2 +44 3 3 0 4 224 187 28 78 +45 3 3 0 4 49 151 225 152 +46 3 3 0 4 151 24 107 225 +47 3 3 0 4 152 225 188 63 +48 3 3 0 4 225 107 46 188 +49 3 3 0 4 14 90 226 88 +50 3 3 0 4 90 49 152 226 +51 3 3 0 4 88 226 124 31 +52 3 3 0 4 226 152 63 124 +53 3 3 0 4 24 108 227 109 +54 3 3 0 4 108 47 147 227 +55 3 3 0 4 109 227 150 48 +56 3 3 0 4 227 147 64 150 +57 3 3 0 4 49 153 228 151 +58 3 3 0 4 153 65 191 228 +59 3 3 0 4 151 228 108 24 +60 3 3 0 4 228 191 47 108 +61 3 3 0 4 14 89 229 90 +62 3 3 0 4 89 32 126 229 +63 3 3 0 4 90 229 153 49 +64 3 3 0 4 229 126 65 153 +65 3 3 0 4 65 190 230 191 +66 3 3 0 4 190 40 138 230 +67 3 3 0 4 191 230 146 47 +68 3 3 0 4 230 138 20 146 +69 3 3 0 4 32 125 231 126 +70 3 3 0 4 125 7 81 231 +71 3 3 0 4 126 231 190 65 +72 3 3 0 4 231 81 40 190 +73 3 3 0 4 66 193 232 192 +74 3 3 0 4 193 31 123 232 +75 3 3 0 4 192 232 121 29 +76 3 3 0 4 232 123 2 121 +77 3 3 0 4 50 155 233 154 +78 3 3 0 4 155 66 192 233 +79 3 3 0 4 154 233 87 13 +80 3 3 0 4 233 192 29 87 +81 3 3 0 4 52 158 234 159 +82 3 3 0 4 158 14 88 234 +83 3 3 0 4 159 234 193 66 +84 3 3 0 4 234 88 31 193 +85 3 3 0 4 25 112 235 110 +86 3 3 0 4 112 52 159 235 +87 3 3 0 4 110 235 155 50 +88 3 3 0 4 235 159 66 155 +89 3 3 0 4 67 195 236 194 +90 3 3 0 4 195 50 154 236 +91 3 3 0 4 194 236 122 30 +92 3 3 0 4 236 154 13 122 +93 3 3 0 4 35 130 237 129 +94 3 3 0 4 130 67 194 237 +95 3 3 0 4 129 237 79 3 +96 3 3 0 4 237 194 30 79 +97 3 3 0 4 53 161 238 162 +98 3 3 0 4 161 25 110 238 +99 3 3 0 4 162 238 195 67 +100 3 3 0 4 238 110 50 195 +101 3 3 0 4 16 94 239 92 +102 3 3 0 4 94 53 162 239 +103 3 3 0 4 92 239 130 35 +104 3 3 0 4 239 162 67 130 +105 3 3 0 4 52 160 240 158 +106 3 3 0 4 160 68 196 240 +107 3 3 0 4 158 240 89 14 +108 3 3 0 4 240 196 32 89 +109 3 3 0 4 25 111 241 112 +110 3 3 0 4 111 51 157 241 +111 3 3 0 4 112 241 160 52 +112 3 3 0 4 241 157 68 160 +113 3 3 0 4 68 197 242 196 +114 3 3 0 4 197 41 139 242 +115 3 3 0 4 196 242 125 32 +116 3 3 0 4 242 139 7 125 +117 3 3 0 4 51 156 243 157 +118 3 3 0 4 156 21 103 243 +119 3 3 0 4 157 243 197 68 +120 3 3 0 4 243 103 41 197 +121 3 3 0 4 53 163 244 161 +122 3 3 0 4 163 69 199 244 +123 3 3 0 4 161 244 111 25 +124 3 3 0 4 244 199 51 111 +125 3 3 0 4 16 93 245 94 +126 3 3 0 4 93 36 132 245 +127 3 3 0 4 94 245 163 53 +128 3 3 0 4 245 132 69 163 +129 3 3 0 4 69 198 246 199 +130 3 3 0 4 198 42 140 246 +131 3 3 0 4 199 246 156 51 +132 3 3 0 4 246 140 21 156 +133 3 3 0 4 36 131 247 132 +134 3 3 0 4 131 8 82 247 +135 3 3 0 4 132 247 198 69 +136 3 3 0 4 247 82 42 198 +137 3 3 0 4 70 201 248 200 +138 3 3 0 4 201 35 129 248 +139 3 3 0 4 200 248 127 33 +140 3 3 0 4 248 129 3 127 +141 3 3 0 4 54 165 249 164 +142 3 3 0 4 165 70 200 249 +143 3 3 0 4 164 249 91 15 +144 3 3 0 4 249 200 33 91 +145 3 3 0 4 56 168 250 169 +146 3 3 0 4 168 16 92 250 +147 3 3 0 4 169 250 201 70 +148 3 3 0 4 250 92 35 201 +149 3 3 0 4 26 115 251 113 +150 3 3 0 4 115 56 169 251 +151 3 3 0 4 113 251 165 54 +152 3 3 0 4 251 169 70 165 +153 3 3 0 4 71 203 252 202 +154 3 3 0 4 203 54 164 252 +155 3 3 0 4 202 252 128 34 +156 3 3 0 4 252 164 15 128 +157 3 3 0 4 38 135 253 134 +158 3 3 0 4 135 71 202 253 +159 3 3 0 4 134 253 80 4 +160 3 3 0 4 253 202 34 80 +161 3 3 0 4 57 171 254 172 +162 3 3 0 4 171 26 113 254 +163 3 3 0 4 172 254 203 71 +164 3 3 0 4 254 113 54 203 +165 3 3 0 4 18 99 255 97 +166 3 3 0 4 99 57 172 255 +167 3 3 0 4 97 255 135 38 +168 3 3 0 4 255 172 71 135 +169 3 3 0 4 56 170 256 168 +170 3 3 0 4 170 72 204 256 +171 3 3 0 4 168 256 93 16 +172 3 3 0 4 256 204 36 93 +173 3 3 0 4 26 114 257 115 +174 3 3 0 4 114 55 167 257 +175 3 3 0 4 115 257 170 56 +176 3 3 0 4 257 167 72 170 +177 3 3 0 4 72 205 258 204 +178 3 3 0 4 205 43 141 258 +179 3 3 0 4 204 258 131 36 +180 3 3 0 4 258 141 8 131 +181 3 3 0 4 55 166 259 167 +182 3 3 0 4 166 22 104 259 +183 3 3 0 4 167 259 205 72 +184 3 3 0 4 259 104 43 205 +185 3 3 0 4 57 173 260 171 +186 3 3 0 4 173 73 207 260 +187 3 3 0 4 171 260 114 26 +188 3 3 0 4 260 207 55 114 +189 3 3 0 4 18 98 261 99 +190 3 3 0 4 98 39 137 261 +191 3 3 0 4 99 261 173 57 +192 3 3 0 4 261 137 73 173 +193 3 3 0 4 73 206 262 207 +194 3 3 0 4 206 44 142 262 +195 3 3 0 4 207 262 166 55 +196 3 3 0 4 262 142 22 166 +197 3 3 0 4 39 136 263 137 +198 3 3 0 4 136 9 83 263 +199 3 3 0 4 137 263 206 73 +200 3 3 0 4 263 83 44 206 +201 3 3 0 4 74 209 264 208 +202 3 3 0 4 209 38 134 264 +203 3 3 0 4 208 264 133 37 +204 3 3 0 4 264 134 4 133 +205 3 3 0 4 58 175 265 174 +206 3 3 0 4 175 74 208 265 +207 3 3 0 4 174 265 96 17 +208 3 3 0 4 265 208 37 96 +209 3 3 0 4 60 180 266 181 +210 3 3 0 4 180 18 97 266 +211 3 3 0 4 181 266 209 74 +212 3 3 0 4 266 97 38 209 +213 3 3 0 4 27 118 267 116 +214 3 3 0 4 118 60 181 267 +215 3 3 0 4 116 267 175 58 +216 3 3 0 4 267 181 74 175 +217 3 3 0 4 27 116 268 119 +218 3 3 0 4 116 58 176 268 +219 3 3 0 4 119 268 184 61 +220 3 3 0 4 268 176 75 184 +221 3 3 0 4 60 182 269 180 +222 3 3 0 4 182 76 211 269 +223 3 3 0 4 180 269 98 18 +224 3 3 0 4 269 211 39 98 +225 3 3 0 4 27 117 270 118 +226 3 3 0 4 117 59 178 270 +227 3 3 0 4 118 270 182 60 +228 3 3 0 4 270 178 76 182 +229 3 3 0 4 76 212 271 211 +230 3 3 0 4 212 45 143 271 +231 3 3 0 4 211 271 136 39 +232 3 3 0 4 271 143 9 136 +233 3 3 0 4 59 177 272 178 +234 3 3 0 4 177 23 106 272 +235 3 3 0 4 178 272 212 76 +236 3 3 0 4 272 106 45 212 +237 3 3 0 4 27 119 273 117 +238 3 3 0 4 119 61 185 273 +239 3 3 0 4 117 273 179 59 +240 3 3 0 4 273 185 77 179 +$ENDELM