Line data Source code
1 : // The libMesh Finite Element Library.
2 : // Copyright (C) 2002-2026 Benjamin S. Kirk, John W. Peterson, Roy H. Stogner
3 :
4 : // This library is free software; you can redistribute it and/or
5 : // modify it under the terms of the GNU Lesser General Public
6 : // License as published by the Free Software Foundation; either
7 : // version 2.1 of the License, or (at your option) any later version.
8 :
9 : // This library is distributed in the hope that it will be useful,
10 : // but WITHOUT ANY WARRANTY; without even the implied warranty of
11 : // MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
12 : // Lesser General Public License for more details.
13 :
14 : // You should have received a copy of the GNU Lesser General Public
15 : // License along with this library; if not, write to the Free Software
16 : // Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
17 :
18 : #include "libmesh/variational_smoother_system.h"
19 :
20 : #include "libmesh/elem.h"
21 : #include "libmesh/face_tri3.h"
22 : #include "libmesh/face_tri6.h"
23 : #include "libmesh/fe_base.h"
24 : #include "libmesh/fe_interface.h"
25 : #include "libmesh/fem_context.h"
26 : #include "libmesh/mesh.h"
27 : #include "libmesh/numeric_vector.h"
28 : #include "libmesh/parallel_ghost_sync.h"
29 : #include "libmesh/quadrature.h"
30 : #include "libmesh/string_to_enum.h"
31 : #include "libmesh/utility.h"
32 : #include "libmesh/enum_to_string.h"
33 : #include <libmesh/reference_elem.h>
34 :
35 : // C++ includes
36 : #include <functional> // std::reference_wrapper
37 :
38 : namespace libMesh
39 : {
40 :
41 : /*
42 : * Gets the dof_id_type value corresponding to the minimum of the Real value.
43 : */
44 13104 : void communicate_pair_min(std::pair<Real, dof_id_type> & pair, const Parallel::Communicator & comm)
45 : {
46 : // Get rank where minimum occurs
47 : unsigned int rank;
48 13104 : comm.minloc(pair.first, rank);
49 13104 : comm.broadcast(pair.second, rank);
50 13104 : }
51 :
52 : /*
53 : * Gets the dof_id_type value corresponding to the maximum of the Real value.
54 : */
55 13104 : void communicate_pair_max(std::pair<Real, dof_id_type> & pair, const Parallel::Communicator & comm)
56 : {
57 : // Get rank where minimum occurs
58 : unsigned int rank;
59 13104 : comm.maxloc(pair.first, rank);
60 13104 : comm.broadcast(pair.second, rank);
61 13104 : }
62 :
63 : /**
64 : * Function to prevent dividing by zero for degenerate elements
65 : */
66 2250224 : Real chi_epsilon(const Real & x, const Real epsilon_squared)
67 : {
68 2812780 : return 0.5 * (x + std::sqrt(epsilon_squared + Utility::pow<2>(x)));
69 : }
70 :
71 : /**
72 : * Given an fe_map, element dimension, and quadrature point index, returns the
73 : * Jacobian of the physical-to-reference mapping.
74 : */
75 2253864 : RealTensor get_jacobian_at_qp(const FEMap & fe_map,
76 : const unsigned int & dim,
77 : const unsigned int & qp)
78 : {
79 2253864 : libmesh_error_msg_if(dim > 3, "Unsupported dimension.");
80 :
81 : // RealTensors are always 3x3, so we will fill any dimensions above dim
82 : // with 1s on the diagonal. This indicates a 1 to 1 relationship between
83 : // the physical and reference elements in these extra dimensions.
84 :
85 2253864 : const auto & dxyzdxi = dim >= 1 ? fe_map.get_dxyzdxi()[qp] : RealGradient(1, 0, 0);
86 2253864 : const auto & dxyzdeta = dim >= 2 ? fe_map.get_dxyzdeta()[qp] : RealGradient(0, 1, 0);
87 2253864 : const auto & dxyzdzeta = dim >= 3 ? fe_map.get_dxyzdzeta()[qp] : RealGradient(0, 0, 1);
88 :
89 2253864 : return RealTensor(dxyzdxi, dxyzdeta, dxyzdzeta).transpose(); // Note the transposition!
90 : }
91 :
92 : /**
93 : * Compute the trace of a dim-dimensional matrix.
94 : */
95 2250224 : Real trace(const RealTensor & A, const unsigned int & dim)
96 : {
97 562556 : Real tr = 0.0;
98 8889088 : for (const auto i : make_range(dim))
99 6638864 : tr += A(i, i);
100 :
101 2250224 : return tr;
102 : }
103 :
104 306 : VariationalSmootherSystem::~VariationalSmootherSystem () = default;
105 :
106 3038 : void VariationalSmootherSystem::assembly (bool get_residual,
107 : bool get_jacobian,
108 : bool apply_heterogeneous_constraints,
109 : bool apply_no_constraints)
110 : {
111 : // Update the mesh based on the current variable values
112 1736 : auto & mesh = this->get_mesh();
113 3038 : this->solution->close();
114 :
115 238632 : for (auto * node : mesh.local_node_ptr_range())
116 : {
117 603904 : for (const auto d : make_range(mesh.mesh_dimension()))
118 : {
119 448288 : const auto dof_id = node->dof_number(this->number(), d, 0);
120 : // Update mesh
121 448288 : (*node)(d) = libmesh_real((*current_local_solution)(dof_id));
122 : }
123 1302 : }
124 :
125 3038 : SyncNodalPositions sync_object(mesh);
126 5208 : Parallel::sync_dofobject_data_by_id (mesh.comm(), mesh.nodes_begin(), mesh.nodes_end(), sync_object);
127 :
128 : // Compute and update mesh quality information
129 3038 : compute_mesh_quality_info();
130 3038 : const bool is_tangled = _mesh_info.mesh_is_tangled;
131 :
132 : // Update _epsilon_squared_assembly based on whether we are untangling or
133 : // smoothing
134 3038 : if (_untangling_solve)
135 : {
136 172 : const Real & min_S = _mesh_info.min_qp_det_S;
137 : // This component is based on what Larisa did in the original code.
138 602 : const Real variable_component = 100. * Utility::pow<2>(_ref_vol * min_S);
139 602 : _epsilon_squared_assembly = _epsilon_squared + variable_component;
140 : }
141 :
142 : else
143 2436 : _epsilon_squared_assembly = 0.;
144 :
145 3038 : FEMSystem::assembly(get_residual, get_jacobian, apply_heterogeneous_constraints, apply_no_constraints);
146 :
147 3038 : if (_untangling_solve && !is_tangled)
148 : {
149 : // Mesh is untangled, artificially reduce residual by factor 0.9 to tell
150 : // the solver we are on the right track. The mesh may become re-tangled in
151 : // subsequent nonlinear and line search iterations, so we will not use
152 : // this reduction factor in the case of re-tangulation. This approach
153 : // should drive the solver to eventually favor untangled solutions.
154 420 : rhs->close();
155 420 : (*rhs) *= 0.9;
156 : }
157 3038 : }
158 :
159 238 : void VariationalSmootherSystem::solve()
160 : {
161 238 : const auto & mesh_info = get_mesh_info();
162 238 : if (_verbosity > 10)
163 4 : libMesh::out << "Initial " << mesh_info << std::endl;
164 238 : if (mesh_info.mesh_is_tangled)
165 : {
166 : // Untangling solve
167 14 : _untangling_solve = true;
168 :
169 : // Untangling seems to work better using only the distortion metric.
170 : // For a mixed metric, I've seen it *technically* untangle a mesh to a
171 : // still-suboptimal mesh (i.e., a local minima), but not been able to
172 : // smooth it well because the smoothest solution is on the other side of
173 : // a tangulation barrier.
174 14 : const auto dilation_weight = _dilation_weight;
175 14 : _dilation_weight = 0.;
176 :
177 14 : if (_verbosity > 10)
178 0 : libMesh::out << "Untangling the mesh" << std::endl;
179 14 : FEMSystem::solve();
180 :
181 : // Reset the dilation weight
182 14 : _dilation_weight = dilation_weight;
183 :
184 14 : if (_verbosity > 10)
185 0 : libMesh::out << "Untangled " << mesh_info << std::endl;
186 : }
187 :
188 : // Smoothing solve
189 238 : _untangling_solve = false;
190 238 : if (_verbosity > 10)
191 4 : libMesh::out << "Smoothing the mesh" << std::endl;
192 238 : FEMSystem::solve();
193 238 : libmesh_error_msg_if(mesh_info.mesh_is_tangled, "The smoothing solve tangled the mesh!");
194 238 : if (_verbosity > 10)
195 4 : libMesh::out << "Smoothed " << mesh_info << std::endl;
196 238 : }
197 :
198 238 : void VariationalSmootherSystem::init_data ()
199 : {
200 136 : auto & mesh = this->get_mesh();
201 68 : const auto & elem_orders = mesh.elem_default_orders();
202 238 : libmesh_error_msg_if(elem_orders.size() != 1,
203 : "The variational smoother cannot be used for mixed-order meshes!");
204 238 : const auto fe_order = *elem_orders.begin();
205 : // Add a variable for each dimension of the mesh
206 : // "r0" for x, "r1" for y, "r2" for z
207 840 : for (const auto & d : make_range(mesh.mesh_dimension()))
208 1032 : this->add_variable ("r" + std::to_string(d), fe_order);
209 :
210 : // Do the parent's initialization after variables are defined
211 238 : FEMSystem::init_data();
212 :
213 : // Set the current_local_solution to the current mesh for the initial guess
214 238 : this->solution->close();
215 :
216 19824 : for (auto * node : mesh.local_node_ptr_range())
217 : {
218 50604 : for (const auto d : make_range(mesh.mesh_dimension()))
219 : {
220 37660 : const auto dof_id = node->dof_number(this->number(), d, 0);
221 : // Update solution
222 37660 : (*solution).set(dof_id, (*node)(d));
223 : }
224 102 : }
225 :
226 238 : this->prepare_for_smoothing();
227 238 : }
228 :
229 238 : void VariationalSmootherSystem::prepare_for_smoothing()
230 : {
231 : // If this method has already been called to set _ref_vol, just return.
232 306 : if (std::abs(_ref_vol) > TOLERANCE * TOLERANCE)
233 0 : return;
234 :
235 306 : std::unique_ptr<DiffContext> con = this->build_context();
236 68 : FEMContext & femcontext = cast_ref<FEMContext &>(*con);
237 238 : this->init_context(femcontext);
238 :
239 136 : const auto & mesh = this->get_mesh();
240 :
241 238 : Real elem_averaged_det_S_sum = 0.;
242 :
243 : // Make pre-requests before reinit() for efficiency in
244 : // --enable-deprecated builds, and to avoid errors in
245 : // --disable-deprecated builds.
246 68 : const auto & fe_map = femcontext.get_element_fe(0)->get_fe_map();
247 68 : const auto & JxW = fe_map.get_JxW();
248 :
249 12168 : for (const auto * elem : mesh.active_local_element_ptr_range())
250 : {
251 11760 : femcontext.pre_fe_reinit(*this, elem);
252 11760 : femcontext.elem_fe_reinit();
253 :
254 : // Add target element info, if applicable
255 14700 : if (_target_jacobians.find(elem->type()) == _target_jacobians.end())
256 : {
257 246 : const auto [target_elem, target_nodes] = get_target_elem(elem->type());
258 246 : get_target_to_reference_jacobian(target_elem.get(),
259 : femcontext,
260 246 : _target_jacobians[elem->type()],
261 492 : _target_jacobian_dets[elem->type()]);
262 : }// if find == end()
263 :
264 : // Reference volume computation
265 2940 : Real elem_integrated_det_S = 0.;
266 111488 : for (const auto qp : index_range(JxW))
267 124660 : elem_integrated_det_S += JxW[qp] / _target_jacobian_dets[elem->type()][qp];
268 11760 : const auto ref_elem_vol = elem->reference_elem()->volume();
269 11760 : elem_averaged_det_S_sum += elem_integrated_det_S / ref_elem_vol;
270 :
271 102 : } // for elem
272 :
273 : // Get contributions from elements on other processors
274 238 : mesh.comm().sum(elem_averaged_det_S_sum);
275 :
276 238 : _ref_vol = elem_averaged_det_S_sum / mesh.n_active_elem();
277 238 : if (_verbosity > 10)
278 8 : libMesh::out << "Reference volume: " << _ref_vol << std::endl;
279 102 : }
280 :
281 6552 : void VariationalSmootherSystem::init_context(DiffContext & context)
282 : {
283 1872 : FEMContext & c = cast_ref<FEMContext &>(context);
284 :
285 1872 : FEBase * my_fe = nullptr;
286 :
287 : // Now make sure we have requested all the data
288 : // we need to build the system.
289 :
290 : // We might have a multi-dimensional mesh
291 : const std::set<unsigned char> & elem_dims =
292 1872 : c.elem_dimensions();
293 :
294 13104 : for (const auto & dim : elem_dims)
295 : {
296 6552 : c.get_element_fe( 0, my_fe, dim );
297 1872 : my_fe->get_nothing();
298 :
299 1872 : auto & fe_map = my_fe->get_fe_map();
300 1872 : fe_map.get_dxyzdxi();
301 1872 : fe_map.get_dxyzdeta();
302 1872 : fe_map.get_dxyzdzeta();
303 1872 : fe_map.get_JxW();
304 :
305 : // Mesh may be tangled, allow negative Jacobians
306 1872 : fe_map.set_jacobian_tolerance(std::numeric_limits<Real>::lowest());
307 :
308 6552 : c.get_side_fe( 0, my_fe, dim );
309 1872 : my_fe->get_nothing();
310 : }
311 :
312 6552 : FEMSystem::init_context(context);
313 6552 : }
314 :
315 :
316 125624 : bool VariationalSmootherSystem::element_time_derivative (bool request_jacobian,
317 : DiffContext & context)
318 : {
319 31406 : FEMContext & c = cast_ref<FEMContext &>(context);
320 :
321 62812 : const Elem & elem = c.get_elem();
322 :
323 125624 : unsigned int dim = c.get_dim();
324 :
325 31406 : unsigned int x_var = 0, y_var = 1, z_var = 2;
326 : // In the case of lower dimensions, we will not access z (1D/2D) or y (1D)
327 125624 : if (dim < 3)
328 : {
329 3132 : z_var = 0;
330 12528 : if (dim < 2)
331 100 : y_var = 0;
332 : }
333 :
334 : // The subvectors and submatrices we need to fill:
335 : // system residual
336 : std::reference_wrapper<DenseSubVector<Number>> F[3] =
337 : {
338 : c.get_elem_residual(x_var),
339 : c.get_elem_residual(y_var),
340 : c.get_elem_residual(z_var)
341 31406 : };
342 : // system jacobian
343 : std::reference_wrapper<DenseSubMatrix<Number>> K[3][3] =
344 : {
345 : {c.get_elem_jacobian(x_var, x_var), c.get_elem_jacobian(x_var, y_var), c.get_elem_jacobian(x_var, z_var)},
346 : {c.get_elem_jacobian(y_var, x_var), c.get_elem_jacobian(y_var, y_var), c.get_elem_jacobian(y_var, z_var)},
347 : {c.get_elem_jacobian(z_var, x_var), c.get_elem_jacobian(z_var, y_var), c.get_elem_jacobian(z_var, z_var)}
348 31406 : };
349 :
350 : // Quadrature info
351 31406 : const auto & quad_weights = c.get_element_qrule().get_weights();
352 :
353 125624 : const auto distortion_weight = 1. - _dilation_weight;
354 :
355 : // Get some references to cell-specific data that
356 : // will be used to assemble the linear system.
357 :
358 31406 : const auto & fe_map = c.get_element_fe(0)->get_fe_map();
359 :
360 31406 : const auto & dphidxi_map = fe_map.get_dphidxi_map();
361 31406 : const auto & dphideta_map = fe_map.get_dphideta_map();
362 31406 : const auto & dphidzeta_map = fe_map.get_dphidzeta_map();
363 :
364 125624 : const auto & target_jacobian_dets = _target_jacobian_dets[elem.type()];
365 125624 : const auto & target_jacobians = _target_jacobians[elem.type()];
366 :
367 : // Integrate the distortion-dilation metric over the reference element
368 1200872 : for (const auto qp : index_range(quad_weights))
369 : {
370 : // Compute quantities needed to evaluate the distortion-dilation metric
371 : // and its gradient and Hessian.
372 : // The metric will be minimized when it's gradient with respect to the node
373 : // locations (R) is zero. For Newton's method, minimizing the gradient
374 : // requires computation of the gradient's Jacobian with respect to R.
375 : // The Jacobian of the distortion-dilation metric's gradientis the Hessian
376 : // of the metric.
377 :
378 : // Transform quad weight from reference element to quad weight for target element
379 1344060 : const auto quad_weight = quad_weights[qp] / target_jacobian_dets[qp];
380 :
381 : // Note that the term "Jacobian" has two meanings in this mesh smoothing
382 : // application. The first meaning refers to the Jacobian w.r.t R of the
383 : // gradient w.r.t. R, (i.e., the Hessian of the metric). This is the K
384 : // variable defined above. This is also the Jacobian the 'request_jacobian'
385 : // variable refers to. The second meaning refers to the Jacobian of the
386 : // physical-to-target element mapping. This is the Jacobian used to
387 : // compute the distorion-dilation metric.
388 : //
389 : // Grab the physical-to-reference mapping Jacobian matrix (i.e., "S") at this qp
390 1344060 : RealTensor S = get_jacobian_at_qp(fe_map, dim, qp);
391 :
392 : // Apply target element transformation to get the physical-to-target jacobian
393 1075248 : S *= _target_jacobians[elem.type()][qp];
394 :
395 : // Compute quantities needed for the smoothing algorithm
396 :
397 : // determinant
398 1075248 : const Real det = S.det();
399 1075248 : const Real det_sq = det * det;
400 1075248 : const Real det_cube = det_sq * det;
401 :
402 1075248 : const Real ref_vol_sq = _ref_vol * _ref_vol;
403 :
404 : // trace of S^T * S
405 : // DO NOT USE RealTensor.tr for the trace, it will NOT be correct for
406 : // 1D and 2D meshes because of our hack of putting 1s in the diagonal of
407 : // S for the extra dimensions (see get_jacobian_at_qp)
408 1075248 : const auto tr = trace(S.transpose() * S, dim);
409 1075248 : const Real tr_div_dim = tr / dim;
410 :
411 : // Precompute pow(tr_div_dim, 0.5 * dim - x) for x = 0, 1, 2
412 1075248 : const Real half_dim = 0.5 * dim;
413 : const std::vector<Real> trace_powers{
414 1075248 : std::pow(tr_div_dim, half_dim),
415 1075248 : std::pow(tr_div_dim, half_dim - 1.),
416 1075248 : std::pow(tr_div_dim, half_dim - 2.),
417 1344060 : };
418 :
419 : // inverse of S
420 1344060 : const RealTensor S_inv = S.inverse();
421 : // inverse transpose of S
422 537624 : const RealTensor S_inv_T = S_inv.transpose();
423 :
424 : // Identity matrix
425 1075248 : const RealTensor I(1, 0, 0,
426 1075248 : 0, 1, 0,
427 1075248 : 0, 0, 1);
428 :
429 : // The chi function allows us to handle degenerate elements
430 1075248 : const auto chi = chi_epsilon(det, _epsilon_squared_assembly);
431 1075248 : const Real chi_sq = chi * chi;
432 1075248 : const Real sqrt_term = std::sqrt(_epsilon_squared_assembly + det_sq);
433 : // dchi(x) / dx
434 1075248 : const Real chi_prime = 0.5 * (1. + det / sqrt_term);
435 1075248 : const Real chi_prime_sq = chi_prime * chi_prime;
436 : // d2chi(x) / dx2
437 1075248 : const Real chi_2prime = 0.5 * (1. / sqrt_term - det_sq / Utility::pow<3>(sqrt_term));
438 :
439 : // Distortion metric (beta)
440 : //const Real beta = trace_powers[0] / chi;
441 2150496 : const RealTensor dbeta_dS = (trace_powers[1] / chi) * S - (trace_powers[0] / chi_sq * chi_prime * det) * S_inv_T;
442 :
443 : // Dilation metric (mu)
444 : //const Real mu = 0.5 * (_ref_vol + det_sq / _ref_vol) / chi;
445 : // We represent d mu / dS as alpha(S) * S^-T, where alpha is a scalar function
446 1075248 : const Real alpha = (-chi_prime * det_cube + 2. * det_sq * chi - ref_vol_sq * det * chi_prime) / (2. * _ref_vol * chi_sq);
447 1075248 : const RealTensor dmu_dS = alpha * S_inv_T;
448 :
449 : // Combined metric (E)
450 : //const Real E = distortion_weight * beta + _dilation_weight * mu;
451 1881684 : const RealTensor dE_dS = distortion_weight * dbeta_dS + _dilation_weight * dmu_dS;
452 :
453 : // This vector is useful in computing dS/dR below
454 5376240 : std::vector<std::vector<std::vector<Real>>> dphi_maps = {dphidxi_map, dphideta_map, dphidzeta_map};
455 :
456 : // Compute residual (i.e., the gradient of the combined metric w.r.t node locations)
457 : // Recall that when the gradient (residual) is zero, the combined metric is minimized
458 10864896 : for (const auto l : elem.node_index_range())
459 : {
460 38939488 : for (const auto var_id : make_range(dim))
461 : {
462 : // Build dS/dR, the derivative of the physical-to-target mapping Jacobin w.r.t. the mesh
463 : // node locations
464 21862380 : RealTensor dS_dR = RealTensor(0);
465 116166432 : for (const auto jj : make_range(dim))
466 174033184 : dS_dR(var_id, jj) = dphi_maps[jj][l][qp];
467 36437300 : dS_dR *= target_jacobians[qp];
468 :
469 : // Residual contribution. The contraction of dE/dS and dS/dR gives us
470 : // the gradient we are looking for, dE/dR
471 36437300 : F[var_id](l) += quad_weight * dE_dS.contract(dS_dR);
472 : }// for var_id
473 : }// for l
474 :
475 :
476 1075248 : if (request_jacobian)
477 : {
478 : // Compute jacobian of the smoothing system (i.e., the Hessian of the
479 : // combined metric w.r.t. the mesh node locations)
480 :
481 : // Precompute coefficients to be applied to each component of tensor
482 : // products in the loops below. At first glance, these coefficients look
483 : // like gibberish, but everything in the Hessian has been verified by
484 : // taking finite differences of the gradient. We should probably write
485 : // down the derivations of the gradient and Hessian somewhere...
486 :
487 : // Recall that above, dbeta_dS takes the form:
488 : // d(beta)/dS = c1(S) * S - c2(S) * S_inv_T,
489 : // where c1 and c2 are scalar-valued functions.
490 : const std::vector<Real> d2beta_dS2_coefs_times_distortion_weight = {
491 : //Part 1: scaler coefficients of d(c1 * S) / dS
492 : //
493 : // multiplies I[i,a] x I[j,b]
494 529288 : (trace_powers[1] / chi) * distortion_weight,
495 : // multiplies S[a,b] x S[i,j]
496 661610 : (((dim - 2.) / dim) * trace_powers[2] / chi) * distortion_weight,
497 : // multiplies S_inv[b,a] * S[i,j]
498 661610 : (-(trace_powers[1] / chi_sq) * chi_prime * det) * distortion_weight,
499 : //
500 : //Part 2: scaler coefficients of d(-c2 * S_inv_T) / dS
501 : //
502 : // multiplies S[a,b] x S_inv[j,i]
503 661610 : (-(trace_powers[1] / chi_sq) * chi_prime * det) * distortion_weight,
504 : // multiplies S_inv[b,a] x S_inv[j,i]
505 661610 : (trace_powers[0] * (det / chi_sq)
506 529288 : * ((2. * chi_prime_sq / chi - chi_2prime) * det - chi_prime)) * distortion_weight,
507 : // multiplies S_inv[b,i] x S_inv[j,a]
508 661610 : ((trace_powers[0] / chi_sq) * chi_prime * det) * distortion_weight,
509 661610 : };
510 :
511 : // d alpha / dS has the form c(S) * S^-T, where c is the scalar coefficient defined below
512 793932 : const Real dalpha_dS_coef_times_dilation_weight = ((det / (2. * _ref_vol * chi_sq))
513 793932 : * (-4. * _ref_vol * alpha * chi * chi_prime - chi_2prime * det_cube
514 793932 : - chi_prime * det_sq + 4 * chi * det - ref_vol_sq * (chi_prime + det * chi_2prime))) * _dilation_weight;
515 :
516 : // This is also useful to precompute
517 529288 : const Real alpha_times_dilation_weight = alpha * _dilation_weight;
518 :
519 : /*
520 :
521 : To increase the efficiency of the Jacobian computation, we take
522 : advantage of l-p symmetry, ij-ab symmetry, and the sparsity pattern of
523 : dS_dR. We also factor all possible multipliers out of inner loops for
524 : efficiency. The result is code that is more difficult to read. For
525 : clarity, consult the pseudo-code below in this comment.
526 :
527 : for (const auto l: elem.node_index_range()) // Contribution to Hessian
528 : from node l
529 : {
530 : for (const auto var_id1 : make_range(dim)) // Contribution from each
531 : x/y/z component of node l
532 : {
533 : // Build dS/dR_l, the derivative of the physical-to-target
534 : mapping Jacobin w.r.t.
535 : // the l-th node
536 : RealTensor dS_dR_l = RealTensor(0);
537 : for (const auto ii : make_range(dim))
538 : dS_dR_l(var_id1, ii) = dphi_maps[ii][l][qp];
539 : dS_dR_l *= target_jacobians[qp];
540 :
541 : for (const auto p: elem.node_index_range()) // Contribution to
542 : Hessian from node p
543 : {
544 : for (const auto var_id2 : make_range(dim)) // Contribution from
545 : each x/y/z component of node p
546 : {
547 : // Build dS/dR_l, the derivative of the physical-to-target
548 : mapping Jacobin w.r.t.
549 : // the p-th node
550 : RealTensor dS_dR_p = RealTensor(0);
551 : for (const auto jj : make_range(dim))
552 : dS_dR_p(var_id2, jj) = dphi_maps[jj][p][qp];
553 : dS_dR_p *= target_jacobians[qp];
554 :
555 : Real d2beta_dR2 = 0.;
556 : Real d2mu_dR2 = 0.;
557 : // Perform tensor contraction
558 : for (const auto i : make_range(dim))
559 : {
560 : for (const auto j : make_range(dim))
561 : {
562 : for (const auto a : make_range(dim))
563 : {
564 : for (const auto b : make_range(dim))
565 : {
566 : // Nasty tensor products to be multiplied by
567 : d2beta_dS2_coefs to get d2(beta) / dS2 const std::vector<Real>
568 : d2beta_dS2_tensor_contributions =
569 : {
570 : I(i,a) * I(j,b),
571 : S(a,b) * S(i,j),
572 : S_inv(b,a) * S(i,j),
573 : S(a,b) * S_inv(j,i),
574 : S_inv(b,a) * S_inv(j,i),
575 : S_inv(j,a) * S_inv(b,i),
576 : };
577 :
578 : // Combine precomputed coefficients with tensor products
579 : to get d2(beta) / dS2 Real d2beta_dS2 = 0.; for (const auto comp_id :
580 : index_range(d2beta_dS2_coefs))
581 : {
582 : const Real contribution = d2beta_dS2_coefs[comp_id] *
583 : d2beta_dS2_tensor_contributions[comp_id]; d2beta_dS2 += contribution;
584 : }
585 :
586 : // Incorporate tensor product portion to get d2(mu) /
587 : dS2 const Real d2mu_dS2 = dalpha_dS_coef * S_inv(b,a) * S_inv(j,i) -
588 : alpha * S_inv(b,i) * S_inv(j,a);
589 :
590 : // Chain rule to change d/dS to d/dR
591 : d2beta_dR2 += d2beta_dS2 * dS_dR_l(a, b) * dS_dR_p(i,
592 : j); d2mu_dR2 += d2mu_dS2 * dS_dR_l(a, b) * dS_dR_p(i, j);
593 :
594 : }// for b
595 : }// for a
596 : }// for j
597 : }// for i, end tensor contraction
598 :
599 : // Jacobian contribution
600 : K[var_id1][var_id2](l, p) += quad_weight * ((1. -
601 : _dilation_weight) * d2beta_dR2 + _dilation_weight * d2mu_dR2);
602 :
603 : }// for var_id2
604 : }// for p
605 : }// for var_id1
606 : }// for l
607 :
608 : End pseudo-code, begin efficient code
609 :
610 : */
611 :
612 5313680 : for (const auto l: elem.node_index_range()) // Contribution to Hessian from node l
613 : {
614 19045136 : for (const auto var_id1 : make_range(dim)) // Contribution from each x/y/z component of node l
615 : {
616 : // Build dS/dR_l, the derivative of the physical-to-target mapping Jacobin w.r.t.
617 : // the l-th node
618 14260744 : RealTensor dS_dR_l = RealTensor(0);
619 56860752 : for (const auto ii : make_range(dim))
620 85200016 : dS_dR_l(var_id1, ii) = dphi_maps[ii][l][qp];
621 17825930 : dS_dR_l *= target_jacobians[qp];
622 :
623 : // Jacobian is symmetric, only need to loop over lower triangular portion
624 114299288 : for (const auto p: make_range(l + 1)) // Contribution to Hessian from node p
625 : {
626 399617184 : for (const auto var_id2 : make_range(dim)) // Contribution from each x/y/z component of node p
627 : {
628 : // Build dS/dR_l, the derivative of the physical-to-target mapping Jacobin w.r.t.
629 : // the p-th node
630 299578640 : RealTensor dS_dR_p = RealTensor(0);
631 1197245472 : for (const auto jj : make_range(dim))
632 1795333664 : dS_dR_p(var_id2, jj) = dphi_maps[jj][p][qp];
633 374473300 : dS_dR_p *= target_jacobians[qp];
634 :
635 74894660 : Real d2E_dR2 = 0.;
636 : // Perform tensor contraction
637 1197245472 : for (const auto i : make_range(dim))
638 : {
639 :
640 3588534048 : for (const auto j : make_range(dim))
641 : {
642 :
643 2690867216 : const auto S_ij = S(i,j);
644 2690867216 : const auto S_inv_ji = S_inv(j,i);
645 :
646 : // Apply the stuff only depending on i and j before entering a, b loops
647 : // beta
648 : const std::vector<Real> d2beta_dS2_coefs_ij_applied{
649 2690867216 : d2beta_dS2_coefs_times_distortion_weight[1] * S_ij,
650 3363584020 : d2beta_dS2_coefs_times_distortion_weight[2] * S_ij,
651 3363584020 : d2beta_dS2_coefs_times_distortion_weight[3] * S_inv_ji,
652 3363584020 : d2beta_dS2_coefs_times_distortion_weight[4] * S_inv_ji,
653 2690867216 : };
654 : // mu
655 2690867216 : const auto dalpha_dS_coef_ij_applied = dalpha_dS_coef_times_dilation_weight * S_inv_ji;
656 :
657 672716804 : Real d2E_dSdR_l = 0.;
658 672716804 : Real d2E_dSdR_p = 0.;
659 :
660 8070470816 : for (const auto a : make_range(i + 1))
661 : {
662 :
663 : // If this condition is met, both the ijab and abij
664 : // contributions to the Jacobian are zero due to the
665 : // spasity patterns of dS_dR_l and dS_dR_p and this
666 : // iteration may be skipped
667 5379603600 : if (!(a == var_id1 && i == var_id2) &&
668 3959407744 : !(a == var_id2 && i == var_id1))
669 3361452576 : continue;
670 :
671 897666832 : const auto S_inv_ja = S_inv(j,a);
672 :
673 897666832 : const Real d2beta_dS2_coef_ia_applied = d2beta_dS2_coefs_times_distortion_weight[0] * I(i,a);
674 897666832 : const Real d2beta_dS2_coef_ja_applied = d2beta_dS2_coefs_times_distortion_weight[5] * S_inv_ja;
675 897666832 : const Real alpha_ja_applied = alpha_times_dilation_weight * S_inv_ja;
676 :
677 897666832 : const auto b_limit = (a == i) ? j + 1 : dim;
678 3289489952 : for (const auto b : make_range(b_limit))
679 : {
680 :
681 : // Combine precomputed coefficients with tensor products
682 : // to get d2(beta) / dS2
683 : Real d2beta_dS2_times_distortion_weight = (
684 2989778900 : d2beta_dS2_coef_ia_applied * I(j,b) +
685 2391823120 : d2beta_dS2_coefs_ij_applied[0] * S(a,b) +
686 2391823120 : d2beta_dS2_coefs_ij_applied[1] * S_inv(b,a) +
687 2989778900 : d2beta_dS2_coefs_ij_applied[2] * S(a,b) +
688 2391823120 : d2beta_dS2_coefs_ij_applied[3] * S_inv(b,a) +
689 2391823120 : d2beta_dS2_coef_ja_applied * S_inv(b,i)
690 2391823120 : );
691 :
692 : // Incorporate tensor product portion to get d2(mu) /
693 : // dS2
694 2391823120 : const Real d2mu_dS2_times_dilation_weight = dalpha_dS_coef_ij_applied * S_inv(b,a) - alpha_ja_applied * S_inv(b,i);
695 :
696 :
697 : // Chain rule to change d/dS to d/dR
698 2391823120 : const auto d2E_dS2 =
699 : d2beta_dS2_times_distortion_weight +
700 : d2mu_dS2_times_dilation_weight;
701 :
702 : // if !(a == var_id1 (next line) && i == var_id2
703 : // (outside 'a' loop)), dS_dR_l(p) multiplier is zero
704 2391823120 : d2E_dSdR_l += d2E_dS2 * dS_dR_l(a, b);
705 :
706 2391823120 : if (!(i == a && j == b))
707 : // if !(a == var_id2 (next line) && i == var_id1
708 : // (outside 'a' loop)), dS_dR_p(l) multiplier is zero
709 2092244480 : d2E_dSdR_p += d2E_dS2 * dS_dR_p(a, b);
710 :
711 : } // for b
712 : } // for a
713 2690867216 : d2E_dR2 +=
714 2690867216 : d2E_dSdR_l * dS_dR_p(i, j) + d2E_dSdR_p * dS_dR_l(i, j);
715 : }// for j
716 : }// for i, end tensor contraction
717 :
718 : // Jacobian contribution
719 299578640 : const Real jacobian_contribution = quad_weight * d2E_dR2;
720 299578640 : K[var_id1][var_id2](l, p) += jacobian_contribution;
721 : // Jacobian is symmetric, add contribution to p,l entry
722 : // Don't get the diagonal twice!
723 299578640 : if (p < l)
724 : // Note the transposition of var_id1 and var_id2 as these are also jacobian indices
725 321223290 : K[var_id2][var_id1](p, l) += jacobian_contribution;
726 :
727 : }// for var_id2
728 : }// for p
729 : }// for var_id1
730 : }// for l
731 : }
732 537624 : } // end of the quadrature point qp-loop
733 :
734 125624 : return request_jacobian;
735 537624 : }
736 :
737 252 : const MeshQualityInfo & VariationalSmootherSystem::get_mesh_info()
738 : {
739 252 : if (!_mesh_info.initialized)
740 238 : compute_mesh_quality_info();
741 :
742 252 : return _mesh_info;
743 : }
744 :
745 3276 : void VariationalSmootherSystem::compute_mesh_quality_info()
746 : {
747 : // If the reference volume has not yet been computed, compute it.
748 4212 : if (std::abs(_ref_vol) < TOLERANCE * TOLERANCE)
749 0 : prepare_for_smoothing();
750 :
751 4212 : std::unique_ptr<DiffContext> con = this->build_context();
752 936 : FEMContext & femcontext = cast_ref<FEMContext &>(*con);
753 3276 : this->init_context(femcontext);
754 :
755 1872 : const auto & mesh = this->get_mesh();
756 3276 : const auto dim = mesh.mesh_dimension();
757 3276 : const Real half_dim = 0.5 * dim;
758 3276 : const auto distortion_weight = 1. - _dilation_weight;
759 :
760 : // Make pre-requests before reinit() for efficiency in
761 : // --enable-deprecated builds, and to avoid errors in
762 : // --disable-deprecated builds.
763 936 : const auto & fe_map = femcontext.get_element_fe(0)->get_fe_map();
764 936 : const auto & quad_weights = femcontext.get_element_qrule().get_weights();
765 936 : const auto & JxW = fe_map.get_JxW();
766 936 : fe_map.get_dxyzdxi();
767 936 : fe_map.get_dxyzdeta();
768 936 : fe_map.get_dxyzdzeta();
769 :
770 3276 : MeshQualityInfo info;
771 :
772 211692 : for (const auto * elem : mesh.active_local_element_ptr_range())
773 : {
774 137384 : femcontext.pre_fe_reinit(*this, elem);
775 137384 : femcontext.elem_fe_reinit();
776 :
777 : // Element-integrated quantities
778 34346 : Real det_S_int = 0.;
779 34346 : Real beta_int = 0.;
780 34346 : Real mu_int = 0.;
781 34346 : Real combined_int = 0.;
782 :
783 137384 : const auto & target_jacobian_dets = _target_jacobian_dets[elem->type()];
784 :
785 1312360 : for (const auto qp : index_range(JxW))
786 : {
787 1468720 : det_S_int += JxW[qp] / _target_jacobian_dets[elem->type()][qp];
788 1468720 : const auto quad_weight = quad_weights[qp] / target_jacobian_dets[qp];
789 :
790 : // Grab the physical-to-reference mapping Jacobian matrix (i.e., "S") at this qp
791 1174976 : RealTensor S = get_jacobian_at_qp(fe_map, dim, qp);
792 :
793 : // Apply target element transformation to get the physical-to-target jacobian
794 1174976 : S *= _target_jacobians[elem->type()][qp];
795 :
796 : // Determinant of S
797 1174976 : const auto det = S.det();
798 1174976 : const auto det_sq = det * det;
799 :
800 1174976 : if (det > info.max_qp_det_S)
801 15952 : info.max_qp_det_S = det;
802 1159024 : else if (det < info.min_qp_det_S)
803 21135 : info.min_qp_det_S = det;
804 :
805 1174976 : if (det < TOLERANCE * TOLERANCE)
806 444 : info.mesh_is_tangled = true;
807 :
808 : // trace of S^T * S
809 1174976 : const auto tr = trace(S.transpose() * S, dim);
810 :
811 : // The chi function allows us to handle degenerate elements
812 1174976 : const auto chi = chi_epsilon(det, _epsilon_squared_assembly);
813 :
814 : // distortion
815 1174976 : const Real beta = std::pow(tr / dim, half_dim) / chi;
816 1174976 : beta_int += beta * quad_weight;
817 :
818 : // dilation
819 1174976 : const Real mu = 0.5 * (_ref_vol + det_sq / _ref_vol) / chi;
820 1174976 : mu_int += mu * quad_weight;
821 :
822 : // combined
823 1174976 : const Real E = distortion_weight * beta + _dilation_weight * mu;
824 1174976 : combined_int += E * quad_weight;
825 : }
826 :
827 137384 : info.total_det_S += det_S_int;
828 137384 : if (det_S_int > info.max_elem_det_S.first)
829 5193 : info.max_elem_det_S = std::make_pair(det_S_int, elem->id());
830 137384 : if (det_S_int < info.min_elem_det_S.first)
831 6430 : info.min_elem_det_S = std::make_pair(det_S_int, elem->id());
832 :
833 137384 : info.total_distortion += beta_int;
834 137384 : if (beta_int > info.max_elem_distortion.first)
835 5239 : info.max_elem_distortion = std::make_pair(beta_int, elem->id());
836 137384 : if (beta_int < info.min_elem_distortion.first)
837 5937 : info.min_elem_distortion = std::make_pair(beta_int, elem->id());
838 :
839 137384 : info.total_dilation += mu_int;
840 137384 : if (mu_int > info.max_elem_dilation.first)
841 4643 : info.max_elem_dilation = std::make_pair(mu_int, elem->id());
842 137384 : if (mu_int < info.min_elem_dilation.first)
843 5707 : info.min_elem_dilation = std::make_pair(mu_int, elem->id());
844 :
845 137384 : info.total_combined += combined_int;
846 137384 : if (combined_int > info.max_elem_combined.first)
847 5048 : info.max_elem_combined = std::make_pair(combined_int, elem->id());
848 137384 : if (combined_int < info.min_elem_combined.first)
849 5927 : info.min_elem_combined = std::make_pair(combined_int, elem->id());
850 :
851 137384 : if (_verbosity > 90)
852 : {
853 51216 : libMesh::out << "Elem " << elem->id() << " quality:" << std::endl
854 17072 : << " distortion-dilation metric: " << combined_int << std::endl
855 17072 : << " distortion metric: " << beta_int << std::endl
856 17072 : << " dilation metric: " << mu_int << std::endl
857 17072 : << " det(S): " << det_S_int << std::endl;
858 : }
859 :
860 1404 : } // for elem
861 :
862 : // Get contributions from elements on other processors
863 3276 : communicate_pair_max(info.max_elem_det_S, mesh.comm());
864 3276 : communicate_pair_min(info.min_elem_det_S, mesh.comm());
865 3276 : mesh.comm().max(info.max_qp_det_S);
866 3276 : mesh.comm().min(info.min_qp_det_S);
867 3276 : mesh.comm().sum(info.total_det_S);
868 :
869 3276 : communicate_pair_max(info.max_elem_distortion, mesh.comm());
870 3276 : communicate_pair_min(info.min_elem_distortion, mesh.comm());
871 3276 : mesh.comm().sum(info.total_distortion);
872 :
873 3276 : communicate_pair_max(info.max_elem_dilation, mesh.comm());
874 3276 : communicate_pair_min(info.min_elem_dilation, mesh.comm());
875 3276 : mesh.comm().sum(info.total_dilation);
876 :
877 3276 : communicate_pair_max(info.max_elem_combined, mesh.comm());
878 3276 : communicate_pair_min(info.min_elem_combined, mesh.comm());
879 3276 : mesh.comm().sum(info.total_combined);
880 :
881 3276 : mesh.comm().max(info.mesh_is_tangled);
882 :
883 3276 : info.initialized = true;
884 :
885 3276 : _mesh_info = info;
886 :
887 3276 : if (_verbosity > 50)
888 44 : libMesh::out << info;
889 3276 : }
890 :
891 : std::pair<std::unique_ptr<Elem>, std::vector<std::unique_ptr<Node>>>
892 246 : VariationalSmootherSystem::get_target_elem(const ElemType & type)
893 : {
894 : // Build target element
895 316 : auto target_elem = Elem::build(type);
896 :
897 : // Volume of reference element
898 246 : const auto ref_vol = target_elem->reference_elem()->volume();
899 :
900 : // Update the nodes of the target element, depending on type
901 70 : const Real sqrt_2 = std::sqrt(Real(2));
902 70 : const Real sqrt_3 = std::sqrt(Real(3));
903 210 : std::vector<std::unique_ptr<Node>> owned_nodes;
904 :
905 316 : const auto type_str = Utility::enum_to_string(type);
906 :
907 : // Elems deriving from Tri
908 246 : if (type_str.compare(0, 3, "TRI") == 0)
909 : {
910 :
911 : // The target element will be an equilateral triangle with area equal to
912 : // the area of the reference element.
913 :
914 : // Equilateral triangle side length preserving area of the reference element
915 28 : const auto side_length = std::sqrt(4. / sqrt_3 * ref_vol);
916 :
917 : // Define the nodal locations of the vertices
918 8 : const auto & s = side_length;
919 : // x y node_id
920 36 : owned_nodes.emplace_back(Node::build(Point(0., 0.), 0));
921 28 : owned_nodes.emplace_back(Node::build(Point(s, 0.), 1));
922 36 : owned_nodes.emplace_back(Node::build(Point(0.5 * s, 0.5 * sqrt_3 * s), 2));
923 :
924 28 : switch (type)
925 : {
926 6 : case TRI3: {
927 : // Nothing to do here, vertices already added above
928 6 : break;
929 : }
930 :
931 7 : case TRI6: {
932 : // Define the midpoint nodes of the equilateral triangle
933 : // x y node_id
934 7 : owned_nodes.emplace_back(Node::build(Point(0.50 * s, 0.00), 3));
935 9 : owned_nodes.emplace_back(Node::build(Point(0.75 * s, 0.25 * sqrt_3 * s), 4));
936 9 : owned_nodes.emplace_back(Node::build(Point(0.25 * s, 0.25 * sqrt_3 * s), 5));
937 :
938 7 : break;
939 : }
940 :
941 0 : default:
942 0 : libmesh_error_msg("Unsupported triangular element: " << type_str);
943 : break;
944 : }
945 : } // if Tri
946 :
947 : // Elems deriving from Prism
948 218 : else if (type_str.compare(0, 5, "PRISM") == 0)
949 : {
950 :
951 : // The target element will be a prism with an equilateral triangular
952 : // base with volume equal to the volume of the reference element.
953 :
954 : // For an equilateral triangular base with side length s, the
955 : // base area is s^2 * sqrt(3) / 4.
956 : // The prism height that will result in equal face areas is
957 : // s * sqrt(3) / 4. We choose s such that the target element has
958 : // the same volume as the reference element:
959 : // v = (s^2 * sqrt(3) / 4) * (s * sqrt(3) / 4) = 3 * s^3 / 4
960 : // --> s = (16 * v / 3)^(1/3)
961 : // I have no particular motivation for imposing equal face areas,
962 : // so this can be updated if a more `optimal` target prism is
963 : // identified.
964 :
965 : // Side length that preserves the volume of the reference element
966 49 : const auto side_length = std::cbrt(16. * ref_vol / 3.);
967 : // Prism height with the property that all faces have equal area
968 49 : const auto target_height = 0.25 * side_length * sqrt_3;
969 :
970 14 : const auto & s = side_length;
971 14 : const auto & h = target_height;
972 : // x y z node_id
973 63 : owned_nodes.emplace_back(Node::build(Point(0., 0., 0.), 0));
974 49 : owned_nodes.emplace_back(Node::build(Point(s, 0., 0.), 1));
975 77 : owned_nodes.emplace_back(Node::build(Point(0.5 * s, 0.5 * sqrt_3 * s, 0.), 2));
976 63 : owned_nodes.emplace_back(Node::build(Point(0., 0., h), 3));
977 63 : owned_nodes.emplace_back(Node::build(Point(s, 0., h), 4));
978 49 : owned_nodes.emplace_back(Node::build(Point(0.5 * s, 0.5 * sqrt_3 * s, h), 5));
979 :
980 49 : if (type == PRISM15 || type == PRISM18 || type == PRISM20 || type == PRISM21)
981 : {
982 : // Define the edge midpoint nodes of the prism
983 10 : const auto & on = owned_nodes;
984 45 : owned_nodes.emplace_back(Node::build(Point((*on[0] + *on[1]) / 2.), 6));
985 45 : owned_nodes.emplace_back(Node::build(Point((*on[1] + *on[2]) / 2.), 7));
986 45 : owned_nodes.emplace_back(Node::build(Point((*on[2] + *on[0]) / 2.), 8));
987 45 : owned_nodes.emplace_back(Node::build(Point((*on[0] + *on[3]) / 2.), 9));
988 45 : owned_nodes.emplace_back(Node::build(Point((*on[1] + *on[4]) / 2.), 10));
989 45 : owned_nodes.emplace_back(Node::build(Point((*on[2] + *on[5]) / 2.), 11));
990 45 : owned_nodes.emplace_back(Node::build(Point((*on[3] + *on[4]) / 2.), 12));
991 45 : owned_nodes.emplace_back(Node::build(Point((*on[4] + *on[5]) / 2.), 13));
992 45 : owned_nodes.emplace_back(Node::build(Point((*on[5] + *on[3]) / 2.), 14));
993 :
994 35 : if (type == PRISM18 || type == PRISM20 || type == PRISM21)
995 : {
996 : // Define the rectangular face midpoint nodes of the prism
997 36 : owned_nodes.emplace_back(Node::build(Point((*on[0] + *on[1] + *on[3] + *on[4]) / 4.), 15));
998 36 : owned_nodes.emplace_back(Node::build(Point((*on[1] + *on[2] + *on[4] + *on[5]) / 4.), 16));
999 36 : owned_nodes.emplace_back(Node::build(Point((*on[0] + *on[2] + *on[3] + *on[5]) / 4.), 17));
1000 :
1001 28 : if (type == PRISM20 || type == PRISM21)
1002 : {
1003 : // Define the triangular face midpoint nodes of the prism
1004 27 : owned_nodes.emplace_back(Node::build(Point((*on[0] + *on[1] + *on[2]) / 3.), 18));
1005 27 : owned_nodes.emplace_back(Node::build(Point((*on[3] + *on[4] + *on[5]) / 3.), 19));
1006 :
1007 21 : if (type == PRISM21)
1008 : // Define the interior point of the prism
1009 22 : owned_nodes.emplace_back(Node::build(Point((*on[9] + *on[10] + *on[11]) / 3.), 20));
1010 :
1011 : }
1012 10 : }
1013 : }
1014 :
1015 14 : else if (type != PRISM6)
1016 0 : libmesh_error_msg("Unsupported prism element: " << type_str);
1017 :
1018 : } // if Prism
1019 :
1020 : // Elems deriving from Pyramid
1021 169 : else if (type_str.compare(0, 7, "PYRAMID") == 0)
1022 : {
1023 :
1024 : // The target element is a pyramid with an square base and
1025 : // equilateral triangular sides with volume equal to the volume of the
1026 : // reference element.
1027 :
1028 : // A pyramid with square base sidelength s and equilateral triangular
1029 : // sides has height h = s / sqrt(2).
1030 : // The volume is v = s^2 h / 3 = s^3 / ( 3 sqrt(2)).
1031 : // Solving for s: s = (3 sqrt(2) v)^(1/3), where v is the volume of the
1032 : // non-optimal reference element.
1033 :
1034 : // Side length that preserves the volume of the reference element
1035 35 : const auto side_length = std::cbrt(3. * sqrt_2 * ref_vol);
1036 : // Pyramid height with the property that all faces are equilateral triangles
1037 35 : const auto target_height = side_length / sqrt_2;
1038 :
1039 10 : const auto & s = side_length;
1040 10 : const auto & h = target_height;
1041 :
1042 : // x y z node_id
1043 45 : owned_nodes.emplace_back(Node::build(Point(0., 0., 0.), 0));
1044 45 : owned_nodes.emplace_back(Node::build(Point(s, 0., 0.), 1));
1045 45 : owned_nodes.emplace_back(Node::build(Point(s, s, 0.), 2));
1046 35 : owned_nodes.emplace_back(Node::build(Point(0., s, 0.), 3));
1047 45 : owned_nodes.emplace_back(Node::build(Point(0.5 * s, 0.5 * s, h), 4));
1048 :
1049 35 : if (type == PYRAMID13 || type == PYRAMID14 || type == PYRAMID18)
1050 : {
1051 8 : const auto & on = owned_nodes;
1052 : // Define the edge midpoint nodes of the pyramid
1053 :
1054 : // Base node to base node midpoint nodes
1055 36 : owned_nodes.emplace_back(Node::build((*on[0] + *on[1]) / 2., 5));
1056 36 : owned_nodes.emplace_back(Node::build((*on[1] + *on[2]) / 2., 6));
1057 36 : owned_nodes.emplace_back(Node::build((*on[2] + *on[3]) / 2., 7));
1058 36 : owned_nodes.emplace_back(Node::build((*on[3] + *on[0]) / 2., 8));
1059 :
1060 : // Base node to apex node midpoint nodes
1061 36 : owned_nodes.emplace_back(Node::build(Point((*on[0] + *on[4]) / 2.), 9));
1062 36 : owned_nodes.emplace_back(Node::build(Point((*on[1] + *on[4]) / 2.), 10));
1063 36 : owned_nodes.emplace_back(Node::build(Point((*on[2] + *on[4]) / 2.), 11));
1064 36 : owned_nodes.emplace_back(Node::build(Point((*on[3] + *on[4]) / 2.), 12));
1065 :
1066 28 : if (type == PYRAMID14 || type == PYRAMID18)
1067 : {
1068 : // Define the square face midpoint node of the pyramid
1069 9 : owned_nodes.emplace_back(
1070 27 : Node::build(Point((*on[0] + *on[1] + *on[2] + *on[3]) / 4.), 13));
1071 :
1072 21 : if (type == PYRAMID18)
1073 : {
1074 : // Define the triangular face nodes
1075 18 : owned_nodes.emplace_back(Node::build(Point((*on[0] + *on[1] + *on[4]) / 3.), 14));
1076 18 : owned_nodes.emplace_back(Node::build(Point((*on[1] + *on[2] + *on[4]) / 3.), 15));
1077 18 : owned_nodes.emplace_back(Node::build(Point((*on[2] + *on[3] + *on[4]) / 3.), 16));
1078 22 : owned_nodes.emplace_back(Node::build(Point((*on[3] + *on[0] + *on[4]) / 3.), 17));
1079 : }
1080 8 : }
1081 : }
1082 :
1083 7 : else if (type != PYRAMID5)
1084 0 : libmesh_error_msg("Unsupported pyramid element: " << type_str);
1085 :
1086 : } // if Pyramid
1087 :
1088 : // Elems deriving from Tet
1089 134 : else if (type_str.compare(0, 3, "TET") == 0)
1090 : {
1091 :
1092 : // The ideal target element is a a regular tet with equilateral
1093 : // triangles for all faces, with volume equal to the volume of the
1094 : // reference element.
1095 :
1096 : // The volume of a tet is given by v = b * h / 3, where b is the area of
1097 : // the base face and h is the height of the apex node. The area of an
1098 : // equilateral triangle with side length s is b = sqrt(3) s^2 / 4.
1099 : // For all faces to have side length s, the height of the apex node is
1100 : // h = sqrt(2/3) * s. Then the volume is v = sqrt(2) * s^3 / 12.
1101 : // Solving for s, the side length that will preserve the volume of the
1102 : // reference element is s = (6 * sqrt(2) * v)^(1/3), where v is the volume
1103 : // of the non-optimal reference element (i.e., a right tet).
1104 :
1105 : // Side length that preserves the volume of the reference element
1106 28 : const auto side_length = std::cbrt(6. * sqrt_2 * ref_vol);
1107 : // tet height with the property that all faces are equilateral triangles
1108 28 : const auto target_height = sqrt_2 / sqrt_3 * side_length;
1109 :
1110 8 : const auto & s = side_length;
1111 8 : const auto & h = target_height;
1112 :
1113 : // For regular tet
1114 : // x y z node_id
1115 36 : owned_nodes.emplace_back(Node::build(Point(0., 0., 0.), 0));
1116 28 : owned_nodes.emplace_back(Node::build(Point(s, 0., 0.), 1));
1117 36 : owned_nodes.emplace_back(Node::build(Point(0.5 * s, 0.5 * sqrt_3 * s, 0.), 2));
1118 36 : owned_nodes.emplace_back(Node::build(Point(0.5 * s, sqrt_3 / 6. * s, h), 3));
1119 :
1120 28 : if (type == TET10 || type == TET14)
1121 : {
1122 6 : const auto & on = owned_nodes;
1123 : // Define the edge midpoint nodes of the tet
1124 :
1125 : // Base node to base node midpoint nodes
1126 27 : owned_nodes.emplace_back(Node::build(Point((*on[0] + *on[1]) / 2.), 4));
1127 27 : owned_nodes.emplace_back(Node::build(Point((*on[1] + *on[2]) / 2.), 5));
1128 27 : owned_nodes.emplace_back(Node::build(Point((*on[2] + *on[0]) / 2.), 6));
1129 : // Base node to apex node midpoint nodes
1130 27 : owned_nodes.emplace_back(Node::build(Point((*on[0] + *on[3]) / 2.), 7));
1131 27 : owned_nodes.emplace_back(Node::build(Point((*on[1] + *on[3]) / 2.), 8));
1132 27 : owned_nodes.emplace_back(Node::build(Point((*on[2] + *on[3]) / 2.), 9));
1133 :
1134 21 : if (type == TET14)
1135 : {
1136 : // Define the face midpoint nodes of the tet
1137 18 : owned_nodes.emplace_back(Node::build(Point((*on[0] + *on[1] + *on[2]) / 3.), 10));
1138 18 : owned_nodes.emplace_back(Node::build(Point((*on[0] + *on[1] + *on[3]) / 3.), 11));
1139 18 : owned_nodes.emplace_back(Node::build(Point((*on[1] + *on[2] + *on[3]) / 3.), 12));
1140 22 : owned_nodes.emplace_back(Node::build(Point((*on[0] + *on[2] + *on[3]) / 3.), 13));
1141 6 : }
1142 : }
1143 :
1144 7 : else if (type != TET4)
1145 0 : libmesh_error_msg("Unsupported tet element: " << type_str);
1146 :
1147 : } // if Tet
1148 :
1149 : // Set the target_elem equal to the reference elem
1150 : else
1151 1269 : for (const auto & node : target_elem->reference_elem()->node_ref_range())
1152 1777 : owned_nodes.emplace_back(Node::build(node, node.id()));
1153 :
1154 : // Set nodes of target element
1155 3003 : for (const auto & node_ptr : owned_nodes)
1156 3543 : target_elem->set_node(node_ptr->id(), node_ptr.get());
1157 :
1158 70 : libmesh_assert(relative_fuzzy_equals(target_elem->volume(), ref_vol, TOLERANCE));
1159 :
1160 316 : return std::make_pair(std::move(target_elem), std::move(owned_nodes));
1161 106 : }
1162 :
1163 246 : void VariationalSmootherSystem::get_target_to_reference_jacobian(
1164 : const Elem * const target_elem,
1165 : const FEMContext & femcontext,
1166 : std::vector<RealTensor> & jacobians,
1167 : std::vector<Real> & jacobian_dets)
1168 : {
1169 :
1170 246 : const auto dim = target_elem->dim();
1171 :
1172 70 : const auto & qrule_points = femcontext.get_element_qrule().get_points();
1173 70 : const auto & qrule_weights = femcontext.get_element_qrule().get_weights();
1174 70 : const auto nq_points = femcontext.get_element_qrule().n_points();
1175 :
1176 : // If the target element is the reference element, Jacobian matrix is
1177 : // identity, det of inverse is 1. These will only be overwritten if a
1178 : // different target element is explicitly specified.
1179 316 : jacobians = std::vector<RealTensor>(nq_points, RealTensor(
1180 : 1., 0., 0.,
1181 : 0., 1., 0.,
1182 70 : 0., 0., 1.));
1183 246 : jacobian_dets = std::vector<Real>(nq_points, 1.0);
1184 :
1185 : // Don't use "if (*target_elem == *(target_elem->reference_elem()))" here, it
1186 : // only compares global node ids, not the node locations themselves.
1187 70 : bool target_equals_reference = true;
1188 246 : const auto * ref_elem = target_elem->reference_elem();
1189 3003 : for (const auto local_id : make_range(target_elem->n_nodes()))
1190 2757 : target_equals_reference &= target_elem->node_ref(local_id) == ref_elem->node_ref(local_id);
1191 246 : if (target_equals_reference)
1192 106 : return;
1193 :
1194 : // Create FEMap to compute target_element mapping information
1195 220 : FEMap fe_map_target;
1196 :
1197 : // pre-request mapping derivatives
1198 40 : fe_map_target.get_dxyzdxi();
1199 40 : fe_map_target.get_dxyzdeta();
1200 40 : fe_map_target.get_dxyzdzeta();
1201 :
1202 : // build map
1203 140 : fe_map_target.init_reference_to_physical_map(dim, qrule_points, target_elem);
1204 140 : fe_map_target.compute_map(dim, qrule_weights, target_elem, /*d2phi=*/false);
1205 :
1206 3780 : for (const auto qp : make_range(nq_points))
1207 : {
1208 : // We use Larisa's H notation to denote the reference-to-target jacobian
1209 3640 : RealTensor H = get_jacobian_at_qp(fe_map_target, dim, qp);
1210 :
1211 : // The target-to-reference jacobian is the inverse of the
1212 : // reference-to-target jacobian
1213 3640 : jacobians[qp] = H.inverse();
1214 4680 : jacobian_dets[qp] = jacobians[qp].det();
1215 : }
1216 60 : }
1217 :
1218 182 : std::ostream &operator<<(std::ostream &os, const MeshQualityInfo & info)
1219 : {
1220 130 : os << "Mesh quality info:" << std::endl
1221 130 : << " Mesh distortion-dilation metric: "
1222 182 : << info.total_combined << std::endl
1223 130 : << " Mesh distortion metric: "
1224 182 : << info.total_distortion << std::endl
1225 130 : << " Mesh dilation metric: "
1226 182 : << info.total_dilation << std::endl
1227 130 : << " Max distortion-dilation is in elem "
1228 182 : << info.max_elem_combined.second << ": "
1229 182 : << info.max_elem_combined.first << std::endl
1230 130 : << " Max distortion is in elem "
1231 182 : << info.max_elem_distortion.second << ": "
1232 182 : << info.max_elem_distortion.first << std::endl
1233 130 : << " Max dilation is in elem "
1234 182 : << info.max_elem_dilation.second << ": "
1235 182 : << info.max_elem_dilation.first << std::endl
1236 130 : << " Max det(S) is in elem "
1237 182 : << info.max_elem_det_S.second << ": "
1238 182 : << info.max_elem_det_S.first << std::endl
1239 130 : << " Min distortion-dilation is in elem "
1240 182 : << info.min_elem_combined.second << ": "
1241 182 : << info.min_elem_combined.first << std::endl
1242 130 : << " Min distortion is in elem "
1243 182 : << info.min_elem_distortion.second << ": "
1244 182 : << info.min_elem_distortion.first << std::endl
1245 130 : << " Min dilation is in elem "
1246 182 : << info.min_elem_dilation.second << ": "
1247 182 : << info.min_elem_dilation.first << std::endl
1248 130 : << " Min det(S) is in elem "
1249 182 : << info.min_elem_det_S.second << ": "
1250 182 : << info.min_elem_det_S.first << std::endl
1251 182 : << " Max qp det(S): " << info.max_qp_det_S << std::endl
1252 182 : << " Min qp det(S): " << info.min_qp_det_S << std::endl
1253 182 : << " Mesh-integrated det(S): " << info.total_det_S << std::endl
1254 182 : << " Tangled: " << info.mesh_is_tangled << std::endl;
1255 :
1256 182 : return os;
1257 : }
1258 :
1259 : } // namespace libMesh
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