Intrepid
test_02.cpp
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43 
49 #include "Intrepid_FieldContainer.hpp"
50 #include "Intrepid_HGRAD_QUAD_C1_FEM.hpp"
51 #include "Intrepid_DefaultCubatureFactory.hpp"
52 #include "Intrepid_RealSpaceTools.hpp"
53 #include "Intrepid_ArrayTools.hpp"
54 #include "Intrepid_FunctionSpaceTools.hpp"
55 #include "Intrepid_CellTools.hpp"
56 #include "Teuchos_oblackholestream.hpp"
57 #include "Teuchos_RCP.hpp"
58 #include "Teuchos_GlobalMPISession.hpp"
59 #include "Teuchos_SerialDenseMatrix.hpp"
60 #include "Teuchos_SerialDenseVector.hpp"
61 #include "Teuchos_LAPACK.hpp"
62 
63 using namespace std;
64 using namespace Intrepid;
65 
66 void rhsFunc(FieldContainer<double> &, const FieldContainer<double> &, int, int);
67 void neumann(FieldContainer<double> & ,
68  const FieldContainer<double> & ,
69  const FieldContainer<double> & ,
70  const shards::CellTopology & ,
71  int, int, int);
72 void u_exact(FieldContainer<double> &, const FieldContainer<double> &, int, int);
73 
75 void rhsFunc(FieldContainer<double> & result,
76  const FieldContainer<double> & points,
77  int xd,
78  int yd) {
79 
80  int x = 0, y = 1;
81 
82  // second x-derivatives of u
83  if (xd > 1) {
84  for (int cell=0; cell<result.dimension(0); cell++) {
85  for (int pt=0; pt<result.dimension(1); pt++) {
86  result(cell,pt) = - xd*(xd-1)*std::pow(points(cell,pt,x), xd-2) * std::pow(points(cell,pt,y), yd);
87  }
88  }
89  }
90 
91  // second y-derivatives of u
92  if (yd > 1) {
93  for (int cell=0; cell<result.dimension(0); cell++) {
94  for (int pt=0; pt<result.dimension(1); pt++) {
95  result(cell,pt) -= yd*(yd-1)*std::pow(points(cell,pt,y), yd-2) * std::pow(points(cell,pt,x), xd);
96  }
97  }
98  }
99 
100  // add u
101  for (int cell=0; cell<result.dimension(0); cell++) {
102  for (int pt=0; pt<result.dimension(1); pt++) {
103  result(cell,pt) += std::pow(points(cell,pt,x), xd) * std::pow(points(cell,pt,y), yd);
104  }
105  }
106 
107 }
108 
109 
111 void neumann(FieldContainer<double> & result,
112  const FieldContainer<double> & points,
113  const FieldContainer<double> & jacs,
114  const shards::CellTopology & parentCell,
115  int sideOrdinal, int xd, int yd) {
116 
117  int x = 0, y = 1;
118 
119  int numCells = result.dimension(0);
120  int numPoints = result.dimension(1);
121 
122  FieldContainer<double> grad_u(numCells, numPoints, 2);
123  FieldContainer<double> side_normals(numCells, numPoints, 2);
124  FieldContainer<double> normal_lengths(numCells, numPoints);
125 
126  // first x-derivatives of u
127  if (xd > 0) {
128  for (int cell=0; cell<numCells; cell++) {
129  for (int pt=0; pt<numPoints; pt++) {
130  grad_u(cell,pt,x) = xd*std::pow(points(cell,pt,x), xd-1) * std::pow(points(cell,pt,y), yd);
131  }
132  }
133  }
134 
135  // first y-derivatives of u
136  if (yd > 0) {
137  for (int cell=0; cell<numCells; cell++) {
138  for (int pt=0; pt<numPoints; pt++) {
139  grad_u(cell,pt,y) = yd*std::pow(points(cell,pt,y), yd-1) * std::pow(points(cell,pt,x), xd);
140  }
141  }
142  }
143 
144  CellTools<double>::getPhysicalSideNormals(side_normals, jacs, sideOrdinal, parentCell);
145 
146  // scale normals
147  RealSpaceTools<double>::vectorNorm(normal_lengths, side_normals, NORM_TWO);
148  FunctionSpaceTools::scalarMultiplyDataData<double>(side_normals, normal_lengths, side_normals, true);
149 
150  FunctionSpaceTools::dotMultiplyDataData<double>(result, grad_u, side_normals);
151 
152 }
153 
155 void u_exact(FieldContainer<double> & result, const FieldContainer<double> & points, int xd, int yd) {
156  int x = 0, y = 1;
157  for (int cell=0; cell<result.dimension(0); cell++) {
158  for (int pt=0; pt<result.dimension(1); pt++) {
159  result(cell,pt) = std::pow(points(pt,x), xd)*std::pow(points(pt,y), yd);
160  }
161  }
162 }
163 
164 
165 
166 
167 int main(int argc, char *argv[]) {
168 
169  Teuchos::GlobalMPISession mpiSession(&argc, &argv);
170 
171  // This little trick lets us print to std::cout only if
172  // a (dummy) command-line argument is provided.
173  int iprint = argc - 1;
174  Teuchos::RCP<std::ostream> outStream;
175  Teuchos::oblackholestream bhs; // outputs nothing
176  if (iprint > 0)
177  outStream = Teuchos::rcp(&std::cout, false);
178  else
179  outStream = Teuchos::rcp(&bhs, false);
180 
181  // Save the format state of the original std::cout.
182  Teuchos::oblackholestream oldFormatState;
183  oldFormatState.copyfmt(std::cout);
184 
185  *outStream \
186  << "===============================================================================\n" \
187  << "| |\n" \
188  << "| Unit Test (Basis_HGRAD_QUAD_C1_FEM) |\n" \
189  << "| |\n" \
190  << "| 1) Patch test involving mass and stiffness matrices, |\n" \
191  << "| for the Neumann problem on a physical parallelogram |\n" \
192  << "| AND a reference quad Omega with boundary Gamma. |\n" \
193  << "| |\n" \
194  << "| - div (grad u) + u = f in Omega, (grad u) . n = g on Gamma |\n" \
195  << "| |\n" \
196  << "| For a generic parallelogram, the basis recovers a complete |\n" \
197  << "| polynomial space of order 1. On a (scaled and/or translated) |\n" \
198  << "| reference quad, the basis recovers a complete tensor product |\n" \
199  << "| space of order 1 (i.e. incl. the xy term). |\n" \
200  << "| |\n" \
201  << "| Questions? Contact Pavel Bochev (pbboche@sandia.gov), |\n" \
202  << "| Denis Ridzal (dridzal@sandia.gov), |\n" \
203  << "| Kara Peterson (kjpeter@sandia.gov). |\n" \
204  << "| |\n" \
205  << "| Intrepid's website: http://trilinos.sandia.gov/packages/intrepid |\n" \
206  << "| Trilinos website: http://trilinos.sandia.gov |\n" \
207  << "| |\n" \
208  << "===============================================================================\n"\
209  << "| TEST 1: Patch test |\n"\
210  << "===============================================================================\n";
211 
212 
213  int errorFlag = 0;
214 
215  outStream -> precision(16);
216 
217 
218  try {
219 
220  int max_order = 1; // max total order of polynomial solution
221  DefaultCubatureFactory<double> cubFactory; // create cubature factory
222  shards::CellTopology cell(shards::getCellTopologyData< shards::Quadrilateral<> >()); // create parent cell topology
223  shards::CellTopology side(shards::getCellTopologyData< shards::Line<> >()); // create relevant subcell (side) topology
224  int cellDim = cell.getDimension();
225  int sideDim = side.getDimension();
226 
227  // Define array containing points at which the solution is evaluated, in reference cell.
228  int numIntervals = 10;
229  int numInterpPoints = (numIntervals + 1)*(numIntervals + 1);
230  FieldContainer<double> interp_points_ref(numInterpPoints, 2);
231  int counter = 0;
232  for (int j=0; j<=numIntervals; j++) {
233  for (int i=0; i<=numIntervals; i++) {
234  interp_points_ref(counter,0) = i*(2.0/numIntervals)-1.0;
235  interp_points_ref(counter,1) = j*(2.0/numIntervals)-1.0;
236  counter++;
237  }
238  }
239 
240  /* Parent cell definition. */
241  FieldContainer<double> cell_nodes[2];
242  cell_nodes[0].resize(1, 4, cellDim);
243  cell_nodes[1].resize(1, 4, cellDim);
244 
245  // Generic parallelogram.
246  cell_nodes[0](0, 0, 0) = -5.0;
247  cell_nodes[0](0, 0, 1) = -1.0;
248  cell_nodes[0](0, 1, 0) = 4.0;
249  cell_nodes[0](0, 1, 1) = 1.0;
250  cell_nodes[0](0, 2, 0) = 8.0;
251  cell_nodes[0](0, 2, 1) = 3.0;
252  cell_nodes[0](0, 3, 0) = -1.0;
253  cell_nodes[0](0, 3, 1) = 1.0;
254  // Reference quad.
255  cell_nodes[1](0, 0, 0) = -1.0;
256  cell_nodes[1](0, 0, 1) = -1.0;
257  cell_nodes[1](0, 1, 0) = 1.0;
258  cell_nodes[1](0, 1, 1) = -1.0;
259  cell_nodes[1](0, 2, 0) = 1.0;
260  cell_nodes[1](0, 2, 1) = 1.0;
261  cell_nodes[1](0, 3, 0) = -1.0;
262  cell_nodes[1](0, 3, 1) = 1.0;
263 
264  std::stringstream mystream[2];
265  mystream[0].str("\n>> Now testing basis on a generic parallelogram ...\n");
266  mystream[1].str("\n>> Now testing basis on the reference quad ...\n");
267 
268  for (int pcell = 0; pcell < 2; pcell++) {
269  *outStream << mystream[pcell].str();
270  FieldContainer<double> interp_points(1, numInterpPoints, cellDim);
271  CellTools<double>::mapToPhysicalFrame(interp_points, interp_points_ref, cell_nodes[pcell], cell);
272  interp_points.resize(numInterpPoints, cellDim);
273 
274  for (int x_order=0; x_order <= max_order; x_order++) {
275  int max_y_order = max_order;
276  if (pcell == 0) {
277  max_y_order -= x_order;
278  }
279  for (int y_order=0; y_order <= max_y_order; y_order++) {
280 
281  // evaluate exact solution
282  FieldContainer<double> exact_solution(1, numInterpPoints);
283  u_exact(exact_solution, interp_points, x_order, y_order);
284 
285  int basis_order = 1;
286 
287  // set test tolerance
288  double zero = basis_order*basis_order*100*INTREPID_TOL;
289 
290  //create basis
291  Teuchos::RCP<Basis<double,FieldContainer<double> > > basis =
292  Teuchos::rcp(new Basis_HGRAD_QUAD_C1_FEM<double,FieldContainer<double> >() );
293  int numFields = basis->getCardinality();
294 
295  // create cubatures
296  Teuchos::RCP<Cubature<double> > cellCub = cubFactory.create(cell, 2*basis_order);
297  Teuchos::RCP<Cubature<double> > sideCub = cubFactory.create(side, 2*basis_order);
298  int numCubPointsCell = cellCub->getNumPoints();
299  int numCubPointsSide = sideCub->getNumPoints();
300 
301  /* Computational arrays. */
302  /* Section 1: Related to parent cell integration. */
303  FieldContainer<double> cub_points_cell(numCubPointsCell, cellDim);
304  FieldContainer<double> cub_points_cell_physical(1, numCubPointsCell, cellDim);
305  FieldContainer<double> cub_weights_cell(numCubPointsCell);
306  FieldContainer<double> jacobian_cell(1, numCubPointsCell, cellDim, cellDim);
307  FieldContainer<double> jacobian_inv_cell(1, numCubPointsCell, cellDim, cellDim);
308  FieldContainer<double> jacobian_det_cell(1, numCubPointsCell);
309  FieldContainer<double> weighted_measure_cell(1, numCubPointsCell);
310 
311  FieldContainer<double> value_of_basis_at_cub_points_cell(numFields, numCubPointsCell);
312  FieldContainer<double> transformed_value_of_basis_at_cub_points_cell(1, numFields, numCubPointsCell);
313  FieldContainer<double> weighted_transformed_value_of_basis_at_cub_points_cell(1, numFields, numCubPointsCell);
314  FieldContainer<double> grad_of_basis_at_cub_points_cell(numFields, numCubPointsCell, cellDim);
315  FieldContainer<double> transformed_grad_of_basis_at_cub_points_cell(1, numFields, numCubPointsCell, cellDim);
316  FieldContainer<double> weighted_transformed_grad_of_basis_at_cub_points_cell(1, numFields, numCubPointsCell, cellDim);
317  FieldContainer<double> fe_matrix(1, numFields, numFields);
318 
319  FieldContainer<double> rhs_at_cub_points_cell_physical(1, numCubPointsCell);
320  FieldContainer<double> rhs_and_soln_vector(1, numFields);
321 
322  /* Section 2: Related to subcell (side) integration. */
323  unsigned numSides = 4;
324  FieldContainer<double> cub_points_side(numCubPointsSide, sideDim);
325  FieldContainer<double> cub_weights_side(numCubPointsSide);
326  FieldContainer<double> cub_points_side_refcell(numCubPointsSide, cellDim);
327  FieldContainer<double> cub_points_side_physical(1, numCubPointsSide, cellDim);
328  FieldContainer<double> jacobian_side_refcell(1, numCubPointsSide, cellDim, cellDim);
329  FieldContainer<double> jacobian_det_side_refcell(1, numCubPointsSide);
330  FieldContainer<double> weighted_measure_side_refcell(1, numCubPointsSide);
331 
332  FieldContainer<double> value_of_basis_at_cub_points_side_refcell(numFields, numCubPointsSide);
333  FieldContainer<double> transformed_value_of_basis_at_cub_points_side_refcell(1, numFields, numCubPointsSide);
334  FieldContainer<double> weighted_transformed_value_of_basis_at_cub_points_side_refcell(1, numFields, numCubPointsSide);
335  FieldContainer<double> neumann_data_at_cub_points_side_physical(1, numCubPointsSide);
336  FieldContainer<double> neumann_fields_per_side(1, numFields);
337 
338  /* Section 3: Related to global interpolant. */
339  FieldContainer<double> value_of_basis_at_interp_points(numFields, numInterpPoints);
340  FieldContainer<double> transformed_value_of_basis_at_interp_points(1, numFields, numInterpPoints);
341  FieldContainer<double> interpolant(1, numInterpPoints);
342 
343  FieldContainer<int> ipiv(numFields);
344 
345 
346 
347  /******************* START COMPUTATION ***********************/
348 
349  // get cubature points and weights
350  cellCub->getCubature(cub_points_cell, cub_weights_cell);
351 
352  // compute geometric cell information
353  CellTools<double>::setJacobian(jacobian_cell, cub_points_cell, cell_nodes[pcell], cell);
354  CellTools<double>::setJacobianInv(jacobian_inv_cell, jacobian_cell);
355  CellTools<double>::setJacobianDet(jacobian_det_cell, jacobian_cell);
356 
357  // compute weighted measure
358  FunctionSpaceTools::computeCellMeasure<double>(weighted_measure_cell, jacobian_det_cell, cub_weights_cell);
359 
361  // Computing mass matrices:
362  // tabulate values of basis functions at (reference) cubature points
363  basis->getValues(value_of_basis_at_cub_points_cell, cub_points_cell, OPERATOR_VALUE);
364 
365  // transform values of basis functions
366  FunctionSpaceTools::HGRADtransformVALUE<double>(transformed_value_of_basis_at_cub_points_cell,
367  value_of_basis_at_cub_points_cell);
368 
369  // multiply with weighted measure
370  FunctionSpaceTools::multiplyMeasure<double>(weighted_transformed_value_of_basis_at_cub_points_cell,
371  weighted_measure_cell,
372  transformed_value_of_basis_at_cub_points_cell);
373 
374  // compute mass matrices
375  FunctionSpaceTools::integrate<double>(fe_matrix,
376  transformed_value_of_basis_at_cub_points_cell,
377  weighted_transformed_value_of_basis_at_cub_points_cell,
378  COMP_BLAS);
380 
382  // Computing stiffness matrices:
383  // tabulate gradients of basis functions at (reference) cubature points
384  basis->getValues(grad_of_basis_at_cub_points_cell, cub_points_cell, OPERATOR_GRAD);
385 
386  // transform gradients of basis functions
387  FunctionSpaceTools::HGRADtransformGRAD<double>(transformed_grad_of_basis_at_cub_points_cell,
388  jacobian_inv_cell,
389  grad_of_basis_at_cub_points_cell);
390 
391  // multiply with weighted measure
392  FunctionSpaceTools::multiplyMeasure<double>(weighted_transformed_grad_of_basis_at_cub_points_cell,
393  weighted_measure_cell,
394  transformed_grad_of_basis_at_cub_points_cell);
395 
396  // compute stiffness matrices and sum into fe_matrix
397  FunctionSpaceTools::integrate<double>(fe_matrix,
398  transformed_grad_of_basis_at_cub_points_cell,
399  weighted_transformed_grad_of_basis_at_cub_points_cell,
400  COMP_BLAS,
401  true);
403 
405  // Computing RHS contributions:
406  // map cell (reference) cubature points to physical space
407  CellTools<double>::mapToPhysicalFrame(cub_points_cell_physical, cub_points_cell, cell_nodes[pcell], cell);
408 
409  // evaluate rhs function
410  rhsFunc(rhs_at_cub_points_cell_physical, cub_points_cell_physical, x_order, y_order);
411 
412  // compute rhs
413  FunctionSpaceTools::integrate<double>(rhs_and_soln_vector,
414  rhs_at_cub_points_cell_physical,
415  weighted_transformed_value_of_basis_at_cub_points_cell,
416  COMP_BLAS);
417 
418  // compute neumann b.c. contributions and adjust rhs
419  sideCub->getCubature(cub_points_side, cub_weights_side);
420  for (unsigned i=0; i<numSides; i++) {
421  // compute geometric cell information
422  CellTools<double>::mapToReferenceSubcell(cub_points_side_refcell, cub_points_side, sideDim, (int)i, cell);
423  CellTools<double>::setJacobian(jacobian_side_refcell, cub_points_side_refcell, cell_nodes[pcell], cell);
424  CellTools<double>::setJacobianDet(jacobian_det_side_refcell, jacobian_side_refcell);
425 
426  // compute weighted edge measure
427  FunctionSpaceTools::computeEdgeMeasure<double>(weighted_measure_side_refcell,
428  jacobian_side_refcell,
429  cub_weights_side,
430  i,
431  cell);
432 
433  // tabulate values of basis functions at side cubature points, in the reference parent cell domain
434  basis->getValues(value_of_basis_at_cub_points_side_refcell, cub_points_side_refcell, OPERATOR_VALUE);
435  // transform
436  FunctionSpaceTools::HGRADtransformVALUE<double>(transformed_value_of_basis_at_cub_points_side_refcell,
437  value_of_basis_at_cub_points_side_refcell);
438 
439  // multiply with weighted measure
440  FunctionSpaceTools::multiplyMeasure<double>(weighted_transformed_value_of_basis_at_cub_points_side_refcell,
441  weighted_measure_side_refcell,
442  transformed_value_of_basis_at_cub_points_side_refcell);
443 
444  // compute Neumann data
445  // map side cubature points in reference parent cell domain to physical space
446  CellTools<double>::mapToPhysicalFrame(cub_points_side_physical, cub_points_side_refcell, cell_nodes[pcell], cell);
447  // now compute data
448  neumann(neumann_data_at_cub_points_side_physical, cub_points_side_physical, jacobian_side_refcell,
449  cell, (int)i, x_order, y_order);
450 
451  FunctionSpaceTools::integrate<double>(neumann_fields_per_side,
452  neumann_data_at_cub_points_side_physical,
453  weighted_transformed_value_of_basis_at_cub_points_side_refcell,
454  COMP_BLAS);
455 
456  // adjust RHS
457  RealSpaceTools<double>::add(rhs_and_soln_vector, neumann_fields_per_side);;
458  }
460 
462  // Solution of linear system:
463  int info = 0;
464  Teuchos::LAPACK<int, double> solver;
465  solver.GESV(numFields, 1, &fe_matrix[0], numFields, &ipiv(0), &rhs_and_soln_vector[0], numFields, &info);
467 
469  // Building interpolant:
470  // evaluate basis at interpolation points
471  basis->getValues(value_of_basis_at_interp_points, interp_points_ref, OPERATOR_VALUE);
472  // transform values of basis functions
473  FunctionSpaceTools::HGRADtransformVALUE<double>(transformed_value_of_basis_at_interp_points,
474  value_of_basis_at_interp_points);
475  FunctionSpaceTools::evaluate<double>(interpolant, rhs_and_soln_vector, transformed_value_of_basis_at_interp_points);
477 
478  /******************* END COMPUTATION ***********************/
479 
480  RealSpaceTools<double>::subtract(interpolant, exact_solution);
481 
482  *outStream << "\nRelative norm-2 error between exact solution polynomial of order ("
483  << x_order << ", " << y_order << ") and finite element interpolant of order " << basis_order << ": "
484  << RealSpaceTools<double>::vectorNorm(&interpolant[0], interpolant.dimension(1), NORM_TWO) /
485  RealSpaceTools<double>::vectorNorm(&exact_solution[0], exact_solution.dimension(1), NORM_TWO) << "\n";
486 
487  if (RealSpaceTools<double>::vectorNorm(&interpolant[0], interpolant.dimension(1), NORM_TWO) /
488  RealSpaceTools<double>::vectorNorm(&exact_solution[0], exact_solution.dimension(1), NORM_TWO) > zero) {
489  *outStream << "\n\nPatch test failed for solution polynomial order ("
490  << x_order << ", " << y_order << ") and basis order " << basis_order << "\n\n";
491  errorFlag++;
492  }
493  } // end for y_order
494  } // end for x_order
495  } // end for pcell
496 
497  }
498  // Catch unexpected errors
499  catch (const std::logic_error & err) {
500  *outStream << err.what() << "\n\n";
501  errorFlag = -1000;
502  };
503 
504  if (errorFlag != 0)
505  std::cout << "End Result: TEST FAILED\n";
506  else
507  std::cout << "End Result: TEST PASSED\n";
508 
509  // reset format state of std::cout
510  std::cout.copyfmt(oldFormatState);
511 
512  return errorFlag;
513 }
rhsFunc
void rhsFunc(FieldContainer< double > &, const FieldContainer< double > &, int, int, int)
right-hand side function
Definition: test_02.cpp:73
u_exact
void u_exact(FieldContainer< double > &, const FieldContainer< double > &, int, int, int)
exact solution
Definition: test_02.cpp:99
neumann
void neumann(FieldContainer< double > &, const FieldContainer< double > &, const FieldContainer< double > &, const shards::CellTopology &, int, int, int, int)
neumann boundary conditions
Definition: test_02.cpp:124
main
int main(int argc, char *argv[])
outdated tests for orthogonal bases
Definition: test_02.cpp:63
Intrepid_ArrayTools.hpp
Header file for utility class to provide array tools, such as tensor contractions,...
Intrepid_CellTools.hpp
Header file for the Intrepid::CellTools class.
Intrepid_DefaultCubatureFactory.hpp
Header file for the abstract base class Intrepid::DefaultCubatureFactory.
Intrepid_FieldContainer.hpp
Header file for utility class to provide multidimensional containers.
Intrepid_FunctionSpaceTools.hpp
Header file for the Intrepid::FunctionSpaceTools class.
Intrepid_RealSpaceTools.hpp
Header file for classes providing basic linear algebra functionality in 1D, 2D and 3D.
Intrepid::Basis_HGRAD_QUAD_C1_FEM
Implementation of the default H(grad)-compatible FEM basis of degree 1 on Quadrilateral cell.
Definition: Intrepid_HGRAD_QUAD_C1_FEM.hpp:81
Intrepid::CellTools
A stateless class for operations on cell data. Provides methods for:
Definition: Intrepid_CellTools.hpp:109
Intrepid::DefaultCubatureFactory
A factory class that generates specific instances of cubatures.
Definition: Intrepid_DefaultCubatureFactory.hpp:77
Intrepid::DefaultCubatureFactory::create
Teuchos::RCP< Cubature< Scalar, ArrayPoint, ArrayWeight > > create(const shards::CellTopology &cellTopology, const std::vector< int > &degree)
Factory method.
Definition: Intrepid_DefaultCubatureFactoryDef.hpp:53
Intrepid::FieldContainer::resize
void resize(const int dim0)
Resizes FieldContainer to a rank-1 container with the specified dimension, initialized by 0.
Definition: Intrepid_FieldContainerDef.hpp:1137
Intrepid::FieldContainer::dimension
int dimension(const int whichDim) const
Returns the specified dimension.
Definition: Intrepid_FieldContainerDef.hpp:658
Intrepid::RealSpaceTools
Implementation of basic linear algebra functionality in Euclidean space.
Definition: Intrepid_RealSpaceTools.hpp:65
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