Intrepid
test_02.cpp
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43 
49 #include "Intrepid_FieldContainer.hpp"
50 #include "Intrepid_HGRAD_QUAD_Cn_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 "Intrepid_PointTools.hpp"
57 #include "Teuchos_oblackholestream.hpp"
58 #include "Teuchos_RCP.hpp"
59 #include "Teuchos_GlobalMPISession.hpp"
60 #include "Teuchos_SerialDenseMatrix.hpp"
61 #include "Teuchos_SerialDenseVector.hpp"
62 #include "Teuchos_LAPACK.hpp"
63 
64 using namespace std;
65 using namespace Intrepid;
66 
67 void rhsFunc(FieldContainer<double> &, const FieldContainer<double> &, int, int);
68 void neumann(FieldContainer<double> & ,
69  const FieldContainer<double> & ,
70  const FieldContainer<double> & ,
71  const shards::CellTopology & ,
72  int, int, int);
73 void u_exact(FieldContainer<double> &, const FieldContainer<double> &, int, int);
74 
76 void rhsFunc(FieldContainer<double> & result,
77  const FieldContainer<double> & points,
78  int xd,
79  int yd) {
80 
81  int x = 0, y = 1;
82 
83  // second x-derivatives of u
84  if (xd > 1) {
85  for (int cell=0; cell<result.dimension(0); cell++) {
86  for (int pt=0; pt<result.dimension(1); pt++) {
87  result(cell,pt) = - xd*(xd-1)*std::pow(points(cell,pt,x), xd-2) * std::pow(points(cell,pt,y), yd);
88  }
89  }
90  }
91 
92  // second y-derivatives of u
93  if (yd > 1) {
94  for (int cell=0; cell<result.dimension(0); cell++) {
95  for (int pt=0; pt<result.dimension(1); pt++) {
96  result(cell,pt) -= yd*(yd-1)*std::pow(points(cell,pt,y), yd-2) * std::pow(points(cell,pt,x), xd);
97  }
98  }
99  }
100 
101  // add u
102  for (int cell=0; cell<result.dimension(0); cell++) {
103  for (int pt=0; pt<result.dimension(1); pt++) {
104  result(cell,pt) += std::pow(points(cell,pt,x), xd) * std::pow(points(cell,pt,y), yd);
105  }
106  }
107 
108 }
109 
110 
112 void neumann(FieldContainer<double> & result,
113  const FieldContainer<double> & points,
114  const FieldContainer<double> & jacs,
115  const shards::CellTopology & parentCell,
116  int sideOrdinal, int xd, int yd) {
117 
118  int x = 0, y = 1;
119 
120  int numCells = result.dimension(0);
121  int numPoints = result.dimension(1);
122 
123  FieldContainer<double> grad_u(numCells, numPoints, 2);
124  FieldContainer<double> side_normals(numCells, numPoints, 2);
125  FieldContainer<double> normal_lengths(numCells, numPoints);
126 
127  // first x-derivatives of u
128  if (xd > 0) {
129  for (int cell=0; cell<numCells; cell++) {
130  for (int pt=0; pt<numPoints; pt++) {
131  grad_u(cell,pt,x) = xd*std::pow(points(cell,pt,x), xd-1) * std::pow(points(cell,pt,y), yd);
132  }
133  }
134  }
135 
136  // first y-derivatives of u
137  if (yd > 0) {
138  for (int cell=0; cell<numCells; cell++) {
139  for (int pt=0; pt<numPoints; pt++) {
140  grad_u(cell,pt,y) = yd*std::pow(points(cell,pt,y), yd-1) * std::pow(points(cell,pt,x), xd);
141  }
142  }
143  }
144 
145  CellTools<double>::getPhysicalSideNormals(side_normals, jacs, sideOrdinal, parentCell);
146 
147  // scale normals
148  RealSpaceTools<double>::vectorNorm(normal_lengths, side_normals, NORM_TWO);
149  FunctionSpaceTools::scalarMultiplyDataData<double>(side_normals, normal_lengths, side_normals, true);
150 
151  FunctionSpaceTools::dotMultiplyDataData<double>(result, grad_u, side_normals);
152 
153 }
154 
156 void u_exact(FieldContainer<double> & result, const FieldContainer<double> & points, int xd, int yd) {
157  int x = 0, y = 1;
158  for (int cell=0; cell<result.dimension(0); cell++) {
159  for (int pt=0; pt<result.dimension(1); pt++) {
160  result(cell,pt) = std::pow(points(pt,x), xd)*std::pow(points(pt,y), yd);
161  }
162  }
163 }
164 
165 
166 
167 
168 int main(int argc, char *argv[]) {
169 
170  Teuchos::GlobalMPISession mpiSession(&argc, &argv);
171 
172  // This little trick lets us print to std::cout only if
173  // a (dummy) command-line argument is provided.
174  int iprint = argc - 1;
175  Teuchos::RCP<std::ostream> outStream;
176  Teuchos::oblackholestream bhs; // outputs nothing
177  if (iprint > 0)
178  outStream = Teuchos::rcp(&std::cout, false);
179  else
180  outStream = Teuchos::rcp(&bhs, false);
181 
182  // Save the format state of the original std::cout.
183  Teuchos::oblackholestream oldFormatState;
184  oldFormatState.copyfmt(std::cout);
185 
186  *outStream \
187  << "===============================================================================\n" \
188  << "| |\n" \
189  << "| Unit Test (Basis_HGRAD_QUAD_Cn_FEM) |\n" \
190  << "| |\n" \
191  << "| 1) Patch test involving mass and stiffness matrices, |\n" \
192  << "| for the Neumann problem on a physical parallelogram |\n" \
193  << "| AND a reference quad Omega with boundary Gamma. |\n" \
194  << "| |\n" \
195  << "| - div (grad u) + u = f in Omega, (grad u) . n = g on Gamma |\n" \
196  << "| |\n" \
197  << "| For a generic parallelogram, the basis recovers a complete |\n" \
198  << "| polynomial space of order 2. On a (scaled and/or translated) |\n" \
199  << "| reference quad, the basis recovers a complete tensor product |\n" \
200  << "| space of order 2 (i.e. incl. the x^2*y, x*y^2, x^2*y^2 terms). |\n" \
201  << "| |\n" \
202  << "| Questions? Contact Pavel Bochev (pbboche@sandia.gov), |\n" \
203  << "| Denis Ridzal (dridzal@sandia.gov), |\n" \
204  << "| Kara Peterson (kjpeter@sandia.gov). |\n" \
205  << "| |\n" \
206  << "| Intrepid's website: http://trilinos.sandia.gov/packages/intrepid |\n" \
207  << "| Trilinos website: http://trilinos.sandia.gov |\n" \
208  << "| |\n" \
209  << "===============================================================================\n"\
210  << "| TEST 1: Patch test |\n"\
211  << "===============================================================================\n";
212 
213 
214  int errorFlag = 0;
215 
216  outStream -> precision(16);
217 
218 
219  try {
220 
221  int max_order = 2; // max total order of polynomial solution
222  DefaultCubatureFactory<double> cubFactory; // create cubature factory
223  shards::CellTopology cell(shards::getCellTopologyData< shards::Quadrilateral<> >()); // create parent cell topology
224  shards::CellTopology side(shards::getCellTopologyData< shards::Line<> >()); // create relevant subcell (side) topology
225  int cellDim = cell.getDimension();
226  int sideDim = side.getDimension();
227 
228  // Define array containing points at which the solution is evaluated, in reference cell.
229  int numIntervals = 10;
230  int numInterpPoints = (numIntervals + 1)*(numIntervals + 1);
231  FieldContainer<double> interp_points_ref(numInterpPoints, 2);
232  int counter = 0;
233  for (int j=0; j<=numIntervals; j++) {
234  for (int i=0; i<=numIntervals; i++) {
235  interp_points_ref(counter,0) = i*(2.0/numIntervals)-1.0;
236  interp_points_ref(counter,1) = j*(2.0/numIntervals)-1.0;
237  counter++;
238  }
239  }
240 
241  /* Parent cell definition. */
242  FieldContainer<double> cell_nodes[2];
243  cell_nodes[0].resize(1, 4, cellDim);
244  cell_nodes[1].resize(1, 4, cellDim);
245 
246  // Generic parallelogram.
247  cell_nodes[0](0, 0, 0) = -5.0;
248  cell_nodes[0](0, 0, 1) = -1.0;
249  cell_nodes[0](0, 1, 0) = 4.0;
250  cell_nodes[0](0, 1, 1) = 1.0;
251  cell_nodes[0](0, 2, 0) = 8.0;
252  cell_nodes[0](0, 2, 1) = 3.0;
253  cell_nodes[0](0, 3, 0) = -1.0;
254  cell_nodes[0](0, 3, 1) = 1.0;
255  // Reference quad.
256  cell_nodes[1](0, 0, 0) = -1.0;
257  cell_nodes[1](0, 0, 1) = -1.0;
258  cell_nodes[1](0, 1, 0) = 1.0;
259  cell_nodes[1](0, 1, 1) = -1.0;
260  cell_nodes[1](0, 2, 0) = 1.0;
261  cell_nodes[1](0, 2, 1) = 1.0;
262  cell_nodes[1](0, 3, 0) = -1.0;
263  cell_nodes[1](0, 3, 1) = 1.0;
264 
265  std::stringstream mystream[2];
266  mystream[0].str("\n>> Now testing basis on a generic parallelogram ...\n");
267  mystream[1].str("\n>> Now testing basis on the reference quad ...\n");
268 
269  for (int pcell = 0; pcell < 2; pcell++) {
270  *outStream << mystream[pcell].str();
271  FieldContainer<double> interp_points(1, numInterpPoints, cellDim);
272  CellTools<double>::mapToPhysicalFrame(interp_points, interp_points_ref, cell_nodes[pcell], cell);
273  interp_points.resize(numInterpPoints, cellDim);
274 
275  for (int x_order=0; x_order <= max_order; x_order++) {
276  int max_y_order = max_order;
277  if (pcell == 0) {
278  max_y_order -= x_order;
279  }
280  for (int y_order=0; y_order <= max_y_order; y_order++) {
281 
282  // evaluate exact solution
283  FieldContainer<double> exact_solution(1, numInterpPoints);
284  u_exact(exact_solution, interp_points, x_order, y_order);
285 
286  int basis_order = 2;
287 
288  // set test tolerance
289  double zero = basis_order*basis_order*100*INTREPID_TOL;
290 
291  //create basis
292  Teuchos::RCP<Basis<double,FieldContainer<double> > > basis =
293  Teuchos::rcp(new Basis_HGRAD_QUAD_Cn_FEM<double,FieldContainer<double> >(2,POINTTYPE_SPECTRAL) );
294  int numFields = basis->getCardinality();
295 
296  // create cubatures
297  Teuchos::RCP<Cubature<double> > cellCub = cubFactory.create(cell, 2*basis_order);
298  Teuchos::RCP<Cubature<double> > sideCub = cubFactory.create(side, 2*basis_order);
299  int numCubPointsCell = cellCub->getNumPoints();
300  int numCubPointsSide = sideCub->getNumPoints();
301 
302  /* Computational arrays. */
303  /* Section 1: Related to parent cell integration. */
304  FieldContainer<double> cub_points_cell(numCubPointsCell, cellDim);
305  FieldContainer<double> cub_points_cell_physical(1, numCubPointsCell, cellDim);
306  FieldContainer<double> cub_weights_cell(numCubPointsCell);
307  FieldContainer<double> jacobian_cell(1, numCubPointsCell, cellDim, cellDim);
308  FieldContainer<double> jacobian_inv_cell(1, numCubPointsCell, cellDim, cellDim);
309  FieldContainer<double> jacobian_det_cell(1, numCubPointsCell);
310  FieldContainer<double> weighted_measure_cell(1, numCubPointsCell);
311 
312  FieldContainer<double> value_of_basis_at_cub_points_cell(numFields, numCubPointsCell);
313  FieldContainer<double> transformed_value_of_basis_at_cub_points_cell(1, numFields, numCubPointsCell);
314  FieldContainer<double> weighted_transformed_value_of_basis_at_cub_points_cell(1, numFields, numCubPointsCell);
315  FieldContainer<double> grad_of_basis_at_cub_points_cell(numFields, numCubPointsCell, cellDim);
316  FieldContainer<double> transformed_grad_of_basis_at_cub_points_cell(1, numFields, numCubPointsCell, cellDim);
317  FieldContainer<double> weighted_transformed_grad_of_basis_at_cub_points_cell(1, numFields, numCubPointsCell, cellDim);
318  FieldContainer<double> fe_matrix(1, numFields, numFields);
319 
320  FieldContainer<double> rhs_at_cub_points_cell_physical(1, numCubPointsCell);
321  FieldContainer<double> rhs_and_soln_vector(1, numFields);
322 
323  /* Section 2: Related to subcell (side) integration. */
324  unsigned numSides = 4;
325  FieldContainer<double> cub_points_side(numCubPointsSide, sideDim);
326  FieldContainer<double> cub_weights_side(numCubPointsSide);
327  FieldContainer<double> cub_points_side_refcell(numCubPointsSide, cellDim);
328  FieldContainer<double> cub_points_side_physical(1, numCubPointsSide, cellDim);
329  FieldContainer<double> jacobian_side_refcell(1, numCubPointsSide, cellDim, cellDim);
330  FieldContainer<double> jacobian_det_side_refcell(1, numCubPointsSide);
331  FieldContainer<double> weighted_measure_side_refcell(1, numCubPointsSide);
332 
333  FieldContainer<double> value_of_basis_at_cub_points_side_refcell(numFields, numCubPointsSide);
334  FieldContainer<double> transformed_value_of_basis_at_cub_points_side_refcell(1, numFields, numCubPointsSide);
335  FieldContainer<double> weighted_transformed_value_of_basis_at_cub_points_side_refcell(1, numFields, numCubPointsSide);
336  FieldContainer<double> neumann_data_at_cub_points_side_physical(1, numCubPointsSide);
337  FieldContainer<double> neumann_fields_per_side(1, numFields);
338 
339  /* Section 3: Related to global interpolant. */
340  FieldContainer<double> value_of_basis_at_interp_points(numFields, numInterpPoints);
341  FieldContainer<double> transformed_value_of_basis_at_interp_points(1, numFields, numInterpPoints);
342  FieldContainer<double> interpolant(1, numInterpPoints);
343 
344  FieldContainer<int> ipiv(numFields);
345 
346 
347 
348  /******************* START COMPUTATION ***********************/
349 
350  // get cubature points and weights
351  cellCub->getCubature(cub_points_cell, cub_weights_cell);
352 
353  // compute geometric cell information
354  CellTools<double>::setJacobian(jacobian_cell, cub_points_cell, cell_nodes[pcell], cell);
355  CellTools<double>::setJacobianInv(jacobian_inv_cell, jacobian_cell);
356  CellTools<double>::setJacobianDet(jacobian_det_cell, jacobian_cell);
357 
358  // compute weighted measure
359  FunctionSpaceTools::computeCellMeasure<double>(weighted_measure_cell, jacobian_det_cell, cub_weights_cell);
360 
362  // Computing mass matrices:
363  // tabulate values of basis functions at (reference) cubature points
364  basis->getValues(value_of_basis_at_cub_points_cell, cub_points_cell, OPERATOR_VALUE);
365 
366  // transform values of basis functions
367  FunctionSpaceTools::HGRADtransformVALUE<double>(transformed_value_of_basis_at_cub_points_cell,
368  value_of_basis_at_cub_points_cell);
369 
370  // multiply with weighted measure
371  FunctionSpaceTools::multiplyMeasure<double>(weighted_transformed_value_of_basis_at_cub_points_cell,
372  weighted_measure_cell,
373  transformed_value_of_basis_at_cub_points_cell);
374 
375  // compute mass matrices
376  FunctionSpaceTools::integrate<double>(fe_matrix,
377  transformed_value_of_basis_at_cub_points_cell,
378  weighted_transformed_value_of_basis_at_cub_points_cell,
379  COMP_BLAS);
381 
383  // Computing stiffness matrices:
384  // tabulate gradients of basis functions at (reference) cubature points
385  basis->getValues(grad_of_basis_at_cub_points_cell, cub_points_cell, OPERATOR_GRAD);
386 
387  // transform gradients of basis functions
388  FunctionSpaceTools::HGRADtransformGRAD<double>(transformed_grad_of_basis_at_cub_points_cell,
389  jacobian_inv_cell,
390  grad_of_basis_at_cub_points_cell);
391 
392  // multiply with weighted measure
393  FunctionSpaceTools::multiplyMeasure<double>(weighted_transformed_grad_of_basis_at_cub_points_cell,
394  weighted_measure_cell,
395  transformed_grad_of_basis_at_cub_points_cell);
396 
397  // compute stiffness matrices and sum into fe_matrix
398  FunctionSpaceTools::integrate<double>(fe_matrix,
399  transformed_grad_of_basis_at_cub_points_cell,
400  weighted_transformed_grad_of_basis_at_cub_points_cell,
401  COMP_BLAS,
402  true);
404 
406  // Computing RHS contributions:
407  // map cell (reference) cubature points to physical space
408  CellTools<double>::mapToPhysicalFrame(cub_points_cell_physical, cub_points_cell, cell_nodes[pcell], cell);
409 
410  // evaluate rhs function
411  rhsFunc(rhs_at_cub_points_cell_physical, cub_points_cell_physical, x_order, y_order);
412 
413  // compute rhs
414  FunctionSpaceTools::integrate<double>(rhs_and_soln_vector,
415  rhs_at_cub_points_cell_physical,
416  weighted_transformed_value_of_basis_at_cub_points_cell,
417  COMP_BLAS);
418 
419  // compute neumann b.c. contributions and adjust rhs
420  sideCub->getCubature(cub_points_side, cub_weights_side);
421  for (unsigned i=0; i<numSides; i++) {
422  // compute geometric cell information
423  CellTools<double>::mapToReferenceSubcell(cub_points_side_refcell, cub_points_side, sideDim, (int)i, cell);
424  CellTools<double>::setJacobian(jacobian_side_refcell, cub_points_side_refcell, cell_nodes[pcell], cell);
425  CellTools<double>::setJacobianDet(jacobian_det_side_refcell, jacobian_side_refcell);
426 
427  // compute weighted edge measure
428  FunctionSpaceTools::computeEdgeMeasure<double>(weighted_measure_side_refcell,
429  jacobian_side_refcell,
430  cub_weights_side,
431  i,
432  cell);
433 
434  // tabulate values of basis functions at side cubature points, in the reference parent cell domain
435  basis->getValues(value_of_basis_at_cub_points_side_refcell, cub_points_side_refcell, OPERATOR_VALUE);
436  // transform
437  FunctionSpaceTools::HGRADtransformVALUE<double>(transformed_value_of_basis_at_cub_points_side_refcell,
438  value_of_basis_at_cub_points_side_refcell);
439 
440  // multiply with weighted measure
441  FunctionSpaceTools::multiplyMeasure<double>(weighted_transformed_value_of_basis_at_cub_points_side_refcell,
442  weighted_measure_side_refcell,
443  transformed_value_of_basis_at_cub_points_side_refcell);
444 
445  // compute Neumann data
446  // map side cubature points in reference parent cell domain to physical space
447  CellTools<double>::mapToPhysicalFrame(cub_points_side_physical, cub_points_side_refcell, cell_nodes[pcell], cell);
448  // now compute data
449  neumann(neumann_data_at_cub_points_side_physical, cub_points_side_physical, jacobian_side_refcell,
450  cell, (int)i, x_order, y_order);
451 
452  FunctionSpaceTools::integrate<double>(neumann_fields_per_side,
453  neumann_data_at_cub_points_side_physical,
454  weighted_transformed_value_of_basis_at_cub_points_side_refcell,
455  COMP_BLAS);
456 
457  // adjust RHS
458  RealSpaceTools<double>::add(rhs_and_soln_vector, neumann_fields_per_side);;
459  }
461 
463  // Solution of linear system:
464  int info = 0;
465  Teuchos::LAPACK<int, double> solver;
466  solver.GESV(numFields, 1, &fe_matrix[0], numFields, &ipiv(0), &rhs_and_soln_vector[0], numFields, &info);
468 
470  // Building interpolant:
471  // evaluate basis at interpolation points
472  basis->getValues(value_of_basis_at_interp_points, interp_points_ref, OPERATOR_VALUE);
473  // transform values of basis functions
474  FunctionSpaceTools::HGRADtransformVALUE<double>(transformed_value_of_basis_at_interp_points,
475  value_of_basis_at_interp_points);
476  FunctionSpaceTools::evaluate<double>(interpolant, rhs_and_soln_vector, transformed_value_of_basis_at_interp_points);
478 
479  /******************* END COMPUTATION ***********************/
480 
481  RealSpaceTools<double>::subtract(interpolant, exact_solution);
482 
483  *outStream << "\nRelative norm-2 error between exact solution polynomial of order ("
484  << x_order << ", " << y_order << ") and finite element interpolant of order " << basis_order << ": "
485  << RealSpaceTools<double>::vectorNorm(&interpolant[0], interpolant.dimension(1), NORM_TWO) /
486  RealSpaceTools<double>::vectorNorm(&exact_solution[0], exact_solution.dimension(1), NORM_TWO) << "\n";
487 
488  if (RealSpaceTools<double>::vectorNorm(&interpolant[0], interpolant.dimension(1), NORM_TWO) /
489  RealSpaceTools<double>::vectorNorm(&exact_solution[0], exact_solution.dimension(1), NORM_TWO) > zero) {
490  *outStream << "\n\nPatch test failed for solution polynomial order ("
491  << x_order << ", " << y_order << ") and basis order " << basis_order << "\n\n";
492  errorFlag++;
493  }
494  } // end for y_order
495  } // end for x_order
496  } // end for pcell
497 
498  }
499  // Catch unexpected errors
500  catch (const std::logic_error & err) {
501  *outStream << err.what() << "\n\n";
502  errorFlag = -1000;
503  };
504 
505  if (errorFlag != 0)
506  std::cout << "End Result: TEST FAILED\n";
507  else
508  std::cout << "End Result: TEST PASSED\n";
509 
510  // reset format state of std::cout
511  std::cout.copyfmt(oldFormatState);
512 
513  return errorFlag;
514 }
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_HGRAD_QUAD_Cn_FEM.hpp
Header file for the Intrepid::HGRAD_QUAD_Cn_FEM class.
Intrepid_PointTools.hpp
Header file for utility class to provide point tools, such as barycentric coordinates,...
Intrepid_RealSpaceTools.hpp
Header file for classes providing basic linear algebra functionality in 1D, 2D and 3D.
Intrepid::Basis_HGRAD_QUAD_Cn_FEM
Definition: Intrepid_HGRAD_QUAD_Cn_FEM.hpp:69
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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