Intrepid
test_02.cpp
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43 
49 #include "Intrepid_FieldContainer.hpp"
50 #include "Intrepid_HGRAD_TRI_Cn_FEM_ORTH.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_TRI_Cn_FEM_ORTH) |\n" \
189  << "| |\n" \
190  << "| 1) Patch test involving mass and stiffness matrices, |\n" \
191  << "| for the Neumann problem on a triangular patch |\n" \
192  << "| 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  << "| Questions? Contact Pavel Bochev (pbboche@sandia.gov), |\n" \
197  << "| Denis Ridzal (dridzal@sandia.gov), |\n" \
198  << "| Kara Peterson (kjpeter@sandia.gov). |\n" \
199  << "| |\n" \
200  << "| Intrepid's website: http://trilinos.sandia.gov/packages/intrepid |\n" \
201  << "| Trilinos website: http://trilinos.sandia.gov |\n" \
202  << "| |\n" \
203  << "===============================================================================\n"\
204  << "| TEST 1: Patch test |\n"\
205  << "===============================================================================\n";
206 
207 
208  int errorFlag = 0;
209 
210  outStream -> precision(16);
211 
212 
213  try {
214 
215  int max_order = 10; // max total order of polynomial solution
216  DefaultCubatureFactory<double> cubFactory; // create cubature factory
217  shards::CellTopology cell(shards::getCellTopologyData< shards::Triangle<> >()); // create parent cell topology
218  shards::CellTopology side(shards::getCellTopologyData< shards::Line<> >()); // create relevant subcell (side) topology
219  int cellDim = cell.getDimension();
220  int sideDim = side.getDimension();
221 
222  // Define array containing points at which the solution is evaluated, in reference cell.
223  int numIntervals = 10;
224  int numInterpPoints = ((numIntervals + 1)*(numIntervals + 2))/2;
225  FieldContainer<double> interp_points_ref(numInterpPoints, 2);
226  int counter = 0;
227  for (int j=0; j<=numIntervals; j++) {
228  for (int i=0; i<=numIntervals; i++) {
229  if (i <= numIntervals-j) {
230  interp_points_ref(counter,0) = i*(1.0/numIntervals);
231  interp_points_ref(counter,1) = j*(1.0/numIntervals);
232  counter++;
233  }
234  }
235  }
236 
237  /* Parent cell definition. */
238  FieldContainer<double> cell_nodes(1, 3, cellDim);
239  // Perturbed reference triangle.
240  cell_nodes(0, 0, 0) = 0.1;
241  cell_nodes(0, 0, 1) = -0.1;
242  cell_nodes(0, 1, 0) = 1.1;
243  cell_nodes(0, 1, 1) = -0.1;
244  cell_nodes(0, 2, 0) = 0.1;
245  cell_nodes(0, 2, 1) = 0.9;
246  // Reference triangle.
247  /*cell_nodes(0, 0, 0) = 0.0;
248  cell_nodes(0, 0, 1) = 0.0;
249  cell_nodes(0, 1, 0) = 1.0;
250  cell_nodes(0, 1, 1) = 0.0;
251  cell_nodes(0, 2, 0) = 0.0;
252  cell_nodes(0, 2, 1) = 1.0;*/
253 
254  FieldContainer<double> interp_points(1, numInterpPoints, cellDim);
255  CellTools<double>::mapToPhysicalFrame(interp_points, interp_points_ref, cell_nodes, cell);
256  interp_points.resize(numInterpPoints, cellDim);
257 
258  for (int x_order=0; x_order <= max_order; x_order++) {
259  for (int y_order=0; y_order <= max_order-x_order; y_order++) {
260 
261  // evaluate exact solution
262  FieldContainer<double> exact_solution(1, numInterpPoints);
263  u_exact(exact_solution, interp_points, x_order, y_order);
264 
265  int total_order = std::max(x_order + y_order, 1);
266 
267  for (int basis_order=total_order; basis_order <= max_order; basis_order++) {
268 
269  // set test tolerance
270  double zero = basis_order*basis_order*100*INTREPID_TOL;
271 
272  //create basis
273  Teuchos::RCP<Basis<double,FieldContainer<double> > > basis =
274  Teuchos::rcp(new Basis_HGRAD_TRI_Cn_FEM_ORTH<double,FieldContainer<double> >(basis_order) );
275  int numFields = basis->getCardinality();
276 
277  // create cubatures
278  Teuchos::RCP<Cubature<double> > cellCub = cubFactory.create(cell, 2*basis_order);
279  Teuchos::RCP<Cubature<double> > sideCub = cubFactory.create(side, 2*basis_order);
280  int numCubPointsCell = cellCub->getNumPoints();
281  int numCubPointsSide = sideCub->getNumPoints();
282 
283  /* Computational arrays. */
284  /* Section 1: Related to parent cell integration. */
285  FieldContainer<double> cub_points_cell(numCubPointsCell, cellDim);
286  FieldContainer<double> cub_points_cell_physical(1, numCubPointsCell, cellDim);
287  FieldContainer<double> cub_weights_cell(numCubPointsCell);
288  FieldContainer<double> jacobian_cell(1, numCubPointsCell, cellDim, cellDim);
289  FieldContainer<double> jacobian_inv_cell(1, numCubPointsCell, cellDim, cellDim);
290  FieldContainer<double> jacobian_det_cell(1, numCubPointsCell);
291  FieldContainer<double> weighted_measure_cell(1, numCubPointsCell);
292 
293  FieldContainer<double> value_of_basis_at_cub_points_cell(numFields, numCubPointsCell);
294  FieldContainer<double> transformed_value_of_basis_at_cub_points_cell(1, numFields, numCubPointsCell);
295  FieldContainer<double> weighted_transformed_value_of_basis_at_cub_points_cell(1, numFields, numCubPointsCell);
296  FieldContainer<double> grad_of_basis_at_cub_points_cell(numFields, numCubPointsCell, cellDim);
297  FieldContainer<double> transformed_grad_of_basis_at_cub_points_cell(1, numFields, numCubPointsCell, cellDim);
298  FieldContainer<double> weighted_transformed_grad_of_basis_at_cub_points_cell(1, numFields, numCubPointsCell, cellDim);
299  FieldContainer<double> fe_matrix(1, numFields, numFields);
300 
301  FieldContainer<double> rhs_at_cub_points_cell_physical(1, numCubPointsCell);
302  FieldContainer<double> rhs_and_soln_vector(1, numFields);
303 
304  /* Section 2: Related to subcell (side) integration. */
305  unsigned numSides = 3;
306  FieldContainer<double> cub_points_side(numCubPointsSide, sideDim);
307  FieldContainer<double> cub_weights_side(numCubPointsSide);
308  FieldContainer<double> cub_points_side_refcell(numCubPointsSide, cellDim);
309  FieldContainer<double> cub_points_side_physical(1, numCubPointsSide, cellDim);
310  FieldContainer<double> jacobian_side_refcell(1, numCubPointsSide, cellDim, cellDim);
311  FieldContainer<double> jacobian_det_side_refcell(1, numCubPointsSide);
312  FieldContainer<double> weighted_measure_side_refcell(1, numCubPointsSide);
313 
314  FieldContainer<double> value_of_basis_at_cub_points_side_refcell(numFields, numCubPointsSide);
315  FieldContainer<double> transformed_value_of_basis_at_cub_points_side_refcell(1, numFields, numCubPointsSide);
316  FieldContainer<double> weighted_transformed_value_of_basis_at_cub_points_side_refcell(1, numFields, numCubPointsSide);
317  FieldContainer<double> neumann_data_at_cub_points_side_physical(1, numCubPointsSide);
318  FieldContainer<double> neumann_fields_per_side(1, numFields);
319 
320  /* Section 3: Related to global interpolant. */
321  FieldContainer<double> value_of_basis_at_interp_points(numFields, numInterpPoints);
322  FieldContainer<double> transformed_value_of_basis_at_interp_points(1, numFields, numInterpPoints);
323  FieldContainer<double> interpolant(1, numInterpPoints);
324 
325  FieldContainer<int> ipiv(numFields);
326 
327 
328 
329  /******************* START COMPUTATION ***********************/
330 
331  // get cubature points and weights
332  cellCub->getCubature(cub_points_cell, cub_weights_cell);
333 
334  // compute geometric cell information
335  CellTools<double>::setJacobian(jacobian_cell, cub_points_cell, cell_nodes, cell);
336  CellTools<double>::setJacobianInv(jacobian_inv_cell, jacobian_cell);
337  CellTools<double>::setJacobianDet(jacobian_det_cell, jacobian_cell);
338 
339  // compute weighted measure
340  FunctionSpaceTools::computeCellMeasure<double>(weighted_measure_cell, jacobian_det_cell, cub_weights_cell);
341 
343  // Computing mass matrices:
344  // tabulate values of basis functions at (reference) cubature points
345  basis->getValues(value_of_basis_at_cub_points_cell, cub_points_cell, OPERATOR_VALUE);
346 
347  // transform values of basis functions
348  FunctionSpaceTools::HGRADtransformVALUE<double>(transformed_value_of_basis_at_cub_points_cell,
349  value_of_basis_at_cub_points_cell);
350 
351  // multiply with weighted measure
352  FunctionSpaceTools::multiplyMeasure<double>(weighted_transformed_value_of_basis_at_cub_points_cell,
353  weighted_measure_cell,
354  transformed_value_of_basis_at_cub_points_cell);
355 
356  // compute mass matrices
357  FunctionSpaceTools::integrate<double>(fe_matrix,
358  transformed_value_of_basis_at_cub_points_cell,
359  weighted_transformed_value_of_basis_at_cub_points_cell,
360  COMP_BLAS);
362 
364  // Computing stiffness matrices:
365  // tabulate gradients of basis functions at (reference) cubature points
366  basis->getValues(grad_of_basis_at_cub_points_cell, cub_points_cell, OPERATOR_GRAD);
367 
368  // transform gradients of basis functions
369  FunctionSpaceTools::HGRADtransformGRAD<double>(transformed_grad_of_basis_at_cub_points_cell,
370  jacobian_inv_cell,
371  grad_of_basis_at_cub_points_cell);
372 
373  // multiply with weighted measure
374  FunctionSpaceTools::multiplyMeasure<double>(weighted_transformed_grad_of_basis_at_cub_points_cell,
375  weighted_measure_cell,
376  transformed_grad_of_basis_at_cub_points_cell);
377 
378  // compute stiffness matrices and sum into fe_matrix
379  FunctionSpaceTools::integrate<double>(fe_matrix,
380  transformed_grad_of_basis_at_cub_points_cell,
381  weighted_transformed_grad_of_basis_at_cub_points_cell,
382  COMP_BLAS,
383  true);
385 
387  // Computing RHS contributions:
388  // map cell (reference) cubature points to physical space
389  CellTools<double>::mapToPhysicalFrame(cub_points_cell_physical, cub_points_cell, cell_nodes, cell);
390 
391  // evaluate rhs function
392  rhsFunc(rhs_at_cub_points_cell_physical, cub_points_cell_physical, x_order, y_order);
393 
394  // compute rhs
395  FunctionSpaceTools::integrate<double>(rhs_and_soln_vector,
396  rhs_at_cub_points_cell_physical,
397  weighted_transformed_value_of_basis_at_cub_points_cell,
398  COMP_BLAS);
399 
400  // compute neumann b.c. contributions and adjust rhs
401  sideCub->getCubature(cub_points_side, cub_weights_side);
402  for (unsigned i=0; i<numSides; i++) {
403  // compute geometric cell information
404  CellTools<double>::mapToReferenceSubcell(cub_points_side_refcell, cub_points_side, sideDim, (int)i, cell);
405  CellTools<double>::setJacobian(jacobian_side_refcell, cub_points_side_refcell, cell_nodes, cell);
406  CellTools<double>::setJacobianDet(jacobian_det_side_refcell, jacobian_side_refcell);
407 
408  // compute weighted edge measure
409  FunctionSpaceTools::computeEdgeMeasure<double>(weighted_measure_side_refcell,
410  jacobian_side_refcell,
411  cub_weights_side,
412  i,
413  cell);
414 
415  // tabulate values of basis functions at side cubature points, in the reference parent cell domain
416  basis->getValues(value_of_basis_at_cub_points_side_refcell, cub_points_side_refcell, OPERATOR_VALUE);
417  // transform
418  FunctionSpaceTools::HGRADtransformVALUE<double>(transformed_value_of_basis_at_cub_points_side_refcell,
419  value_of_basis_at_cub_points_side_refcell);
420 
421  // multiply with weighted measure
422  FunctionSpaceTools::multiplyMeasure<double>(weighted_transformed_value_of_basis_at_cub_points_side_refcell,
423  weighted_measure_side_refcell,
424  transformed_value_of_basis_at_cub_points_side_refcell);
425 
426  // compute Neumann data
427  // map side cubature points in reference parent cell domain to physical space
428  CellTools<double>::mapToPhysicalFrame(cub_points_side_physical, cub_points_side_refcell, cell_nodes, cell);
429  // now compute data
430  neumann(neumann_data_at_cub_points_side_physical, cub_points_side_physical, jacobian_side_refcell,
431  cell, (int)i, x_order, y_order);
432 
433  FunctionSpaceTools::integrate<double>(neumann_fields_per_side,
434  neumann_data_at_cub_points_side_physical,
435  weighted_transformed_value_of_basis_at_cub_points_side_refcell,
436  COMP_BLAS);
437 
438  // adjust RHS
439  RealSpaceTools<double>::add(rhs_and_soln_vector, neumann_fields_per_side);;
440  }
442 
444  // Solution of linear system:
445  int info = 0;
446  Teuchos::LAPACK<int, double> solver;
447  solver.GESV(numFields, 1, &fe_matrix[0], numFields, &ipiv(0), &rhs_and_soln_vector[0], numFields, &info);
449 
451  // Building interpolant:
452  // evaluate basis at interpolation points
453  basis->getValues(value_of_basis_at_interp_points, interp_points_ref, OPERATOR_VALUE);
454  // transform values of basis functions
455  FunctionSpaceTools::HGRADtransformVALUE<double>(transformed_value_of_basis_at_interp_points,
456  value_of_basis_at_interp_points);
457  FunctionSpaceTools::evaluate<double>(interpolant, rhs_and_soln_vector, transformed_value_of_basis_at_interp_points);
459 
460  /******************* END COMPUTATION ***********************/
461 
462  RealSpaceTools<double>::subtract(interpolant, exact_solution);
463 
464  *outStream << "\nRelative norm-2 error between exact solution polynomial of order ("
465  << x_order << ", " << y_order << ") and finite element interpolant of order " << basis_order << ": "
466  << RealSpaceTools<double>::vectorNorm(&interpolant[0], interpolant.dimension(1), NORM_TWO) /
467  RealSpaceTools<double>::vectorNorm(&exact_solution[0], exact_solution.dimension(1), NORM_TWO) << "\n";
468 
469  if (RealSpaceTools<double>::vectorNorm(&interpolant[0], interpolant.dimension(1), NORM_TWO) /
470  RealSpaceTools<double>::vectorNorm(&exact_solution[0], exact_solution.dimension(1), NORM_TWO) > zero) {
471  *outStream << "\n\nPatch test failed for solution polynomial order ("
472  << x_order << ", " << y_order << ") and basis order " << basis_order << "\n\n";
473  errorFlag++;
474  }
475  } // end for basis_order
476  } // end for y_order
477  } // end for x_order
478 
479  }
480  // Catch unexpected errors
481  catch (const std::logic_error & err) {
482  *outStream << err.what() << "\n\n";
483  errorFlag = -1000;
484  };
485 
486  if (errorFlag != 0)
487  std::cout << "End Result: TEST FAILED\n";
488  else
489  std::cout << "End Result: TEST PASSED\n";
490 
491  // reset format state of std::cout
492  std::cout.copyfmt(oldFormatState);
493 
494  return errorFlag;
495 }
Intrepid::RealSpaceTools
Implementation of basic linear algebra functionality in Euclidean space.
Definition: Intrepid_RealSpaceTools.hpp:65
Intrepid::Basis_HGRAD_TRI_Cn_FEM_ORTH
Implementation of the default H(grad)-compatible orthogonal basis (Dubiner) of arbitrary degree on tr...
Definition: Intrepid_HGRAD_TRI_Cn_FEM_ORTH.hpp:66
Intrepid_CellTools.hpp
Header file for the Intrepid::CellTools class.
Intrepid::FieldContainer::dimension
int dimension(const int whichDim) const
Returns the specified dimension.
Definition: Intrepid_FieldContainerDef.hpp:658
Intrepid_FieldContainer.hpp
Header file for utility class to provide multidimensional containers.
Intrepid_ArrayTools.hpp
Header file for utility class to provide array tools, such as tensor contractions, etc.
Intrepid_DefaultCubatureFactory.hpp
Header file for the abstract base class Intrepid::DefaultCubatureFactory.
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::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::CellTools
A stateless class for operations on cell data. Provides methods for:
Definition: Intrepid_CellTools.hpp:109
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