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Intrepid
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00001 // @HEADER 00002 // ************************************************************************ 00003 // 00004 // Intrepid Package 00005 // Copyright (2007) Sandia Corporation 00006 // 00007 // Under terms of Contract DE-AC04-94AL85000, there is a non-exclusive 00008 // license for use of this work by or on behalf of the U.S. Government. 00009 // 00010 // This library is free software; you can redistribute it and/or modify 00011 // it under the terms of the GNU Lesser General Public License as 00012 // published by the Free Software Foundation; either version 2.1 of the 00013 // License, or (at your option) any later version. 00014 // 00015 // This library is distributed in the hope that it will be useful, but 00016 // WITHOUT ANY WARRANTY; without even the implied warranty of 00017 // MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU 00018 // Lesser General Public License for more details. 00019 // 00020 // You should have received a copy of the GNU Lesser General Public 00021 // License along with this library; if not, write to the Free Software 00022 // Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 00023 // USA 00024 // Questions? Contact Pavel Bochev (pbboche@sandia.gov), 00025 // Denis Ridzal (dridzal@sandia.gov), 00026 // Kara Peterson (kjpeter@sandia.gov). 00027 // 00028 // ************************************************************************ 00029 // @HEADER 00030 00036 #include "Intrepid_FieldContainer.hpp" 00037 #include "Intrepid_HGRAD_QUAD_Cn_FEM.hpp" 00038 #include "Intrepid_DefaultCubatureFactory.hpp" 00039 #include "Intrepid_RealSpaceTools.hpp" 00040 #include "Intrepid_ArrayTools.hpp" 00041 #include "Intrepid_FunctionSpaceTools.hpp" 00042 #include "Intrepid_CellTools.hpp" 00043 #include "Intrepid_PointTools.hpp" 00044 #include "Teuchos_oblackholestream.hpp" 00045 #include "Teuchos_RCP.hpp" 00046 #include "Teuchos_GlobalMPISession.hpp" 00047 #include "Teuchos_SerialDenseMatrix.hpp" 00048 #include "Teuchos_SerialDenseVector.hpp" 00049 #include "Teuchos_LAPACK.hpp" 00050 00051 using namespace std; 00052 using namespace Intrepid; 00053 00054 void rhsFunc(FieldContainer<double> &, const FieldContainer<double> &, int, int); 00055 void neumann(FieldContainer<double> & , 00056 const FieldContainer<double> & , 00057 const FieldContainer<double> & , 00058 const shards::CellTopology & , 00059 int, int, int); 00060 void u_exact(FieldContainer<double> &, const FieldContainer<double> &, int, int); 00061 00063 void rhsFunc(FieldContainer<double> & result, 00064 const FieldContainer<double> & points, 00065 int xd, 00066 int yd) { 00067 00068 int x = 0, y = 1; 00069 00070 // second x-derivatives of u 00071 if (xd > 1) { 00072 for (int cell=0; cell<result.dimension(0); cell++) { 00073 for (int pt=0; pt<result.dimension(1); pt++) { 00074 result(cell,pt) = - xd*(xd-1)*std::pow(points(cell,pt,x), xd-2) * std::pow(points(cell,pt,y), yd); 00075 } 00076 } 00077 } 00078 00079 // second y-derivatives of u 00080 if (yd > 1) { 00081 for (int cell=0; cell<result.dimension(0); cell++) { 00082 for (int pt=0; pt<result.dimension(1); pt++) { 00083 result(cell,pt) -= yd*(yd-1)*std::pow(points(cell,pt,y), yd-2) * std::pow(points(cell,pt,x), xd); 00084 } 00085 } 00086 } 00087 00088 // add u 00089 for (int cell=0; cell<result.dimension(0); cell++) { 00090 for (int pt=0; pt<result.dimension(1); pt++) { 00091 result(cell,pt) += std::pow(points(cell,pt,x), xd) * std::pow(points(cell,pt,y), yd); 00092 } 00093 } 00094 00095 } 00096 00097 00099 void neumann(FieldContainer<double> & result, 00100 const FieldContainer<double> & points, 00101 const FieldContainer<double> & jacs, 00102 const shards::CellTopology & parentCell, 00103 int sideOrdinal, int xd, int yd) { 00104 00105 int x = 0, y = 1; 00106 00107 int numCells = result.dimension(0); 00108 int numPoints = result.dimension(1); 00109 00110 FieldContainer<double> grad_u(numCells, numPoints, 2); 00111 FieldContainer<double> side_normals(numCells, numPoints, 2); 00112 FieldContainer<double> normal_lengths(numCells, numPoints); 00113 00114 // first x-derivatives of u 00115 if (xd > 0) { 00116 for (int cell=0; cell<numCells; cell++) { 00117 for (int pt=0; pt<numPoints; pt++) { 00118 grad_u(cell,pt,x) = xd*std::pow(points(cell,pt,x), xd-1) * std::pow(points(cell,pt,y), yd); 00119 } 00120 } 00121 } 00122 00123 // first y-derivatives of u 00124 if (yd > 0) { 00125 for (int cell=0; cell<numCells; cell++) { 00126 for (int pt=0; pt<numPoints; pt++) { 00127 grad_u(cell,pt,y) = yd*std::pow(points(cell,pt,y), yd-1) * std::pow(points(cell,pt,x), xd); 00128 } 00129 } 00130 } 00131 00132 CellTools<double>::getPhysicalSideNormals(side_normals, jacs, sideOrdinal, parentCell); 00133 00134 // scale normals 00135 RealSpaceTools<double>::vectorNorm(normal_lengths, side_normals, NORM_TWO); 00136 FunctionSpaceTools::scalarMultiplyDataData<double>(side_normals, normal_lengths, side_normals, true); 00137 00138 FunctionSpaceTools::dotMultiplyDataData<double>(result, grad_u, side_normals); 00139 00140 } 00141 00143 void u_exact(FieldContainer<double> & result, const FieldContainer<double> & points, int xd, int yd) { 00144 int x = 0, y = 1; 00145 for (int cell=0; cell<result.dimension(0); cell++) { 00146 for (int pt=0; pt<result.dimension(1); pt++) { 00147 result(cell,pt) = std::pow(points(pt,x), xd)*std::pow(points(pt,y), yd); 00148 } 00149 } 00150 } 00151 00152 00153 00154 00155 int main(int argc, char *argv[]) { 00156 00157 Teuchos::GlobalMPISession mpiSession(&argc, &argv); 00158 00159 // This little trick lets us print to std::cout only if 00160 // a (dummy) command-line argument is provided. 00161 int iprint = argc - 1; 00162 Teuchos::RCP<std::ostream> outStream; 00163 Teuchos::oblackholestream bhs; // outputs nothing 00164 if (iprint > 0) 00165 outStream = Teuchos::rcp(&std::cout, false); 00166 else 00167 outStream = Teuchos::rcp(&bhs, false); 00168 00169 // Save the format state of the original std::cout. 00170 Teuchos::oblackholestream oldFormatState; 00171 oldFormatState.copyfmt(std::cout); 00172 00173 *outStream \ 00174 << "===============================================================================\n" \ 00175 << "| |\n" \ 00176 << "| Unit Test (Basis_HGRAD_QUAD_Cn_FEM) |\n" \ 00177 << "| |\n" \ 00178 << "| 1) Patch test involving mass and stiffness matrices, |\n" \ 00179 << "| for the Neumann problem on a physical parallelogram |\n" \ 00180 << "| AND a reference quad Omega with boundary Gamma. |\n" \ 00181 << "| |\n" \ 00182 << "| - div (grad u) + u = f in Omega, (grad u) . n = g on Gamma |\n" \ 00183 << "| |\n" \ 00184 << "| For a generic parallelogram, the basis recovers a complete |\n" \ 00185 << "| polynomial space of order 2. On a (scaled and/or translated) |\n" \ 00186 << "| reference quad, the basis recovers a complete tensor product |\n" \ 00187 << "| space of order 2 (i.e. incl. the x^2*y, x*y^2, x^2*y^2 terms). |\n" \ 00188 << "| |\n" \ 00189 << "| Questions? Contact Pavel Bochev (pbboche@sandia.gov), |\n" \ 00190 << "| Denis Ridzal (dridzal@sandia.gov), |\n" \ 00191 << "| Kara Peterson (kjpeter@sandia.gov). |\n" \ 00192 << "| |\n" \ 00193 << "| Intrepid's website: http://trilinos.sandia.gov/packages/intrepid |\n" \ 00194 << "| Trilinos website: http://trilinos.sandia.gov |\n" \ 00195 << "| |\n" \ 00196 << "===============================================================================\n"\ 00197 << "| TEST 1: Patch test |\n"\ 00198 << "===============================================================================\n"; 00199 00200 00201 int errorFlag = 0; 00202 00203 outStream -> precision(16); 00204 00205 00206 try { 00207 00208 int max_order = 2; // max total order of polynomial solution 00209 DefaultCubatureFactory<double> cubFactory; // create cubature factory 00210 shards::CellTopology cell(shards::getCellTopologyData< shards::Quadrilateral<> >()); // create parent cell topology 00211 shards::CellTopology side(shards::getCellTopologyData< shards::Line<> >()); // create relevant subcell (side) topology 00212 int cellDim = cell.getDimension(); 00213 int sideDim = side.getDimension(); 00214 00215 // Define array containing points at which the solution is evaluated, in reference cell. 00216 int numIntervals = 10; 00217 int numInterpPoints = (numIntervals + 1)*(numIntervals + 1); 00218 FieldContainer<double> interp_points_ref(numInterpPoints, 2); 00219 int counter = 0; 00220 for (int j=0; j<=numIntervals; j++) { 00221 for (int i=0; i<=numIntervals; i++) { 00222 interp_points_ref(counter,0) = i*(2.0/numIntervals)-1.0; 00223 interp_points_ref(counter,1) = j*(2.0/numIntervals)-1.0; 00224 counter++; 00225 } 00226 } 00227 00228 /* Parent cell definition. */ 00229 FieldContainer<double> cell_nodes[2]; 00230 cell_nodes[0].resize(1, 4, cellDim); 00231 cell_nodes[1].resize(1, 4, cellDim); 00232 00233 // Generic parallelogram. 00234 cell_nodes[0](0, 0, 0) = -5.0; 00235 cell_nodes[0](0, 0, 1) = -1.0; 00236 cell_nodes[0](0, 1, 0) = 4.0; 00237 cell_nodes[0](0, 1, 1) = 1.0; 00238 cell_nodes[0](0, 2, 0) = 8.0; 00239 cell_nodes[0](0, 2, 1) = 3.0; 00240 cell_nodes[0](0, 3, 0) = -1.0; 00241 cell_nodes[0](0, 3, 1) = 1.0; 00242 // Reference quad. 00243 cell_nodes[1](0, 0, 0) = -1.0; 00244 cell_nodes[1](0, 0, 1) = -1.0; 00245 cell_nodes[1](0, 1, 0) = 1.0; 00246 cell_nodes[1](0, 1, 1) = -1.0; 00247 cell_nodes[1](0, 2, 0) = 1.0; 00248 cell_nodes[1](0, 2, 1) = 1.0; 00249 cell_nodes[1](0, 3, 0) = -1.0; 00250 cell_nodes[1](0, 3, 1) = 1.0; 00251 00252 std::stringstream mystream[2]; 00253 mystream[0].str("\n>> Now testing basis on a generic parallelogram ...\n"); 00254 mystream[1].str("\n>> Now testing basis on the reference quad ...\n"); 00255 00256 for (int pcell = 0; pcell < 2; pcell++) { 00257 *outStream << mystream[pcell].str(); 00258 FieldContainer<double> interp_points(1, numInterpPoints, cellDim); 00259 CellTools<double>::mapToPhysicalFrame(interp_points, interp_points_ref, cell_nodes[pcell], cell); 00260 interp_points.resize(numInterpPoints, cellDim); 00261 00262 for (int x_order=0; x_order <= max_order; x_order++) { 00263 int max_y_order = max_order; 00264 if (pcell == 0) { 00265 max_y_order -= x_order; 00266 } 00267 for (int y_order=0; y_order <= max_y_order; y_order++) { 00268 00269 // evaluate exact solution 00270 FieldContainer<double> exact_solution(1, numInterpPoints); 00271 u_exact(exact_solution, interp_points, x_order, y_order); 00272 00273 int basis_order = 2; 00274 00275 // set test tolerance 00276 double zero = basis_order*basis_order*100*INTREPID_TOL; 00277 00278 //create basis 00279 Teuchos::RCP<Basis<double,FieldContainer<double> > > basis = 00280 Teuchos::rcp(new Basis_HGRAD_QUAD_Cn_FEM<double,FieldContainer<double> >(2,POINTTYPE_SPECTRAL) ); 00281 int numFields = basis->getCardinality(); 00282 00283 // create cubatures 00284 Teuchos::RCP<Cubature<double> > cellCub = cubFactory.create(cell, 2*basis_order); 00285 Teuchos::RCP<Cubature<double> > sideCub = cubFactory.create(side, 2*basis_order); 00286 int numCubPointsCell = cellCub->getNumPoints(); 00287 int numCubPointsSide = sideCub->getNumPoints(); 00288 00289 /* Computational arrays. */ 00290 /* Section 1: Related to parent cell integration. */ 00291 FieldContainer<double> cub_points_cell(numCubPointsCell, cellDim); 00292 FieldContainer<double> cub_points_cell_physical(1, numCubPointsCell, cellDim); 00293 FieldContainer<double> cub_weights_cell(numCubPointsCell); 00294 FieldContainer<double> jacobian_cell(1, numCubPointsCell, cellDim, cellDim); 00295 FieldContainer<double> jacobian_inv_cell(1, numCubPointsCell, cellDim, cellDim); 00296 FieldContainer<double> jacobian_det_cell(1, numCubPointsCell); 00297 FieldContainer<double> weighted_measure_cell(1, numCubPointsCell); 00298 00299 FieldContainer<double> value_of_basis_at_cub_points_cell(numFields, numCubPointsCell); 00300 FieldContainer<double> transformed_value_of_basis_at_cub_points_cell(1, numFields, numCubPointsCell); 00301 FieldContainer<double> weighted_transformed_value_of_basis_at_cub_points_cell(1, numFields, numCubPointsCell); 00302 FieldContainer<double> grad_of_basis_at_cub_points_cell(numFields, numCubPointsCell, cellDim); 00303 FieldContainer<double> transformed_grad_of_basis_at_cub_points_cell(1, numFields, numCubPointsCell, cellDim); 00304 FieldContainer<double> weighted_transformed_grad_of_basis_at_cub_points_cell(1, numFields, numCubPointsCell, cellDim); 00305 FieldContainer<double> fe_matrix(1, numFields, numFields); 00306 00307 FieldContainer<double> rhs_at_cub_points_cell_physical(1, numCubPointsCell); 00308 FieldContainer<double> rhs_and_soln_vector(1, numFields); 00309 00310 /* Section 2: Related to subcell (side) integration. */ 00311 unsigned numSides = 4; 00312 FieldContainer<double> cub_points_side(numCubPointsSide, sideDim); 00313 FieldContainer<double> cub_weights_side(numCubPointsSide); 00314 FieldContainer<double> cub_points_side_refcell(numCubPointsSide, cellDim); 00315 FieldContainer<double> cub_points_side_physical(1, numCubPointsSide, cellDim); 00316 FieldContainer<double> jacobian_side_refcell(1, numCubPointsSide, cellDim, cellDim); 00317 FieldContainer<double> jacobian_det_side_refcell(1, numCubPointsSide); 00318 FieldContainer<double> weighted_measure_side_refcell(1, numCubPointsSide); 00319 00320 FieldContainer<double> value_of_basis_at_cub_points_side_refcell(numFields, numCubPointsSide); 00321 FieldContainer<double> transformed_value_of_basis_at_cub_points_side_refcell(1, numFields, numCubPointsSide); 00322 FieldContainer<double> weighted_transformed_value_of_basis_at_cub_points_side_refcell(1, numFields, numCubPointsSide); 00323 FieldContainer<double> neumann_data_at_cub_points_side_physical(1, numCubPointsSide); 00324 FieldContainer<double> neumann_fields_per_side(1, numFields); 00325 00326 /* Section 3: Related to global interpolant. */ 00327 FieldContainer<double> value_of_basis_at_interp_points(numFields, numInterpPoints); 00328 FieldContainer<double> transformed_value_of_basis_at_interp_points(1, numFields, numInterpPoints); 00329 FieldContainer<double> interpolant(1, numInterpPoints); 00330 00331 FieldContainer<int> ipiv(numFields); 00332 00333 00334 00335 /******************* START COMPUTATION ***********************/ 00336 00337 // get cubature points and weights 00338 cellCub->getCubature(cub_points_cell, cub_weights_cell); 00339 00340 // compute geometric cell information 00341 CellTools<double>::setJacobian(jacobian_cell, cub_points_cell, cell_nodes[pcell], cell); 00342 CellTools<double>::setJacobianInv(jacobian_inv_cell, jacobian_cell); 00343 CellTools<double>::setJacobianDet(jacobian_det_cell, jacobian_cell); 00344 00345 // compute weighted measure 00346 FunctionSpaceTools::computeCellMeasure<double>(weighted_measure_cell, jacobian_det_cell, cub_weights_cell); 00347 00349 // Computing mass matrices: 00350 // tabulate values of basis functions at (reference) cubature points 00351 basis->getValues(value_of_basis_at_cub_points_cell, cub_points_cell, OPERATOR_VALUE); 00352 00353 // transform values of basis functions 00354 FunctionSpaceTools::HGRADtransformVALUE<double>(transformed_value_of_basis_at_cub_points_cell, 00355 value_of_basis_at_cub_points_cell); 00356 00357 // multiply with weighted measure 00358 FunctionSpaceTools::multiplyMeasure<double>(weighted_transformed_value_of_basis_at_cub_points_cell, 00359 weighted_measure_cell, 00360 transformed_value_of_basis_at_cub_points_cell); 00361 00362 // compute mass matrices 00363 FunctionSpaceTools::integrate<double>(fe_matrix, 00364 transformed_value_of_basis_at_cub_points_cell, 00365 weighted_transformed_value_of_basis_at_cub_points_cell, 00366 COMP_BLAS); 00368 00370 // Computing stiffness matrices: 00371 // tabulate gradients of basis functions at (reference) cubature points 00372 basis->getValues(grad_of_basis_at_cub_points_cell, cub_points_cell, OPERATOR_GRAD); 00373 00374 // transform gradients of basis functions 00375 FunctionSpaceTools::HGRADtransformGRAD<double>(transformed_grad_of_basis_at_cub_points_cell, 00376 jacobian_inv_cell, 00377 grad_of_basis_at_cub_points_cell); 00378 00379 // multiply with weighted measure 00380 FunctionSpaceTools::multiplyMeasure<double>(weighted_transformed_grad_of_basis_at_cub_points_cell, 00381 weighted_measure_cell, 00382 transformed_grad_of_basis_at_cub_points_cell); 00383 00384 // compute stiffness matrices and sum into fe_matrix 00385 FunctionSpaceTools::integrate<double>(fe_matrix, 00386 transformed_grad_of_basis_at_cub_points_cell, 00387 weighted_transformed_grad_of_basis_at_cub_points_cell, 00388 COMP_BLAS, 00389 true); 00391 00393 // Computing RHS contributions: 00394 // map cell (reference) cubature points to physical space 00395 CellTools<double>::mapToPhysicalFrame(cub_points_cell_physical, cub_points_cell, cell_nodes[pcell], cell); 00396 00397 // evaluate rhs function 00398 rhsFunc(rhs_at_cub_points_cell_physical, cub_points_cell_physical, x_order, y_order); 00399 00400 // compute rhs 00401 FunctionSpaceTools::integrate<double>(rhs_and_soln_vector, 00402 rhs_at_cub_points_cell_physical, 00403 weighted_transformed_value_of_basis_at_cub_points_cell, 00404 COMP_BLAS); 00405 00406 // compute neumann b.c. contributions and adjust rhs 00407 sideCub->getCubature(cub_points_side, cub_weights_side); 00408 for (unsigned i=0; i<numSides; i++) { 00409 // compute geometric cell information 00410 CellTools<double>::mapToReferenceSubcell(cub_points_side_refcell, cub_points_side, sideDim, (int)i, cell); 00411 CellTools<double>::setJacobian(jacobian_side_refcell, cub_points_side_refcell, cell_nodes[pcell], cell); 00412 CellTools<double>::setJacobianDet(jacobian_det_side_refcell, jacobian_side_refcell); 00413 00414 // compute weighted edge measure 00415 FunctionSpaceTools::computeEdgeMeasure<double>(weighted_measure_side_refcell, 00416 jacobian_side_refcell, 00417 cub_weights_side, 00418 i, 00419 cell); 00420 00421 // tabulate values of basis functions at side cubature points, in the reference parent cell domain 00422 basis->getValues(value_of_basis_at_cub_points_side_refcell, cub_points_side_refcell, OPERATOR_VALUE); 00423 // transform 00424 FunctionSpaceTools::HGRADtransformVALUE<double>(transformed_value_of_basis_at_cub_points_side_refcell, 00425 value_of_basis_at_cub_points_side_refcell); 00426 00427 // multiply with weighted measure 00428 FunctionSpaceTools::multiplyMeasure<double>(weighted_transformed_value_of_basis_at_cub_points_side_refcell, 00429 weighted_measure_side_refcell, 00430 transformed_value_of_basis_at_cub_points_side_refcell); 00431 00432 // compute Neumann data 00433 // map side cubature points in reference parent cell domain to physical space 00434 CellTools<double>::mapToPhysicalFrame(cub_points_side_physical, cub_points_side_refcell, cell_nodes[pcell], cell); 00435 // now compute data 00436 neumann(neumann_data_at_cub_points_side_physical, cub_points_side_physical, jacobian_side_refcell, 00437 cell, (int)i, x_order, y_order); 00438 00439 FunctionSpaceTools::integrate<double>(neumann_fields_per_side, 00440 neumann_data_at_cub_points_side_physical, 00441 weighted_transformed_value_of_basis_at_cub_points_side_refcell, 00442 COMP_BLAS); 00443 00444 // adjust RHS 00445 RealSpaceTools<double>::add(rhs_and_soln_vector, neumann_fields_per_side);; 00446 } 00448 00450 // Solution of linear system: 00451 int info = 0; 00452 Teuchos::LAPACK<int, double> solver; 00453 solver.GESV(numFields, 1, &fe_matrix[0], numFields, &ipiv(0), &rhs_and_soln_vector[0], numFields, &info); 00455 00457 // Building interpolant: 00458 // evaluate basis at interpolation points 00459 basis->getValues(value_of_basis_at_interp_points, interp_points_ref, OPERATOR_VALUE); 00460 // transform values of basis functions 00461 FunctionSpaceTools::HGRADtransformVALUE<double>(transformed_value_of_basis_at_interp_points, 00462 value_of_basis_at_interp_points); 00463 FunctionSpaceTools::evaluate<double>(interpolant, rhs_and_soln_vector, transformed_value_of_basis_at_interp_points); 00465 00466 /******************* END COMPUTATION ***********************/ 00467 00468 RealSpaceTools<double>::subtract(interpolant, exact_solution); 00469 00470 *outStream << "\nRelative norm-2 error between exact solution polynomial of order (" 00471 << x_order << ", " << y_order << ") and finite element interpolant of order " << basis_order << ": " 00472 << RealSpaceTools<double>::vectorNorm(&interpolant[0], interpolant.dimension(1), NORM_TWO) / 00473 RealSpaceTools<double>::vectorNorm(&exact_solution[0], exact_solution.dimension(1), NORM_TWO) << "\n"; 00474 00475 if (RealSpaceTools<double>::vectorNorm(&interpolant[0], interpolant.dimension(1), NORM_TWO) / 00476 RealSpaceTools<double>::vectorNorm(&exact_solution[0], exact_solution.dimension(1), NORM_TWO) > zero) { 00477 *outStream << "\n\nPatch test failed for solution polynomial order (" 00478 << x_order << ", " << y_order << ") and basis order " << basis_order << "\n\n"; 00479 errorFlag++; 00480 } 00481 } // end for y_order 00482 } // end for x_order 00483 } // end for pcell 00484 00485 } 00486 // Catch unexpected errors 00487 catch (std::logic_error err) { 00488 *outStream << err.what() << "\n\n"; 00489 errorFlag = -1000; 00490 }; 00491 00492 if (errorFlag != 0) 00493 std::cout << "End Result: TEST FAILED\n"; 00494 else 00495 std::cout << "End Result: TEST PASSED\n"; 00496 00497 // reset format state of std::cout 00498 std::cout.copyfmt(oldFormatState); 00499 00500 return errorFlag; 00501 }
1.7.4