BOOST_AUTO_TEST_CASE_TEMPLATE(symmetric_matches_nonsymmetric_in_aca_mode,
ValueType, result_types)
{
typedef ValueType RT;
typedef typename Fiber::ScalarTraits<ValueType>::RealType RealType;
typedef RealType BFT;
if (boost::is_same<RT, std::complex<float> >::value) {
// The AHMED support for single-precision complex symmetric matrices
// is broken
BOOST_CHECK(true);
return;
}
GridParameters params;
params.topology = GridParameters::TRIANGULAR;
shared_ptr<Grid> grid = GridFactory::importGmshGrid(
params, "../../examples/meshes/sphere-h-0.4.msh",
false /* verbose */);
PiecewiseLinearContinuousScalarSpace<BFT> pwiseLinears(grid);
PiecewiseConstantScalarSpace<BFT> pwiseConstants(grid);
AssemblyOptions assemblyOptions;
assemblyOptions.setVerbosityLevel(VerbosityLevel::LOW);
AcaOptions acaOptions;
acaOptions.minimumBlockSize = 4;
assemblyOptions.switchToAcaMode(acaOptions);
AccuracyOptions accuracyOptions;
accuracyOptions.doubleRegular.setRelativeQuadratureOrder(4);
accuracyOptions.doubleSingular.setRelativeQuadratureOrder(2);
NumericalQuadratureStrategy<BFT, RT> quadStrategy(accuracyOptions);
Context<BFT, RT> context(make_shared_from_ref(quadStrategy), assemblyOptions);
const RT waveNumber = initWaveNumber<RT>();
BoundaryOperator<BFT, RT> opNonsymmetric =
modifiedHelmholtz3dSingleLayerBoundaryOperator<BFT, RT, RT>(
make_shared_from_ref(context),
make_shared_from_ref(pwiseLinears),
make_shared_from_ref(pwiseConstants),
make_shared_from_ref(pwiseLinears),
waveNumber,
"", NO_SYMMETRY);
BoundaryOperator<BFT, RT> opSymmetric =
modifiedHelmholtz3dSingleLayerBoundaryOperator<BFT, RT, RT>(
make_shared_from_ref(context),
make_shared_from_ref(pwiseLinears),
make_shared_from_ref(pwiseConstants),
make_shared_from_ref(pwiseLinears),
waveNumber,
"", SYMMETRIC);
arma::Mat<RT> matNonsymmetric = opNonsymmetric.weakForm()->asMatrix();
arma::Mat<RT> matSymmetric = opSymmetric.weakForm()->asMatrix();
BOOST_CHECK(check_arrays_are_close<RT>(
matNonsymmetric, matSymmetric, 2 * acaOptions.eps));
}
BOOST_AUTO_TEST_CASE_TEMPLATE(aca_of_synthetic_maxwell_single_layer_operator_agrees_with_dense_assembly_in_asymmetric_case,
ValueType, complex_result_types)
{
typedef ValueType RT;
typedef typename ScalarTraits<ValueType>::RealType RealType;
typedef RealType BFT;
GridParameters params;
params.topology = GridParameters::TRIANGULAR;
shared_ptr<Grid> grid = GridFactory::importGmshGrid(
params, "../../meshes/sphere-h-0.4.msh", false /* verbose */);
RT waveNumber = initWaveNumber<RT>();
shared_ptr<Space<BFT> > vectorPwiseLinears(
new RaviartThomas0VectorSpace<BFT>(grid));
shared_ptr<Space<BFT> > vectorPwiseLinears2(
new RaviartThomas0VectorSpace<BFT>(grid));
AccuracyOptions accuracyOptions;
accuracyOptions.doubleRegular.setRelativeQuadratureOrder(2);
accuracyOptions.singleRegular.setRelativeQuadratureOrder(2);
shared_ptr<NumericalQuadratureStrategy<BFT, RT> > quadStrategy(
new NumericalQuadratureStrategy<BFT, RT>(accuracyOptions));
AssemblyOptions assemblyOptionsDense;
assemblyOptionsDense.setVerbosityLevel(VerbosityLevel::LOW);
shared_ptr<Context<BFT, RT> > contextDense(
new Context<BFT, RT>(quadStrategy, assemblyOptionsDense));
BoundaryOperator<BFT, RT> opDense =
maxwell3dSingleLayerBoundaryOperator<BFT>(
contextDense,
vectorPwiseLinears, vectorPwiseLinears, vectorPwiseLinears,
waveNumber);
arma::Mat<RT> weakFormDense = opDense.weakForm()->asMatrix();
AssemblyOptions assemblyOptionsAca;
assemblyOptionsAca.setVerbosityLevel(VerbosityLevel::LOW);
AcaOptions acaOptions;
acaOptions.mode = AcaOptions::LOCAL_ASSEMBLY;
assemblyOptionsAca.switchToAcaMode(acaOptions);
shared_ptr<Context<BFT, RT> > contextAca(
new Context<BFT, RT>(quadStrategy, assemblyOptionsAca));
// Internal domain different from dualToRange
BoundaryOperator<BFT, RT> opAca =
maxwell3dSingleLayerBoundaryOperator<BFT>(
contextAca,
vectorPwiseLinears, vectorPwiseLinears, vectorPwiseLinears2,
waveNumber);
arma::Mat<RT> weakFormAca = opAca.weakForm()->asMatrix();
BOOST_CHECK(check_arrays_are_close<ValueType>(
weakFormDense, weakFormAca, 2. * acaOptions.eps));
}
BOOST_AUTO_TEST_CASE_TEMPLATE(interpolated_matches_noniterpolated,
BasisFunctionType, basis_function_types)
{
typedef BasisFunctionType BFT;
typedef typename Fiber::ScalarTraits<BFT>::ComplexType RT;
typedef typename Fiber::ScalarTraits<BFT>::ComplexType KT;
typedef typename Fiber::ScalarTraits<BFT>::RealType CT;
GridParameters params;
params.topology = GridParameters::TRIANGULAR;
shared_ptr<Grid> grid = GridFactory::importGmshGrid(
params, "meshes/two_disjoint_triangles.msh",
false /* verbose */);
PiecewiseLinearContinuousScalarSpace<BFT> pwiseLinears(grid);
PiecewiseConstantScalarSpace<BFT> pwiseConstants(grid);
AssemblyOptions assemblyOptions;
assemblyOptions.setVerbosityLevel(VerbosityLevel::LOW);
AccuracyOptions accuracyOptions;
accuracyOptions.doubleRegular.setAbsoluteQuadratureOrder(5);
accuracyOptions.doubleSingular.setAbsoluteQuadratureOrder(5);
NumericalQuadratureStrategy<BFT, RT> quadStrategy(accuracyOptions);
Context<BFT, RT> context(make_shared_from_ref(quadStrategy), assemblyOptions);
const KT waveNumber(3.23, 0.31);
BoundaryOperator<BFT, RT> opNoninterpolated =
modifiedHelmholtz3dSingleLayerBoundaryOperator<BFT, KT, RT>(
make_shared_from_ref(context),
make_shared_from_ref(pwiseLinears),
make_shared_from_ref(pwiseLinears),
make_shared_from_ref(pwiseLinears),
waveNumber,
"", NO_SYMMETRY,
false);
BoundaryOperator<BFT, RT> opInterpolated =
modifiedHelmholtz3dSingleLayerBoundaryOperator<BFT, KT, RT>(
make_shared_from_ref(context),
make_shared_from_ref(pwiseLinears),
make_shared_from_ref(pwiseLinears),
make_shared_from_ref(pwiseLinears),
waveNumber,
"", NO_SYMMETRY,
true);
arma::Mat<RT> matNoninterpolated = opNoninterpolated.weakForm()->asMatrix();
arma::Mat<RT> matInterpolated = opInterpolated.weakForm()->asMatrix();
const CT eps = std::numeric_limits<CT>::epsilon();
BOOST_CHECK(check_arrays_are_close<RT>(
matNoninterpolated, matInterpolated, 100 * eps));
}
BOOST_AUTO_TEST_CASE_TEMPLATE(aca_of_synthetic_modified_helmholtz_hypersingular_operator_agrees_with_dense_assembly_in_symmetric_case,
ValueType, result_types)
{
typedef ValueType RT;
typedef typename ScalarTraits<ValueType>::RealType RealType;
typedef RealType BFT;
GridParameters params;
params.topology = GridParameters::TRIANGULAR;
shared_ptr<Grid> grid = GridFactory::importGmshGrid(
params, "../../meshes/sphere-h-0.4.msh", false /* verbose */);
RT waveNumber = initWaveNumber<RT>();
shared_ptr<Space<BFT> > pwiseConstants(
new PiecewiseConstantScalarSpace<BFT>(grid));
shared_ptr<Space<BFT> > pwiseLinears(
new PiecewiseLinearContinuousScalarSpace<BFT>(grid));
AccuracyOptions accuracyOptions;
accuracyOptions.doubleRegular.setRelativeQuadratureOrder(2);
accuracyOptions.singleRegular.setRelativeQuadratureOrder(2);
shared_ptr<NumericalQuadratureStrategy<BFT, RT> > quadStrategy(
new NumericalQuadratureStrategy<BFT, RT>(accuracyOptions));
AssemblyOptions assemblyOptionsDense;
assemblyOptionsDense.setVerbosityLevel(VerbosityLevel::LOW);
shared_ptr<Context<BFT, RT> > contextDense(
new Context<BFT, RT>(quadStrategy, assemblyOptionsDense));
BoundaryOperator<BFT, RT> opDense =
modifiedHelmholtz3dHypersingularBoundaryOperator<BFT, RT, RT>(
contextDense, pwiseLinears, pwiseConstants, pwiseLinears,
waveNumber);
arma::Mat<RT> weakFormDense = opDense.weakForm()->asMatrix();
AssemblyOptions assemblyOptionsAca;
assemblyOptionsAca.setVerbosityLevel(VerbosityLevel::LOW);
AcaOptions acaOptions;
acaOptions.mode = AcaOptions::LOCAL_ASSEMBLY;
assemblyOptionsAca.switchToAcaMode(acaOptions);
shared_ptr<Context<BFT, RT> > contextAca(
new Context<BFT, RT>(quadStrategy, assemblyOptionsAca));
BoundaryOperator<BFT, RT> opAca =
modifiedHelmholtz3dHypersingularBoundaryOperator<BFT, RT, RT>(
contextAca, pwiseLinears, pwiseConstants, pwiseLinears,
waveNumber);
arma::Mat<RT> weakFormAca = opAca.weakForm()->asMatrix();
BOOST_CHECK(check_arrays_are_close<ValueType>(
weakFormDense, weakFormAca, 2. * acaOptions.eps));
}
BOOST_AUTO_TEST_CASE_TEMPLATE(works, BasisFunctionType, basis_function_types)
{
typedef BasisFunctionType BFT;
typedef typename Fiber::ScalarTraits<BFT>::ComplexType RT;
typedef typename Fiber::ScalarTraits<BFT>::RealType CT;
GridParameters params;
params.topology = GridParameters::TRIANGULAR;
shared_ptr<Grid> grid = GridFactory::importGmshGrid(
params, "meshes/two_disjoint_triangles.msh",
false /* verbose */);
PiecewiseLinearContinuousScalarSpace<BFT> pwiseLinears(grid);
PiecewiseConstantScalarSpace<BFT> pwiseConstants(grid);
AssemblyOptions assemblyOptions;
assemblyOptions.setVerbosityLevel(VerbosityLevel::LOW);
AccuracyOptions accuracyOptions;
accuracyOptions.doubleRegular.setAbsoluteQuadratureOrder(5);
accuracyOptions.doubleSingular.setAbsoluteQuadratureOrder(5);
NumericalQuadratureStrategy<BFT, RT> quadStrategy(accuracyOptions);
Context<BFT, RT> context(make_shared_from_ref(quadStrategy), assemblyOptions);
const RT waveNumber(1.23, 0.31);
BoundaryOperator<BFT, RT> slpOpConstants = Bempp::helmholtz3dSingleLayerBoundaryOperator<BFT>(
make_shared_from_ref(context),
make_shared_from_ref(pwiseConstants),
make_shared_from_ref(pwiseConstants),
make_shared_from_ref(pwiseConstants),
waveNumber);
BoundaryOperator<BFT, RT> slpOpLinears = helmholtz3dSingleLayerBoundaryOperator<BFT>(
make_shared_from_ref(context),
make_shared_from_ref(pwiseLinears),
make_shared_from_ref(pwiseLinears),
make_shared_from_ref(pwiseLinears),
waveNumber);
BoundaryOperator<BFT, RT> hypOp = helmholtz3dHypersingularBoundaryOperator<BFT>(
make_shared_from_ref(context),
make_shared_from_ref(pwiseLinears),
make_shared_from_ref(pwiseLinears),
make_shared_from_ref(pwiseLinears),
waveNumber);
// Get the matrix repr. of the hypersingular operator
arma::Mat<RT> hypMat = hypOp.weakForm()->asMatrix();
// Construct the expected hypersingular operator matrix. For this, we need:
// * the surface curls of all basis functions (which are constant)
typedef Fiber::SurfaceCurl3dElementaryFunctor<CT> ElementaryFunctor;
typedef Fiber::ElementaryBasisTransformationFunctorWrapper<ElementaryFunctor> Functor;
Functor functor;
size_t basisDeps = 0, geomDeps = 0;
functor.addDependencies(basisDeps, geomDeps);
arma::Mat<CT> points(2, 1);
points.fill(0.);
typedef Fiber::PiecewiseLinearContinuousScalarBasis<3, BFT> Basis;
Basis basis;
Fiber::BasisData<BFT> basisData;
basis.evaluate(basisDeps, points, Fiber::ALL_DOFS, basisData);
Fiber::GeometricalData<CT> geomData[2];
std::unique_ptr<GridView> view = grid->leafView();
std::unique_ptr<EntityIterator<0> > it = view->entityIterator<0>();
it->entity().geometry().getData(geomDeps, points, geomData[0]);
it->next();
it->entity().geometry().getData(geomDeps, points, geomData[1]);
Fiber::DefaultCollectionOfBasisTransformations<Functor> transformations(functor);
Fiber::CollectionOf3dArrays<BFT> surfaceCurl[2];
transformations.evaluate(basisData, geomData[0], surfaceCurl[0]);
transformations.evaluate(basisData, geomData[1], surfaceCurl[1]);
// * the single-layer potential matrix for constant basis functions
arma::Mat<RT> slpMatConstants = slpOpConstants.weakForm()->asMatrix();
// * the single-layer potential matrix for linear basis functions
arma::Mat<RT> slpMatLinears = slpOpLinears.weakForm()->asMatrix();
arma::Mat<RT> expectedHypMat(6, 6);
for (size_t testElement = 0; testElement < 2; ++testElement)
for (size_t trialElement = 0; trialElement < 2; ++trialElement)
for (size_t r = 0; r < 3; ++r)
for (size_t c = 0; c < 3; ++c) {
RT curlMultiplier = 0.;
for (size_t dim = 0; dim < 3; ++dim)
curlMultiplier += surfaceCurl[testElement][0](dim, r, 0) *
surfaceCurl[trialElement][0](dim, c, 0);
RT valueMultiplier = 0.;
for (size_t dim = 0; dim < 3; ++dim)
valueMultiplier += geomData[testElement].normals(dim, 0) *
geomData[trialElement].normals(dim, 0);
expectedHypMat(3 * testElement + r, 3 * trialElement + c) =
curlMultiplier * slpMatConstants(testElement, trialElement) -
waveNumber * waveNumber * valueMultiplier *
slpMatLinears(3 * testElement + r, 3 * trialElement + c);
}
BOOST_CHECK(check_arrays_are_close<RT>(expectedHypMat, hypMat, 1e-6));
}
int main(int argc, char* argv[])
{
// Process command-line args
if (argc < 7 || argc % 2 != 1) {
std::cout << "Solve a Maxwell Dirichlet problem in an exterior domain.\n"
<< "Usage: " << argv[0]
<< " <mesh_file> <n_threads> <aca_eps> <solver_tol>"
<< " <singular_order_increment>"
<< " [<regular_order_increment_1> <min_relative_distance_1>]"
<< " [<regular_order_increment_2> <min_relative_distance_2>]"
<< " [...] <regular_order_increment_n>"
<< std::endl;
return 1;
}
int maxThreadCount = atoi(argv[2]);
double acaEps = atof(argv[3]);
double solverTol = atof(argv[4]);
int singOrderIncrement = atoi(argv[5]);
if (acaEps > 1. || acaEps < 0.) {
std::cout << "Invalid aca_eps: " << acaEps << std::endl;
return 1;
}
if (solverTol > 1. || solverTol < 0.) {
std::cout << "Invalid solver_tol: " << solverTol << std::endl;
return 1;
}
AccuracyOptionsEx accuracyOptions;
std::vector<double> maxNormDists;
std::vector<int> orderIncrements;
for (int i = 6; i < argc - 1; i += 2) {
orderIncrements.push_back(atoi(argv[i]));
maxNormDists.push_back(atof(argv[i + 1]));
}
orderIncrements.push_back(atoi(argv[argc - 1]));
accuracyOptions.setDoubleRegular(maxNormDists, orderIncrements);
accuracyOptions.setDoubleSingular(singOrderIncrement);
accuracyOptions.setSingleRegular(2);
// Load mesh
GridParameters params;
params.topology = GridParameters::TRIANGULAR;
shared_ptr<Grid> grid = GridFactory::importGmshGrid(params, argv[1]);
// Initialize the space
RaviartThomas0VectorSpace<BFT> HdivSpace(grid);
// Set assembly mode and options
AssemblyOptions assemblyOptions;
assemblyOptions.setMaxThreadCount(maxThreadCount);
if (acaEps > 0.) {
AcaOptions acaOptions;
acaOptions.eps = acaEps;
assemblyOptions.switchToAcaMode(acaOptions);
}
NumericalQuadratureStrategy<BFT, RT> quadStrategy(accuracyOptions);
Context<BFT, RT> context(make_shared_from_ref(quadStrategy), assemblyOptions);
// Construct operators
BoundaryOperator<BFT, RT> slpOp = maxwell3dSingleLayerBoundaryOperator<BFT>(
make_shared_from_ref(context),
make_shared_from_ref(HdivSpace),
make_shared_from_ref(HdivSpace),
make_shared_from_ref(HdivSpace),
k,
"SLP");
BoundaryOperator<BFT, RT> dlpOp = maxwell3dDoubleLayerBoundaryOperator<BFT>(
make_shared_from_ref(context),
make_shared_from_ref(HdivSpace),
make_shared_from_ref(HdivSpace),
make_shared_from_ref(HdivSpace),
k,
"DLP");
BoundaryOperator<BFT, RT> idOp = maxwell3dIdentityOperator<BFT, RT>(
make_shared_from_ref(context),
make_shared_from_ref(HdivSpace),
make_shared_from_ref(HdivSpace),
make_shared_from_ref(HdivSpace),
"Id");
// Construct a grid function representing the Dirichlet data
GridFunction<BFT, RT> dirichletData(
make_shared_from_ref(context),
make_shared_from_ref(HdivSpace),
make_shared_from_ref(HdivSpace),
surfaceNormalDependentFunction(DirichletData()));
dirichletData.exportToVtk(VtkWriter::CELL_DATA, "Dirichlet_data",
"input_dirichlet_data_cell");
dirichletData.exportToVtk(VtkWriter::VERTEX_DATA, "Dirichlet_data",
"input_dirichlet_data_vertex");
// Construct a grid function representing the right-hand side
GridFunction<BFT, RT> rhs = -((0.5 * idOp + dlpOp) * dirichletData);
// Solve the equation
Solution<BFT, RT> solution;
#ifdef WITH_AHMED
if (solverTol > 0.) {
DefaultIterativeSolver<BFT, RT> solver(slpOp);
AcaApproximateLuInverse<RT> slpOpLu(
DiscreteAcaBoundaryOperator<RT>::castToAca(
*slpOp.weakForm()),
/* LU factorisation accuracy */ 0.01);
Preconditioner<RT> prec =
discreteOperatorToPreconditioner<RT>(
make_shared_from_ref(slpOpLu));
solver.initializeSolver(defaultGmresParameterList(solverTol, 10000), prec);
solution = solver.solve(rhs);
} else {
DefaultDirectSolver<BFT, RT> solver(slpOp);
solution = solver.solve(rhs);
}
#else // WITH_AHMED
DefaultDirectSolver<BFT, RT> solver(slpOp);
solution = solver.solve(rhs);
#endif
std::cout << solution.solverMessage() << std::endl;
// Extract the solution
const GridFunction<BFT, RT>& neumannData = solution.gridFunction();
neumannData.exportToVtk(VtkWriter::CELL_DATA, "Neumann_data",
"calculated_neumann_data_cell");
neumannData.exportToVtk(VtkWriter::VERTEX_DATA, "Neumann_data",
"calculated_neumann_data_vertex");
// Compare the solution against the analytical result
GridFunction<BFT, RT> exactNeumannData(
make_shared_from_ref(context),
make_shared_from_ref(HdivSpace),
make_shared_from_ref(HdivSpace),
surfaceNormalDependentFunction(ExactNeumannData()));
exactNeumannData.exportToVtk(VtkWriter::CELL_DATA, "Neumann_data",
"exact_neumann_data_cell");
exactNeumannData.exportToVtk(VtkWriter::VERTEX_DATA, "Neumann_data",
"exact_neumann_data_vertex");
EvaluationOptions evaluationOptions;
CT absoluteError = L2NormOfDifference(
neumannData, surfaceNormalDependentFunction(ExactNeumannData()),
quadStrategy, evaluationOptions);
CT exactSolNorm = L2NormOfDifference(
0. * neumannData, surfaceNormalDependentFunction(ExactNeumannData()),
quadStrategy, evaluationOptions);
std::cout << "Relative L^2 error: " << absoluteError / exactSolNorm
<< "\nAbsolute L^2 error: " << absoluteError << std::endl;
// Evaluate the solution at a few points
Maxwell3dSingleLayerPotentialOperator<BFT> slpPotOp(k);
Maxwell3dDoubleLayerPotentialOperator<BFT> dlpPotOp(k);
const int dimWorld = 3;
const int pointCount = 3;
arma::Mat<CT> points(dimWorld, pointCount * pointCount);
for (int i = 0; i < pointCount; ++i)
for (int j = 0; j < pointCount; ++j) {
points(0, i * pointCount + j) = 3. + i / CT(pointCount - 1);
points(1, i * pointCount + j) = 2. + j / CT(pointCount - 1);
points(2, i * pointCount + j) = 0.5;
}
arma::Mat<RT> slpContrib = slpPotOp.evaluateAtPoints(
neumannData, points, quadStrategy, evaluationOptions);
arma::Mat<RT> dlpContrib = dlpPotOp.evaluateAtPoints(
dirichletData, points, quadStrategy, evaluationOptions);
arma::Mat<RT> values = -slpContrib - dlpContrib;
// Evaluate the analytical solution at the same points
ExactSolution exactSolution;
arma::Mat<RT> exactValues(dimWorld, pointCount * pointCount);
for (int i = 0; i < pointCount * pointCount; ++i) {
arma::Col<RT> activeResultColumn = exactValues.unsafe_col(i);
exactSolution.evaluate(points.unsafe_col(i), activeResultColumn);
}
// Compare the numerical and analytical solutions
std::cout << "Numerical | analytical\n";
for (int i = 0; i < pointCount * pointCount; ++i)
std::cout << values(0, i) << " "
<< values(1, i) << " "
<< values(2, i) << " | "
<< exactValues(0, i) << " "
<< exactValues(1, i) << " "
<< exactValues(2, i) << "\n";
}