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(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));
}
int main()
{
// Import symbols from namespace Bempp to the global namespace
using namespace Bempp;
// Load mesh
const char* meshFile = "meshes/sphere-h-0.2.msh";
GridParameters params;
params.topology = GridParameters::TRIANGULAR;
shared_ptr<Grid> grid = GridFactory::importGmshGrid(params, meshFile);
// Initialize the spaces
PiecewiseLinearContinuousScalarSpace<BFT> pwiseLinears(grid);
PiecewiseConstantScalarSpace<BFT> pwiseConstants(grid);
// Define the quadrature strategy
AccuracyOptions accuracyOptions;
// Increase by 2 the order of quadrature rule used to approximate
// integrals of regular functions on pairs on elements
accuracyOptions.doubleRegular.setRelativeQuadratureOrder(2);
// Increase by 2 the order of quadrature rule used to approximate
// integrals of regular functions on single elements
accuracyOptions.singleRegular.setRelativeQuadratureOrder(2);
NumericalQuadratureStrategy<BFT, RT> quadStrategy(accuracyOptions);
// Specify the assembly method. We want to use ACA
AssemblyOptions assemblyOptions;
AcaOptions acaOptions; // Default parameters for ACA
assemblyOptions.switchToAcaMode(acaOptions);
// Create the assembly context
Context<BFT, RT> context(make_shared_from_ref(quadStrategy), assemblyOptions);
// Construct elementary operators
BoundaryOperator<BFT, RT> slpOp =
laplace3dSingleLayerBoundaryOperator<BFT, RT>(
make_shared_from_ref(context),
make_shared_from_ref(pwiseConstants),
make_shared_from_ref(pwiseLinears),
make_shared_from_ref(pwiseConstants));
BoundaryOperator<BFT, RT> dlpOp =
laplace3dDoubleLayerBoundaryOperator<BFT, RT>(
make_shared_from_ref(context),
make_shared_from_ref(pwiseLinears),
make_shared_from_ref(pwiseLinears),
make_shared_from_ref(pwiseConstants));
BoundaryOperator<BFT, RT> idOp =
identityOperator<BFT, RT>(
make_shared_from_ref(context),
make_shared_from_ref(pwiseLinears),
make_shared_from_ref(pwiseLinears),
make_shared_from_ref(pwiseConstants));
// Form the right-hand side sum
BoundaryOperator<BFT, RT> rhsOp = -0.5 * idOp + dlpOp;
// Construct the grid function representing the (input) Dirichlet data
GridFunction<BFT, RT> dirichletData(
make_shared_from_ref(context),
make_shared_from_ref(pwiseLinears),
make_shared_from_ref(pwiseLinears),
surfaceNormalIndependentFunction(DirichletData()));
// Construct the right-hand-side grid function
GridFunction<BFT, RT> rhs = rhsOp * dirichletData;
// Initialize the solver
DefaultIterativeSolver<BFT, RT> solver(slpOp);
solver.initializeSolver(defaultGmresParameterList(1e-5));
// Solve the equation
Solution<BFT, RT> solution = solver.solve(rhs);
std::cout << solution.solverMessage() << std::endl;
// Extract the solution in the form of a grid function
// and export it in VTK format
const GridFunction<BFT, RT>& solFun = solution.gridFunction();
solFun.exportToVtk(VtkWriter::CELL_DATA, "Neumann_data", "solution");
// Compare the numerical and analytical solution on the grid
// GridFunction<BFT, RT> exactSolFun(
// make_shared_from_ref(context),
// make_shared_from_ref(pwiseConstants),
// make_shared_from_ref(pwiseConstants),
// surfaceNormalIndependentFunction(ExactNeumannData()));
CT absoluteError, relativeError;
estimateL2Error(
solFun, surfaceNormalIndependentFunction(ExactNeumannData()),
quadStrategy, absoluteError, relativeError);
std::cout << "Relative L^2 error: " << relativeError << std::endl;
// GridFunction<BFT, RT> diff = solFun - exactSolFun;
// double relativeError = diff.L2Norm() / exactSolFun.L2Norm();
// std::cout << "Relative L^2 error: " << relativeError << std::endl;
// Prepare to evaluate the solution on an annulus outside the sphere
// Create potential operators
Laplace3dSingleLayerPotentialOperator<BFT, RT> slPotOp;
Laplace3dDoubleLayerPotentialOperator<BFT, RT> dlPotOp;
// Construct the array 'evaluationPoints' containing the coordinates
// of points where the solution should be evaluated
const int rCount = 51;
const int thetaCount = 361;
const CT minTheta = 0., maxTheta = 2. * M_PI;
const CT minR = 1., maxR = 2.;
const int dimWorld = 3;
arma::Mat<CT> evaluationPoints(dimWorld, rCount * thetaCount);
for (int iTheta = 0; iTheta < thetaCount; ++iTheta) {
CT theta = minTheta + (maxTheta - minTheta) *
iTheta / (thetaCount - 1);
for (int iR = 0; iR < rCount; ++iR) {
CT r = minR + (maxR - minR) * iR / (rCount - 1);
evaluationPoints(0, iR + iTheta * rCount) = r * cos(theta); // x
evaluationPoints(1, iR + iTheta * rCount) = r * sin(theta); // y
evaluationPoints(2, iR + iTheta * rCount) = 0.; // z
}
}
// Use the Green's representation formula to evaluate the solution
EvaluationOptions evaluationOptions;
arma::Mat<RT> field =
-slPotOp.evaluateAtPoints(solFun, evaluationPoints,
quadStrategy, evaluationOptions) +
dlPotOp.evaluateAtPoints(dirichletData, evaluationPoints,
quadStrategy, evaluationOptions);
// Export the solution into text file
std::ofstream out("solution.txt");
out << "# x y z u\n";
for (int i = 0; i < rCount * thetaCount; ++i)
out << evaluationPoints(0, i) << ' '
<< evaluationPoints(1, i) << ' '
<< evaluationPoints(2, i) << ' '
<< field(0, i) << '\n';
}
int main(int argc, char* argv[])
{
// Physical parameters, general
const BFT c0 = 0.3; // speed of light in vacuum [mm/ps]
BFT refind = 1.4; // refractive index
BFT alpha = A_Keijzer(refind); // boundary term
BFT c = c0/refind; // speed of light in medium [mm/ps]
BFT freq = 100*1e6; // modulation frequency [Hz]
BFT omega = 2.0*M_PI * freq*1e-12; // modulation frequency [cycles/ps]
// Physical parameters, outer region
BFT mua1 = 0.01; // absorption coefficient
BFT mus1 = 1.0; // scattering coefficient
BFT kappa1 = 1.0/(3.0*(mua1+mus1)); // diffusion coefficient
RT waveNumber1 = sqrt (RT(mua1/kappa1, omega/(c*kappa1))); // outer region
// Physical parameters, inner region
BFT mua2 = 0.02; // absorption coefficient
BFT mus2 = 0.5; // scattering coefficient
BFT kappa2 = 1.0/(3.0*(mua2+mus2)); // diffusion coefficient
RT waveNumber2 = sqrt (RT(mua2/kappa2, omega/(c*kappa2))); // outer region
// Create sphere meshes on the fly
shared_ptr<Grid> grid1 = CreateSphere(25.0, 1.0);
shared_ptr<Grid> grid2 = CreateSphere(15.0, 1.0);
// Initialize the spaces
PiecewiseLinearContinuousScalarSpace<BFT> HplusHalfSpace1(grid1);
PiecewiseLinearContinuousScalarSpace<BFT> HplusHalfSpace2(grid2);
// Define some default options.
AssemblyOptions assemblyOptions;
// We want to use ACA
AcaOptions acaOptions; // Default parameters for ACA
acaOptions.eps = 1e-5;
assemblyOptions.switchToAcaMode(acaOptions);
// Define the standard integration factory
AccuracyOptions accuracyOptions;
accuracyOptions.doubleRegular.setRelativeQuadratureOrder(2);
accuracyOptions.singleRegular.setRelativeQuadratureOrder(1);
NumericalQuadratureStrategy<BFT, RT> quadStrategy(accuracyOptions);
Context<BFT, RT> context(make_shared_from_ref(quadStrategy), assemblyOptions);
// We need the single layer, double layer, and the identity operator
// mesh1 x mesh1
BoundaryOperator<BFT, RT> slp11 =
modifiedHelmholtz3dSingleLayerBoundaryOperator<BFT, RT, RT>(
SHARED(context), SHARED(HplusHalfSpace1),
SHARED(HplusHalfSpace1), SHARED(HplusHalfSpace1), waveNumber1);
BoundaryOperator<BFT, RT> dlp11 =
modifiedHelmholtz3dDoubleLayerBoundaryOperator<BFT, RT, RT>(
SHARED(context), SHARED(HplusHalfSpace1),
SHARED(HplusHalfSpace1), SHARED(HplusHalfSpace1), waveNumber1);
BoundaryOperator<BFT, RT> id11 =
identityOperator<BFT, RT>(
SHARED(context), SHARED(HplusHalfSpace1),
SHARED(HplusHalfSpace1), SHARED(HplusHalfSpace1));
// mesh2 x mesh2, wavenumber 1
BoundaryOperator<BFT, RT> slp22_w1 =
modifiedHelmholtz3dSingleLayerBoundaryOperator<BFT, RT, RT>(
SHARED(context), SHARED(HplusHalfSpace2),
SHARED(HplusHalfSpace2), SHARED(HplusHalfSpace2), waveNumber1);
BoundaryOperator<BFT, RT> dlp22_w1 =
modifiedHelmholtz3dDoubleLayerBoundaryOperator<BFT, RT, RT>(
SHARED(context), SHARED(HplusHalfSpace2),
SHARED(HplusHalfSpace2), SHARED(HplusHalfSpace2), waveNumber1);
BoundaryOperator<BFT, RT> id22 =
identityOperator<BFT, RT>(
SHARED(context), SHARED(HplusHalfSpace2),
SHARED(HplusHalfSpace2), SHARED(HplusHalfSpace2));
// mesh2 x mesh2, wavenumber 2
BoundaryOperator<BFT, RT> slp22_w2 =
modifiedHelmholtz3dSingleLayerBoundaryOperator<BFT, RT, RT>(
SHARED(context), SHARED(HplusHalfSpace2),
SHARED(HplusHalfSpace2), SHARED(HplusHalfSpace2), waveNumber2);
BoundaryOperator<BFT, RT> dlp22_w2 =
modifiedHelmholtz3dDoubleLayerBoundaryOperator<BFT, RT, RT>(
SHARED(context), SHARED(HplusHalfSpace2),
SHARED(HplusHalfSpace2), SHARED(HplusHalfSpace2), waveNumber2);
// mesh1 x mesh2
BoundaryOperator<BFT, RT> slp12 =
modifiedHelmholtz3dSingleLayerBoundaryOperator<BFT, RT, RT>(
SHARED(context), SHARED(HplusHalfSpace2),
SHARED(HplusHalfSpace1), SHARED(HplusHalfSpace1), waveNumber1);
BoundaryOperator<BFT, RT> dlp12 =
modifiedHelmholtz3dDoubleLayerBoundaryOperator<BFT, RT, RT>(
SHARED(context), SHARED(HplusHalfSpace2),
SHARED(HplusHalfSpace1), SHARED(HplusHalfSpace1), waveNumber1);
// mesh2 x mesh1
BoundaryOperator<BFT, RT> slp21 =
modifiedHelmholtz3dSingleLayerBoundaryOperator<BFT, RT, RT>(
SHARED(context), SHARED(HplusHalfSpace1),
SHARED(HplusHalfSpace2), SHARED(HplusHalfSpace2), waveNumber1);
BoundaryOperator<BFT, RT> dlp21 =
modifiedHelmholtz3dDoubleLayerBoundaryOperator<BFT, RT, RT>(
SHARED(context), SHARED(HplusHalfSpace1),
SHARED(HplusHalfSpace2), SHARED(HplusHalfSpace2), waveNumber1);
BFT scale = 1.0/(2.0*alpha*kappa1);
BoundaryOperator<BFT, RT> lhs_k11 = 0.5*id11 + dlp11 + scale*slp11;
BoundaryOperator<BFT, RT> lhs_k12 = -1.0*dlp12; // sign flipped to accommodate normal direction
BoundaryOperator<BFT, RT> lhs_k13 = -(1.0/kappa1)*slp12;
BoundaryOperator<BFT, RT> lhs_k21 = dlp21 + scale*slp21;
BoundaryOperator<BFT, RT> lhs_k22 = 0.5*id22 - dlp22_w1; // sign flipped to accommodate normal direction
BoundaryOperator<BFT, RT> lhs_k23 = -(1.0/kappa1)*slp22_w1;
// BoundaryOperator<BFT, RT> lhs_k31 -- empty
BoundaryOperator<BFT, RT> lhs_k32 = 0.5*id22 + dlp22_w2;
BoundaryOperator<BFT, RT> lhs_k33 = (1.0/kappa2) * slp22_w2;
BlockedOperatorStructure<BFT, RT> structure;
structure.setBlock(0, 0, lhs_k11);
structure.setBlock(0, 1, lhs_k12);
structure.setBlock(0, 2, lhs_k13);
structure.setBlock(1, 0, lhs_k21);
structure.setBlock(1, 1, lhs_k22);
structure.setBlock(1, 2, lhs_k23);
// structure.setBlock(2, 0, ...); -- empty
structure.setBlock(2, 1, lhs_k32);
structure.setBlock(2, 2, lhs_k33);
BlockedBoundaryOperator<BFT, RT> blockedOp(structure);
// Grid functions for the RHS
// TODO: remove the necessity of creating "dummy" functions
// corresponding to zero blocks.
BoundaryOperator<BFT, RT> rhs1 = scale*slp11;
BoundaryOperator<BFT, RT> rhs2 = scale*slp21;
std::vector<GridFunction<BFT, RT> > blockedRhs(3);
blockedRhs[0] = rhs1 * GridFunction<BFT, RT>(
SHARED(context),
SHARED(HplusHalfSpace1), SHARED(HplusHalfSpace1),
//surfaceNormalIndependentFunction(NullFunctor()));
surfaceNormalIndependentFunction(MyFunctor(waveNumber1)));
blockedRhs[1] = rhs2 * GridFunction<BFT, RT>(
SHARED(context),
SHARED(HplusHalfSpace1), SHARED(HplusHalfSpace1),
//surfaceNormalIndependentFunction(NullFunctor()));
surfaceNormalIndependentFunction(MyFunctor(waveNumber1)));
blockedRhs[2] = GridFunction<BFT, RT>(
SHARED(context),
SHARED(HplusHalfSpace2), SHARED(HplusHalfSpace2),
//surfaceNormalIndependentFunction(MyFunctor(waveNumber2)));
surfaceNormalIndependentFunction(NullFunctor()));
// Initialize the solver
const double solverTol = 1e-10;
DefaultIterativeSolver<BFT, RT> solver(blockedOp);
solver.initializeSolver(defaultGmresParameterList(solverTol));
// Solve
BlockedSolution<BFT, RT> solution = solver.solve(blockedRhs);
std::cout << solution.solverMessage() << std::endl;
arma::Col<RT> solutionVectorBlock1 = solution.gridFunction(0).coefficients();
arma::Col<RT> solutionVectorBlock2 = solution.gridFunction(1).coefficients();
arma::Col<RT> solutionVectorBlock3 = solution.gridFunction(2).coefficients();
arma::diskio::save_raw_ascii(solutionVectorBlock1, "sol1.txt");
arma::diskio::save_raw_ascii(solutionVectorBlock2, "sol2.txt");
arma::diskio::save_raw_ascii(solutionVectorBlock3, "sol3.txt");
solution.gridFunction(0).exportToVtk(VtkWriter::VERTEX_DATA, "gf1", "gf1");
solution.gridFunction(1).exportToVtk(VtkWriter::VERTEX_DATA, "gf2", "gf2");
solution.gridFunction(2).exportToVtk(VtkWriter::VERTEX_DATA, "gf3", "gf3");
}
int main(int argc, char* argv[])
{
// Load mesh
if (argc != 2) {
std::cout << "Solve a Dirichlet problem for the Laplace equation.\n"
"Usage: " << argv[0] << " <mesh_file>" << std::endl;
return 1;
}
shared_ptr<Grid> grid = loadTriangularMeshFromFile(argv[1]);
//std::cout << grid->gridTopology() << std::endl;
// Initialize the spaces
PiecewiseLinearContinuousScalarSpace<BFT> HplusHalfSpace(grid);
PiecewiseConstantScalarSpace<BFT> HminusHalfSpace(grid);
// Define some default options.
AssemblyOptions assemblyOptions;
// We want to use ACA
AcaOptions acaOptions; // Default parameters for ACA
acaOptions.eps = 1e-5;
assemblyOptions.switchToAcaMode(acaOptions);
// Define the standard integration factory
AccuracyOptions accuracyOptions;
accuracyOptions.doubleRegular.setRelativeQuadratureOrder(1);
NumericalQuadratureStrategy<BFT, RT> quadStrategy(accuracyOptions);
Context<BFT, RT> context(make_shared_from_ref(quadStrategy), assemblyOptions);
// We need the single layer, double layer, and the identity operator
BoundaryOperator<BFT, RT> slpOp = laplace3dSingleLayerBoundaryOperator<BFT, RT >(
make_shared_from_ref(context),
make_shared_from_ref(HminusHalfSpace),
make_shared_from_ref(HplusHalfSpace),
make_shared_from_ref(HminusHalfSpace),
"SLP");
BoundaryOperator<BFT, RT> dlpOp = laplace3dDoubleLayerBoundaryOperator<BFT, RT >(
make_shared_from_ref(context),
make_shared_from_ref(HplusHalfSpace),
make_shared_from_ref(HplusHalfSpace),
make_shared_from_ref(HminusHalfSpace),
"DLP");
BoundaryOperator<BFT, RT> id = identityOperator<BFT, RT>(
make_shared_from_ref(context),
make_shared_from_ref(HplusHalfSpace),
make_shared_from_ref(HplusHalfSpace),
make_shared_from_ref(HminusHalfSpace),
"I");
// Form the right-hand side sum
BoundaryOperator<BFT, RT> rhsOp = -0.5 * id + dlpOp;
// We also want a grid function
GridFunction<BFT, RT> u(
make_shared_from_ref(context),
make_shared_from_ref(HplusHalfSpace),
make_shared_from_ref(HplusHalfSpace), // is this the right choice?
surfaceNormalIndependentFunction(MyFunctor()));
// Assemble the rhs
std::cout << "Assemble rhs" << std::endl;
GridFunction<BFT, RT> rhs = rhsOp * u;
// Initialize the solver
std::cout << "Initialize solver" << std::endl;
#ifdef WITH_TRILINOS
DefaultIterativeSolver<BFT, RT> solver(slpOp,ConvergenceTestMode::TEST_CONVERGENCE_IN_DUAL_TO_RANGE);
solver.initializeSolver(defaultGmresParameterList(1e-5));
Solution<BFT, RT> solution = solver.solve(rhs);
std::cout << solution.solverMessage() << std::endl;
#else
DefaultDirectSolver<BFT, RT> solver(slp, rhs);
solver.solve();
#endif
// Extract the solution
const GridFunction<BFT, RT>& solFun = solution.gridFunction();
// Write out as VTK
solFun.exportToVtk(VtkWriter::CELL_DATA, "Neumann_data",
"calculated_neumann_data_cell");
solFun.exportToVtk(VtkWriter::VERTEX_DATA, "Neumann_data",
"calculated_neumann_data_vertex");
// Uncomment the block below if you are solving the problem on a sphere and
// you want to compare the numerical and analytical solution.
// arma::Col<RT> solutionCoefficients = solFun.coefficients();
// std::cout << solutionCoefficients << std::endl;
// arma::Col<RT> deviation = solutionCoefficients - static_cast<RT>(-1.);
// // % in Armadillo -> elementwise multiplication
// RT stdDev = sqrt(arma::accu(deviation % deviation) /
// static_cast<RT>(solutionCoefficients.n_rows));
// std::cout << "Standard deviation: " << stdDev << std::endl;
}
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";
}