1034 lines
30 KiB
C++
1034 lines
30 KiB
C++
// MFEM Example 25
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//
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// Compile with: make ex25
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//
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// Sample runs: ex25 -o 2 -f 1.0 -ref 2 -prob 0
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// ex25 -o 3 -f 10.0 -ref 2 -prob 1
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// ex25 -o 2 -f 5.0 -ref 4 -prob 2
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// ex25 -o 2 -f 1.0 -ref 2 -prob 3
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// ex25 -o 2 -f 1.0 -ref 2 -prob 0 -m ../data/beam-quad.mesh
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// ex25 -o 2 -f 8.0 -ref 3 -prob 4 -m ../data/inline-quad.mesh
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// ex25 -o 2 -f 2.0 -ref 1 -prob 4 -m ../data/inline-hex.mesh
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//
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// Device sample runs:
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// ex25 -o 2 -f 8.0 -ref 3 -prob 4 -m ../data/inline-quad.mesh -pa -d cuda
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// ex25 -o 2 -f 2.0 -ref 1 -prob 4 -m ../data/inline-hex.mesh -pa -d cuda
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//
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// Description: This example code solves a simple electromagnetic wave
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// propagation problem corresponding to the second order
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// indefinite Maxwell equation
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//
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// (1/mu) * curl curl E - \omega^2 * epsilon E = f
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//
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// with a Perfectly Matched Layer (PML).
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//
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// The example demonstrates discretization with Nedelec finite
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// elements in 2D or 3D, as well as the use of complex-valued
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// bilinear and linear forms. Several test problems are included,
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// with prob = 0-3 having known exact solutions, see "On perfectly
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// matched layers for discontinuous Petrov-Galerkin methods" by
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// Vaziri Astaneh, Keith, Demkowicz, Comput Mech 63, 2019.
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//
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// We recommend viewing Example 22 before viewing this example.
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#include "mfem.hpp"
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#include <memory>
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#include <fstream>
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#include <iostream>
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#ifdef _WIN32
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#define jn(n, x) _jn(n, x)
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#define yn(n, x) _yn(n, x)
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#endif
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using namespace std;
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using namespace mfem;
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// Class for setting up a simple Cartesian PML region
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class PML
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{
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private:
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Mesh *mesh;
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int dim;
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// Length of the PML Region in each direction
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Array2D<real_t> length;
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// Computational Domain Boundary
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Array2D<real_t> comp_dom_bdr;
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// Domain Boundary
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Array2D<real_t> dom_bdr;
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// Integer Array identifying elements in the PML
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// 0: in the PML, 1: not in the PML
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Array<int> elems;
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// Compute Domain and Computational Domain Boundaries
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void SetBoundaries();
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public:
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// Constructor
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PML(Mesh *mesh_,Array2D<real_t> length_);
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// Return Computational Domain Boundary
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Array2D<real_t> GetCompDomainBdr() {return comp_dom_bdr;}
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// Return Domain Boundary
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Array2D<real_t> GetDomainBdr() {return dom_bdr;}
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// Return Markers list for elements
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Array<int> * GetMarkedPMLElements() {return &elems;}
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// Mark elements in the PML region
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void SetAttributes(Mesh *mesh_);
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// PML complex stretching function
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void StretchFunction(const Vector &x, vector<complex<real_t>> &dxs);
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};
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// Class for returning the PML coefficients of the bilinear form
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class PMLDiagMatrixCoefficient : public VectorCoefficient
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{
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private:
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PML * pml = nullptr;
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void (*Function)(const Vector &, PML *, Vector &);
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public:
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PMLDiagMatrixCoefficient(int dim, void(*F)(const Vector &, PML *,
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Vector &),
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PML * pml_)
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: VectorCoefficient(dim), pml(pml_), Function(F)
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{}
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using VectorCoefficient::Eval;
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void Eval(Vector &K, ElementTransformation &T,
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const IntegrationPoint &ip) override
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{
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real_t x[3];
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Vector transip(x, 3);
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T.Transform(ip, transip);
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K.SetSize(vdim);
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(*Function)(transip, pml, K);
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}
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};
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void maxwell_solution(const Vector &x, vector<complex<real_t>> &Eval);
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void E_bdr_data_Re(const Vector &x, Vector &E);
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void E_bdr_data_Im(const Vector &x, Vector &E);
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void E_exact_Re(const Vector &x, Vector &E);
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void E_exact_Im(const Vector &x, Vector &E);
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void source(const Vector &x, Vector & f);
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// Functions for computing the necessary coefficients after PML stretching.
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// J is the Jacobian matrix of the stretching function
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void detJ_JT_J_inv_Re(const Vector &x, PML * pml, Vector &D);
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void detJ_JT_J_inv_Im(const Vector &x, PML * pml, Vector &D);
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void detJ_JT_J_inv_abs(const Vector &x, PML * pml, Vector &D);
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void detJ_inv_JT_J_Re(const Vector &x, PML * pml, Vector &D);
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void detJ_inv_JT_J_Im(const Vector &x, PML * pml, Vector &D);
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void detJ_inv_JT_J_abs(const Vector &x, PML * pml, Vector &D);
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Array2D<real_t> comp_domain_bdr;
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Array2D<real_t> domain_bdr;
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real_t mu = 1.0;
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real_t epsilon = 1.0;
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real_t omega;
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int dim;
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bool exact_known = false;
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template <typename T> T pow2(const T &x) { return x*x; }
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enum prob_type
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{
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beam, // Wave propagating in a beam-like domain
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disc, // Point source propagating in the square-disc domain
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lshape, // Point source propagating in the L-shape domain
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fichera, // Point source propagating in the fichera domain
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load_src // Approximated point source with PML all around
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};
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prob_type prob;
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int main(int argc, char *argv[])
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{
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// 1. Parse command-line options.
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const char *mesh_file = nullptr;
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int order = 1;
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int ref_levels = 3;
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int iprob = 4;
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real_t freq = 5.0;
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bool herm_conv = true;
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bool umf_solver = false;
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bool visualization = 1;
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bool pa = false;
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const char *device_config = "cpu";
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OptionsParser args(argc, argv);
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args.AddOption(&mesh_file, "-m", "--mesh",
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"Mesh file to use.");
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args.AddOption(&order, "-o", "--order",
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"Finite element order (polynomial degree).");
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args.AddOption(&iprob, "-prob", "--problem", "Problem case"
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" 0: beam, 1: disc, 2: lshape, 3: fichera, 4: General");
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args.AddOption(&ref_levels, "-ref", "--refinements",
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"Number of refinements");
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args.AddOption(&mu, "-mu", "--permeability",
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"Permeability of free space (or 1/(spring constant)).");
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args.AddOption(&epsilon, "-eps", "--permittivity",
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"Permittivity of free space (or mass constant).");
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args.AddOption(&freq, "-f", "--frequency",
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"Frequency (in Hz).");
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args.AddOption(&herm_conv, "-herm", "--hermitian", "-no-herm",
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"--no-hermitian", "Use convention for Hermitian operators.");
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#ifdef MFEM_USE_SUITESPARSE
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args.AddOption(&umf_solver, "-umf", "--umfpack", "-no-umf",
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"--no-umfpack", "Use the UMFPack Solver.");
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#endif
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args.AddOption(&visualization, "-vis", "--visualization", "-no-vis",
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"--no-visualization",
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"Enable or disable GLVis visualization.");
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args.AddOption(&pa, "-pa", "--partial-assembly", "-no-pa",
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"--no-partial-assembly", "Enable Partial Assembly.");
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args.AddOption(&device_config, "-d", "--device",
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"Device configuration string, see Device::Configure().");
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args.Parse();
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if (iprob > 4) { iprob = 4; }
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prob = (prob_type)iprob;
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// 2. Enable hardware devices such as GPUs, and programming models such as
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// CUDA, OCCA, RAJA and OpenMP based on command line options.
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Device device(device_config);
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device.Print();
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// 3. Setup the mesh
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if (!mesh_file)
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{
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exact_known = true;
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switch (prob)
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{
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case beam:
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mesh_file = "../data/beam-hex.mesh";
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break;
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case disc:
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mesh_file = "../data/square-disc.mesh";
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break;
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case lshape:
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mesh_file = "../data/l-shape.mesh";
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break;
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case fichera:
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mesh_file = "../data/fichera.mesh";
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break;
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default:
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exact_known = false;
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mesh_file = "../data/inline-quad.mesh";
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break;
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}
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}
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if (!args.Good())
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{
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args.PrintUsage(cout);
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return 1;
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}
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args.PrintOptions(cout);
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Mesh * mesh = new Mesh(mesh_file, 1, 1);
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dim = mesh->Dimension();
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// Angular frequency
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omega = real_t(2.0 * M_PI) * freq;
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// Setup PML length
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Array2D<real_t> length(dim, 2); length = 0.0;
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// 4. Setup the Cartesian PML region.
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switch (prob)
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{
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case disc:
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length = 0.2;
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break;
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case lshape:
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length(0, 0) = 0.1;
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length(1, 0) = 0.1;
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break;
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case fichera:
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length(0, 1) = 0.5;
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length(1, 1) = 0.5;
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length(2, 1) = 0.5;
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break;
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case beam:
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length(0, 1) = 2.0;
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break;
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default:
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length = 0.25;
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break;
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}
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PML * pml = new PML(mesh,length);
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comp_domain_bdr = pml->GetCompDomainBdr();
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domain_bdr = pml->GetDomainBdr();
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// 5. Refine the mesh to increase the resolution.
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for (int l = 0; l < ref_levels; l++)
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{
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mesh->UniformRefinement();
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}
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// 6. Set element attributes in order to distinguish elements in the
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// PML region
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pml->SetAttributes(mesh);
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// 7. Define a finite element space on the mesh. Here we use the Nedelec
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// finite elements of the specified order.
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FiniteElementCollection *fec = new ND_FECollection(order, dim);
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FiniteElementSpace *fespace = new FiniteElementSpace(mesh, fec);
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int size = fespace->GetTrueVSize();
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cout << "Number of finite element unknowns: " << size << endl;
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// 8. Determine the list of true essential boundary dofs. In this example,
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// the boundary conditions are defined based on the specific mesh and the
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// problem type.
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Array<int> ess_tdof_list;
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Array<int> ess_bdr;
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if (mesh->bdr_attributes.Size())
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{
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ess_bdr.SetSize(mesh->bdr_attributes.Max());
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ess_bdr = 1;
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if (prob == lshape || prob == fichera)
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{
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ess_bdr = 0;
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for (int j = 0; j < mesh->GetNBE(); j++)
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{
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Vector center(dim);
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int bdrgeom = mesh->GetBdrElementGeometry(j);
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ElementTransformation * tr = mesh->GetBdrElementTransformation(j);
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tr->Transform(Geometries.GetCenter(bdrgeom),center);
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int k = mesh->GetBdrAttribute(j);
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switch (prob)
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{
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case lshape:
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if (center[0] == 1_r || center[0] == 0.5_r ||
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center[1] == 0.5_r)
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{
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ess_bdr[k - 1] = 1;
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}
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break;
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case fichera:
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if (center[0] == -1_r || center[0] == 0_r ||
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center[1] == 0_r || center[2] == 0_r)
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{
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ess_bdr[k - 1] = 1;
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}
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break;
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default:
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break;
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}
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}
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}
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}
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fespace->GetEssentialTrueDofs(ess_bdr, ess_tdof_list);
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// 9. Setup Complex Operator convention
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ComplexOperator::Convention conv =
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herm_conv ? ComplexOperator::HERMITIAN : ComplexOperator::BLOCK_SYMMETRIC;
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// 10. Set up the linear form b(.) which corresponds to the right-hand side of
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// the FEM linear system.
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VectorFunctionCoefficient f(dim, source);
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ComplexLinearForm b(fespace, conv);
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if (prob == load_src)
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{
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b.AddDomainIntegrator(NULL, new VectorFEDomainLFIntegrator(f));
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}
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b = 0.0;
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b.Assemble();
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// 11. Define the solution vector x as a complex finite element grid function
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// corresponding to fespace.
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ComplexGridFunction x(fespace);
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x = 0.0;
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VectorFunctionCoefficient E_Re(dim, E_bdr_data_Re);
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VectorFunctionCoefficient E_Im(dim, E_bdr_data_Im);
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x.ProjectBdrCoefficientTangent(E_Re, E_Im, ess_bdr);
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// 12. Set up the sesquilinear form a(.,.)
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//
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// In Comp
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// Domain: 1/mu (Curl E, Curl F) - omega^2 * epsilon (E,F)
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//
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// In PML: 1/mu (1/det(J) J^T J Curl E, Curl F)
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// - omega^2 * epsilon (det(J) * (J^T J)^-1 * E, F)
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//
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// where J denotes the Jacobian Matrix of the PML Stretching function
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Array<int> attr;
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Array<int> attrPML;
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if (mesh->attributes.Size())
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{
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attr.SetSize(mesh->attributes.Max());
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attrPML.SetSize(mesh->attributes.Max());
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attr = 0; attr[0] = 1;
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attrPML = 0;
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if (mesh->attributes.Max() > 1)
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{
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attrPML[1] = 1;
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}
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}
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ConstantCoefficient muinv(1_r / mu);
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ConstantCoefficient omeg(-pow2(omega) * epsilon);
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RestrictedCoefficient restr_muinv(muinv,attr);
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RestrictedCoefficient restr_omeg(omeg,attr);
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// Integrators inside the computational domain (excluding the PML region)
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SesquilinearForm a(fespace, conv);
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a.AddDomainIntegrator(new CurlCurlIntegrator(restr_muinv),NULL);
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a.AddDomainIntegrator(new VectorFEMassIntegrator(restr_omeg),NULL);
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int cdim = (dim == 2) ? 1 : dim;
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PMLDiagMatrixCoefficient pml_c1_Re(cdim,detJ_inv_JT_J_Re, pml);
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PMLDiagMatrixCoefficient pml_c1_Im(cdim,detJ_inv_JT_J_Im, pml);
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ScalarVectorProductCoefficient c1_Re(muinv,pml_c1_Re);
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ScalarVectorProductCoefficient c1_Im(muinv,pml_c1_Im);
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VectorRestrictedCoefficient restr_c1_Re(c1_Re,attrPML);
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VectorRestrictedCoefficient restr_c1_Im(c1_Im,attrPML);
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PMLDiagMatrixCoefficient pml_c2_Re(dim, detJ_JT_J_inv_Re,pml);
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PMLDiagMatrixCoefficient pml_c2_Im(dim, detJ_JT_J_inv_Im,pml);
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ScalarVectorProductCoefficient c2_Re(omeg,pml_c2_Re);
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ScalarVectorProductCoefficient c2_Im(omeg,pml_c2_Im);
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VectorRestrictedCoefficient restr_c2_Re(c2_Re,attrPML);
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VectorRestrictedCoefficient restr_c2_Im(c2_Im,attrPML);
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// Integrators inside the PML region
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a.AddDomainIntegrator(new CurlCurlIntegrator(restr_c1_Re),
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new CurlCurlIntegrator(restr_c1_Im));
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a.AddDomainIntegrator(new VectorFEMassIntegrator(restr_c2_Re),
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new VectorFEMassIntegrator(restr_c2_Im));
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// 13. Assemble the bilinear form and the corresponding linear system,
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// applying any necessary transformations such as: assembly, eliminating
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// boundary conditions, applying conforming constraints for
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// non-conforming AMR, etc.
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if (pa) { a.SetAssemblyLevel(AssemblyLevel::PARTIAL); }
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a.Assemble(0);
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OperatorPtr A;
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Vector B, X;
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a.FormLinearSystem(ess_tdof_list, x, b, A, X, B);
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// 14. Solve using a direct or an iterative solver
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#ifdef MFEM_USE_SUITESPARSE
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if (!pa && umf_solver)
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{
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ComplexUMFPackSolver csolver(*A.As<ComplexSparseMatrix>());
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csolver.Control[UMFPACK_ORDERING] = UMFPACK_ORDERING_METIS;
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csolver.SetPrintLevel(1);
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csolver.Mult(B, X);
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}
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#endif
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// 14a. Set up the Bilinear form a(.,.) for the preconditioner
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//
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// In Comp
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// Domain: 1/mu (Curl E, Curl F) + omega^2 * epsilon (E,F)
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//
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// In PML: 1/mu (abs(1/det(J) J^T J) Curl E, Curl F)
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// + omega^2 * epsilon (abs(det(J) * (J^T J)^-1) * E, F)
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if (pa || !umf_solver)
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{
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ConstantCoefficient absomeg(pow2(omega) * epsilon);
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RestrictedCoefficient restr_absomeg(absomeg,attr);
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BilinearForm prec(fespace);
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prec.AddDomainIntegrator(new CurlCurlIntegrator(restr_muinv));
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prec.AddDomainIntegrator(new VectorFEMassIntegrator(restr_absomeg));
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PMLDiagMatrixCoefficient pml_c1_abs(cdim,detJ_inv_JT_J_abs, pml);
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ScalarVectorProductCoefficient c1_abs(muinv,pml_c1_abs);
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VectorRestrictedCoefficient restr_c1_abs(c1_abs,attrPML);
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PMLDiagMatrixCoefficient pml_c2_abs(dim, detJ_JT_J_inv_abs,pml);
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ScalarVectorProductCoefficient c2_abs(absomeg,pml_c2_abs);
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VectorRestrictedCoefficient restr_c2_abs(c2_abs,attrPML);
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prec.AddDomainIntegrator(new CurlCurlIntegrator(restr_c1_abs));
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prec.AddDomainIntegrator(new VectorFEMassIntegrator(restr_c2_abs));
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if (pa) { prec.SetAssemblyLevel(AssemblyLevel::PARTIAL); }
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prec.Assemble();
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// 14b. Define and apply a GMRES solver for AU=B with a block diagonal
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// preconditioner based on the Gauss-Seidel or Jacobi sparse smoother.
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Array<int> offsets(3);
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offsets[0] = 0;
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offsets[1] = fespace->GetTrueVSize();
|
|
offsets[2] = fespace->GetTrueVSize();
|
|
offsets.PartialSum();
|
|
|
|
std::unique_ptr<Operator> pc_r;
|
|
std::unique_ptr<Operator> pc_i;
|
|
real_t s = (conv == ComplexOperator::HERMITIAN) ? -1_r : 1_r;
|
|
if (pa)
|
|
{
|
|
// Jacobi Smoother
|
|
pc_r.reset(new OperatorJacobiSmoother(prec, ess_tdof_list));
|
|
pc_i.reset(new ScaledOperator(pc_r.get(), s));
|
|
}
|
|
else
|
|
{
|
|
OperatorPtr PCOpAh;
|
|
prec.SetDiagonalPolicy(mfem::Operator::DIAG_ONE);
|
|
prec.FormSystemMatrix(ess_tdof_list, PCOpAh);
|
|
|
|
// Gauss-Seidel Smoother
|
|
pc_r.reset(new GSSmoother(*PCOpAh.As<SparseMatrix>()));
|
|
pc_i.reset(new ScaledOperator(pc_r.get(), s));
|
|
}
|
|
|
|
BlockDiagonalPreconditioner BlockDP(offsets);
|
|
BlockDP.SetDiagonalBlock(0, pc_r.get());
|
|
BlockDP.SetDiagonalBlock(1, pc_i.get());
|
|
|
|
GMRESSolver gmres;
|
|
gmres.SetPrintLevel(1);
|
|
gmres.SetKDim(200);
|
|
gmres.SetMaxIter(pa ? 5000 : 2000);
|
|
gmres.SetRelTol(1e-5);
|
|
gmres.SetAbsTol(0.0);
|
|
gmres.SetOperator(*A);
|
|
gmres.SetPreconditioner(BlockDP);
|
|
gmres.Mult(B, X);
|
|
}
|
|
|
|
// 15. Recover the solution as a finite element grid function and compute the
|
|
// errors if the exact solution is known.
|
|
a.RecoverFEMSolution(X, b, x);
|
|
|
|
// If exact is known compute the error
|
|
if (exact_known)
|
|
{
|
|
VectorFunctionCoefficient E_ex_Re(dim, E_exact_Re);
|
|
VectorFunctionCoefficient E_ex_Im(dim, E_exact_Im);
|
|
int order_quad = max(2, 2 * order + 1);
|
|
const IntegrationRule *irs[Geometry::NumGeom];
|
|
for (int i = 0; i < Geometry::NumGeom; ++i)
|
|
{
|
|
irs[i] = &(IntRules.Get(i, order_quad));
|
|
}
|
|
|
|
real_t L2Error_Re = x.real().ComputeL2Error(E_ex_Re, irs,
|
|
pml->GetMarkedPMLElements());
|
|
real_t L2Error_Im = x.imag().ComputeL2Error(E_ex_Im, irs,
|
|
pml->GetMarkedPMLElements());
|
|
|
|
ComplexGridFunction x_gf0(fespace);
|
|
x_gf0 = 0.0;
|
|
real_t norm_E_Re, norm_E_Im;
|
|
norm_E_Re = x_gf0.real().ComputeL2Error(E_ex_Re, irs,
|
|
pml->GetMarkedPMLElements());
|
|
norm_E_Im = x_gf0.imag().ComputeL2Error(E_ex_Im, irs,
|
|
pml->GetMarkedPMLElements());
|
|
|
|
cout << "\n Relative Error (Re part): || E_h - E || / ||E|| = "
|
|
<< L2Error_Re / norm_E_Re
|
|
<< "\n Relative Error (Im part): || E_h - E || / ||E|| = "
|
|
<< L2Error_Im / norm_E_Im
|
|
<< "\n Total Error: "
|
|
<< sqrt(L2Error_Re*L2Error_Re + L2Error_Im*L2Error_Im) << "\n\n";
|
|
}
|
|
|
|
// 16. Save the refined mesh and the solution. This output can be viewed
|
|
// later using GLVis: "glvis -m mesh -g sol".
|
|
{
|
|
ofstream mesh_ofs("ex25.mesh");
|
|
mesh_ofs.precision(8);
|
|
mesh->Print(mesh_ofs);
|
|
|
|
ofstream sol_r_ofs("ex25-sol_r.gf");
|
|
ofstream sol_i_ofs("ex25-sol_i.gf");
|
|
sol_r_ofs.precision(8);
|
|
sol_i_ofs.precision(8);
|
|
x.real().Save(sol_r_ofs);
|
|
x.imag().Save(sol_i_ofs);
|
|
}
|
|
|
|
// 17. Send the solution by socket to a GLVis server.
|
|
if (visualization)
|
|
{
|
|
// Define visualization keys for GLVis (see GLVis documentation)
|
|
string keys;
|
|
keys = (dim == 3) ? "keys macF\n" : keys = "keys amrRljcUUuu\n";
|
|
if (prob == beam && dim == 3) {keys = "keys macFFiYYYYYYYYYYYYYYYYYY\n";}
|
|
if (prob == beam && dim == 2) {keys = "keys amrRljcUUuuu\n"; }
|
|
|
|
char vishost[] = "localhost";
|
|
int visport = 19916;
|
|
|
|
socketstream sol_sock_re(vishost, visport);
|
|
sol_sock_re.precision(8);
|
|
sol_sock_re << "solution\n"
|
|
<< *mesh << x.real() << keys
|
|
<< "window_title 'Solution real part'" << flush;
|
|
|
|
socketstream sol_sock_im(vishost, visport);
|
|
sol_sock_im.precision(8);
|
|
sol_sock_im << "solution\n"
|
|
<< *mesh << x.imag() << keys
|
|
<< "window_title 'Solution imag part'" << flush;
|
|
|
|
GridFunction x_t(fespace);
|
|
x_t = x.real();
|
|
socketstream sol_sock(vishost, visport);
|
|
sol_sock.precision(8);
|
|
sol_sock << "solution\n"
|
|
<< *mesh << x_t << keys << "autoscale off\n"
|
|
<< "window_title 'Harmonic Solution (t = 0.0 T)'"
|
|
<< "pause\n" << flush;
|
|
cout << "GLVis visualization paused."
|
|
<< " Press space (in the GLVis window) to resume it.\n";
|
|
int num_frames = 32;
|
|
int i = 0;
|
|
while (sol_sock)
|
|
{
|
|
real_t t = (real_t)(i % num_frames) / num_frames;
|
|
ostringstream oss;
|
|
oss << "Harmonic Solution (t = " << t << " T)";
|
|
|
|
add(cos(real_t(2.0 * M_PI) * t), x.real(),
|
|
sin(real_t(2.0 * M_PI) * t), x.imag(), x_t);
|
|
sol_sock << "solution\n"
|
|
<< *mesh << x_t
|
|
<< "window_title '" << oss.str() << "'" << flush;
|
|
i++;
|
|
}
|
|
}
|
|
|
|
// 18. Free the used memory.
|
|
delete pml;
|
|
delete fespace;
|
|
delete fec;
|
|
delete mesh;
|
|
return 0;
|
|
}
|
|
|
|
void source(const Vector &x, Vector &f)
|
|
{
|
|
Vector center(dim);
|
|
real_t r = 0.0;
|
|
for (int i = 0; i < dim; ++i)
|
|
{
|
|
center(i) = 0.5_r * (comp_domain_bdr(i, 0) + comp_domain_bdr(i, 1));
|
|
r += pow2(x[i] - center[i]);
|
|
}
|
|
real_t n = 5_r * omega * sqrt(epsilon * mu) / real_t(M_PI);
|
|
real_t coeff = pow2(n) / real_t(M_PI);
|
|
real_t alpha = -pow2(n) * r;
|
|
f = 0.0;
|
|
f[0] = coeff * exp(alpha);
|
|
}
|
|
|
|
void maxwell_solution(const Vector &x, vector<complex<real_t>> &E)
|
|
{
|
|
// Initialize
|
|
for (int i = 0; i < dim; ++i)
|
|
{
|
|
E[i] = 0.0;
|
|
}
|
|
|
|
constexpr complex<real_t> zi = complex<real_t>(0., 1.);
|
|
real_t k = omega * sqrt(epsilon * mu);
|
|
switch (prob)
|
|
{
|
|
case disc:
|
|
case lshape:
|
|
case fichera:
|
|
{
|
|
Vector shift(dim);
|
|
shift = 0.0;
|
|
if (prob == fichera) { shift = 1.0; }
|
|
if (prob == disc) { shift = -0.5; }
|
|
if (prob == lshape) { shift = -1.0; }
|
|
|
|
if (dim == 2)
|
|
{
|
|
real_t x0 = x(0) + shift(0);
|
|
real_t x1 = x(1) + shift(1);
|
|
real_t r = sqrt(x0 * x0 + x1 * x1);
|
|
real_t beta = k * r;
|
|
|
|
// Bessel functions
|
|
complex<real_t> Ho, Ho_r, Ho_rr;
|
|
Ho = real_t(jn(0, beta)) + zi * real_t(yn(0, beta));
|
|
Ho_r = -k * (real_t(jn(1, beta)) + zi * real_t(yn(1, beta)));
|
|
Ho_rr = -k * k * (1_r / beta *
|
|
(real_t(jn(1, beta)) + zi * real_t(yn(1, beta))) -
|
|
(real_t(jn(2, beta)) + zi * real_t(yn(2, beta))));
|
|
|
|
// First derivatives
|
|
real_t r_x = x0 / r;
|
|
real_t r_y = x1 / r;
|
|
real_t r_xy = -(r_x / r) * r_y;
|
|
real_t r_xx = (1_r / r) * (1_r - r_x * r_x);
|
|
|
|
complex<real_t> val, val_xx, val_xy;
|
|
val = real_t(0.25) * zi * Ho;
|
|
val_xx = real_t(0.25) * zi * (r_xx * Ho_r + r_x * r_x * Ho_rr);
|
|
val_xy = real_t(0.25) * zi * (r_xy * Ho_r + r_x * r_y * Ho_rr);
|
|
E[0] = zi / k * (k * k * val + val_xx);
|
|
E[1] = zi / k * val_xy;
|
|
}
|
|
else if (dim == 3)
|
|
{
|
|
real_t x0 = x(0) + shift(0);
|
|
real_t x1 = x(1) + shift(1);
|
|
real_t x2 = x(2) + shift(2);
|
|
real_t r = sqrt(x0 * x0 + x1 * x1 + x2 * x2);
|
|
|
|
real_t r_x = x0 / r;
|
|
real_t r_y = x1 / r;
|
|
real_t r_z = x2 / r;
|
|
real_t r_xx = (1_r / r) * (1_r - r_x * r_x);
|
|
real_t r_yx = -(r_y / r) * r_x;
|
|
real_t r_zx = -(r_z / r) * r_x;
|
|
|
|
complex<real_t> val, val_r, val_rr;
|
|
val = exp(zi * k * r) / r;
|
|
val_r = val / r * (zi * k * r - 1_r);
|
|
val_rr = val / (r * r) * (-k * k * r * r
|
|
- real_t(2) * zi * k * r + real_t(2));
|
|
|
|
complex<real_t> val_xx, val_yx, val_zx;
|
|
val_xx = val_rr * r_x * r_x + val_r * r_xx;
|
|
val_yx = val_rr * r_x * r_y + val_r * r_yx;
|
|
val_zx = val_rr * r_x * r_z + val_r * r_zx;
|
|
|
|
complex<real_t> alpha = zi * k / real_t(4) / (real_t) M_PI / k / k;
|
|
E[0] = alpha * (k * k * val + val_xx);
|
|
E[1] = alpha * val_yx;
|
|
E[2] = alpha * val_zx;
|
|
}
|
|
break;
|
|
}
|
|
case beam:
|
|
{
|
|
// T_10 mode
|
|
if (dim == 3)
|
|
{
|
|
real_t k10 = sqrt(k * k - real_t(M_PI * M_PI));
|
|
E[1] = -zi * k / (real_t) M_PI *
|
|
sin((real_t) M_PI*x(2))*exp(zi * k10 * x(0));
|
|
}
|
|
else if (dim == 2)
|
|
{
|
|
E[1] = -zi * k / (real_t) M_PI * exp(zi * k * x(0));
|
|
}
|
|
break;
|
|
}
|
|
default:
|
|
break;
|
|
}
|
|
}
|
|
|
|
void E_exact_Re(const Vector &x, Vector &E)
|
|
{
|
|
vector<complex<real_t>> Eval(E.Size());
|
|
maxwell_solution(x, Eval);
|
|
for (int i = 0; i < dim; ++i)
|
|
{
|
|
E[i] = Eval[i].real();
|
|
}
|
|
}
|
|
|
|
void E_exact_Im(const Vector &x, Vector &E)
|
|
{
|
|
vector<complex<real_t>> Eval(E.Size());
|
|
maxwell_solution(x, Eval);
|
|
for (int i = 0; i < dim; ++i)
|
|
{
|
|
E[i] = Eval[i].imag();
|
|
}
|
|
}
|
|
|
|
void E_bdr_data_Re(const Vector &x, Vector &E)
|
|
{
|
|
E = 0.0;
|
|
bool in_pml = false;
|
|
|
|
for (int i = 0; i < dim; ++i)
|
|
{
|
|
// check if in PML
|
|
if (x(i) - comp_domain_bdr(i, 0) < 0_r ||
|
|
x(i) - comp_domain_bdr(i, 1) > 0_r)
|
|
{
|
|
in_pml = true;
|
|
break;
|
|
}
|
|
}
|
|
if (!in_pml)
|
|
{
|
|
vector<complex<real_t>> Eval(E.Size());
|
|
maxwell_solution(x, Eval);
|
|
for (int i = 0; i < dim; ++i)
|
|
{
|
|
E[i] = Eval[i].real();
|
|
}
|
|
}
|
|
}
|
|
|
|
// Define bdr_data solution
|
|
void E_bdr_data_Im(const Vector &x, Vector &E)
|
|
{
|
|
E = 0.0;
|
|
bool in_pml = false;
|
|
|
|
for (int i = 0; i < dim; ++i)
|
|
{
|
|
// check if in PML
|
|
if (x(i) - comp_domain_bdr(i, 0) < 0_r ||
|
|
x(i) - comp_domain_bdr(i, 1) > 0_r)
|
|
{
|
|
in_pml = true;
|
|
break;
|
|
}
|
|
}
|
|
if (!in_pml)
|
|
{
|
|
vector<complex<real_t>> Eval(E.Size());
|
|
maxwell_solution(x, Eval);
|
|
for (int i = 0; i < dim; ++i)
|
|
{
|
|
E[i] = Eval[i].imag();
|
|
}
|
|
}
|
|
}
|
|
|
|
void detJ_JT_J_inv_Re(const Vector &x, PML * pml, Vector &D)
|
|
{
|
|
vector<complex<real_t>> dxs(dim);
|
|
complex<real_t> det(1.0, 0.0);
|
|
pml->StretchFunction(x, dxs);
|
|
|
|
for (int i = 0; i < dim; ++i)
|
|
{
|
|
det *= dxs[i];
|
|
}
|
|
|
|
for (int i = 0; i < dim; ++i)
|
|
{
|
|
D(i) = (det / pow2(dxs[i])).real();
|
|
}
|
|
}
|
|
|
|
void detJ_JT_J_inv_Im(const Vector &x, PML * pml, Vector &D)
|
|
{
|
|
vector<complex<real_t>> dxs(dim);
|
|
complex<real_t> det = 1.0;
|
|
pml->StretchFunction(x, dxs);
|
|
|
|
for (int i = 0; i < dim; ++i)
|
|
{
|
|
det *= dxs[i];
|
|
}
|
|
|
|
for (int i = 0; i < dim; ++i)
|
|
{
|
|
D(i) = (det / pow2(dxs[i])).imag();
|
|
}
|
|
}
|
|
|
|
void detJ_JT_J_inv_abs(const Vector &x, PML * pml, Vector &D)
|
|
{
|
|
vector<complex<real_t>> dxs(dim);
|
|
complex<real_t> det = 1.0;
|
|
pml->StretchFunction(x, dxs);
|
|
|
|
for (int i = 0; i < dim; ++i)
|
|
{
|
|
det *= dxs[i];
|
|
}
|
|
|
|
for (int i = 0; i < dim; ++i)
|
|
{
|
|
D(i) = abs(det / pow2(dxs[i]));
|
|
}
|
|
}
|
|
|
|
void detJ_inv_JT_J_Re(const Vector &x, PML * pml, Vector &D)
|
|
{
|
|
vector<complex<real_t>> dxs(dim);
|
|
complex<real_t> det(1.0, 0.0);
|
|
pml->StretchFunction(x, dxs);
|
|
|
|
for (int i = 0; i < dim; ++i)
|
|
{
|
|
det *= dxs[i];
|
|
}
|
|
|
|
// in the 2D case the coefficient is scalar 1/det(J)
|
|
if (dim == 2)
|
|
{
|
|
D = (1_r / det).real();
|
|
}
|
|
else
|
|
{
|
|
for (int i = 0; i < dim; ++i)
|
|
{
|
|
D(i) = (pow2(dxs[i]) / det).real();
|
|
}
|
|
}
|
|
}
|
|
|
|
void detJ_inv_JT_J_Im(const Vector &x, PML * pml, Vector &D)
|
|
{
|
|
vector<complex<real_t>> dxs(dim);
|
|
complex<real_t> det = 1.0;
|
|
pml->StretchFunction(x, dxs);
|
|
|
|
for (int i = 0; i < dim; ++i)
|
|
{
|
|
det *= dxs[i];
|
|
}
|
|
|
|
if (dim == 2)
|
|
{
|
|
D = (1_r / det).imag();
|
|
}
|
|
else
|
|
{
|
|
for (int i = 0; i < dim; ++i)
|
|
{
|
|
D(i) = (pow2(dxs[i]) / det).imag();
|
|
}
|
|
}
|
|
}
|
|
|
|
void detJ_inv_JT_J_abs(const Vector &x, PML * pml, Vector &D)
|
|
{
|
|
vector<complex<real_t>> dxs(dim);
|
|
complex<real_t> det = 1.0;
|
|
pml->StretchFunction(x, dxs);
|
|
|
|
for (int i = 0; i < dim; ++i)
|
|
{
|
|
det *= dxs[i];
|
|
}
|
|
|
|
if (dim == 2)
|
|
{
|
|
D = abs(1_r / det);
|
|
}
|
|
else
|
|
{
|
|
for (int i = 0; i < dim; ++i)
|
|
{
|
|
D(i) = abs(pow2(dxs[i]) / det);
|
|
}
|
|
}
|
|
}
|
|
|
|
PML::PML(Mesh *mesh_, Array2D<real_t> length_)
|
|
: mesh(mesh_), length(length_)
|
|
{
|
|
dim = mesh->Dimension();
|
|
SetBoundaries();
|
|
}
|
|
|
|
void PML::SetBoundaries()
|
|
{
|
|
comp_dom_bdr.SetSize(dim, 2);
|
|
dom_bdr.SetSize(dim, 2);
|
|
Vector pmin, pmax;
|
|
mesh->GetBoundingBox(pmin, pmax);
|
|
for (int i = 0; i < dim; i++)
|
|
{
|
|
dom_bdr(i, 0) = pmin(i);
|
|
dom_bdr(i, 1) = pmax(i);
|
|
comp_dom_bdr(i, 0) = dom_bdr(i, 0) + length(i, 0);
|
|
comp_dom_bdr(i, 1) = dom_bdr(i, 1) - length(i, 1);
|
|
}
|
|
}
|
|
|
|
void PML::SetAttributes(Mesh *mesh_)
|
|
{
|
|
// Initialize bdr attributes
|
|
for (int i = 0; i < mesh_->GetNBE(); ++i)
|
|
{
|
|
mesh_->GetBdrElement(i)->SetAttribute(i+1);
|
|
}
|
|
|
|
int nrelem = mesh_->GetNE();
|
|
|
|
elems.SetSize(nrelem);
|
|
|
|
// Loop through the elements and identify which of them are in the PML
|
|
for (int i = 0; i < nrelem; ++i)
|
|
{
|
|
elems[i] = 1;
|
|
bool in_pml = false;
|
|
Element *el = mesh_->GetElement(i);
|
|
Array<int> vertices;
|
|
|
|
// Initialize attribute
|
|
el->SetAttribute(1);
|
|
el->GetVertices(vertices);
|
|
int nrvert = vertices.Size();
|
|
|
|
// Check if any vertex is in the PML
|
|
for (int iv = 0; iv < nrvert; ++iv)
|
|
{
|
|
int vert_idx = vertices[iv];
|
|
real_t *coords = mesh_->GetVertex(vert_idx);
|
|
for (int comp = 0; comp < dim; ++comp)
|
|
{
|
|
if (coords[comp] > comp_dom_bdr(comp, 1) ||
|
|
coords[comp] < comp_dom_bdr(comp, 0))
|
|
{
|
|
in_pml = true;
|
|
break;
|
|
}
|
|
}
|
|
}
|
|
if (in_pml)
|
|
{
|
|
elems[i] = 0;
|
|
el->SetAttribute(2);
|
|
}
|
|
}
|
|
mesh_->SetAttributes();
|
|
}
|
|
|
|
void PML::StretchFunction(const Vector &x,
|
|
vector<complex<real_t>> &dxs)
|
|
{
|
|
constexpr complex<real_t> zi = complex<real_t>(0., 1.);
|
|
|
|
real_t n = 2.0;
|
|
real_t c = 5.0;
|
|
real_t coeff;
|
|
real_t k = omega * sqrt(epsilon * mu);
|
|
|
|
// Stretch in each direction independently
|
|
for (int i = 0; i < dim; ++i)
|
|
{
|
|
dxs[i] = 1.0;
|
|
if (x(i) >= comp_domain_bdr(i, 1))
|
|
{
|
|
coeff = n * c / k / pow(length(i, 1), n);
|
|
dxs[i] = 1_r + zi * coeff *
|
|
abs(pow(x(i) - comp_domain_bdr(i, 1), n - 1_r));
|
|
}
|
|
if (x(i) <= comp_domain_bdr(i, 0))
|
|
{
|
|
coeff = n * c / k / pow(length(i, 0), n);
|
|
dxs[i] = 1_r + zi * coeff *
|
|
abs(pow(x(i) - comp_domain_bdr(i, 0), n - 1_r));
|
|
}
|
|
}
|
|
}
|