1057 lines
31 KiB
C++
1057 lines
31 KiB
C++
// MFEM Example 25 - Parallel Version
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//
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// Compile with: make ex25p
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//
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// Sample runs: mpirun -np 4 ex25p -o 2 -f 1.0 -rs 1 -rp 1 -prob 0
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// mpirun -np 4 ex25p -o 3 -f 10.0 -rs 1 -rp 1 -prob 1
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// mpirun -np 4 ex25p -o 3 -f 5.0 -rs 3 -rp 1 -prob 2
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// mpirun -np 4 ex25p -o 2 -f 1.0 -rs 1 -rp 1 -prob 3
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// mpirun -np 4 ex25p -o 2 -f 1.0 -rs 2 -rp 2 -prob 0 -m ../data/beam-quad.mesh
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// mpirun -np 4 ex25p -o 2 -f 8.0 -rs 2 -rp 2 -prob 4 -m ../data/inline-quad.mesh
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// mpirun -np 4 ex25p -o 2 -f 2.0 -rs 1 -rp 1 -prob 4 -m ../data/inline-hex.mesh
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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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// (1/mu) * curl curl E - \omega^2 * epsilon E = f
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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 <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 CartesianPML
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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<double> length;
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// Computational Domain Boundary
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Array2D<double> comp_dom_bdr;
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// Domain Boundary
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Array2D<double> 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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CartesianPML(Mesh *mesh_,Array2D<double> length_);
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// Return Computational Domain Boundary
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Array2D<double> GetCompDomainBdr() {return comp_dom_bdr;}
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// Return Domain Boundary
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Array2D<double> 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(ParMesh *pmesh);
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// PML complex stretching function
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void StretchFunction(const Vector &x, vector<complex<double>> &dxs);
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};
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// Class for returning the PML coefficients of the bilinear form
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class PMLMatrixCoefficient : public MatrixCoefficient
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{
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private:
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CartesianPML * pml = nullptr;
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void (*Function)(const Vector &, CartesianPML * , DenseMatrix &);
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public:
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PMLMatrixCoefficient(int dim, void(*F)(const Vector &, CartesianPML *,
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DenseMatrix &),
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CartesianPML * pml_)
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: MatrixCoefficient(dim), pml(pml_), Function(F)
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{}
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virtual void Eval(DenseMatrix &K, ElementTransformation &T,
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const IntegrationPoint &ip)
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{
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double x[3];
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Vector transip(x, 3);
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T.Transform(ip, transip);
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K.SetSize(height, width);
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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<double>> &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, CartesianPML * pml, DenseMatrix &M);
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void detJ_JT_J_inv_Im(const Vector &x, CartesianPML * pml, DenseMatrix &M);
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void detJ_JT_J_inv_abs(const Vector &x, CartesianPML * pml, DenseMatrix &M);
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void detJ_inv_JT_J_Re(const Vector &x, CartesianPML * pml, DenseMatrix &M);
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void detJ_inv_JT_J_Im(const Vector &x, CartesianPML * pml, DenseMatrix &M);
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void detJ_inv_JT_J_abs(const Vector &x, CartesianPML * pml, DenseMatrix &M);
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Array2D<double> comp_domain_bdr;
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Array2D<double> domain_bdr;
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double mu = 1.0;
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double epsilon = 1.0;
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double omega;
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int dim;
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bool exact_known = false;
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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. Initialize MPI.
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int num_procs, myid;
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MPI_Init(&argc, &argv);
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MPI_Comm_size(MPI_COMM_WORLD, &num_procs);
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MPI_Comm_rank(MPI_COMM_WORLD, &myid);
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// 2. 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 = 1;
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int par_ref_levels = 2;
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int iprob = 4;
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double freq = 5.0;
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bool herm_conv = true;
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bool visualization = 1;
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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, "-rs", "--refinements-serial",
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"Number of serial refinements");
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args.AddOption(&par_ref_levels, "-rp", "--refinements-parallel",
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"Number of parallel 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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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.Parse();
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if (iprob > 4) { iprob = 4; }
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prob = (prob_type)iprob;
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// 3. Setup the (serial) mesh on all processors.
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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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if (myid == 0)
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{
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args.PrintUsage(cout);
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}
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MPI_Finalize();
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return 1;
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}
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if (myid == 0)
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{
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args.PrintOptions(cout);
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}
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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 = 2.0 * M_PI * freq;
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// Setup PML length
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Array2D<double> 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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CartesianPML * pml = new CartesianPML(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 serial mesh on all processors 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. Define a parallel mesh by a partitioning of the serial mesh.
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ParMesh *pmesh = new ParMesh(MPI_COMM_WORLD, *mesh);
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delete mesh;
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{
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for (int l = 0; l < par_ref_levels; l++)
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{
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pmesh->UniformRefinement();
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}
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}
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// 6a. Reorient mesh in case of a tet mesh
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pmesh->ReorientTetMesh();
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// 7. Set element attributes in order to distinguish elements in the PML
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pml->SetAttributes(pmesh);
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// 8. Define a parallel finite element space on the parallel mesh. Here we
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// use the Nedelec finite elements of the specified order.
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FiniteElementCollection *fec = new ND_FECollection(order, dim);
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ParFiniteElementSpace *fespace = new ParFiniteElementSpace(pmesh, fec);
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HYPRE_Int size = fespace->GlobalTrueVSize();
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if (myid == 0)
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{
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cout << "Number of finite element unknowns: " << size << endl;
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}
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// 9. Determine the list of true (i.e. parallel conforming) essential
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// boundary dofs. In this example, the boundary conditions are defined
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// based on the specific mesh and the problem type.
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Array<int> ess_tdof_list;
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Array<int> ess_bdr;
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if (pmesh->bdr_attributes.Size())
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{
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ess_bdr.SetSize(pmesh->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 < pmesh->GetNBE(); j++)
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{
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Vector center(dim);
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int bdrgeom = pmesh->GetBdrElementBaseGeometry(j);
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ElementTransformation * tr = pmesh->GetBdrElementTransformation(j);
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tr->Transform(Geometries.GetCenter(bdrgeom),center);
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int k = pmesh->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] == 1.0 || center[0] == 0.5 || center[1] == 0.5)
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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] == -1.0 || center[0] == 0.0 ||
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center[1] == 0.0 || center[2] == 0.0)
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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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// 10. 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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// 11. Set up the parallel linear form b(.) which corresponds to the
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// right-hand side of the FEM linear system.
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VectorFunctionCoefficient f(dim, source);
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ParComplexLinearForm 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.Vector::operator=(0.0);
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b.Assemble();
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// 12. Define the solution vector x as a parallel complex finite element grid
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// function corresponding to fespace.
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ParComplexGridFunction 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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// 13. Set up the parallel 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 (pmesh->attributes.Size())
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{
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attr.SetSize(pmesh->attributes.Max());
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attrPML.SetSize(pmesh->attributes.Max());
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attr = 0; attr[0] = 1;
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attrPML = 0;
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if (pmesh->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(1.0/mu);
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ConstantCoefficient omeg(-pow(omega, 2) * 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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ParSesquilinearForm 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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PMLMatrixCoefficient pml_c1_Re(cdim,detJ_inv_JT_J_Re, pml);
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PMLMatrixCoefficient pml_c1_Im(cdim,detJ_inv_JT_J_Im, pml);
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ScalarMatrixProductCoefficient c1_Re(muinv,pml_c1_Re);
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ScalarMatrixProductCoefficient c1_Im(muinv,pml_c1_Im);
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MatrixRestrictedCoefficient restr_c1_Re(c1_Re,attrPML);
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MatrixRestrictedCoefficient restr_c1_Im(c1_Im,attrPML);
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PMLMatrixCoefficient pml_c2_Re(dim, detJ_JT_J_inv_Re,pml);
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PMLMatrixCoefficient pml_c2_Im(dim, detJ_JT_J_inv_Im,pml);
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ScalarMatrixProductCoefficient c2_Re(omeg,pml_c2_Re);
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ScalarMatrixProductCoefficient c2_Im(omeg,pml_c2_Im);
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MatrixRestrictedCoefficient restr_c2_Re(c2_Re,attrPML);
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MatrixRestrictedCoefficient 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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// 14. Assemble the parallel bilinear form and the corresponding linear
|
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// system, applying any necessary transformations such as: parallel
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// assembly, eliminating boundary conditions, applying conforming
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// constraints for non-conforming AMR, etc.
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a.Assemble();
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OperatorHandle Ah;
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Vector B, X;
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a.FormLinearSystem(ess_tdof_list, x, b, Ah, X, B);
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// 15. Transform to monolithic HypreParMatrix
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HypreParMatrix *A = Ah.As<ComplexHypreParMatrix>()->GetSystemMatrix();
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if (myid == 0)
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{
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cout << "Size of linear system: " << A->GetGlobalNumRows() << endl;
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}
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// 16. Solve using a direct or an iterative solver
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#ifdef MFEM_USE_SUPERLU
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{
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SuperLURowLocMatrix SA(*A);
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SuperLUSolver superlu(MPI_COMM_WORLD);
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superlu.SetPrintStatistics(false);
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superlu.SetSymmetricPattern(false);
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superlu.SetColumnPermutation(superlu::PARMETIS);
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superlu.SetOperator(SA);
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superlu.Mult(B, X);
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}
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#else
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// 16a. Set up the parallel 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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{
|
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ConstantCoefficient absomeg(pow(omega, 2) * epsilon);
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RestrictedCoefficient restr_absomeg(absomeg,attr);
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|
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ParBilinearForm 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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|
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PMLMatrixCoefficient pml_c1_abs(cdim,detJ_inv_JT_J_abs, pml);
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ScalarMatrixProductCoefficient c1_abs(muinv,pml_c1_abs);
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MatrixRestrictedCoefficient restr_c1_abs(c1_abs,attrPML);
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PMLMatrixCoefficient pml_c2_abs(dim, detJ_JT_J_inv_abs,pml);
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ScalarMatrixProductCoefficient c2_abs(absomeg,pml_c2_abs);
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MatrixRestrictedCoefficient 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));
|
||
|
||
prec.Assemble();
|
||
|
||
OperatorHandle PCOpAh;
|
||
prec.FormSystemMatrix(ess_tdof_list, PCOpAh);
|
||
|
||
// 16b. Define and apply a parallel GMRES solver for AU=B with a block
|
||
// diagonal preconditioner based on hypre's AMS preconditioner.
|
||
Array<int> offsets(3);
|
||
offsets[0] = 0;
|
||
offsets[1] = fespace->GetTrueVSize();
|
||
offsets[2] = fespace->GetTrueVSize();
|
||
offsets.PartialSum();
|
||
|
||
HypreAMS ams00(*PCOpAh.As<HypreParMatrix>(),fespace);
|
||
BlockDiagonalPreconditioner BlockAMS(offsets);
|
||
ScaledOperator ams11(&ams00,
|
||
(conv == ComplexOperator::HERMITIAN) ? -1.0 : 1.0);
|
||
BlockAMS.SetDiagonalBlock(0,&ams00);
|
||
BlockAMS.SetDiagonalBlock(1,&ams11);
|
||
|
||
GMRESSolver gmres(MPI_COMM_WORLD);
|
||
gmres.SetPrintLevel(1);
|
||
gmres.SetKDim(200);
|
||
gmres.SetMaxIter(2000);
|
||
gmres.SetRelTol(1e-5);
|
||
gmres.SetAbsTol(0.0);
|
||
gmres.SetOperator(*A);
|
||
gmres.SetPreconditioner(BlockAMS);
|
||
gmres.Mult(B, X);
|
||
}
|
||
#endif
|
||
|
||
// 17. Recover the parallel grid function corresponding to X. This is the
|
||
// local finite element solution on each processor.
|
||
a.RecoverFEMSolution(X, b, x);
|
||
|
||
// If exact is known compute the error
|
||
if (exact_known)
|
||
{
|
||
ParComplexGridFunction x_gf(fespace);
|
||
VectorFunctionCoefficient E_ex_Re(dim, E_exact_Re);
|
||
VectorFunctionCoefficient E_ex_Im(dim, E_exact_Im);
|
||
x_gf.ProjectCoefficient(E_ex_Re, E_ex_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));
|
||
}
|
||
|
||
double L2Error_Re = x.real().ComputeL2Error(E_ex_Re, irs,
|
||
pml->GetMarkedPMLElements());
|
||
double L2Error_Im = x.imag().ComputeL2Error(E_ex_Im, irs,
|
||
pml->GetMarkedPMLElements());
|
||
|
||
ParComplexGridFunction x_gf0(fespace);
|
||
x_gf0 = 0.0;
|
||
double 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());
|
||
|
||
if (myid == 0)
|
||
{
|
||
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";
|
||
}
|
||
}
|
||
|
||
// 18. Save the refined mesh and the solution in parallel. This output can be
|
||
// viewed later using GLVis: "glvis -np <np> -m mesh -g sol".
|
||
{
|
||
ostringstream mesh_name, sol_r_name, sol_i_name;
|
||
mesh_name << "mesh." << setfill('0') << setw(6) << myid;
|
||
sol_r_name << "ex25p-sol_r." << setfill('0') << setw(6) << myid;
|
||
sol_i_name << "ex25p-sol_i." << setfill('0') << setw(6) << myid;
|
||
|
||
ofstream mesh_ofs(mesh_name.str().c_str());
|
||
mesh_ofs.precision(8);
|
||
pmesh->Print(mesh_ofs);
|
||
|
||
ofstream sol_r_ofs(sol_r_name.str().c_str());
|
||
ofstream sol_i_ofs(sol_i_name.str().c_str());
|
||
sol_r_ofs.precision(8);
|
||
sol_i_ofs.precision(8);
|
||
x.real().Save(sol_r_ofs);
|
||
x.imag().Save(sol_i_ofs);
|
||
}
|
||
|
||
// 19. 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 << "parallel " << num_procs << " " << myid << "\n"
|
||
<< "solution\n" << *pmesh << x.real() << keys
|
||
<< "window_title 'Solution real part'" << flush;
|
||
MPI_Barrier(MPI_COMM_WORLD); // try to prevent streams from mixing
|
||
}
|
||
|
||
{
|
||
socketstream sol_sock_im(vishost, visport);
|
||
sol_sock_im.precision(8);
|
||
sol_sock_im << "parallel " << num_procs << " " << myid << "\n"
|
||
<< "solution\n" << *pmesh << x.imag() << keys
|
||
<< "window_title 'Solution imag part'" << flush;
|
||
MPI_Barrier(MPI_COMM_WORLD); // try to prevent streams from mixing
|
||
}
|
||
|
||
{
|
||
ParGridFunction x_t(fespace);
|
||
x_t = x.real();
|
||
|
||
socketstream sol_sock(vishost, visport);
|
||
sol_sock.precision(8);
|
||
sol_sock << "parallel " << num_procs << " " << myid << "\n"
|
||
<< "solution\n" << *pmesh << x_t << keys << "autoscale off\n"
|
||
<< "window_title 'Harmonic Solution (t = 0.0 T)'"
|
||
<< "pause\n" << flush;
|
||
|
||
if (myid == 0)
|
||
{
|
||
cout << "GLVis visualization paused."
|
||
<< " Press space (in the GLVis window) to resume it.\n";
|
||
}
|
||
|
||
int num_frames = 32;
|
||
int i = 0;
|
||
while (sol_sock)
|
||
{
|
||
double t = (double)(i % num_frames) / num_frames;
|
||
ostringstream oss;
|
||
oss << "Harmonic Solution (t = " << t << " T)";
|
||
|
||
add(cos(2.0*M_PI*t), x.real(), sin(2.0*M_PI*t), x.imag(), x_t);
|
||
sol_sock << "parallel " << num_procs << " " << myid << "\n";
|
||
sol_sock << "solution\n" << *pmesh << x_t
|
||
<< "window_title '" << oss.str() << "'" << flush;
|
||
i++;
|
||
}
|
||
}
|
||
}
|
||
|
||
// 20. Free the used memory.
|
||
delete A;
|
||
delete pml;
|
||
delete fespace;
|
||
delete fec;
|
||
delete pmesh;
|
||
MPI_Finalize();
|
||
return 0;
|
||
}
|
||
|
||
void source(const Vector &x, Vector &f)
|
||
{
|
||
Vector center(dim);
|
||
double r = 0.0;
|
||
for (int i = 0; i < dim; ++i)
|
||
{
|
||
center(i) = 0.5 * (comp_domain_bdr(i, 0) + comp_domain_bdr(i, 1));
|
||
r += pow(x[i] - center[i], 2.);
|
||
}
|
||
double n = 5.0 * omega * sqrt(epsilon * mu) / M_PI;
|
||
double coeff = pow(n, 2) / M_PI;
|
||
double alpha = -pow(n, 2) * r;
|
||
f = 0.0;
|
||
f[0] = coeff * exp(alpha);
|
||
}
|
||
|
||
void maxwell_solution(const Vector &x, vector<complex<double>> &E)
|
||
{
|
||
// Initialize
|
||
for (int i = 0; i < dim; ++i)
|
||
{
|
||
E[i] = 0.0;
|
||
}
|
||
|
||
complex<double> zi = complex<double>(0., 1.);
|
||
double 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)
|
||
{
|
||
double x0 = x(0) + shift(0);
|
||
double x1 = x(1) + shift(1);
|
||
double r = sqrt(x0 * x0 + x1 * x1);
|
||
double beta = k * r;
|
||
|
||
// Bessel functions
|
||
complex<double> Ho, Ho_r, Ho_rr;
|
||
Ho = jn(0, beta) + zi * yn(0, beta);
|
||
Ho_r = -k * (jn(1, beta) + zi * yn(1, beta));
|
||
Ho_rr = -k * k * (1.0 / beta *
|
||
(jn(1, beta) + zi * yn(1, beta)) -
|
||
(jn(2, beta) + zi * yn(2, beta)));
|
||
|
||
// First derivatives
|
||
double r_x = x0 / r;
|
||
double r_y = x1 / r;
|
||
double r_xy = -(r_x / r) * r_y;
|
||
double r_xx = (1.0 / r) * (1.0 - r_x * r_x);
|
||
|
||
complex<double> val, val_xx, val_xy;
|
||
val = 0.25 * zi * Ho;
|
||
val_xx = 0.25 * zi * (r_xx * Ho_r + r_x * r_x * Ho_rr);
|
||
val_xy = 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)
|
||
{
|
||
double x0 = x(0) + shift(0);
|
||
double x1 = x(1) + shift(1);
|
||
double x2 = x(2) + shift(2);
|
||
double r = sqrt(x0 * x0 + x1 * x1 + x2 * x2);
|
||
|
||
double r_x = x0 / r;
|
||
double r_y = x1 / r;
|
||
double r_z = x2 / r;
|
||
double r_xx = (1.0 / r) * (1.0 - r_x * r_x);
|
||
double r_yx = -(r_y / r) * r_x;
|
||
double r_zx = -(r_z / r) * r_x;
|
||
|
||
complex<double> val, val_r, val_rr;
|
||
val = exp(zi * k * r) / r;
|
||
val_r = val / r * (zi * k * r - 1.0);
|
||
val_rr = val / (r * r) * (-k * k * r * r
|
||
- 2.0 * zi * k * r + 2.0);
|
||
|
||
complex<double> 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<double> alpha = zi * k / 4.0 / 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)
|
||
{
|
||
double k10 = sqrt(k * k - M_PI * M_PI);
|
||
E[1] = -zi * k / M_PI * sin(M_PI*x(2))*exp(zi * k10 * x(0));
|
||
}
|
||
else if (dim == 2)
|
||
{
|
||
E[1] = -zi * k / M_PI * exp(zi * k * x(0));
|
||
}
|
||
break;
|
||
}
|
||
default:
|
||
break;
|
||
}
|
||
}
|
||
|
||
void E_exact_Re(const Vector &x, Vector &E)
|
||
{
|
||
vector<complex<double>> 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<double>> 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) < 0.0 ||
|
||
x(i) - comp_domain_bdr(i, 1) > 0.0)
|
||
{
|
||
in_pml = true;
|
||
break;
|
||
}
|
||
}
|
||
if (!in_pml)
|
||
{
|
||
vector<complex<double>> 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) < 0.0 ||
|
||
x(i) - comp_domain_bdr(i, 1) > 0.0)
|
||
{
|
||
in_pml = true;
|
||
break;
|
||
}
|
||
}
|
||
if (!in_pml)
|
||
{
|
||
vector<complex<double>> 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, CartesianPML * pml, DenseMatrix &M)
|
||
{
|
||
vector<complex<double>> dxs(dim);
|
||
complex<double> det(1.0, 0.0);
|
||
pml->StretchFunction(x, dxs);
|
||
|
||
for (int i = 0; i < dim; ++i)
|
||
{
|
||
det *= dxs[i];
|
||
}
|
||
|
||
M = 0.0;
|
||
for (int i = 0; i < dim; ++i)
|
||
{
|
||
M(i, i) = (det / pow(dxs[i], 2)).real();
|
||
}
|
||
}
|
||
|
||
void detJ_JT_J_inv_Im(const Vector &x, CartesianPML * pml, DenseMatrix &M)
|
||
{
|
||
vector<complex<double>> dxs(dim);
|
||
complex<double> det = 1.0;
|
||
pml->StretchFunction(x, dxs);
|
||
|
||
for (int i = 0; i < dim; ++i)
|
||
{
|
||
det *= dxs[i];
|
||
}
|
||
|
||
M = 0.0;
|
||
for (int i = 0; i < dim; ++i)
|
||
{
|
||
M(i, i) = (det / pow(dxs[i], 2)).imag();
|
||
}
|
||
}
|
||
|
||
void detJ_JT_J_inv_abs(const Vector &x, CartesianPML * pml, DenseMatrix &M)
|
||
{
|
||
vector<complex<double>> dxs(dim);
|
||
complex<double> det = 1.0;
|
||
pml->StretchFunction(x, dxs);
|
||
|
||
for (int i = 0; i < dim; ++i)
|
||
{
|
||
det *= dxs[i];
|
||
}
|
||
|
||
M = 0.0;
|
||
for (int i = 0; i < dim; ++i)
|
||
{
|
||
M(i, i) = abs(det / pow(dxs[i], 2));
|
||
}
|
||
}
|
||
|
||
void detJ_inv_JT_J_Re(const Vector &x, CartesianPML * pml, DenseMatrix &M)
|
||
{
|
||
vector<complex<double>> dxs(dim);
|
||
complex<double> 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)
|
||
{
|
||
M = (1.0 / det).real();
|
||
}
|
||
else
|
||
{
|
||
M = 0.0;
|
||
for (int i = 0; i < dim; ++i)
|
||
{
|
||
M(i, i) = (pow(dxs[i], 2) / det).real();
|
||
}
|
||
}
|
||
}
|
||
|
||
void detJ_inv_JT_J_Im(const Vector &x, CartesianPML * pml, DenseMatrix &M)
|
||
{
|
||
vector<complex<double>> dxs(dim);
|
||
complex<double> det = 1.0;
|
||
pml->StretchFunction(x, dxs);
|
||
|
||
for (int i = 0; i < dim; ++i)
|
||
{
|
||
det *= dxs[i];
|
||
}
|
||
|
||
if (dim == 2)
|
||
{
|
||
M = (1.0 / det).imag();
|
||
}
|
||
else
|
||
{
|
||
M = 0.0;
|
||
for (int i = 0; i < dim; ++i)
|
||
{
|
||
M(i, i) = (pow(dxs[i], 2) / det).imag();
|
||
}
|
||
}
|
||
}
|
||
|
||
void detJ_inv_JT_J_abs(const Vector &x, CartesianPML * pml, DenseMatrix &M)
|
||
{
|
||
vector<complex<double>> dxs(dim);
|
||
complex<double> det = 1.0;
|
||
pml->StretchFunction(x, dxs);
|
||
|
||
for (int i = 0; i < dim; ++i)
|
||
{
|
||
det *= dxs[i];
|
||
}
|
||
|
||
if (dim == 2)
|
||
{
|
||
M = abs(1.0 / det);
|
||
}
|
||
else
|
||
{
|
||
M = 0.0;
|
||
for (int i = 0; i < dim; ++i)
|
||
{
|
||
M(i, i) = abs(pow(dxs[i], 2) / det);
|
||
}
|
||
}
|
||
}
|
||
|
||
CartesianPML::CartesianPML(Mesh *mesh_, Array2D<double> length_)
|
||
: mesh(mesh_), length(length_)
|
||
{
|
||
dim = mesh->Dimension();
|
||
SetBoundaries();
|
||
}
|
||
|
||
void CartesianPML::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 CartesianPML::SetAttributes(ParMesh *pmesh)
|
||
{
|
||
int myid;
|
||
MPI_Comm_rank(MPI_COMM_WORLD,&myid);
|
||
int nrelem = pmesh->GetNE();
|
||
|
||
// Initialize list with 1
|
||
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 = pmesh->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];
|
||
double *coords = pmesh->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);
|
||
}
|
||
}
|
||
pmesh->SetAttributes();
|
||
}
|
||
|
||
void CartesianPML::StretchFunction(const Vector &x,
|
||
vector<complex<double>> &dxs)
|
||
{
|
||
complex<double> zi = complex<double>(0., 1.);
|
||
|
||
double n = 2.0;
|
||
double c = 5.0;
|
||
double coeff;
|
||
double 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] = 1.0 + zi * coeff *
|
||
abs(pow(x(i) - comp_domain_bdr(i, 1), n - 1.0));
|
||
}
|
||
if (x(i) <= comp_domain_bdr(i, 0))
|
||
{
|
||
coeff = n * c / k / pow(length(i, 0), n);
|
||
dxs[i] = 1.0 + zi * coeff *
|
||
abs(pow(x(i) - comp_domain_bdr(i, 0), n - 1.0));
|
||
}
|
||
}
|
||
}
|