790 lines
25 KiB
C++
790 lines
25 KiB
C++
// Copyright (c) 2010-2025, Lawrence Livermore National Security, LLC. Produced
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// at the Lawrence Livermore National Laboratory. All Rights reserved. See files
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// LICENSE and NOTICE for details. LLNL-CODE-806117.
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//
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// This file is part of the MFEM library. For more information and source code
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// availability visit https://mfem.org.
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//
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// MFEM is free software; you can redistribute it and/or modify it under the
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// terms of the BSD-3 license. We welcome feedback and contributions, see file
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// CONTRIBUTING.md for details.
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//
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// MFEM Ultraweak DPG example for acoustics (Helmholtz)
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//
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// Compile with: make acoustics
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//
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// Sample runs
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//
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// acoustics -ref 4 -o 1 -rnum 1.0
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// acoustics -m ../../data/inline-tri.mesh -ref 4 -o 2 -sc -rnum 3.0
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// acoustics -m ../../data/amr-quad.mesh -ref 3 -o 3 -sc -rnum 4.5 -prob 1
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// acoustics -m ../../data/inline-quad.mesh -ref 2 -o 4 -sc -rnum 11.5 -prob 1
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// acoustics -m ../../data/inline-hex.mesh -ref 1 -o 2 -sc -rnum 1.0
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// Description:
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// This example code demonstrates the use of MFEM to define and solve
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// the "ultraweak" (UW) DPG formulation for the Helmholtz problem
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// - Δ p - ω² p = f̃ , in Ω
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// p = p₀, on ∂Ω
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// It solves two kinds of problems
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// a) f̃ = 0 and p₀ is a plane wave
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// b) A manufactured solution problem where p_exact is a gaussian beam
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// This example computes and prints out convergence rates for the L² error.
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// The DPG UW deals with the First Order System
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// ∇ p + i ω u = 0, in Ω
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// ∇⋅u + i ω p = f, in Ω (1)
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// p = p_0, in ∂Ω
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// where f:=f̃/(i ω)
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// Ultraweak-DPG is obtained by integration by parts of both equations and the
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// introduction of trace unknowns on the mesh skeleton
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//
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// p ∈ L²(Ω), u ∈ (L²(Ω))ᵈⁱᵐ
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// p̂ ∈ H^1/2(Ω), û ∈ H^-1/2(Ω)
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// -(p, ∇⋅v) + i ω (u , v) + < p̂, v⋅n> = 0, ∀ v ∈ H(div,Ω)
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// -(u , ∇ q) + i ω (p , q) + < û, q > = (f,q) ∀ q ∈ H¹(Ω)
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// p̂ = p₀ on ∂Ω
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// Note:
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// p̂ := p, û := u on the mesh skeleton
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// For more information see https://doi.org/10.1016/j.camwa.2017.06.044
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// -------------------------------------------------------------
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// | | p | u | p̂ | û | RHS |
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// -------------------------------------------------------------
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// | v | -(p, ∇⋅v) | i ω (u,v) | < p̂, v⋅n> | | |
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// | | | | | | |
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// | q | i ω (p,q) |-(u , ∇ q) | | < û,q > | (f,q) |
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// where (q,v) ∈ H¹(Ω) × H(div,Ω)
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// Here we use the "Adjoint Graph" norm on the test space i.e.,
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// ||(q,v)||²ᵥ = ||A^*(q,v)||² + ||(q,v)||² where A is the
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// acoustics operator defined by (1)
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#include "mfem.hpp"
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#include "util/complexweakform.hpp"
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#include "../common/mfem-common.hpp"
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#include <fstream>
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#include <iostream>
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using namespace std;
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using namespace mfem;
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using namespace mfem::common;
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complex<real_t> acoustics_solution(const Vector & X);
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void acoustics_solution_grad(const Vector & X,vector<complex<real_t>> &dp);
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complex<real_t> acoustics_solution_laplacian(const Vector & X);
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real_t p_exact_r(const Vector &x);
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real_t p_exact_i(const Vector &x);
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void u_exact_r(const Vector &x, Vector & u);
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void u_exact_i(const Vector &x, Vector & u);
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real_t rhs_func_r(const Vector &x);
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real_t rhs_func_i(const Vector &x);
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void gradp_exact_r(const Vector &x, Vector &gradu);
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void gradp_exact_i(const Vector &x, Vector &gradu);
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real_t divu_exact_r(const Vector &x);
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real_t divu_exact_i(const Vector &x);
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real_t d2_exact_r(const Vector &x);
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real_t d2_exact_i(const Vector &x);
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real_t hatp_exact_r(const Vector & X);
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real_t hatp_exact_i(const Vector & X);
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void hatu_exact_r(const Vector & X, Vector & hatu);
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void hatu_exact_i(const Vector & X, Vector & hatu);
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int dim;
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real_t omega;
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enum prob_type
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{
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plane_wave,
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gaussian_beam
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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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const char *mesh_file = "../../data/inline-quad.mesh";
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int order = 1;
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int delta_order = 1;
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bool visualization = true;
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real_t rnum=1.0;
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int ref = 0;
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bool static_cond = false;
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int visport = 19916;
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int iprob = 0;
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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(&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(&rnum, "-rnum", "--number_of_wavelengths",
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"Number of wavelengths");
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args.AddOption(&iprob, "-prob", "--problem", "Problem case"
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" 0: plane wave, 1: Gaussian beam");
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args.AddOption(&delta_order, "-do", "--delta_order",
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"Order enrichment for DPG test space.");
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args.AddOption(&ref, "-ref", "--refinements",
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"Number of serial refinements.");
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args.AddOption(&static_cond, "-sc", "--static-condensation", "-no-sc",
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"--no-static-condensation", "Enable static condensation.");
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args.AddOption(&visport, "-p", "--send-port", "Socket for GLVis.");
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args.Parse();
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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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if (iprob > 1) { iprob = 0; }
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prob = (prob_type)iprob;
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omega = 2.*M_PI*rnum;
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Mesh mesh(mesh_file, 1, 1);
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dim = mesh.Dimension();
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MFEM_VERIFY(dim > 1, "Dimension = 1 is not supported in this example");
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// Define spaces
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enum TrialSpace
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{
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p_space = 0,
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u_space = 1,
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hatp_space = 2,
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hatu_space = 3
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};
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enum TestSpace
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{
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q_space = 0,
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v_space = 1
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};
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// L2 space for p
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FiniteElementCollection *p_fec = new L2_FECollection(order-1,dim);
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FiniteElementSpace *p_fes = new FiniteElementSpace(&mesh,p_fec);
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// Vector L2 space for u
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FiniteElementCollection *u_fec = new L2_FECollection(order-1,dim);
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FiniteElementSpace *u_fes = new FiniteElementSpace(&mesh,u_fec, dim);
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// H^1/2 space for p̂
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FiniteElementCollection * hatp_fec = new H1_Trace_FECollection(order,dim);
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FiniteElementSpace *hatp_fes = new FiniteElementSpace(&mesh,hatp_fec);
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// H^-1/2 space for û
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FiniteElementCollection * hatu_fec = new RT_Trace_FECollection(order-1,dim);
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FiniteElementSpace *hatu_fes = new FiniteElementSpace(&mesh,hatu_fec);
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// testspace fe collections
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int test_order = order+delta_order;
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FiniteElementCollection * q_fec = new H1_FECollection(test_order, dim);
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FiniteElementCollection * v_fec = new RT_FECollection(test_order-1, dim);
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// Coefficients
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ConstantCoefficient one(1.0);
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ConstantCoefficient zero(0.0);
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Vector vec0(dim); vec0 = 0.;
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VectorConstantCoefficient vzero(vec0);
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ConstantCoefficient negone(-1.0);
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ConstantCoefficient omeg(omega);
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ConstantCoefficient omeg2(omega*omega);
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ConstantCoefficient negomeg(-omega);
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Array<FiniteElementSpace * > trial_fes;
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Array<FiniteElementCollection * > test_fec;
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trial_fes.Append(p_fes);
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trial_fes.Append(u_fes);
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trial_fes.Append(hatp_fes);
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trial_fes.Append(hatu_fes);
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test_fec.Append(q_fec);
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test_fec.Append(v_fec);
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ComplexDPGWeakForm * a = new ComplexDPGWeakForm(trial_fes,test_fec);
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// i ω (p,q)
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a->AddTrialIntegrator(nullptr,new MixedScalarMassIntegrator(omeg),
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TrialSpace::p_space,TestSpace::q_space);
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// -(u , ∇ q)
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a->AddTrialIntegrator(new TransposeIntegrator(new GradientIntegrator(negone)),
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nullptr,TrialSpace::u_space,TestSpace::q_space);
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// -(p, ∇⋅v)
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a->AddTrialIntegrator(new MixedScalarWeakGradientIntegrator(one),nullptr,
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TrialSpace::p_space,TestSpace::v_space);
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// i ω (u,v)
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a->AddTrialIntegrator(nullptr,
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new TransposeIntegrator(new VectorFEMassIntegrator(omeg)),
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TrialSpace::u_space,TestSpace::v_space);
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// < p̂, v⋅n>
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a->AddTrialIntegrator(new NormalTraceIntegrator,nullptr,
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TrialSpace::hatp_space,TestSpace::v_space);
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// < û,q >
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a->AddTrialIntegrator(new TraceIntegrator,nullptr,
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TrialSpace::hatu_space,TestSpace::q_space);
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// test space integrators (Adjoint graph norm)
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// (∇q,∇δq)
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a->AddTestIntegrator(new DiffusionIntegrator(one),nullptr,
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TestSpace::q_space, TestSpace::q_space);
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// (q,δq)
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a->AddTestIntegrator(new MassIntegrator(one),nullptr,
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TestSpace::q_space, TestSpace::q_space);
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// (∇⋅v,∇⋅δv)
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a->AddTestIntegrator(new DivDivIntegrator(one),nullptr,
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TestSpace::v_space, TestSpace::v_space);
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// (v,δv)
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a->AddTestIntegrator(new VectorFEMassIntegrator(one),nullptr,
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TestSpace::v_space, TestSpace::v_space);
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// -i ω (∇q,δv)
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a->AddTestIntegrator(nullptr,new MixedVectorGradientIntegrator(negomeg),
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TestSpace::q_space, TestSpace::v_space);
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// i ω (v,∇ δq)
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a->AddTestIntegrator(nullptr,new MixedVectorWeakDivergenceIntegrator(negomeg),
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TestSpace::v_space, TestSpace::q_space);
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// ω^2 (v,δv)
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a->AddTestIntegrator(new VectorFEMassIntegrator(omeg2),nullptr,
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TestSpace::v_space, TestSpace::v_space);
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// - i ω (∇⋅v,δq)
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a->AddTestIntegrator(nullptr,new VectorFEDivergenceIntegrator(negomeg),
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TestSpace::v_space, TestSpace::q_space);
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// i ω (q,∇⋅v)
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a->AddTestIntegrator(nullptr,new MixedScalarWeakGradientIntegrator(negomeg),
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TestSpace::q_space, TestSpace::v_space);
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// ω^2 (q,δq)
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a->AddTestIntegrator(new MassIntegrator(omeg2),nullptr,
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TestSpace::q_space, TestSpace::q_space);
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// RHS
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FunctionCoefficient f_rhs_r(rhs_func_r);
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FunctionCoefficient f_rhs_i(rhs_func_i);
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if (prob == prob_type::gaussian_beam)
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{
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a->AddDomainLFIntegrator(new DomainLFIntegrator(f_rhs_r),
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new DomainLFIntegrator(f_rhs_i), TestSpace::q_space);
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}
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FunctionCoefficient hatpex_r(hatp_exact_r);
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FunctionCoefficient hatpex_i(hatp_exact_i);
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VectorFunctionCoefficient hatuex_r(dim,hatu_exact_r);
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VectorFunctionCoefficient hatuex_i(dim,hatu_exact_i);
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socketstream p_out_r;
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socketstream p_out_i;
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real_t err0 = 0.;
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int dof0 = 0; // init to suppress gcc warning
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std::cout << "\n Ref |"
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<< " Dofs |"
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<< " ω |"
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<< " L2 Error |"
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<< " Rate |"
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<< " PCG it |" << endl;
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std::cout << std::string(60,'-')
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<< endl;
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for (int it = 0; it<=ref; it++)
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{
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if (static_cond) { a->EnableStaticCondensation(); }
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a->Assemble();
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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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hatp_fes->GetEssentialTrueDofs(ess_bdr, ess_tdof_list);
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}
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// shift the ess_tdofs
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for (int j = 0; j < ess_tdof_list.Size(); j++)
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{
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ess_tdof_list[j] += p_fes->GetTrueVSize() + u_fes->GetTrueVSize();
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}
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Array<int> offsets(5);
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offsets[0] = 0;
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offsets[1] = p_fes->GetVSize();
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offsets[2] = u_fes->GetVSize();
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offsets[3] = hatp_fes->GetVSize();
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offsets[4] = hatu_fes->GetVSize();
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offsets.PartialSum();
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Vector x(2*offsets.Last());
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x = 0.;
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GridFunction hatp_gf_r(hatp_fes, x, offsets[2]);
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GridFunction hatp_gf_i(hatp_fes, x, offsets.Last()+ offsets[2]);
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hatp_gf_r.ProjectBdrCoefficient(hatpex_r, ess_bdr);
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hatp_gf_i.ProjectBdrCoefficient(hatpex_i, ess_bdr);
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OperatorPtr Ah;
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Vector X,B;
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a->FormLinearSystem(ess_tdof_list,x,Ah, X,B);
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// Setup operator and preconditioner
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ComplexOperator * Ahc = Ah.As<ComplexOperator>();
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BlockMatrix * A_r = dynamic_cast<BlockMatrix *>(&Ahc->real());
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BlockMatrix * A_i = dynamic_cast<BlockMatrix *>(&Ahc->imag());
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int num_blocks = A_r->NumRowBlocks();
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Array<int> tdof_offsets(2*num_blocks+1);
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tdof_offsets[0] = 0;
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int k = (static_cond) ? 2 : 0;
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for (int i=0; i<num_blocks; i++)
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{
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tdof_offsets[i+1] = trial_fes[i+k]->GetTrueVSize();
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tdof_offsets[num_blocks+i+1] = trial_fes[i+k]->GetTrueVSize();
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}
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tdof_offsets.PartialSum();
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BlockOperator A(tdof_offsets);
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for (int i = 0; i<num_blocks; i++)
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{
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for (int j = 0; j<num_blocks; j++)
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{
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A.SetBlock(i,j,&A_r->GetBlock(i,j));
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A.SetBlock(i,j+num_blocks,&A_i->GetBlock(i,j), -1.0);
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A.SetBlock(i+num_blocks,j+num_blocks,&A_r->GetBlock(i,j));
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A.SetBlock(i+num_blocks,j,&A_i->GetBlock(i,j));
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}
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}
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BlockDiagonalPreconditioner M(tdof_offsets);
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M.owns_blocks = 1;
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for (int i = 0; i<num_blocks; i++)
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{
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M.SetDiagonalBlock(i, new GSSmoother((SparseMatrix&)A_r->GetBlock(i,i)));
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M.SetDiagonalBlock(num_blocks+i, new GSSmoother((SparseMatrix&)A_r->GetBlock(i,
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i)));
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}
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CGSolver cg;
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cg.SetRelTol(1e-10);
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cg.SetMaxIter(2000);
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cg.SetPrintLevel(0);
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cg.SetPreconditioner(M);
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cg.SetOperator(A);
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cg.Mult(B, X);
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a->RecoverFEMSolution(X,x);
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GridFunction p_r(p_fes, x, 0);
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GridFunction p_i(p_fes, x, offsets.Last());
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GridFunction u_r(u_fes, x, offsets[1]);
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GridFunction u_i(u_fes, x, offsets.Last() + offsets[1]);
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FunctionCoefficient p_ex_r(p_exact_r);
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FunctionCoefficient p_ex_i(p_exact_i);
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VectorFunctionCoefficient u_ex_r(dim,u_exact_r);
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VectorFunctionCoefficient u_ex_i(dim,u_exact_i);
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int dofs = 0;
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for (int i = 0; i<trial_fes.Size(); i++)
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{
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dofs += trial_fes[i]->GetTrueVSize();
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}
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real_t p_err_r = p_r.ComputeL2Error(p_ex_r);
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real_t p_err_i = p_i.ComputeL2Error(p_ex_i);
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real_t u_err_r = u_r.ComputeL2Error(u_ex_r);
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real_t u_err_i = u_i.ComputeL2Error(u_ex_i);
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real_t L2Error = sqrt(p_err_r*p_err_r + p_err_i*p_err_i
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+u_err_r*u_err_r + u_err_i*u_err_i);
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real_t rate_err = (it) ? dim*log(err0/L2Error)/log((real_t)dof0/dofs) : 0.0;
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err0 = L2Error;
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dof0 = dofs;
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std::ios oldState(nullptr);
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oldState.copyfmt(std::cout);
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std::cout << std::right << std::setw(5) << it << " | "
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<< std::setw(10) << dof0 << " | "
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<< std::setprecision(1) << std::fixed
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<< std::setw(4) << 2*rnum << " π | "
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<< std::setprecision(3)
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<< std::setw(10) << std::scientific << err0 << " | "
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<< std::setprecision(2)
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<< std::setw(6) << std::fixed << rate_err << " | "
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<< std::setw(6) << std::fixed << cg.GetNumIterations() << " | "
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<< std::endl;
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std::cout.copyfmt(oldState);
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if (visualization)
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{
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const char * keys = (it == 0 && dim == 2) ? "jRcml\n" : nullptr;
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char vishost[] = "localhost";
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VisualizeField(p_out_r,vishost, visport, p_r,
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"Numerical presure (real part)", 0, 0, 500, 500, keys);
|
||
VisualizeField(p_out_i,vishost, visport, p_i,
|
||
"Numerical presure (imaginary part)", 501, 0, 500, 500, keys);
|
||
}
|
||
|
||
if (it == ref)
|
||
{
|
||
break;
|
||
}
|
||
|
||
mesh.UniformRefinement();
|
||
for (int i =0; i<trial_fes.Size(); i++)
|
||
{
|
||
trial_fes[i]->Update(false);
|
||
}
|
||
a->Update();
|
||
}
|
||
|
||
delete a;
|
||
delete q_fec;
|
||
delete v_fec;
|
||
delete hatp_fes;
|
||
delete hatp_fec;
|
||
delete hatu_fes;
|
||
delete hatu_fec;
|
||
delete u_fec;
|
||
delete p_fec;
|
||
delete u_fes;
|
||
delete p_fes;
|
||
|
||
return 0;
|
||
}
|
||
|
||
real_t p_exact_r(const Vector &x)
|
||
{
|
||
return acoustics_solution(x).real();
|
||
}
|
||
|
||
real_t p_exact_i(const Vector &x)
|
||
{
|
||
return acoustics_solution(x).imag();
|
||
}
|
||
|
||
real_t hatp_exact_r(const Vector & X)
|
||
{
|
||
return p_exact_r(X);
|
||
}
|
||
|
||
real_t hatp_exact_i(const Vector & X)
|
||
{
|
||
return p_exact_i(X);
|
||
}
|
||
|
||
void gradp_exact_r(const Vector &x, Vector &grad_r)
|
||
{
|
||
grad_r.SetSize(x.Size());
|
||
vector<complex<real_t>> grad;
|
||
acoustics_solution_grad(x,grad);
|
||
for (unsigned i = 0; i < grad.size(); i++)
|
||
{
|
||
grad_r[i] = grad[i].real();
|
||
}
|
||
}
|
||
|
||
void gradp_exact_i(const Vector &x, Vector &grad_i)
|
||
{
|
||
grad_i.SetSize(x.Size());
|
||
vector<complex<real_t>> grad;
|
||
acoustics_solution_grad(x,grad);
|
||
for (unsigned i = 0; i < grad.size(); i++)
|
||
{
|
||
grad_i[i] = grad[i].imag();
|
||
}
|
||
}
|
||
|
||
real_t d2_exact_r(const Vector &x)
|
||
{
|
||
|
||
return acoustics_solution_laplacian(x).real();
|
||
}
|
||
|
||
real_t d2_exact_i(const Vector &x)
|
||
{
|
||
return acoustics_solution_laplacian(x).imag();
|
||
}
|
||
|
||
// u = - ∇ p / (i ω )
|
||
// = i (∇ p_r + i * ∇ p_i) / ω
|
||
// = - ∇ p_i / ω + i ∇ p_r / ω
|
||
void u_exact_r(const Vector &x, Vector & u)
|
||
{
|
||
gradp_exact_i(x,u);
|
||
u *= -1./omega;
|
||
}
|
||
|
||
void u_exact_i(const Vector &x, Vector & u)
|
||
{
|
||
gradp_exact_r(x,u);
|
||
u *= 1./omega;
|
||
}
|
||
|
||
void hatu_exact_r(const Vector & X, Vector & hatu)
|
||
{
|
||
u_exact_r(X,hatu);
|
||
}
|
||
void hatu_exact_i(const Vector & X, Vector & hatu)
|
||
{
|
||
u_exact_i(X,hatu);
|
||
}
|
||
|
||
// ∇⋅u = i Δ p / ω
|
||
// = i (Δ p_r + i * Δ p_i) / ω
|
||
// = - Δ p_i / ω + i Δ p_r / ω
|
||
real_t divu_exact_r(const Vector &x)
|
||
{
|
||
return -d2_exact_i(x)/omega;
|
||
}
|
||
|
||
real_t divu_exact_i(const Vector &x)
|
||
{
|
||
return d2_exact_r(x)/omega;
|
||
}
|
||
|
||
// f = ∇⋅u + i ω p
|
||
// f_r = ∇⋅u_r - ω p_i
|
||
real_t rhs_func_r(const Vector &x)
|
||
{
|
||
real_t p = p_exact_i(x);
|
||
real_t divu = divu_exact_r(x);
|
||
return divu - omega * p;
|
||
}
|
||
|
||
// f_i = ∇⋅u_i + ω p_r
|
||
real_t rhs_func_i(const Vector &x)
|
||
{
|
||
real_t p = p_exact_r(x);
|
||
real_t divu = divu_exact_i(x);
|
||
return divu + omega * p;
|
||
}
|
||
|
||
complex<real_t> acoustics_solution(const Vector & X)
|
||
{
|
||
complex<real_t> zi = complex<real_t>(0., 1.);
|
||
switch (prob)
|
||
{
|
||
case plane_wave:
|
||
{
|
||
real_t beta = omega/std::sqrt((real_t)X.Size());
|
||
complex<real_t> alpha = beta * zi * X.Sum();
|
||
return exp(alpha);
|
||
}
|
||
break;
|
||
default:
|
||
{
|
||
real_t rk = omega;
|
||
real_t degrees = 45;
|
||
real_t alpha = (180+degrees) * M_PI/180.;
|
||
real_t sina = sin(alpha);
|
||
real_t cosa = cos(alpha);
|
||
// shift the origin
|
||
real_t xprim=X(0) + 0.1;
|
||
real_t yprim=X(1) + 0.1;
|
||
|
||
real_t x = xprim*sina - yprim*cosa;
|
||
real_t y = xprim*cosa + yprim*sina;
|
||
//wavelength
|
||
real_t rl = 2.*M_PI/rk;
|
||
// beam waist radius
|
||
real_t w0 = 0.05;
|
||
// function w
|
||
real_t fact = rl/M_PI/(w0*w0);
|
||
real_t aux = 1. + (fact*y)*(fact*y);
|
||
real_t w = w0*sqrt(aux);
|
||
real_t phi0 = atan(fact*y);
|
||
real_t r = y + 1./y/(fact*fact);
|
||
|
||
// pressure
|
||
complex<real_t> ze = - x*x/(w*w) - zi*rk*y - zi * real_t(M_PI) * x * x/rl/r +
|
||
zi*phi0/2_r;
|
||
|
||
real_t pf = pow(2.0/M_PI/(w*w),0.25);
|
||
return pf*exp(ze);
|
||
}
|
||
break;
|
||
}
|
||
}
|
||
|
||
void acoustics_solution_grad(const Vector & X, vector<complex<real_t>> & dp)
|
||
{
|
||
dp.resize(X.Size());
|
||
complex<real_t> zi = complex<real_t>(0., 1.);
|
||
switch (prob)
|
||
{
|
||
case plane_wave:
|
||
{
|
||
real_t beta = omega/std::sqrt((real_t)X.Size());
|
||
complex<real_t> alpha = beta * zi * X.Sum();
|
||
complex<real_t> p = exp(alpha);
|
||
for (int i = 0; i<X.Size(); i++)
|
||
{
|
||
dp[i] = zi * beta * p;
|
||
}
|
||
}
|
||
break;
|
||
default:
|
||
{
|
||
real_t rk = omega;
|
||
real_t degrees = 45;
|
||
real_t alpha = (180+degrees) * M_PI/180.;
|
||
real_t sina = sin(alpha);
|
||
real_t cosa = cos(alpha);
|
||
// shift the origin
|
||
real_t xprim=X(0) + 0.1;
|
||
real_t yprim=X(1) + 0.1;
|
||
|
||
real_t x = xprim*sina - yprim*cosa;
|
||
real_t y = xprim*cosa + yprim*sina;
|
||
real_t dxdxprim = sina, dxdyprim = -cosa;
|
||
real_t dydxprim = cosa, dydyprim = sina;
|
||
//wavelength
|
||
real_t rl = 2.*M_PI/rk;
|
||
|
||
// beam waist radius
|
||
real_t w0 = 0.05;
|
||
|
||
// function w
|
||
real_t fact = rl/M_PI/(w0*w0);
|
||
real_t aux = 1. + (fact*y)*(fact*y);
|
||
|
||
real_t w = w0*sqrt(aux);
|
||
real_t dwdy = w0*fact*fact*y/sqrt(aux);
|
||
|
||
real_t phi0 = atan(fact*y);
|
||
real_t dphi0dy = cos(phi0)*cos(phi0)*fact;
|
||
|
||
real_t r = y + 1./y/(fact*fact);
|
||
real_t drdy = 1. - 1./(y*y)/(fact*fact);
|
||
|
||
constexpr real_t r2 = 2.0;
|
||
const real_t rPI = M_PI;
|
||
|
||
// pressure
|
||
complex<real_t> ze = - x*x/(w*w) - zi*rk*y - zi * rPI * x * x/rl/r +
|
||
zi*phi0/r2;
|
||
|
||
complex<real_t> zdedx = -r2*x/(w*w) - r2*zi*rPI*x/rl/r;
|
||
complex<real_t> zdedy = r2*x*x/(w*w*w)*dwdy - zi*rk + zi*rPI*x*x/rl/
|
||
(r*r)*drdy + zi*dphi0dy/r2;
|
||
|
||
real_t pf = pow(2.0/M_PI/(w*w),0.25);
|
||
real_t dpfdy = -pow(2./M_PI/(w*w),-0.75)/M_PI/(w*w*w)*dwdy;
|
||
|
||
complex<real_t> zp = pf*exp(ze);
|
||
complex<real_t> zdpdx = zp*zdedx;
|
||
complex<real_t> zdpdy = dpfdy*exp(ze)+zp*zdedy;
|
||
|
||
dp[0] = (zdpdx*dxdxprim + zdpdy*dydxprim);
|
||
dp[1] = (zdpdx*dxdyprim + zdpdy*dydyprim);
|
||
if (dim == 3) { dp[2] = 0.0; }
|
||
}
|
||
break;
|
||
}
|
||
}
|
||
|
||
complex<real_t> acoustics_solution_laplacian(const Vector & X)
|
||
{
|
||
complex<real_t> zi = complex<real_t>(0., 1.);
|
||
switch (prob)
|
||
{
|
||
case plane_wave:
|
||
{
|
||
real_t beta = omega/std::sqrt((real_t)X.Size());
|
||
complex<real_t> alpha = beta * zi * X.Sum();
|
||
complex<real_t> p = exp(alpha);
|
||
return dim * beta * beta * p;
|
||
}
|
||
break;
|
||
default:
|
||
{
|
||
real_t rk = omega;
|
||
real_t degrees = 45;
|
||
real_t alpha = (180+degrees) * M_PI/180.;
|
||
real_t sina = sin(alpha);
|
||
real_t cosa = cos(alpha);
|
||
// shift the origin
|
||
real_t xprim=X(0) + 0.1;
|
||
real_t yprim=X(1) + 0.1;
|
||
|
||
real_t x = xprim*sina - yprim*cosa;
|
||
real_t y = xprim*cosa + yprim*sina;
|
||
real_t dxdxprim = sina, dxdyprim = -cosa;
|
||
real_t dydxprim = cosa, dydyprim = sina;
|
||
//wavelength
|
||
real_t rl = 2.*M_PI/rk;
|
||
|
||
// beam waist radius
|
||
real_t w0 = 0.05;
|
||
|
||
// function w
|
||
real_t fact = rl/M_PI/(w0*w0);
|
||
real_t aux = 1. + (fact*y)*(fact*y);
|
||
|
||
real_t w = w0*sqrt(aux);
|
||
real_t dwdy = w0*fact*fact*y/sqrt(aux);
|
||
real_t d2wdydy = w0*fact*fact*(1. - (fact*y)*(fact*y)/aux)/sqrt(aux);
|
||
|
||
real_t phi0 = atan(fact*y);
|
||
real_t dphi0dy = cos(phi0)*cos(phi0)*fact;
|
||
real_t d2phi0dydy = -2.*cos(phi0)*sin(phi0)*fact*dphi0dy;
|
||
|
||
real_t r = y + 1./y/(fact*fact);
|
||
real_t drdy = 1. - 1./(y*y)/(fact*fact);
|
||
real_t d2rdydy = 2./(y*y*y)/(fact*fact);
|
||
|
||
constexpr real_t r2 = 2.0;
|
||
const real_t rPI = M_PI;
|
||
|
||
// pressure
|
||
complex<real_t> ze = - x*x/(w*w) - zi*rk*y - zi * rPI * x * x/rl/r +
|
||
zi*phi0/r2;
|
||
|
||
complex<real_t> zdedx = -r2*x/(w*w) - r2*zi*rPI*x/rl/r;
|
||
complex<real_t> zdedy = r2*x*x/(w*w*w)*dwdy - zi*rk + zi*rPI*x*x/rl/
|
||
(r*r)*drdy + zi*dphi0dy/r2;
|
||
complex<real_t> zd2edxdx = -r2/(w*w) - r2*zi*rPI/rl/r;
|
||
complex<real_t> zd2edxdy = 4_r*x/(w*w*w)*dwdy + r2*zi*rPI*x/rl/(r*r)*drdy;
|
||
complex<real_t> zd2edydx = zd2edxdy;
|
||
complex<real_t> zd2edydy = -6_r*x*x/(w*w*w*w)*dwdy*dwdy + r2*x*x/
|
||
complex<real_t>(w*w*w)*d2wdydy - r2*zi*rPI*x*x/rl/(r*r*r)*drdy*drdy
|
||
+ zi*rPI*x*x/rl/(r*r)*d2rdydy + zi/r2*d2phi0dydy;
|
||
|
||
real_t pf = pow(2.0/M_PI/(w*w),0.25);
|
||
real_t dpfdy = -pow(2./M_PI/(w*w),-0.75)/M_PI/(w*w*w)*dwdy;
|
||
real_t d2pfdydy = -1./M_PI*pow(2./M_PI,-0.75)*(-1.5*pow(w,-2.5)
|
||
*dwdy*dwdy + pow(w,-1.5)*d2wdydy);
|
||
|
||
complex<real_t> zp = pf*exp(ze);
|
||
complex<real_t> zdpdx = zp*zdedx;
|
||
complex<real_t> zdpdy = dpfdy*exp(ze)+zp*zdedy;
|
||
complex<real_t> zd2pdxdx = zdpdx*zdedx + zp*zd2edxdx;
|
||
complex<real_t> zd2pdxdy = zdpdy*zdedx + zp*zd2edxdy;
|
||
complex<real_t> zd2pdydx = dpfdy*exp(ze)*zdedx + zdpdx*zdedy + zp*zd2edydx;
|
||
complex<real_t> zd2pdydy = d2pfdydy*exp(ze) + dpfdy*exp(
|
||
ze)*zdedy + zdpdy*zdedy + zp*zd2edydy;
|
||
|
||
|
||
return (zd2pdxdx*dxdxprim + zd2pdydx*dydxprim)*dxdxprim
|
||
+ (zd2pdxdy*dxdxprim + zd2pdydy*dydxprim)*dydxprim
|
||
+ (zd2pdxdx*dxdyprim + zd2pdydx*dydyprim)*dxdyprim
|
||
+ (zd2pdxdy*dxdyprim + zd2pdydy*dydyprim)*dydyprim;
|
||
}
|
||
break;
|
||
}
|
||
}
|