635 lines
20 KiB
C++
635 lines
20 KiB
C++
// MFEM Example 5
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//
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// Compile with: make ex5
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//
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// Sample runs: ex5 -m ../data/square-disc.mesh
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// ex5 -m ../data/star.mesh
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// ex5 -m ../data/star.mesh -pa
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// ex5 -m ../data/beam-tet.mesh
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// ex5 -m ../data/beam-hex.mesh
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// ex5 -m ../data/beam-hex.mesh -pa
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// ex5 -m ../data/escher.mesh
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// ex5 -m ../data/fichera.mesh
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//
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// Device sample runs:
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// ex5 -m ../data/star.mesh -pa -d cuda
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// ex5 -m ../data/star.mesh -pa -d raja-cuda
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// ex5 -m ../data/star.mesh -pa -d raja-omp
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// ex5 -m ../data/beam-hex.mesh -pa -d cuda
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//
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// Description: This example code solves a simple 2D/3D asymptotic heat diffusion
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// problem in the mixed formulation corresponding to the system
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//
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// k^-1.q + grad T = f
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// div q + a*T = -g
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//
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// with natural boundary condition q.n = 0, where n is the outer
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// normal. The tensor k represents the heat conductivity, where its
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// symmetric and antisymmetric parts can be adjusted. The scalar a
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// is then the heat capacity, which can be zero, changing the problem
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// to steady-state, indefinite, saddle-point. The r.h.s. is f = 0 and
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// g = -a * <initial temperature> for the definite problem and
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// g = -<initial temperature> for the indefinite one. As a reference,
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// we use the exact solution (q,T) for the asymptote a -> infinity.
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// We discretize with Raviart-Thomas finite elements (heat flux q)
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// and piecewise discontinuous polynomials (temperature T). Alternatively,
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// the piecewise discontinuous polynomials are used for both quantities.
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//
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// The example demonstrates the use of the DarcyForm class, as
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// well as hybridization of mixed systems and the collective saving
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// of several grid functions in VisIt (visit.llnl.gov) and ParaView
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// (paraview.org) formats.
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//
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// We recommend viewing examples 1-4 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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#include <algorithm>
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using namespace std;
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using namespace mfem;
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// Define the analytical solution and forcing terms / boundary conditions
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typedef std::function<real_t(const Vector &, real_t t)> TDFunc;
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typedef std::function<void(const Vector &, Vector &)> VecFunc;
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typedef std::function<void(const Vector &, DenseMatrix &)> MatFunc;
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TDFunc GetTFun(real_t t_0, real_t a, const MatFunc &kFun);
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VecFunc GetQFun(real_t t_0, real_t a, const MatFunc &kFun);
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MatFunc GetKFun(real_t k, real_t ks, real_t ka);
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int main(int argc, char *argv[])
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{
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StopWatch chrono;
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// 1. Parse command-line options.
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const char *mesh_file = "";
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int nx = 0;
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int ny = 0;
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int order = 1;
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bool dg = false;
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real_t ks = 1.;
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real_t ka = 0.;
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real_t a = 0.;
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real_t td = 0.5;
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bool hybridization = false;
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bool pa = false;
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const char *device_config = "cpu";
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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(&nx, "-nx", "--ncells-x",
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"Number of cells in x.");
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args.AddOption(&ny, "-ny", "--ncells-y",
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"Number of cells in y.");
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args.AddOption(&order, "-o", "--order",
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"Finite element order (polynomial degree).");
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args.AddOption(&dg, "-dg", "--discontinuous", "-no-dg",
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"--no-discontinuous", "Enable DG elements for fluxes.");
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args.AddOption(&ks, "-ks", "--kappa_sym",
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"Symmetric anisotropy of the heat conductivity tensor");
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args.AddOption(&ka, "-ka", "--kappa_anti",
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"Antisymmetric anisotropy of the heat conductivity tensor");
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args.AddOption(&a, "-a", "--heat_capacity",
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"Heat capacity coefficient (0=indefinite problem)");
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args.AddOption(&td, "-td", "--stab_diff",
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"Diffusion stabilization factor (1/2=default)");
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args.AddOption(&hybridization, "-hb", "--hybridization", "-no-hb",
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"--no-hybridization", "Enable hybridization.");
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args.AddOption(&pa, "-pa", "--partial-assembly", "-no-pa",
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"--no-partial-assembly", "Enable Partial Assembly.");
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args.AddOption(&device_config, "-d", "--device",
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"Device configuration string, see Device::Configure().");
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args.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 (!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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// 2. Enable hardware devices such as GPUs, and programming models such as
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// CUDA, OCCA, RAJA and OpenMP based on command line options.
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Device device(device_config);
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device.Print();
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// 3. Read the mesh from the given mesh file. We can handle triangular,
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// quadrilateral, tetrahedral, hexahedral, surface and volume meshes with
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// the same code.
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if (ny <= 0)
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{
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ny = nx;
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}
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Mesh *mesh = NULL;
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if (strlen(mesh_file) > 0)
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{
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mesh = new Mesh(mesh_file, 1, 1);
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}
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else
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{
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mesh = new Mesh(Mesh::MakeCartesian2D(nx, ny, Element::QUADRILATERAL));
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}
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int dim = mesh->Dimension();
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// 4. Refine the mesh to increase the resolution. In this example we do
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// 'ref_levels' of uniform refinement. We choose 'ref_levels' to be the
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// largest number that gives a final mesh with no more than 10,000
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// elements.
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if (strlen(mesh_file) > 0)
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{
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int ref_levels =
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(int)floor(log(10000./mesh->GetNE())/log(2.)/dim);
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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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}
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// 5. Define a finite element space on the mesh. Here we use the
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// Raviart-Thomas finite elements of the specified order.
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FiniteElementCollection *V_coll;
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if (dg)
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{
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// In the case of LDG formulation, we chose a closed basis as it
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// is customary for HDG to match trace DOFs, but an open basis can
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// be used instead.
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V_coll = new L2_FECollection(order, dim, BasisType::GaussLobatto);
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}
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else
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{
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V_coll = new RT_FECollection(order, dim);
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}
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FiniteElementCollection *W_coll = new L2_FECollection(order, dim);
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FiniteElementSpace *V_space = new FiniteElementSpace(mesh, V_coll,
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(dg)?(dim):(1));
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FiniteElementSpace *W_space = new FiniteElementSpace(mesh, W_coll);
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DarcyForm *darcy = new DarcyForm(V_space, W_space);
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// 6. Define the BlockStructure of the problem, i.e. define the array of
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// offsets for each variable. The last component of the Array is the sum
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// of the dimensions of each block.
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const Array<int> &block_offsets = darcy->GetOffsets();
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std::cout << "***********************************************************\n";
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std::cout << "dim(R) = " << block_offsets[1] - block_offsets[0] << "\n";
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std::cout << "dim(W) = " << block_offsets[2] - block_offsets[1] << "\n";
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std::cout << "dim(R+W) = " << block_offsets.Last() << "\n";
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std::cout << "***********************************************************\n";
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// 7. Define the coefficients, analytical solution, and rhs of the PDE.
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const real_t t_0 = 1.; //base temperature
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const real_t k = 1.; //base heat conductivity
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ConstantCoefficient acoeff(a); //heat capacity
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auto kFun = GetKFun(k, ks, ka);
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MatrixFunctionCoefficient kcoeff(dim, kFun); //tensor conductivity
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InverseMatrixCoefficient ikcoeff(kcoeff); //inverse tensor conductivity
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auto tFun = GetTFun(t_0, a, kFun);
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FunctionCoefficient tcoeff(tFun); //temperature
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SumCoefficient gcoeff(0, tcoeff, 1., //boundary heat flux rhs
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-((a>0.)?(a):(1.)));//<-- due to symmetrization, the sign is opposite
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auto qFun = GetQFun(t_0, a, kFun);
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VectorFunctionCoefficient qcoeff(dim, qFun); //heat flux
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// 8. Allocate memory (x, rhs) for the analytical solution and the right hand
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// side. Define the GridFunction q,t for the finite element solution and
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// linear forms fform and gform for the right hand side. The data
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// allocated by x and rhs are passed as a reference to the grid functions
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// (q,t) and the linear forms (fform, gform).
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MemoryType mt = device.GetMemoryType();
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BlockVector x(block_offsets, mt), rhs(block_offsets, mt);
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LinearForm *fform(new LinearForm);
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fform->Update(V_space, rhs.GetBlock(0), 0);
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fform->Assemble();
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fform->SyncAliasMemory(rhs);
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LinearForm *gform(new LinearForm);
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gform->Update(W_space, rhs.GetBlock(1), 0);
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gform->AddDomainIntegrator(new DomainLFIntegrator(gcoeff));
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/*Vector v ({-1., 0.});
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VectorConstantCoefficient vc(v);
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ConstantCoefficient one;
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gform->AddBdrFaceIntegrator(new BoundaryFlowIntegrator(one, vc, 1.));*/
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gform->Assemble();
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gform->SyncAliasMemory(rhs);
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// 9. Assemble the finite element matrices for the Darcy operator
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//
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// D = [ M B^T ]
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// [ B 0 ]
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// where:
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//
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// M = \int_\Omega k u_h \cdot v_h d\Omega q_h, v_h \in V_h
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// B = -\int_\Omega \div u_h q_h d\Omega q_h \in V_h, w_h \in W_h
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BilinearForm *Mq = darcy->GetFluxMassForm();
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MixedBilinearForm *B = darcy->GetFluxDivForm();
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BilinearForm *Mt = (a > 0. || dg)?(darcy->GetPotentialMassForm()):(NULL);
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if (dg)
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{
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Mq->AddDomainIntegrator(new VectorMassIntegrator(ikcoeff));
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B->AddDomainIntegrator(new VectorDivergenceIntegrator());
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B->AddInteriorFaceIntegrator(new TransposeIntegrator(
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new DGNormalTraceIntegrator(-1.)));
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if (td > 0.)
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{
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Mt->AddInteriorFaceIntegrator(new HDGDiffusionIntegrator(kcoeff, td));
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}
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}
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else
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{
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Mq->AddDomainIntegrator(new VectorFEMassIntegrator(ikcoeff));
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B->AddDomainIntegrator(new VectorFEDivergenceIntegrator());
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}
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if (Mt)
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{
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Mt->AddDomainIntegrator(new MassIntegrator(acoeff));
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}
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//set hybridization / assembly level
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Array<int> ess_flux_tdofs_list;
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/*Array<int> bdr_is_ess(mesh->bdr_attributes.Max());
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bdr_is_ess = 0;
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bdr_is_ess[3] = -1;
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V_space->GetEssentialTrueDofs(bdr_is_ess, ess_flux_tdofs_list);*/
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FiniteElementCollection *trace_coll = NULL;
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FiniteElementSpace *trace_space = NULL;
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chrono.Clear();
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chrono.Start();
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if (hybridization)
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{
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trace_coll = new DG_Interface_FECollection(order, dim);
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trace_space = new FiniteElementSpace(mesh, trace_coll);
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darcy->EnableHybridization(trace_space,
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new NormalTraceJumpIntegrator(),
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ess_flux_tdofs_list);
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}
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if (pa) { darcy->SetAssemblyLevel(AssemblyLevel::PARTIAL); }
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darcy->Assemble();
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OperatorHandle pDarcyOp;
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Vector X, RHS;
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//x = 1./ny;
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darcy->FormLinearSystem(ess_flux_tdofs_list, x, rhs,
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pDarcyOp, X, RHS);
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chrono.Stop();
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std::cout << "Assembly took " << chrono.RealTime() << "s.\n";
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int maxIter(1000);
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real_t rtol(1.e-6);
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real_t atol(1.e-10);
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if (hybridization)
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{
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// 10. Construct the preconditioner
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GSSmoother prec(*pDarcyOp.As<SparseMatrix>());
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// 11. Solve the linear system with GMRES.
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// Check the norm of the unpreconditioned residual.
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chrono.Clear();
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chrono.Start();
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GMRESSolver solver;
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solver.SetAbsTol(atol);
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solver.SetRelTol(rtol);
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solver.SetMaxIter(maxIter);
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solver.SetOperator(*pDarcyOp);
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solver.SetPreconditioner(prec);
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solver.SetPrintLevel(1);
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solver.Mult(RHS, X);
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darcy->RecoverFEMSolution(X, rhs, x);
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chrono.Stop();
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if (solver.GetConverged())
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{
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std::cout << "GMRES converged in " << solver.GetNumIterations()
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<< " iterations with a residual norm of "
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<< solver.GetFinalNorm() << ".\n";
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}
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else
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{
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std::cout << "GMRES did not converge in " << solver.GetNumIterations()
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<< " iterations. Residual norm is " << solver.GetFinalNorm()
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<< ".\n";
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}
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std::cout << "GMRES solver took " << chrono.RealTime() << "s.\n";
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}
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else
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{
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// 10. Construct the operators for preconditioner
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//
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// P = [ diag(M) 0 ]
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// [ 0 B diag(M)^-1 B^T ]
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//
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// Here we use Symmetric Gauss-Seidel to approximate the inverse of the
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// temperature Schur Complement
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SparseMatrix *MinvBt = NULL;
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Vector Md(Mq->Height());
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BlockDiagonalPreconditioner darcyPrec(block_offsets);
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Solver *invM, *invS;
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SparseMatrix *S = NULL;
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if (pa)
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{
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Mq->AssembleDiagonal(Md);
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auto Md_host = Md.HostRead();
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Vector invMd(Mq->Height());
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for (int i=0; i<Mq->Height(); ++i)
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{
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invMd(i) = 1.0 / Md_host[i];
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}
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Vector BMBt_diag(B->Height());
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B->AssembleDiagonal_ADAt(invMd, BMBt_diag);
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Array<int> ess_tdof_list; // empty
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invM = new OperatorJacobiSmoother(Md, ess_tdof_list);
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invS = new OperatorJacobiSmoother(BMBt_diag, ess_tdof_list);
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}
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else
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{
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SparseMatrix &Mqm(Mq->SpMat());
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Mqm.GetDiag(Md);
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Md.HostReadWrite();
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SparseMatrix &Bm(B->SpMat());
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MinvBt = Transpose(Bm);
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for (int i = 0; i < Md.Size(); i++)
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{
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MinvBt->ScaleRow(i, 1./Md(i));
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}
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S = Mult(Bm, *MinvBt);
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if (Mt)
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{
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SparseMatrix &Mtm(Mt->SpMat());
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SparseMatrix *Snew = Add(Mtm, *S);
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delete S;
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S = Snew;
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}
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invM = new DSmoother(Mqm);
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#ifndef MFEM_USE_SUITESPARSE
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invS = new GSSmoother(*S);
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#else
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invS = new UMFPackSolver(*S);
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#endif
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}
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invM->iterative_mode = false;
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invS->iterative_mode = false;
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darcyPrec.SetDiagonalBlock(0, invM);
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darcyPrec.SetDiagonalBlock(1, invS);
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// 11. Solve the linear system with MINRES.
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// Check the norm of the unpreconditioned residual.
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chrono.Clear();
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chrono.Start();
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MINRESSolver solver;
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solver.SetAbsTol(atol);
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solver.SetRelTol(rtol);
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solver.SetMaxIter(maxIter);
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solver.SetOperator(*pDarcyOp);
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solver.SetPreconditioner(darcyPrec);
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solver.SetPrintLevel(1);
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solver.Mult(RHS, X);
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darcy->RecoverFEMSolution(X, rhs, x);
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if (device.IsEnabled()) { x.HostRead(); }
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chrono.Stop();
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if (solver.GetConverged())
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{
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std::cout << "MINRES converged in " << solver.GetNumIterations()
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<< " iterations with a residual norm of "
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<< solver.GetFinalNorm() << ".\n";
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}
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else
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{
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std::cout << "MINRES did not converge in " << solver.GetNumIterations()
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<< " iterations. Residual norm is " << solver.GetFinalNorm()
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<< ".\n";
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}
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std::cout << "MINRES solver took " << chrono.RealTime() << "s.\n";
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delete invM;
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delete invS;
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delete S;
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//delete Bt;
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delete MinvBt;
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}
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// 12. Create the grid functions q and t. Compute the L2 error norms.
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GridFunction q, t;
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q.MakeRef(V_space, x.GetBlock(0), 0);
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t.MakeRef(W_space, x.GetBlock(1), 0);
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int order_quad = max(2, 2*order+1);
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const IntegrationRule *irs[Geometry::NumGeom];
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for (int i=0; i < Geometry::NumGeom; ++i)
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{
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irs[i] = &(IntRules.Get(i, order_quad));
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}
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qcoeff.SetTime(1.);
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tcoeff.SetTime(1.);
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real_t err_q = q.ComputeL2Error(qcoeff, irs);
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real_t norm_q = ComputeLpNorm(2., qcoeff, *mesh, irs);
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real_t err_t = t.ComputeL2Error(tcoeff, irs);
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real_t norm_t = ComputeLpNorm(2., tcoeff, *mesh, irs);
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std::cout << "|| q_h - q_ex || / || q_ex || = " << err_q / norm_q << "\n";
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std::cout << "|| t_h - t_ex || / || t_ex || = " << err_t / norm_t << "\n";
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// 13. Save the mesh and the solution. This output can be viewed later using
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// GLVis: "glvis -m ex5.mesh -g sol_q.gf" or "glvis -m ex5.mesh -g
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// sol_t.gf".
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{
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ofstream mesh_ofs("ex5.mesh");
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mesh_ofs.precision(8);
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|
mesh->Print(mesh_ofs);
|
|
|
|
ofstream q_ofs("sol_q.gf");
|
|
q_ofs.precision(8);
|
|
q.Save(q_ofs);
|
|
|
|
ofstream t_ofs("sol_t.gf");
|
|
t_ofs.precision(8);
|
|
t.Save(t_ofs);
|
|
}
|
|
|
|
// 14. Save data in the VisIt format
|
|
VisItDataCollection visit_dc("Example5", mesh);
|
|
visit_dc.RegisterField("heat flux", &q);
|
|
visit_dc.RegisterField("temperature", &t);
|
|
visit_dc.Save();
|
|
|
|
// 15. Save data in the ParaView format
|
|
ParaViewDataCollection paraview_dc("Example5", mesh);
|
|
paraview_dc.SetPrefixPath("ParaView");
|
|
paraview_dc.SetLevelsOfDetail(order);
|
|
paraview_dc.SetCycle(0);
|
|
paraview_dc.SetDataFormat(VTKFormat::BINARY);
|
|
paraview_dc.SetHighOrderOutput(true);
|
|
paraview_dc.SetTime(0.0); // set the time
|
|
paraview_dc.RegisterField("heat flux",&q);
|
|
paraview_dc.RegisterField("temperature",&t);
|
|
paraview_dc.Save();
|
|
|
|
// 16. Send the solution by socket to a GLVis server.
|
|
if (visualization)
|
|
{
|
|
char vishost[] = "localhost";
|
|
int visport = 19916;
|
|
socketstream q_sock(vishost, visport);
|
|
q_sock.precision(8);
|
|
q_sock << "solution\n" << *mesh << q << "window_title 'Heat flux'" << endl;
|
|
q_sock << "keys Rljvvvvvmmc" << endl;
|
|
socketstream t_sock(vishost, visport);
|
|
t_sock.precision(8);
|
|
t_sock << "solution\n" << *mesh << t << "window_title 'Temperature'" << endl;
|
|
t_sock << "keys Rljmmc" << endl;
|
|
}
|
|
|
|
// 17. Free the used memory.
|
|
delete fform;
|
|
delete gform;
|
|
//delete Mq;
|
|
//delete B;
|
|
delete darcy;
|
|
delete W_space;
|
|
delete V_space;
|
|
delete trace_space;
|
|
delete W_coll;
|
|
delete V_coll;
|
|
delete trace_coll;
|
|
delete mesh;
|
|
|
|
return 0;
|
|
}
|
|
|
|
TDFunc GetTFun(real_t t_0, real_t a, const MatFunc &kFun)
|
|
{
|
|
return [=](const Vector &x, real_t t) -> real_t
|
|
{
|
|
const int ndim = x.Size();
|
|
real_t t0 = t_0 * sin(M_PI*x(0)) * sin(M_PI*x(1));
|
|
if (ndim > 2)
|
|
{
|
|
t0 *= sin(M_PI*x(2));
|
|
}
|
|
|
|
if (a <= 0.) { return t0; }
|
|
|
|
Vector ddT((ndim<=2)?(2):(4));
|
|
ddT(0) = -t_0 * M_PI*M_PI * sin(M_PI*x(0)) * sin(M_PI*x(1));//xx,yy
|
|
ddT(1) = +t_0 * M_PI*M_PI * cos(M_PI*x(0)) * cos(M_PI*x(1));//xy
|
|
if (ndim > 2)
|
|
{
|
|
ddT(0) *= sin(M_PI*x(2));//xx,yy,zz
|
|
ddT(1) *= sin(M_PI*x(2));//xy
|
|
ddT(2) = +t_0 * M_PI*M_PI * cos(M_PI*x(0)) * sin(M_PI*x(1)) * cos(M_PI*x(
|
|
2));//xz
|
|
ddT(3) = +t_0 * M_PI*M_PI * sin(M_PI*x(0)) * cos(M_PI*x(1)) * cos(M_PI*x(
|
|
2));//yz
|
|
|
|
}
|
|
|
|
DenseMatrix kappa;
|
|
kFun(x, kappa);
|
|
|
|
real_t div = -(kappa(0,0) + kappa(1,1)) * ddT(0) - (kappa(0,1) + kappa(1,0)) * ddT(1);
|
|
if (ndim > 2)
|
|
{
|
|
div += -kappa(2,2) * ddT(0) - (kappa(0,2) + kappa(2,0)) * ddT(2) - (kappa(1,
|
|
2) + kappa(2,1)) * ddT(3);
|
|
}
|
|
return t0 - div / a * t;
|
|
};
|
|
}
|
|
|
|
VecFunc GetQFun(real_t t_0, real_t a, const MatFunc &kFun)
|
|
{
|
|
return [=](const Vector &x, Vector &v)
|
|
{
|
|
const int vdim = x.Size();
|
|
v.SetSize(vdim);
|
|
|
|
Vector gT(vdim);
|
|
gT = 0.;
|
|
gT(0) = t_0 * M_PI * cos(M_PI*x(0)) * sin(M_PI*x(1));
|
|
gT(1) = t_0 * M_PI * sin(M_PI*x(0)) * cos(M_PI*x(1));
|
|
if (vdim > 2)
|
|
{
|
|
gT(0) *= sin(M_PI*x(2));
|
|
gT(1) *= sin(M_PI*x(2));
|
|
gT(2) = t_0 * M_PI * sin(M_PI*x(0)) * sin(M_PI*x(1)) * cos(M_PI*x(2));
|
|
}
|
|
|
|
DenseMatrix kappa;
|
|
kFun(x, kappa);
|
|
|
|
if (vdim <= 2)
|
|
{
|
|
v(0) = -kappa(0,0) * gT(0) -kappa(0,1) * gT(1);
|
|
v(1) = -kappa(1,0) * gT(0) -kappa(1,1) * gT(1);
|
|
}
|
|
else
|
|
{
|
|
kappa.Mult(gT, v);
|
|
v.Neg();
|
|
}
|
|
};
|
|
}
|
|
|
|
MatFunc GetKFun(real_t k, real_t ks, real_t ka)
|
|
{
|
|
return [=](const Vector &x, DenseMatrix &kappa)
|
|
{
|
|
const int ndim = x.Size();
|
|
kappa.Diag(k, ndim);
|
|
kappa(0,0) *= ks;
|
|
kappa(0,1) = +ka * k;
|
|
kappa(1,0) = -ka * k;
|
|
if (ndim > 2)
|
|
{
|
|
kappa(0,2) = +ka * k;
|
|
kappa(2,0) = -ka * k;
|
|
}
|
|
};
|
|
}
|
|
|
|
|