1215 lines
39 KiB
C++
1215 lines
39 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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// 1/k*q + grad T = g
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// div q + div(T*c) + dT/dt = -f
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//
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// with natural boundary condition -T = <given temperature> and/or
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// essential (RT) / natural (DG) boundary condition qT.n = (q + T*c).n
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// = <given total flux>. The scalar k is the heat conductivity and c the
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// given velocity field. Multiple problems are offered based on the paper:
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// N.C. Nguyen et al., Journal of Computational Physics 228 (2009) 3232–3254.
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// In particular, they are (corresponding to the subsections of section 5):
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// 1) steady-state diffusion - with zero Dirichlet temperature BCs
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// 2) steady-state advection-diffusion - with zero Dirichlet temperature BCs
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// 3) steady-state advection - with Dirichlet temperature inflow BC and
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// Neumann total flux outflow BC
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// 4) non-steady advection(-diffusion) - with Dirichlet temperature BCs
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// 5) Kovasznay flow - with Dirichlet temperature inflow BC and Neumann
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// total flux outflow BCs
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// 6) steady-state Burgers flow - with zero Dirichlet temperature BCs
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// 7) non-steady Burgers flow - with zero Dirichlet temperature BCs
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// Here, we use a given exact solution (q,T) and compute the
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// corresponding r.h.s. (f,g). We discretize with Raviart-Thomas
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// finite elements (heat flux q) and piecewise discontinuous
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// polynomials (temperature T).
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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 "darcyform.hpp"
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#include "darcyop.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)> TFunc;
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typedef std::function<void(const Vector &, Vector &)> VecFunc;
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typedef std::function<void(const Vector &, real_t, Vector &)> VecTFunc;
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typedef std::function<void(const Vector &, DenseMatrix &)> MatFunc;
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enum Problem
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{
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SteadyDiffusion = 1,
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MFEMLogo,
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};
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constexpr real_t epsilon = numeric_limits<real_t>::epsilon();
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MatFunc GetKFun(Problem prob, real_t k, real_t ks, real_t ka);
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TFunc GetTFun(Problem prob, real_t t_0, real_t a, const MatFunc &kFun,
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real_t c);
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VecTFunc GetQFun(Problem prob, real_t t_0, real_t a, const MatFunc &kFun,
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real_t c);
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VecFunc GetCFun(Problem prob, real_t c);
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TFunc GetFFun(Problem prob, real_t t_0, real_t a, const MatFunc &kFun,
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real_t c);
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FluxFunction* GetFluxFun(Problem prob, VectorCoefficient &ccoeff);
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MixedFluxFunction* GetHeatFluxFun(Problem prob, real_t k, int dim);
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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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real_t sx = 1.;
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real_t sy = 1.;
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int order = 1;
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bool dg = false;
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bool upwinded = false;
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int iproblem = Problem::SteadyDiffusion;
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real_t tf = 1.;
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int nt = 0;
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int ode = 1;
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real_t k = 1.;
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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 c = 1.;
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real_t td = 0.5;
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bool bc_neumann = false;
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bool reduction = false;
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bool hybridization = false;
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bool nonlinear = false;
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bool nonlinear_conv = false;
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bool nonlinear_diff = false;
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int hdg_scheme = 1;
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int solver_type = (int)DarcyOperator::SolverType::LBFGS;
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bool pa = false;
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const char *device_config = "cpu";
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bool mfem = false;
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bool visit = false;
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bool paraview = false;
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bool visualization = true;
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bool analytic = false;
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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(&sx, "-sx", "--size-x",
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"Size along x axis.");
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args.AddOption(&sy, "-sy", "--size-y",
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"Size along y axis.");
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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(&upwinded, "-up", "--upwinded", "-ce", "--centered",
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"Switches between upwinded (1) and centered (0=default) stabilization.");
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args.AddOption(&iproblem, "-p", "--problem",
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"Problem to solve:\n\t\t"
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"1=steady diff\n\t\t"
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"2=steady adv-diff\n\t\t"
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"3=steady adv\n\t\t"
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"4=nonsteady adv-diff\n\t\t"
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"5=Kovasznay flow\n\t\t"
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"6=steady Burgers\n\t\t"
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"7=nonsteady Burgers\n\t\t");
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args.AddOption(&tf, "-tf", "--time-final",
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"Final time.");
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args.AddOption(&nt, "-nt", "--ntimesteps",
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"Number of time steps.");
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args.AddOption(&ode, "-ode", "--ode-solver",
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"ODE time solver (1=Bacward Euler, 2=RK23L, 3=RK23A, 4=RK34).");
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args.AddOption(&k, "-k", "--kappa",
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"Heat conductivity");
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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(&c, "-c", "--velocity",
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"Convection velocity");
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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(&bc_neumann, "-bcn", "--bc-neumann", "-no-bcn",
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"--no-bc-neumann", "Enable Neumann outflow boundary condition.");
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args.AddOption(&reduction, "-rd", "--reduction", "-no-rd",
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"--no-reduction", "Enable reduction.");
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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(&nonlinear, "-nl", "--nonlinear", "-no-nl",
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"--no-nonlinear", "Enable non-linear regime.");
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args.AddOption(&nonlinear_conv, "-nlc", "--nonlinear-convection", "-no-nlc",
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"--no-nonlinear-convection", "Enable non-linear convection regime.");
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args.AddOption(&nonlinear_diff, "-nld", "--nonlinear-diffusion", "-no-nld",
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"--no-nonlinear-diffusion", "Enable non-linear diffusion regime.");
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args.AddOption(&hdg_scheme, "-hdg", "--hdg_scheme",
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"HDG scheme (1=HDG-I, 2=HDG-II, 3=Rusanov, 4=Godunov).");
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args.AddOption(&solver_type, "-nls", "--nonlinear-solver",
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"Nonlinear solver type (1=LBFGS, 2=LBB, 3=Newton).");
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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(&mfem, "-mfem", "--mfem", "-no-mfem",
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"--no-mfem",
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"Enable or disable MFEM output.");
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args.AddOption(&visit, "-visit", "--visit", "-no-visit",
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"--no-visit",
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"Enable or disable Visit output.");
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args.AddOption(¶view, "-paraview", "--paraview", "-no-paraview",
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"--no-paraview",
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"Enable or disable ParaView output.");
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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(&analytic, "-anal", "--analytic", "-no-anal",
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"--no-analytic",
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"Enable or disable analytic solution.");
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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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// Set the problem options
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Problem problem = (Problem)iproblem;
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bool bconv = false, bnlconv = false, bnldiff = nonlinear_diff, btime = false;
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switch (problem)
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{
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case Problem::SteadyDiffusion:
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break;
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case Problem::MFEMLogo:
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break;
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default:
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cerr << "Unknown problem" << endl;
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return 1;
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}
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if (bnldiff && reduction)
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{
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cerr << "Reduction is not possible with non-linear diffusion" << endl;
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return 1;
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}
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if (!bconv && !bnlconv && upwinded)
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{
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cerr << "Upwinded scheme cannot work without advection" << endl;
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return 1;
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}
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if (bnlconv && !nonlinear)
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{
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cerr << "Nonlinear convection can only work in the nonlinear regime" << endl;
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return 1;
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}
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if (nonlinear && !hybridization)
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{
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cerr << "Warning: A linear solver is used" << endl;
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}
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if (btime && nt <= 0)
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{
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cerr << "You must specify the number of time steps for time evolving problems"
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<< endl;
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return 1;
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}
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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, false,
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sx, sy));
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}
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int dim = mesh->Dimension();
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// Mark boundary conditions
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Array<int> bdr_is_dirichlet(mesh->bdr_attributes.Max());
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Array<int> bdr_is_neumann(mesh->bdr_attributes.Max());
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bdr_is_dirichlet = 0;
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bdr_is_neumann = 0;
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switch (problem)
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{
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case Problem::SteadyDiffusion:
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case Problem::MFEMLogo:
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//free (zero Dirichlet)
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if (bc_neumann)
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{
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bdr_is_neumann[1] = -1;//outflow
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bdr_is_neumann[2] = -1;//outflow
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}
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break;
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}
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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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BasisType::GaussLobatto);
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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 coefficients, analytical solution, and rhs of the PDE.
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const real_t t_0 = 1.; //base temperature
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ConstantCoefficient acoeff(a);
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auto kFun = GetKFun(problem, k, ks, ka);
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MatrixFunctionCoefficient kcoeff(dim, kFun);
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InverseMatrixCoefficient ikcoeff(kcoeff);
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auto cFun = GetCFun(problem, c);
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VectorFunctionCoefficient ccoeff(dim, cFun);
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auto tFun = GetTFun(problem, t_0, a, kFun, c);
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FunctionCoefficient tcoeff(tFun);
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SumCoefficient gcoeff(0., tcoeff, 1., -1.);
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auto fFun = GetFFun(problem, t_0, a, kFun, c);
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FunctionCoefficient fcoeff(fFun);
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auto qFun = GetQFun(problem, t_0, a, kFun, c);
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VectorFunctionCoefficient qcoeff(dim, qFun);
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ConstantCoefficient one;
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VectorSumCoefficient qtcoeff_(ccoeff, qcoeff, tcoeff, one);//total flux
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VectorCoefficient &qtcoeff = (bconv)?((VectorCoefficient&)qtcoeff_)
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:((VectorCoefficient&)qcoeff);//<--velocity is undefined
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// 7. 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 =(!nonlinear && !bnldiff)?(darcy->GetFluxMassForm()):(NULL);
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NonlinearForm *Mqnl = (nonlinear && !bnldiff)?
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(darcy->GetFluxMassNonlinearForm()):(NULL);
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BlockNonlinearForm *Mnl = (bnldiff)?(darcy->GetBlockNonlinearForm()):(NULL);
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MixedBilinearForm *B = darcy->GetFluxDivForm();
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BilinearForm *Mt = (!nonlinear && ((dg && td > 0.) || bconv || btime ||
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a > 0.))?
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(darcy->GetPotentialMassForm()):(NULL);
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NonlinearForm *Mtnl = (nonlinear && ((dg && td > 0.) || bconv || bnlconv ||
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a > 0. || btime))?
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(darcy->GetPotentialMassNonlinearForm()):(NULL);
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FluxFunction *FluxFun = NULL;
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RiemannSolver *FluxSolver = NULL;
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MixedFluxFunction *HeatFluxFun = NULL;
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//diffusion
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if (!bnldiff)
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{
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//linear diffusion
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if (dg)
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{
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if (Mq)
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{
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Mq->AddDomainIntegrator(new VectorMassIntegrator(ikcoeff));
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}
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if (Mqnl)
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{
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Mqnl->AddDomainIntegrator(new VectorMassIntegrator(ikcoeff));
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}
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}
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else
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{
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if (Mq)
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{
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Mq->AddDomainIntegrator(new VectorFEMassIntegrator(ikcoeff));
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}
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if (Mqnl)
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{
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Mqnl->AddDomainIntegrator(new VectorFEMassIntegrator(ikcoeff));
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}
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}
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}
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else
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{
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//nonlinear diffusion
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HeatFluxFun = GetHeatFluxFun(problem, k, dim);
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if (dg)
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{
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Mnl->AddDomainIntegrator(new MixedConductionNLFIntegrator(*HeatFluxFun));
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}
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else
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{
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Mnl->AddDomainIntegrator(new MixedConductionNLFIntegrator(*HeatFluxFun));
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}
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}
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//diffusion stabilization
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if (dg)
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{
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if (bnldiff)
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{
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cerr << "Warning: Using linear stabilization for non-linear diffusion" << endl;
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}
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if (upwinded && td > 0. && hybridization)
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{
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if (Mt)
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{
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Mt->AddInteriorFaceIntegrator(new HDGDiffusionIntegrator(ccoeff, kcoeff, td));
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Mt->AddBdrFaceIntegrator(new HDGDiffusionIntegrator(ccoeff, kcoeff, td),
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bdr_is_neumann);
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}
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if (Mtnl)
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{
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Mtnl->AddInteriorFaceIntegrator(new HDGDiffusionIntegrator(ccoeff, kcoeff, td));
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Mtnl->AddBdrFaceIntegrator(new HDGDiffusionIntegrator(ccoeff, kcoeff, td),
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bdr_is_neumann);
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}
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}
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else if (!upwinded && td > 0.)
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{
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if (Mt)
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{
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Mt->AddInteriorFaceIntegrator(new HDGDiffusionIntegrator(kcoeff, td));
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Mt->AddBdrFaceIntegrator(new HDGDiffusionIntegrator(kcoeff, td),
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bdr_is_neumann);
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}
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if (Mtnl)
|
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{
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Mtnl->AddInteriorFaceIntegrator(new HDGDiffusionIntegrator(kcoeff, td));
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Mtnl->AddBdrFaceIntegrator(new HDGDiffusionIntegrator(kcoeff, td),
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bdr_is_neumann);
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}
|
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}
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||
}
|
||
|
||
//divergence/weak gradient
|
||
|
||
if (dg)
|
||
{
|
||
B->AddDomainIntegrator(new VectorDivergenceIntegrator());
|
||
if (upwinded)
|
||
{
|
||
B->AddInteriorFaceIntegrator(new TransposeIntegrator(
|
||
new DGNormalTraceIntegrator(ccoeff, -1.)));
|
||
B->AddBdrFaceIntegrator(new TransposeIntegrator(new DGNormalTraceIntegrator(
|
||
ccoeff, -1.)), bdr_is_neumann);
|
||
}
|
||
else
|
||
{
|
||
B->AddInteriorFaceIntegrator(new TransposeIntegrator(
|
||
new DGNormalTraceIntegrator(-1.)));
|
||
B->AddBdrFaceIntegrator(new TransposeIntegrator(new DGNormalTraceIntegrator(
|
||
-1.)), bdr_is_neumann);
|
||
}
|
||
}
|
||
else
|
||
{
|
||
B->AddDomainIntegrator(new VectorFEDivergenceIntegrator());
|
||
}
|
||
|
||
//linear convection in the linear regime
|
||
|
||
if (bconv && Mt)
|
||
{
|
||
Mt->AddDomainIntegrator(new ConservativeConvectionIntegrator(ccoeff));
|
||
if (upwinded)
|
||
{
|
||
Mt->AddInteriorFaceIntegrator(new HDGConvectionUpwindedIntegrator(ccoeff));
|
||
Mt->AddBdrFaceIntegrator(new HDGConvectionUpwindedIntegrator(ccoeff));
|
||
}
|
||
else
|
||
{
|
||
Mt->AddInteriorFaceIntegrator(new HDGConvectionCenteredIntegrator(ccoeff));
|
||
if (hybridization)
|
||
{
|
||
//centered scheme does not work with Dirichlet when hybridized,
|
||
//giving an diverging system, we use the full BC flux here
|
||
Mt->AddBdrFaceIntegrator(new HDGConvectionCenteredIntegrator(ccoeff),
|
||
bdr_is_neumann);
|
||
}
|
||
else
|
||
{
|
||
Mt->AddBdrFaceIntegrator(new HDGConvectionCenteredIntegrator(ccoeff));
|
||
}
|
||
}
|
||
}
|
||
|
||
//linear convection in the nonlinear regime
|
||
|
||
if (bconv && Mtnl)
|
||
{
|
||
Mtnl->AddDomainIntegrator(new ConservativeConvectionIntegrator(ccoeff));
|
||
if (upwinded)
|
||
{
|
||
Mtnl->AddInteriorFaceIntegrator(new HDGConvectionUpwindedIntegrator(ccoeff));
|
||
Mtnl->AddBdrFaceIntegrator(new HDGConvectionUpwindedIntegrator(ccoeff));
|
||
}
|
||
else
|
||
{
|
||
Mtnl->AddInteriorFaceIntegrator(new HDGConvectionCenteredIntegrator(ccoeff));
|
||
if (hybridization)
|
||
{
|
||
//centered scheme does not work with Dirichlet when hybridized,
|
||
//giving an diverging system, we use the full BC flux here
|
||
Mtnl->AddBdrFaceIntegrator(new HDGConvectionCenteredIntegrator(ccoeff),
|
||
bdr_is_neumann);
|
||
}
|
||
else
|
||
{
|
||
Mtnl->AddBdrFaceIntegrator(new HDGConvectionCenteredIntegrator(ccoeff));
|
||
}
|
||
}
|
||
}
|
||
|
||
//nonlinear convection in the nonlinear regime
|
||
|
||
if (bnlconv && Mtnl)
|
||
{
|
||
FluxFun = GetFluxFun(problem, ccoeff);
|
||
switch (hdg_scheme)
|
||
{
|
||
case 1: FluxSolver = new HDGFlux(*FluxFun, HDGFlux::HDGScheme::HDG_1); break;
|
||
case 2: FluxSolver = new HDGFlux(*FluxFun, HDGFlux::HDGScheme::HDG_2); break;
|
||
case 3: FluxSolver = new RusanovFlux(*FluxFun); break;
|
||
case 4: FluxSolver = new GodunovFlux(*FluxFun); break;
|
||
default:
|
||
cerr << "Unknown HDG scheme" << endl;
|
||
exit(1);
|
||
}
|
||
Mtnl->AddDomainIntegrator(new HyperbolicFormIntegrator(*FluxSolver, 0, -1.));
|
||
Mtnl->AddInteriorFaceIntegrator(new HyperbolicFormIntegrator(
|
||
*FluxSolver, 0, -1.));
|
||
Mtnl->AddBdrFaceIntegrator(new HyperbolicFormIntegrator(
|
||
*FluxSolver, 0, -1.));
|
||
}
|
||
|
||
//inertial term
|
||
|
||
if (a > 0.)
|
||
{
|
||
if (Mt)
|
||
{
|
||
Mt->AddDomainIntegrator(new MassIntegrator(acoeff));
|
||
}
|
||
else
|
||
{
|
||
Mtnl->AddDomainIntegrator(new MassIntegrator(acoeff));
|
||
}
|
||
}
|
||
|
||
//set hybridization / assembly level
|
||
|
||
Array<int> ess_flux_tdofs_list;
|
||
if (!dg)
|
||
{
|
||
V_space->GetEssentialTrueDofs(bdr_is_neumann, ess_flux_tdofs_list);
|
||
}
|
||
|
||
FiniteElementCollection *trace_coll = NULL;
|
||
FiniteElementSpace *trace_space = NULL;
|
||
|
||
|
||
if (hybridization)
|
||
{
|
||
chrono.Clear();
|
||
chrono.Start();
|
||
|
||
trace_coll = new RT_Trace_FECollection(order, dim, 0);
|
||
//trace_coll = new DG_Interface_FECollection(order, dim, 0);
|
||
trace_space = new FiniteElementSpace(mesh, trace_coll);
|
||
darcy->EnableHybridization(trace_space,
|
||
new NormalTraceJumpIntegrator(),
|
||
ess_flux_tdofs_list);
|
||
|
||
chrono.Stop();
|
||
std::cout << "Hybridization init took " << chrono.RealTime() << "s.\n";
|
||
}
|
||
else if (reduction)
|
||
{
|
||
chrono.Clear();
|
||
chrono.Start();
|
||
|
||
if (dg)
|
||
{
|
||
darcy->EnableFluxReduction();
|
||
}
|
||
else if (!bconv && !bnlconv)
|
||
{
|
||
darcy->EnablePotentialReduction(ess_flux_tdofs_list);
|
||
}
|
||
else
|
||
{
|
||
std::cerr << "No possible reduction!" << std::endl;
|
||
return 1;
|
||
}
|
||
|
||
chrono.Stop();
|
||
std::cout << "Reduction init took " << chrono.RealTime() << "s.\n";
|
||
}
|
||
|
||
if (pa) { darcy->SetAssemblyLevel(AssemblyLevel::PARTIAL); }
|
||
|
||
// 8. Define the BlockStructure of the problem, i.e. define the array of
|
||
// offsets for each variable. The last component of the Array is the sum
|
||
// of the dimensions of each block.
|
||
const Array<int> block_offsets(DarcyOperator::ConstructOffsets(*darcy));
|
||
|
||
std::cout << "***********************************************************\n";
|
||
if (!reduction || (reduction && !dg))
|
||
{
|
||
std::cout << "dim(V) = " << block_offsets[1] - block_offsets[0] << "\n";
|
||
}
|
||
if (!reduction || (reduction && dg))
|
||
{
|
||
std::cout << "dim(W) = " << block_offsets[2] - block_offsets[1] << "\n";
|
||
}
|
||
if (!reduction)
|
||
{
|
||
if (hybridization)
|
||
{
|
||
std::cout << "dim(M) = " << block_offsets[3] - block_offsets[2] << "\n";
|
||
std::cout << "dim(V+W+M) = " << block_offsets.Last() << "\n";
|
||
}
|
||
else
|
||
{
|
||
std::cout << "dim(V+W) = " << block_offsets.Last() << "\n";
|
||
}
|
||
}
|
||
std::cout << "***********************************************************\n";
|
||
|
||
// 9. Allocate memory (x, rhs) for the analytical solution and the right hand
|
||
// side. Define the GridFunction q,t for the finite element solution and
|
||
// linear forms fform and gform for the right hand side. The data
|
||
// allocated by x and rhs are passed as a reference to the grid functions
|
||
// (q,t) and the linear forms (fform, gform).
|
||
MemoryType mt = device.GetMemoryType();
|
||
BlockVector x(block_offsets, mt), rhs(block_offsets, mt);
|
||
|
||
x = 0.;
|
||
GridFunction q_h, t_h;
|
||
q_h.MakeRef(V_space, x.GetBlock(0), 0);
|
||
t_h.MakeRef(W_space, x.GetBlock(1), 0);
|
||
|
||
if (btime)
|
||
{
|
||
t_h.ProjectCoefficient(tcoeff); //initial condition
|
||
}
|
||
|
||
if (!dg)
|
||
{
|
||
q_h.ProjectBdrCoefficientNormal(qcoeff,
|
||
bdr_is_neumann); //essential Neumann BC
|
||
}
|
||
|
||
LinearForm *gform(new LinearForm);
|
||
gform->Update(V_space, rhs.GetBlock(0), 0);
|
||
if (dg)
|
||
{
|
||
gform->AddBdrFaceIntegrator(new VectorBoundaryFluxLFIntegrator(gcoeff),
|
||
bdr_is_dirichlet);
|
||
}
|
||
else
|
||
{
|
||
gform->AddBoundaryIntegrator(new VectorFEBoundaryFluxLFIntegrator(gcoeff),
|
||
bdr_is_dirichlet);
|
||
}
|
||
|
||
LinearForm *fform(new LinearForm);
|
||
fform->Update(W_space, rhs.GetBlock(1), 0);
|
||
fform->AddDomainIntegrator(new DomainLFIntegrator(fcoeff));
|
||
if (!hybridization)
|
||
{
|
||
if (upwinded)
|
||
fform->AddBdrFaceIntegrator(new BoundaryFlowIntegrator(one, qtcoeff, +1.),
|
||
bdr_is_neumann);
|
||
else
|
||
fform->AddBdrFaceIntegrator(new BoundaryFlowIntegrator(one, qtcoeff, +1., 0.),
|
||
bdr_is_neumann);
|
||
}
|
||
if (bconv)
|
||
{
|
||
if (upwinded)
|
||
fform->AddBdrFaceIntegrator(new BoundaryFlowIntegrator(tcoeff, ccoeff, +1.),
|
||
bdr_is_dirichlet);
|
||
else
|
||
{
|
||
if (hybridization)
|
||
fform->AddBdrFaceIntegrator(new BoundaryFlowIntegrator(tcoeff, ccoeff, +2., 0.),
|
||
bdr_is_dirichlet);//<-- full BC flux, see above
|
||
else
|
||
fform->AddBdrFaceIntegrator(new BoundaryFlowIntegrator(tcoeff, ccoeff, +1., 0.),
|
||
bdr_is_dirichlet);
|
||
}
|
||
}
|
||
|
||
//prepare (reduced) solution and rhs vectors
|
||
|
||
LinearForm *hform = NULL;
|
||
|
||
//Neumann BC for the hybridized system
|
||
|
||
if (hybridization)
|
||
{
|
||
hform = new LinearForm();
|
||
hform->Update(trace_space, rhs.GetBlock(2), 0);
|
||
//note that Neumann BC must be applied only for the heat flux
|
||
//and not the total flux for stability reasons
|
||
hform->AddBoundaryIntegrator(new BoundaryNormalLFIntegrator(qcoeff, 2),
|
||
bdr_is_neumann);
|
||
}
|
||
|
||
//construct the operator
|
||
|
||
Array<Coefficient*> coeffs({(Coefficient*)&gcoeff,
|
||
(Coefficient*)&fcoeff,
|
||
(Coefficient*)&qtcoeff});
|
||
|
||
DarcyOperator op(ess_flux_tdofs_list, darcy, gform, fform, hform, coeffs,
|
||
(DarcyOperator::SolverType) solver_type, false, btime);
|
||
|
||
//construct the time solver
|
||
|
||
ODESolver *ode_solver;
|
||
|
||
switch (ode)
|
||
{
|
||
case 1: ode_solver = new BackwardEulerSolver(); break;
|
||
case 2: ode_solver = new SDIRK23Solver(2); break;
|
||
case 3: ode_solver = new SDIRK23Solver(); break;
|
||
case 4: ode_solver = new SDIRK34Solver(); break;
|
||
default:
|
||
MFEM_ABORT("Unknown solver");
|
||
return 1;
|
||
}
|
||
|
||
ode_solver->Init(op);
|
||
|
||
//iterate in time
|
||
|
||
if (!btime) { nt = 1; }
|
||
|
||
const real_t dt = tf / nt; //time step
|
||
|
||
for (int ti = 0; ti < nt; ti++)
|
||
{
|
||
//set current time
|
||
|
||
real_t t = tf * ti / nt;
|
||
|
||
//perform time step
|
||
|
||
real_t dt_ = dt;//<---ignore time step changes
|
||
ode_solver->Step(x, t, dt_);
|
||
|
||
// 12. Compute the L2 error norms.
|
||
|
||
int order_quad = max(2, 2*order+1);
|
||
const IntegrationRule *irs[Geometry::NumGeom];
|
||
for (int i=0; i < Geometry::NumGeom; ++i)
|
||
{
|
||
irs[i] = &(IntRules.Get(i, order_quad));
|
||
}
|
||
|
||
real_t err_q = q_h.ComputeL2Error(qcoeff, irs);
|
||
real_t norm_q = ComputeLpNorm(2., qcoeff, *mesh, irs);
|
||
real_t err_t = t_h.ComputeL2Error(tcoeff, irs);
|
||
real_t norm_t = ComputeLpNorm(2., tcoeff, *mesh, irs);
|
||
|
||
if (btime)
|
||
{
|
||
cout << "iter:\t" << ti
|
||
<< "\ttime:\t" << t
|
||
<< "\tq_err:\t" << err_q / norm_q
|
||
<< "\tt_err:\t" << err_t / norm_t
|
||
<< endl;
|
||
}
|
||
else
|
||
{
|
||
cout << "|| q_h - q_ex || / || q_ex || = " << err_q / norm_q << "\n";
|
||
cout << "|| t_h - t_ex || / || t_ex || = " << err_t / norm_t << "\n";
|
||
}
|
||
|
||
// Project the analytic solution
|
||
|
||
static GridFunction q_a, qt_a, t_a, c_gf;
|
||
|
||
q_a.SetSpace(V_space);
|
||
q_a.ProjectCoefficient(qcoeff);
|
||
|
||
qt_a.SetSpace(V_space);
|
||
qt_a.ProjectCoefficient(qtcoeff);
|
||
|
||
t_a.SetSpace(W_space);
|
||
t_a.ProjectCoefficient(tcoeff);
|
||
|
||
if (bconv)
|
||
{
|
||
c_gf.SetSpace(V_space);
|
||
c_gf.ProjectCoefficient(ccoeff);
|
||
}
|
||
|
||
// 13. Save the mesh and the solution. This output can be viewed later using
|
||
// GLVis: "glvis -m ex5.mesh -g sol_q.gf" or "glvis -m ex5.mesh -g
|
||
// sol_t.gf".
|
||
if (mfem)
|
||
{
|
||
stringstream ss;
|
||
ss.str("");
|
||
ss << "ex5";
|
||
if (btime) { ss << "_" << ti; }
|
||
ss << ".mesh";
|
||
ofstream mesh_ofs(ss.str());
|
||
mesh_ofs.precision(8);
|
||
mesh->Print(mesh_ofs);
|
||
|
||
ss.str("");
|
||
ss << "sol_q";
|
||
if (btime) { ss << "_" << ti; }
|
||
ss << ".gf";
|
||
ofstream q_ofs(ss.str());
|
||
q_ofs.precision(8);
|
||
q_h.Save(q_ofs);
|
||
|
||
ss.str("");
|
||
ss << "sol_t";
|
||
if (btime) { ss << "_" << ti; }
|
||
ss << ".gf";
|
||
ofstream t_ofs(ss.str());
|
||
t_ofs.precision(8);
|
||
t_h.Save(t_ofs);
|
||
}
|
||
|
||
// 14. Save data in the VisIt format
|
||
if (visit)
|
||
{
|
||
static VisItDataCollection visit_dc("Example5", mesh);
|
||
if (ti == 0)
|
||
{
|
||
visit_dc.RegisterField("heat flux", &q_h);
|
||
visit_dc.RegisterField("temperature", &t_h);
|
||
if (analytic)
|
||
{
|
||
visit_dc.RegisterField("heat flux analytic", &q_a);
|
||
visit_dc.RegisterField("temperature analytic", &t_a);
|
||
}
|
||
}
|
||
visit_dc.SetCycle(ti);
|
||
visit_dc.SetTime(t); // set the time
|
||
visit_dc.Save();
|
||
}
|
||
|
||
// 15. Save data in the ParaView format
|
||
if (paraview)
|
||
{
|
||
static ParaViewDataCollection paraview_dc("Example5", mesh);
|
||
if (ti == 0)
|
||
{
|
||
paraview_dc.SetPrefixPath("ParaView");
|
||
paraview_dc.SetLevelsOfDetail(order);
|
||
paraview_dc.SetDataFormat(VTKFormat::BINARY);
|
||
paraview_dc.SetHighOrderOutput(true);
|
||
paraview_dc.RegisterField("heat flux",&q_h);
|
||
paraview_dc.RegisterField("temperature",&t_h);
|
||
if (analytic)
|
||
{
|
||
paraview_dc.RegisterField("heat flux analytic", &q_a);
|
||
paraview_dc.RegisterField("temperature analytic", &t_a);
|
||
}
|
||
}
|
||
paraview_dc.SetCycle(ti);
|
||
paraview_dc.SetTime(t); // set the time
|
||
paraview_dc.Save();
|
||
}
|
||
|
||
// 16. Send the solution by socket to a GLVis server.
|
||
if (visualization)
|
||
{
|
||
const char vishost[] = "localhost";
|
||
const int visport = 19916;
|
||
static socketstream q_sock(vishost, visport);
|
||
q_sock.precision(8);
|
||
q_sock << "solution\n" << *mesh << q_h << endl;
|
||
if (ti == 0)
|
||
{
|
||
q_sock << "window_title 'Heat flux'" << endl;
|
||
q_sock << "keys Rljvvvvvmmc" << endl;
|
||
}
|
||
static socketstream t_sock(vishost, visport);
|
||
t_sock.precision(8);
|
||
t_sock << "solution\n" << *mesh << t_h << endl;
|
||
if (ti == 0)
|
||
{
|
||
t_sock << "window_title 'Temperature'" << endl;
|
||
t_sock << "keys Rljmmc" << endl;
|
||
}
|
||
if (analytic)
|
||
{
|
||
static socketstream qa_sock(vishost, visport);
|
||
qa_sock.precision(8);
|
||
qa_sock << "solution\n" << *mesh << q_a << endl;
|
||
if (ti == 0)
|
||
{
|
||
qa_sock << "window_title 'Heat flux analytic'" << endl;
|
||
qa_sock << "keys Rljvvvvvmmc" << endl;
|
||
}
|
||
if (bconv || bnlconv)
|
||
{
|
||
static socketstream qta_sock(vishost, visport);
|
||
qta_sock.precision(8);
|
||
qta_sock << "solution\n" << *mesh << qt_a << endl;
|
||
if (ti == 0)
|
||
{
|
||
qta_sock << "window_title 'Total flux analytic'" << endl;
|
||
qta_sock << "keys Rljvvvvvmmc" << endl;
|
||
}
|
||
}
|
||
static socketstream ta_sock(vishost, visport);
|
||
ta_sock.precision(8);
|
||
ta_sock << "solution\n" << *mesh << t_a << endl;
|
||
if (ti == 0)
|
||
{
|
||
ta_sock << "window_title 'Temperature analytic'" << endl;
|
||
ta_sock << "keys Rljmmc" << endl;
|
||
}
|
||
if (bconv)
|
||
{
|
||
static socketstream c_sock(vishost, visport);
|
||
c_sock.precision(8);
|
||
c_sock << "solution\n" << *mesh << c_gf << endl;
|
||
if (ti == 0)
|
||
{
|
||
c_sock << "window_title 'Velocity'" << endl;
|
||
c_sock << "keys Rljvvvvvmmc" << endl;
|
||
}
|
||
}
|
||
}
|
||
}
|
||
}
|
||
|
||
// 17. Free the used memory.
|
||
|
||
delete ode_solver;
|
||
delete HeatFluxFun;
|
||
delete FluxFun;
|
||
delete FluxSolver;
|
||
delete fform;
|
||
delete gform;
|
||
delete hform;
|
||
delete darcy;
|
||
delete W_space;
|
||
delete V_space;
|
||
delete trace_space;
|
||
delete W_coll;
|
||
delete V_coll;
|
||
delete trace_coll;
|
||
delete mesh;
|
||
|
||
return 0;
|
||
}
|
||
|
||
MatFunc GetKFun(Problem prob, real_t k, real_t ks, real_t ka)
|
||
{
|
||
switch (prob)
|
||
{
|
||
case Problem::SteadyDiffusion:
|
||
case Problem::MFEMLogo:
|
||
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;
|
||
}
|
||
};
|
||
}
|
||
return MatFunc();
|
||
}
|
||
|
||
TFunc GetTFun(Problem prob, real_t t_0, real_t a, const MatFunc &kFun, real_t c)
|
||
{
|
||
switch (prob)
|
||
{
|
||
case Problem::SteadyDiffusion:
|
||
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
|
||
//xz
|
||
ddT(2) = +t_0 * M_PI*M_PI * cos(M_PI*x(0)) * sin(M_PI*x(1)) * cos(M_PI*x(2));
|
||
//yz
|
||
ddT(3) = +t_0 * M_PI*M_PI * sin(M_PI*x(0)) * cos(M_PI*x(1)) * cos(M_PI*x(2));
|
||
|
||
}
|
||
|
||
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;
|
||
};
|
||
case Problem::MFEMLogo:
|
||
return [=](const Vector &x, real_t t) -> real_t
|
||
{
|
||
#if 1
|
||
constexpr int iw = 38;
|
||
constexpr int ih = 7;
|
||
static const unsigned char logo[ih][iw] = {
|
||
"## ## ######## ######## ## ##",
|
||
"### ### ## ## ### ###",
|
||
"#### #### ## ## #### ####",
|
||
"## ### ## ###### ###### ## ### ##",
|
||
"## ## ## ## ## ##",
|
||
"## ## ## ## ## ##",
|
||
"## ## ## ######## ## ##",
|
||
};
|
||
#else
|
||
constexpr int iw = 50;
|
||
constexpr int ih = 8;
|
||
static const unsigned char logo[ih][iw] = {
|
||
"888b d888 8888888888 8888888888 888b d888",
|
||
"8888b d8888 888 888 8888b d8888",
|
||
"88888b.d88888 888 888 88888b.d88888",
|
||
"888Y88888P888 8888888 8888888 888Y88888P888",
|
||
"888 Y888P 888 888 888 888 Y888P 888",
|
||
"888 Y8P 888 888 888 888 Y8P 888",
|
||
"888 8 888 888 888 888 8 888",
|
||
"888 888 888 8888888888 888 888",
|
||
};
|
||
#endif
|
||
|
||
constexpr real_t w = 0.8;
|
||
constexpr real_t h = (w * ih) / iw;
|
||
constexpr real_t xo = 0.5;
|
||
constexpr real_t yo = 0.5;
|
||
const real_t dx = x(0) - xo;
|
||
const real_t dy = x(1) - yo;
|
||
|
||
const int ix = (dx/w + 0.5) * iw;
|
||
const int iy = (dy/h + 0.5) * ih;
|
||
|
||
if (ix < 0 || ix >= iw || iy < 0 || iy >= ih)
|
||
{
|
||
return 0.;
|
||
}
|
||
|
||
const real_t T = (logo[ih-1-iy][ix] != ' ')?(t_0):(0.);
|
||
return T;
|
||
};
|
||
|
||
}
|
||
return TFunc();
|
||
}
|
||
|
||
VecTFunc GetQFun(Problem prob, real_t t_0, real_t a, const MatFunc &kFun,
|
||
real_t c)
|
||
{
|
||
switch (prob)
|
||
{
|
||
case Problem::SteadyDiffusion:
|
||
return [=](const Vector &x, real_t, 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();
|
||
}
|
||
};
|
||
case Problem::MFEMLogo:
|
||
return [=](const Vector &x, real_t, Vector &v)
|
||
{
|
||
const int vdim = x.Size();
|
||
v.SetSize(vdim);
|
||
v = 0.;
|
||
};
|
||
}
|
||
return VecTFunc();
|
||
}
|
||
|
||
VecFunc GetCFun(Problem prob, real_t c)
|
||
{
|
||
switch (prob)
|
||
{
|
||
case Problem::SteadyDiffusion:
|
||
case Problem::MFEMLogo:
|
||
// null
|
||
break;
|
||
}
|
||
return VecFunc();
|
||
}
|
||
|
||
TFunc GetFFun(Problem prob, real_t t_0, real_t a, const MatFunc &kFun, real_t c)
|
||
{
|
||
auto TFun = GetTFun(prob, t_0, a, kFun, c);
|
||
|
||
switch (prob)
|
||
{
|
||
case Problem::SteadyDiffusion:
|
||
case Problem::MFEMLogo:
|
||
return [=](const Vector &x, real_t) -> real_t
|
||
{
|
||
const real_t T = TFun(x, 0);
|
||
return -((a > 0.)?(a):(1.)) * T;
|
||
};
|
||
}
|
||
return TFunc();
|
||
}
|
||
|
||
FluxFunction* GetFluxFun(Problem prob, VectorCoefficient &ccoef)
|
||
{
|
||
switch (prob)
|
||
{
|
||
case Problem::SteadyDiffusion:
|
||
case Problem::MFEMLogo:
|
||
//null
|
||
break;
|
||
}
|
||
|
||
return NULL;
|
||
}
|
||
|
||
MixedFluxFunction* GetHeatFluxFun(Problem prob, real_t k, int dim)
|
||
{
|
||
switch (prob)
|
||
{
|
||
case Problem::SteadyDiffusion:
|
||
case Problem::MFEMLogo:
|
||
static FunctionCoefficient ikappa([=](const Vector &x) -> real_t { return 1./k; });
|
||
return new LinearDiffusionFlux(dim, &ikappa);
|
||
}
|
||
|
||
return NULL;
|
||
}
|