Will Pazner
dc13e67e87
Currently just a cleaned-up version of ex9. Will add diffusion and IMEX time integration.
417 lines
12 KiB
C++
417 lines
12 KiB
C++
// MFEM Example 41
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//
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// Compile with: make ex41
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//
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// Sample runs:
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//
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// Device sample runs:
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//
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// Description: This example code solves the time-dependent advection-diffusion
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// equation du/dt + v.grad(u) + a div(grad(u)) = 0, where v is a
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// given fluid velocity, a is the diffusion coefficient, and
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// u0(x)=u(0,x) is a given initial condition.
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//
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// The example demonstrates the use of Discontinuous Galerkin (DG)
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// bilinear forms in MFEM (face integrators), the use of implicit,
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// explicit, and IMEX ODE time integrators.
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#include "mfem.hpp"
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using namespace std;
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using namespace mfem;
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// Choice for the problem setup. The fluid velocity, initial condition and
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// inflow boundary condition are chosen based on this parameter.
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int problem;
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// Velocity coefficient
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void velocity_function(const Vector &x, Vector &v);
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// Initial condition
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real_t u0_function(const Vector &x);
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// Inflow boundary condition
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real_t inflow_function(const Vector &x);
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// Mesh bounding box
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Vector bb_min, bb_max;
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class DG_Solver : public Solver
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{
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private:
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SparseMatrix &M, &K, A;
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GMRESSolver linear_solver;
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BlockILU prec;
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real_t dt;
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public:
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DG_Solver(SparseMatrix &M_, SparseMatrix &K_, const FiniteElementSpace &fes)
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: M(M_),
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K(K_),
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prec(fes.GetTypicalFE()->GetDof(),
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BlockILU::Reordering::MINIMUM_DISCARDED_FILL),
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dt(-1.0)
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{
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linear_solver.iterative_mode = false;
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linear_solver.SetRelTol(1e-9);
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linear_solver.SetAbsTol(0.0);
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linear_solver.SetMaxIter(100);
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linear_solver.SetPrintLevel(0);
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linear_solver.SetPreconditioner(prec);
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}
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void SetTimeStep(real_t dt_)
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{
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if (dt_ != dt)
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{
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dt = dt_;
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// Form operator A = M - dt*K
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A = K;
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A *= -dt;
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A += M;
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// this will also call SetOperator on the preconditioner
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linear_solver.SetOperator(A);
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}
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}
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void SetOperator(const Operator &op) override
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{
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linear_solver.SetOperator(op);
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}
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void Mult(const Vector &x, Vector &y) const override
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{
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linear_solver.Mult(x, y);
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}
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};
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/** A time-dependent operator for the right-hand side of the ODE. The DG weak
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form of du/dt = -v.grad(u) is M du/dt = K u + b, where M and K are the mass
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and advection matrices, and b describes the flow on the boundary. This can
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be written as a general ODE, du/dt = M^{-1} (K u + b), and this class is
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used to evaluate the right-hand side. */
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class FE_Evolution : public TimeDependentOperator
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{
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private:
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BilinearForm &M, &K;
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const Vector &b;
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unique_ptr<Solver> M_prec;
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CGSolver M_solver;
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unique_ptr<DG_Solver> dg_solver;
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mutable Vector z;
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public:
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FE_Evolution(BilinearForm &M_, BilinearForm &K_, const Vector &b_);
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void Mult(const Vector &x, Vector &y) const override;
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void ImplicitSolve(const real_t dt, const Vector &x, Vector &k) override;
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};
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int main(int argc, char *argv[])
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{
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// 1. Parse command-line options.
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problem = 0;
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const char *mesh_file = "../data/periodic-hexagon.mesh";
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int ref_levels = 2;
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int order = 3;
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const char *device_config = "cpu";
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int ode_solver_type = 4;
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real_t t_final = 10.0;
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real_t dt = 0.01;
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bool paraview = false;
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int vis_steps = 5;
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OptionsParser args(argc, argv);
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args.AddOption(&mesh_file, "-m", "--mesh", "Mesh file to use.");
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args.AddOption(&problem, "-p", "--problem",
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"Problem setup to use. See options in velocity_function().");
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args.AddOption(&ref_levels, "-r", "--refine",
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"Number of times to refine the mesh uniformly.");
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args.AddOption(&order, "-o", "--order", "Order of the finite elements.");
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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(&ode_solver_type, "-s", "--ode-solver",
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ODESolver::Types.c_str());
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args.AddOption(&t_final, "-tf", "--t-final", "Final time; start time is 0.");
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args.AddOption(&dt, "-dt", "--time-step", "Time step.");
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args.AddOption(¶view, "-paraview", "--paraview-datafiles", "-no-paraview",
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"--no-paraview-datafiles",
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"Save data files for ParaView (paraview.org) visualization.");
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args.AddOption(&vis_steps, "-vs", "--visualization-steps",
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"Visualize every n-th timestep.");
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args.ParseCheck();
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Device device(device_config);
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device.Print();
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// 2. Read the mesh from the given mesh file. We can handle geometrically
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// periodic meshes in this code.
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Mesh mesh(mesh_file);
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const int dim = mesh.Dimension();
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// 3. Define the ODE solver used for time integration. May be explicit,
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// implicit, or IMEX.
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unique_ptr<ODESolver> ode_solver = ODESolver::Select(ode_solver_type);
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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, where 'ref_levels' is a
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// command-line parameter.
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for (int lev = 0; lev < ref_levels; lev++) { mesh.UniformRefinement(); }
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mesh.GetBoundingBox(bb_min, bb_max, max(order, 1));
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// 5. Define the discontinuous DG finite element space of the given
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// polynomial order on the refined mesh.
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DG_FECollection fec(order, dim, BasisType::GaussLobatto);
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FiniteElementSpace fes(&mesh, &fec);
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cout << "Number of unknowns: " << fes.GetVSize() << endl;
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// 6. Set up and assemble the bilinear and linear forms corresponding to the
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// DG discretization. The DGTraceIntegrator involves integrals over mesh
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// interior faces.
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VectorFunctionCoefficient velocity(dim, velocity_function);
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FunctionCoefficient inflow(inflow_function);
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FunctionCoefficient u0(u0_function);
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BilinearForm m(&fes);
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BilinearForm k(&fes);
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m.AddDomainIntegrator(new MassIntegrator);
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constexpr real_t alpha = -1.0;
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k.AddDomainIntegrator(new ConvectionIntegrator(velocity, alpha));
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k.AddInteriorFaceIntegrator(
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new NonconservativeDGTraceIntegrator(velocity, alpha));
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k.AddBdrFaceIntegrator(
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new NonconservativeDGTraceIntegrator(velocity, alpha));
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LinearForm b(&fes);
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b.AddBdrFaceIntegrator(
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new BoundaryFlowIntegrator(inflow, velocity, alpha));
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m.Assemble();
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int skip_zeros = 0;
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k.Assemble(skip_zeros);
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b.Assemble();
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m.Finalize();
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k.Finalize(skip_zeros);
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// 7. Define the initial conditions.
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GridFunction u(&fes);
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u.ProjectCoefficient(u0);
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unique_ptr<ParaViewDataCollection> pv;
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if (paraview)
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{
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pv = make_unique<ParaViewDataCollection>("Example41", &mesh);
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pv->SetPrefixPath("ParaView");
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pv->RegisterField("solution", &u);
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pv->SetLevelsOfDetail(order);
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pv->SetDataFormat(VTKFormat::BINARY);
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pv->SetHighOrderOutput(true);
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pv->SetCycle(0);
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pv->SetTime(0.0);
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pv->Save();
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}
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// 8. Define the time-dependent evolution operator describing the ODE
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// right-hand side, and perform time-integration (looping over the time
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// iterations, ti, with a time-step dt).
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FE_Evolution adv(m, k, b);
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real_t t = 0.0;
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adv.SetTime(t);
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ode_solver->Init(adv);
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bool done = false;
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for (int ti = 0; !done; )
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{
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real_t dt_real = min(dt, t_final - t);
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ode_solver->Step(u, t, dt_real);
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ti++;
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done = (t >= t_final - 1e-8*dt);
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if (done || ti % vis_steps == 0)
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{
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cout << "time step: " << ti << ", time: " << t << endl;
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if (paraview)
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{
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pv->SetCycle(ti);
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pv->SetTime(t);
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pv->Save();
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}
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}
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}
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return 0;
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}
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// Implementation of class FE_Evolution
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FE_Evolution::FE_Evolution(BilinearForm &M_, BilinearForm &K_, const Vector &b_)
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: TimeDependentOperator(M_.FESpace()->GetTrueVSize()),
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M(M_), K(K_), b(b_), z(height)
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{
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Array<int> ess_tdof_list;
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if (M.GetAssemblyLevel() == AssemblyLevel::LEGACY)
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{
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M_prec = make_unique<DSmoother>(M.SpMat());
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M_solver.SetOperator(M.SpMat());
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dg_solver = make_unique<DG_Solver>(M.SpMat(), K.SpMat(), *M.FESpace());
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}
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else
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{
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M_prec = make_unique<OperatorJacobiSmoother>(M, ess_tdof_list);
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M_solver.SetOperator(M);
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dg_solver = NULL;
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}
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M_solver.SetPreconditioner(*M_prec);
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M_solver.iterative_mode = false;
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M_solver.SetRelTol(1e-9);
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M_solver.SetAbsTol(0.0);
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M_solver.SetMaxIter(100);
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M_solver.SetPrintLevel(0);
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}
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void FE_Evolution::Mult(const Vector &x, Vector &y) const
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{
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// y = M^{-1} (K x + b)
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K.Mult(x, z);
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z += b;
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M_solver.Mult(z, y);
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}
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void FE_Evolution::ImplicitSolve(const real_t dt, const Vector &x, Vector &k)
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{
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MFEM_VERIFY(dg_solver != NULL,
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"Implicit time integration is not supported with partial assembly");
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K.Mult(x, z);
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z += b;
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dg_solver->SetTimeStep(dt);
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dg_solver->Mult(z, k);
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}
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// Velocity coefficient
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void velocity_function(const Vector &x, Vector &v)
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{
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int dim = x.Size();
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// map to the reference [-1,1] domain
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Vector X(dim);
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for (int i = 0; i < dim; i++)
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{
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real_t center = (bb_min[i] + bb_max[i]) * 0.5;
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X(i) = 2 * (x(i) - center) / (bb_max[i] - bb_min[i]);
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}
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switch (problem)
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{
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case 0:
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{
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// Translations in 1D, 2D, and 3D
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switch (dim)
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{
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case 1: v(0) = 1.0; break;
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case 2: v(0) = sqrt(2./3.); v(1) = sqrt(1./3.); break;
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case 3: v(0) = sqrt(3./6.); v(1) = sqrt(2./6.); v(2) = sqrt(1./6.);
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break;
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}
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break;
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}
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case 1:
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case 2:
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{
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// Clockwise rotation in 2D around the origin
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const real_t w = M_PI/2;
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switch (dim)
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{
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case 1: v(0) = 1.0; break;
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case 2: v(0) = w*X(1); v(1) = -w*X(0); break;
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case 3: v(0) = w*X(1); v(1) = -w*X(0); v(2) = 0.0; break;
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}
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break;
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}
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case 3:
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{
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// Clockwise twisting rotation in 2D around the origin
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const real_t w = M_PI/2;
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real_t d = max((X(0)+1.)*(1.-X(0)),0.) * max((X(1)+1.)*(1.-X(1)),0.);
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d = d*d;
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switch (dim)
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{
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case 1: v(0) = 1.0; break;
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case 2: v(0) = d*w*X(1); v(1) = -d*w*X(0); break;
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case 3: v(0) = d*w*X(1); v(1) = -d*w*X(0); v(2) = 0.0; break;
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}
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break;
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}
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}
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}
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// Initial condition
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real_t u0_function(const Vector &x)
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{
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int dim = x.Size();
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// map to the reference [-1,1] domain
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Vector X(dim);
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for (int i = 0; i < dim; i++)
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{
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real_t center = (bb_min[i] + bb_max[i]) * 0.5;
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X(i) = 2 * (x(i) - center) / (bb_max[i] - bb_min[i]);
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}
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switch (problem)
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{
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case 0:
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case 1:
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{
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switch (dim)
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{
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case 1:
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return exp(-40.*pow(X(0)-0.5,2));
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case 2:
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case 3:
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{
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real_t rx = 0.45, ry = 0.25, cx = 0., cy = -0.2, w = 10.;
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if (dim == 3)
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{
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const real_t s = (1. + 0.25*cos(2*M_PI*X(2)));
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rx *= s;
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ry *= s;
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}
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return ( std::erfc(w*(X(0)-cx-rx))*std::erfc(-w*(X(0)-cx+rx)) *
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std::erfc(w*(X(1)-cy-ry))*std::erfc(-w*(X(1)-cy+ry)) )/16;
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}
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}
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}
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case 2:
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{
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real_t x_ = X(0), y_ = X(1), rho, phi;
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rho = std::hypot(x_, y_);
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phi = atan2(y_, x_);
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return pow(sin(M_PI*rho),2)*sin(3*phi);
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}
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case 3:
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{
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const real_t f = M_PI;
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return sin(f*X(0))*sin(f*X(1));
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}
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}
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return 0.0;
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}
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// Inflow boundary condition (zero for the problems considered in this example)
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real_t inflow_function(const Vector &x)
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{
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switch (problem)
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{
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case 0:
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case 1:
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case 2:
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case 3: return 0.0;
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}
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return 0.0;
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}
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