447 lines
13 KiB
C++
447 lines
13 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 mixed Darcy problem
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// corresponding to the saddle point system
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//
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// k*u + grad p = f
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// - div u = g
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//
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// with natural boundary condition -p = <given pressure>.
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// Here, we use a given exact solution (u,p) and compute the
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// corresponding r.h.s. (f,g). We discretize with Raviart-Thomas
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// finite elements (velocity u) and piecewise discontinuous
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// polynomials (pressure p).
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//
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// The example demonstrates the use of the BlockOperator class, as
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// well as the collective saving of several grid functions in
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// VisIt (visit.llnl.gov) and ParaView (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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void uFun_ex(const Vector & x, Vector & u);
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real_t pFun_ex(const Vector & x);
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void fFun(const Vector & x, Vector & f);
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real_t gFun(const Vector & x);
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real_t f_natural(const Vector & x);
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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 = "../data/star.mesh";
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int order = 1;
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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(&order, "-o", "--order",
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"Finite element order (polynomial degree).");
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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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Mesh *mesh = new Mesh(mesh_file, 1, 1);
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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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{
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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 *hdiv_coll(new RT_FECollection(order, dim));
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FiniteElementCollection *l2_coll(new L2_FECollection(order, dim));
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FiniteElementSpace *R_space = new FiniteElementSpace(mesh, hdiv_coll);
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FiniteElementSpace *W_space = new FiniteElementSpace(mesh, l2_coll);
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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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Array<int> block_offsets(3); // number of variables + 1
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block_offsets[0] = 0;
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block_offsets[1] = R_space->GetVSize();
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block_offsets[2] = W_space->GetVSize();
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block_offsets.PartialSum();
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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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ConstantCoefficient k(1.0);
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VectorFunctionCoefficient fcoeff(dim, fFun);
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FunctionCoefficient fnatcoeff(f_natural);
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FunctionCoefficient gcoeff(gFun);
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VectorFunctionCoefficient ucoeff(dim, uFun_ex);
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FunctionCoefficient pcoeff(pFun_ex);
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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 u,p 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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// (u,p) 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(R_space, rhs.GetBlock(0), 0);
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fform->AddDomainIntegrator(new VectorFEDomainLFIntegrator(fcoeff));
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fform->AddBoundaryIntegrator(new VectorFEBoundaryFluxLFIntegrator(fnatcoeff));
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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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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 u_h, v_h \in R_h
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// B = -\int_\Omega \div u_h q_h d\Omega u_h \in R_h, q_h \in W_h
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BilinearForm *mVarf(new BilinearForm(R_space));
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MixedBilinearForm *bVarf(new MixedBilinearForm(R_space, W_space));
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if (pa) { mVarf->SetAssemblyLevel(AssemblyLevel::PARTIAL); }
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mVarf->AddDomainIntegrator(new VectorFEMassIntegrator(k));
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mVarf->Assemble();
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if (!pa) { mVarf->Finalize(); }
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if (pa) { bVarf->SetAssemblyLevel(AssemblyLevel::PARTIAL); }
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bVarf->AddDomainIntegrator(new VectorFEDivergenceIntegrator);
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bVarf->Assemble();
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if (!pa) { bVarf->Finalize(); }
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BlockOperator darcyOp(block_offsets);
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TransposeOperator *Bt = NULL;
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if (pa)
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{
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Bt = new TransposeOperator(bVarf);
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darcyOp.SetBlock(0,0, mVarf);
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darcyOp.SetBlock(0,1, Bt, -1.0);
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darcyOp.SetBlock(1,0, bVarf, -1.0);
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}
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else
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{
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SparseMatrix &M(mVarf->SpMat());
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SparseMatrix &B(bVarf->SpMat());
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B *= -1.;
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Bt = new TransposeOperator(&B);
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darcyOp.SetBlock(0,0, &M);
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darcyOp.SetBlock(0,1, Bt);
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darcyOp.SetBlock(1,0, &B);
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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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// pressure Schur Complement
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SparseMatrix *MinvBt = NULL;
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Vector Md(mVarf->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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mVarf->AssembleDiagonal(Md);
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auto Md_host = Md.HostRead();
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Vector invMd(mVarf->Height());
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for (int i=0; i<mVarf->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(bVarf->Height());
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bVarf->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 &M(mVarf->SpMat());
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M.GetDiag(Md);
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Md.HostReadWrite();
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SparseMatrix &B(bVarf->SpMat());
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MinvBt = Transpose(B);
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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(B, *MinvBt);
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invM = new DSmoother(M);
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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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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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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(darcyOp);
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solver.SetPreconditioner(darcyPrec);
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solver.SetPrintLevel(1);
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x = 0.0;
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solver.Mult(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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// 12. Create the grid functions u and p. Compute the L2 error norms.
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GridFunction u, p;
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u.MakeRef(R_space, x.GetBlock(0), 0);
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p.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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real_t err_u = u.ComputeL2Error(ucoeff, irs);
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real_t norm_u = ComputeLpNorm(2., ucoeff, *mesh, irs);
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real_t err_p = p.ComputeL2Error(pcoeff, irs);
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real_t norm_p = ComputeLpNorm(2., pcoeff, *mesh, irs);
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std::cout << "|| u_h - u_ex || / || u_ex || = " << err_u / norm_u << "\n";
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std::cout << "|| p_h - p_ex || / || p_ex || = " << err_p / norm_p << "\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_u.gf" or "glvis -m ex5.mesh -g
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// sol_p.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);
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ofstream u_ofs("sol_u.gf");
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u_ofs.precision(8);
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u.Save(u_ofs);
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ofstream p_ofs("sol_p.gf");
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p_ofs.precision(8);
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p.Save(p_ofs);
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}
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// 14. Save data in the VisIt format
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VisItDataCollection visit_dc("Example5", mesh);
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visit_dc.RegisterField("velocity", &u);
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visit_dc.RegisterField("pressure", &p);
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visit_dc.Save();
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// 15. Save data in the ParaView format
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ParaViewDataCollection paraview_dc("Example5", mesh);
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paraview_dc.SetPrefixPath("ParaView");
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paraview_dc.SetLevelsOfDetail(order);
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paraview_dc.SetCycle(0);
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paraview_dc.SetDataFormat(VTKFormat::BINARY);
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paraview_dc.SetHighOrderOutput(true);
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paraview_dc.SetTime(0.0); // set the time
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paraview_dc.RegisterField("velocity",&u);
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paraview_dc.RegisterField("pressure",&p);
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paraview_dc.Save();
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// 16. Send the solution by socket to a GLVis server.
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if (visualization)
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{
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char vishost[] = "localhost";
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int visport = 19916;
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socketstream u_sock(vishost, visport);
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u_sock.precision(8);
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u_sock << "solution\n" << *mesh << u << "window_title 'Velocity'" << endl;
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socketstream p_sock(vishost, visport);
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p_sock.precision(8);
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p_sock << "solution\n" << *mesh << p << "window_title 'Pressure'" << endl;
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}
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// 17. Free the used memory.
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delete fform;
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delete gform;
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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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delete mVarf;
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delete bVarf;
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delete W_space;
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delete R_space;
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delete l2_coll;
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delete hdiv_coll;
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delete mesh;
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return 0;
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}
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void uFun_ex(const Vector & x, Vector & u)
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{
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real_t xi(x(0));
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real_t yi(x(1));
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real_t zi(0.0);
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if (x.Size() == 3)
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{
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zi = x(2);
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}
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u(0) = - exp(xi)*sin(yi)*cos(zi);
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u(1) = - exp(xi)*cos(yi)*cos(zi);
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if (x.Size() == 3)
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{
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u(2) = exp(xi)*sin(yi)*sin(zi);
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}
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}
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// Change if needed
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real_t pFun_ex(const Vector & x)
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{
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real_t xi(x(0));
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real_t yi(x(1));
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real_t zi(0.0);
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if (x.Size() == 3)
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{
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zi = x(2);
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}
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return exp(xi)*sin(yi)*cos(zi);
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}
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void fFun(const Vector & x, Vector & f)
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{
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f = 0.0;
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}
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real_t gFun(const Vector & x)
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{
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if (x.Size() == 3)
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{
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return -pFun_ex(x);
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}
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else
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{
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return 0;
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}
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}
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real_t f_natural(const Vector & x)
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{
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return (-pFun_ex(x));
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}
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