756 lines
28 KiB
C++
756 lines
28 KiB
C++
// Copyright (c) 2010, Lawrence Livermore National Security, LLC. Produced at
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// the Lawrence Livermore National Laboratory. LLNL-CODE-443211. All Rights
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// reserved. See file COPYRIGHT for details.
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//
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// This file is part of the MFEM library. For more information and source code
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// availability see http://mfem.org.
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//
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// MFEM is free software; you can redistribute it and/or modify it under the
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// terms of the GNU Lesser General Public License (as published by the Free
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// Software Foundation) version 2.1 dated February 1999.
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//
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// --------------------------------------------------
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// Mesh Optimizer Miniapp: Optimize high-order meshes
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// --------------------------------------------------
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//
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// This miniapp performs mesh optimization using the Target-Matrix Optimization
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// Paradigm (TMOP) by P.Knupp et al., and a global variational minimization
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// approach. It minimizes the quantity sum_T int_T mu(J(x)), where T are the
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// target (ideal) elements, J is the Jacobian of the transformation from the
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// target to the physical element, and mu is the mesh quality metric. This
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// metric can measure shape, size or alignment of the region around each
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// quadrature point. The combination of targets & quality metrics is used to
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// optimize the physical node positions, i.e., they must be as close as possible
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// to the shape / size / alignment of their targets. This code also demonstrates
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// a possible use of nonlinear operators (the class TMOP_QualityMetric, defining
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// mu(J), and the class TMOP_Integrator, defining int mu(J)), as well as their
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// coupling to Newton methods for solving minimization problems. Note that the
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// utilized Newton methods are oriented towards avoiding invalid meshes with
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// negative Jacobian determinants. Each Newton step requires the inversion of a
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// Jacobian matrix, which is done through an inner linear solver.
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//
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// Compile with: make mesh-optimizer
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//
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// Sample runs:
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// Blade shape:
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// mesh-optimizer -m blade.mesh -o 4 -rs 0 -mid 2 -tid 1 -ni 200 -ls 2 -li 100 -bnd -qt 1 -qo 8
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// Blade limited shape:
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// mesh-optimizer -m blade.mesh -o 4 -rs 0 -mid 2 -tid 1 -ni 200 -ls 2 -li 100 -bnd -qt 1 -qo 8 -lc 5000
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// ICF shape and equal size:
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// mesh-optimizer -o 3 -rs 0 -mid 9 -tid 2 -ni 200 -ls 2 -li 100 -bnd -qt 1 -qo 8
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// ICF shape and initial size:
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// mesh-optimizer -o 3 -rs 0 -mid 9 -tid 3 -ni 100 -ls 2 -li 100 -bnd -qt 1 -qo 8
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// ICF shape:
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// mesh-optimizer -o 3 -rs 0 -mid 1 -tid 1 -ni 100 -ls 2 -li 100 -bnd -qt 1 -qo 8
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// ICF limited shape:
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// mesh-optimizer -o 3 -rs 0 -mid 1 -tid 1 -ni 100 -ls 2 -li 100 -bnd -qt 1 -qo 8 -lc 10
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// ICF combo shape + size (rings, slow convergence):
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// mesh-optimizer -o 3 -rs 0 -mid 1 -tid 1 -ni 1000 -ls 2 -li 100 -bnd -qt 1 -qo 8 -cmb
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// 3D pinched sphere shape (the mesh is in the mfem/data GitHub repository):
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// * mesh-optimizer -m ../../../mfem_data/ball-pert.mesh -o 4 -rs 0 -mid 303 -tid 1 -ni 20 -ls 2 -li 500 -fix-bnd
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#include "mfem.hpp"
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#include <fstream>
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#include <iostream>
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using namespace mfem;
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using namespace std;
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double weight_fun(const Vector &x);
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// Metric values are visualized by creating an L2 finite element functions and
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// computing the metric values at the nodes.
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void vis_metric(int order, TMOP_QualityMetric &qm, const TargetConstructor &tc,
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Mesh &mesh, char *title, int position)
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{
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L2_FECollection fec(order, mesh.Dimension(), BasisType::GaussLobatto);
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FiniteElementSpace fes(&mesh, &fec, 1);
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GridFunction metric(&fes);
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InterpolateTMOP_QualityMetric(qm, tc, mesh, metric);
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osockstream sock(19916, "localhost");
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sock << "solution\n";
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mesh.Print(sock);
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metric.Save(sock);
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sock.send();
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sock << "window_title '"<< title << "'\n"
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<< "window_geometry "
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<< position << " " << 0 << " " << 600 << " " << 600 << "\n"
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<< "keys jRmclA" << endl;
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}
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class RelaxedNewtonSolver : public NewtonSolver
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{
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private:
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// Quadrature points that are checked for negative Jacobians etc.
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const IntegrationRule &ir;
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FiniteElementSpace *fes;
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mutable GridFunction x_gf;
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public:
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RelaxedNewtonSolver(const IntegrationRule &irule, FiniteElementSpace *f)
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: ir(irule), fes(f) { }
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virtual double ComputeScalingFactor(const Vector &x, const Vector &b) const;
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};
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double RelaxedNewtonSolver::ComputeScalingFactor(const Vector &x,
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const Vector &b) const
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{
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const NonlinearForm *nlf = dynamic_cast<const NonlinearForm *>(oper);
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MFEM_VERIFY(nlf != NULL, "invalid Operator subclass");
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const bool have_b = (b.Size() == Height());
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const int NE = fes->GetMesh()->GetNE(), dim = fes->GetFE(0)->GetDim(),
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dof = fes->GetFE(0)->GetDof(), nsp = ir.GetNPoints();
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Array<int> xdofs(dof * dim);
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DenseMatrix Jpr(dim), dshape(dof, dim), pos(dof, dim);
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Vector posV(pos.Data(), dof * dim);
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Vector x_out(x.Size());
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bool x_out_ok = false;
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const double energy_in = nlf->GetEnergy(x);
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double scale = 1.0, energy_out;
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double norm0 = Norm(r);
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x_gf.MakeTRef(fes, x_out, 0);
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// Decreases the scaling of the update until the new mesh is valid.
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for (int i = 0; i < 12; i++)
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{
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add(x, -scale, c, x_out);
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x_gf.SetFromTrueVector();
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energy_out = nlf->GetGridFunctionEnergy(x_gf);
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if (energy_out > 1.2*energy_in || isnan(energy_out) != 0)
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{
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if (print_level >= 0)
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{ cout << "Scale = " << scale << " Increasing energy." << endl; }
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scale *= 0.5; continue;
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}
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int jac_ok = 1;
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for (int i = 0; i < NE; i++)
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{
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fes->GetElementVDofs(i, xdofs);
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x_gf.GetSubVector(xdofs, posV);
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for (int j = 0; j < nsp; j++)
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{
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fes->GetFE(i)->CalcDShape(ir.IntPoint(j), dshape);
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MultAtB(pos, dshape, Jpr);
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if (Jpr.Det() <= 0.0) { jac_ok = 0; goto break2; }
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}
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}
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break2:
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if (jac_ok == 0)
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{
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if (print_level >= 0)
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{ cout << "Scale = " << scale << " Neg det(J) found." << endl; }
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scale *= 0.5; continue;
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}
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oper->Mult(x_out, r);
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if (have_b) { r -= b; }
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double norm = Norm(r);
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if (norm > 1.2*norm0)
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{
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if (print_level >= 0)
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{ cout << "Scale = " << scale << " Norm increased." << endl; }
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scale *= 0.5; continue;
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}
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else { x_out_ok = true; break; }
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}
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if (print_level >= 0)
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{
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cout << "Energy decrease: "
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<< (energy_in - energy_out) / energy_in * 100.0
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<< "% with " << scale << " scaling." << endl;
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}
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if (x_out_ok == false) { scale = 0.0; }
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return scale;
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}
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// Allows negative Jacobians. Used in untangling metrics.
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class DescentNewtonSolver : public NewtonSolver
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{
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private:
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// Quadrature points that are checked for negative Jacobians etc.
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const IntegrationRule &ir;
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FiniteElementSpace *fes;
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mutable GridFunction x_gf;
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public:
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DescentNewtonSolver(const IntegrationRule &irule, FiniteElementSpace *f)
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: ir(irule), fes(f) { }
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virtual double ComputeScalingFactor(const Vector &x, const Vector &b) const;
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};
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double DescentNewtonSolver::ComputeScalingFactor(const Vector &x,
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const Vector &b) const
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{
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const NonlinearForm *nlf = dynamic_cast<const NonlinearForm *>(oper);
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MFEM_VERIFY(nlf != NULL, "invalid Operator subclass");
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const int NE = fes->GetMesh()->GetNE(), dim = fes->GetFE(0)->GetDim(),
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dof = fes->GetFE(0)->GetDof(), nsp = ir.GetNPoints();
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Array<int> xdofs(dof * dim);
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DenseMatrix Jpr(dim), dshape(dof, dim), pos(dof, dim);
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Vector posV(pos.Data(), dof * dim);
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x_gf.MakeTRef(fes, x.GetData());
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x_gf.SetFromTrueVector();
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double min_detJ = infinity();
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for (int i = 0; i < NE; i++)
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{
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fes->GetElementVDofs(i, xdofs);
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x_gf.GetSubVector(xdofs, posV);
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for (int j = 0; j < nsp; j++)
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{
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fes->GetFE(i)->CalcDShape(ir.IntPoint(j), dshape);
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MultAtB(pos, dshape, Jpr);
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min_detJ = min(min_detJ, Jpr.Det());
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}
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}
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cout << "Minimum det(J) = " << min_detJ << endl;
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Vector x_out(x.Size());
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bool x_out_ok = false;
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const double energy_in = nlf->GetGridFunctionEnergy(x_gf);
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double scale = 1.0, energy_out;
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for (int i = 0; i < 7; i++)
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{
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add(x, -scale, c, x_out);
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energy_out = nlf->GetEnergy(x_out);
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if (energy_out > energy_in || isnan(energy_out) != 0)
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{
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scale *= 0.5;
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}
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else { x_out_ok = true; break; }
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}
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cout << "Energy decrease: " << (energy_in - energy_out) / energy_in * 100.0
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<< "% with " << scale << " scaling." << endl;
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if (x_out_ok == false) { return 0.0;}
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return scale;
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}
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// Additional IntegrationRules that can be used with the --quad-type option.
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IntegrationRules IntRulesLo(0, Quadrature1D::GaussLobatto);
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IntegrationRules IntRulesCU(0, Quadrature1D::ClosedUniform);
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int main (int argc, char *argv[])
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{
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// 0. Set the method's default parameters.
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const char *mesh_file = "icf.mesh";
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int mesh_poly_deg = 1;
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int rs_levels = 0;
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double jitter = 0.0;
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int metric_id = 1;
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int target_id = 1;
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double lim_const = 0.0;
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int quad_type = 1;
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int quad_order = 8;
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int newton_iter = 10;
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double newton_rtol = 1e-12;
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int lin_solver = 2;
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int max_lin_iter = 100;
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bool move_bnd = true;
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bool combomet = 0;
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bool normalization = false;
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bool visualization = true;
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int verbosity_level = 0;
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// 1. Parse command-line options.
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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(&mesh_poly_deg, "-o", "--order",
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"Polynomial degree of mesh finite element space.");
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args.AddOption(&rs_levels, "-rs", "--refine-serial",
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"Number of times to refine the mesh uniformly in serial.");
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args.AddOption(&jitter, "-ji", "--jitter",
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"Random perturbation scaling factor.");
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args.AddOption(&metric_id, "-mid", "--metric-id",
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"Mesh optimization metric:\n\t"
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"1 : |T|^2 -- 2D shape\n\t"
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"2 : 0.5|T|^2/tau-1 -- 2D shape (condition number)\n\t"
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"7 : |T-T^-t|^2 -- 2D shape+size\n\t"
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"9 : tau*|T-T^-t|^2 -- 2D shape+size\n\t"
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"22 : 0.5(|T|^2-2*tau)/(tau-tau_0) -- 2D untangling\n\t"
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"50 : 0.5|T^tT|^2/tau^2-1 -- 2D shape\n\t"
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"55 : (tau-1)^2 -- 2D size\n\t"
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"56 : 0.5(sqrt(tau)-1/sqrt(tau))^2 -- 2D size\n\t"
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"58 : |T^tT|^2/(tau^2)-2*|T|^2/tau+2 -- 2D shape\n\t"
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"77 : 0.5(tau-1/tau)^2 -- 2D size\n\t"
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"211: (tau-1)^2-tau+sqrt(tau^2) -- 2D untangling\n\t"
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"252: 0.5(tau-1)^2/(tau-tau_0) -- 2D untangling\n\t"
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"301: (|T||T^-1|)/3-1 -- 3D shape\n\t"
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"302: (|T|^2|T^-1|^2)/9-1 -- 3D shape\n\t"
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"303: (|T|^2)/3*tau^(2/3)-1 -- 3D shape\n\t"
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"315: (tau-1)^2 -- 3D size\n\t"
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"316: 0.5(sqrt(tau)-1/sqrt(tau))^2 -- 3D size\n\t"
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"321: |T-T^-t|^2 -- 3D shape+size\n\t"
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"352: 0.5(tau-1)^2/(tau-tau_0) -- 3D untangling");
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args.AddOption(&target_id, "-tid", "--target-id",
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"Target (ideal element) type:\n\t"
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"1: Ideal shape, unit size\n\t"
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"2: Ideal shape, equal size\n\t"
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"3: Ideal shape, initial size");
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args.AddOption(&lim_const, "-lc", "--limit-const", "Limiting constant.");
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args.AddOption(&quad_type, "-qt", "--quad-type",
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"Quadrature rule type:\n\t"
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"1: Gauss-Lobatto\n\t"
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"2: Gauss-Legendre\n\t"
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"3: Closed uniform points");
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args.AddOption(&quad_order, "-qo", "--quad_order",
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"Order of the quadrature rule.");
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args.AddOption(&newton_iter, "-ni", "--newton-iters",
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"Maximum number of Newton iterations.");
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args.AddOption(&newton_rtol, "-rtol", "--newton-rel-tolerance",
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"Relative tolerance for the Newton solver.");
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args.AddOption(&lin_solver, "-ls", "--lin-solver",
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"Linear solver: 0 - l1-Jacobi, 1 - CG, 2 - MINRES.");
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args.AddOption(&max_lin_iter, "-li", "--lin-iter",
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"Maximum number of iterations in the linear solve.");
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args.AddOption(&move_bnd, "-bnd", "--move-boundary", "-fix-bnd",
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"--fix-boundary",
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"Enable motion along horizontal and vertical boundaries.");
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args.AddOption(&combomet, "-cmb", "--combo-met", "-no-cmb", "--no-combo-met",
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"Combination of metrics.");
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args.AddOption(&normalization, "-nor", "--normalization", "-no-nor",
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"--no-normalization",
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"Make all terms in the optimization functional unitless.");
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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(&verbosity_level, "-vl", "--verbosity-level",
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"Set the verbosity level - 0, 1, or 2.");
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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. Initialize and refine the starting mesh.
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Mesh *mesh = new Mesh(mesh_file, 1, 1, false);
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for (int lev = 0; lev < rs_levels; lev++) { mesh->UniformRefinement(); }
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const int dim = mesh->Dimension();
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cout << "Mesh curvature: ";
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if (mesh->GetNodes()) { cout << mesh->GetNodes()->OwnFEC()->Name(); }
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else { cout << "(NONE)"; }
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cout << endl;
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// 3. Define a finite element space on the mesh. Here we use vector finite
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// elements which are tensor products of quadratic finite elements. The
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// number of components in the vector finite element space is specified by
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// the last parameter of the FiniteElementSpace constructor.
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FiniteElementCollection *fec;
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if (mesh_poly_deg <= 0)
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{
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fec = new QuadraticPosFECollection;
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mesh_poly_deg = 2;
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}
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else { fec = new H1_FECollection(mesh_poly_deg, dim); }
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FiniteElementSpace *fespace = new FiniteElementSpace(mesh, fec, dim);
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// 4. Make the mesh curved based on the above finite element space. This
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// means that we define the mesh elements through a fespace-based
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// transformation of the reference element.
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mesh->SetNodalFESpace(fespace);
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// 5. Set up an empty right-hand side vector b, which is equivalent to b=0.
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Vector b(0);
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// 6. Get the mesh nodes (vertices and other degrees of freedom in the finite
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// element space) as a finite element grid function in fespace. Note that
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// changing x automatically changes the shapes of the mesh elements.
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GridFunction *x = mesh->GetNodes();
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// 7. Define a vector representing the minimal local mesh size in the mesh
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// nodes. We index the nodes using the scalar version of the degrees of
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// freedom in pfespace. Note: this is partition-dependent.
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//
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// In addition, compute average mesh size and total volume.
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Vector h0(fespace->GetNDofs());
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h0 = infinity();
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double volume = 0.0;
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Array<int> dofs;
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for (int i = 0; i < mesh->GetNE(); i++)
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{
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// Get the local scalar element degrees of freedom in dofs.
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fespace->GetElementDofs(i, dofs);
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// Adjust the value of h0 in dofs based on the local mesh size.
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const double hi = mesh->GetElementSize(i);
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for (int j = 0; j < dofs.Size(); j++)
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{
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h0(dofs[j]) = min(h0(dofs[j]), hi);
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}
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volume += mesh->GetElementVolume(i);
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}
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const double small_phys_size = pow(volume, 1.0 / dim) / 100.0;
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// 8. Add a random perturbation to the nodes in the interior of the domain.
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// We define a random grid function of fespace and make sure that it is
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// zero on the boundary and its values are locally of the order of h0.
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// The latter is based on the DofToVDof() method which maps the scalar to
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// the vector degrees of freedom in fespace.
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GridFunction rdm(fespace);
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rdm.Randomize();
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rdm -= 0.25; // Shift to random values in [-0.5,0.5].
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rdm *= jitter;
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// Scale the random values to be of order of the local mesh size.
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for (int i = 0; i < fespace->GetNDofs(); i++)
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{
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for (int d = 0; d < dim; d++)
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{
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rdm(fespace->DofToVDof(i,d)) *= h0(i);
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}
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}
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Array<int> vdofs;
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for (int i = 0; i < fespace->GetNBE(); i++)
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{
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// Get the vector degrees of freedom in the boundary element.
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fespace->GetBdrElementVDofs(i, vdofs);
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// Set the boundary values to zero.
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for (int j = 0; j < vdofs.Size(); j++) { rdm(vdofs[j]) = 0.0; }
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}
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*x -= rdm;
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// Set the perturbation of all nodes from the true nodes.
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x->SetTrueVector();
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x->SetFromTrueVector();
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// 9. Save the starting (prior to the optimization) mesh to a file. This
|
|
// output can be viewed later using GLVis: "glvis -m perturbed.mesh".
|
|
{
|
|
ofstream mesh_ofs("perturbed.mesh");
|
|
mesh->Print(mesh_ofs);
|
|
}
|
|
|
|
// 10. Store the starting (prior to the optimization) positions.
|
|
GridFunction x0(fespace);
|
|
x0 = *x;
|
|
|
|
// 11. Form the integrator that uses the chosen metric and target.
|
|
double tauval = -0.1;
|
|
TMOP_QualityMetric *metric = NULL;
|
|
switch (metric_id)
|
|
{
|
|
case 1: metric = new TMOP_Metric_001; break;
|
|
case 2: metric = new TMOP_Metric_002; break;
|
|
case 7: metric = new TMOP_Metric_007; break;
|
|
case 9: metric = new TMOP_Metric_009; break;
|
|
case 22: metric = new TMOP_Metric_022(tauval); break;
|
|
case 50: metric = new TMOP_Metric_050; break;
|
|
case 55: metric = new TMOP_Metric_055; break;
|
|
case 56: metric = new TMOP_Metric_056; break;
|
|
case 58: metric = new TMOP_Metric_058; break;
|
|
case 77: metric = new TMOP_Metric_077; break;
|
|
case 211: metric = new TMOP_Metric_211; break;
|
|
case 252: metric = new TMOP_Metric_252(tauval); break;
|
|
case 301: metric = new TMOP_Metric_301; break;
|
|
case 302: metric = new TMOP_Metric_302; break;
|
|
case 303: metric = new TMOP_Metric_303; break;
|
|
case 315: metric = new TMOP_Metric_315; break;
|
|
case 316: metric = new TMOP_Metric_316; break;
|
|
case 321: metric = new TMOP_Metric_321; break;
|
|
case 352: metric = new TMOP_Metric_352(tauval); break;
|
|
default: cout << "Unknown metric_id: " << metric_id << endl; return 3;
|
|
}
|
|
TargetConstructor::TargetType target_t;
|
|
switch (target_id)
|
|
{
|
|
case 1: target_t = TargetConstructor::IDEAL_SHAPE_UNIT_SIZE; break;
|
|
case 2: target_t = TargetConstructor::IDEAL_SHAPE_EQUAL_SIZE; break;
|
|
case 3: target_t = TargetConstructor::IDEAL_SHAPE_GIVEN_SIZE; break;
|
|
default: cout << "Unknown target_id: " << target_id << endl;
|
|
delete metric; return 3;
|
|
}
|
|
TargetConstructor *target_c = new TargetConstructor(target_t);
|
|
target_c->SetNodes(x0);
|
|
TMOP_Integrator *he_nlf_integ = new TMOP_Integrator(metric, target_c);
|
|
|
|
// 12. Setup the quadrature rule for the non-linear form integrator.
|
|
const IntegrationRule *ir = NULL;
|
|
const int geom_type = fespace->GetFE(0)->GetGeomType();
|
|
switch (quad_type)
|
|
{
|
|
case 1: ir = &IntRulesLo.Get(geom_type, quad_order); break;
|
|
case 2: ir = &IntRules.Get(geom_type, quad_order); break;
|
|
case 3: ir = &IntRulesCU.Get(geom_type, quad_order); break;
|
|
default: cout << "Unknown quad_type: " << quad_type << endl;
|
|
delete he_nlf_integ; return 3;
|
|
}
|
|
cout << "Quadrature points per cell: " << ir->GetNPoints() << endl;
|
|
he_nlf_integ->SetIntegrationRule(*ir);
|
|
|
|
if (normalization) { he_nlf_integ->EnableNormalization(x0); }
|
|
|
|
// 13. Limit the node movement.
|
|
// The limiting distances can be given by a general function of space.
|
|
GridFunction dist(fespace);
|
|
dist = 1.0;
|
|
// The small_phys_size is relevant only with proper normalization.
|
|
if (normalization) { dist = small_phys_size; }
|
|
ConstantCoefficient lim_coeff(lim_const);
|
|
if (lim_const != 0.0) { he_nlf_integ->EnableLimiting(x0, dist, lim_coeff); }
|
|
|
|
// 14. Setup the final NonlinearForm (which defines the integral of interest,
|
|
// its first and second derivatives). Here we can use a combination of
|
|
// metrics, i.e., optimize the sum of two integrals, where both are
|
|
// scaled by used-defined space-dependent weights. Note that there are no
|
|
// command-line options for the weights and the type of the second
|
|
// metric; one should update those in the code.
|
|
NonlinearForm a(fespace);
|
|
ConstantCoefficient *coeff1 = NULL;
|
|
TMOP_QualityMetric *metric2 = NULL;
|
|
TargetConstructor *target_c2 = NULL;
|
|
FunctionCoefficient coeff2(weight_fun);
|
|
|
|
if (combomet == 1)
|
|
{
|
|
// TODO normalization of combinations.
|
|
// We will probably drop this example and replace it with adaptivity.
|
|
if (normalization) { MFEM_ABORT("Not implemented."); }
|
|
|
|
// Weight of the original metric.
|
|
coeff1 = new ConstantCoefficient(1.0);
|
|
he_nlf_integ->SetCoefficient(*coeff1);
|
|
a.AddDomainIntegrator(he_nlf_integ);
|
|
|
|
metric2 = new TMOP_Metric_077;
|
|
target_c2 = new TargetConstructor(
|
|
TargetConstructor::IDEAL_SHAPE_EQUAL_SIZE);
|
|
target_c2->SetVolumeScale(0.01);
|
|
target_c2->SetNodes(x0);
|
|
TMOP_Integrator *he_nlf_integ2 = new TMOP_Integrator(metric2, target_c2);
|
|
he_nlf_integ2->SetIntegrationRule(*ir);
|
|
|
|
// Weight of metric2.
|
|
he_nlf_integ2->SetCoefficient(coeff2);
|
|
a.AddDomainIntegrator(he_nlf_integ2);
|
|
}
|
|
else { a.AddDomainIntegrator(he_nlf_integ); }
|
|
|
|
const double init_energy = a.GetGridFunctionEnergy(*x);
|
|
|
|
// 15. Visualize the starting mesh and metric values.
|
|
if (visualization)
|
|
{
|
|
char title[] = "Initial metric values";
|
|
vis_metric(mesh_poly_deg, *metric, *target_c, *mesh, title, 0);
|
|
}
|
|
|
|
// 16. Fix all boundary nodes, or fix only a given component depending on the
|
|
// boundary attributes of the given mesh. Attributes 1/2/3 correspond to
|
|
// fixed x/y/z components of the node. Attribute 4 corresponds to an
|
|
// entirely fixed node. Other boundary attributes do not affect the node
|
|
// movement boundary conditions.
|
|
if (move_bnd == false)
|
|
{
|
|
Array<int> ess_bdr(mesh->bdr_attributes.Max());
|
|
ess_bdr = 1;
|
|
a.SetEssentialBC(ess_bdr);
|
|
}
|
|
else
|
|
{
|
|
const int nd = fespace->GetBE(0)->GetDof();
|
|
int n = 0;
|
|
for (int i = 0; i < mesh->GetNBE(); i++)
|
|
{
|
|
const int attr = mesh->GetBdrElement(i)->GetAttribute();
|
|
MFEM_VERIFY(!(dim == 2 && attr == 3),
|
|
"Boundary attribute 3 must be used only for 3D meshes. "
|
|
"Adjust the attributes (1/2/3/4 for fixed x/y/z/all "
|
|
"components, rest for free nodes), or use -fix-bnd.");
|
|
if (attr == 1 || attr == 2 || attr == 3) { n += nd; }
|
|
if (attr == 4) { n += nd * dim; }
|
|
}
|
|
Array<int> ess_vdofs(n), vdofs;
|
|
n = 0;
|
|
for (int i = 0; i < mesh->GetNBE(); i++)
|
|
{
|
|
const int attr = mesh->GetBdrElement(i)->GetAttribute();
|
|
fespace->GetBdrElementVDofs(i, vdofs);
|
|
if (attr == 1) // Fix x components.
|
|
{
|
|
for (int j = 0; j < nd; j++)
|
|
{ ess_vdofs[n++] = vdofs[j]; }
|
|
}
|
|
else if (attr == 2) // Fix y components.
|
|
{
|
|
for (int j = 0; j < nd; j++)
|
|
{ ess_vdofs[n++] = vdofs[j+nd]; }
|
|
}
|
|
else if (attr == 3) // Fix z components.
|
|
{
|
|
for (int j = 0; j < nd; j++)
|
|
{ ess_vdofs[n++] = vdofs[j+2*nd]; }
|
|
}
|
|
else if (attr == 4) // Fix all components.
|
|
{
|
|
for (int j = 0; j < vdofs.Size(); j++)
|
|
{ ess_vdofs[n++] = vdofs[j]; }
|
|
}
|
|
}
|
|
a.SetEssentialVDofs(ess_vdofs);
|
|
}
|
|
|
|
// 17. As we use the Newton method to solve the resulting nonlinear system,
|
|
// here we setup the linear solver for the system's Jacobian.
|
|
Solver *S = NULL;
|
|
const double linsol_rtol = 1e-12;
|
|
if (lin_solver == 0)
|
|
{
|
|
S = new DSmoother(1, 1.0, max_lin_iter);
|
|
}
|
|
else if (lin_solver == 1)
|
|
{
|
|
CGSolver *cg = new CGSolver;
|
|
cg->SetMaxIter(max_lin_iter);
|
|
cg->SetRelTol(linsol_rtol);
|
|
cg->SetAbsTol(0.0);
|
|
cg->SetPrintLevel(verbosity_level >= 2 ? 3 : -1);
|
|
S = cg;
|
|
}
|
|
else
|
|
{
|
|
MINRESSolver *minres = new MINRESSolver;
|
|
minres->SetMaxIter(max_lin_iter);
|
|
minres->SetRelTol(linsol_rtol);
|
|
minres->SetAbsTol(0.0);
|
|
minres->SetPrintLevel(verbosity_level >= 2 ? 3 : -1);
|
|
S = minres;
|
|
}
|
|
|
|
// 18. Compute the minimum det(J) of the starting mesh.
|
|
tauval = infinity();
|
|
const int NE = mesh->GetNE();
|
|
for (int i = 0; i < NE; i++)
|
|
{
|
|
ElementTransformation *transf = mesh->GetElementTransformation(i);
|
|
for (int j = 0; j < ir->GetNPoints(); j++)
|
|
{
|
|
transf->SetIntPoint(&ir->IntPoint(j));
|
|
tauval = min(tauval, transf->Jacobian().Det());
|
|
}
|
|
}
|
|
cout << "Minimum det(J) of the original mesh is " << tauval << endl;
|
|
|
|
// 19. Finally, perform the nonlinear optimization.
|
|
NewtonSolver *newton = NULL;
|
|
if (tauval > 0.0)
|
|
{
|
|
tauval = 0.0;
|
|
newton = new RelaxedNewtonSolver(*ir, fespace);
|
|
cout << "The RelaxedNewtonSolver is used (as all det(J)>0)." << endl;
|
|
}
|
|
else
|
|
{
|
|
if ( (dim == 2 && metric_id != 22 && metric_id != 252) ||
|
|
(dim == 3 && metric_id != 352) )
|
|
{
|
|
cout << "The mesh is inverted. Use an untangling metric." << endl;
|
|
return 3;
|
|
}
|
|
tauval -= 0.01 * h0.Min(); // Slightly below minJ0 to avoid div by 0.
|
|
newton = new DescentNewtonSolver(*ir, fespace);
|
|
cout << "The DescentNewtonSolver is used (as some det(J)<0)." << endl;
|
|
}
|
|
newton->SetPreconditioner(*S);
|
|
newton->SetMaxIter(newton_iter);
|
|
newton->SetRelTol(newton_rtol);
|
|
newton->SetAbsTol(0.0);
|
|
newton->SetPrintLevel(verbosity_level >= 1 ? 1 : -1);
|
|
newton->SetOperator(a);
|
|
newton->Mult(b, x->GetTrueVector());
|
|
x->SetFromTrueVector();
|
|
if (newton->GetConverged() == false)
|
|
{
|
|
cout << "NewtonIteration: rtol = " << newton_rtol << " not achieved."
|
|
<< endl;
|
|
}
|
|
delete newton;
|
|
|
|
// 20. Save the optimized mesh to a file. This output can be viewed later
|
|
// using GLVis: "glvis -m optimized.mesh".
|
|
{
|
|
ofstream mesh_ofs("optimized.mesh");
|
|
mesh_ofs.precision(14);
|
|
mesh->Print(mesh_ofs);
|
|
}
|
|
|
|
// 21. Compute the amount of energy decrease.
|
|
const double fin_energy = a.GetGridFunctionEnergy(*x);
|
|
double metric_part = fin_energy;
|
|
if (lim_const != 0.0)
|
|
{
|
|
lim_coeff.constant = 0.0;
|
|
metric_part = a.GetGridFunctionEnergy(*x);
|
|
lim_coeff.constant = lim_const;
|
|
}
|
|
cout << "Initial strain energy: " << init_energy
|
|
<< " = metrics: " << init_energy
|
|
<< " + limiting term: " << 0.0 << endl;
|
|
cout << " Final strain energy: " << fin_energy
|
|
<< " = metrics: " << metric_part
|
|
<< " + limiting term: " << fin_energy - metric_part << endl;
|
|
cout << "The strain energy decreased by: " << setprecision(12)
|
|
<< (init_energy - fin_energy) * 100.0 / init_energy << " %." << endl;
|
|
|
|
// 22. Visualize the final mesh and metric values.
|
|
if (visualization)
|
|
{
|
|
char title[] = "Final metric values";
|
|
vis_metric(mesh_poly_deg, *metric, *target_c, *mesh, title, 600);
|
|
}
|
|
|
|
// 23. Visualize the mesh displacement.
|
|
if (visualization)
|
|
{
|
|
x0 -= *x;
|
|
osockstream sock(19916, "localhost");
|
|
sock << "solution\n";
|
|
mesh->Print(sock);
|
|
x0.Save(sock);
|
|
sock.send();
|
|
sock << "window_title 'Displacements'\n"
|
|
<< "window_geometry "
|
|
<< 1200 << " " << 0 << " " << 600 << " " << 600 << "\n"
|
|
<< "keys jRmclA" << endl;
|
|
}
|
|
|
|
// 24. Free the used memory.
|
|
delete S;
|
|
delete target_c2;
|
|
delete metric2;
|
|
delete coeff1;
|
|
delete target_c;
|
|
delete metric;
|
|
delete fespace;
|
|
delete fec;
|
|
delete mesh;
|
|
|
|
return 0;
|
|
}
|
|
|
|
// Defined with respect to the icf mesh.
|
|
double weight_fun(const Vector &x)
|
|
{
|
|
const double r = sqrt(x(0)*x(0) + x(1)*x(1) + 1e-12);
|
|
const double den = 0.002;
|
|
double l2 = 0.2 + 0.5*std::tanh((r-0.16)/den) - 0.5*std::tanh((r-0.17)/den)
|
|
+ 0.5*std::tanh((r-0.23)/den) - 0.5*std::tanh((r-0.24)/den);
|
|
return l2;
|
|
}
|