seqheat.cpp

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// Copyright (c) 2010-2025, Lawrence Livermore National Security, LLC. Produced
// at the Lawrence Livermore National Laboratory. All Rights reserved. See files
// LICENSE and NOTICE for details. LLNL-CODE-806117.
//
// This file is part of the MFEM library. For more information and source code
// availability visit https://mfem.org.
//
// MFEM is free software; you can redistribute it and/or modify it under the
// terms of the BSD-3 license. We welcome feedback and contributions, see file
// CONTRIBUTING.md for details.
//
//     ----------------------------------------------------------------
//     SeqHeat Miniapp: Gradients of PDE constrained objective function
//     ----------------------------------------------------------------
//                          (Sequential Version)
//
// The following example computes the gradients of a specified objective
// function with respect to parametric fields. The objective function is having
// the following form f(u(\rho)) where u(\rho) is a solution of a specific state
// problem (in the example that is the diffusion equation), and \rho is a
// parametric field discretized by finite elements. The parametric field (also
// called density in topology optimization) controls the coefficients of the
// state equation. For the considered case, the density controls the diffusion
// coefficient within the computational domain.
//
// For more information, the users are referred to:
//
//    Hinze, M.; Pinnau, R.; Ulbrich, M. & Ulbrich, S.
//    Optimization with PDE Constraints
//    Springer Netherlands, 2009
//
//    Bendsøe, M. P. & Sigmund, O.
//    Topology Optimization - Theory, Methods and Applications
//    Springer Verlag, Berlin Heidelberg, 2003
//
// Compile with: make seqheat
//
// Sample runs:
//
//    seqheat -m ../../data/star-mixed.mesh
//    seqheat --visualization
 
#include "mfem.hpp"
#include <fstream>
#include <iostream>
 
#include "mtop_integrators.hpp"
 
using namespace mfem;
 
int main(int argc, char *argv[])
{
   const char *mesh_file = "../../data/star.vtk";
   int ser_ref_levels = 1;
   int order = 2;
   bool visualization = false;
   real_t newton_rel_tol = 1e-4;
   real_t newton_abs_tol = 1e-6;
   int newton_iter = 10;
   int print_level = 0;
 
   mfem::OptionsParser args(argc, argv);
   args.AddOption(&mesh_file, "-m", "--mesh", "Mesh file to use.");
   args.AddOption(&ser_ref_levels,
                  "-rs",
                  "--refine-serial",
                  "Number of times to refine the mesh uniformly in serial.");
   args.AddOption(&order,
                  "-o",
                  "--order",
                  "Order (degree) of the finite elements.");
   args.AddOption(&visualization,
                  "-vis",
                  "--visualization",
                  "-no-vis",
                  "--no-visualization",
                  "Enable or disable GLVis visualization.");
   args.AddOption(&newton_rel_tol,
                  "-rel",
                  "--relative-tolerance",
                  "Relative tolerance for the Newton solve.");
   args.AddOption(&newton_abs_tol,
                  "-abs",
                  "--absolute-tolerance",
                  "Absolute tolerance for the Newton solve.");
   args.AddOption(&newton_iter,
                  "-it",
                  "--newton-iterations",
                  "Maximum iterations for the Newton solve.");
   args.Parse();
   if (!args.Good())
   {
      args.PrintUsage(std::cout);
      return 1;
   }
   args.PrintOptions(std::cout);
 
   // Read the (serial) mesh from the given mesh file on all processors. We
   // can handle triangular, quadrilateral, tetrahedral and hexahedral meshes
   // with the same code.
   mfem::Mesh *mesh = new mfem::Mesh(mesh_file, 1, 1);
   int dim = mesh->Dimension();
 
   // Refine the mesh in serial to increase the resolution. In this example
   // we do 'ser_ref_levels' of uniform refinement, where 'ser_ref_levels' is
   // a command-line parameter.
   for (int lev = 0; lev < ser_ref_levels; lev++)
   {
      mesh->UniformRefinement();
   }
 
   // Diffusion coefficient
   mfem::ConstantCoefficient* diffco=new mfem::ConstantCoefficient(1.0);
   // Heat source
   mfem::ConstantCoefficient* loadco=new mfem::ConstantCoefficient(1.0);
   // Define the q-function
   mfem::QLinearDiffusion* qfun=new mfem::QLinearDiffusion(*diffco,*loadco,1.0,
                                                           1e-7,4.0,0.5);
 
   // Define FE collection and space for the state solution
   mfem::H1_FECollection sfec(order, dim);
   mfem::FiniteElementSpace* sfes=new mfem::FiniteElementSpace(mesh,&sfec,1);
   // Define FE collection and space for the density field
   mfem::L2_FECollection pfec(order, dim);
   mfem::FiniteElementSpace* pfes=new mfem::FiniteElementSpace(mesh,&pfec,1);
 
   // Define the arrays for the nonlinear form
   mfem::Array<mfem::FiniteElementSpace*> asfes;
   mfem::Array<mfem::FiniteElementSpace*> apfes;
 
   asfes.Append(sfes);
   apfes.Append(pfes);
   // Define parametric block nonlinear form using single scalar H1 field
   // and L2 scalar density field
   mfem::ParametricBNLForm* nf=new mfem::ParametricBNLForm(asfes,apfes);
   // Add the parametric integrator
   nf->AddDomainIntegrator(new mfem::ParametricLinearDiffusion(*qfun));
 
   // Define true block vectors for state, adjoint, residual
   mfem::BlockVector solbv; solbv.Update(nf->GetBlockTrueOffsets());    solbv=0.0;
   mfem::BlockVector adjbv; adjbv.Update(nf->GetBlockTrueOffsets());    adjbv=0.0;
   mfem::BlockVector resbv; resbv.Update(nf->GetBlockTrueOffsets());    resbv=0.0;
   // Define true block vectors for parametric field and gradients
   mfem::BlockVector prmbv; prmbv.Update(nf->ParamGetBlockTrueOffsets());
   prmbv=0.0;
   mfem::BlockVector grdbv; grdbv.Update(nf->ParamGetBlockTrueOffsets());
   grdbv=0.0;
 
   // Set the BC for the physics
   mfem::Array<mfem::Array<int> *> ess_bdr;
   mfem::Array<mfem::Vector*>      ess_rhs;
   ess_bdr.Append(new mfem::Array<int>(mesh->bdr_attributes.Max()));
   ess_rhs.Append(nullptr);
   (*ess_bdr[0]) = 1;
   nf->SetEssentialBC(ess_bdr,ess_rhs);
   delete ess_bdr[0];
 
   // Define the linear solvers
   mfem::GMRESSolver *gmres;
   gmres = new mfem::GMRESSolver();
   gmres->SetAbsTol(newton_abs_tol/10);
   gmres->SetRelTol(newton_rel_tol/10);
   gmres->SetMaxIter(300);
   gmres->SetPrintLevel(print_level);
 
   // Define the Newton solver
   mfem::NewtonSolver *ns;
   ns = new mfem::NewtonSolver();
   ns->iterative_mode = true;
   ns->SetSolver(*gmres);
   ns->SetOperator(*nf);
   ns->SetPrintLevel(print_level);
   ns->SetRelTol(newton_rel_tol);
   ns->SetAbsTol(newton_abs_tol);
   ns->SetMaxIter(newton_iter);
 
   // Solve the problem
   // Set the density to 0.5
   prmbv=0.5;
   nf->SetParamFields(prmbv); // Set the density
   // Define the RHS
   mfem::Vector b;
   solbv=0.0;
   // Newton solve
   ns->Mult(b, solbv);
 
   // Compute the residual
   nf->Mult(solbv,resbv);
   std::cout<<"Norm residual="<<resbv.Norml2()<<std::endl;
 
   // Compute the energy of the state system
   real_t energy = nf->GetEnergy(solbv);
   std::cout<<"energy ="<< energy<<std::endl;
 
   // Define the block nonlinear form utilized for representing the
   // objective. The input is the state array asfes defined earlier.
   mfem::BlockNonlinearForm* ob=new mfem::BlockNonlinearForm(asfes);
 
   // Add the integrator for the objective
   ob->AddDomainIntegrator(new mfem::DiffusionObjIntegrator());
 
   // Compute the objective
   real_t obj=ob->GetEnergy(solbv);
   std::cout<<"Objective ="<<obj<<std::endl;
 
   // Solve the adjoint
   {
      mfem::BlockVector adjrhs; adjrhs.Update(nf->GetBlockTrueOffsets());  adjrhs=0.0;
      // Compute the RHS for the adjoint
      ob->Mult(solbv, adjrhs);
      // Get the tangent matrix from the state problem
      mfem::BlockOperator& A=nf->GetGradient(solbv);
      // We do not need to transpose the operator for diffusion
      gmres->SetOperator(A.GetBlock(0,0));
      // Compute the adjoint solution
      gmres->Mult(adjrhs.GetBlock(0), adjbv.GetBlock(0));
   }
 
   // Compute gradients
   nf->SetAdjointFields(adjbv);
   nf->SetStateFields(solbv);
   nf->ParamMult(prmbv, grdbv);
 
   // Dump out the data
   if (visualization)
   {
      mfem::ParaViewDataCollection *dacol=new mfem::ParaViewDataCollection("SeqHeat",
                                                                           mesh);
      mfem::GridFunction gfgrd(pfes); gfgrd.SetFromTrueDofs(grdbv.GetBlock(0));
      mfem::GridFunction gfdns(pfes); gfdns.SetFromTrueDofs(prmbv.GetBlock(0));
      // Define state grid function
      mfem::GridFunction gfsol(sfes); gfsol.SetFromTrueDofs(solbv.GetBlock(0));
      mfem::GridFunction gfadj(sfes); gfadj.SetFromTrueDofs(adjbv.GetBlock(0));
 
      dacol->SetLevelsOfDetail(order);
      dacol->RegisterField("sol", &gfsol);
      dacol->RegisterField("adj", &gfadj);
      dacol->RegisterField("dns", &gfdns);
      dacol->RegisterField("grd", &gfgrd);
 
      dacol->SetTime(1.0);
      dacol->SetCycle(1);
      dacol->Save();
 
      delete dacol;
   }
 
   // FD check
   {
      // Perturbation vector
      mfem::BlockVector prtbv;
      mfem::BlockVector tmpbv;
      prtbv.Update(nf->ParamGetBlockTrueOffsets());
      tmpbv.Update(nf->ParamGetBlockTrueOffsets());
      // Generate the perturbation
      prtbv.GetBlock(0).Randomize();
      prtbv*=1.0;
      // Scaling parameter
      real_t lsc=1.0;
 
      // Compute initial objective
      real_t gQoI=ob->GetEnergy(solbv);
      real_t lQoI;
 
      // Norm of the perturbation
      real_t nd=mfem::InnerProduct(prtbv,prtbv);
      // Projection of the adjoint gradient on the perturbation
      real_t td=mfem::InnerProduct(prtbv,grdbv);
      // Normalize the directional derivative
      td=td/nd;
 
      for (int l = 0; l < 10; l++)
      {
         lsc/=10.0;
         // Scale the perturbation
         prtbv/=10.0;
         // Add the perturbation to the original density
         add(prmbv,prtbv,tmpbv);
         nf->SetParamFields(tmpbv);
         // Solve the physics
         ns->Mult(b,solbv);
         // Compute the objective
         lQoI=ob->GetEnergy(solbv);
         // FD approximation
         real_t ld=(lQoI-gQoI)/lsc;
         std::cout << "dx=" << lsc << " FD gradient=" << ld/nd
                   << " adjoint gradient=" << td
                   << " err=" << std::fabs(ld/nd-td) << std::endl;
      }
   }
 
   delete ob;
 
   delete ns;
   delete gmres;
 
   delete nf;
   delete pfes;
   delete sfes;
 
   delete qfun;
   delete loadco;
   delete diffco;
 
   delete mesh;
 
   return 0;
}