ex36.cpp

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//                                MFEM Example 36
//
// Compile with: make ex36
//
// Sample runs: ex36 -o 2
//              ex36 -o 2 -r 4
//
// Description: This example code demonstrates the use of MFEM to solve the
//              bound-constrained energy minimization problem
//
//                      minimize ||∇u||² subject to u ≥ ϕ in H¹₀.
//
//              This is known as the obstacle problem, and it is a simple
//              mathematical model for contact mechanics.
//
//              In this example, the obstacle ϕ is a half-sphere centered
//              at the origin of a circular domain Ω. After solving to a
//              specified tolerance, the numerical solution is compared to
//              a closed-form exact solution to assess accuracy.
//
//              The problem is discretized and solved using the proximal
//              Galerkin finite element method, introduced by Keith and
//              Surowiec [1].
//
//              This example highlights the ability of MFEM to deliver high-
//              order solutions to variation inequality problems and
//              showcases how to set up and solve nonlinear mixed methods.
//
// [1] Keith, B. and Surowiec, T. (2023) Proximal Galerkin: A structure-
//     preserving finite element method for pointwise bound constraints.
//     arXiv:2307.12444 [math.NA]
 
#include "mfem.hpp"
#include <fstream>
#include <iostream>
 
using namespace std;
using namespace mfem;
 
real_t spherical_obstacle(const Vector &pt);
real_t exact_solution_obstacle(const Vector &pt);
void exact_solution_gradient_obstacle(const Vector &pt, Vector &grad);
 
class LogarithmGridFunctionCoefficient : public Coefficient
{
protected:
   GridFunction *u; // grid function
   Coefficient *obstacle;
   real_t min_val;
 
public:
   LogarithmGridFunctionCoefficient(GridFunction &u_, Coefficient &obst_,
                                    real_t min_val_=-36)
      : u(&u_), obstacle(&obst_), min_val(min_val_) { }
 
   virtual real_t Eval(ElementTransformation &T, const IntegrationPoint &ip);
};
 
class ExponentialGridFunctionCoefficient : public Coefficient
{
protected:
   GridFunction *u;
   Coefficient *obstacle;
   real_t min_val;
   real_t max_val;
 
public:
   ExponentialGridFunctionCoefficient(GridFunction &u_, Coefficient &obst_,
                                      real_t min_val_=0.0, real_t max_val_=1e6)
      : u(&u_), obstacle(&obst_), min_val(min_val_), max_val(max_val_) { }
 
   virtual real_t Eval(ElementTransformation &T, const IntegrationPoint &ip);
};
 
int main(int argc, char *argv[])
{
   // 1. Parse command-line options.
   int order = 1;
   int max_it = 10;
   int ref_levels = 3;
   real_t alpha = 1.0;
   real_t tol = 1e-5;
   bool visualization = true;
 
   OptionsParser args(argc, argv);
   args.AddOption(&order, "-o", "--order",
                  "Finite element order (polynomial degree).");
   args.AddOption(&ref_levels, "-r", "--refs",
                  "Number of h-refinements.");
   args.AddOption(&max_it, "-mi", "--max-it",
                  "Maximum number of iterations");
   args.AddOption(&tol, "-tol", "--tol",
                  "Stopping criteria based on the difference between"
                  "successive solution updates");
   args.AddOption(&alpha, "-step", "--step",
                  "Step size alpha");
   args.AddOption(&visualization, "-vis", "--visualization", "-no-vis",
                  "--no-visualization",
                  "Enable or disable GLVis visualization.");
   args.Parse();
   if (!args.Good())
   {
      args.PrintUsage(cout);
      return 1;
   }
   args.PrintOptions(cout);
 
   // 2. Read the mesh from the mesh file.
   const char *mesh_file = "../data/disc-nurbs.mesh";
   Mesh mesh(mesh_file, 1, 1);
   int dim = mesh.Dimension();
 
   // 3. Postprocess the mesh.
   // 3A. Refine the mesh to increase the resolution.
   for (int l = 0; l < ref_levels; l++)
   {
      mesh.UniformRefinement();
   }
 
   // 3B. Interpolate the geometry after refinement to control geometry error.
   // NOTE: Minimum second-order interpolation is used to improve the accuracy.
   int curvature_order = max(order,2);
   mesh.SetCurvature(curvature_order);
 
   // 3C. Rescale the domain to a unit circle (radius = 1).
   GridFunction *nodes = mesh.GetNodes();
   real_t scale = 2*sqrt(2);
   *nodes /= scale;
 
   // 4. Define the necessary finite element spaces on the mesh.
   H1_FECollection H1fec(order+1, dim);
   FiniteElementSpace H1fes(&mesh, &H1fec);
 
   L2_FECollection L2fec(order-1, dim);
   FiniteElementSpace L2fes(&mesh, &L2fec);
 
   cout << "Number of H1 finite element unknowns: "
        << H1fes.GetTrueVSize() << endl;
   cout << "Number of L2 finite element unknowns: "
        << L2fes.GetTrueVSize() << endl;
 
   Array<int> offsets(3);
   offsets[0] = 0;
   offsets[1] = H1fes.GetVSize();
   offsets[2] = L2fes.GetVSize();
   offsets.PartialSum();
 
   BlockVector x(offsets), rhs(offsets);
   x = 0.0; rhs = 0.0;
 
   // 5. Determine the list of true (i.e. conforming) essential boundary dofs.
   Array<int> ess_bdr;
   if (mesh.bdr_attributes.Size())
   {
      ess_bdr.SetSize(mesh.bdr_attributes.Max());
      ess_bdr = 1;
   }
 
   // 6. Define an initial guess for the solution.
   auto IC_func = [](const Vector &x)
   {
      real_t r0 = 1.0;
      real_t rr = 0.0;
      for (int i=0; i<x.Size(); i++)
      {
         rr += x(i)*x(i);
      }
      return r0*r0 - rr;
   };
   ConstantCoefficient one(1.0);
   ConstantCoefficient zero(0.0);
 
   // 7. Define the solution vectors as a finite element grid functions
   //    corresponding to the fespaces.
   GridFunction u_gf, delta_psi_gf;
 
   u_gf.MakeRef(&H1fes,x,offsets[0]);
   delta_psi_gf.MakeRef(&L2fes,x,offsets[1]);
   delta_psi_gf = 0.0;
 
   GridFunction u_old_gf(&H1fes);
   GridFunction psi_old_gf(&L2fes);
   GridFunction psi_gf(&L2fes);
   u_old_gf = 0.0;
   psi_old_gf = 0.0;
 
   // 8. Define the function coefficients for the solution and use them to
   //    initialize the initial guess
   FunctionCoefficient exact_coef(exact_solution_obstacle);
   VectorFunctionCoefficient exact_grad_coef(dim,exact_solution_gradient_obstacle);
   FunctionCoefficient IC_coef(IC_func);
   ConstantCoefficient f(0.0);
   FunctionCoefficient obstacle(spherical_obstacle);
   u_gf.ProjectCoefficient(IC_coef);
   u_old_gf = u_gf;
 
   // 9. Initialize the slack variable ψₕ = ln(uₕ)
   LogarithmGridFunctionCoefficient ln_u(u_gf, obstacle);
   psi_gf.ProjectCoefficient(ln_u);
   psi_old_gf = psi_gf;
 
   char vishost[] = "localhost";
   int  visport   = 19916;
   socketstream sol_sock;
   if (visualization)
   {
      sol_sock.open(vishost,visport);
      sol_sock.precision(8);
   }
 
   // 10. Iterate
   int k;
   int total_iterations = 0;
   real_t increment_u = 0.1;
   for (k = 0; k < max_it; k++)
   {
      GridFunction u_tmp(&H1fes);
      u_tmp = u_old_gf;
 
      mfem::out << "\nOUTER ITERATION " << k+1 << endl;
 
      int j;
      for ( j = 0; j < 10; j++)
      {
         total_iterations++;
 
         ConstantCoefficient alpha_cf(alpha);
 
         LinearForm b0,b1;
         b0.Update(&H1fes,rhs.GetBlock(0),0);
         b1.Update(&L2fes,rhs.GetBlock(1),0);
 
         ExponentialGridFunctionCoefficient exp_psi(psi_gf, zero);
         ProductCoefficient neg_exp_psi(-1.0,exp_psi);
         GradientGridFunctionCoefficient grad_u_old(&u_old_gf);
         ProductCoefficient alpha_f(alpha, f);
         GridFunctionCoefficient psi_cf(&psi_gf);
         GridFunctionCoefficient psi_old_cf(&psi_old_gf);
         SumCoefficient psi_old_minus_psi(psi_old_cf, psi_cf, 1.0, -1.0);
 
         b0.AddDomainIntegrator(new DomainLFIntegrator(alpha_f));
         b0.AddDomainIntegrator(new DomainLFIntegrator(psi_old_minus_psi));
         b0.Assemble();
 
         b1.AddDomainIntegrator(new DomainLFIntegrator(exp_psi));
         b1.AddDomainIntegrator(new DomainLFIntegrator(obstacle));
         b1.Assemble();
 
         BilinearForm a00(&H1fes);
         a00.SetDiagonalPolicy(mfem::Operator::DIAG_ONE);
         a00.AddDomainIntegrator(new DiffusionIntegrator(alpha_cf));
         a00.Assemble();
         a00.EliminateEssentialBC(ess_bdr,x.GetBlock(0),rhs.GetBlock(0),
                                  mfem::Operator::DIAG_ONE);
         a00.Finalize();
         SparseMatrix &A00 = a00.SpMat();
 
         MixedBilinearForm a10(&H1fes,&L2fes);
         a10.AddDomainIntegrator(new MixedScalarMassIntegrator());
         a10.Assemble();
         a10.EliminateTrialDofs(ess_bdr, x.GetBlock(0), rhs.GetBlock(1));
         a10.Finalize();
         SparseMatrix &A10 = a10.SpMat();
 
         SparseMatrix *A01 = Transpose(A10);
 
         BilinearForm a11(&L2fes);
         a11.AddDomainIntegrator(new MassIntegrator(neg_exp_psi));
         // NOTE: Shift the spectrum of the Hessian matrix for additional
         //       stability (Quasi-Newton).
         ConstantCoefficient eps_cf(-1e-6);
         if (order == 1)
         {
            // NOTE: ∇ₕuₕ = 0 for constant functions.
            //       Therefore, we use the mass matrix to shift the spectrum
            a11.AddDomainIntegrator(new MassIntegrator(eps_cf));
         }
         else
         {
            a11.AddDomainIntegrator(new DiffusionIntegrator(eps_cf));
         }
         a11.Assemble();
         a11.Finalize();
         SparseMatrix &A11 = a11.SpMat();
 
         BlockOperator A(offsets);
         A.SetBlock(0,0,&A00);
         A.SetBlock(1,0,&A10);
         A.SetBlock(0,1,A01);
         A.SetBlock(1,1,&A11);
 
         BlockDiagonalPreconditioner prec(offsets);
         prec.SetDiagonalBlock(0,new GSSmoother(A00));
         prec.SetDiagonalBlock(1,new GSSmoother(A11));
         prec.owns_blocks = 1;
 
         GMRES(A,prec,rhs,x,0,10000,500,1e-12,0.0);
 
         u_gf.MakeRef(&H1fes, x.GetBlock(0), 0);
         delta_psi_gf.MakeRef(&L2fes, x.GetBlock(1), 0);
 
         u_tmp -= u_gf;
         real_t Newton_update_size = u_tmp.ComputeL2Error(zero);
         u_tmp = u_gf;
 
         real_t gamma = 1.0;
         delta_psi_gf *= gamma;
         psi_gf += delta_psi_gf;
 
         if (visualization)
         {
            sol_sock << "solution\n" << mesh << u_gf << "window_title 'Discrete solution'"
                     << flush;
            mfem::out << "Newton_update_size = " << Newton_update_size << endl;
         }
 
         delete A01;
 
         if (Newton_update_size < increment_u)
         {
            break;
         }
      }
 
      u_tmp = u_gf;
      u_tmp -= u_old_gf;
      increment_u = u_tmp.ComputeL2Error(zero);
 
      mfem::out << "Number of Newton iterations = " << j+1 << endl;
      mfem::out << "Increment (|| uₕ - uₕ_prvs||) = " << increment_u << endl;
 
      u_old_gf = u_gf;
      psi_old_gf = psi_gf;
 
      if (increment_u < tol || k == max_it-1)
      {
         break;
      }
 
      real_t H1_error = u_gf.ComputeH1Error(&exact_coef,&exact_grad_coef);
      mfem::out << "H1-error  (|| u - uₕᵏ||)       = " << H1_error << endl;
 
   }
 
   mfem::out << "\n Outer iterations: " << k+1
             << "\n Total iterations: " << total_iterations
             << "\n Total dofs:       " << H1fes.GetTrueVSize() + L2fes.GetTrueVSize()
             << endl;
 
   // 11. Exact solution.
   if (visualization)
   {
      socketstream err_sock(vishost, visport);
      err_sock.precision(8);
 
      GridFunction error_gf(&H1fes);
      error_gf.ProjectCoefficient(exact_coef);
      error_gf -= u_gf;
 
      err_sock << "solution\n" << mesh << error_gf << "window_title 'Error'"  <<
               flush;
   }
 
   {
      real_t L2_error = u_gf.ComputeL2Error(exact_coef);
      real_t H1_error = u_gf.ComputeH1Error(&exact_coef,&exact_grad_coef);
 
      ExponentialGridFunctionCoefficient u_alt_cf(psi_gf,obstacle);
      GridFunction u_alt_gf(&L2fes);
      u_alt_gf.ProjectCoefficient(u_alt_cf);
      real_t L2_error_alt = u_alt_gf.ComputeL2Error(exact_coef);
 
      mfem::out << "\n Final L2-error (|| u - uₕ||)          = " << L2_error <<
                endl;
      mfem::out << " Final H1-error (|| u - uₕ||)          = " << H1_error << endl;
      mfem::out << " Final L2-error (|| u - ϕ - exp(ψₕ)||) = " << L2_error_alt <<
                endl;
   }
 
   return 0;
}
 
real_t LogarithmGridFunctionCoefficient::Eval(ElementTransformation &T,
                                              const IntegrationPoint &ip)
{
   MFEM_ASSERT(u != NULL, "grid function is not set");
 
   real_t val = u->GetValue(T, ip) - obstacle->Eval(T, ip);
   return max(min_val, log(val));
}
 
real_t ExponentialGridFunctionCoefficient::Eval(ElementTransformation &T,
                                                const IntegrationPoint &ip)
{
   MFEM_ASSERT(u != NULL, "grid function is not set");
 
   real_t val = u->GetValue(T, ip);
   return min(max_val, max(min_val, exp(val) + obstacle->Eval(T, ip)));
}
 
real_t spherical_obstacle(const Vector &pt)
{
   real_t x = pt(0), y = pt(1);
   real_t r = sqrt(x*x + y*y);
   real_t r0 = 0.5;
   real_t beta = 0.9;
 
   real_t b = r0*beta;
   real_t tmp = sqrt(r0*r0 - b*b);
   real_t B = tmp + b*b/tmp;
   real_t C = -b/tmp;
 
   if (r > b)
   {
      return B + r * C;
   }
   else
   {
      return sqrt(r0*r0 - r*r);
   }
}
 
real_t exact_solution_obstacle(const Vector &pt)
{
   real_t x = pt(0), y = pt(1);
   real_t r = sqrt(x*x + y*y);
   real_t r0 = 0.5;
   real_t a =  0.348982574111686;
   real_t A = -0.340129705945858;
 
   if (r > a)
   {
      return A * log(r);
   }
   else
   {
      return sqrt(r0*r0-r*r);
   }
}
 
void exact_solution_gradient_obstacle(const Vector &pt, Vector &grad)
{
   real_t x = pt(0), y = pt(1);
   real_t r = sqrt(x*x + y*y);
   real_t r0 = 0.5;
   real_t a =  0.348982574111686;
   real_t A = -0.340129705945858;
 
   if (r > a)
   {
      grad(0) =  A * x / (r*r);
      grad(1) =  A * y / (r*r);
   }
   else
   {
      grad(0) = - x / sqrt( r0*r0 - r*r );
      grad(1) = - y / sqrt( r0*r0 - r*r );
   }
}