SCL-5: Minimal surface equation

Table of Contents

• Introduction
• Plain program
  • Implementation of the main program (*.cpp)

Note

Prerequisite of this tutorial is SCL-4: Nonlinear Poisson's equation

Note

Intended learning outcome:

• practice of solving a nonlinear problem in MoFEM

Introduction

Introduction

Note

A different tutorial written in an old way solving the same problem can be found at Soap film spanned on wire

Plain program

The plain program for both the implementation of the UDOs (*.cpp) and the main program (*.cpp) are as follows

Implementation of the main program (*.cpp)

#include <stdlib.h>
#include <BasicFiniteElements.hpp>
using namespace MoFEM;
 
static char help[] = "...\n\n";
 
using DomainEle = PipelineManager::FaceEle;
using DomainEleOp = DomainEle::UserDataOperator;
using BoundaryEle = PipelineManager::EdgeEle;
using BoundaryEleOp = BoundaryEle::UserDataOperator;
using PostProcEle = PostProcFaceOnRefinedMesh;
 
using OpBoundaryMass = FormsIntegrators<BoundaryEleOp>::Assembly<
    PETSC>::BiLinearForm<GAUSS>::OpMass<1, 1>;
using OpBoundaryTimeScalarField = FormsIntegrators<BoundaryEleOp>::Assembly<
    PETSC>::LinearForm<GAUSS>::OpBaseTimesScalarField<1>;
using OpBoundarySource = FormsIntegrators<BoundaryEleOp>::Assembly<
    PETSC>::LinearForm<GAUSS>::OpSource<1, 1>;
 
using AssemblyDomainEleOp =
    FormsIntegrators<DomainEleOp>::Assembly<PETSC>::OpBase;
using AssemblyBoundaryEleOp =
    FormsIntegrators<BoundaryEleOp>::Assembly<PETSC>::OpBase;
 
FTensor::Index<'i', 2> i;
 
/** \brief Integrate the domain tangent matrix (LHS)
 
\f[
\sum\limits_j {\left[ {\int\limits_{{\Omega _e}} {\left( {{a_n}\nabla {\phi _i}
\cdot \nabla {\phi _j} - a_n^3\nabla {\phi _i}\left( {\nabla u \cdot \nabla
{\phi _j}} \right)\nabla u} \right)d{\Omega _e}} } \right]\delta {U_j}}  =
\int\limits_{{\Omega _e}} {{\phi _i}fd{\Omega _e}}  - \int\limits_{{\Omega _e}}
{\nabla {\phi _i}{a_n}\nabla ud{\Omega _e}} \\
{a_n} = \frac{1}{{{{\left( {1 +
{{\left| {\nabla u} \right|}^2}} \right)}^{\frac{1}{2}}}}}
\f]
 
*/
struct OpDomainTangentMatrix : public AssemblyDomainEleOp {
public:
  OpDomainTangentMatrix(std::string row_field_name, std::string col_field_name,
                        boost::shared_ptr<MatrixDouble> field_grad_mat)
      : AssemblyDomainEleOp(row_field_name, col_field_name,
                            DomainEleOp::OPROWCOL),
        fieldGradMat(field_grad_mat) {}
 
  MoFEMErrorCode iNtegrate(EntData &row_data, EntData &col_data) {
    MoFEMFunctionBegin;
 
    auto &locLhs = AssemblyDomainEleOp::locMat;
 
    // get element area
    const double area = getMeasure();
 
    // get number of integration points
    const int nb_integration_points = getGaussPts().size2();
    // get integration weights
    auto t_w = getFTensor0IntegrationWeight();
    // get gradient of the field at integration points
    auto t_field_grad = getFTensor1FromMat<2>(*fieldGradMat);
 
    // get derivatives of base functions on row
    auto t_row_diff_base = row_data.getFTensor1DiffN<2>();
 
    // START THE LOOP OVER INTEGRATION POINTS TO CALCULATE LOCAL MATRIX
    for (int gg = 0; gg != nb_integration_points; gg++) {
 
      const double a = t_w * area;
      const double an = 1. / std::sqrt(1 + t_field_grad(i) * t_field_grad(i));
 
      for (int rr = 0; rr != AssemblyDomainEleOp::nbRows; ++rr) {
        // get derivatives of base functions on column
        auto t_col_diff_base = col_data.getFTensor1DiffN<2>(gg, 0);
 
        for (int cc = 0; cc != AssemblyDomainEleOp::nbRows; cc++) {
 
          // calculate components of the local matrix
          locLhs(rr, cc) += (t_row_diff_base(i) * t_col_diff_base(i)) * an * a -
                            t_row_diff_base(i) *
                                (t_field_grad(i) * t_col_diff_base(i)) *
                                t_field_grad(i) * an * an * an * a;
 
          // move to the derivatives of the next base functions on column
          ++t_col_diff_base;
        }
 
        // move to the derivatives of the next base functions on row
        ++t_row_diff_base;
      }
 
      // move to the weight of the next integration point
      ++t_w;
      // move to the gradient of the field at the next integration point
      ++t_field_grad;
    }
 
    MoFEMFunctionReturn(0);
  }
 
private:
  boost::shared_ptr<MatrixDouble> fieldGradMat;
};
 
/** \brief Integrate the domain residual vector (RHS)
 
\f[
\sum\limits_j {\left[ {\int\limits_{{\Omega _e}} {\left( {{a_n}\nabla {\phi _i}
\cdot \nabla {\phi _j} - a_n^3\nabla {\phi _i}\left( {\nabla u \cdot \nabla
{\phi _j}} \right)\nabla u} \right)d{\Omega _e}} } \right]\delta {U_j}}  =
\int\limits_{{\Omega _e}} {{\phi _i}fd{\Omega _e}}  - \int\limits_{{\Omega _e}}
{\nabla {\phi _i}{a_n}\nabla ud{\Omega _e}} \\
{a_n} = \frac{1}{{{{\left( {1 +
{{\left| {\nabla u} \right|}^2}} \right)}^{\frac{1}{2}}}}}
\f]
 
*/
struct OpDomainResidualVector : public AssemblyDomainEleOp {
public:
  OpDomainResidualVector(std::string field_name,
                         boost::shared_ptr<MatrixDouble> field_grad_mat)
      : AssemblyDomainEleOp(field_name, field_name, AssemblyDomainEleOp::OPROW),
        fieldGradMat(field_grad_mat) {}
 
  MoFEMErrorCode iNtegrate(EntData &data) {
    MoFEMFunctionBegin;
 
    auto &nf = AssemblyDomainEleOp::locF;
 
    // get element area
    const double area = getMeasure();
 
    // get number of integration points
    const int nb_integration_points = getGaussPts().size2();
    // get integration weights
    auto t_w = getFTensor0IntegrationWeight();
    // get gradient of the field at integration points
    auto t_field_grad = getFTensor1FromMat<2>(*fieldGradMat);
 
    // get base functions
    auto t_base = data.getFTensor0N();
    // get derivatives of base functions
    auto t_diff_base = data.getFTensor1DiffN<2>();
 
    // START THE LOOP OVER INTEGRATION POINTS TO CALCULATE LOCAL VECTOR
    for (int gg = 0; gg != nb_integration_points; gg++) {
 
      const double a = t_w * area;
      const double an = 1. / std::sqrt(1 + t_field_grad(i) * t_field_grad(i));
 
      for (int rr = 0; rr != AssemblyDomainEleOp::nbRows; rr++) {
 
        // calculate components of the local vector
        // remember to use -= here due to PETSc consideration of Residual Vec
        nf[rr] += (t_diff_base(i) * t_field_grad(i)) * an * a;
 
        // move to the next base function
        ++t_base;
        // move to the derivatives of the next base function
        ++t_diff_base;
      }
 
      // move to the weight of the next integration point
      ++t_w;
      // move to the gradient of the field at the next integration point
      ++t_field_grad;
    }
 
    MoFEMFunctionReturn(0);
  }
 
private:
  boost::shared_ptr<MatrixDouble> fieldGradMat;
};
 
struct MinimalSurfaceEqn {
public:
  MinimalSurfaceEqn(MoFEM::Interface &m_field);
 
  // Declaration of the main function to run analysis
  MoFEMErrorCode runProgram();
 
private:
  // Declaration of other main functions called in runProgram()
  MoFEMErrorCode readMesh();
  MoFEMErrorCode setupProblem();
  MoFEMErrorCode setIntegrationRules();
  MoFEMErrorCode boundaryCondition();
  MoFEMErrorCode assembleSystem();
  MoFEMErrorCode solveSystem();
  MoFEMErrorCode outputResults();
 
  // Function to calculate the Boundary term
  static double boundaryFunction(const double x, const double y,
                                 const double z) {
    return sin(2 * M_PI * (x + y));
  }
 
  // Main interfaces
  MoFEM::Interface &mField;
 
  // Object to mark boundary entities for the assembling of domain elements
  boost::shared_ptr<std::vector<unsigned char>> boundaryMarker;
  Range boundaryEnts;
};
 
MinimalSurfaceEqn::MinimalSurfaceEqn(MoFEM::Interface &m_field)
    : mField(m_field) {}
 
MoFEMErrorCode MinimalSurfaceEqn::runProgram() {
  MoFEMFunctionBegin;
 
  readMesh();
  setupProblem();
  setIntegrationRules();
  boundaryCondition();
  assembleSystem();
  solveSystem();
  outputResults();
 
  MoFEMFunctionReturn(0);
}
 
MoFEMErrorCode MinimalSurfaceEqn::readMesh() {
  MoFEMFunctionBegin;
 
  auto *simple = mField.getInterface<Simple>();
  CHKERR mField.getInterface(simple);
  CHKERR simple->getOptions();
  CHKERR simple->loadFile();
 
  MoFEMFunctionReturn(0);
}
 
MoFEMErrorCode MinimalSurfaceEqn::setupProblem() {
  MoFEMFunctionBegin;
 
  auto *simple = mField.getInterface<Simple>();
  CHKERR simple->addDomainField("U", H1, AINSWORTH_BERNSTEIN_BEZIER_BASE, 1);
  CHKERR simple->addBoundaryField("U", H1, AINSWORTH_BERNSTEIN_BEZIER_BASE, 1);
 
  int order = 3;
  CHKERR PetscOptionsGetInt(PETSC_NULL, "", "-order", &order, PETSC_NULL);
  CHKERR simple->setFieldOrder("U", order);
  CHKERR simple->setUp();
 
  MoFEMFunctionReturn(0);
}
 
MoFEMErrorCode MinimalSurfaceEqn::setIntegrationRules() {
  MoFEMFunctionBegin;
 
  auto integration_rule = [](int o_row, int o_col, int approx_order) {
    return 2 * approx_order;
  };
 
  auto *pipeline_mng = mField.getInterface<PipelineManager>();
  CHKERR pipeline_mng->setDomainRhsIntegrationRule(integration_rule);
  CHKERR pipeline_mng->setDomainLhsIntegrationRule(integration_rule);
  CHKERR pipeline_mng->setBoundaryLhsIntegrationRule(integration_rule);
  CHKERR pipeline_mng->setBoundaryRhsIntegrationRule(integration_rule);
 
  MoFEMFunctionReturn(0);
}
 
MoFEMErrorCode MinimalSurfaceEqn::boundaryCondition() {
  MoFEMFunctionBegin;
 
  auto *simple = mField.getInterface<Simple>();
 
  auto get_ents_on_mesh_skin = [&]() {
    Range body_ents;
    CHKERR mField.get_moab().get_entities_by_dimension(0, 2, body_ents);
    Skinner skin(&mField.get_moab());
    Range skin_ents;
    CHKERR skin.find_skin(0, body_ents, false, skin_ents);
    // filter not owned entities, those are not on boundary
    Range boundary_ents;
    ParallelComm *pcomm =
        ParallelComm::get_pcomm(&mField.get_moab(), MYPCOMM_INDEX);
    pcomm->filter_pstatus(skin_ents, PSTATUS_SHARED | PSTATUS_MULTISHARED,
                          PSTATUS_NOT, -1, &boundary_ents);
 
    Range skin_verts;
    mField.get_moab().get_connectivity(boundary_ents, skin_verts, true);
    boundary_ents.merge(skin_verts);
    boundaryEnts = boundary_ents;
    return boundary_ents;
  };
 
  auto mark_boundary_dofs = [&](Range &&skin_edges) {
    auto problem_manager = mField.getInterface<ProblemsManager>();
    auto marker_ptr = boost::make_shared<std::vector<unsigned char>>();
    problem_manager->markDofs(simple->getProblemName(), ROW,
                              ProblemsManager::OR, skin_edges, *marker_ptr);
    return marker_ptr;
  };
 
  // Get global local vector of marked DOFs. Is global, since is set for all
  // DOFs on processor. Is local since only DOFs on processor are in the
  // vector. To access DOFs use local indices.
  boundaryMarker = mark_boundary_dofs(get_ents_on_mesh_skin());
 
  MoFEMFunctionReturn(0);
}
 
MoFEMErrorCode MinimalSurfaceEqn::assembleSystem() {
  MoFEMFunctionBegin;
  auto add_domain_base_ops = [&](auto &pipeline) {
    auto det_ptr = boost::make_shared<VectorDouble>();
    auto jac_ptr = boost::make_shared<MatrixDouble>();
    auto inv_jac_ptr = boost::make_shared<MatrixDouble>();
    pipeline.push_back(new OpCalculateHOJacForFace(jac_ptr));
    pipeline.push_back(new OpInvertMatrix<2>(jac_ptr, det_ptr, inv_jac_ptr));
    pipeline.push_back(new OpSetInvJacH1ForFace(inv_jac_ptr));
    pipeline.push_back(new OpSetHOWeightsOnFace());
  };
 
  auto add_domain_lhs_ops = [&](auto &pipeline) {
    pipeline.push_back(new OpSetBc("U", true, boundaryMarker));
    auto grad_u_at_gauss_pts = boost::make_shared<MatrixDouble>();
    pipeline.push_back(
        new OpCalculateScalarFieldGradient<2>("U", grad_u_at_gauss_pts));
    pipeline.push_back(
        new OpDomainTangentMatrix("U", "U", grad_u_at_gauss_pts));
    pipeline.push_back(new OpUnSetBc("U"));
  };
 
  auto add_domain_rhs_ops = [&](auto &pipeline) {
    pipeline.push_back(new OpSetBc("U", true, boundaryMarker));
    auto grad_u_at_gauss_pts = boost::make_shared<MatrixDouble>();
    pipeline.push_back(
        new OpCalculateScalarFieldGradient<2>("U", grad_u_at_gauss_pts));
    pipeline.push_back(new OpDomainResidualVector("U", grad_u_at_gauss_pts));
    pipeline.push_back(new OpUnSetBc("U"));
  };
 
  auto add_boundary_base_ops = [&](auto &pipeline) {};
 
  auto add_lhs_base_ops = [&](auto &pipeline) {
    pipeline.push_back(new OpSetBc("U", false, boundaryMarker));
    pipeline.push_back(new OpBoundaryMass(
        "U", "U", [](const double, const double, const double) { return 1; }));
    pipeline.push_back(new OpUnSetBc("U"));
  };
  auto add_rhs_base_ops = [&](auto &pipeline) {
    pipeline.push_back(new OpSetBc("U", false, boundaryMarker));
    auto u_at_gauss_pts = boost::make_shared<VectorDouble>();
    pipeline.push_back(new OpCalculateScalarFieldValues("U", u_at_gauss_pts));
    pipeline.push_back(new OpBoundaryTimeScalarField(
        "U", u_at_gauss_pts,
        [](const double, const double, const double) { return 1; }));
    pipeline.push_back(new OpBoundarySource("U", boundaryFunction));
    pipeline.push_back(new OpUnSetBc("U"));
  };
 
  auto pipeline_mng = mField.getInterface<PipelineManager>();
  add_domain_base_ops(pipeline_mng->getOpDomainLhsPipeline());
  add_domain_base_ops(pipeline_mng->getOpDomainRhsPipeline());
  add_domain_lhs_ops(pipeline_mng->getOpDomainLhsPipeline());
  add_domain_rhs_ops(pipeline_mng->getOpDomainRhsPipeline());
 
  add_boundary_base_ops(pipeline_mng->getOpBoundaryLhsPipeline());
  add_boundary_base_ops(pipeline_mng->getOpBoundaryRhsPipeline());
  add_lhs_base_ops(pipeline_mng->getOpBoundaryLhsPipeline());
  add_rhs_base_ops(pipeline_mng->getOpBoundaryRhsPipeline());
 
  MoFEMFunctionReturn(0);
}
 
MoFEMErrorCode MinimalSurfaceEqn::solveSystem() {
  MoFEMFunctionBegin;
 
  auto *simple = mField.getInterface<Simple>();
 
  auto set_fieldsplit_preconditioner = [&](auto snes) {
    MoFEMFunctionBeginHot;
    KSP ksp;
    CHKERR SNESGetKSP(snes, &ksp);
    PC pc;
    CHKERR KSPGetPC(ksp, &pc);
    PetscBool is_pcfs = PETSC_FALSE;
    PetscObjectTypeCompare((PetscObject)pc, PCFIELDSPLIT, &is_pcfs);
 
    if (is_pcfs == PETSC_TRUE) {
      auto bc_mng = mField.getInterface<BcManager>();
      auto name_prb = simple->getProblemName();
      SmartPetscObj<IS> is_all_bc;
      CHKERR mField.getInterface<ISManager>()->isCreateProblemFieldAndRank(
          name_prb, ROW, "U", 0, 1, is_all_bc, &boundaryEnts);
      int is_all_bc_size;
      CHKERR ISGetSize(is_all_bc, &is_all_bc_size);
      MOFEM_LOG("EXAMPLE", Sev::inform)
          << "Field split block size " << is_all_bc_size;
      CHKERR PCFieldSplitSetIS(pc, PETSC_NULL,
                               is_all_bc); // boundary block
    }
    MoFEMFunctionReturnHot(0);
  };
 
  // Create global RHS and solution vectors
  auto dm = simple->getDM();
  SmartPetscObj<Vec> global_rhs, global_solution;
  CHKERR DMCreateGlobalVector_MoFEM(dm, global_rhs);
  global_solution = smartVectorDuplicate(global_rhs);
 
  // Create nonlinear solver (SNES)
  auto pipeline_mng = mField.getInterface<PipelineManager>();
  auto solver = pipeline_mng->createSNES();
  CHKERR SNESSetFromOptions(solver);
  CHKERR set_fieldsplit_preconditioner(solver);
  CHKERR SNESSetUp(solver);
 
  // Solve the system
  CHKERR SNESSolve(solver, global_rhs, global_solution);
 
  // Scatter result data on the mesh
  CHKERR DMoFEMMeshToGlobalVector(dm, global_solution, INSERT_VALUES,
                                  SCATTER_REVERSE);
 
  MoFEMFunctionReturn(0);
}
 
MoFEMErrorCode MinimalSurfaceEqn::outputResults() {
  MoFEMFunctionBegin;
 
  auto post_proc = boost::make_shared<PostProcEle>(mField);
  CHKERR post_proc->generateReferenceElementMesh();
  CHKERR post_proc->addFieldValuesPostProc("U");
 
  auto *simple = mField.getInterface<Simple>();
  auto dm = simple->getDM();
  CHKERR DMoFEMLoopFiniteElements(dm, simple->getDomainFEName(), post_proc);
 
  CHKERR post_proc->writeFile("out_result.h5m");
 
  MoFEMFunctionReturn(0);
}
 
int main(int argc, char *argv[]) {
 
  // Initialisation of MoFEM/PETSc and MOAB data structures
  const char param_file[] = "param_file.petsc";
  MoFEM::Core::Initialize(&argc, &argv, param_file, help);
 
  // Add logging channel for example
  auto core_log = logging::core::get();
  core_log->add_sink(
      LogManager::createSink(LogManager::getStrmWorld(), "EXAMPLE"));
  LogManager::setLog("EXAMPLE");
  MOFEM_LOG_TAG("EXAMPLE", "example")
 
  // Error handling
  try {
    // Register MoFEM discrete manager in PETSc
    DMType dm_name = "DMMOFEM";
    CHKERR DMRegister_MoFEM(dm_name);
 
    // Create MOAB instance
    moab::Core mb_instance;              // mesh database
    moab::Interface &moab = mb_instance; // mesh database interface
 
    // Create MoFEM instance
    MoFEM::Core core(moab);           // finite element database
    MoFEM::Interface &m_field = core; // finite element interface
 
    // Run the main analysis
    MinimalSurfaceEqn minimal_surface_problem(m_field);
    CHKERR minimal_surface_problem.runProgram();
  }
  CATCH_ERRORS;
 
  // Finish work: cleaning memory, getting statistics, etc.
  MoFEM::Core::Finalize();
 
  return 0;
}