incompressible_elasticity.cpp

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/**
 * @file incompressible_elasticity.cpp
 * @brief Incompressible elasticity problem
 */
 
#include <MoFEM.hpp>
 
using namespace MoFEM;
 
constexpr int SPACE_DIM =
    EXECUTABLE_DIMENSION; //< Space dimension of problem, mesh
 
constexpr AssemblyType AT =
    (SCHUR_ASSEMBLE) ? AssemblyType::SCHUR
                     : AssemblyType::PETSC; //< selected assembly type
 
constexpr IntegrationType IT =
    IntegrationType::GAUSS; //< selected integration type
constexpr CoordinateTypes coord_type = CARTESIAN;
 
using EntData = EntitiesFieldData::EntData;
using DomainEle = PipelineManager::ElementsAndOpsByDim<SPACE_DIM>::DomainEle;
using DomainEleOp = DomainEle::UserDataOperator;
using BoundaryEle =
    PipelineManager::ElementsAndOpsByDim<SPACE_DIM>::BoundaryEle;
using BoundaryEleOp = BoundaryEle::UserDataOperator;
using PostProcEle = PostProcBrokenMeshInMoab<DomainEle>;
using SkinPostProcEle = PostProcBrokenMeshInMoab<BoundaryEle>;
using SetPtsData = FieldEvaluatorInterface::SetPtsData;
 
// struct DomainBCs {};
// struct BoundaryBCs {};
 
// using DomainRhsBCs = NaturalBC<DomainEleOp>::Assembly<AT>::LinearForm<IT>;
// using OpDomainRhsBCs = DomainRhsBCs::OpFlux<DomainBCs, 1, SPACE_DIM>;
// using BoundaryRhsBCs = NaturalBC<BoundaryEleOp>::Assembly<AT>::LinearForm<IT>;
// using OpBoundaryRhsBCs = BoundaryRhsBCs::OpFlux<BoundaryBCs, 1, SPACE_DIM>;
// using BoundaryLhsBCs = NaturalBC<BoundaryEleOp>::Assembly<AT>::BiLinearForm<IT>;
// using OpBoundaryLhsBCs = BoundaryLhsBCs::OpFlux<BoundaryBCs, 1, SPACE_DIM>;
 
PetscBool doEvalField;
struct MonitorIncompressible : public FEMethod {
 
  MonitorIncompressible(
      SmartPetscObj<DM> dm,
      std::pair<boost::shared_ptr<PostProcEle>,
                boost::shared_ptr<SkinPostProcEle>>
          pair_post_proc_fe,
      std::tuple<SmartPetscObj<Vec>, SmartPetscObj<VecScatter>> ux_scatter,
      std::tuple<SmartPetscObj<Vec>, SmartPetscObj<VecScatter>> uy_scatter,
      std::tuple<SmartPetscObj<Vec>, SmartPetscObj<VecScatter>> uz_scatter,
      std::array<double, SPACE_DIM> pass_field_eval_coords,
      boost::shared_ptr<SetPtsData> pass_field_eval_data,
      boost::shared_ptr<MatrixDouble> vec_field_ptr)
      : dM(dm), uXScatter(ux_scatter), uYScatter(uy_scatter),
        uZScatter(uz_scatter), fieldEvalCoords(pass_field_eval_coords),
        fieldEvalData(pass_field_eval_data), vecFieldPtr(vec_field_ptr) {
    postProcFe = pair_post_proc_fe.first;
    skinPostProcFe = pair_post_proc_fe.second;
  };
 
  MoFEMErrorCode preProcess() { return 0; }
  MoFEMErrorCode operator()() { return 0; }
 
  MoFEMErrorCode postProcess() {
    MoFEMFunctionBegin;
 
    MoFEM::Interface *m_field_ptr;
    CHKERR DMoFEMGetInterfacePtr(dM, &m_field_ptr);
    auto *simple = m_field_ptr->getInterface<Simple>();
 
    if (doEvalField) {
      if (SPACE_DIM == 3) {
        CHKERR m_field_ptr->getInterface<FieldEvaluatorInterface>()
            ->evalFEAtThePoint3D(
                fieldEvalCoords.data(), 1e-12, simple->getProblemName(),
                simple->getDomainFEName(), fieldEvalData,
                m_field_ptr->get_comm_rank(), m_field_ptr->get_comm_rank(),
                getCacheWeakPtr().lock(), MF_EXIST, QUIET);
      } else {
        CHKERR m_field_ptr->getInterface<FieldEvaluatorInterface>()
            ->evalFEAtThePoint2D(
                fieldEvalCoords.data(), 1e-12, simple->getProblemName(),
                simple->getDomainFEName(), fieldEvalData,
                m_field_ptr->get_comm_rank(), m_field_ptr->get_comm_rank(),
                getCacheWeakPtr().lock(), MF_EXIST, QUIET);
      }
 
      if (vecFieldPtr->size1()) {
        auto t_disp = getFTensor1FromMat<SPACE_DIM>(*vecFieldPtr);
        if constexpr (SPACE_DIM == 2)
          MOFEM_LOG("SYNC", Sev::inform)
              << "U_X: " << t_disp(0) << " U_Y: " << t_disp(1);
        else
          MOFEM_LOG("SYNC", Sev::inform)
              << "U_X: " << t_disp(0) << " U_Y: " << t_disp(1)
              << " U_Z: " << t_disp(2);
      }
    }
    MOFEM_LOG_SEVERITY_SYNC(m_field_ptr->get_comm(), Sev::inform);
 
    auto make_vtk = [&]() {
      MoFEMFunctionBegin;
      if (postProcFe) {
        CHKERR DMoFEMLoopFiniteElements(dM, "dFE", postProcFe,
                                        getCacheWeakPtr());
        CHKERR postProcFe->writeFile("out_incomp_elasticity" +
                                     boost::lexical_cast<std::string>(ts_step) +
                                     ".h5m");
      }
      if (skinPostProcFe) {
        CHKERR DMoFEMLoopFiniteElements(dM, "bFE", skinPostProcFe,
                                        getCacheWeakPtr());
        CHKERR skinPostProcFe->writeFile(
            "out_skin_incomp_elasticity_" +
            boost::lexical_cast<std::string>(ts_step) + ".h5m");
      }
      MoFEMFunctionReturn(0);
    };
 
    auto print_max_min = [&](auto &tuple, const std::string msg) {
      MoFEMFunctionBegin;
      CHKERR VecScatterBegin(std::get<1>(tuple), ts_u, std::get<0>(tuple),
                             INSERT_VALUES, SCATTER_FORWARD);
      CHKERR VecScatterEnd(std::get<1>(tuple), ts_u, std::get<0>(tuple),
                           INSERT_VALUES, SCATTER_FORWARD);
      double max, min;
      CHKERR VecMax(std::get<0>(tuple), PETSC_NULL, &max);
      CHKERR VecMin(std::get<0>(tuple), PETSC_NULL, &min);
      MOFEM_LOG_C("INCOMP_ELASTICITY", Sev::inform,
                  "%s time %3.4e min %3.4e max %3.4e", msg.c_str(), ts_t, min,
                  max);
      MoFEMFunctionReturn(0);
    };
 
    CHKERR make_vtk();
    CHKERR print_max_min(uXScatter, "Ux");
    CHKERR print_max_min(uYScatter, "Uy");
    if constexpr (SPACE_DIM == 3)
      CHKERR print_max_min(uZScatter, "Uz");
 
    MoFEMFunctionReturn(0);
  }
 
private:
  SmartPetscObj<DM> dM;
  boost::shared_ptr<PostProcEle> postProcFe;
  boost::shared_ptr<SkinPostProcEle> skinPostProcFe;
  std::tuple<SmartPetscObj<Vec>, SmartPetscObj<VecScatter>> uXScatter;
  std::tuple<SmartPetscObj<Vec>, SmartPetscObj<VecScatter>> uYScatter;
  std::tuple<SmartPetscObj<Vec>, SmartPetscObj<VecScatter>> uZScatter;
  std::array<double, SPACE_DIM> fieldEvalCoords;
  boost::shared_ptr<SetPtsData> fieldEvalData;
  boost::shared_ptr<MatrixDouble> vecFieldPtr;
};
 
// Assemble to A matrix, by default, however, some terms are assembled only to
// preconditioning.
 
template <>
typename MoFEM::OpBaseImpl<AT, DomainEleOp>::MatSetValuesHook
    MoFEM::OpBaseImpl<AT, DomainEleOp>::matSetValuesHook =
        [](ForcesAndSourcesCore::UserDataOperator *op_ptr,
           const EntitiesFieldData::EntData &row_data,
           const EntitiesFieldData::EntData &col_data, MatrixDouble &m) {
          return MatSetValues<AssemblyTypeSelector<AT>>(
              op_ptr->getKSPA(), row_data, col_data, m, ADD_VALUES);
        };
 
/**
 * @brief Element used to specialise assembly
 *
 */
struct DomainEleOpStab : public DomainEleOp {
  using DomainEleOp::DomainEleOp;
};
 
/**
 * @brief Specialise assembly for Stabilised matrix
 *
 * @tparam
 */
template <>
typename MoFEM::OpBaseImpl<AT, DomainEleOpStab>::MatSetValuesHook
    MoFEM::OpBaseImpl<AT, DomainEleOpStab>::matSetValuesHook =
        [](ForcesAndSourcesCore::UserDataOperator *op_ptr,
           const EntitiesFieldData::EntData &row_data,
           const EntitiesFieldData::EntData &col_data, MatrixDouble &m) {
          return MatSetValues<AssemblyTypeSelector<AT>>(
              op_ptr->getKSPB(), row_data, col_data, m, ADD_VALUES);
        };
//! [Specialisation for assembly]
 
int order = 2;
int geom_order = 1;
inline static double young_modulus = 100;
inline static double poisson_ratio = 0.25;
inline static double mu;
inline static double lambda;
 
PetscBool isDiscontinuousPressure = PETSC_FALSE;
 
struct Incompressible {
 
  Incompressible(MoFEM::Interface &m_field) : mField(m_field) {}
 
  MoFEMErrorCode runProblem();
 
private:
  MoFEM::Interface &mField;
 
  MoFEMErrorCode setupProblem();
  MoFEMErrorCode createCommonData();
  MoFEMErrorCode bC();
  MoFEMErrorCode OPs();
  MoFEMErrorCode tsSolve();
  MoFEMErrorCode checkResults();
 
  std::tuple<SmartPetscObj<Vec>, SmartPetscObj<VecScatter>> uXScatter;
  std::tuple<SmartPetscObj<Vec>, SmartPetscObj<VecScatter>> uYScatter;
  std::tuple<SmartPetscObj<Vec>, SmartPetscObj<VecScatter>> uZScatter;
};
 
template <int DIM>
struct OpCalculateLameStress : public ForcesAndSourcesCore::UserDataOperator {
  OpCalculateLameStress(double m_u, boost::shared_ptr<MatrixDouble> stress_ptr,
                        boost::shared_ptr<MatrixDouble> strain_ptr,
                        boost::shared_ptr<VectorDouble> pressure_ptr)
      : ForcesAndSourcesCore::UserDataOperator(NOSPACE, OPLAST), mU(m_u),
        stressPtr(stress_ptr), strainPtr(strain_ptr),
        pressurePtr(pressure_ptr) {}
 
  MoFEMErrorCode doWork(int side, EntityType type,
                        EntitiesFieldData::EntData &data) {
    MoFEMFunctionBegin;
    // Define Indicies
    FTensor::Index<'i', SPACE_DIM> i;
    FTensor::Index<'j', SPACE_DIM> j;
 
    // Define Kronecker Delta
    constexpr auto t_kd = FTensor::Kronecker_Delta_symmetric<double>();
 
    // Number of Gauss points
    const size_t nb_gauss_pts = getGaussPts().size2();
 
    stressPtr->resize((DIM * (DIM + 1)) / 2, nb_gauss_pts);
    auto t_stress = getFTensor2SymmetricFromMat<DIM>(*(stressPtr));
    auto t_strain = getFTensor2SymmetricFromMat<DIM>(*(strainPtr));
    auto t_pressure = getFTensor0FromVec(*(pressurePtr));
 
    const double l_mu = mU;
    // Extract matrix from data matrix
    for (auto gg = 0; gg != nb_gauss_pts; ++gg) {
 
      t_stress(i, j) = t_pressure * t_kd(i, j) + 2. * l_mu * t_strain(i, j);
 
      ++t_strain;
      ++t_stress;
      ++t_pressure;
    }
 
    MoFEMFunctionReturn(0);
  }
 
private:
  double mU;
  boost::shared_ptr<MatrixDouble> stressPtr;
  boost::shared_ptr<MatrixDouble> strainPtr;
  boost::shared_ptr<VectorDouble> pressurePtr;
};
 
//! [Run problem]
MoFEMErrorCode Incompressible::runProblem() {
  MoFEMFunctionBegin;
  CHKERR setupProblem();
  CHKERR createCommonData();
  CHKERR bC();
  CHKERR OPs();
  CHKERR tsSolve();
  CHKERR checkResults();
  MoFEMFunctionReturn(0);
}
//! [Run problem]
 
//! [Set up problem]
MoFEMErrorCode Incompressible::setupProblem() {
  MoFEMFunctionBegin;
  Simple *simple = mField.getInterface<Simple>();
 
  CHKERR PetscOptionsGetInt(PETSC_NULL, "", "-order", &order, PETSC_NULL);
  CHKERR PetscOptionsGetInt(PETSC_NULL, "", "-geom_order", &geom_order,
                            PETSC_NULL);
  CHKERR PetscOptionsGetBool(PETSC_NULL, "", "-is_discontinuous_pressure",
                             &isDiscontinuousPressure, PETSC_NULL);
 
  MOFEM_LOG("INCOMP_ELASTICITY", Sev::inform) << "Order " << order;
  MOFEM_LOG("INCOMP_ELASTICITY", Sev::inform) << "Geom order " << geom_order;
 
  // Select base
  enum bases { AINSWORTH, DEMKOWICZ, LASBASETOPT };
  const char *list_bases[LASBASETOPT] = {"ainsworth", "demkowicz"};
  PetscInt choice_base_value = AINSWORTH;
  CHKERR PetscOptionsGetEList(PETSC_NULL, NULL, "-base", list_bases,
                              LASBASETOPT, &choice_base_value, PETSC_NULL);
 
  FieldApproximationBase base;
  switch (choice_base_value) {
  case AINSWORTH:
    base = AINSWORTH_LEGENDRE_BASE;
    MOFEM_LOG("INCOMP_ELASTICITY", Sev::inform)
        << "Set AINSWORTH_LEGENDRE_BASE for displacements";
    break;
  case DEMKOWICZ:
    base = DEMKOWICZ_JACOBI_BASE;
    MOFEM_LOG("INCOMP_ELASTICITY", Sev::inform)
        << "Set DEMKOWICZ_JACOBI_BASE for displacements";
    break;
  default:
    base = LASTBASE;
    break;
  }
 
  // Note: For tets we have only H1 Ainsworth base, for Hex we have only H1
  // Demkowicz base. We need to implement Demkowicz H1 base on tet.
  CHKERR simple->addDataField("GEOMETRY", H1, base, SPACE_DIM);
  CHKERR simple->setFieldOrder("GEOMETRY", geom_order);
 
  // Adding fields related to incompressible elasticity
  // Add displacement domain and boundary fields
  CHKERR simple->addDomainField("U", H1, base, SPACE_DIM);
  CHKERR simple->addBoundaryField("U", H1, base, SPACE_DIM);
  CHKERR simple->setFieldOrder("U", order);
 
  // Add pressure domain and boundary fields
  // Choose either Crouzeix-Raviart element:
  if (isDiscontinuousPressure) {
    CHKERR simple->addDomainField("P", L2, base, 1);
    CHKERR simple->setFieldOrder("P", order - 2);
  } else {
    // ... or Taylor-Hood element:
    CHKERR simple->addDomainField("P", H1, base, 1);
    CHKERR simple->setFieldOrder("P", order - 1);
  }
 
  // Add geometry data field
  CHKERR simple->addDataField("GEOMETRY", H1, base, SPACE_DIM);
  CHKERR simple->setFieldOrder("GEOMETRY", geom_order);
 
  CHKERR simple->setUp();
 
  auto project_ho_geometry = [&]() {
    Projection10NodeCoordsOnField ent_method(mField, "GEOMETRY");
    return mField.loop_dofs("GEOMETRY", ent_method);
  };
  CHKERR project_ho_geometry();
 
  MoFEMFunctionReturn(0);
} //! [Set up problem]
 
//! [Create common data]
MoFEMErrorCode Incompressible::createCommonData() {
  MoFEMFunctionBegin;
 
  auto get_options = [&]() {
    MoFEMFunctionBegin;
    CHKERR PetscOptionsGetScalar(PETSC_NULL, "", "-young_modulus",
                                 &young_modulus, PETSC_NULL);
    CHKERR PetscOptionsGetScalar(PETSC_NULL, "", "-poisson_ratio",
                                 &poisson_ratio, PETSC_NULL);
 
    MOFEM_LOG("INCOMP_ELASTICITY", Sev::inform)
        << "Young modulus " << young_modulus;
    MOFEM_LOG("INCOMP_ELASTICITY", Sev::inform)
        << "Poisson_ratio " << poisson_ratio;
 
    mu = young_modulus / (2. + 2. * poisson_ratio);
    const double lambda_denom =
        (1. + poisson_ratio) * (1. - 2. * poisson_ratio);
    lambda = young_modulus * poisson_ratio / lambda_denom;
 
    MoFEMFunctionReturn(0);
  };
 
  CHKERR get_options();
 
  MoFEMFunctionReturn(0);
}
//! [Create common data]
 
//! [Boundary condition]
MoFEMErrorCode Incompressible::bC() {
  MoFEMFunctionBegin;
 
  auto pipeline_mng = mField.getInterface<PipelineManager>();
  auto simple = mField.getInterface<Simple>();
  auto bc_mng = mField.getInterface<BcManager>();
  auto time_scale = boost::make_shared<TimeScale>();
 
  auto integration_rule = [](int, int, int approx_order) {
    return 2 * (approx_order - 1);
  };
 
  using DomainNaturalBC =
      NaturalBC<DomainEleOp>::Assembly<PETSC>::LinearForm<GAUSS>;
  using OpBodyForce =
      DomainNaturalBC::OpFlux<NaturalMeshsetType<BLOCKSET>, 1, SPACE_DIM>;
  using BoundaryNaturalBC =
      NaturalBC<BoundaryEleOp>::Assembly<PETSC>::LinearForm<GAUSS>;
  using OpForce = BoundaryNaturalBC::OpFlux<NaturalForceMeshsets, 1, SPACE_DIM>;
  
  CHKERR pipeline_mng->setBoundaryRhsIntegrationRule(integration_rule);
  CHKERR AddHOOps<SPACE_DIM - 1, SPACE_DIM, SPACE_DIM>::add(
      pipeline_mng->getOpBoundaryRhsPipeline(), {NOSPACE}, "GEOMETRY");
  //! [Natural boundary condition]
  CHKERR BoundaryNaturalBC::AddFluxToPipeline<OpForce>::add(
      pipeline_mng->getOpBoundaryRhsPipeline(), mField, "U", {time_scale},
      "FORCE", Sev::inform);
 
  //! [Define gravity vector]
  CHKERR DomainNaturalBC::AddFluxToPipeline<OpBodyForce>::add(
      pipeline_mng->getOpDomainRhsPipeline(), mField, "U", {time_scale},
      "BODY_FORCE", Sev::inform);
 
  // Essential BC
  CHKERR bc_mng->removeBlockDOFsOnEntities(simple->getProblemName(), "REMOVE_X",
                                           "U", 0, 0);
  CHKERR bc_mng->removeBlockDOFsOnEntities(simple->getProblemName(), "REMOVE_Y",
                                           "U", 1, 1);
  CHKERR bc_mng->removeBlockDOFsOnEntities(simple->getProblemName(), "REMOVE_Z",
                                           "U", 2, 2);
  CHKERR bc_mng->pushMarkDOFsOnEntities<DisplacementCubitBcData>(
      simple->getProblemName(), "U");
 
  MoFEMFunctionReturn(0);
}
//! [Boundary condition]
 
//! [Push operators to pip]
MoFEMErrorCode Incompressible::OPs() {
  MoFEMFunctionBegin;
  auto simple = mField.getInterface<Simple>();
  auto pip_mng = mField.getInterface<PipelineManager>();
  auto bc_mng = mField.getInterface<BcManager>();
 
  auto integration_rule_vol = [](int, int, int approx_order) {
    return 2 * approx_order + geom_order - 1;
  };
 
  auto add_domain_base_ops = [&](auto &pip) {
    MoFEMFunctionBegin;
    CHKERR AddHOOps<SPACE_DIM, SPACE_DIM, SPACE_DIM>::add(pip, {H1, L2},
                                                          "GEOMETRY");
    MoFEMFunctionReturn(0);
  };
 
  auto add_domain_ops_lhs = [&](auto &pip) {
    MoFEMFunctionBegin;
 
    // This assemble A-matrix
    using OpMassPressure = FormsIntegrators<DomainEleOp>::Assembly<
        AT>::BiLinearForm<GAUSS>::OpMass<1, 1>;
    // This assemble B-matrix (preconditioned)
    using OpMassPressureStab = FormsIntegrators<DomainEleOpStab>::Assembly<
        AT>::BiLinearForm<GAUSS>::OpMass<1, 1>;
    //! [Operators used for RHS incompressible elasticity]
 
    //! [Operators used for incompressible elasticity]
    using OpGradSymTensorGrad = FormsIntegrators<DomainEleOp>::Assembly<
        AT>::BiLinearForm<IT>::OpGradSymTensorGrad<1, SPACE_DIM, SPACE_DIM, 0>;
    using OpMixScalarTimesDiv = FormsIntegrators<DomainEleOp>::Assembly<
        AT>::BiLinearForm<IT>::OpMixScalarTimesDiv<SPACE_DIM, coord_type>;
    //! [Operators used for incompressible elasticity]
 
    auto mat_D_ptr = boost::make_shared<MatrixDouble>();
    constexpr auto size_symm = (SPACE_DIM * (SPACE_DIM + 1)) / 2;
    mat_D_ptr->resize(size_symm * size_symm, 1);
 
    FTensor::Index<'i', SPACE_DIM> i;
    FTensor::Index<'j', SPACE_DIM> j;
    FTensor::Index<'k', SPACE_DIM> k;
    FTensor::Index<'l', SPACE_DIM> l;
 
    constexpr auto t_kd = FTensor::Kronecker_Delta_symmetric<int>();
    auto t_mat = getFTensor4DdgFromMat<SPACE_DIM, SPACE_DIM, 0>(*mat_D_ptr);
    t_mat(i, j, k, l) = -2. * mu * ((t_kd(i, k) ^ t_kd(j, l)) / 4.);
 
    pip.push_back(new OpMixScalarTimesDiv(
        "P", "U",
        [](const double, const double, const double) constexpr { return -1.; },
        true, false));
    pip.push_back(new OpGradSymTensorGrad("U", "U", mat_D_ptr));
 
    auto get_lambda_reciprocal = [](const double, const double, const double) {
      return 1. / lambda;
    };
    if (poisson_ratio < 0.5)
      pip.push_back(new OpMassPressure("P", "P", get_lambda_reciprocal));
 
    double eps_stab = 0;
    CHKERR PetscOptionsGetScalar(PETSC_NULL, "", "-eps_stab", &eps_stab,
                                 PETSC_NULL);
    if (eps_stab > 0)
      pip.push_back(new OpMassPressureStab(
          "P", "P", [eps_stab](double, double, double) { return eps_stab; }));
 
    MoFEMFunctionReturn(0);
  };
 
  auto add_domain_ops_rhs = [&](auto &pip) {
    MoFEMFunctionBegin;
 
    //! [Operators used for RHS incompressible elasticity]
    using OpDomainGradTimesTensor = FormsIntegrators<DomainEleOp>::Assembly<
        AT>::LinearForm<GAUSS>::OpGradTimesSymTensor<1, SPACE_DIM, SPACE_DIM>;
 
    using OpDivDeltaUTimesP = FormsIntegrators<DomainEleOp>::Assembly<
        AT>::LinearForm<GAUSS>::OpMixDivTimesU<1, SPACE_DIM, SPACE_DIM>;
 
    using OpBaseTimesScalarValues = FormsIntegrators<DomainEleOp>::Assembly<
        AT>::LinearForm<GAUSS>::OpBaseTimesScalar<1>;
 
    //! [Operators used for RHS incompressible elasticity]
 
    auto pressure_ptr = boost::make_shared<VectorDouble>();
    pip.push_back(new OpCalculateScalarFieldValues("P", pressure_ptr));
 
    auto div_u_ptr = boost::make_shared<VectorDouble>();
    pip.push_back(
        new OpCalculateDivergenceVectorFieldValues<SPACE_DIM>("U", div_u_ptr));
 
    auto grad_u_ptr = boost::make_shared<MatrixDouble>();
    pip.push_back(new OpCalculateVectorFieldGradient<SPACE_DIM, SPACE_DIM>(
        "U", grad_u_ptr));
 
    auto strain_ptr = boost::make_shared<MatrixDouble>();
    pip.push_back(new OpSymmetrizeTensor<SPACE_DIM>(grad_u_ptr, strain_ptr));
 
    auto get_four_mu = [](const double, const double, const double) {
      return -2. * mu;
    };
    auto minus_one = [](const double, const double, const double) constexpr {
      return -1.;
    };
 
    pip.push_back(new OpDomainGradTimesTensor("U", strain_ptr, get_four_mu));
 
    pip.push_back(new OpDivDeltaUTimesP("U", pressure_ptr, minus_one));
 
    pip.push_back(new OpBaseTimesScalarValues("P", div_u_ptr, minus_one));
 
    auto get_lambda_reciprocal = [](const double, const double, const double) {
      return 1. / lambda;
    };
    if (poisson_ratio < 0.5) {
      pip.push_back(new OpBaseTimesScalarValues("P", pressure_ptr,
                                                get_lambda_reciprocal));
    }
 
    MoFEMFunctionReturn(0);
  };
 
  CHKERR add_domain_base_ops(pip_mng->getOpDomainLhsPipeline());
  CHKERR add_domain_base_ops(pip_mng->getOpDomainRhsPipeline());
  CHKERR add_domain_ops_lhs(pip_mng->getOpDomainLhsPipeline());
  CHKERR add_domain_ops_rhs(pip_mng->getOpDomainRhsPipeline());
 
  CHKERR pip_mng->setDomainRhsIntegrationRule(integration_rule_vol);
  CHKERR pip_mng->setDomainLhsIntegrationRule(integration_rule_vol);
 
  MoFEMFunctionReturn(0);
}
//! [Push operators to pip]
 
//! [Solve]
struct SetUpSchur {
  static boost::shared_ptr<SetUpSchur>
  createSetUpSchur(MoFEM::Interface &m_field);
  virtual MoFEMErrorCode setUp(SmartPetscObj<TS> solver) = 0;
 
protected:
  SetUpSchur() = default;
};
 
MoFEMErrorCode Incompressible::tsSolve() {
  MoFEMFunctionBegin;
 
  auto simple = mField.getInterface<Simple>();
  auto pip_mng = mField.getInterface<PipelineManager>();
  auto is_manager = mField.getInterface<ISManager>();
 
  auto set_section_monitor = [&](auto solver) {
    MoFEMFunctionBegin;
    SNES snes;
    CHKERR TSGetSNES(solver, &snes);
    PetscViewerAndFormat *vf;
    CHKERR PetscViewerAndFormatCreate(PETSC_VIEWER_STDOUT_WORLD,
                                      PETSC_VIEWER_DEFAULT, &vf);
    CHKERR SNESMonitorSet(
        snes,
        (MoFEMErrorCode(*)(SNES, PetscInt, PetscReal, void *))SNESMonitorFields,
        vf, (MoFEMErrorCode(*)(void **))PetscViewerAndFormatDestroy);
    MoFEMFunctionReturn(0);
  };
 
  auto scatter_create = [&](auto D, auto coeff) {
    SmartPetscObj<IS> is;
    CHKERR is_manager->isCreateProblemFieldAndRank(simple->getProblemName(),
                                                   ROW, "U", coeff, coeff, is);
    int loc_size;
    CHKERR ISGetLocalSize(is, &loc_size);
    Vec v;
    CHKERR VecCreateMPI(mField.get_comm(), loc_size, PETSC_DETERMINE, &v);
    VecScatter scatter;
    CHKERR VecScatterCreate(D, is, v, PETSC_NULL, &scatter);
    return std::make_tuple(SmartPetscObj<Vec>(v),
                           SmartPetscObj<VecScatter>(scatter));
  };
 
  auto create_post_process_elements = [&]() {
    auto pp_fe = boost::make_shared<PostProcEle>(mField);
    auto &pip = pp_fe->getOpPtrVector();
 
    auto push_vol_ops = [this](auto &pip) {
      MoFEMFunctionBegin;
      CHKERR AddHOOps<SPACE_DIM, SPACE_DIM, SPACE_DIM>::add(pip, {H1},
                                                            "GEOMETRY");
 
      MoFEMFunctionReturn(0);
    };
 
    auto push_vol_post_proc_ops = [this](auto &pp_fe, auto &&p) {
      MoFEMFunctionBegin;
 
      auto &pip = pp_fe->getOpPtrVector();
 
      using OpPPMap = OpPostProcMapInMoab<SPACE_DIM, SPACE_DIM>;
 
      auto x_ptr = boost::make_shared<MatrixDouble>();
      pip.push_back(
          new OpCalculateVectorFieldValues<SPACE_DIM>("GEOMETRY", x_ptr));
      auto u_ptr = boost::make_shared<MatrixDouble>();
      pip.push_back(new OpCalculateVectorFieldValues<SPACE_DIM>("U", u_ptr));
 
      auto pressure_ptr = boost::make_shared<VectorDouble>();
      pip.push_back(new OpCalculateScalarFieldValues("P", pressure_ptr));
 
      auto div_u_ptr = boost::make_shared<VectorDouble>();
      pip.push_back(new OpCalculateDivergenceVectorFieldValues<SPACE_DIM>(
          "U", div_u_ptr));
 
      auto grad_u_ptr = boost::make_shared<MatrixDouble>();
      pip.push_back(new OpCalculateVectorFieldGradient<SPACE_DIM, SPACE_DIM>(
          "U", grad_u_ptr));
 
      auto strain_ptr = boost::make_shared<MatrixDouble>();
      pip.push_back(new OpSymmetrizeTensor<SPACE_DIM>(grad_u_ptr, strain_ptr));
 
      auto stress_ptr = boost::make_shared<MatrixDouble>();
      pip.push_back(new OpCalculateLameStress<SPACE_DIM>(
          mu, stress_ptr, strain_ptr, pressure_ptr));
 
      pip.push_back(
 
          new OpPPMap(
 
              pp_fe->getPostProcMesh(), pp_fe->getMapGaussPts(),
 
              {{"P", pressure_ptr}},
 
              {{"U", u_ptr}, {"GEOMETRY", x_ptr}},
 
              {},
 
              {{"STRAIN", strain_ptr}, {"STRESS", stress_ptr}}
 
              )
 
      );
 
      MoFEMFunctionReturn(0);
    };
 
    auto vol_post_proc = [this, push_vol_post_proc_ops, push_vol_ops]() {
      PetscBool post_proc_vol = PETSC_FALSE;
      CHKERR PetscOptionsGetBool(PETSC_NULL, "", "-post_proc_vol",
                                 &post_proc_vol, PETSC_NULL);
      if (post_proc_vol == PETSC_FALSE)
        return boost::shared_ptr<PostProcEle>();
      auto pp_fe = boost::make_shared<PostProcEle>(mField);
      CHK_MOAB_THROW(
          push_vol_post_proc_ops(pp_fe, push_vol_ops(pp_fe->getOpPtrVector())),
          "push_vol_post_proc_ops");
      return pp_fe;
    };
 
    auto skin_post_proc = [this, push_vol_post_proc_ops, push_vol_ops]() {
      PetscBool post_proc_skin = PETSC_TRUE;
      CHKERR PetscOptionsGetBool(PETSC_NULL, "", "-post_proc_skin",
                                 &post_proc_skin, PETSC_NULL);
      if (post_proc_skin == PETSC_FALSE)
        return boost::shared_ptr<SkinPostProcEle>();
 
      auto simple = mField.getInterface<Simple>();
      auto pp_fe = boost::make_shared<SkinPostProcEle>(mField);
      auto op_side = new OpLoopSide<DomainEle>(
          mField, simple->getDomainFEName(), SPACE_DIM, Sev::verbose);
      pp_fe->getOpPtrVector().push_back(op_side);
      CHK_MOAB_THROW(push_vol_post_proc_ops(
                         pp_fe, push_vol_ops(op_side->getOpPtrVector())),
                     "push_vol_post_proc_ops");
      return pp_fe;
    };
 
    return std::make_pair(vol_post_proc(), skin_post_proc());
  };
 
  boost::shared_ptr<SetPtsData> field_eval_data;
  boost::shared_ptr<MatrixDouble> vector_field_ptr;
 
  std::array<double, SPACE_DIM> field_eval_coords;
  int coords_dim = SPACE_DIM;
  CHKERR PetscOptionsGetRealArray(NULL, NULL, "-field_eval_coords",
                                  field_eval_coords.data(), &coords_dim,
                                  &doEvalField);
 
  if (doEvalField) {
    vector_field_ptr = boost::make_shared<MatrixDouble>();
    field_eval_data =
        mField.getInterface<FieldEvaluatorInterface>()->getData<DomainEle>();
    if constexpr (SPACE_DIM == 3) {
      CHKERR mField.getInterface<FieldEvaluatorInterface>()->buildTree3D(
          field_eval_data, simple->getDomainFEName());
    } else {
      CHKERR mField.getInterface<FieldEvaluatorInterface>()->buildTree2D(
          field_eval_data, simple->getDomainFEName());
    }
 
    field_eval_data->setEvalPoints(field_eval_coords.data(), 1);
    auto no_rule = [](int, int, int) { return -1; };
    auto field_eval_fe_ptr = field_eval_data->feMethodPtr.lock();
    field_eval_fe_ptr->getRuleHook = no_rule;
    field_eval_fe_ptr->getOpPtrVector().push_back(
        new OpCalculateVectorFieldValues<SPACE_DIM>("U", vector_field_ptr));
  }
 
  auto set_time_monitor = [&](auto dm, auto solver) {
    MoFEMFunctionBegin;
    boost::shared_ptr<MonitorIncompressible> monitor_ptr(
        new MonitorIncompressible(SmartPetscObj<DM>(dm, true),
                                  create_post_process_elements(), uXScatter,
                                  uYScatter, uZScatter, field_eval_coords,
                                  field_eval_data, vector_field_ptr));
    boost::shared_ptr<ForcesAndSourcesCore> null;
    CHKERR DMMoFEMTSSetMonitor(dm, solver, simple->getDomainFEName(),
                               monitor_ptr, null, null);
    MoFEMFunctionReturn(0);
  };
 
  auto set_essential_bc = [&]() {
    MoFEMFunctionBegin;
    // This is low level pushing finite elements (pipelines) to solver
    auto ts_ctx_ptr = getDMTsCtx(simple->getDM());
    auto pre_proc_ptr = boost::make_shared<FEMethod>();
    auto post_proc_rhs_ptr = boost::make_shared<FEMethod>();
 
    // Add boundary condition scaling
    auto time_scale = boost::make_shared<TimeScale>();
 
    pre_proc_ptr->preProcessHook = EssentialPreProc<DisplacementCubitBcData>(
        mField, pre_proc_ptr, {time_scale}, false);
    post_proc_rhs_ptr->postProcessHook =
        EssentialPostProcRhs<DisplacementCubitBcData>(mField, post_proc_rhs_ptr,
                                                      1.);
    ts_ctx_ptr->getPreProcessIFunction().push_front(pre_proc_ptr);
    ts_ctx_ptr->getPreProcessIJacobian().push_front(pre_proc_ptr);
    ts_ctx_ptr->getPostProcessIFunction().push_back(post_proc_rhs_ptr);
    MoFEMFunctionReturn(0);
  };
 
  auto set_schur_pc = [&](auto solver) {
    SNES snes;
    CHKERR TSGetSNES(solver, &snes);
    KSP ksp;
    CHKERR SNESGetKSP(snes, &ksp);
    PC pc;
    CHKERR KSPGetPC(ksp, &pc);
    PetscBool is_pcfs = PETSC_FALSE;
    PetscObjectTypeCompare((PetscObject)pc, PCFIELDSPLIT, &is_pcfs);
    boost::shared_ptr<SetUpSchur> schur_ptr;
    auto ts_ctx_ptr = getDMTsCtx(simple->getDM());
 
    auto A = createDMMatrix(simple->getDM());
    auto B = matDuplicate(A, MAT_DO_NOT_COPY_VALUES);
 
    CHK_MOAB_THROW(
        TSSetIJacobian(solver, A, B, TsSetIJacobian, ts_ctx_ptr.get()),
        "set operators");
    auto pre_proc_schur_lhs_ptr = boost::make_shared<FEMethod>();
    auto post_proc_schur_lhs_ptr = boost::make_shared<FEMethod>();
    pre_proc_schur_lhs_ptr->preProcessHook = [pre_proc_schur_lhs_ptr]() {
      MoFEMFunctionBegin;
      MOFEM_LOG("INCOMP_ELASTICITY", Sev::verbose) << "Lhs Zero matrices";
      CHKERR MatZeroEntries(pre_proc_schur_lhs_ptr->A);
      CHKERR MatZeroEntries(pre_proc_schur_lhs_ptr->B);
      MoFEMFunctionReturn(0);
    };
    post_proc_schur_lhs_ptr->postProcessHook = [this,
                                                post_proc_schur_lhs_ptr]() {
      MoFEMFunctionBegin;
      MOFEM_LOG("INCOMP_ELASTICITY", Sev::verbose) << "Lhs Assemble Begin";
      *(post_proc_schur_lhs_ptr->matAssembleSwitch) = false;
      CHKERR MatAssemblyBegin(post_proc_schur_lhs_ptr->A, MAT_FINAL_ASSEMBLY);
      CHKERR MatAssemblyEnd(post_proc_schur_lhs_ptr->A, MAT_FINAL_ASSEMBLY);
      CHKERR EssentialPostProcLhs<DisplacementCubitBcData>(
          mField, post_proc_schur_lhs_ptr, 1.,
          SmartPetscObj<Mat>(post_proc_schur_lhs_ptr->A, true))();
 
      CHKERR MatAssemblyBegin(post_proc_schur_lhs_ptr->B, MAT_FINAL_ASSEMBLY);
      CHKERR MatAssemblyEnd(post_proc_schur_lhs_ptr->B, MAT_FINAL_ASSEMBLY);
      CHKERR MatAXPY(post_proc_schur_lhs_ptr->B, 1, post_proc_schur_lhs_ptr->A,
                     SAME_NONZERO_PATTERN);
      MOFEM_LOG("INCOMP_ELASTICITY", Sev::verbose) << "Lhs Assemble End";
      MoFEMFunctionReturn(0);
    };
    ts_ctx_ptr->getPreProcessIJacobian().push_front(pre_proc_schur_lhs_ptr);
    ts_ctx_ptr->getPostProcessIJacobian().push_back(post_proc_schur_lhs_ptr);
 
    if (is_pcfs == PETSC_TRUE) {
      if (AT == AssemblyType::SCHUR) {
        schur_ptr = SetUpSchur::createSetUpSchur(mField);
        CHK_MOAB_THROW(schur_ptr->setUp(solver), "setup schur preconditioner");
      } else {
        auto set_pcfieldsplit_preconditioned_ts = [&](auto solver) {
          MoFEMFunctionBegin;
          auto bc_mng = mField.getInterface<BcManager>();
          auto name_prb = simple->getProblemName();
          SmartPetscObj<IS> is_p;
          CHKERR mField.getInterface<ISManager>()->isCreateProblemFieldAndRank(
              name_prb, ROW, "P", 0, 1, is_p);
          CHKERR PCFieldSplitSetIS(pc, PETSC_NULL, is_p);
          MoFEMFunctionReturn(0);
        };
        CHK_MOAB_THROW(set_pcfieldsplit_preconditioned_ts(solver),
                       "set pcfieldsplit preconditioned");
      }
      return boost::make_tuple(schur_ptr, A, B);
    }
 
    return boost::make_tuple(schur_ptr, A, B);
  };
 
  auto dm = simple->getDM();
  auto D = createDMVector(dm);
 
  uXScatter = scatter_create(D, 0);
  uYScatter = scatter_create(D, 1);
  if (SPACE_DIM == 3)
    uZScatter = scatter_create(D, 2);
 
  // Add extra finite elements to SNES solver pipelines to resolve essential
  // boundary conditions
  CHKERR set_essential_bc();
 
  auto solver = pip_mng->createTSIM();
  CHKERR TSSetFromOptions(solver);
 
  CHKERR set_section_monitor(solver);
  CHKERR set_time_monitor(dm, solver);
  auto [schur_pc_ptr, A, B] = set_schur_pc(solver);
 
  CHKERR TSSetSolution(solver, D);
  CHKERR TSSetUp(solver);
  CHKERR TSSolve(solver, NULL);
 
  MoFEMFunctionReturn(0);
}
//! [Solve]
 
//! [Check]
MoFEMErrorCode Incompressible::checkResults() { return 0; }
//! [Check]
 
static char help[] = "...\n\n";
 
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 CONTACT
  auto core_log = logging::core::get();
  core_log->add_sink(
      LogManager::createSink(LogManager::getStrmWorld(), "INCOMP_ELASTICITY"));
  LogManager::setLog("INCOMP_ELASTICITY");
  MOFEM_LOG_TAG("INCOMP_ELASTICITY", "Indent");
 
  try {
 
    //! [Register MoFEM discrete manager in PETSc]
    DMType dm_name = "DMMOFEM";
    CHKERR DMRegister_MoFEM(dm_name);
    //! [Register MoFEM discrete manager in PETSc
 
    //! [Create MoAB]
    moab::Core mb_instance;              ///< mesh database
    moab::Interface &moab = mb_instance; ///< mesh database interface
    //! [Create MoAB]
 
    //! [Create MoFEM]
    MoFEM::Core core(moab);           ///< finite element database
    MoFEM::Interface &m_field = core; ///< finite element database interface
    //! [Create MoFEM]
 
    //! [Load mesh]
    Simple *simple = m_field.getInterface<Simple>();
    CHKERR simple->getOptions();
    CHKERR simple->loadFile("");
    //! [Load mesh]
 
    //! [CONTACT]
    Incompressible ex(m_field);
    CHKERR ex.runProblem();
    //! [CONTACT]
  }
  CATCH_ERRORS;
 
  CHKERR MoFEM::Core::Finalize();
 
  return 0;
}
 
struct SetUpSchurImpl : public SetUpSchur {
 
  SetUpSchurImpl(MoFEM::Interface &m_field) : SetUpSchur(), mField(m_field) {}
  virtual ~SetUpSchurImpl() { S.reset(); }
  MoFEMErrorCode setUp(SmartPetscObj<TS> solver);
 
private:
  MoFEMErrorCode setOperator();
  MoFEMErrorCode setPC(PC pc);
 
  SmartPetscObj<DM> createSubDM();
 
  SmartPetscObj<Mat> S;
 
  MoFEM::Interface &mField;
 
  SmartPetscObj<DM> subDM;
};
 
MoFEMErrorCode SetUpSchurImpl::setUp(SmartPetscObj<TS> solver) {
  MoFEMFunctionBegin;
  auto simple = mField.getInterface<Simple>();
  auto pip = mField.getInterface<PipelineManager>();
  auto dm = simple->getDM();
 
  SNES snes;
  CHKERR TSGetSNES(solver, &snes);
  KSP ksp;
  CHKERR SNESGetKSP(snes, &ksp);
  PC pc;
  CHKERR KSPGetPC(ksp, &pc);
 
  MOFEM_LOG("INCOMP_ELASTICITY", Sev::inform) << "Setup Schur pc";
 
  if (S) {
    CHK_THROW_MESSAGE(MOFEM_DATA_INCONSISTENCY,
                      "Is expected that schur matrix is not allocated. This is "
                      "possible only is PC is set up twice");
  }
 
  auto create_sub_dm = [&]() {
    auto sub_dm = createDM(mField.get_comm(), "DMMOFEM");
    auto set_up = [&]() {
      MoFEMFunctionBegin;
      CHKERR DMMoFEMCreateSubDM(sub_dm, simple->getDM(), "SUB");
      CHKERR DMMoFEMSetSquareProblem(sub_dm, PETSC_TRUE);
      CHKERR DMMoFEMAddElement(sub_dm, simple->getDomainFEName());
      CHKERR DMMoFEMAddSubFieldRow(sub_dm, "U");
      CHKERR DMSetUp(sub_dm);
      MoFEMFunctionReturn(0);
    };
    CHK_THROW_MESSAGE(set_up(), "sey up dm");
    return sub_dm;
  };
 
  subDM = create_sub_dm();
  S = createDMMatrix(subDM);
  CHKERR MatSetBlockSize(S, SPACE_DIM);
 
  auto dm_is = getDMSubData(subDM)->getSmartRowIs();
  auto ao_up = createAOMappingIS(dm_is, PETSC_NULL);
  // Domain
  pip->getOpDomainLhsPipeline().push_front(createOpSchurAssembleBegin());
  pip->getOpDomainLhsPipeline().push_back(
      createOpSchurAssembleEnd({"P"}, {nullptr}, ao_up, S, false, false));
 
  auto pre_proc_schur_lhs_ptr = boost::make_shared<FEMethod>();
  auto post_proc_schur_lhs_ptr = boost::make_shared<FEMethod>();
 
  pre_proc_schur_lhs_ptr->preProcessHook = [this]() {
    MoFEMFunctionBegin;
    CHKERR MatZeroEntries(S);
    MoFEMFunctionReturn(0);
  };
 
  post_proc_schur_lhs_ptr->postProcessHook = [this, ao_up,
                                              post_proc_schur_lhs_ptr]() {
    MoFEMFunctionBegin;
 
    auto print_mat_norm = [this](auto a, std::string prefix) {
      MoFEMFunctionBegin;
      double nrm;
      CHKERR MatNorm(a, NORM_FROBENIUS, &nrm);
      MOFEM_LOG("INCOMP_ELASTICITY", Sev::verbose)
          << prefix << " norm = " << nrm;
      MoFEMFunctionReturn(0);
    };
 
    CHKERR MatAssemblyBegin(S, MAT_FINAL_ASSEMBLY);
    CHKERR MatAssemblyEnd(S, MAT_FINAL_ASSEMBLY);
    CHKERR EssentialPostProcLhs<DisplacementCubitBcData>(
        mField, post_proc_schur_lhs_ptr, 1, S, ao_up)();
 
#ifndef NDEBUG
    CHKERR print_mat_norm(S, "S");
#endif // NDEBUG
    MoFEMFunctionReturn(0);
  };
 
  auto ts_ctx_ptr = getDMTsCtx(simple->getDM());
  ts_ctx_ptr->getPreProcessIJacobian().push_front(pre_proc_schur_lhs_ptr);
  ts_ctx_ptr->getPostProcessIJacobian().push_back(post_proc_schur_lhs_ptr);
 
  SmartPetscObj<IS> is_p;
  CHKERR mField.getInterface<ISManager>()->isCreateProblemFieldAndRank(
      simple->getProblemName(), ROW, "P", 0, 1, is_p);
  CHKERR PCFieldSplitSetIS(pc, NULL, is_p);
  CHKERR PCFieldSplitSetSchurPre(pc, PC_FIELDSPLIT_SCHUR_PRE_USER, S);
 
  MoFEMFunctionReturn(0);
}
 
boost::shared_ptr<SetUpSchur>
SetUpSchur::createSetUpSchur(MoFEM::Interface &m_field) {
  return boost::shared_ptr<SetUpSchur>(new SetUpSchurImpl(m_field));
}