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mfem/examples/dfem/plasticity.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.
#include <mfem.hpp>
// TODO: Do we want this to be included from mfem.hpp automatically now?
#include <fem/dfem/doperator.hpp>
#include <linalg/tensor.hpp>
#include <fstream>
using namespace mfem;
using mfem::internal::tensor;
constexpr int DIMENSION = 2;
template <typename T, int dim>
MFEM_HOST_DEVICE inline
tensor<T, 3, 3> tensor_to_3D(const tensor<T, dim, dim>& A)
{
tensor<T, 3, 3> A3D{};
for (int i = 0; i < dim; i++)
{
for (int j = 0; j < dim; j++)
{
A3D[i][j] = A[i][j];
}
}
return A3D;
}
template <typename Material, int dim = DIMENSION>
struct InternalStateQFunction
{
InternalStateQFunction() = default;
MFEM_HOST_DEVICE inline
auto operator()(
const tensor<real_t, dim, dim> &dudxi,
const tensor<real_t, dim, dim> &J,
const tensor<real_t, 10> &internal_state,
const double &w) const
{
auto invJ = inv(J);
auto dudX = dudxi * invJ;
auto dudX3D = tensor_to_3D(dudX);
//auto internal_state_new = get<1>(material(dudX3D, internal_state));
auto [stress, internal_state_new] = material(dudX3D, internal_state);
// real_t vm = sqrt(1.5)*norm(dev(stress));
// out << vm << " " << internal_state_new[9] << std::endl;
return mfem::tuple{internal_state_new};
}
Material material;
};
template <typename Material, int dim = DIMENSION>
struct MomentumRefStateQFunction
{
MomentumRefStateQFunction() = default;
MFEM_HOST_DEVICE inline
auto operator()(
const tensor<real_t, dim, dim> &dudxi,
const tensor<real_t, dim, dim> &J,
const tensor<real_t, 10> &internal_state,
const double &w) const
{
auto invJ = inv(J);
auto dudX = dudxi * invJ;
auto dudX3D = tensor_to_3D(dudX);
auto [P3D, Qnew] = material(dudX3D, internal_state);
auto P = mfem::internal::make_tensor<dim, dim>([&P3D](int i, int j) { return P3D[i][j]; });
auto JxW = det(J) * w * transpose(invJ);
return mfem::tuple{P * JxW};
}
Material material;
};
struct J2SmallStrain
{
static constexpr int dim = 3; ///< spatial dimension
static constexpr int n_internal_states = 10;
static constexpr double tol =
1e-10; ///< relative tolerance on residual mag to judge convergence of return map
real_t E; ///< Young's modulus
real_t nu; ///< Poisson's ratio
real_t sigma_y; ///< Yield strength
real_t Hi; ///< Isotropic hardening modulus
real_t density; ///< Mass density
/// @brief variables required to characterize the hysteresis response
struct InternalState
{
tensor<double, dim, dim> plastic_strain; ///< plastic strain
double accumulated_plastic_strain; ///< uniaxial equivalent plastic strain
};
MFEM_HOST_DEVICE inline
InternalState unpack_internal_state(const tensor<real_t, n_internal_states> &
packed_state) const
{
// we could use type punning here to avoid copies
auto plastic_strain = mfem::internal::make_tensor<dim, dim>(
[&packed_state](int i, int j) { return packed_state[dim*i + j]; });
real_t accumulated_plastic_strain = packed_state[n_internal_states - 1];
return {plastic_strain, accumulated_plastic_strain};
}
MFEM_HOST_DEVICE inline
tensor<real_t, n_internal_states> pack_internal_state(const
tensor<real_t, dim, dim> & plastic_strain,
real_t accumulated_plastic_strain) const
{
tensor<real_t, n_internal_states> packed_state{};
for (int i = 0, ij = 0; i < dim; i++)
{
for (int j = 0; j < dim; j++, ij++)
{
packed_state[ij] = plastic_strain[i][j];
}
}
packed_state[n_internal_states - 1] = accumulated_plastic_strain;
return packed_state;
}
MFEM_HOST_DEVICE inline
tuple<tensor<real_t, dim, dim>, tensor<real_t, n_internal_states>>
operator()(const tensor<real_t, dim, dim> & dudX,
const tensor<real_t, n_internal_states> & internal_state) const
{
auto I = mfem::internal::Identity<dim>();
const real_t K = E / (3.0 * (1.0 - 2.0 * nu));
const real_t G = 0.5 * E / (1.0 + nu);
auto [plastic_strain, accumulated_plastic_strain] = unpack_internal_state(
internal_state);
// (i) elastic predictor
auto el_strain = sym(dudX) - plastic_strain;
auto p = K * tr(el_strain);
auto s = 2.0 * G * dev(el_strain);
auto q = sqrt(1.5) * norm(s);
[[maybe_unused]] real_t delta_eqps = 0.0;
[[maybe_unused]] auto flow_strength = [this](real_t eqps) { return this->sigma_y + this->Hi*eqps; };
// (ii) admissibility
if (q - (sigma_y + Hi*accumulated_plastic_strain) > tol*sigma_y)
{
// (iii) return mapping
real_t delta_eqps = (q - sigma_y - Hi*accumulated_plastic_strain)/(3*G + Hi);
auto Np = 1.5 * s / q;
s -= 2.0 * G * delta_eqps * Np;
plastic_strain += delta_eqps * Np;
accumulated_plastic_strain += delta_eqps;
}
auto stress = s + p * I;
auto internal_state_new = pack_internal_state(plastic_strain,
accumulated_plastic_strain);
return {stress, internal_state_new};
}
};
class ElasticityOperator : public Operator
{
static constexpr int Displacement = 0;
static constexpr int Coordinates = 1;
static constexpr int InternalState = 2;
public:
class ElasticityJacobianOperator : public Operator
{
public:
ElasticityJacobianOperator(const ElasticityOperator *elasticity,
const Vector &x) :
Operator(elasticity->Height()),
elasticity(elasticity),
z(elasticity->Height())
{
ParGridFunction u(&elasticity->displacement_fes);
u.SetFromTrueDofs(x);
auto mesh_nodes = static_cast<ParGridFunction*>
(elasticity->displacement_fes.GetParMesh()->GetNodes());
momentum_du = elasticity->momentum->GetDerivative(Displacement, {&u}, {mesh_nodes, &elasticity->internal_state});
}
void Mult(const Vector &x, Vector &y) const override
{
z = x;
z.SetSubVector(elasticity->displacement_ess_tdof, 0.0);
momentum_du->Mult(z, y);
for (int i = 0; i < elasticity->displacement_ess_tdof.Size(); i++)
{
y[elasticity->displacement_ess_tdof[i]] =
x[elasticity->displacement_ess_tdof[i]];
}
}
const ElasticityOperator *elasticity;
std::shared_ptr<DerivativeOperator> momentum_du;
mutable Vector z;
};
template <typename Material>
ElasticityOperator(ParFiniteElementSpace &displacement_fes,
Array<int> &vel_ess_tdofs,
const IntegrationRule &displacement_ir,
ParametricFunction &internal_state,
Material material) :
Operator(displacement_fes.GetTrueVSize()),
density(1.0e3),
body_force(displacement_fes.GetTrueVSize()),
displacement_ess_tdof(vel_ess_tdofs),
displacement_fes(displacement_fes),
displacement_ir(displacement_ir),
internal_state(internal_state)
{
auto mesh = displacement_fes.GetParMesh();
mesh_nodes = static_cast<ParGridFunction*>(mesh->GetNodes());
ParFiniteElementSpace& mesh_fes = *mesh_nodes->ParFESpace();
{
auto solutions = std::vector
{
FieldDescriptor{Displacement, &displacement_fes},
};
auto parameters = std::vector
{
FieldDescriptor{Coordinates, &mesh_fes},
FieldDescriptor{InternalState, &internal_state.space}
};
momentum =
std::make_shared<DifferentiableOperator>(solutions, parameters, *mesh);
momentum->DisableTensorProductStructure();
mfem::tuple inputs{Gradient<Displacement>{}, Gradient<Coordinates>{}, None<InternalState>{}, Weight{}};
mfem::tuple outputs{Gradient<Displacement>{}};
auto momentum_qf = MomentumRefStateQFunction<Material, DIMENSION> {.material = material};
auto derivatives = std::integer_sequence<size_t, Displacement> {};
Array<int> solid_domain_attr(mesh->attributes.Max());
solid_domain_attr[0] = 1;
momentum->AddDomainIntegrator(
momentum_qf, inputs, outputs, displacement_ir, solid_domain_attr, derivatives);
}
{
Vector g(DIMENSION);
g = 0.0;
ParLinearForm body_force_lf(&displacement_fes);
body_force_coef = new VectorConstantCoefficient(g);
auto integ = new VectorDomainLFIntegrator(*body_force_coef);
integ->SetIntRule(&displacement_ir);
body_force_lf.AddDomainIntegrator(integ);
body_force_lf.Assemble();
body_force_lf.ParallelAssemble(body_force);
}
}
void Mult(const Vector &displacement, Vector &r) const override
{
momentum->SetParameters({mesh_nodes, &internal_state});
momentum->Mult(displacement, r);
r -= body_force;
r.SetSubVector(displacement_ess_tdof, 0.0);
}
void Reaction(const Vector &displacement, Vector &r) const
{
momentum->SetParameters({mesh_nodes, &internal_state});
momentum->Mult(displacement, r);
r -= body_force;
r.Neg();
}
Operator &GetGradient(const Vector &x) const override
{
jacobian_operator = std::make_shared<ElasticityJacobianOperator>(this, x);
return *jacobian_operator;
// fd_jacobian = std::make_shared<FDJacobian>(*this, x);
// return *fd_jacobian;
}
real_t density;
std::shared_ptr<DifferentiableOperator> momentum;
mutable std::shared_ptr<HypreParMatrix> A;
VectorConstantCoefficient *body_force_coef = nullptr;
Vector body_force;
ParGridFunction *mesh_nodes;
const Array<int> displacement_ess_tdof;
ParFiniteElementSpace &displacement_fes;
IntegrationRule displacement_ir;
ParametricFunction& internal_state;
mutable std::shared_ptr<ElasticityJacobianOperator> jacobian_operator;
mutable std::shared_ptr<FDJacobian> fd_jacobian;
};
class InternalStateUpdater : public Operator
{
public:
static constexpr int Displacement = 0;
static constexpr int Coordinates = 1;
static constexpr int InternalState = 2;
template <typename Material>
InternalStateUpdater(ParFiniteElementSpace &displacement_fes,
const IntegrationRule &displacement_ir,
ParametricFunction &internal_state,
Material material) :
Operator(displacement_fes.GetTrueVSize()),
displacement_fes(displacement_fes),
displacement_ir(displacement_ir),
internal_state(internal_state)
{
auto mesh = displacement_fes.GetParMesh();
mesh_nodes = static_cast<ParGridFunction*>(mesh->GetNodes());
ParFiniteElementSpace& mesh_fes = *mesh_nodes->ParFESpace();
auto solutions = std::vector
{
FieldDescriptor{Displacement, &displacement_fes}
};
auto parameters = std::vector
{
FieldDescriptor{Coordinates, &mesh_fes},
FieldDescriptor{InternalState, &internal_state.space}
};
op = std::make_shared<DifferentiableOperator>(solutions, parameters, *mesh);
op->DisableTensorProductStructure();
mfem::tuple inputs{Gradient<Displacement>{}, Gradient<Coordinates>{}, None<InternalState>{}, Weight{}};
mfem::tuple outputs{None<InternalState>{}};
auto qfunction = InternalStateQFunction<Material, DIMENSION> {.material = material};
// just a placeholder for now. We want vjps wrt both displacement and old internal state eventually
auto derivatives = std::integer_sequence<size_t, Displacement> {};
Array<int> solid_domain_attr(mesh->attributes.Max());
solid_domain_attr[0] = 1;
op->AddDomainIntegrator(
qfunction, inputs, outputs, displacement_ir, solid_domain_attr, derivatives);
}
void Mult(const Vector &displacement, Vector& internal_state_new) const override
{
op->SetParameters({mesh_nodes, &internal_state});
op->Mult(displacement, internal_state_new);
}
void VjpDisplacement(ParGridFunction &u, Vector& internal_state_old,
Vector& internal_state_new_bar, Vector& displacement_bar) const
{
// u, internal_state_old, internal_state_new_bar should be const
out << "Sizes " << "u " << u.Size() << ", qold " << internal_state_old.Size() <<
", qbar " << internal_state_new_bar.Size() << ", ubar " <<
displacement_bar.Size() << std::endl;
auto grad_op = op->GetDerivative(Displacement, {&u}, {mesh_nodes, &internal_state_old});
out << "grad_op " << grad_op->Height() << " " << grad_op->Width() << std::endl;
out << "grad_op^T " << grad_op->Width() << " " << grad_op->Height() <<
std::endl;
grad_op->MultTranspose(internal_state_new_bar, displacement_bar);
}
ParGridFunction *mesh_nodes;
ParFiniteElementSpace &displacement_fes;
std::shared_ptr<DifferentiableOperator> op;
IntegrationRule displacement_ir;
ParametricFunction& internal_state;
};
int main(int argc, char* argv[])
{
constexpr int dim = 2;
Mpi::Init();
const char* device_config = "cpu";
int polynomial_order = 1;
int ir_order = 2;
int refinements = 0;
int nonlinear_solver_type = 0;
OptionsParser args(argc, argv);
args.AddOption(&polynomial_order, "-o", "--order", "");
args.AddOption(&refinements, "-r", "--refinements", "");
args.AddOption(&ir_order, "-iro", "--integration-rule-order", "");
args.AddOption(&device_config, "-d", "--device",
"Device configuration string, see Device::Configure().");
args.AddOption(&nonlinear_solver_type, "-nls", "--nonlinear-solver", "");
args.ParseCheck();
Device device(device_config);
if (Mpi::Root() == 0)
{
device.Print();
}
out << std::setprecision(8);
Mesh mesh_serial = Mesh::MakeCartesian2D(1, 1, Element::QUADRILATERAL,
false, 1.0, 0.1);
mesh_serial.EnsureNodes();
auto mesh_beam = ParMesh(MPI_COMM_WORLD, mesh_serial);
out << "#el: " << mesh_beam.GetNE() << "\n";
H1_FECollection displacement_fec(polynomial_order, dim);
ParFiniteElementSpace displacement_fes(&mesh_beam, &displacement_fec, dim);
HYPRE_BigInt global_size = displacement_fes.GlobalTrueVSize();
if (Mpi::Root())
{
out << "Number of unknowns: " << global_size << "\n";
}
const IntegrationRule &displacement_ir =
IntRules.Get(displacement_fes.GetFE(0)->GetGeomType(),
2 * ir_order + displacement_fes.GetFE(0)->GetOrder());
constexpr int n_internal_state_variables = 10;
ParametricSpace internal_state_space(dim, n_internal_state_variables,
displacement_ir.GetNPoints(),
n_internal_state_variables*displacement_ir.GetNPoints()*mesh_beam.GetNE());
ParametricFunction internal_state(internal_state_space);
internal_state = 0.0;
ParametricFunction internal_state_old(internal_state_space);
internal_state_old = 0.0;
Array<int> bdr_attr_is_ess(mesh_beam.bdr_attributes.Max());
Array<int> displacement_ess_tdof;
Array<int> bc_tdof;
bdr_attr_is_ess = 0;
bdr_attr_is_ess[0] = 1;
displacement_fes.GetEssentialTrueDofs(bdr_attr_is_ess, bc_tdof, 1);
for (auto td : bc_tdof) { displacement_ess_tdof.Append(td); };
bdr_attr_is_ess = 0;
bdr_attr_is_ess[3] = 1;
displacement_fes.GetEssentialTrueDofs(bdr_attr_is_ess, bc_tdof, 0);
for (auto td : bc_tdof) { displacement_ess_tdof.Append(td); };
bdr_attr_is_ess = 0;
bdr_attr_is_ess[1] = 1;
displacement_fes.GetEssentialTrueDofs(bdr_attr_is_ess, bc_tdof, 0);
for (auto td : bc_tdof) { displacement_ess_tdof.Append(td); };
ParGridFunction u(&displacement_fes);
u = 0.0;
using Material = J2SmallStrain; // StVenantKirchhoff
Material material{.E = 1000.0, .nu = 0.25, .sigma_y = 0.53333, .Hi = 40.0, .density = 1.0};
// Material material{.mu = 0.5e6, .nu = 0.4};
ElasticityOperator elasticity(displacement_fes, displacement_ess_tdof,
displacement_ir, internal_state, material);
CGSolver solver(MPI_COMM_WORLD);
solver.SetAbsTol(0.0);
solver.SetRelTol(1e-10);
solver.SetMaxIter(1000);
solver.SetPrintLevel(2);
std::shared_ptr<NewtonSolver> nonlinear_solver;
if (nonlinear_solver_type == 0)
{
nonlinear_solver = std::make_shared<NewtonSolver>(MPI_COMM_WORLD);
}
// else if (nonlinear_solver_type == 1)
// {
// nonlinear_solver = std::make_shared<KINSolver>(MPI_COMM_WORLD, KIN_LINESEARCH);
// }
else
{
MFEM_ABORT("invalid nonlinear solver type");
}
nonlinear_solver->SetOperator(elasticity);
nonlinear_solver->SetRelTol(1e-9);
nonlinear_solver->SetMaxIter(25);
nonlinear_solver->SetSolver(solver);
nonlinear_solver->SetPrintLevel(1);
// variables for output
QuadratureSpace output_internal_state_space(mesh_beam, displacement_ir);
QuadratureFunction output_internal_state(&output_internal_state_space,
internal_state.GetData(), material.n_internal_states);
Vector r(displacement_fes.GetTrueVSize());
ParGridFunction reaction(&displacement_fes);
Vector end_forces_x(bc_tdof.Size());
ParaViewDataCollection dc("dfem_plasticity", &mesh_beam);
dc.SetHighOrderOutput(true);
dc.SetLevelsOfDetail(1);
dc.RegisterField("displacement", &u);
dc.RegisterField("reaction", &reaction);
dc.RegisterQField("internal_state", &output_internal_state);
dc.SetCycle(0);
dc.Save();
InternalStateUpdater internal_state_update(displacement_fes, displacement_ir,
internal_state, material);
//Vector q(internal_state_space.GetTotalSize());
auto applied_displacement = [](double t) { return 1.2e-2*t; };
real_t time = 0.0;
std::ofstream history_file("history_output.csv");
history_file << applied_displacement(time) << " " << 0.0 << std::endl;
Vector zero, x(displacement_fes.GetTrueVSize());
constexpr int max_cycles = 3;
const real_t dt = 1.0/(max_cycles - 1);
for (int cycle = 1; cycle < max_cycles; cycle++)
{
time += dt;
out << "-------------------------------------------" << std::endl;
out << "TIME STEP " << cycle << std::endl;
out << "t = " << time << std::endl;
real_t ubc = applied_displacement(time);
u.SetSubVector(bc_tdof, ubc);
u.GetTrueDofs(x);
nonlinear_solver->Mult(zero, x);
u.SetFromTrueDofs(x);
// update internal variables
internal_state_old.Set(1.0, internal_state);
internal_state_update.Mult(u, internal_state);
// Compute reactions
elasticity.Reaction(x, r);
reaction.SetFromTrueDofs(r);
reaction.GetSubVector(bc_tdof, end_forces_x);
real_t force = -end_forces_x.Sum();
out << "u = " << applied_displacement(time) << ", Force = " << force <<
std::endl;
history_file << applied_displacement(time) << " " << force << std::endl;
output_internal_state = internal_state;
dc.SetCycle(cycle);
dc.SetTime(time);
dc.Save();
}
// try to use the derivative to see if it works
ParametricFunction internal_state_bar(internal_state_space);
internal_state_bar = 1.0;
//ParGridFunction u_bar(displacement_fes);
Vector u_bar(displacement_fes.GetTrueVSize());
internal_state_update.VjpDisplacement(u, internal_state_old, internal_state_bar,
u_bar);
pretty_print(u_bar);
history_file.close();
return 0;
}