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Deformation of an incompressible rubber block

Description

iks003_incompressible_LinearElasticity.cpp uses finite element technology with displacement and pressure as independent degrees of freedom to simulate the deformation of an incompressible rubber block. The potential energy for such a system is defined in the Solid struct by the function calculateScalarImpl(const FERequirementType &par, const Eigen::VectorX<ScalarType> &dx). This function uses the principles of automatic differentiation to provide the stiffness matrix and other necessary quantities to perform a static structural analysis.

Code highlights

The struct named Solid is created which is not inherited from any class. It is constructed as shown below: 

Solid(const Basis &basis, const typename LocalView::Element &element, double emod, double nu)
    : localView_{basis.flat().localView()}, emod_{emod}, nu_{nu} {
  localView_.bind(element);
  mu_       = emod_ / (2 * (1 + nu_));
  lambdaMat = convertLameConstants({.emodul = emod_, .nu = nu_}).toLamesFirstParameter();
}
It takes a reference to the basis function (&basis), the element (&element), and the material parameters, namely Young's modulus (emod) and Poisson's ratio (nu), as arguments during construction. The function convertLameConstants() is a helper function to switch between the Lame parameters.

ScalarType calculateScalarImpl(const FERequirementType &par, const Eigen::VectorX<ScalarType> &dx) is then defined, returning a scalar value, in this case the energy. The energy is then calculated as follows: 

energy += (0.5 * (2 * mu_ * symgradu.squaredNorm() - 1 / lambdaMat * Dune::power(pressure, 2)) + pressure * divU
           - x.dot(fext))
          * geo.integrationElement(gp.position()) * gp.weight();  // plane strain for 2D
Here:

  • symgradu is the symmetric part of the gradient of displacements
  • lambdaMat is the first Lame parameter
  • pressure and x are the nodal pressure and current position, respectively
  • divU is the divergence of the displacement vector
  • fext is the external force vector
  • gp.position() and gp.weight() are the positions and weights from the quadrature rule
  • geo.integrationElement() returns the determinant of Jacobian required from the iso-parametric concept

A Yasp 2D grid1 is created of the size \(L\) x \(h\) with 20 elements in both directions, as shown below: 

using Grid        = Dune::YaspGrid<gridDim>;
const double L    = 1;
const double h    = 1;
const size_t elex = 20;
const size_t eley = 20;

Dune::FieldVector<double, 2> bbox       = {L, h};
std::array<int, 2> elementsPerDirection = {elex, eley};
auto grid                               = std::make_shared<Grid>(bbox, elementsPerDirection);
auto gridView                           = grid->leafGridView();
A linear Lagrangian basis is opted for the displacements and a constant basis for the pressure degrees of freedom using the composite basis feature from Dune, as shown below: 
auto basis = Ikarus::makeBasis(
gridView, composite(power<2>(lagrange<1>()), lagrange<0>()));
Here, power<2> is used to approximate the displacement field in both \(x\) and \(y\) directions. A vector of Solid finite elements that are decorated by AutoDiffFE are then constructed as shown below: 
std::vector<AutoDiffFE<Solid<decltype(basis)>>> fes;
for (auto &ele : elements(gridView))
  fes.emplace_back(basis, ele, Emod, nu);
The displacement degrees of freedom at position \(y=0\) are fixed using the following snippet: 
auto basisP = std::make_shared<const decltype(basis)>(basis);
Ikarus::DirichletValues dirichletValues(basisP->flat());
dirichletValues.fixDOFs([](auto &basis_, auto &dirichletFlags) {
  Dune::Functions::forEachBoundaryDOF(subspaceBasis(basis_, _0),
                   [&](auto &&localIndex, auto &&localView, auto &&intersection) {
                       if (std::abs(intersection.geometry().center()[1]) < 1e-8)
                         dirichletFlags[localView.index(localIndex)] = true;
  });
});
Here, all the element edges lying on the boundary of the physical domain are looped over and checked to see if the first index of the center of the edge (intersection.geometry().center()[1]) is closer to zero. If this is the case, the corresponding displacement degrees of freedom (obtained via subspaceBasis(basis_, _0)) are set to true and used by the assembler later.

A sparse and a dense assembler are used to arrive at the stiffness matrix and the external load vector using the finite element requirements as described here.

auto sparseFlatAssembler = SparseFlatAssembler(fes, dirichletValues);
auto denseFlatAssembler  = DenseFlatAssembler(fes, dirichletValues);
auto req = FErequirements().addAffordance(Ikarus::AffordanceCollections::elastoStatics);

auto fextFunction = [&](auto &&lambdaLocal, auto &&dLocal) -> auto & {
  req.insertGlobalSolution(Ikarus::FESolutions::displacement, dLocal)
      .insertParameter(Ikarus::FEParameter::loadfactor, lambdaLocal);
  return denseFlatAssembler.getReducedVector(req);
};
auto KFunction = [&](auto &&lambdaLocal, auto &&dLocal) -> auto & {
  req.insertGlobalSolution(Ikarus::FESolutions::displacement, dLocal)
      .insertParameter(Ikarus::FEParameter::loadfactor, lambdaLocal);
  return sparseFlatAssembler.getReducedMatrix(req);
};
The SparseLU package from the Eigen library is used to solve the linear system of equations.

For post-processing, the function Dune::Functions::makeDiscreteGlobalBasisFunction() is used to create a function for the displacements and pressure using the basis functions and the nodal values. Dune::VTKWriter is used to write the *.vtu files. The results can then be plotted, for example, using Paraview. 

auto disp
    = Dune::Functions::makeDiscreteGlobalBasisFunction<Dune::FieldVector<double, 2>>(subspaceBasis(basis.flat(), _0), d);
auto pressure = Dune::Functions::makeDiscreteGlobalBasisFunction<double>(subspaceBasis(basis.flat(), _1), d);
Dune::VTKWriter vtkWriter(gridView, Dune::VTK::nonconforming);
vtkWriter.addVertexData(disp, Dune::VTK::FieldInfo("displacement", Dune::VTK::FieldInfo::Type::vector, 2));
vtkWriter.addVertexData(pressure, Dune::VTK::FieldInfo("pressure", Dune::VTK::FieldInfo::Type::scalar, 1));
vtkWriter.write("iks003_incompressibleLinearElasticity");

Takeaways

  • Ikarus::AutoDiffFE can be used to arrive at the stiffness matrix and external load vector from the energy function.
  • Easier implementation of mixed finite elements is possible due to the composite basis feature from Dune.
  • Helper functions are included to switch between the Lame parameters.
  • Grids from Dune can be directly incorporated within the Ikarus framework.
  • Sparse and dense assemblers can be used to construct the global stiffness matrices and load vectors.
  • Solvers from the Eigen library can be used to solve the linear system of equations.
  • Post-processing can be done via Paraview after writing the *.vtu files using Dune::VTKWriter.

  1. Oliver Sander. DUNE—The Distributed and Unified Numerics Environment. Volume 140. Springer Nature, 2020. doi:10.1007/978-3-030-59702-3. ↩