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Von Mises truss calculation

Description

iks007_vonMisesTruss.cpp utilizes the tools and features mentioned in the previous examples to solve the standard Von-Mises truss example found in literature (refer to Section 21).

Code highlights

The struct named Truss is created such that it inherits from PowerBasisFE. It must be decorated with AutoDiffFE as well to compute the stiffness matrix and load vectors during construction. It is constructed as shown below: 

Truss(const Basis &basis, const typename LocalView::Element &element, double p_EA)
    : BaseDisp(basis.flat(), element), EA{p_EA} {
  this->localView().bind(element);
}
It takes a reference to the basis function (&basis), the element (&element), and the axial stiffness of the truss structure (p_EA) as arguments during construction.

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 defined as 0.5 * EA / sqrt(LRefsquared) * Egl * Egl with Egl being the Green-Lagrange strain defined as 

const Scalar Egl = 0.5 * (lsquared - LRefsquared) / LRefsquared;
The grid in this example is created explicitly. The grid is created here from the dune-foamgrid module. This allows the user to embed a one- or two-dimensional grid in a physical space of any dimension. Thus, here we embed a one-dimensional truss system in a 2D plane. The height (h) and length (L) of the truss system are defined, which is followed by the addition of the vertices and elements to create a grid as shown below: 
Dune::GridFactory<Dune::FoamGrid<1, 2, double>> gridFactory;
const double h = 1.0;
const double L = 1.0;
gridFactory.insertVertex({0, 0});
gridFactory.insertVertex({L, h});
gridFactory.insertVertex({2 * L, 0});
gridFactory.insertElement(Dune::GeometryTypes::line, {0, 1});
gridFactory.insertElement(Dune::GeometryTypes::line, {1, 2});
auto grid     = gridFactory.createGrid();
auto gridView = grid->leafGridView();
The Lagrange basis is used to approximate the displacement field. The Truss elements are then created, followed by the fixing of the degrees of freedom at the boundaries ({0,0} and {2 * L,0}). A vertical downward load is applied to the center node. The non-linear operator is then constructed. The Newton-Raphson method is used as the non-linear solver, and an nonLinearSolverObserver is created to write messages as desired by the non-linear solver. An additional lvkObserver is created using the Ikarus::GenericControlObserver feature. This observer helps to fill up the matrix lambdaAndDisp with the load factor lambda and the two unconstrained degrees of freedom whenever the solution is changed (ControlMessages::SOLUTION_CHANGED), which means that the Newton-Raphson method has converged to a solution. This is implemented as depicted in the following: 
const int loadSteps = 10;
Eigen::Matrix3Xd lambdaAndDisp;
lambdaAndDisp.setZero(Eigen::NoChange, loadSteps + 1);
auto lvkObserver = std::make_shared<Ikarus::GenericControlObserver>(ControlMessages::SOLUTION_CHANGED, [&](int step) {
  lambdaAndDisp(0, step) = lambda;
  lambdaAndDisp(1, step) = d[2];
  lambdaAndDisp(2, step) = d[3];
});
The load control method is used as the path-following strategy, and it is subscribed to both vtkWriter and lvkObserver. The features from Matplot++ are then used to plot the load-displacement curve from the matrix lambdaAndDisp.

Takeaways

  • Dune::FoamGrid can be used to embed one or two-dimensional grid entities into a multi-dimensional physical space.
  • A simple truss element can be constructed using the automatic differentiation procedure.
  • Ikarus::GenericControlObserver can be used to perform user-desired tasks at any desired point by observing a non-linear solver procedure.

  1. R. V. Mises. Über die stabilitätsprobleme der elastizitätstheorie. ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, 3(6):406–422, 1923. doi:10.1002/zamm.19230030602. ↩