2D Elasticity problem#

Keywords: PDE:Elasticity Method:FEM FE:Lagrange Solver:Direct

The elasticity problem illustrates how to use vector unknown in XLiFE++:

\begin{array}{r} {\begin{cases} - div (σ u) - ω^{2} u = f & in Ω \\ σ u * n = 0 & on \partial Ω \end{cases} \end{array}

For homogeneous isotropic material:

σ u = λ (div u) I + 2 μ ε u, ε_{i j} u = \partial_{i} u_{j} .

The variational formulation in $V = (H^{1} (Ω))^{3}$ is to find:

u \in V ∣ \forall v \in V, λ \int_{Ω} div u div \bar{v} + 2 μ \int_{Ω} ε u : ε \bar{v} - ω^{2} \int_{Ω} u \cdot \bar{v} = \int_{Ω} f \cdot \bar{v}

This is implemented as follows:

#include "xlife++.h"
using namespace xlifepp;

// data function
Vector<Real> f(const Point& P, Parameters& pa = defaultParameters)
{  Vector<Real> F(2, 0.); F(2)=-0.005; return F;}

int main(int argc, char** argv)
{
  init(argc, argv, _lang=en); // mandatory initialization of xlifepp

  // mesh rectangle
  Rectangle rect(_center=Point(0., 0.), _xlength=20, _ylength=2, _nnodes=Numbers(50, 5), _domain_name="Omega", _side_names=Strings("", "", "", "Gamma"));
  Mesh mesh2d(rect, _shape=triangle, _generator=gmsh);
  Domain omega=mesh2d.domain("Omega"), Gamma=mesh2d.domain("Gamma");
  // create P1 Lagrange interpolation
  Space V(_domain=omega, _interpolation=P1, _name="V");
  Unknown u(V, _name="u", _dim=2);  TestFunction v(u, _name="v");
  // create bilinear form and linear form
  Real lambda=167.06, mu=56,67, omg2=0, rho=7.86;
  BilinearForm auv = lambda*intg(omega, div(u)*div(v)) + 2*mu*intg(omega, epsilon(u) % epsilon(v)) - omg2*intg(omega, u|v);
  LinearForm fv=intg(omega, f|v);
  EssentialConditions ecs= (u|Gamma=0);
  TermMatrix A(auv, ecs, _name="A");
  TermVector B(fv, _name="B");
  //solve linear system AX=B using direct method
  TermVector U=directSolve(A, B);
  thePrintStream<<U;
  saveToFile("U", U, _format=vtu);

  // create the deformation of the mesh
  for (number_t i=0;i<mesh2d.nbOfNodes();i++)
    mesh2d.nodes[i] += U.evaluate(mesh2d.nodes[i]).value<std::vector<Real> >();
  saveToFile("Ud", mesh2d, _format=msh);

  return 0;
}

../_images/ex_2d_elasticity1.png — Fig. 31 Displacement and modulus of the solution of the elasticity 2D problem#