Unfitted Finite Element Method -- UNFEM

The idea is to discretize problems on complicated and/or time dependent domains by a standard finite element approach, but without fitting the boundary of the mesh to the boundary of the computational domain. Instead, the mesh elements are clipped to the domain of integration during the assembly of the discrete systems of equations. Care is needed to keep the condition numbers of the associated matrices under control.

click to enlarge

click to enlarge

Dirichlet boundary conditions can be prescribed by means of a weight function which vanishes on the boundary. The pictures show examples for Dirichlet and Neumann boundary conditions for a 3d toy-problem, a Poisson equation.

Of course, the goal is to use UNFEM for more than just a mere Poissoin equation. The movie on the left shows a 2d-simulation of the incompressible Navier Stokes equations with free capillary boundaries, a falling drop which recombines with the fluid filling the bottom of the container.

The simulation prescribes slip boundary condtions on the bottom and periodic boundary conditions on the left and right side of the container.

G=(-4,4)×(0,8), Re=100, surface tension 1/20, ball's initial velocity (0,-2.2), gravity 0.05 (downward).

The simulations where performed by a recent version of

, adhanced by an add-on library implementing the unfitted finite element method, the pictures where produced using

, the movie was generated with the help of Paraview.

Claus-Justus Heine

_{Last modified: Fri Aug 22 15:35:27 CEST 2008}

Deutsche Version