Author:
Dr. Sebastien Court | Karl-Franzens-Universität Graz | Austria
Addressing PDE problems that involve moving boundaries is a delicate issue since the geometry of the domain can make part of the unknowns and can generate strong nonlinearities.
Simulating this kind of situations could lead to consider moving computational domains; Instead of that, we use finite element formulations with a fictitious domain approach for which the boundary does not fit the mesh: The goal is to change the less things we need when the boundary moves through the time.
The method we present here has been first introduced in the paper of J. Haslinger and Y. Renard in 2009 [1] for the Poisson problem. We adapted this method for the Stokes problem [2], and next for Navier-Stokes problem [3]. A first illustration is the unsteady displacement of a solid immersed in a viscous incompressible fluid. An other illustration of this class of methods lies in an algorithmic framework. For instance, for solving numerically an inverse problem, the iterative update of an interface does not necessitate to re-mesh the whole domain. Then re-assembling the whole system is avoided, and so we spare time computation and resources. We will take as an example the problem of a crack inducing displacement discontinuities inside a volcano (see [4]).
[1] J. Haslinger and Y.Renard, A new fictitious domain approach inspired by the extended finite element method, SIAM J. Numer. Anal., 47 (2009), no. 2, pp. 1474--1499.
[2] S. Court, M. Fournié and A. Lozinski, A fictitious domain approach for the Stokes problem based on the extended finite element method, Int. J. Numer. Meth. Fluids., 74 (2014), pp.73--99.
[3] S. Court, M. Fournié, A fictitious domain finite element method for simulations of fluid-structure interactions: The Navier-Stokes equations coupled with a moving solid, Journal of Fluids and Structures, no. 55 (2015), pp.398--408.
[4] O. Bodart, V. Cayol, S. Court and J. Koko, XFEM based fictitious domain method for linear elasticity model with crack, to appear in SIAM Scientific Computing.