The Fixed-Mesh ALE method applied to multiphysics problems using stabilized formulations

Abstract

The finite element method is a tool very often employed to deal with the numerical simulation of multiphysics problems.Many times each of these problems can be attached to a subdomain in space which evolves in time. Fixed grid methods appear in order to avoid the drawbacks of remeshing in ALE (Arbitrary Lagrangian-Eulerian) methods when the domain undergoes very large deformations. Instead of having one mesh attached to each of the subdomains, one has a single mesh which covers the whole computational domain. Equations arising from the finite element analysis are solved in an Eulerian manner in this background mesh. In this work we present our particular approach to fixed mesh methods, which we call FM-ALE (Fixed-Mesh ALE). Our main concern is to properly account for the advection of information as the domain boundary evolves. To achieve this, we use an arbitrary Lagrangian-Eulerian framework, the distinctive feature being that at each time step results are projected onto a fixed, background mesh, that is where the problem is actually solved.We analyze several possibilities to prescribe boundary conditions in the context of immersed boundary methods. When dealing with certain physical problems, and depending on the finite element space used, the standard Galerkin finite element method fails and leads to unstable solutions. The variational multiscale method is often used to deal with this instability. We introduce a way to approximate the subgrid scales on the boundaries of the elements in a variational twoscale finite element approximation to flow problems. The key idea is that the subscales on the element boundaries must be such that the transmission conditions for the unknown, split as its finite element contribution and the subscale, hold. We then use the subscales on the element boundaries to improve transmition conditions between subdomains by introducing the subgrid scales between the interfaces in homogeneous domain interaction problems and at the interface between the fluid and the solid in fluid-structure interaction problems. The benefits in each case are respectively a stronger enforcement of the stress continuity in homogeneous domain decomposition problems and a considerable improvement of the behaviour of the iterative algorithm to couple the fluid and the solid in fluid-structure interaction problems. We develop FELAP, a linear systems of equations solver package for problems arising from finite element analysis. The main features of the package are its capability to work with symmetric and unsymmetric systems of equations, direct and iterative solvers and various renumbering techniques. Performance is enhanced by considering the finite element mesh graph instead of the matrix graph, which allows to perform highly efficient block computations.