# Jansson 200802

### From NA-Wiki

General Galerkin ALE computation of fluid-structure interaction using a unified continuum formulation

The typical approach to fluid-structure interaction (FSI) problems is based on coupling separate discretizations of the fluid and the structure, by passing information over the fluid-structure boundary as the computation progresses in time. Such a coupling typically suffers from instability or expense. Monolithic methods are based on forming a full continuum equation for both fluid and structure and then discretizing. Such methods have proven to be very robust [1], but also so far rather complex as equations and in the discretization.

We present a Unified Continuum (UC) formulation of fluid-structure interaction where the basic conservation laws for mass, momentum and enrgy are unified for the full continuum of the combined fluid-structure domain, with only the constitutive laws being different. The structure constitutive equations are based on a stress rate formulation, computing the Cauchy stress directly. An Arbitrary Euler Lagrange (ALE) stabilized finite element method is used, which we refer to as a General Galerkin (G2) method.

Mesh smoothing is necessary for an ALE fluid formulation with large displacement or deformation of the domain. We briefly present an elastic mesh smoothing method based on a close variant of the structure part of the UC. We solve for the deformation gradient F as an unknown. This allows us to control the shape and size of each cell.

The FEniCS open source framework provides automated computation of the discrete system from the weak form of the UC and the mesh smoothing equations. By using published benchmarks, we compare the UC formulation against other monolithic methods. The UC implementation is published as open source in the Unicorn component of FEniCS, ensuring repeatability as part of the scientific method.

[1]: J. Hron, S. Turek. "A Monolithic FEM Solver for ALE Formulation of Fluid Structure Interaction with Configurations for Numerical Benchmarking". Computational Methods for Coupled Problems in Science and Engineering, Proceedings First International Conference on Computational Methods for Coupled Problems in Science and Engineering, 2005.