Embedding Domain Technique for a Fluid-Structure Interaction Problem

Abstract

Abstract. We present a weak formulation for a steady fluid-structure interaction problem using an embedding domain technique with penalization. Except of the penalizing term, the coefficients of the fluid problem are constant and independent of the deformation of the structure, which represents an advantage of this approach. A second advantage of this model is the fact that the continuity of the stress at the fluid-structure interface does not appear explicitly. Numerical results are presented.Keywords: fluid-structure interaction, embedding domain. A Steady Fluid-Structure Interaction ProblemThe present paper is devoted to the study of the numerical behavior of an elastic structure immersed in a viscous incompressible fluid. We use Stokes equation to model the flow motion. The displacement of the structure under the flow motion will be modeled by linear elasticity equations, under the small deformations assumption. In this paper, we study the steady case.Let D ⊂ R 2 be a bounded open domain with boundary ∂D. Let Ω S 0 be the undeformed structure domain, and suppose that its boundary admits the decomposition ∂Ω S 0 = Γ D ∪ Γ 0 , where Γ 0 is a relatively open subset of the boundary. On Γ D we impose zero displacement for the structure. We assume that Ω S 0 ⊂ D. Suppose that the structure is elastic and denote by u = (u 1 , u 2 ) : Ω S 0 → R 2 its displacement. A particle of the structure with initial position at the point X will occupy the position x = ϕ (X) = X + u (X) in the deformed domainWe set Γ u = ϕ (Γ 0 ) and we suppose that Γ u does not touch the container wall, i.e. ∂D ∩ Γ u = ∅. We recall that Γ 0 is a relatively open subset. The boundary Γ u represents the moving fluid-structure interface. The boundary of the deformed D.

Steady Fluid-Structure Interaction Using Fictitious Domain

“…The equations (1), (2) concern the structure, while (3)-(6) concern the fluid. The equations (7), (8) represent the boundary conditions on the moving fluidstructure interface. The fluid and the structure domains Ω F u , Ω S u depend on the structure displacement u which is unknown.…”