An energy stable monolithic Eulerian fluid‐structure finite element method

Abstract

Summary When written in an Eulerian frame, the conservation laws of continuum mechanics are similar for fluids and solids leading to a single set of variables for a monolithic formulation. Such formulations are well adapted to large displacement fluid‐structure configurations, but stability is a challenging problem because of moving geometries. In this article, the method is presented; time implicit discretizations are proposed with iterative algorithms well posed at each step, at least for small displacements; stability is discussed for an implicit in time finite element method in space by showing that energy decreases with time. The key numerical ingredient is the Characterics‐Galerkin method coupled with a powerful mesh generator. A numerical section discusses implementation issues and presents a few simple tests. It is also shown that contacts are easily handled by extending the method to variational inequalities. This paper deals only with incompressible neo‐Hookean Mooney‐Rivlin hyperelastic material in 2 dimensions in a Newtonian fluid, but the method is not limited to these; compressible and 3D cases will be presented later.

Numerical Study of a 3D Eulerian Monolithic Formulation for Incompressible Fluid-Structures Systems

“…It was argued in [18] that the same proof holds for scheme discretized with P 1 triangular elements for velocity and pressure because then Y transforms a tetrahedron into a tetrahedron and so (27) holds also in the discrete case. Compatible linear elements are either the P 1 − P 1 stabilized element or the P 1 element for the pressure with P 1 also for the velocity but on a mesh where each tetrahedron is divided in sub-tetrahedra from an inner additional vertex.…”

“…In [17,18] a similar monolithic numerical method was proposed for the fluid-structure equations in two dimensions. It was also shown to be energy-stable.…”

“…It was also shown to be energy-stable. For more details on the motivations behind such a radically different approach to FSI, see [18]. The basic idea is to write the equations in terms of velocity, pressure, and displacement, as usual but in an Eulerian frame, and then eliminate the displacement in the time-discretized equations; the result is a monolithic formulation which is well-posed at each time step.…”

“…The basic idea is to write the equations in terms of velocity, pressure, and displacement, as usual but in an Eulerian frame, and then eliminate the displacement in the time-discretized equations; the result is a monolithic formulation which is well-posed at each time step. Furthermore, to obtain energy stability it was suggested in [18] to update the structure with its own velocity and remesh the fluid domain at every time step.…”

“…• In Section 2 the equations are discretized implicitly in time as in [18,19]; then the displacements are eliminated and a non-linear system for the velocities and pressures remains. Spatial discretization is obtained by using stable finite element spaces like linear pressures and quadratic velocities on a tetrahedral mesh.…”