On the existence and the uniqueness of the solution to a fluid-structure interaction problem

In this paper we consider the linearized version of a system of partial differential equations arising from a fluid-structure interaction model. We prove the existence and the uniqueness of the solution under natural regularity assumptions

Higher-order time-stepping schemes for fluid-structure interaction problems

We consider a recently introduced formulation for fluid-structure interaction problems which makes use of a distributed Lagrange multiplier in the spirit of the fictitious domain method. In this paper we focus on time integration methods of second order based on backward differentiation formulae and on the Crank-Nicolson method. We show the stability properties of the resulting method; numerical tests confirm the theoretical results

Unfitted mixed finite element methods for elliptic interface problems

In this paper, new unfitted mixed finite elements are presented for elliptic interface problems with jump coefficients. Our model is based on a fictitious domain formulation with distributed Lagrange multiplier. The relevance of our investigations is better seen when applied to the framework of fluid structure interaction problems. Two finite elements schemes with piecewise constant Lagrange multiplier are proposed and their stability is proved theoretically. Numerical results compare the performance of those elements, confirming the theoretical proofs and verifying that the schemes converge with optimal rate.Comment: 29 pages, 16 figures, 18 table

Immersed boundary method: performance analysis of popular finite element spaces

The aim of this paper is to understand the performances of different finite elements in the space discretization of the Finite Element Immersed Boundary Method. In this exploration we will analyze two popular solution spaces: Hood-Taylor and Bercovier- Pironneau (P1-iso-P2). Immersed boundary solution is characterized by pressure discontinuities at fluid structure interface. Due to such a discontinuity a natural enrichment choice is to add piecewise constant functions to the pressure space. Results show that P1 + P0 pressure spaces are a significant cure for the well known “boundary leakage” affecting IBM. Convergence analysis is performed, showing how the discontinuity in the pressure is affecting the convergence rate for our finite element approximation

A parallel solver for FSI problems with fictitious domain approach

We present and analyze a parallel solver for the solution of fluid structure interaction problems described by a fictitious domain approach. In particular, the fluid is modeled by the non-stationary incompressible Navier-Stokes equations, while the solid evolution is represented by the elasticity equations. The parallel implementation is based on the PETSc library and the solver has been tested in terms of robustness with respect to mesh refinement and weak scalability by running simulations on a Linux cluster.Comment: Contribution to the 5th African Conference on Computational Mechanic

Mass preserving distributed langrage multiplier approach to immersed boundary method

This research is devoted to mass conservation and CFL properties of the Finite Elements Immersed Boundary Method. We first explore an enhanced higher order scheme applied to the Finite Element Immersed Boundary Method technique introduced by Boffi and Gastaldi. This technique is based on a Pointwise (PW) formulation of the kinematic condition, and higher order elements show better conservation properties than the original scheme. A further improvement with respect to the classical PW formulation is achieved introducing a fully variational Distributed Lagrange Multiplier (DLM) formulation. Numerical experiments show that DLM is not affected by any CFL condition. Furthermore the mass conservation properties of this method are extremely competitive

A parallel solver for fluid structure interaction problems with Lagrange multiplier

The aim of this work is to present a parallel solver for a formulation of fluid-structure interaction (FSI) problems which makes use of a distributed Lagrange multiplier in the spirit of the fictitious domain method. The fluid subproblem, consisting of the non-stationary Stokes equations, is discretized in space by $\mathcal{Q}_2$-$\mathcal{P}_1$ finite elements, whereas the structure subproblem, consisting of the linear or finite incompressible elasticity equations, is discretized in space by $\mathcal{Q}_1$ finite elements. A first order semi-implicit finite difference scheme is employed for time discretization. The resulting linear system at each time step is solved by a parallel GMRES solver, accelerated by block diagonal or triangular preconditioners. The parallel implementation is based on the PETSc library. Several numerical tests have been performed on Linux clusters to investigate the effectiveness of the proposed FSI solver.Comment: 27 pages, 8 figures, 14 table