Numerical Simulation and Benchmarking of Fluid-Structure Interaction With Application to Hemodynamics

Abstract

Numerical techniques for solving the problem of fluid-structure interaction with an elastic material in a laminar incompressible viscous flow are described. An Arbitrary Lagrangian-Eulerian (ALE) formulation is employed in a fully coupled monolithic way, considering the problem as one continuum. The mathematical description and the numerical schemes are designed in such a way that more complicated constitutive relations (and more realistic for biomechanics applications) for the fluid as well as the structural part can be easily incorporated. We utilize the well-known Q 2 P 1 finite element pair for discretization in space to gain high accuracy and perform as time-stepping the 2nd order Crank-Nicholson, resp., Fractional-Step-θ-scheme for both solid and fluid parts. The resulting nonlinear discretized algebraic system is solved by a Newton method which approximates the Jacobian matrices by a divided differences approach, and the resulting linear systems are solved by iterative solvers, preferably of Krylov-multigrid type. For validation and evaluation of the accuracy of the proposed methodology, we present corresponding results for a new set of FSI benchmarking configurations which describe the self-induced elastic deformation of a beam attached to a cylinder in laminar channel flow, allowing stationary as well as periodically oscillating deformations. Then, as an example for fluid-structure interaction (FSI) in biomedical problems, the influence of endovascular stent implantation onto cerebral aneurysm hemodynamics is numerically investigated. The aim is to study the interaction of the elastic walls of the aneurysm with the geometrical shape of the implanted stent structure for prototypical 2D configurations. This study can be seen as a basic step towards the understanding of the resulting complex flow phenomena so that in future aneurysm rupture shall be suppressed by an optimal setting for the implanted stent geometry.

Proposal for Numerical Benchmarking of Fluid-Structure Interaction between an Elastic Object and Laminar Incompressible Flow

“…These benchmarks have been developed as a collaborative project in the DFG Research Unit 493. The configurations have been carefully chosen and validated via extensive numerical tests (see also [6,3]) with various CFD codes so that, as a main result, characteristic flow quantities can be provided which allow a quantitative validation and comparison of different numerical methods and software tools. As an extension, corresponding 3D simulations are planned as well as the embedding into outer optimization tools (see http://jucri.jyu.fi/?q=node/14 for a first attempt towards optimal control on the basis of the presented FSI1 configuration).…”

A Parallel Partitioned Approach On Fluid-Structure Interaction Simulations Using the Multiscale Universal Interface Coupling Library

“…Hooke's law is limited to small deformations of the structure, while the St. Vernant-Kirchhoff model is able to handle large deformations but remains valid for small strains [10,11]. The onestep scheme is implemented for time-stepping of both Hooke's law and the St. Vernant-Kirchhoff model.…”

Numerical investigation of the impact of inclined baffles and an elastic vibrating beam on the thermo-fluid behavior in a rectangular channel

“…where F represents the force exerted to the fluid-structure boundary, n denotes the normal vector on the boundary outwards, and I is the unit matrix. In this paper, the FSI solution method is used based on the method presented by Turek and Hron [19] as well as Razzaq et al [24].…”