Finite element simulation techniques for incompressible fluid structure interaction with applications to bioengineering and optimization
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Date
2011-08-05
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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 Q2P1 finite element
pair for discretization in space to gain high accuracy and perform as time-stepping
the 2nd order Crank-Nicholson, resp., Fractional-Step-q -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 Krylovmultigrid
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.
Keywords: Fluid-structure interaction (FSI), monolithic FEM, ALE, multigrid, incompressible
laminar flow, bio-engineering, optimization, benchmarking.
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Keywords
ALE, FENI, Fluid structure interaction