An ALE approach to mechano‐chemical processes in fluid–structure interactions

Abstract

Summary Mathematical modeling and simulation of fluid–structure interaction problems are in the focus of research already for a longer period. However, taking into account also chemical reactions, leading to structural changes, including changes of mechanical properties of the solid phase, is rather new but for many applications is highly important area. This paper formulates a model system for reactive flow and transport in a vessel system, the penetration of chemical substances into the solid wall. Inside the wall, reactions take place that lead to changes of volume and of the mechanical properties of the wall. Numerical algorithms are developed and used to simulate the dynamics of such a mechano‐chemical fluid–structure interaction problem. As a proof of concept scenario, plaque formation in blood vessels is chosen. The arbitrary Lagrangian Eulerian approach (ALE) is chosen to solve the systems numerically. Temporal discretization of the fully coupled monolithic model is accomplished by backward Euler scheme and spatial discretization by stabilized finite elements. The numerical approach is verified by numerical tests, and effective methods to maintain mesh qualities under large deformations are described. For realistic system parameters, the simulations show that the plaque formation in blood vessel is a long‐time effect. The time scale of the formation is in the simulation of comparable order as in reality. Copyright © 2016 John Wiley & Sons, Ltd.

Long-term simulation of large deformation, mechano-chemical fluid-structure interactions in ALE and fully Eulerian coordinates

“…ALE formulations are well-established and can be regarded as standard formulation for fluid-structure interaction problems (see [18,9,8] and the many references cited therein). They have also been applied to plaque growth problems [51].…”

A multiphysics approach for modeling early atherosclerosis

“…The cardiovascular mechanics is modeled by a FSI method [19,70,61,107,57] coupling an incompressible non-Newtonian fluid including embedded three-element Windkessels with a hyperelastic solid undergoing finite deformations, anisotropic growth and a change of its constitutive equation. Fig.…”

“…It is frequently stated that pulsatile flow should not be neglected [58,66,93] lying in contradiction to many models, which commonly assume stationary blood flows. The influence of the compliance of the artery is usually investigated using fluid-structure interaction (FSI) models allowing for a physiologically more realistic deformation of the artery wall [19,58,71,107,31]. However, the influence of the compliance on the atherosclerotic process has not been considered much [21].…”

An Immersed Method Based on Cut-Cells for the Simulation of 2D Incompressible Fluid Flows Past Solid Structures

“…Similar techniques have also been investigated by Maury et al[17] and justified from a mathematical point of view in [21]. These methods have been shown to be efficient in the context of several particles in a flow [19], and when considering possible collisions between them [32].Arbitrary Lagrangian Eulerian (ALE) methods have been developped for flows in geometries which vary in time (see [28,29,33] where authors use some of the ideas of [5]). The aim is to formulate the equation in a fixed reference domain, by using a mapping from the reference domain Ω(0) to the domain Ω(t) occupied by the fluid at time t. The position of the moving bodies, which correspond to the boundary of Ω(t), being available, the velocity field of these bodies defined on ∂Ω(t) have to be extended to Ω(t).…”

“…Arbitrary Lagrangian Eulerian (ALE) methods have been developped for flows in geometries which vary in time (see [28,29,33] where authors use some of the ideas of [5]). The aim is to formulate the equation in a fixed reference domain, by using a mapping from the reference domain Ω(0) to the domain Ω(t) occupied by the fluid at time t. The position of the moving bodies, which correspond to the boundary of Ω(t), being available, the velocity field of these bodies defined on ∂Ω(t) have to be extended to Ω(t).…”