Splitting Methods Based on Algebraic Factorization for Fluid-Structure Interaction

Abstract

Abstract. We discuss in this paper the numerical approximation of fluid-structure interaction (FSI) problems dealing with strong added-mass effect. We propose new semi-implicit algorithms based on inexact block-LU factorization of the linear system obtained after the space-time discretization and linearization of the FSI problem. As a result, the fluid velocity is computed separately from the coupled pressure-structure velocity system at each iteration, reducing the computational cost. We investigate explicit-implicit decomposition through algebraic splitting techniques originally designed for the FSI problem. This approach leads to two different families of methods which extend to FSI the algebraic pressure correction method and the Yosida method, two schemes that were previously adopted for pure fluid problems. Furthermore, we have considered the inexact factorization of the fluid-structure system as a preconditioner. The numerical properties of these methods have been tested on a model problem representing a blood-vessel system. Key words. fluid-structure interaction, semi-implicit coupling, inexact factorization, algebraic splitting methods, added-mass effect, blood flow AMS subject classifications. 74F10, 65N30, 76D05 DOI. 10.1137/070680497 1. Introduction. We are interested in the numerical approximation of the heterogeneous mechanical system which couples the equations governing a fluid flow and the deformation of a structure; this situation arises in many engineering problems.A great variety of strategies have been proposed to solve fluid-structure interaction (FSI) problems. A first issue is how to deal with the nonlinearity of the problem. In fact, not only are the fluid (and in some cases the structure) equations nonlinear, but the structure displacement also modifies the fluid domain generating geometrical nonlinearities. The fixed point technique (e.g., [3,20,21,10]) is the simplest to linearize the FSI problem; however, Newton (e.g., [13]) and quasi-Newton (e.g., [30,15,19,17]) methods have also been considered.A classical restriction for fluid-structure algorithms is modularity. Most of the time the codes for the pure fluid problem and for the pure structure problem already exist, and they are optimized for the specific mathematical features of the two different problems. Then the best way to solve the FSI problem would be to design algorithms

Modeling and simulations for fluid and rotating structure interactions

“…We refer to [24,1,2,22,46,16,3,4,5,48,49] for details. In particular, when stable mixed finite element pairs are used, we can develop robust block preconditioners based on the well-posedness shown in Theorem 1.…”

“…the mass equation, in order to obtain a stable solution of velocity and pressure. The stabilization parameter δh 2 can be chosen as δ0h 2 µ f , and 0 < δ 0 < 1 is an appropriately tuned parameter. Comparing with other stable mixed elements such as P 2 -P 1 (Taylor-Hood) element, P 1 -P 1 element with pressure stabilization can save a great deal of computational cost with an acceptable accuracy, especially for the realistic three-dimensional FSI problems.…”

Some Numerical Approaches to Solve Fluid Structure Interaction Problems in Blood Flow

“…Badia et al [8] proposed a similar previous semi-implicit approach which was based on the inexact block-LU factorization of the linear system. The linear system was obtained after the space-time discretization and linearization of the fluid structure interaction problems.…”

“…From the theoretical point of view, such problems are complex and challenging due to the high nonlinearity of the problem. Not only the fluid equation exhibits nonlinearity, the structure displacement modifies the fluid domain which generates geometrical nonlinearities as well [8].…”

“…In addition, the implicit partitioned algorithms are also affected by the added mass effect in terms of convergence. Special treatment of the interface conditions needs to be considered [8,16,[28][29][30]. To date, it seems that only the monolithic and implicit schemes are applicable in blood flow simulation involving fluid structure interaction.…”