Computer Methods in Applied Mechanics and Engineering
Added-mass effect in the design of partitioned algorithms for fluid–structure problems
Introduction
In this work we focus our attention on the numerical simulation of an incompressible fluid interacting with a thin elastic structure. The target application is the simulation of blood flow in large arteries, although the analysis presented here applies to a much wider class of fluid–structure interaction (FSI) problems.
Among the most used numerical techniques in this context are the so-called “partitioned” time marching algorithms, which are based on subsequent solutions of the fluid and structure subproblems and allow one to reuse existing computational codes (see e.g. [22], [21], [18], [5], [4], [11], [13]).
Although loosely coupled (explicit) algorithms are successfully used in aeroelasticity (see [5] and the references therein), it is common experience in other applications [20], [10], [17] that they feature instabilities under certain choices of the physical parameters, typically when the densities of the fluid and the structure are comparable or when the domain has a slender shape, irrespectively of the choice of the time step. Conversely, strongly coupled (implicit) algorithms feature convergence problems under the same conditions. These issues are still not well understood and, in particular, very few mathematical explanations and mathematically formulated stability or convergence conditions are available, so far, also because of the high complexity and non-linearity of FSI problems.
The goal of this work is to provide a tool for the study of the stability and/or the convergence analysis of partitioned coupled algorithms, which is based on a “toy FSI model” that represents the interaction between a potential fluid and a linear elastic thin tube. We do not claim that this model is relevant to properly describe complex situations, like fluid–structure interaction in arteries since, for example, it does not include non-linearities and dissipation phenomena. Nevertheless, it retains important physical features common to more complex models: in particular, it reproduces propagation phenomena and takes into account the added-mass effect of the fluid on the structure, which is known to induce numerical difficulties [10]. This model problem is simple enough to perform mathematical and numerical studies but, at the same time, complex enough to mimic more realistic situations, at least in the case of incompressible fluids. As it will be shown in Sections 5 Stability of explicit time-marching schemes, 6 Implicit time-marching schemes, starting from this simple FSI model, we are able to derive stability and convergence conditions that are in excellent agreement with the numerical observations collected in much more complex situations. Moreover, this toy FSI model, and the type of analysis proposed in this paper, may be helpful also in devising new and, possibly, more efficient coupled algorithms.
The work is organized as follows. In Section 2 we give some details on empirical observations and results concerning the stability of partitioned algorithms. In Section 3 we present the simplified fluid–structure model on which we are going to carry out our analysis. In Section 4 we introduce the functional setting and we present the FSI problem as a structure equation modified by the introduction of an added-mass operator that represents the effect of the fluid on the structure itself. In Sections 5 Stability of explicit time-marching schemes, 6 Implicit time-marching schemes we consider different explicit and implicit schemes to advance in time the interface FSI problem and we propose mathematical criteria to individuate the range of values of the physical parameters leading to numerical instabilities. In Section 7 we briefly address the space discretization of the FSI problem and we present numerical results that validate the mathematical analysis of the preceding section. In Section 8 we use scaling arguments to reproduce and extend to a wider spectrum of situations, even if in a more qualitative way, the results of Sections 5 Stability of explicit time-marching schemes, 6 Implicit time-marching schemes. Conclusions are eventually drawn in Section 9.
Section snippets
Motivations
In order to motivate the main results of this article, we recall some empirical observations made on a basic FSI test case proposed in previous studies (see [6], [7], [17]). The goal of this test case is to simulate, in a very idealized framework, the mechanical interaction between blood and arterial wall. The geometry at rest is a cylinder. The fluid (see [12] for a comprehensive discussion on blood rheology) is described by the incompressible Navier–Stokes equations in Arbitrary Lagrangian
A simplified fluid–structure problem
We consider a rectangular domain whose boundary is split into , , and Σ (see Fig. 2). The part Σ corresponds to the fluid–structure interface. In this simplified model, the domain ΩS occupied by the structure is such that . We set . We denote by n the unit outward normal vector on ∂ΩF.
Functional setting
We introduce the functional spacesWe denote by (· , ·) the L2(ΩF) or L2(ΩS) inner products and we define the bilinear forms:In all this section, the Hilbert space V will be endowed with the scalar product defined by aS(· , ·). We make the following regularity assumptions on the boundary data for the fluidand on the initial data for the structureIn the following,
Stability of explicit time-marching schemes
We present in this section a stability analysis of an explicit time-marching scheme for the temporal discretization of the FSI problem presented in Section 3. The space discretization will be addressed in the next section. Moreover, for the sake of simplicity and under the scaling considerations presented in Section 8 for the problem at hand, we assume that the coefficient b in (1) is zero. The differential structural operator defined in (10) therefore reduces to . The results obtained in
Implicit time-marching schemes
Proposition 3 suggests that explicit coupling algorithms might not work for fluid–structure interaction problems in certain conditions. An obvious remedy consists in switching to implicit couplings. As a prototype of implicit coupling algorithms we choose here the one obtained by combining the Implicit Euler scheme for the fluid with the first order backward difference scheme for the structure (IE-BDF scheme). The time-discrete problem reads:
Numerical results
In the following we present a series of numerical tests carried out on a code implementing the toy FSI problem and using the different time-marching schemes discussed in the previous sections. The aim of these tests is to provide a numerical validation of the above theoretical results. Before presenting the results, we need to introduce some further notation.
Characterization of the numerical properties of partitioned schemes through dimensional analysis
The analysis carried out in the preceding sections shows the dependence of the numerical stability of the solution of the added-mass problem on the values of the physical parameters. To highlight this fact, another possible way to proceed is proposed in [16], [10], where scaling arguments are used to provide indications about the influence of the physical parameters. This analysis applies to the linear case. In the present section we will pursue a similar approach: this will allow us to put the
Conclusions
In this work we have presented a mathematical contribution to explain the numerical instabilities encountered using time-partitioned schemes in presence of certain combinations of physical and geometrical parameters in the simulation of a class of fluid–structure interaction problems. Our target application is the simulation of blood flow in large human arteries, but the discussion applies as well to FSI problems encountered in other applications in which an incompressible fluid interacts with
Acknowledgement
This work has been partially supported by the Research Training Network “Mathematical Modelling of the Cardiovascular System” (HaeMOdel), contract HPRN-CT-2002-00270 of the European Community.
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