Artificial added mass instabilities in sequential staggered coupling of nonlinear structures and incompressible viscous flows

doi:10.1016/j.cma.2006.09.002

Computer Methods in Applied Mechanics and Engineering

Volume 196, Issue 7, 10 January 2007, Pages 1278-1293

https://doi.org/10.1016/j.cma.2006.09.002 Get rights and content

Abstract

Within this paper the so-called artificial added mass effect is investigated which is responsible for devastating instabilities within sequentially staggered Fluid–structure Interaction (FSI) simulations where incompressible fluids are considered.

A discrete representation of the added mass operator $M_{A}$ is given and ‘instability conditions’ are evaluated for different temporal discretisation schemes. It is proven that for every sequentially staggered scheme and given spatial discretisation of a problem, a mass ratio between fluid and structural mass density can be found at which the coupled system becomes unstable. The analysis is quite general and does not depend upon the particular spatial discretisation schemes used. However here special attention is given to stabilised finite elements employed on the fluid partition. Numerical investigations further highlight the results.

Introduction

Partitioned approaches are rather popular for the simulation of surface coupled problems as they allow to use specifically designed codes on the different fields and offer significant benefits in terms of efficiency. Smaller and better conditioned subsystems are solved instead of one overall problem. We distinguish loose and strong coupling partitioned schemes. Loose coupling requires only one solution of either field per time step in a sequentially staggered manner and is thus particularly appealing in terms of efficiency. However sequentially staggered schemes exhibit an inherent instability when used on fluid–structure interaction problems where incompressible flows are considered. The instability can be overcome by iteratively staggered partitioned schemes also called strongly coupled which utilise subiterations over the single fields to converge to the solution of the monolithic system.

The present work aims at further clarifying the destabilising property of the so-called added mass effect [1], [2], [3] on sequentially staggered fluid–structure calculations. It has been observed that so-called loose coupling of fluid and structural part in the context of incompressible flow and slender structures frequently yields unstable computations. Surprisingly the instability depends upon the densities of fluid and structure and also on the geometry of the domain [1], [2], [3], [4]. Clearly sequential coupling introduces an explicit flavour into the computation even if both partitions themselves are solved implicitly. Thus restrictions on the time step have to be expected. Observations however show that decreasing of the time step results in an increased instability. The instability is inherent in the scheme itself and has been named ‘artificial added mass effect’ since major parts of the fluid act as an extra mass on the structural degrees of freedom at the coupling interface. In sequentially staggered schemes the fluid forces depend upon predicted structural interface displacements rather than the correct ones and thus contain a portion of incorrect coupling forces. It is this ‘artificial’ contribution to the coupling which yields the instability.

An upper limit on the time step size for staggered analysis of acoustic FSI depending on the ratio of structural and fluid mass has been obtained in [5]. For the incompressible limit case which comes along with an infinite speed of sound the result obtained there predicts instability irrespective of the time step size.

While the artificial added mass effect causes an instability of sequentially staggered schemes it also harms the convergence of iterative methods [4]. Thus in [6] it is proposed to increase the structural mass artificially on the left-hand-side of structural equations while leaving the residual on the right-hand-side unchanged ensuring that the converged solution is correct. A more sophisticated way to circumvent the instability has recently been proposed by Fernández et al. [25] who suggest to restrict the implicit coupling to the pressure in the context of a projection scheme.

The added mass effect already mentioned in [3], [7] has been investigated by means of a reduced model problem in [4] where it is shown that the onset of the instability can be predicted well within the simplified problem. Here we wish to investigate the effect of these results on different time discretisation schemes and want to understand the influence of small time steps on the onset of the instability of sequentially staggered schemes. Further we study the particular influence of a spatial discretisation of the fluid domain by stabilised finite elements on the artificial added mass effect.

A detailed analysis shows why more accurate schemes tend to be even more unstable. We employ generalised-α time integration on the structural part and consider backward Euler, the trapezoidal rule and second order backward differencing (BDF2) on the fluid domain. Further different predictors and ways to obtain the Dirichlet boundary condition on the fluid partition along the coupling interface are used.

In [8], [9] modifications at the load- and motion transfer in the context of coupled aeroelastic problems are suggested to improve the accuracy and stability of the overall scheme which show beneficial in particular for compressible flow. However when incompressible flow is considered these modifications might postpone the onset of the instability while being unable to actually prevent it. Within this paper we show that every sequentially staggered scheme for incompressible flow will get unstable provided that the mass density ratio of fluid and structure is large enough.

The paper is organised as follows. In Section 2 we introduce the coupled problem and the sequentially staggered algorithm under consideration. Further the details of the temporal discretisation employed within the two fields are given. We derive discrete representations of the added mass operator in the regime of very small time steps for inf–sup stable and stabilised finite element discretisations of the fluid domain in Section 3. The stability properties of the overall scheme are investigated in Section 4. To this end the influence of the actual time discretisation scheme is considered. Here we distinguish between temporal discretisation schemes with limited recursion and fully recursive characteristics. While the former yield a constant instability limit in terms of a maximal eigenvalue of the respective added mass operator the latter results in a stability constant which gets more and more restrictive with every time step. Thus we show why schemes with instationary characteristics will definitely fail after a few time steps irrespective of the eigenvalues of the added mass operator. We further show that every sequential staggered coupling scheme has a limiting mass density ratio at which the scheme becomes unstable. A detailed numerical investigation reported in Section 5 confirms the theoretical findings.

Section snippets

Governing equations

We consider problems consisting of a fluid (occupying Ω^F) and a structural part (in Ω^S) which interact at the common boundary Γ. This interface Γ is represented by a curved line or surface in two- or three-dimensional problems, respectively. The possibly large motion of the structure is governed by $ρ^{S} \frac{D^{2} d}{D t^{2}} - \nabla \cdot (F \cdot S (d)) = ρ^{S} b^{S} in Ω^{S} \times (0, T),$ where d represents the structural displacement and b^S body forces applied on the structure. The second order tensor S denotes the second Piola–Kirchhoff stress

Spatial discretisation with LBB stable fluid elements

The fluid Eqs. (2), (3) are discretised in space by means of finite elements. The spatial domain Ω^F is divided into non-overlapping patches, the elements. The spatial discretisation maintains its topology while following the deformation of the domain. Within the present investigation of the instability due to the sequentially staggered coupling scheme the motion of the domain is a secondary matter and not taken into account subsequently.

To define the Galerkin form we select C⁰ Lagrangian finite

The influence of the discretisation in time

Introducing the coupling force (28), (45) into the discrete linearised structural Eq. (23) yields $[\begin{matrix} M_{II}^{S} & M_{I Γ}^{S} \\ M_{Γ I}^{S} & M_{Γ Γ}^{S} \end{matrix}] [\begin{matrix} {\ddot{d}}_{I} \\ {\ddot{d}}_{Γ} \end{matrix}] + [\begin{matrix} K_{II}^{S} & K_{I Γ}^{S} \\ K_{Γ I}^{S} & K_{Γ Γ}^{S} \end{matrix}] [\begin{matrix} d_{I} \\ d_{Γ} \end{matrix}] = [\begin{matrix} 0 \\ - m^{F} M_{A} {\dot{u}}_{Γ} \end{matrix}],$ where within the staggered scheme the fluid interface acceleration ${\dot{u}}_{Γ}$ is obtained from a structural prediction of the new interface displacement. The matrix K^S denotes the structural tangent stiffness obtained from a linearisation of the internal structural forces N^S(d).

Eq. (49) reveals why $M_{A}$ is named ‘added mass operator’. Identifying the

Numerical investigation

The classical driven cavity problem equipped with a thin flexible bottom is used to numerically investigate the added mass instability within a full fluid–structure interaction environment. The example which is taken from [1], [2], [3] is depicted in Fig. 1. The fluid domain is discretised by 32 × 32 stabilised bilinear elements. We wish to examine the influence of different parameters and discretisation schemes on the onset of the instability within the time interval t ∈ [0 s; 100 s]. To diagnose

Summary and conclusions

The destabilising added mass effect in the context of fluid–structure interaction computations is not an artifact of a particular discretisation scheme. It rather is an inherent property of sequentially staggered scheme itself and cannot be decreased by increasing the accuracy. Nevertheless there are time discretisation schemes for the fluid equations which allow for a stable calculation at high ratios between structural and fluid density. Such schemes which exhibit characteristics with a

Acknowledgements

The present study is supported by a grant of the German science foundation “Deutsche Forschungsgemeinschaft” (DFG) under project B4 of the collaborative research centre SFB 404 ‘Multifield Problems in Continuum Mechanics’. This support is gratefully acknowledged.

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