Stability and Convergence Analysis of the Extensions of the Kinematically Coupled Scheme for the Fluid-Structure Interaction

Abstract

In this work we analyze the stability and convergence properties of a loosely-coupled scheme, called the kinematically coupled scheme, and its extensions for the interaction between an incompressible, viscous fluid and a thin, elastic structure. We consider a benchmark problem where the structure is modeled using a general thin structure model, and the coupling between the fluid and structure is linear. We derive the energy estimates associated with the unconditional stability of an extension of the kinematically coupled scheme, called the β -scheme. Furthermore, for the first time we present a priori estimates showing optimal, first-order in time convergence in the case when β = 1. We further discuss the extensions of our results to other fluid-structure interaction problems, in particular the fluid-thick structure interaction problem. The theoretical stability and convergence results are supported with numerical examples.1. Introduction. The interaction between an incompressible viscous fluid and an elastic structure has been of great interest due to various applications in different areas (see e.g. [8]). This problem is characterized by highly nonlinear coupling between two different physical phenomena. As a result, a comprehensive study of such problems remains a challenge [34]. The solution strategies for fluid-structure interaction (FSI) problems can be roughly classified as monolithic schemes and loosely or strongly coupled partitioned schemes. Monolithic algorithms, see for example [7,29,41,27,43,35], consist of solving the entire coupled problem as one system of algebraic equations. They, however, require well-designed preconditioners [27,3,33] and are generally quite expensive in terms of computational time and memory requirements. Hence, to obtain smaller and better conditioned sub-problems, reduce the computational cost and treat each physical phenomenon separately, partitioned numerical schemes that solve the fluid problem separately from the structure problem have been a popular choice. The development of partitioned numerical methods for FSI problems has been extensively studied [21,20,22,13,2,39,23,42,32,37,6,5,24], but the design of efficient schemes to produce stable, accurate results remains a challenge. Moreover, despite the recent developments, there are just a few works where the convergence is proved rigorously [40,39,23,24].A classical partitioned scheme, particularly popular in aerodynamics, is known as the Dirichlet-Neumann (DN) partitioned scheme [17,42,26]. The DN scheme consists of solving the fluid problem with a Dirichlet boundary condition (structure velocity) at the fluid-structure interface, and the structure problem with a Neumann boundary condition (fluid stress) at the interface. While the DN scheme features appealing properties such as modularity, simple implementation and fast computational time, it has been shown to be stable only if the structure density is much larger than the fluid density. This requirement is easily achieved in some applications like aerodynamics, but n...

Compressible Fluids Interacting with a Linear-Elastic Shell

“…The paper [38] deals with a cylindrical linear elastic/viscoelastic Koiter shell in two dimensions (the shell is prescribed by a one-dimensional curve). The papers [6,39] extend this to cylindrical three-dimensional fluid flows. Note that in [39] even nonlinear elastic behavior of the shell is allowed.…”

“…The results from [32] have been extended to some incompressible non-Newtonian cases in [31]; see also [24]. Results for incompressible fluids in cylindrical domains have been shown in [38,39] and [6]. The paper [38] deals with a cylindrical linear elastic/viscoelastic Koiter shell in two dimensions (the shell is prescribed by a one-dimensional curve).…”

Multi-component model of intramural hematoma

“…Hence, we solve the full problem using a partitioned approach so that the fluid dynamics of blood flow are solved separately from the aortic wall and IMH mechanics. To separate the fluid sub-problem from the composite structure sub-problem describing the aortic wall, we use a stable, partitioned, non-iterative scheme called the kinematically coupled β scheme, presented and analyzed by Bukač et al (2015) and Bukač and Muha (2016). It was shown in Bukač and Muha (2016) that the scheme is unconditionally stable and first-order accurate in time.…”

Energy-based operator splitting approach for the time discretization of coupled systems of partial and ordinary differential equations for fluid flows: The Stokes case

“…In addition, the framework presented here could serve as a basis for extensions to (i) higher order time-discretization methods via, for example, symmetrization [17] or time-extrapolation [21,22]; (ii) different PDE systems in each Ω l region deriving, for example, from Navier-Stokes equations [18], fluid-structure interactions [19,32], non-Newtonian fluid flows [33] or porous media flows [34,35]; and (iii) different spatial discretization methods for each PDE problem including, for example, higher order finite element methods [36] or hybridizable discontinuous Galerkin methods [34,37,38]. The extensions mentioned above are relatively straight-forward, since the algorithms described in the corresponding references are directly implementable on that presented in this work.…”