An energy stable one-field monolithic arbitrary Lagrangian–Eulerian formulation for fluid–structure interaction

Abstract

In this article we present a one-field monolithic finite element method in the Arbitrary Lagrangian-Eulerian (ALE) formulation for Fluid-Structure Interaction (FSI) problems. The method only solves for one velocity field in the whole FSI domain, and it solves in a monolithic manner so that the fluid solid interface conditions are satisfied automatically. We prove that the proposed scheme is unconditionally stable, through energy analysis, by utilising a conservative formulation and an exact quadrature rule. We implement the algorithm using both F-scheme and d-scheme, and demonstrate that the former has the same formulation in two and three dimensions. Finally several numerical examples are presented to validate this methodology, including combination with remesh techniques to handle the case of very large solid displacement.

An optimal control method for time-dependent fluid-structure interaction problems

“…These figures have a good agreement with the reference values given in Turek and Hron (2006) with a period and amplitude being 0.530 and 0.034 respectively. We note here that although our neo-Hookean solid model is different from the Saint Venant-Kirchhoff model in Turek and Hron (2006), however, these two models are equivalent when applying to solving this FSI benchmark problem (Kadapa et al 2018;Hecht and Pironneau 2017;Wang et al 2020), in the sense that they present the same numerical results as first presented in Turek and Hron (2006).…”

“…When solving the mesh equation, the mesh velocity follows the fluid velocity only at the interface. Afterwards, both the fluid and solid meshes are updated based on the mesh velocity, and this can improve the mesh quality for the fluid as well as the solid (Wang et al 2020).…”

“…Numerical methods for FSI problem have been intensively studied during the past decades. Based upon mesh types, the FSI numerical methods can be broadly categorised into fitted-mesh methods (Heil 2004;Hecht and Pironneau 2017;Tanaka and Kashiyama 2006) and unfittedmesh methods (Peskin 2002;Zhang et al 2004;Baaijens 2001;Boffi and Gastaldi 2016;Kreissl and Maute 2012); based upon solving strategies, these numerical methods include partitioned/segregated methods (Küttler and Wall 2008;Degroote et al 2009Degroote et al , 2013Bazilevs et al 2013) and monolithic/fully coupled methods (Heil 2004(Heil , 2008Muddle et al 2012;Wang et al 2017Wang et al , 2020; and based upon solving variables, there are one-field (one velocity) methods (Wang et al 2017(Wang et al , 2019aHecht and Pironneau 2017) and multi-field (velocity, displacement and Lagrange multiplier) methods (Muddle et al 2012;Boffi et al 2015Boffi et al , 2016. In a recent study (Wang et al 2020), we proposed an energystable one-field monolithic method based on an ALE fitted mesh, which is adopted to derive the optimality condition for the FSI control formulation in this article.…”

“…The time-coupled primal and adjoint problems can also be decoupled and solved in an iterative manner as done in Heners et al (2018), which also solves the FSI subproblems using a partitioned/decoupled method. In this article, we decouple the primal and adjoint equations in time; however, we solve both the FSI system and the adjoint FSI system using a one-field monolithic approach (Wang et al 2020).…”

Interaction of finitely-strained viscoelastic multipolar solids and fluids by an Eulerian approach

“…Yet, RMT goes just opposite when mapping the solid into actual Eulerian framework to be compatible with fluid, cf. [30,Ch.6] or also [40] for a comparison.…”