Adaptive finite element approximation of multiphysics problems: A fluid–structure interaction model problem

Abstract

SUMMARYIn this paper we consider finite element simulation of the mechanical response of an elastic solid immersed into a viscous incompressible fluid flow. For simplicity, we assume that the mechanics of the solid is governed by linear elasticity and the motion of the fluid by the Stokes equation. For this one-way coupled multiphysics problem we derive an a posteriori error estimate using duality techniques. Based on the estimate we propose an adaptive algorithm that automatically constructs a suitable mesh for the fluid and solid computational domains given a specific goal quantity for the elastic problem.

Duality‐based adaptive model reduction for one‐way coupled thermoelastic problems

“…A general framework for adaptive multiphysics solvers is presented by Larson and Bengzon in , where an application to microelectromechanical systems (MEMSs)involving electrostatics, heat conduction, and elasticity is presented. This methodology has also been applied to pressure‐driven contaminant transport and fluid structure interaction . Related works on multiphysics include that of Carey et al for operator decomposition of elliptic systems and van der Zee et al for fluid structure interaction.…”

Worst-case multi-objective error estimation and adaptivity

“…Consider the Galerkin approximation u h = u h 0 + h g ∈ V h to u according to (6). The dual solution enables us to express the error in the objective functional, (u) − (u h ), without direct reference to the error in the approximation.…”

“…The first term in the error representation can be conceived of as the error induced by a discrepancy in the data of the weak formulation (3) and its Galerkin finite-element approximation (6). Typically, this discrepancy results from an incompatibility of the boundary data, viz.…”

“…It is noteworthy, however, that the use of dual solutions in a-posteriori error estimates, which forms the basis of goal-oriented error estimation, had already been pursued before by Johnson, Eriksson and Hansbo [10,21,22]. Goal-oriented adaptivity has been successfully applied to a wide variety of problems including compressible and incompressible flow problems [17,31,32], elasticity and plasticity [37,33,34], variational inequalities [38], fluid-structure interaction [45,13,36,6], free-boundary problems [46,47], phase-field models [42,44], and kinetic equations [19]. This overview is in fact far from exhaustive, and many more applications and contributions can be found in the literature.…”

Adaptive Model Reduction for Thermoelastic Problems

“…cf. [6,7,8], and the error estimate measures the difference between the reduced and the full finite element solution in a given quantity of interest. The results presented herein extends the results in [5] by allowing temperature dependendent elastic parameters, leading to a linearized thermal dual problem.…”