Flow simulation on moving boundary-fitted grids and application to fluid-structure interaction problems

Abstract

SUMMARYWe present a method for the parallel numerical simulation of transient three-dimensional fluidstructure interaction problems. Here, we consider the interaction of incompressible flow in the fluid domain and linear elastic deformation in the solid domain. The coupled problem is tackled by an approach based on the classical alternating Schwarz method with non-overlapping subdomains, the subproblems are solved alternatingly and the coupling conditions are realized via the exchange of boundary conditions. The elasticity problem is solved by a standard linear finite element method. A main issue is that the flow solver has to be able to handle time-dependent domains. To this end, we present a technique to solve the incompressible Navier-Stokes equation in three-dimensional domains with moving boundaries. This numerical method is a generalization of a finite volume discretization using curvilinear coordinates to time-dependent coordinate transformations. It corresponds to a discretization of the Arbitrary Lagrangian-Eulerian formulation of the Navier-Stokes equations. Here the grid velocity is treated in such a way that the so-called Geometric Conservation Law is implicitly satisfied. Altogether, our approach results in a scheme which is an extension of the well-known MACmethod to a staggered mesh in moving boundary-fitted coordinates which uses grid-dependent velocity components as the primary variables.To validate our method, we present some numerical results which show that second order convergence in space is obtained on moving grids. Finally, we give the results of a fully coupled fluidstructure interaction problem. It turns out that already a simple explicit coupling with one iteration of the Schwarz method, i.e. one solution of the fluid problem and one solution of the elasticity problem per time step, yields a convergent, simple, yet efficient overall method for fluid-structure interaction problems.

The Schwarz alternating method in solid mechanics

“…In these works, different numerical schemes are applied in the atomistic and continuum regions, each restricted to its own subdomain, and the transfer of information from the continuum to the atomistic scale occurs in the overlap region. Schwarz-like methods have also been proposed to couple distinct physics in different subdomains, for example, fluid-structure interaction (Engel and Griebel [18]), and computational fluid dynamics with aero-acoustics (Borrel, Halpern, and Ryan [13] and Ryan, Halpern, and Borrel [52]). In these works, the governing equations are solved alternatively in each subdomain, and the necessary information, such as structure-fluid loading, is transferred through boundary conditions at the interface.…”

Optimal Solvers for PDE-Constrained Optimization

“…where the matrix K = { ∇φ i • ∇φ j } i,j=1...n is the discrete Laplacian (the stiffness matrix) and d contains the terms coming from the boundary values of u h . Thus (7) and (8) together are equivalent to (5) and (6). One way to solve this minimization problem is by considering the Lagrangian…”

An improved three-dimensional coupled fluid–structure model for Coriolis flowmeters

“…The latter proved to be more favourable regarding the computational time (or inner iterations per time step) needed for the cases analysed. Besides the fluid load zero-order predictor (already used by Engel and Griebel, 2006), a three-point prediction of the fluid load was also introduced.…”