Abstract
We present a parallel time-stepping method for fluid–structure interactions. The interaction between the incompressible Navier-Stokes equations and a hyperelastic solid is formulated in a fully monolithic framework. Discretization in space is based on equal order finite element for all variables and a variant of the Crank-Nicolson scheme is used as second order time integrator. To accelerate the solution of the systems, we analyze a parallel-in time method. For different numerical test cases in 2d and in 3d we present the efficiency of the resulting solution approach. We also discuss some challenges and limitations that are connected to the special structure of fluid–structure interaction problem. In particular, we will investigate stability and dissipation effects of the time integration and their influence on the convergence of the parareal method. It turns out that especially processes based on an internal dynamics (e.g.driven by the vortex street around an elastic obstacle) cause great difficulties. Configurations however, which are driven by oscillatory problem data, are well-suited for parallel time stepping and allow for substantial speedups.
Highlights
Fluid structure interactions appear in various problems ranging from classical applications in engineering like the design of ships or aircrafts, the design of wind turbines, but they are present in bio/medical systems describing the blood flow in the heart or in general problems involving the cardiovascular system
Numerical approaches can usually be classified into monolithic approaches, where the coupled fluid–structure interaction system is taken as one entity and into partitioned approaches, where two separate problems – for fluid and solid – are formulated and where the coupling between them is incorporated in terms of an outer algorithm
This test case corresponds to the 2d case discussed in Section 4.1 and since the dynamics is driven by the oscillatory inflow profile, it should be better suited for the Parareal algorithm than the previously discussed fsi-3 problem
Summary
Fluid structure interactions appear in various problems ranging from classical applications in engineering like the design of ships or aircrafts, the design of wind turbines, but they are present in bio/medical systems describing the blood flow in the heart or in general problems involving the cardiovascular system. Numerical approaches can usually be classified into monolithic approaches, where the coupled fluid–structure interaction system is taken as one entity and into partitioned approaches, where two separate problems – for fluid and solid – are formulated and where the coupling between them is incorporated in terms of an outer (iterative) algorithm. This second approach has the advantage that difficulties are isolated and that perfectly. Accurate results and efficient approaches for 3d problems are still very rare [16, 39] In this contribution we exploit the perspectives (and limitations) of parallel time-stepping schemes for fluid– structure interaction systems.
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