A Newton multigrid framework for optimal control of fluid–structure interactions

Abstract

In this paper we consider optimal control of nonlinear time-dependent fluid structure interactions. To determine a time-dependent control variable a BFGS algorithm is used, whereby gradient information is computed via a dual problem. To solve the resulting ill conditioned linear problems occurring in every time step of state and dual equation, we develop a highly efficient monolithic solver that is based on an approximated Newton scheme for the primal equation and a preconditioned Richardson iteration for the dual problem. The performance of the presented algorithms is tested for one 2d and one 3d example numerically.

On the Impact of Fluid Structure Interaction in Blood Flow Simulations

“…Further, it allows to apply very simple iterations of Vanka-type, as smoother in the geometric multigrid preconditioner, that are easy to parallelize. In [15,16] we have demonstrated the efficiency of the approach in different 3d configurations. The implementation is based on the finite element toolkit Gascoigne3D [7].…”

Parallel time-stepping for fluid–structure interactions

“…We usually consider θ = 1 2 + θ 0 k with a small fixed parameter θ 0 , which results in a second order accurate scheme, see [34,37] and [41] for a numerical study applied to fluid-structure interactions. A proper combination of three substeps with different choices of θ would give the fractional step theta method which is of second order and strongly A-stable, see [49] or [15] for an application to fluid-structure interactions. The discretization of the nonlinear terms including time derivatives (e.g.…”

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

“…However, few studies of optimal FSI control, in particular time-dependent FSI problems, have appeared in the literature to date and there is no recognised benchmark solution for comparison. Although there has been some progress reported in Heners et al (2018), Chirco and Manservisi (2020), Failer and Richter (2020), and Wick and Wollner (2020), these are still very challenging both analytically and numerically. The problem falls into the category of inverse FSI problems of moving shape control (Moubachir and Zolesio 2006).…”

“…For example, Failer et al (2016) enforce the coupling condition weakly to analyse optimal control for a linear FSI problem, whereas Chirco and Manservisi (2020) and Chierici et al (2019) introduce an auxiliary mesh displacement in the solid domain to enforce the coupling condition. In Failer and Richter (2020) and Wick and Wollner (2020), the authors solve both solid velocity and displacement, together with fluid velocity and pressure using a monolithic Newton solver. In our previous work, we formulated the FSI problem using a one-field FEM scheme (Wang et al 2017(Wang et al , 2019a and solved it in a fully coupled system, so that the interface conditions are satisfied automatically in the primal FSI equations and are therefore not present in the adjoint FSI equations.…”