We present a monolithic parallel Newton-multigrid solver for nonlinear three dimensional fluid-structure interactions in Arbitrary Lagrangian Eulerian (ALE) formulation. We start with a finite element discretization of the coupled problem, based on a remapping of the Navier-Stokes equation onto a fixed reference framework. The strongly coupled fluid-structure interaction problem is discretized with finite elements in space and finite differences in time. The resulting nonlinear and linear systems of equations are large and show a very high condition number.We present a novel Newton approach that is based on two essential ideas: First, a condensation of the solid deformation by exploiting the discretized velocity-deformation relation d t u = v. Second, the Jacobian of the fluid-structure interaction system is simplified by neglecting all derivatives with respect to the ALE deformation, an approximation that has shown to have little impact. The resulting system of equations decouples into a joint momentum equation and into two separated equations for the deformation fields in solid and fluid. Besides a reduction of the problem sizes, the approximation has a positive effect on the conditioning of the systems such that multigrid solvers with simple smoothers like a parallel Vanka-iteration can be applied.We demonstrate the efficiency of the resulting solver infrastructure on a wellstudied 2d test-case and we also introduce a challenging 3d problem. For 3d problems we achieve a substantial accelaration as compared to established approaches found in literature.

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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.

In this paper, we consider optimal control problems governed by linear unsteady fluid-structure interaction problems. Based on a novel symmetric monolithic formulation, we derive optimality systems and provide regularity results for optimal solutions. The proposed formulation allows for natural application of gradient-based optimization algorithms and for space-time finite element discretizations.

We study the impact of using fluid-structure interactions (FSI) to simulate blood flow in a stenosed artery. We compare typical flow configurations using Navier–Stokes in a rigid geometry setting to a fully coupled FSI model. The relevance of vascular elasticity is investigated with respect to several questions of clinical importance. Namely, we study the effect of using FSI on the wall shear stress distribution, on the Fractional Flow Reserve and on the damping effect of a stenosis on the pressure amplitude during the pulsatile cycle. The coupled problem is described in a monolithic variational formulation based on Arbitrary Lagrangian Eulerian (ALE) coordinates. For comparison, we perform pure Navier–Stokes simulations on a pre-stressed geometry to give a good matching of both configurations. A series of numerical simulations that cover important hemodynamical factors are presented and discussed.