Multigrid Solver With Domain Decomposition Smoothing for Steady-State Incompressible Fsi Problems

Abstract

Abstract. In this paper we investigate the numerical performance of a monolithic Newtonmultigrid solver with domain decomposition smoothers for the solution of a class of stationary incompressible FSI problems. The physics of the problem is described using a monolithic approach, where mass continuity and stress balance are automatically satisfied across the fluidsolid interface. The deformation of the fluid domain is taken into account within the nonlinear Newton iterations according to an Arbitrary Lagrangian Eulerian (ALE) scheme. Due to the complexity and variety of the operators, the implementation of the Jacobian matrix in the nonlinear iterations is not a trivial task. To this purpose, we make use of automatic differentiation tools for an exact computation of the Jacobian matrix. The numerical solution of steady-state problems is particularly challenging, due to the ill-conditioning of the induced stiffness matrix. Moreover, the enforcement of the incompressibility condition calls for the use of incompressible solvers either of mixed or segregated type. At each nonlinear outer iteration the resulting linearized system is solved with a geometric multigrid solver. We consider a GMRES smoother preconditioned by an Additive Schwarz Method (ASM). The domain decomposition of the preconditioner is driven by the natural splitting between fluid and solid domain. The numerical results of some benchmark tests for steady-state cases show agreement with the literature and an increased robustness with our choice of smoothers with respect to standard ones.

A monolithic ALE Newton–Krylov solver with Multigrid-Richardson–Schwarz preconditioning for incompressible Fluid-Structure Interaction

“…All these tests were validated with respect to the available literature results. We refer to [2] for the validation of FSI1-S-2D. In the time-dependent simulations both profiles are multiplied by α(t) given by…”

“…We leave the testing and profiling of the parallel version of our solver to a future work. average cond 1 9.21 · 10 5 3.70 · 10 6 1.60 · 10 7 5.99 · 10 7 2.42 · 10 8 max cond 1 1.55 · 10 6 6.66 · 10 6 2.68 · 10 7 1.06 · 10 8 4.23 · 10 8 average cond 2 2.99 · 10 5 1.52 · 10 6 1.54 · 10 7 9.02 · 10 8 6.69 · 10 9 max cond 2 3.03 · 10 5 6.22 · 10 6 8.64 · 10 7 5.36 · 10 9 3.82 · 10 10…”

A field‐split preconditioning technique for fluid‐structure interaction problems with applications in biomechanics

“…Recent works showed that the above equation ordering is ill‐conditioned for steady‐state problems, and it becomes unstable for time‐dependent problems as the time step increases. Such a behavior can be easily explained by looking at the diagonal terms and in the first and third rows of (26).…”

“…For steady state problems, such terms are identically zero, and for time dependent problems they become quite small for large time steps, resulting in a Jacobian matrix with zero or small diagonal terms. In Aulisa et al, a stable row/column pivoting alternative has been proposed,

…”

“…For the sake of simplicity, we refer to these two combinations as the AS and the FS preconditioner, respectively. See in Aulisa et al, 47,58 while the FS preconditioner is the novelty of this paper. We remark that the FS preconditioner described in this work is an additive preconditioner.…”