Robust Multisecant Quasi-Newton Variants for Parallel Fluid-Structure Simulations---and Other Multiphysics Applications

Multisecant quasi-Newton methods have been shown to be particularly suited to solve nonlinear fixed-point equations that arise from partitioned multiphysics simulations where the exact Jacobian is inaccessible. In all these methods, the underdetermined multisecant equation for the approximate (inverse) Jacobian is enhanced by a norm minimization condition. The standard choice is the minimization of the Frobenius norm of the approximate inverse Jacobian. In this setting, it is well known that transient fluid-structure simulations typically require the use of secant information also from previous time steps to achieve a small enough number of iterations per implicit time step. The number of these time steps highly depends on the application, the physical parameters, the used solvers, and the mesh resolution. Using too few leads to a relatively high number of iterations, while using too many leads not only to a computational overhead but also to an increase in the number of iterations as well. Determining the optimal number requires a costly trial-and-error process. In this paper, we present results for two different approaches to overcome this issue: The first approach is to use a modified method (presented in [F. Lindner, M. Mehl, K. Scheufele, and B. Uekermann, “A Comparison of Various Quasi-Newton Schemes for Partitioned Fluid-Structure Interaction,” in Proceedings of ECCOMAS Coupled Problems, Venice, 2015, pp. 1--12]) that minimizes the Frobenius norm of the difference between the current approximate (inverse) Jacobian and that of the previous time step. Thus, previous time step information is taken into account in an implicit and automatized way without magic parameters. The second approach is to use an unrestricted number of previous time steps in combination with a suitable filtering algorithm automatically removing secant information that is outdated (thus, slowing down convergence) or contradicting newer information (deteriorating the condition of the multisecant equation system). We present a novel algorithm to realize the first idea with linear complexity in the number of coupling surface unknowns (note that already storing the approximate inverse Jacobian would induce quadratic complexity) and the efficient parallel implementation for both approaches. This results in highly efficient, parallelizable, and robust iterative solvers applicable for surface coupling in many types of multiphysics simulations using black-box solver software. In addition, our numerical results for the fluid-structure benchmark (FSI3) from Turek and Hron [``Proposal for Numerical Benchmarking of Fluid-Structure Interaction between an Elastic Object and Laminar Incompressible Flow,” in Fluid-Structure Interaction, Springer, Berlin, 2006, pp. 371--385] and for a flow through a flexible tube with a large variety of parameter settings prove the robustness and numerical efficiency of the first approach in particular. The second approach can be shown to be highly sensitive to the choice of the filtering method for the secant information and not always robust for the filters we currently use.