Topology optimization of stationary fluid–structure interaction problems including large displacements via the TOBS-GT method

This paper addresses the topology optimization of fluid–structure interaction (FSI) systems considering large displacements. We consider the steady-state analysis of flexible structures in contact with a fluid flow governed by the incompressible Navier–Stokes equations. The optimization method used in this work considers the physical analysis and optimization module in a decoupled form. The decoupled analysis allows the finite element problem to be meshed and solved accordingly to the physics requirements. Optimized geometry is constructed by reading and trimming out from an optimization grid described by a set of binary {0,1}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{0,1\}$$\end{document} design variables. The method is so-called TOBS (Topology Optimization of Binary Structures) with geometry trimming (TOBS-GT). Displacements are resolved using an elastic formulation with geometrical nonlinearities to allow for large deformations. The FSI system is solved by using finite elements and the Arbitrary Lagrangian–Eulerian (ALE) method. Low Reynolds numbers are assumed. The sensitivities are calculated using semi-automatic differentiation and interpolated to optimization grid points. In order to consider large displacements, a mapping between material and spatial coordinates is used to identify and track the deformed configuration of the structure. The optimized binary topology is found by using the standard TOBS approach (Sivapuram and Picelli in Finite Elem Anal Des 139:49–61, 2018) based on sequential integer linear programming. Numerical examples show that the TOBS-GT method can be effectively applied to design 2D and 3D structures in FSI problems including nonlinear structural responses.