Level set topology optimization of stationary fluid-structure interaction problems

This paper introduces a topology optimization approach that combines an explicit level set method (LSM) and the extended finite element method (XFEM) for designing the internal structural layout of fluid-structure interaction (FSI) problems. The FSI response is predicted by a monolithic solver that couples an incompressible Navier-Stokes flow model with a small-deformation solid model. The fluid mesh is modeled as an elastic continuum that deforms with the structure. The fluid model is discretized with stabilized finite elements and the structural model by a generalized formulation of the XFEM. The nodal parameters of the discretized level set field are defined as explicit functions of the optimization variables. The optimization problem is solved by a nonlinear programming method. The LSM-XFEM approach is studied for two- and three-dimensional FSI problems at steady-state and compared against a density topology optimization approach. The numerical examples illustrate that the LSM-XFEM approach convergences to well-defined geometries even on coarse meshes, regardless of the choice of objective and constraints. In contrast, the density method requires refined grids and a mass penalization to yield smooth and crisp designs. The numerical studies show that the LSM-XFEM approach can suffer from a discontinuous evolution of the design in the optimization process as thin structural members disconnect. This issue is caused by the interpolation of the level set field and the inability to represent particular geometric configurations in the XFEM model. While this deficiency is generic to the LSM-XFEM approach used here, it is pronounced by the nonlinear response of FSI problems.