Immersed Boundary Methods for Optimization of Strongly Coupled Fluid-Structure Systems

Conventional methods for design of tightly coupled multidisciplinary systems, such as fluid-structure interaction (FSI) problems, traditionally rely on manual revisions informed by a loosely coupled linearized analysis. These approaches are both inaccurate for a multitude of applications, and they require an intimate understanding of the assumptions and limitations of the procedure in order to soundly optimize the design. Computational optimization, in particular topology optimization, has been shown to yield remarkable results for problems in solid mechanics using density interpolations schemes. In the context of FSI, however, well defined boundaries play a key role in both the design problem and the mechanical model. Density methods neither accurately represent the material boundary, nor provide a suitable platform to apply appropriate interface conditions. This thesis presents a new framework for shape and topology optimization of FSI problems that uses for the design problem the Level Set method (LSM) to describe the geometry evolution in the optimization process. The Extended Finite Element method (XFEM) is combined with a fictitiously deforming fluid domain (stationary arbitrary Lagrangian-Eulerian method) to predict the FSI response. The novelty of the proposed approach lies in the fact that the XFEM explicitly captures the material boundary defined by the level set iso-surface. Moreover, the XFEM provides a means to discretize the governing equations, and weak immersed boundary conditions are applied with Nitsche’s Method to couple the fields. The flow is predicted by the incompressible Navier-Stokes equations, and a finite-deformation solid model is developed and tested for both hyperelastic and linear elastic problems. Transient and stationary numerical examples are presented to validate the FSI model and numerical solver approach. Pertaining to the optimization of FSI problems, the parameters of the discretized level set function are defined as explicit functions of the optimization variables, and the parameteric optimization problem is solved by nonlinear programming methods. The gradients of the objective and constrains are computed by the adjoint method for the global monolithic fluid-solid system. Two types of design problems are explored for optimization of the fluid-structure response: 1) the internal structural topology is varied, preserving the fluid-solid interface geometry, and 2) the fluid-solid interface is manipulated directly, which leads to simultaneously configuring both internal structural topology and outer mold shape. The numerical results show that the LSM-XFEM approach is well suited for designing practical applications, while at the same time reducing the requirement on highly refined mesh resolution compared to traditional density methods. However, these results also emphasize the need for a more robust embedded boundary condition framework. Further, the LSM can exhibit greater dependence on initial design seeding, and can impede design convergence. In particular for the strongly coupled FSI analysis developed here, the thinning and eventual removal of structural members can cause jumps in the evolution of the optimization functions