Distributed Shape Derivatives for Level-Set Topology Optimization and Their Applications to Robust Topology Optimization

In this thesis, we address challenges faced by level-set topology optimization methods for linear elastic structures.We focus on the formulation, analysis, and implementation of distributed shape derivatives which provide accurate approximations. Conventionally used boundary-based shape derivatives have high regularity requirements that are typically not met in practical applications and converge at a slower rate than the associated objective functionals. We provide \textit{a priori} error analysis and numerical comparisons of boundary-based and distributed shape derivatives of linear objective functionals for topology optimization. We analyze the error in the degree-$k$ polynomial finite element approximations of the two expressions; we show that, for sufficiently regular problems, the boundary-based and distributed shape derivatives provide $k$-th and $2k$-th order accurate approximations, respectively, of the true shape derivative. We then assess, through numerical examples, the practical implications of using distributed versus boundary-based shape derivatives in topology optimization problems; we demonstrate that methods based on the distributed shape derivative yield more robust solutions to topology optimization problems. Next, we investigate problems with nonlinear stress-based objective functionals which are highly sensitive to minor changes in geometry. We derive the distributed shape derivative and extend the shape derivative error analysis to stress minimization problems, where the boundary-based and distributed shape derivatives are $k$-th and $2k$-th order accurate approximations under idealized conditions. We also provide numerical comparisons of the shape derivative errors for practical examples. We then use the distributed shape derivative to perform topology optimization for stress minimization problems without the use of problem-specific heuristics or regularization techniques. Finally, we present a non-intrusive approach to robust structural topology optimization for problems with probabilistic uncertainties in the loading and material properties. We approximate the solution to the stochastic linear elasticity equations using an anchored ANOVA Petrov-Galerkin projection scheme and develop a non-intrusive quadrature-based formulation to evaluate the robustness metric and associated shape derivative. This method significantly reduces the computational cost of evaluating the robustness metric where conventional polynomial chaos methods scale exponentially with respect to the number of random variables. We demonstrate the effectiveness of the proposed approach on various problems under loading and material uncertainties.

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