On performance bounds for topology optimization

Abstract

Topology optimization has matured to become a powerful engineering design tool that is capable of designing extraordinary structures and materials taking into account various physical phenomena. Despite the method's great advancements in recent years, several unanswered questions remain. This paper takes a step towards answering one of the larger questions, namely: How far from the global optimum is a given topology optimized design? Typically this is a hard question to answer, as almost all interesting topology optimization problems are non-convex. Unfortunately, this non-convexity implies that local minima may plague the design space, resulting in optimizers ending up in suboptimal designs. In this work, we investigate performance bounds for topology optimization via a computational framework that utilizes Lagrange duality theory. This approach provides a viable measure of how \say{close} a given design is to the global optimum for a subset of optimization formulations. The method's capabilities are exemplified via several numerical examples, including the design of mode converters and resonating plates.

Figures (16)

• Figure 1: The mode converter geometry. The relative permittivity is fixed in white and black regions. The design domain $\Omega_d$ is the centered gray square in which the permittivity is to be determined.
• Figure 2: The topology optimized design when maximizing the normalized mode overlap.
• Figure 3: The normalized output $\vert E_z\vert/\vert\vert E_z\vert\vert_2$ and target mode profiles at the right-hand side boundary for the Fig. \ref{['fig:mc_eff_designs']} design.
• Figure 4: The (top) max normalized $\vert E_z\vert$ and (bottom) real part $\Re(E_z)$ over $\Omega$ corresponding to the Fig. \ref{['fig:mc_eff_designs']} design.
• Figure 5: The topology optimized designs (left) and row corresponding max-normalized field distributions (right) for varying design domain sizes when maximizing the mode overlap magnitude. With reference to the designs starting from the top, the optimization terminated at 235, 684, 1000, 1000 and 1000 design iterations, respectively.