Worst-Case Non-parametric Shape Optimization

The advancements in low-cost manufacturing hardware have enabled inexperienced users to create 3D product prototypes in-house. As a common issue, however, the resulting 3D models, despite their aesthetic appeal, often lack the structural integrity required for practical use. While there has been significant research into automated shape adjustment techniques, these methods rely on additional inputs such as parametric representations and specific external load conditions, making them inaccessible to novice users. To address this challenge, we propose a shape optimization technique that does not require extra inputs. Starting from a target shape represented by a general 3D mesh, our method aims to produce an optimized shape that is both visually consistent and structurally more robust. Our approach formulates the problem as an approximate bilevel optimization: the inner problem solves a compliance-based surrogate to identify a critical load direction, while the outer problem minimizes the maximum stress across the entire shape surface. We develop an approximate bilevel solver that alternates between solving the outer and inner problems, as well as regular mesh refinement. Our empirical results demonstrate the effectiveness of our method in finding 3D shapes that are visually consistent, while also providing the necessary robustness for practical applications.