Abstract
Many design optimization problems include constraints to prevent intersection of the geometric shape being optimized with other objects or with domain boundaries. When applying gradient-based optimization to such problems, the constraint function must provide an accurate representation of an object's boundary and be smooth, amenable to numerical differentiation, and fast-to-evaluate for a large number of points. We propose the use of tensor product Bsplines to generate an efficient-to-evaluate implicit level set function that locally approximates the signed distance function for representing geometric non-interference constraints. Adapting ideas from the field of surface reconstruction, we formulate an energy minimization problem to compute the implicit function values at B-spline control points given an oriented point set sampled over a geometric shape. Unlike previous non-interference constraint formulations, our method requires an initial setup, but results in a more efficient and scalable representation of geometric non-interference constraints. This paper presents the results of accuracy and scaling studies performed on our formulation. We demonstrate our method by solving a medical robot design optimization problem with non-interference constraints. We achieve constraint evaluation times on the order of 10 −6 seconds per point on a modern desktop workstation, and a maximum on-surface error of less than 1.0% of the minimum bounding box diagonal for all examples studied. Overall, our method provides an effective formulation for non-interference constraint enforcement with high computational efficiency for gradient-based design optimization problems whose solutions may require hundreds of evaluations of constraints and their derivatives.