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 the domain 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 B-splines to construct an efficient-to-evaluate level set function that locally approximates the signed distance function for representing geometric non-interference constraints. Adapting ideas from the surface reconstruction methods, we formulate an energy minimization problem to compute the B-spline control points that define the level set function given an oriented point cloud sampled over a geometric shape. Unlike previous explicit non-interference constraint formulations, our method requires an initial setup operation, but results in a more efficient-to-evaluate 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 require at least hundreds of evaluations of constraints and their derivatives.

许多设计优化问题包括防止正在优化的几何形状与其他对象或域边界相交的约束。当将基于梯度的优化应用于此类问题时，约束函数必须提供域边界的准确表示，并且必须平滑、适合数值微分，并且能够快速评估大量点。我们建议使用张量积 B 样条来构造一个高效评估的水平集函数，该函数局部逼近用于表示几何无干扰约束的符号距离函数。采用表面重建方法的思想，我们制定了一个能量最小化问题来计算 B 样条控制点，这些控制点定义给定在几何形状上采样的定向点云的水平集函数。与以前的显式非干涉约束公式不同，我们的方法需要初始设置操作，但可以更有效地评估和扩展几何非干涉约束的表示。本文介绍了对我们的配方进行的准确性和规模研究的结果。我们通过解决具有非干扰约束的医疗机器人设计优化问题来演示我们的方法。我们在现代桌面工作站上实现了每点约\(10^{-6}\)秒的约束评估时间，并且对于所有研究的示例，最大表面误差小于最小边界框对角线的 1.0% 。总体而言，我们的方法为基于梯度的设计优化问题提供了一种具有高计算效率的无干扰约束执行的有效公式，其解决方案需要至少数百次约束及其导数的评估。