Barrier functions are crucial for maintaining an intersection- and inversion-free simulation trajectory but existing methods, which directly use distance can restrict implementation design and performance. We present an approach to rewriting the barrier function for arriving at an efficient and robust approximation of its Hessian. The key idea is to formulate a simplicial geometric measure of contact using mesh boundary elements, from which analytic eigensystems are derived and enhanced with filtering and stiffening terms that ensure robustness with respect to the convergence of a Project-Newton solver. A further advantage of our rewriting of the barrier function is that it naturally caters to the notorious case of nearly parallel edge-edge contacts for which we also present a novel analytic eigensystem. Our approach is thus well suited for standard second-order unconstrained optimization strategies for resolving contacts, minimizing nonlinear nonconvex functions where the Hessian may be indefinite. The efficiency of our eigensystems alone yields a 3× speedup over the standard Incremental Potential Contact (IPC) barrier formulation. We further apply our analytic proxy eigensystems to produce an entirely GPU-based implementation of IPC with significant further acceleration.

障碍函数对于维持无交集和无反演的模拟轨迹至关重要，但直接使用距离的现有方法可能会限制实现设计和性能。我们提出了一种重写势垒函数的方法，以获得其 Hessian 矩阵的高效且稳健的近似。关键思想是使用网格边界元素制定接触的简单几何测量，从中导出分析本征系统并通过过滤和强化项进行增强，以确保 Project-Newton 求解器收敛的鲁棒性。我们重写势垒函数的另一个优点是，它自然地迎合了臭名昭著的几乎平行边缘接触的情况，为此我们还提出了一种新颖的分析本征系统。因此，我们的方法非常适合标准二阶无约束优化策略，用于解析接触、最小化 Hessian 可能不确定的非线性非凸函数。仅我们本征系统的效率就比标准增量电势接触 (IPC) 势垒公式提高了 3 倍。我们进一步应用我们的分析代理特征系统来生成完全基于 GPU 的 IPC 实现，并具有显着的进一步加速。