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 3x speedup over the standard IPC barrier formulation. We further apply our analytic proxy eigensystems to produce an entirely GPU-based implementation of IPC with significant further acceleration.

翻译：屏障函数对于维持无相交及无翻转的仿真轨迹至关重要，但现有直接使用距离的方法会限制实现设计与性能。我们提出一种重写屏障函数的方法，以实现其海森矩阵的高效且鲁棒近似。核心思想是利用网格边界单元构造接触的单纯形几何度量，由此推导解析特征系统，并通过滤波与刚化项增强其鲁棒性，确保Project-Newton求解器的收敛稳定性。我们的屏障函数重写方案另一优势在于能自然处理极具挑战性的近平行边-边接触情形，并为此提出了新颖的解析特征系统。因此，该方法适用于标准二阶无约束优化策略求解接触问题，可最小化海森矩阵可能非定的非线性非凸函数。仅我们的特征系统效率即可较标准IPC屏障公式实现3倍加速。进一步地，我们应用解析代理特征系统实现了完全基于GPU的IPC算法，获得显著加速效果。