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：势垒函数对于维持无交叉且无反转的模拟轨迹至关重要，但现有直接基于距离的方法会限制实现设计与性能。我们提出一种重写势垒函数的方法，以实现其海森矩阵的高效且稳健近似。核心思想是利用网格边界元素构建接触的单纯形几何度量，从中推导出解析特征系统，并通过滤波项和硬化项增强其鲁棒性，从而确保投影牛顿求解器的收敛稳定性。这种势垒函数重写方法的另一优势在于能自然处理近乎平行的边-边接触这一棘手情形——对此我们亦提出新的解析特征系统。因此，我们的方法适用于求解接触问题的标准无约束二阶优化策略，可有效最小化海森矩阵可能非正定的非线性非凸函数。仅特征系统的效率即可为IPC标准势垒公式带来3倍加速。我们进一步应用这些解析代理特征系统，实现了完全基于GPU的IPC计算框架，获得显著的额外加速效果。