Barrier terms for Incremental Potential Contact (IPC) energy are crucial for maintaining an intersection and inversion free simulation trajectory. However, existing formulations which directly use distance for measuring the state of contact can restrict implementation design and performance. This is because numerical eigendecompositions are required for guaranteeing positive semi-definiteness of the Hessian of the barrier energy during optimization with Projected-Newton solvers, and alternative Gauss-Newton methods suffer from significantly reduced convergence rates. We rewrite the barrier function of IPC to derive an efficient approximation its Hessian, which can then be used for simultaneous construction and projection to positive semi-definite state. The key idea is to formulate a simplicial geometric measure of contact using mesh boundary elements, where analytic eigensystems are obtained for minimising the proposed barrier energy with superlinear convergence to retain much of the advantage of Newton-type methods. Our approach is suitable for standard second order unconstrained optimization strategies for IPC based collision processing, minimizing nonlinear nonconvex functions where the Hessian may be indefinite. The result is a 3-times speedup over the standard full-space IPC barrier formulation based on direct Euclidean distance measures. We further apply our analytic proxy eigensystems to produce an entirely GPU-based implementation of IPC with significant further acceleration.

翻译：增量势接触（IPC）能量中的势垒项对于维持无碰撞和逆态反转的仿真轨迹至关重要。然而，现有直接利用距离衡量接触状态的公式会限制实现设计与性能。这是因为在使用投影牛顿求解器优化势垒能量时，数值特征分解是保证Hessian矩阵半正定的必要条件，而替代的高斯-牛顿方法则会显著降低收敛速度。我们重写了IPC的势垒函数，推导出其Hessian矩阵的高效近似形式，从而能够同时构造并投影至半正定状态。关键思想是利用网格边界单元构建接触的单形几何度量，通过解析特征系统以超线性收敛速率最小化所提出的势垒能量，从而保留牛顿类方法的大部分优势。我们的方法适用于基于IPC碰撞处理的标准二阶无约束优化策略，可处理Hessian矩阵可能非定的非线性非凸函数最小化问题。与基于欧氏距离直接度量的标准全空间IPC势垒公式相比，本方法实现了3倍加速。我们进一步应用解析代理特征系统，实现完全基于GPU的IPC计算，并获得了显著的额外加速效果。