(SIG 2020)IPC算法

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Incremental Potential Contact: Intersection-and Inversion-free, Large-Deformation Dynamics(SIG 2020)

Introduction

1.1 基本理论

(1) 优化问题

$$E(x,x^t,v^t)=\frac12(x-\hat x)^TM(x-\hat x)-h^2x^Tf_d+h^2\Psi(x)\\ \hat x=x^t+hv^t+h^2M^{-1}f_e\\ x^{t+1}=\arg\min_xE(x,x^t,v^t),\quad x^{\tau}\in A$$

其中：

• $A$: the set of admissible trajectories（说人话就是，$A$ 表示点在离散时间之间的运动轨迹点的集合，运动过程不会发生碰撞）
• $f_e$: external forces
• $f_d$: dissipative frictional forces
• $x^\tau,\tau\in[t,t+1]$: the linear trajectory between $x^t$ and $x^{t+1}$.

若不考虑dissipative frictional forces

$$E(x)=\frac12||x-x^\star||^2_M+h^2\Psi(x)$$

其中：

• inertial potential: （惯性势能）比较好处理

• $$\frac12||x-x^\star||^2_M$$

• elastic potential: （弹性势能）非线性，非凸优化

• $$h^2\Psi(x)$$

(2) 碰撞, Contact

两种方式：

• 给碰撞对象施加一个惩罚的力/冲量（力/冲量不确定，需要反复调参实验），不稳定，不能完全保证处理好碰撞
• 给优化添加约束
  • IPC计算仍然很慢，依然是一个非线性问题.

Incremental Potential Contact IPC: （本质是内点法）

• $$\arg\min_xE(x)=\frac12||x-x^\star||^2_M+h^2\Psi(x)\\ s.t.\quad d(x)\ge0$$

• 变为：

• $$\arg\min_xE(x)=\frac12||x-x^\star||^2_M+h^2\Psi(x)+\kappa\sum_{k\in C}b(d_k(x))$$

其中：Barrier function $b$ 的作用是当 $0<d<\hat d$ 时，会impose a large internal force.（靠的越近，力越大）

$$b(d_k(x)=\left\{ \begin{aligned} &-(d-\hat d)^2\ln(\frac d{\hat d }),&0<d<\hat d\\ &0,&d\ge\hat d \end{aligned} \right.$$

图像：

性质：

• smooth

1.2. 论文的Contribution

• A contact model based on the unsigned distance function;

• An almost everywhere $C^2$, $C^1$-continuous barrier formulation, approximating the contact problem with arbitrary accuracy, with barrier support localized in the configuration space, enabling efficient time-stepping;

• Contact-aware line search that continuously guarantees penetration-free descent steps with CCD evaluations accelerated by a conservative-bound contact-specific CFL-inspired filter;

• A new variational friction model with smoothed static friction, formulated as a lagged dissipative potential, robustly resolving challenging frictional contact behaviors;

• A new benchmark of simulation test sets with careful evaluation of constraint and time stepping formulations along with an extensive evaluation of existing contact solvers

Chapter 4. Primal Barrier Contact Mechanics

求解带约束（primitive-pair admissibility constrains）的优化问题