Physics-Based Simulation

Friction Force

Analogous to Frictional Contact, maximizing the dissipation rate subject to the Coulomb constraint defines friction forces per unit area variationally:

$$\begin{array}{r}{\mathbf{T}}_F(\mathbf{X},t)=\underset{\mathbf{\beta }\in {\mathbb{R}}^d}{\mathrm{argmin}}{\mathbf{\beta }}^T{\mathbf{V}}_F(\mathbf{X},t)\\ \text{s.t.}\Vert \mathbf{\beta }\Vert \le \mu \Vert {\mathbf{T}}_C(\mathbf{X},t)\Vert \text{and}\mathbf{\beta }\cdot \mathbf{N}(\mathbf{X},t)=0.\end{array}\text{\hspace{1em}(18.4.1)}$$

Here, ${\mathbf{V}}_F(\mathbf{X},t)=\mathbf{V}(\mathbf{X},t)-\mathbf{V}({\mathbf{X}}_2,t)$ is the relative sliding velocity between $\mathbf{X}$ and the closest point ${\mathbf{X}}_2=\text{arg}{\min }_{{\mathbf{X}}_2\in {\Gamma }_C-\mathcal{N}(\mathbf{X})}\Vert \mathbf{X}-{\mathbf{X}}_2\Vert$, $\mu$ is the coefficient of friction, ${\mathbf{T}}_C$ is the normal contact force per unit area, and $\mathbf{N}$ is the normal direction.

This is equivalent to:

$${\mathbf{T}}_F(\mathbf{X},t)=-\mu \Vert {\mathbf{T}}_C(\mathbf{X},t)\Vert f(\Vert {\mathbf{V}}_F(\mathbf{X},t)\Vert )\mathbf{s}({\mathbf{V}}_F(\mathbf{X},t)),\text{\hspace{1em}(18.4.2)}$$

with $\mathbf{s}({\mathbf{V}}_F)=\frac{{\mathbf{V}}_F}{\Vert {\mathbf{V}}_F\Vert }$ when $\Vert {\mathbf{V}}_F\Vert >0$, while $\mathbf{s}({\mathbf{V}}_F)$ takes any unit vector orthogonal to $\mathbf{N}(\mathbf{X},t)$ when $\Vert {\mathbf{V}}_F\Vert =0$.

In addition, the friction scaling function $f$ is also nonsmooth with respect to ${\mathbf{V}}_F$, since $f(\Vert {\mathbf{V}}_F\Vert )=1$ when $\Vert {\mathbf{V}}_F\Vert >0$, and $f(\Vert {\mathbf{V}}_F\Vert )\in [0,1]$ when $\Vert {\mathbf{V}}_F\Vert =0$. These nonsmooth properties can severely hinder or even break the convergence of gradient-based optimization. The mollification of the friction-velocity relationship here follows the same approach as in Frictional Contact.