Physics-Based Simulation

Barrier and Distances

With triangle mesh discretization, the barrier potential in the continuous settings (Equation (18.3.5)) can be approximated as $$\begin{array}{rl}& {\int }_{{\Gamma }_C}\frac{1}{2}b(\underset{{\mathbf{X}}_2\in {\Gamma }_C-\mathcal{N}(\mathbf{X})}{\min }\Vert \mathbf{x}(\mathbf{X},t)-\mathbf{x}({\mathbf{X}}_2,t)\Vert ,\hat{d})ds(\mathbf{X})\\ \approx & {\int }_{{\Gamma }_C}\frac{1}{2}b(\underset{e\in \mathcal{T}-I(\mathbf{X})}{\min }\underset{{\mathbf{X}}_2\in e}{\min }\Vert \mathbf{x}(\mathbf{X},t)-\mathbf{x}({\mathbf{X}}_2,t)\Vert ,\hat{d})ds(\mathbf{X})\\ =& {\int }_{{\Gamma }_C}\frac{1}{2}b(\underset{e\in \mathcal{T}-I(\mathbf{X})}{\min }{d}^{\text{PT}}(\mathbf{x}(\mathbf{X},t),e),\hat{d})ds(\mathbf{X})\\ =& {\int }_{{\Gamma }_C}\frac{1}{2}\underset{e\in \mathcal{T}-I(\mathbf{X})}{\max }b(d^{\text{PT}}(\mathbf{x}(\mathbf{X},t),e),\hat{d})ds(\mathbf{X}),\end{array}\text{\hspace{1em}(24.1.1)}$$ where $\mathcal{T}$ is the set of all surface triangles, $I(\mathbf{X})$ is the set of all surface triangles that hold point $\mathbf{X}$, and $d^{\text{PT}}$ is the point-triangle distance. Further approximating the max operator with summations and use mesh surface nodes $\hat{a}$ as quadrature points, we have $$\begin{array}{rl}& {\int }_{{\Gamma }_C}\frac{1}{2}\underset{e\in \mathcal{T}-I(\mathbf{X})}{\max }b(d^{\text{PT}}(\mathbf{x}(\mathbf{X},t),e),\hat{d})ds(\mathbf{X})\\ \approx & {\int }_{{\Gamma }_C}\frac{1}{2}\sum _{e\in \mathcal{T}-I(\mathbf{X})}b(d^{\text{PT}}(\mathbf{x}(\mathbf{X},t),e),\hat{d})ds(\mathbf{X})\\ \approx & \sum _{\hat{a}}\frac{1}{2}{w}_{\hat{a}}\sum _{e\in \mathcal{T}-I({\mathbf{X}}_{\hat{a}})}b(d^{\text{PT}}({\mathbf{x}}_{\hat{a}},e),\hat{d}),\end{array}\text{\hspace{1em}(24.1.2)}$$ where $w_{\hat{a}}=\hat{d}\frac{1}{3}{\sum }_{j\in I({\mathbf{X}}_{\hat{a}})}{A}_j$ is the integration weight and $A_j$ is the area of node $\hat{a}$'s incident surface triangle $j$.

Now, getting back to the second line of Equation (24.1.1), if we only use points on the edges to approximate the minimum distance, we obtain $$\begin{array}{rl}& {\int }_{{\Gamma }_C}\frac{1}{2}b(\underset{{\mathbf{X}}_2\in {\Gamma }_C-\mathcal{N}(\mathbf{X})}{\min }\Vert \mathbf{x}(\mathbf{X},t)-\mathbf{x}({\mathbf{X}}_2,t)\Vert ,\hat{d})ds(\mathbf{X})\\ \approx & {\int }_{{\Gamma }_C}\frac{1}{2}b(\underset{e\in \mathcal{E}-I(\mathbf{X})}{\min }\underset{{\mathbf{X}}_2\in e}{\min }\Vert \mathbf{x}(\mathbf{X},t)-\mathbf{x}({\mathbf{X}}_2,t)\Vert ,\hat{d})ds(\mathbf{X})\\ =& {\int }_{{\Gamma }_C}\frac{1}{2}\underset{e\in \mathcal{E}-I(\mathbf{X})}{\max }b(\underset{{\mathbf{X}}_2\in e}{\min }\Vert \mathbf{x}(\mathbf{X},t)-\mathbf{x}({\mathbf{X}}_2,t)\Vert ,\hat{d})ds(\mathbf{X}).\end{array}$$ Then if we choose a special quadrature point ${\mathbf{X}}_{e_1}$ per surface edge $e_1$ and approximate the max operators with summations, we get $$\begin{array}{rl}& {\int }_{{\Gamma }_C}\frac{1}{2}\underset{e\in \mathcal{E}-I(\mathbf{X})}{\max }b(\underset{{\mathbf{X}}_2\in e}{\min }\Vert \mathbf{x}(\mathbf{X},t)-\mathbf{x}({\mathbf{X}}_2,t)\Vert ,\hat{d})ds(\mathbf{X})\\ \approx & \sum _{e_1\in \mathcal{E}}\frac{1}{2}{w}_{e_1}\sum _{e\in \mathcal{E}-I({\mathbf{X}}_{e_1})}b(\underset{{\mathbf{X}}_2\in e}{\min }\Vert \mathbf{x}({\mathbf{X}}_{e_1},t)-\mathbf{x}({\mathbf{X}}_2,t)\Vert ,\hat{d}),\end{array}$$ where $w_{e_1}=\hat{d}\frac{1}{3}{\sum }_{j\in I(e_1)}{A}_j$ is the integration weight and $A_j$ is the area of $e_1$'s incident surface triangle $j$. Next, if we always select ${\mathbf{X}}_{e_1}$ to be the closest point to ${\mathbf{X}}_2$ on $e_1$, we will get $$\begin{array}{rl}& \sum _{e_1\in \mathcal{E}}\frac{1}{2}{w}_{e_1}\sum _{e\in \mathcal{E}-I({\mathbf{X}}_{e_1})}b(\underset{{\mathbf{X}}_2\in e}{\min }\Vert \mathbf{x}({\mathbf{X}}_{e_1},t)-\mathbf{x}({\mathbf{X}}_2,t)\Vert ,\hat{d})\\ \approx & \sum _{e_1\in \mathcal{E}}\frac{1}{2}{w}_{e_1}\sum _{e\in \mathcal{E}-\mathcal{N}(e_1)}b(\underset{{\mathbf{X}}_2\in e,\mathbf{X}\in e_1}{\min }\Vert \mathbf{x}(\mathbf{X},t)-\mathbf{x}({\mathbf{X}}_2,t)\Vert ,\hat{d})\\ =& \sum _{e_1\in \mathcal{E}}\frac{1}{2}{w}_{e_1}\sum _{e\in \mathcal{E}-\mathcal{N}(e_1)}b(d^{\text{EE}}(e,e_1),\hat{d}),\end{array}\text{\hspace{1em}(24.1.3)}$$ where $\mathcal{N}(e_1)$ is the set of all the surface edge neighbors of $e_1$ plus itself. For the summation over all surface edges in Equation (24.1.3), if we only account for $(e_1,e)$ with $e_1<e$ or the other way around, then the coefficient $1/2$ can be omitted.

Now we have two kinds of discretizations for the 3D barrier potential energy. To use them together in practice, we can take advantage of a linear combination of them, and the coefficient could usually be set to $1/2$.

For point-triangle and edge-edge distances, they are also both small optimization problems with analytical solutions, which can be represented as piecewise smooth functions like the 2D Point-Edge distance in Equation (21.2.1). For example, in the point-triangle case, the expression can be determined by checking which region the projection of the point onto the triangle plane is located.

Definition 24.1.1 (3D Point-Triangle Distance). The distance between point ${\mathbf{x}}_p$ and triangle with vertices ${\mathbf{x}}_{t1}$, ${\mathbf{x}}_{t2}$, and ${\mathbf{x}}_{t3}$ can be defined as $$\text{arg}\underset{{\beta }_1,{\beta }_2}{\min }\Vert {\mathbf{x}}_p-({\mathbf{x}}_{t1}+{\beta }_1({\mathbf{x}}_{t2}-{\mathbf{x}}_{t1})+{\beta }_2({\mathbf{x}}_{t3}-{\mathbf{x}}_{t1}))\Vert \text{s.t.}{\beta }_1,{\beta }_2\ge 0,{\beta }_1+{\beta }_2\le 1$$

Definition 24.1.2 (3D Edge-Edge Distance). The distance between edge with end nodes ${\mathbf{x}}_{e1a}$ and ${\mathbf{x}}_{e1b}$ and edge with end nodes ${\mathbf{x}}_{e2a}$ and ${\mathbf{x}}_{e2b}$ can be defined as $$\text{arg}\underset{{\beta }_1,{\beta }_2}{\min }\Vert {\mathbf{x}}_{e1a}+{\beta }_1({\mathbf{x}}_{e1b}-{\mathbf{x}}_{e1a})-({\mathbf{x}}_{e2a}+{\beta }_2({\mathbf{x}}_{e2b}-{\mathbf{x}}_{e2a}))\Vert \text{s.t.}{\beta }_1,{\beta }_2\in [0,1]$$

Remark 24.1.1 (Smoothness of 3D Distance Functions). Note that the point-triangle distance is at least $C^1$ continuous everywhere. This means that even when the projected point is located on the borders of the piecewise function, the distance gradient still exists and is continuous. However, for edge-edge distance, when the edges are parallel, the distance function is only $C^0$ continuous, as the gradient of the expressions in adjacent regions do not agree. To address this issue, IPC [Li et al. 2020] proposed multiplying a mollifier to the edge-edge barrier energy density function to make the potential $C^1$ continuous everywhere. This mollifier smoothly decreases to zero when the edges are parallel. This ensures that gradient-based optimization methods can still be applied efficiently to solve the problem.