Physics-Based Simulation

Normal Contact for Non-penetration

To prevent self-interpenetration during simulation, it's essential to enforce a condition ensuring that the deformation map $\varphi (\cdot ,t):{\Omega }^0\to {\Omega }^t$ is bijective for any $t\ge 0$. This bijection is maintained by boundary forces acting on pairs of contacting surface regions, referred to as ${\Gamma }_C$. We can think of these forces as another set of Neumann boundary conditions that exert extra forces on ${\Gamma }_C$ only when necessary to prevent interpenetration. Thus, we can extend the boundary integral term in the weak form as follows:

$$\begin{array}{rl}& {\int }_{\partial {\Omega }^0}{\mathbf{Q}}_i(\mathbf{X},t){\mathbf{T}}_i(\mathbf{X},t)ds(\mathbf{X})\\ & ={\int }_{{\Gamma }_D}{\mathbf{Q}}_i(\mathbf{X},t){\mathbf{T}}_{D|i}(\mathbf{X},t)ds(\mathbf{X})+{\int }_{{\Gamma }_N}{\mathbf{Q}}_i(\mathbf{X},t){\mathbf{T}}_{N|i}(\mathbf{X},t)ds(\mathbf{X})\\ & +{\int }_{{\Gamma }_C}{\mathbf{Q}}_i(\mathbf{X},t){\mathbf{T}}_{C|i}(\mathbf{X},t)ds(\mathbf{X}),\end{array}\text{\hspace{1em}(18.2.1)}$$

where ${\mathbf{T}}_N(\mathbf{X},t)$ is the original Neumann boundary force specified in the problem setup, and ${\mathbf{T}}_C(\mathbf{X},t)$ is the normal contact force arising from the bijectivity constraint.

Similar to Dirichlet boundary conditions, ${\mathbf{T}}_C(\mathbf{X},t)$ can only be determined once we solve the problem. However, enforcing non-interpenetration is more complex than prescribing displacements. Fortunately, we can use the approximate constitutive model of ${\mathbf{T}}_C(\mathbf{X},t)$ in IPC to represent the contact force as a function of $\mathbf{x}$, ensuring non-interpenetration by simply including this additional conservative force.

Remark 18.2.1 (Overlapping Boundaries). Note that here ${\Gamma }_C$ can overlap with both ${\Gamma }_D$ and ${\Gamma }_N$. For a free (Neumann) boundary contacting a Dirichlet boundary, ${\mathbf{T}}_C(\mathbf{X},t)$ on the Dirichlet part will also be ignored when enforcing the Dirichlet boundary conditions. However, if two Dirichlet boundaries interpenetrate each other, the problem will have no solution with the bijectivity constraint.