Physics-Based Simulation

Boundary Conditions

In the Material Point Method, boundary conditions (BCs) are enforced on the background Eulerian grid, because the governing equations—discretized in their weak form—are formulated over grid nodes, which serve as the true degrees of freedom (DOFs) of the system. These nodes must satisfy Newton's Second Law. Thus, all external constraints, including static walls, moving boundaries, or contact forces, must be applied directly on the grid after P2G and before G2P.

As usual, we denote the nodal velocity at a grid node ${\mathbf{x}}_i$ as ${\hat{\mathbf{v}}}_i^{n+1}$. After external force integration, we apply different velocity projections on boundary nodes depending on the type of boundary condition.

Types of Boundary Conditions

Let $\mathbf{n}$ be the surface normal at a boundary grid node ${\mathbf{x}}_i$:

• Sticky: The velocity is fully suppressed: $${\hat{\mathbf{v}}}_i^{n+1}\leftarrow 0.$$

• Slip: Only the normal component is removed; tangential motion is preserved: $${\hat{\mathbf{v}}}_i^{n+1}\leftarrow {\hat{\mathbf{v}}}_i^{n+1}-\mathbf{n}({\mathbf{n}}^{\top }{\hat{\mathbf{v}}}_i^{n+1}).$$

• Separate (One-way wall): Normal inflow is blocked, but outward motion is allowed: $${\hat{\mathbf{v}}}_i^{n+1}\leftarrow {\hat{\mathbf{v}}}_i^{n+1}-\mathbf{n}\cdot \min ({\mathbf{n}}^{\top }{\hat{\mathbf{v}}}_i^{n+1},0).$$

These operations are local, applied independently to each boundary grid node.

External Forces

External forces such as gravity can be added after P2G:

$${\hat{\mathbf{v}}}_i^{n+1}\leftarrow {\hat{\mathbf{v}}}_i^{n+1}+\Delta t\cdot \mathbf{g}.$$

Moving Colliders

For moving or deforming objects, Signed Distance Functions (SDFs) are used to detect whether a grid node ${\mathbf{x}}_i$ lies inside the object. Once detected, the surface normal $\mathbf{n}$ and the relative velocity $\Delta \mathbf{v}$ are computed, and the grid velocity is adjusted accordingly.

Coulomb Friction

To enforce Coulomb friction on embedded colliders (e.g., sphere or capsule), we apply a projected velocity correction in the direction of the surface normal:

Let $\mathbf{n}$ be the contact normal and $\Delta \mathbf{v}$ be the opposing velocity (i.e., $\Delta \mathbf{v}={\mathbf{v}}_{\text{SDF}}-{\hat{\mathbf{v}}}_i^{n+1}$). Define the normal component:

$${\dot{v}}_n={\mathbf{n}}^{\top }\Delta \mathbf{v}.$$

Then the corrected grid velocity is:

$${\hat{\mathbf{v}}}_i^{n+1}\leftarrow {\hat{\mathbf{v}}}_i^{n+1}+\Delta \mathbf{v}\cdot \mu +\mathbf{n}\cdot {\dot{v}}_n\cdot (1-\mu ),$$

where $\mu \in [0,1]$ is the friction coefficient. This operation smoothly blends between full separation ($\mu =0$) and sticking ($\mu =1$).