Physics-Based Simulation

Affine Body Dynamics

Previously, we reviewed rigid body dynamics and implemented a simple solver using explicit integration.

When dealing with many rigid bodies — especially with complex geometry and dense contact — it’s natural to attempt IPC for contact resolution. However, representing a rigid body by its position $\mathbf{x}$ and rotation $\mathbf{q}$ creates a challenge: step size filtering in Newton’s method becomes nonlinear in $\alpha$, due to the nonlinearity of quaternion update.

Let the state of a rigid body be denoted by $\mathbf{y}=(\mathbf{x},\mathbf{q})$, where $\mathbf{x}$ is the center of mass and $\mathbf{q}$ is a unit quaternion representing orientation. During a Newton iteration, let the update direction be $\mathbf{p}=({\mathbf{p}}_{\mathbf{x}},{\mathbf{p}}_{\mathbf{q}})$ with step size $\alpha$, so that the updated state is given by $\tilde{\mathbf{y}}=\mathbf{y}+\alpha \mathbf{p}$. The position of a vertex initially located at ${\mathbf{x}}_v^{(0)}$ is then transformed to:

$${\mathbf{x}}_v=\mathbf{q}({\mathbf{x}}_v^{(0)}-{\mathbf{x}}^{(0)}){\mathbf{q}}^{-1}+\mathbf{x}$$

Expanding this shows ${\tilde{\mathbf{x}}}_v$ is nonlinear in $\alpha$:

$${\mathbf{x}}_v=(\mathbf{q}+\alpha {\mathbf{p}}_{\mathbf{q}})({\mathbf{x}}_v^{(0)}-{\mathbf{x}}^{(0)})(\mathbf{q}+\alpha {\mathbf{p}}_{\mathbf{q}}{)}^{-1}+\mathbf{x}+\alpha {\mathbf{p}}_{\mathbf{x}}$$ and since $\mathbf{q}$ depends nonlinearly on $\alpha$, so does ${\tilde{\mathbf{x}}}_v$. Hence, linear CCD may fail to guarantee intersection-free motion.

We could model rigid bodies as very stiff soft bodies (e.g., mass-spring or hyperelastic), but this is inefficient and negates the DOF-reduction benefit.

Affine Body Dynamics (ABD) [Lan et al. 2022] addresses this: instead of strict rigidity, we allow tiny affine deformations $\bar{\mathbf{x}}\mapsto \mathbf{x}=\mathbf{A}\mathbf{x}+\mathbf{b}$ for each vertex $\mathbf{x}$ of the body where $\bar{\mathbf{x}}$ denotes the rest position. The body state is $\mathbf{y}=({\mathbf{b}}^T,{\mathbf{a}}_1^T,{\mathbf{a}}_2^T,{\mathbf{a}}_3^T)$ where $\mathbf{A}=({\mathbf{a}}_1,{\mathbf{a}}_2,{\mathbf{a}}_3{)}^T$, so the vertex position $\mathbf{x}$ becomes:

$$\begin{array}{c}\mathbf{x}=\mathbf{J}\mathbf{y}\end{array}$$

with $\mathbf{J}=({\mathbb{I}}_{3\times 3},{\mathbb{I}}_{3\times 3}\otimes {\bar{\mathbf{x}}}^T)$ where $\otimes$ denotes the Kronecker product.

This linearity makes $\mathbf{x}$ linearly dependent on the optimization step size $\alpha$, enabling robust linear CCD.

Let $E(\mathbf{x})$ be the augmented IPC energy. Original IPC solves: $$\begin{array}{c}{\nabla }_{\mathbf{x}}E(\mathbf{x})=0\end{array}$$

ABD solves in a reduced subspace:

$$\begin{array}{c}{\nabla }_{\mathbf{y}}E(\mathbf{J}\mathbf{y})=0\end{array}$$

Using the chain rule:

$$\begin{array}{rl}{\nabla }_{\mathbf{y}}E(\mathbf{J}\mathbf{y})& ={\mathbf{J}}^T{\nabla }_{\mathbf{x}}E(\mathbf{x})\\ D_{\mathbf{y}}^{2}E(\mathbf{J}\mathbf{y})& ={\mathbf{J}}^T{D}_{\mathbf{x}}^{2}E(\mathbf{x})\mathbf{J}\end{array}\text{\hspace{1em}(25.3.1)}$$ , we can solve Newton Optimization direction by solving the linear system ${\mathbf{J}}^T{D}_{\mathbf{x}}^{2}E(\mathbf{x})\mathbf{J}=-{\mathbf{J}}^T{\nabla }_{\mathbf{x}}E(\mathbf{x})$.

With these, we can implement ABD using implicit Euler integration for a simple 2D case.

Before implementation, consider this geometric view: in 2D, an affine transformation is determined by 3 points. Choosing a triangle per rigid body, we can interpolate all vertices via barycentric coordinates. The triangle drives the simulation; the original mesh is used for collisions. This yields a reduced DOF system where the triangle controls motion—essentially what ABD does.