Affine Body

Affine body dynamics: fast, stable and intersection-free simulation of stiff materials

State Variable

The state variable for an Affine Body:

\[ \mathbf{q} = \begin{bmatrix} p_x \\ p_y \\ p_z \\ \hline a_{11}\\ a_{12}\\ a_{13}\\ \hdashline a_{21}\\ a_{22}\\ a_{23}\\ \hdashline a_{31}\\ a_{32}\\ a_{33}\\ \end{bmatrix} = \begin{bmatrix} \mathbf{p} \\ \mathbf{a}_1 \\ \mathbf{a}_2 \\ \mathbf{a}_3 \\ \end{bmatrix}, \]

where \(\mathbf{p}\) is the position of the affine body and the \(\mathbf{a}_i\) are the rows of the affine transformation matrix.

\[ \mathbf{A} = [\mathbf{a}_1,\mathbf{a}_2,\mathbf{a}_3]^T = \begin{bmatrix} \mathbf{a}_1^T \\ \mathbf{a}_2^T \\ \mathbf{a}_3^T \\ \end{bmatrix} \]

P.S. The state variable for a Soft Body Vertex:

\[ \mathbf{x} = \begin{bmatrix} x \\ y \\ z \\ \end{bmatrix} \]

Jacobi Matrix

The Jacobi Matrix for Affine Body:

\[ \mathbf{J}(\bar{\mathbf{x}}) = \left[\begin{array}{ccc|ccc:ccc:ccc} 1 & & & \bar{x}_1 & \bar{x}_2 & \bar{x}_3 & & & & & & \\ & 1 & & & & & \bar{x}_1 & \bar{x}_2 & \bar{x}_3 & & & \\ & & 1 & & & & & & & \bar{x}_1 & \bar{x}_2 & \bar{x}_3\\ \end{array}\right] \]

where, \(\bar{\mathbf{x}} = \begin{bmatrix}\bar{x}^1 & \bar{x}^2 & \bar{x}^2\end{bmatrix}^T\), \(\bar{\mathbf{x}}\) is the material point position in the local(model) space. Every material point \(\mathbf{x}\) has a Jacobi Matrix \(\mathbf{J}(\bar{\mathbf{x}})\)

P.S. The Jacobi Matrix for a Soft Body Vertex is an identity matrix:

\[ \mathbf{J} = \mathbf{I} \]

Calculate \(\mathbf{x}(t)\)

For a vertex in an Affine Body:

\[ \mathbf{x}(t) = \mathbf{J}(\bar{\mathbf{x}}) \mathbf{q}(t) \]

To efficiently calculate \(\mathbf{x}(t)\), we just use:

\[ \mathbf{x}(t) = \begin{bmatrix} \mathbf{a}_1(t)\cdot \bar{\mathbf{x}} \\ \mathbf{a}_2(t)\cdot \bar{\mathbf{x}} \\ \mathbf{a}_3(t)\cdot \bar{\mathbf{x}} \\ \end{bmatrix} +\mathbf{p} \]

#1 OrthoPotential

Formulation: Affine body dynamics: fast, stable and intersection-free simulation of stiff materials (eq. 7)

Shape preservation energy per body:

\[ V_{\perp}(\mathbf{q}) = \kappa \bar{v} \|\mathbf{A}\mathbf{A}^T - \mathbf{I}_3\|_F^2, \]

where \(\kappa\) is the stiffness parameter, \(\bar{v}\) is the rest volume of the affine body, \(\mathbf{I}_3\) is the \(3\times3\) identity matrix, and \(\|\cdot\|_F\) is the Frobenius norm.

We'd like to take \(100MPa \le \kappa \le 100GPa\).

#2 ARAP

Dynamic Deformables: Implementation and Production Practicalities (Now With Code!)

As Rigid As Possible (ARAP) deformation energy per body:

\[ V = \kappa \bar{v} \|\mathbf{A}-\mathbf{R}\|_F^2 = \kappa \bar{v} \|\mathbf{S}-\mathbf{I}\|_F^2 \]

where \(\mathbf{R}\mathbf{S}\) is the polar decomposition of \(\mathbf{A}\).