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:

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

Shell (Codim 2D)

Codim 2D variant of AffineBodyConstitution for triangle meshes (dim == 2).

The shell is modelled as a uniform slab of full thickness \(2r\), giving an effective volume per triangle \(i\):

\[\bar{v}_i = A_i \cdot 2r\]

where \(A_i\) is the triangle area and \(r\) is the thickness radius.

Rod (Codim 1D)

Codim 1D variant of AffineBodyConstitution for edge meshes (dim == 1).

The rod is modelled as a cylinder of circular cross-section with thickness \(r\), giving an effective volume per segment \(i\):

\[\bar{v}_i = L_i \cdot \pi r^2\]

where \(L_i\) is the segment length.

Attributes

AffineBodyConstitution::apply_to sets the following attributes on the SimplicialComplex.

Meta Attributes (`sc.meta()`)

Per-geometry, shared by all instances.

Attribute	Type	Description
`constitution_uid`	`U64`	Constitution UID identifying the ABD formulation (e.g. 1 = OrthoPotential, 2 = ARAP).
`volume`	`Float`	Rest volume \(\bar{v}\) of the geometry. For full 3D bodies this is the mesh volume; for shells \(\bar{v} = \sum A_i \cdot 2r\); for rods \(\bar{v} = \sum L_i \cdot \pi r^2\).
`mass_density`	`Float`	Volume mass density \(\rho\) (kg/m³).
`abd_mass`	`Float`	Dyadic mass: total mass \(m = \rho \bar{v}\).
`abd_mass_x_bar`	`Vector3`	Dyadic mass: first moment \(m\bar{\mathbf{x}} = \int \rho\, \bar{\mathbf{x}}\, dV\).
`abd_mass_x_bar_x_bar`	`Matrix3x3`	Dyadic mass: second moment tensor \(\mathbf{S} = \int \rho\, \bar{\mathbf{x}}\bar{\mathbf{x}}^T\, dV\).
`mass`	`Float`	Total mass (same value as `abd_mass`, for user convenience).
`mass_center`	`Vector3`	Center of mass \(\mathbf{c} = m\bar{\mathbf{x}} / m\) in the rest configuration.
`inertia`	`Matrix3x3`	Inertia tensor about the center of mass \(\mathbf{I}_{cm}\). Derived from the dyadic mass via the parallel axis theorem.
`self_collision`	`IndexT`	Whether self-collision detection is enabled. Default: `0` (off).
`dof_offset`	`IndexT`	Degree-of-freedom offset in the global system (filled by the backend).
`dof_count`	`IndexT`	Degree-of-freedom count in the global system (filled by the backend).

Instance Attributes (`sc.instances()`)

Per-instance (each instance is one affine body sharing the same rest geometry).

Attribute	Type	Default	Description
`kappa`	`Float`	user-specified	Stiffness parameter \(\kappa\) for the shape preservation energy.
`is_fixed`	`IndexT`	`0`	`1` = body is fixed (position unchanged, not influenced by constitution/kinetics).
`is_dynamic`	`IndexT`	`1`	`1` = kinetics (inertia, velocity) are considered; `0` = quasi-static.
`velocity`	`Matrix4x4`	`Zero`	Initial velocity as the time derivative of the 4x4 transform matrix.
`external_kinetic`	`IndexT`	`0`	`1` = skip default kinetic computation (used with external constraints).

Additional Attributes for Shell / Rod

AffineBodyShell and AffineBodyRod set extra attributes on top of the above:

Location	Attribute	Type	Description
`sc.meta()`	`is_codim`	`IndexT`	Always `1` for shell/rod, indicating a codimensional body.
`sc.vertices()`	`thickness`	`Float`	Thickness radius \(r\). For shells, the effective slab thickness is \(2r\); for rods, the cross-section radius.

Dyadic Mass and the 12x12 Mass Matrix

The three dyadic mass attributes (abd_mass, abd_mass_x_bar, abd_mass_x_bar_x_bar) define the 12x12 ABD mass matrix:

\[ \mathbf{M} = \begin{bmatrix} m\mathbf{I}_3 & \mathbf{I}_3 \otimes (m\bar{\mathbf{x}})^T \\ \mathbf{I}_3 \otimes (m\bar{\mathbf{x}}) & \mathbf{I}_3 \otimes \mathbf{S} \end{bmatrix} \]

The user-facing attributes (mass, mass_center, inertia) are the equivalent rigid body quantities recovered via geometry::affine_body::to_rigid_body.

#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 \(\mathbf{A}\). Each concrete Affine Body constitution documents only the attributes it introduces and how they map to its formulas.