External Force for Affine Body

#666 AffineBodyExternalForce

The Affine Body External Force applies external 12D generalized forces to affine body instances through a constraint mechanism, supporting both translational forces and affine (rotational) forces.

The external force is applied as acceleration:

\[ \mathbf{a}_{ext} = \mathbf{M}^{-1} \mathbf{F}_{ext} \]

where: - \(\mathbf{F}_{ext} \in \mathbb{R}^{12}\) is the generalized force vector - \(\mathbf{M} \in \mathbb{R}^{12 \times 12}\) is the affine body mass matrix - \(\mathbf{a}_{ext} \in \mathbb{R}^{12}\) is the resulting acceleration

The force vector is structured as:

\[ \mathbf{F}_{ext} = \begin{bmatrix} \mathbf{f} \\ \mathbf{f}_a \end{bmatrix} \in \mathbb{R}^{12} \]

where: - \(\mathbf{f} \in \mathbb{R}^3\) is the translational force - \(\mathbf{f}_a \in \mathbb{R}^{9}\) is the affine force (flattened 3x3 matrix)

The energy function for the Affine Body External Force will be incorporated into the Affine Body Kinetic Term as:

\[ E = \frac{1}{2} \left(\mathbf{q} - \tilde{\mathbf{q}}\right)^T \mathbf{M} \left(\mathbf{q} - \tilde{\mathbf{q}}\right) \]

where \(\tilde{\mathbf{q}}\) is updated each time step by the velocity and acceleration, the formula depends on the time integration scheme used.

Typically, when bdf1 (implicit euler) is used:

\[ \tilde{\mathbf{q}} = \mathbf{q}^t + \Delta t \cdot \mathbf{v}^t + \Delta t^2 \cdot (\mathbf{a}_{ext} + \mathbf{g}), \]

where \(\mathbf{q}^t\), is the previous state vector at time \(t\), \(\mathbf{v}^t\) is the previous velocity vector at time \(t\), \(\Delta t\) is the time step size, and \(\mathbf{g}\) is the gravitational acceleration.

Users are allowed to modify the external force \(\mathbf{F}_{ext}\) using the uipc Animator System interface during the simulation.