Time stepping

We use variational implicit Euler for dynamic time stepping and solve for end-of-time-step position \(\mathbf{x}_t\) $$ E_{\mathrm{total}} = E_{\mathrm{inertia}} + E_{\mathrm{elastic}} + E_{\mathrm{external}} + E_{\mathrm{contact}} \ , $$ where the inertial energy is computed as $$ E_{\mathrm{inertia}} = \frac{1}{2} (x_t - 2_{t-1} + x_{t-2})^{T} \mathbf{M} (x_t - 2x_{t-1} + x_{t-2}) \ . $$

The energy corresponding to constant external forces \(\mathbf{f}\) is simply \(E_{\mathrm{external}} = \mathbf{f}^{T} \mathbf{x}_t \ .\)

The formula for elastic energy can be found in the projects.

We use the contact potential from IPC.