Fluid Solid Coupling

    This project implemented the algorithm in A Fast Variational Framework for Accurate Solid-Fluid Coupling. The simulation results are shown in Figure 1.

    When solving the pressure which enforces the velocity field to be divergence free, this algorithm doesn’t discretize PDE with boundary conditions, which can be tricky at non-grid-aligned boundaries. Instead, it relies on the estimation of the whole system’s kinetic energy. Assuming the system stays at the state with minimum energy, the pressure field \(\textbf{p}\) can be computed by solving \(\begin{equation} \frac{\Delta t}{\rho^2}\textbf{G}^T\textbf{M}_F\textbf{Gp} + \Delta t\textbf{J}^T\textbf{M}_S\textbf{J} = \frac{1}{\rho}\textbf{G}^T\textbf{M}_F\textbf{u} - \textbf{J}^T\textbf{V}^n \end{equation}\) where \(\textbf{G}\) is the gradient finite difference operator, \(\textbf{M}_F\) and \(\textbf{M}_S\) are the mass matrix of fluid and solid respectively, \(\textbf{J}\) is the operator mapping pressure to net force and torgue on the solid, \(\textbf{u}\) is the fluid velocity field and \(\textbf{V}^n\) is the (translational and angular) velocity of the solid.

    We use Jacobi iterations to solve this equation. Due to the large size of the matrices \(\textbf{G}^T\textbf{M}_F\textbf{Gp}\) and \(\textbf{J}^T\textbf{M}_S\textbf{J}\), they are stored and computed in sparse form.