A two-grid iterative approach for real time haptics mediated interactive simulation of deformable objects

Fast and efficient algorithms are paramount in any real-time multimodal interactive simulation involving soft deformable objects. To achieve real time computational rates, it is expedient to adaptively refine the simulation mesh in the vicinity of the interaction region instead of using a uniformly refined mesh. While appealing, such an approach is difficult to implement as the system of linear algebraic equations changes during the course of the simulation as the interaction region is dynamically updated. A direct solution approach for the discretized system of equations is, of course, computationally expensive hence iterative approaches must be pursued. In this paper, we present a novel two-grid computational methodology that uses pre-computed solution on a coarse grid representation of the geometry and a prolongation operator that transfers the coarse grid solution to a locally refined fine grid to generate the initial guess for a Gauss-Seidel type iterative solver. A local relaxation approach is then introduced that preferentially relaxes the local and global residuals and vastly improves computational efficiency, especially with increasing number of degrees of freedom of the mesh. Example problems demonstrate the effectiveness of the method.