A Staggered Grid-Based Variational Approach for Modeling Elastic Deformation

Abstract

Elastic deformation is often simulated in computer graphics using the Finite Element Method on tetrahedral meshes. However, generating a tetrahedral mesh can be complicated and expensive. When a hierarchy of meshes is needed (for example, with a progressive or multigrid method), generating this set of hierarchical meshes is a time-consuming process. However, in other areas of physics simulation, such as fluid simulation, the use of staggered grids and finite differences is much more common. The application of adaptive or multigrid methods to grid-based simulations is trivial in comparison. By applying the variational method of Batty et al. from fluid simulation to the staggered grid-based elasticity simulation method of Zhu et al., we produce a method that accurately solves the linear elasticity partial differential equations with free boundary conditions solved implicitly. In this work, we derive a variational formulation for the linear elasticity partial differential equations on a staggered grid. We derive both the static and dynamic forms of the minimization problem, and their subsequent discretizations. Our method only requires an indicator function that specifies the interior and exterior of the solid to be simulated, and does not require any information about the normals of the object’s surface. Furthermore, our method retains the simplicity and sparsity of the basic staggered grid finite difference scheme, but supports non-axis-aligned boundaries without the need for boundary-conforming meshing. We apply our method to several examples of uniform and nonuniform objects under different deformations, demonstrated with both static and quasi-static simulations. We also compare our results with analytical solutions to the linear elasticity partial differential equations to show the accuracy of our method and that our method converges well with grid refinement.