A Singular Value Decomposition Based Method for Isotropic Invertible Solid Deformation ?

It is a common issue in solid deformation simulation that material models will collapse and even permanently invert. In this paper, we present an algorithm for finite element simulation of isotropic tetrahedral mesh that is capable of robustly restoring from inverted configuration to uninverted state in the deformation process. We perform singular value decomposition of deformation gradient to obtain the diagonal matrix which is rotation invariant in space and material space. Since the inversion in a particular direction will correspondingly result in a negative entry of the diagonal matrix, we provide a mechanism for controlling the entries of the diagonal matrix to recover the element from inversion. With the decomposition solution, we exploit Saint Venant-Kirchhoff material model and derive first PiolaKirchhoff stress to obtain internal forces for recovering smoothly. In fact, the solution can be extended to any isotropic hyperelastic materials. We provide several deformable tetrahedral meshes in inverted configuration to demonstrate the effectiveness of the solution.