Orientation-Preservation Conditions on an Iso-parametric FEM in Cavitation Computation

The orientation-preservation condition, i.e., the Jacobian determinant of the deformation gradient $\det \nabla u$ is required to be positive, is a natural physical constraint in elasticity as well as in many other fields. It is well known that the constraint can often cause serious difficulties in both theoretical analysis and numerical computation, especially when the material is subject to large deformations. In this paper, we derive a set of sufficient and necessary conditions for the quadratic iso-parametric finite element interpolation functions of cavity solutions to be orientation preserving on a class of radially symmetric large expansion accommodating triangulations. The result provides a practical quantitative guide for meshing in the neighborhood of a cavity and shows that the orientation-preservation can be achieved with a reasonable number of total degrees of freedom by the quadratic iso-parametric finite element method.