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

TL;DR

This paper establishes conditions under which quadratic iso-parametric finite element methods preserve orientation in cavity deformation simulations, aiding mesh design for large expansions.

Contribution

It derives necessary and sufficient conditions for orientation preservation in quadratic iso-parametric FEM for cavity problems, improving numerical stability and mesh design.

Findings

Orientation-preservation conditions are derived for quadratic iso-parametric FEM.

Conditions enable effective meshing near cavities with large expansions.

Orientation-preservation is achievable with a reasonable number of degrees of freedom.

Abstract

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.