Orientation-preservation conditions on an iso-parametric FEM in cavitation computation

Abstract

The orientation-preservation condition, i.e., the Jacobian determinant of the deformation gradient det ∇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 isoparametric finite element method.

A meshing strategy for a quadratic iso-parametric FEM in cavitation computation in nonlinear elasticity

“…However, strict analytical results are insufficient. The only practical analytical results for the cavitation computation known to the authors so far are [34], where a sufficient orientation-preservation condition and the interpolation error estimates were given for a dual-parametric bi-quadratic finite element method, and [33], where a set of sufficient and necessary orientation-preservation conditions for the quadratic iso-parametric finite element interpolation functions of radially symmetric cavity deformations are derived.…”

Error Analysis of a Dual-parametric Bi-quadratic FEM in Cavitation Computation in Elasticity

“…Downloaded 07/24/15 to 139.80.123.46. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php A comparison between W 1,p errors of the iso-parametric triangular FEM [24] and the dual-parametric bi-quadratic FEM is also shown in Figure 6, which demonstrates that the latter should be a more efficient method in cavitation computations. Γ0 = (λ 1 x 1 , λ 2 x 2 ).…”

“…As a consequence, the total degrees of freedom of a dual-parametric bi-quadratic finite element approximation, in which a triangulation is introduced on the domain by local polar coordinates maps and the shape functions are bi-quadratic with respect to (r, θ), are significantly less than that of an iso-parametric quadratic finite element approximation, where both the elements in the triangulation and the shape functions are given by quadratic functions defined on a reference triangle and where the orientation-preservation condition plays a leading role in determining N i , especially when i h [24]. …”

“…In [13,14,26], other finite element methods are proposed to overcome this difficulty, and these methods have shown significant numerical success in the cavitation computation. In particular, Su and Li [24] analyzed the iso-parametric quadratic finite element method applied in [14]. Even though limited to the radially symmetric cavitation solutions, the result, the first of its kind to our knowledge, nevertheless quantifies the theoretical as well as practical advantages of the method.…”

“…≤ M is generally satisfied, and there exists a positive constant C such that (3.22) holds whenever ≥ Cτ 2 (see [24]). As a consequence, we see that, for the dual-parametric bi-quadratic finite element method, the orientation preservation adds no further restrictions on the number of elements N on an annular ring, which means that a much smaller number of total degrees of freedom is required as compared with the methods in [24,26].…”