Numerical Solution of Nonlinear Elasticity Problems With Lavrentiev Phenomenon

Abstract

Abstract. A convergence theory is established for a truncation method in solving polyconvex elasticity problems involving the Lavrentiev phenomenon. Numerical results on a recent example by Foss et al, which has a polyconvex integrand and admits continuous singular minimizers, not only verify our convergence theorems but also provid a sharper estimate on the upper bound of a perturbation parameter for the existence of the Lavrentiev phenomenon in the example.

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

“…The perfect model typically exhibits the Lavrentiev phenomenon [18] when there is a cavitation solution, leading to the failure of the conventional finite element methods [1,4]. Though there are existing numerical methods developed to deal with the Lavrentiev phenomenon [1,4,20,28], they do not seem to be suitable to tackle the cavitation problem. In fact, most of the numerical studies on cavitation are based on the defect model, in which one considers to minimize the total energy of the form E(u) = Ω̺ W (∇u(x))dx, (1.1) in the set of admissible functions A = {u ∈ W 1,p (Ω ̺ ; R n ) is one-to-one a.e.…”

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

“…The perfect model typically displays the Lavrentiev phenomenon [8] when there is a cavitation solution, leading to the failure of the conventional finite element methods [1,4]. Though there are existing numerical methods developed to overcome the Lavrentiev phenomenon ( [1,4,9,14]), they do not seem to be powerful and efficient enough to tackle the cavitation problem on their own.…”

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

“…Though there are numerical methods developed to overcome the Lavrentiev phenomenon in some nonlinear elasticity problems [1,3,12,17,18], they do not seem to be powerful enough for the cavitation problem. On the other hand, some numerical methods (see, e.g., [13,14,26]) have been successfully developed for cavitation computation on general domains with single or multiple prescribed defects, based on the defect model or the regularized perfect model [9,19,20].…”

“…Since r(R) is increasing and convex, it is easily seen that C( 1] if and only if C (−1) > 0. Hence, the first part of the theorem follows.…”