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

Abstract

The orientation-preservation conditions and approximation errors of a dual-parametric bi-quadratic finite element method for the computation of both radially symmetric and general nonsymmetric cavity solutions in nonlinear elasticity are analyzed. The analytical results allow us to establish, based on an error equidistribution principle, an optimal meshing strategy for the method in cavitation computation. Numerical results are in good agreement with the analytical results. Introduction. Void formation on nonlinear elastic bodies under hydrostatic tension was observed and analyzed through a defect model by Gent and Lindley [8].Ball [2] established a perfect model and studied a class of bifurcation problems in nonlinear elasticity, in which voids form in an intact body so that the total stored energy of the material is minimized in a class of radially symmetric deformations. The work stimulated an intensive study on various aspects of radially symmetric cavitations (see, e.g., Sivaloganathan [19], Stuart [25], and a review paper by Horgan and Polignone [10], among many others).Müller and Spector [16] later developed a general existence theory in nonlinear elasticity that allows for cavitation, which is not necessarily radially symmetric, by adding a surface energy term. Sivaloganathan and Spector [21] deduced the existence of hole creating deformations without the need for the surface energy term under the assumption that the points (a finite number) at which the cavities can form are prescribed. Optimal locations where cavities can arise are also studied analytically [22,23] and numerically [14].Numerically computing cavities based on the perfect model of Ball is very challenging, due to the so-called Lavrentiev phenomenon [11]. 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]. In these models, one considers minimizing the total elastic energy of the form

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

“…For the simplicity of the finite element coding, we require that either N i = 2N i+1 or N i = N i+1 . By Theorem 3.7 of [33] (see also Remark 3.4), to preserve the orientation of the finite element interpolation functions, ǫ i , τ i , N i must satisfy the conditions τ i ≤ C 1 ǫ For given positive constants C 1 , C 2 , C ≥ (2 − p)2 p−1 , h ≤ min{ 2−p 2 2−p C , 2−p 2 p−1 C } (see [34]), A 1 < A 2 satisfying [(A 2 h) −1 , (A 1 h) −1 ] ∩ Z + = ∅, the analysis above leads to the following meshing strategy. (2) Set k 0 = 0.…”

“…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.…”

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

“…Remark 3.10 Compared to the orientation-preservation condition for the dualparametric bi-quadratic FEM in [22], where the corresponding sufficient and necessary condition for the interpolation of the radially symmetric cavity solution is (3.13) only, while the quadratic iso-parametric FEM imposes additional restrictions on the mesh distribution in the angular direction, which can be more severe a condition. However, when the radius ̺ of the initial defect is very small, to achieve the optimal interpolation error, similar restrictions on the mesh distribution in the angular direction are also required for the dual-parametric bi-quadratic FEM, particularly in the non-radially-symmetric case [21,22]. Thus, to control the total degrees of freedom of the mesh, it is often necessary, for both triangular and rectangular triangulations, to coarsening the mesh layers away from the cavity.…”

“…The only practical analytical result on the orientation preservation condition for the cavitation computation known to the authors so far is [22], where a sufficient condition was given for a dual-parametric bi-quadratic finite element method.…”

Fourier-Chebyshev spectral method for cavitation computation in nonlinear elasticity

“…Since the perfect model is known to be seriously challenged by the Lavrentiev phenomenon [9], the defective model is chosen by most researchers in numerical studies of the cavitation phenomenon, using mainly a variety of the finite element methods (see Xu and Henao [10], Lian and Li [11,12], Su and Li [13] among many others). A spectral collocation method [14], which approximates the cavitation solution with truncated Fourier series in the circumferential direction and finite differences in the radial direction, is also found some success.…”