A numerical method for detecting singular minimizers of multidimensional problems in nonlinear elasticity

1990 ◽  
Vol 58 (1) ◽  
pp. 135-144 ◽  
Author(s):  
Pablo V. Negr�n Marrero
Keyword(s):  
Numerical Method ◽  
Nonlinear Elasticity ◽  
Multidimensional Problems ◽  
Singular Minimizers

scholarly journals NUMERICAL SOLUTION OF NONLINEAR ELASTICITY PROBLEMS WITH LAVRENTIEV PHENOMENON

2007 ◽  
Vol 17 (10) ◽  
pp. 1619-1640 ◽  
Author(s):  
YU BAI ◽  
ZHIPING LI
Keyword(s):  
Numerical Solution ◽  
Nonlinear Elasticity ◽  
Upper Bound ◽  
Perturbation Parameter ◽  
Numerical Results ◽  
Convergence Theory ◽  
Lavrentiev Phenomenon ◽  
Convergence Theorems ◽  
Elasticity Problems ◽  
Singular Minimizers

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 provide a sharper estimate on the upper bound of a perturbation parameter for the existence of the Lavrentiev phenomenon in the example.

Element removal method for singular minimizers in variational problems involving Lavrentiev phenomenon

1992 ◽  
Vol 439 (1905) ◽  
pp. 131-137 ◽  
Keyword(s):  
Numerical Methods ◽  
Numerical Method ◽  
Numerical Experiments ◽  
Variational Problems ◽  
Lavrentiev Phenomenon ◽  
Removal Method ◽  
Element Removal Method ◽  
Element Removal ◽  
Singular Minimizers

A numerical method called element removal method is designed to calculate singular minimizers which cannot be approximated by simple applications of standard numerical methods because of the so-called Lavrentiev phenomenon. The convergence of the method is proved. The results of numerical experiments show that the method is effective.

AN EFFICIENT NUMERICAL METHOD FOR CAVITATION IN NONLINEAR ELASTICITY

2011 ◽  
Vol 21 (08) ◽  
pp. 1733-1760 ◽  
Author(s):  
XIANMIN XU ◽  
DUVAN HENAO
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Numerical Method ◽  
Nonlinear Elasticity ◽  
Elastic Energy ◽  
Finite Element Approximation ◽  
Void Coalescence ◽  
Element Approximation ◽  
Nonconforming Finite Element ◽  
First Time

This paper is concerned with the numerical computation of cavitation in nonlinear elasticity. The Crouzeix–Raviart nonconforming finite element method is shown to prevent the degeneration of the mesh provoked by the conventional finite element approximation of this problem. Upon the addition of a suitable stabilizing term to the elastic energy, the method is used to solve cavitation problems in both radially symmetric and non-radially symmetric settings. While the radially symmetric examples serve to illustrate the efficiency of the method, and for validation purposes, the experiments with non-centered and multiple cavities (carried out for the first time) yield novel observations of situations potentially leading to void coalescence.

ELEMENT REMOVAL METHOD FOR SINGULAR MINIMIZERS IN PROBLEMS OF HYPERELASTICITY

1995 ◽  
Vol 05 (03) ◽  
pp. 387-399 ◽  
Author(s):  
ZHIPING LI
Keyword(s):  
Finite Element ◽  
Numerical Method ◽  
Lavrentiev Phenomenon ◽  
Removal Method ◽  
Finite Element Approximations ◽  
Element Removal Method ◽  
Element Removal ◽  
Admissible Functions ◽  
Singular Minimizers

A numerical method called element removal method is applied to calculate singular minimizers in problems of hyperelasticity. The method overcomes the difficulty in finite element approximations caused by restrictions, such as det (I+∇u)>0, on admissible functions and can avoid Lavrentiev phenomenon if it does occur in the problem. The convergence of the method is proved.

scholarly journals A TRUNCATION METHOD FOR DETECTING SINGULAR MINIMIZERS INVOLVING THE LAVRENTIEV PHENOMENON

2006 ◽  
Vol 16 (06) ◽  
pp. 847-867 ◽  
Author(s):  
YU BAI ◽  
ZHI-PING LI
Keyword(s):  
Numerical Method ◽  
Variational Problems ◽  
Numerical Results ◽  
Two Dimensional ◽  
Dimensional Problem ◽  
Minimizing Sequences ◽  
Lavrentiev Phenomenon ◽  
Absolute Minimizers ◽  
Sobolev Exponent ◽  
Singular Minimizers

A numerical method using the truncation technique on the integrand is developed for computing singular minimizers or singular minimizing sequences in variational problems involving the Lavrentiev phenomenon. It is proved that the method can detect absolute minimizers with various singularities whether the Lavrentiev phenomenon is involved or not. It is also proved that, when the absolute infimum is not attainable, the method can produce minimizing sequences. Numerical results on Manià's example and a two-dimensional problem involving the Lavrentiev phenomenon with continuous Sobolev exponent dependence, are given to show the efficiency of the method.

A numerical method for detecting singular minimizers

1987 ◽  
Vol 51 (2) ◽  
pp. 181-197 ◽  
Author(s):  
J. M. Ball ◽  
G. Knowles
Keyword(s):  
Numerical Method ◽  
Singular Minimizers

An Iterative Numerical Method for Nonlinear Vibrations

1951 ◽  
Vol 18 (1) ◽  
pp. 1-11
Author(s):  
J. E. Brock
Keyword(s):  
Numerical Method ◽  
Nonlinear Elasticity ◽  
Iterative Procedure ◽  
Nonlinear Vibrations ◽  
Initial Approximation ◽  
Forced Vibrations ◽  
Free And Forced Vibrations ◽  
Numerical Integrations

Abstract A simple iterative procedure employing numerical integrations is presented for the analysis of free and forced vibrations of undamped systems having nonlinear elasticity. If the elasticity is symmetrical, it is always possible to make an excellent initial approximation, but the method may also be used in cases of unsymmetrical elasticity. Other applications, such as to systems having parameters which vary with time, are discussed.

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