On the locking phenomenon for a class of elliptic problems

Abstract

In this paper we study the numerical behaviour of elliptic problems in which a small parameter is involved and an example concerning the computation of elastic arches is analyzed using this mathematical framework. At first, the statements of the problem and its Galerkin approximations are defined and an asymptotic analysis is performed. Then we give general conditions ensuring that a numerical scheme will converge uniformly with respect to the small parameter. Finally we study an example in computation of arches working in linear elasticity conditions. We build one finite element scheme giving a locking behaviour, and another one which does not.

Membrane locking in the finite element computation of very thin elastic shells

“…In these conditions, the convergence of the approximation is seriously compromised. It should be noticed that the above reasoning is very close to that of [6], theorem 4, where it is shown that in the case of a circular arch, there is a robust non-classical approximation by finite éléments where v a vol. 32, n° 2,1998 (a -1, 2) and v 3 by polynomials of orders 4 and 3 respectively But we should not conclude that discretizmg i?…”

“…The proof of (3.10) follows exactly the same steps, showing the weak convergence in V. The strong convergence in V needs and ulterior reasoning. As we pointed out before, the proof is classical, and may be seen for instance in [6], Theorem 1.…”

“…It is known that the phenomenon of membrane locking in numerical computation of thin shells consists in an inadequacy of the finite éléments to describe the very peculiar déformations of a shell [15], [10], [6], [9]. Actually, the natural trend of a thin shell is to perform pure bendings, i.e.…”

Numerical approximation of stiff transmission problems by mixed finite element methods

“…Problem (3) fits into the gênerai setting of parameter dependent problems, studied in [4] and equally in [11]. The same gênerai framework is also used for the analysis of arch modélisation problems (see [2]).…”

“…Briefiy, the locking phenomenon can be described as a loss of convergence which stems trom the approximation scheme: although mathematical convergence is secured and computer accuracy is adequate, the approximation does not square with the expected solution. A définition of locking is given in [4] and a criterion for avoiding locking is established in [11].…”