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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.

Abstract. The buckling load of a structure may usually be computed with an eigenvalue problem: it is the eigenvalue of smallest absolute value. In optimizing structures with a constraint on the buckling load, repeated eigenvalues are likely to occur. We prove continuity and differentiability results of eigenelements with respect to design variables using the variational characterization of eigenvalues. We illustrate these results with a classical problem: buckling of a beam. Application to arch buckling is presented in another article.

In this article we study a controllability problem for an elliptic partial differential equation in which the control is the shape of the domain where the equation holds. The quantity to be controlled is the trace of the solution, with a given right hand side source term, into an open subdomain. The mapping that associates this trace to the shape of the domain is nonlinear. We first consider the linearized problem and show an approximate controllability property. We then address the same questions in the context of a finite difference discretization of the elliptic problem. We prove a local controllability result applying the Inverse Function Theorem together with a "unique continuation" property of the underlying adjoint discrete system. Subject Classification (1991): 35J05, 93B03, 65M06 Mathematics