Continuit� et diff�rentiabilit� d'�l�ments propres: Application � l'optimisation de structures

Abstract

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.

Dependence of the buckling load of a nonshallow arch with respect to the shape of its midcurve

“…The differentiability result we are going to give is an application of a gênerai result given in référence [7] that we recall now. …”

“…In référence [7], we have proved a gênerai resuit of dependence of eigenvalues and eigenvectors of a linear operator with respect to a functional parameter. We have used this result to study the dependence of the buckling load of a beam with respect to its thickness.…”

“…abstract frame. Let us recall the result of référence [7] we are going to use : V : is a Hubert space with inner product < . , • ) a : V x V -> IR is bilinear, symmetrie, continuous, coercive, Both are selfadjoint.…”

“…The first one is positive invertible. Référence [7] shows that if B : V -> V is compact, then the set of eigenvalues of…”

“…Paragraph III. 1 recalls the results of référence [7] we use. Then, in paragraph III.2 we prove the directional differentiability of the eigenvalue if it is multiple, its Fréchet differentiability if it is simple.…”