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Continuité et différentiabilité d’éléments propres: application à l’optimisation de structures. (Continuity and differentiability of eigen-elements: Application to the optimization of structures). (French) Zbl 0741.73029

The sensitivity analysis of static structural eigenvalue response with respect to design variation is presented in this important paper. The effect of design variation on buckling of a beam is considered. The buckling load of a structure (computed by an eigenvalue problem, the eigenvalue of the smallest absolute value) is described by a linear, elliptic eigenvalue problem. In optimizing structures with a constraint on the buckling load, repeated eigenvalues are likely to occur.
The main results are: The work brings some new ideas in the proof of continuity and differentiability results of eigen-elements with respect to design variables (by using the variational characterization of eigenvalues). Finally the authors illustrate these results by a classical problem: buckling of a beam.

MSC:

74P99 Optimization problems in solid mechanics
74M05 Control, switches and devices (“smart materials”) in solid mechanics
49Q12 Sensitivity analysis for optimization problems on manifolds
49R50 Variational methods for eigenvalues of operators (MSC2000)
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
74S30 Other numerical methods in solid mechanics (MSC2010)
74P10 Optimization of other properties in solid mechanics
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