Optimal design for uncertain loading condition

Optimal design problems are usually formulated as problems of minimization of the energy, stored in the design under a prescribed loading. Solutions of these problems are unstable to perturbations of the loading. We suggest a new min-max formulation of the optimal design problem which has a stable solution. The loading is not prescribed, but only a set of admissible loadings is given. The stable optimal design problem is formulated as minimization of the stored energy of the project under the most unfavorable loading. This most dangerous loading is one that maximizes the stored energy over the class of admissible functions. The problem is reduced to minimization of Steklov eigenvalues. Several stable solutions of various optimal design problems are demon-strated; among them are the optimal structure of a material stable to variations in uniaxial loading, the optimal specific stiffness of an uncertainly loaded beam, and the stable design of an optimal wheel. Key words: structural optimization, stability, Steklov eigenvalues, optimal composites, minimax