Uncertainties in engineering design. Mathematical theory and numerical experience

Shape optimization in structure mechanics is the center of research and applications. Many papers and books dealing with this subject appeared and special conferences dealt with these problems. The research is directed toward theoretical questions as existence and characterization of the optimal design, bounds for the optimized values, numerical treatment of the optimal design problems, etc. This paper addresses the problem of the reliability of the conclusions based on computational analysis and their relation to the problems of the optimal design. The problem of optimal design consists--in principle--in the comparison of the solutions of states from the set S of admissible states (for example, solution of the problems from the set of admissible domains) and the selection of the optimal state (e.g. the domain) for the engineering design. It is obvious that such a selection can be successful only if the solution of every state is uniformly reliable with respect to the entire set S. This requirement creates a serious difficulty because we are used to solve in practice numerically the simplified mathemtical model formulation of the problem and have experience only with small limited set of practical problems.