15. Probabilistic Approaches for Incomplete Information

Most structural optimization problems that have been investigated use approaches designed for complete data. Nevertheless, some parametric probabilistic problems of optimal structural design with limiting state and buckling constraints have been formulated and investigated in [AB73, ABC84, AC79, Bra84, DFH77, ETB84, AGRZ84, EKS86, Mos77, Mar95, BV07]. Nondeterministic approaches have been used for some optimal design problems when the external conditions (external forces) were taken as a random variables. Fewer probabilistic studies have been devoted to the important class of problems of quasi-brittle elastic body optimization on the basis of modern fracture mechanics criteria [ABC84, BRS99, BRS03a, BRS03b, LM96]. In accordance with fracture mechanics representations it is necessary to consider all possibilities for the appearance of cracks, and to take into account that the crack position, size and mode (opening cracks, shear cracks, etc.) are unknown beforehand. Therefore, the formulation and solution of these optimization problems requires the application of methods which can take into account incomplete information. Minimax methods of game theory (guaranteed approach), probabilistic methods, etc., can be used for this purpose.