Stochastic optimization methods in optimal engineering design under stochastic uncertainty - Marti, K. (Zeitschrift für angewandte Mathematik und Mechanik, 2003)
Problems from optimal plastic design are based on the convex, linear or linearized yield/strength condition and the linear equilibrium equation for the generic stress (state) vector. Having to take into account, in practice, stochastic variations of the model parameters (e.g. yield stresses, plastic capacities) and external loadings, the basic stochastic optimal plastic design problem must be replaced by an appropriate deterministic substitute problem. After considering stochastic optimization methods based on failure/survival probabilities and presenting differentiation techniques and differentiation formulas for probability of failure/survival functions, a direct approach is presented using the construction and failure costs (e.g. costs for damage, repair, compensation for weakness within the structure, etc.). Based on the basic mechanical survival conditions, the failure costs may be represented by the minimum value of a convex and often linear program. Several mathematical properties of this program are shown. Minimizing then the total expected costs subject to the remaining (simple) deterministic constraints, a stochastic optimization problem is obtained which may be represented by a “Stochastic Convex Program (SCP) with recourse”. Working with linearized yield/strength conditions, a “Stochastic Linear Program (SLP) with complete fixed recourse” results. In case of a discretely distributed probability distribution or after the discretization of a more general probability distribution of the random structural parameters and loadings as well as certain random cost factors one has a linear program (LP) with a so‐called “dual decomposition data” structure. For stochastic programs of this type many theoretical results and efficient numerical solution procedures (LP‐solver) are available. The mathematical properties of theses substitute problems are considered, and numerical solution procedures are described.