Discrete–continuous variable structural optimization of systems under stochastic loading

The paper deals with the optimization of structural systems involving discrete and continuous sizing type of design variables. In particular, the reliability-based optimization of non-linear systems subject to stochastic excitation where some or all of the design variables are discrete is considered. The reliability-based optimization problem is formulated as the minimization of an objective function subject to multiple reliability constraints. The probability that design conditions are satisfied within a given time interval is used as measure of system reliability. The basic mathematical programming statement of the structural optimization problem is converted into a sequence of explicit approximate primal problems of separable form. The explicit approximate primal problems are solved by constructing continuous explicit dual functions, which are maximized subject to simple non-negativity constraints on the dual variables. A gradient projection type of algorithm is used to find the solution of each dual problem. The effectiveness of the method is demonstrated by presenting a numerical example of a non-linear system subject to stochastic ground acceleration.