Keywords: stochastic design, stochastic subset optimization, stochastic simulation, response surface methodologies, model uncertainty, structural robustness.

The knowledge about a system in engineering design applications is never complete. Often, a probabilistic quantification of the uncertainty arising from this missing information is warranted in order to efficiently incorporate our partial knowledge about the system and its environment into their respective models. In this stochastic setting, the design objective is related to the expected value of a system performance measure, such as reliability or expected life-cycle cost. For problems involving complex system models, this expected value can rarely be evaluated, or efficiently approximated, analytically and so it is often estimated using stochastic simulation techniques which involve an unavoidable estimation error and significant computational burden, features that make the associated stochastic optimization a challenging task.

An efficient approach, called stochastic subset optimization (SSO) [1,2], is presented for efficiently identifying subsets for the optimal design variables within the original design space. SSO is based on the formulation of an augmented stochastic optimization problem where the design variables are artificially considered as uncertain with uniform probability distribution. The optimization of the initial objective function is then equivalently expressed as the optimization of an auxiliary probability density function (PDF) that is formulated in the context of the augmented stochastic problem. At each iteration samples from this auxiliary density are efficiently simulated and the information in these samples is then used to identify the subset that has the highest likelihood of containing the optimal design variables within some class of admissible subsets. Simultaneously a global sensitivity analysis for the model parameters and the design variables is established, that illustrates which parameters have higher contribution to the overall stochastic performance. Topics are discussed related to stopping criteria for the iterative approach and combination of SSO with other stochastic optimization algorithms for greater efficiency.

Implementation of moving least squares (MLS) response surface methodologies is also discussed for approximation of the system model response. This leads to a significant reduction of the computational cost of the stochastic simulations within the SSO. An approach for adaptive improvement of the MLS approximation is discussed using the sensitivity analysis information available at each iteration of the SSO.

An illustrative example is presented that shows the efficiency of the proposed methodology. In that example, SSO is shown to efficiently identify a small subset that contains the optimal design variables. The use of information from SSO to fine tune the characteristics of MLS is also shown to be highly effective.

1

A.A. Taflanidis, J.L. Beck, "An efficient framework for optimal robust stochastic system design using stochastic simulation", Computer Methods in Applied Mechanics and Engineering, 198 (1), 88-101, 2008. doi:10.1016/j.cma.2008.03.029

2

A.A. Taflanidis, J.L. Beck, "Stochastic Subset Optimization for reliability optimization and sensitivity analysis in system design", Computers and Structures, 87, 318-331, 2009. doi:10.1016/j.compstruc.2008.12.015

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