A Framework for Determination of the Weak Pareto Frontier Design Solutions under Probabilistic Constraints
Mavris, Dimitri N.
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The design of complex systems such as aircraft or missiles requires the synergy of multiple disciplines. The design quality must ultimately be assessed by multiple criteria that often can not be optimized simultaneously. Therefore, in a less restrictive sense the Weak Pareto Frontier (WPF) in the objective space and the corresponding design solutions must be found because the WPF includes more compromised solutions than the conventional Pareto frontier. Real-world decisions are usually made in a state of uncertainty. Most often the effects of uncertainties are embodied in the probabilistic constraints (PC) that usually must be satisfied jointly. The combination of these issues requires a new framework to combine separately developing multidisciplinary optimization, multi-objective optimization, and joint probability assessment methods together, to solve a joint probabilistic constraint, multi-objective, multidisciplinary optimization problem and find the WPF solutions. The purpose of this paper is to provide such a framework. This framework starts with constructing fast and accurate surrogate models of different disciplinary analyses in order to reduce the computational time and expense to a manageable level and obtain trustworthy probabilities of the PC’s and the WPF. A hybrid method is formed here that consists of the second order response surface methodology (RSM) and the support vector regression method (SVR) capturing the global tendency and local nonlinear behavior, respectively. The parameters of SVR to be pre-specified are selected using practical methods and a modified information criterion that makes use of model fitting error, predicting error, and model complexity information. Then a neighborhood search method based on Monte Carlo simulation is provided to find valid designs that are feasible and consistent for the coupling variables featured in a multidisciplinary design problem. Two schemes have been developed. One scheme finds the WPF by finding a large enough number of valid design solutions such that some WPF solutions are included in them. Another scheme finds the WPF by directly finding the WPF of each consistent design zone that is made up of consistent design solutions. Then the probabilities of the PC’s are estimated, and the WPF and corresponding design solutions are found. A simple yet typical aircraft design problem is solved to demonstrate the feasibility of this framework. The results show that the method to select the pre-specified parameters of SVR works well, the hybrid surrogate models are fast and accurate, and both neighborhood search schemes can find the WPF.