Constraint aggregation for large number of constraints in wing surrogate-based optimization
The method of aggregating a large number of constraints into one or few constraints has been successfully applied to wing structural design using gradient-based local optimization. However, numerical difficulties may occur in the case that the local curvatures of the aggregated constraint become extremely large and then ill-conditioned Hessian matrix may be yielded. This paper aims to test different methods of constraint aggregation within the framework of a gradient-free optimization, which makes use of cheap-to-evaluate surrogate models to find the global optimum. Three constraint aggregation approaches are investigated: the maximum constraint approach, the constant parameter Kreisselmeier-Steinhauser (KS) function, and the adaptive KS function. We also explore methods of aggregating constraints over the entire structure and within sub-domains. Examples of structural optimization and aero-structural optimization for a transport aircraft wing are employed and the results show that (1) the KS function with a larger constant parameter ρ can lead to better optimization results than the adaptive method, as the active constraints are approximated more accurately; (2) lumping the constraints within sub-domains instead of all together can improve the accuracy of the aggregated constraint and therefore helps find a better design. Finally, it is concluded from current test cases that the most efficient way of handling large-scale constraints for wing surrogate-based optimization is to aggregate constraints within sub-domains and with a relatively large constant parameter.
KeywordsWing design Surrogate-based optimization Constraint aggregation Kreisselmeier-Steinhauser function Aero-structural optimization
Computational fluid dynamics
Computational structural dynamics
Finite element method
Lower confidence bounding
Latin hypercube sampling
Mean squared error
Minimizing surrogate prediction
Probability of improvement
Sequential quadratic programming
The authors are grateful for the thoughtful comments and valuable suggestions given by the anonymous reviewers.
This research was sponsored by the National Natural Science Foundation of China (NSFC) under grant no. 11772261 and Aeronautical Science Foundation of China under grant no. 2016ZA53011.
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