Kriging Based High Dimensional Model Representation Method With Applications

As a typical kind of multidisciplinary system with high dimensional design variables and many design constraints,the computational burden of directly using numerical simulation models for optimization is heavily high and even unacceptable.To alleviate the computational cost in flight vehicle design optimization,metamodeling has received an increasing attention and application in the past two decades.When dealing with high dimensional problems,those well-known traditional metamodels(techniques such as Polynomial Response Surface method,Radial Basis Function and Kriging)still face some bottlenecks,which means that the computational cost of building a metamodel increases exponentially with the increase of dimensionality.To alleviate the computational cost and tackle the limitation of solving high dimensionality,the main work of this thesis is to develop a Kriging based High Dimensional Model Representation method(KRG-HDMR).With the high global metamodeling accuracy and efficiency,the KRG-HDMR method is applied to two real-world engineering optimization examples in aerospcae engineering field.The main contents are summarized as follows:(1)A comprehensive review on the-state-of-art of metamodel based approximate optimization strategies for flight vehicle design is reported.Then a detailed introduction of the most promising global metamodeling technique,High Dimensional Model Representation,especially Cut-HDMR is given.And it is concluded that the metamodeling accuracy and efficiency still need to be further improved.(2)Based on the concept of RBF-HDMR,a new global metamodeling method combining Kriging and Cut-HDMR,denoted as KRG-HDMR,is proposed.The first order and second order component functions are approximated by Kriging metamodels respectively,which establishes a basic framework for integrating different adaptive sampling methods with KRG-HDMR for better global metamodeling performance.(3)To further improve the metamodeling accuracy,a KRG-HDMR method with Proportional Sampling,denoted as KRG-HDMR-PS is proposed At each sampling iteration of KRG-HDMR-PS,the sub-interval with the strongest nonlinear feature is first identified,where a new sample point is generated in terms of a predefined proportion coefficient.Iterate this sampling process till construction convergence.Via two numerical functions with strongly nonlinear first order terms,comprehensive tests are carried out in terms of different values of the proportion coefficient.The test results reveal that the KRG-HDMR-PS with proportion coefficient set as 1/2 greatly outperforms RBF-HDMR in both approximation accuracy and robustness.It is also demonstrated that the proposed proportional sampling strategy can sequentially generate construction samples with more proper distribution.(4)To fully utilize the metamodeling information provided by component Kriging metamodels,KRG-HDMR method with Variance based Sampling,i.e.,denoted as KRGHDMR-VS,is proposed to achieve higher global approximation accuracy and metamodeling efficiency.KRG-HDMR-VS fully takes the advantage of the prediction variance provided by Kriging,and samples points with the maximal prediction variance on each cut-line and cutplane,which guarantees that the most informative points are chosen in each sampling iteration.Through a series of high dimensional numerical functions,the proposed KRGHDMR-VS method is demonstrated to outperform the well-known Cut-HDMR variations AERBF-HDMR and RBF-HDMR in terms of approximation accuracy and metamodeling efficiency.(5)To solve the limitation that Cut-HDMR cannot utilize existed random points,the error information of these existed points are used to build error models to refine KRG-HDMR model.There are two strategies: error model strategy and allocation error model strategy.The error model strategy is to build one Kriging metamodel with the error information,while the allocation error model strategy is to allocate the error information to each component term and update their corresponding Kriging metamodel.Both of these two strategies are applicable to KRG-HDMR-PS and KRG-HDMR-VS.(6)Based on the aforementioned studies,the most robust and accurate method,i.e.,KRG-HDMR-VS,is successfully applied to a 10-D airfoil aerodynamic and a 16-D satellite structural optimization problem,which respectively need to evaluate time-consuming CFD and FEA simulation models.Then KRG-HDMR-VS is used to approximate aerodynamic performance(i.e.,the lift-drag ratio)and structural behaviors(i.e.,satellite total mass and frequency).The accuracy evaluation results show that KRG-HDMR-VS can build an accurate global metamodel to approximate the original time-consuming simulations with limited samples.Then genetic algorithm is used to carry out optimization based on the KRGHDMR-VS model.The lift-drag ratio of the optimized airfoil has improved by 23.5% and the total mass of the satellite has reduced by 44.8kg.Compared to Kriging under the same computational cost,KRG-HDMR-VS constructs global metamodels with higher approximation accuracy,which makes it more appropriate to approximate high dimensional,expensive simulation models and thus boost the optimization efficiency.