The Key Of Research And Application Of Multi-parameter Nonlinear Optimization

Development of modern CAE technique has boosted wide applications of variousoptimizations to automobile body design. Objective functions of such optimizationproblems are black-box and usually require complicate CAE analysis which is verytime-consuming. For such problems, traditional gradient-based and heuristic methodscan hardly satisfy the application requirement of practical engineering. So far,metamodel technique is an effective approach to solve black-box expensiveoptimizations. The most remarkable advantage of metamodel-based optimization isthat experimental prediction of the objective functions can be drawn by a few functionevaluations, so that the efficiency of optimization can be improved. However, withincreasing number of design variables, the scale of design space will expand rapidly.Accuracy and efficiency become main bottlenecks of the metamodel technique. Thus,constructing an effective metamodel method for multi-variable problems is verynecessary. Moreover, many engineering optimizations involve discrete variables. Thehigh dimensionality of discrete variable limits the advantage of metamodel technique,so a study of discrete variables based multi-variable optimization from anotherperspective is needed. Summarily, this paper conducts research on the basis ofmulti-variable optimization. Details of the research contents are described as thefollows:(1) With increasing number of design variables, the number of sample pointsused to construct the metamodel grows almost exponentially. For nonlinear problems,metamodels with high accuracy can be hardly obtained under acceptablecomputational cost. Moreover, conventional metamodel techniques lacks capability toeffectively identify coupling relationships among the design variables and theircontributions to the objective response, so essence of the problem cannot be wellreflected. Thus, a multi-variable decoupling based adaptive nonlinear metamodeltechnique (Kriging-HDMR) is proposed in this paper. The method uses multi-variabledecoupling model (Cut-HDMR) to decompose a high-dimensional problem into aseries of coupling terms with different hierarchies. The components of Cut-HDMR aredetermined by identifying the relationship of each pair of variables and neglectinghigh-order coupling terms which have weak contributions to the objective response,so that the computational cost can be reduced from exponential growth to polynomial level. On the other hand, the proposed method uses Cut-HDMR model to transformmetamodeling object from a high-dimensional problem into a few lower-dimensionalsub-problems, successfully reducing the metamodeling complexity, so accuracy of themetamodel can be remarkably improved. Research results indicate that based on asame set of sample points, the accuracy of Kriging-HDMR model is much better thanthat of Kriging model.(2) So far, popular methods for continuous variable optimizations are based onmetamodel techniques and intelligent sampling strategies. The performance of thesemethods significantly depends on the accuracy of metamodels. However, formulti-variable problems, traditional metamodels usually have low accuracies, leadingto low efficiency and local optimums. Therefore, a projection-based heuristic globalsearch algorithm (P-HGS) is proposed for multi-variable optimization. The algorithmadopts multi-dimensional projection technique to integrate Kriging-HDMR with MPSseamlessly. In each iteration, new samples are projected to the lines and hyper-planescrossing through the cut center. The projected points are used to update the Krigingmodels of the component terms of Kriging-HDMR. In order to ensure theinterpolation of Kriging-HDMR at the new samples, a error modification term is alsoadded. P-HGS takes advantage of Kriging-HDMR to improve the efficiency andaccuracy of the optimizaition. Tested by several nonlinear functions, P-HGSobviously excels MPS in global search, efficiency and robustness.(3) Engineering optimizations often concern with discrete variables, such as thematerial variable. Existence of discrete variables extraly, even multiply increase thedimensionality of solution vector so that the accuracy of metamodel can hardly beguaranteed, particularly for the multi-variable optimization. Thus, a K-mean Clusterbased Heuristic Sampling with Utopia-Pareto Directing Adaptive Strategy(KCHS-UPDA) is proposed for discrete design variables based multi-objectiveoptimization. The method defines several feature solutions according to the Paretofrontier. In each iteration, the feature solutions are located and correspondingsampling sets are generated, so that space reduction can be easily realized. In order toenhance the convergent rate, in each sampling set, k-mean cluster is employed toconstruct a probabilistic model, according to which new samples are drawnstochastically. The method successfully replaces metamodels so that accuracy defectof metamodel for the discrete variable can be well avoided. Moreover, in a iteration,the locations of the feature solutions are usually disperse and the sampling setsdistribute around each feature solution, so global search ability of the algorithm can be enhanced to some extent. A few benchmark problems are used to test theperformance of the proposed algorithm. Test results indicate that KCHS-UPDA cangenerally converge to the Pareto frontier with a small quantity of number of functionevaluations.