Research On Approximate Modeling And Sampling Strategies Based On High Dimensional Model Representation

With the continuous development of engineering technologies and the increasingcomplexity of system structures, approximation techniques are widely adopted to solvecomplex engineering problems. Traditional approximation methods can achieve the modelaccuracy and modeling efficiency when approximating the low dimensional nonlinearproblems. However, for the high dimensional nonlinear problems, their computational costwill grow exponentially, and their approximation ability will decrease largely, sometimesthese methods may be useless.High dimensional model representation (HDMR) is a specialmethod aimed at solving high dimensional problems. It can not only reflect the mappingrelationship between input and output, but also identify the correlation between the inputvariables through the hierarchical structure of the model.In this paper, a new HDMR, called SVR-HDMR based on DIRECT, is proposed bycombining the technique of support vector regression (SVR) and the strategy of dividingrectangle (DIRECT) sampling. It inherits good hierarchical structures of HDMR throughdividing a high dimensional problem into a sum of several low dimensional problems, andutilizes the adaptive ability of SVR for the small sample size and the intelligence of theDIRECT sampling to achieve a higher accurate model by a certain number of samples.Then, an improved DIRECT sampling method (IDRECT) is proposed to fit for theHDMR. Due to the lack of samples of the boundary of the design space during the samplingprocess when using the DIRECT sampling in the HDMR, the fitting performance of themodel is usually bad near the boundary, which may limit the application of the DIRECTsampling in some engineering problems，therefore，the sampling method is improved. Todemonstrate the effectiveness of the method, ten common numerical examples are tested. Andthe SVR-HDMR based on IDRECT is compared with other approximation methods. Theresults show that SVR-HDMR based on IDRECT is more efficient than other methods, andovercomes the shortcoming of the bad performance near the boundary involved in DIRECTsampling.Based on the above research, the proposed HDMR is also applied in an optimizationdesign of the column structure size parameters successively. Firstly, the software UG is usedto model the column structure. Secondly, a set of samples can be obtained through the finite element simulation by the software Hyperworks. Thirdly, a SVR-HDMR model based onIDIRECT can be built by utilizing these samples, which describes the mapping relationshipbetween column structure size parameters and the rail deflections. Finally, the model isregarded as a constraint in the optimization design for the column structure size parameters,and a more practical column structure product can be obtained.