Exploration of anisotropic design space by using unified Taylor-cokriging method

Abstract

Anisotropy occurs naturally when modelling engineering design spaces and sometimes becomes significant, such as in robust and reliability-based design optimization, where the “noise factor” is often modeled in a different way with the design variables. Hybrid models are often used to handle the anisotropy by using different methods for different types of variables. However, since different models are used, the dimensional interactions cannot be accurately approximated, and the intensive computational cost remains high due to the number of samples needed. This work presents a novel method to tackle this anisotropic design space with a proposed Hessian-enhanced Taylor–cokriging unified model. First, the sampling strategies for the exploration of anisotropic design spaces is presented. Second, the difficulty of modelling with anisotropic design spaces is analyzed. Third, a solution is proposed by introducing additional assumptions of the probability distribution into the modelling process. Furthermore, to reduce the computational cost, the sensitivity method is integrated into the methodology to obtain low-cost derivatives. Finally, the proposed method is assessed for the specific case where only a single sample value exists in certain dimensions. The elementary effect is adopted to quantify the dimensional properties. Three test cases, including two theoretical ones and an industrial one, are used to assess the validity of this method. The first theoretical test case calls for a massive number of multimodal two-dimensional test functions whose spatial properties in each dimension are distinctive. The second theoretical test case uses a ten-dimensional test function. Design optimization is performed for an automotive engine cooling fan whose design space exhibits anisotropic properties. The model built with the proposed Taylor–cokriging method attains a good accuracy. This study is meaningful for the large-scale robust design and reliability-based design optimizations whose design spaces are likely to show anisotropic properties.