基于改进的Co-Kriging模型的高维气动优化设计新方法研究

Surrogate-based optimization via Kriging model is a new methodology which can be used to greatly improve the design efficiency of complex optimization problems in aerospace applications. It has been classified as the third type of optimization algorithm, other than the conventional gradient-based and gradient-free numerical optimization algorithms. However, this method, up to now, is typically limited to the applications with the number of design variables being about 10. For the higher -dimensional aerodynamic design optimization problems (typically about 100) based on high-fidelity computation fluid dynamics (CFD), this method is suffering from the limitations of requiring extremly large number of CFD analysis, thus prohibitive computational cost. The "size of the problem", or the so-called "curse of dimensionality" has become a "bottleneck" before it is further extended to the real-world applications. This project aims to develop the new concept of a variable-fidelity, global optimization framework which is supposed to be able to breakthrough the "bottleneck" of the surrogate-based optimization for higher-dimensional problems. The basic idea is to use an improved co-Kriging surrogate modeling which simultaneously incorporates gradient solution by Adjoint method and the solution by lower-fidelity CFD analysis. This study will address the key issues such as how to efficiently build an accurate co-Kriging model, and how to quickly find the global optimum as well as how to handle with the multiple constraints based the statistical uncertainty estimation of a co-Kriging model. Through developing such an innovative optimzation method based on co-Kriging model, we expect to work out a more efficient and practical optimization algorithm,which can stimulate the scientific research as well as the enginering design applications in this area. The resulting algorithm can be potentially applied to the aerodynamic design of an aircraft or other disciplines such as structural design where numerial analysis methods of varying fidelity are in use.

基于Kriging代理模型的优化方法是一种有望大大提高航空航天领域复杂设计问题效率的新方法，被称为区别于传统的梯度优化与非梯度优化的第三类优化算法。但是，该方法目前适用于直到10维左右的优化问题;对于基于高可信度CFD的高维气动优化设计问题(100维左右)，该方法面临CFD分析次数过多、计算量过大的困难。高维问题成为限制其进一步应用的"瓶颈"。本项目基于一种改进的co-Kriging模型，通过在优化过程中同时引入adjoint梯度求解和低可信度CFD分析，从而提出一种适用于高维优化问题的全局优化框架，并发展出相应的统计学寻优机制和约束处理方法。通过发展基于co-Kriging模型的新的优化理论和算法，有望解决代理模型优化方法在高维优化问题方面面临的困难，为飞行器气动优化设计及其它学科相关研究提供一种比传统的代理模型优化更有效的方法。

目前，代理优化方法在解决基于高可信度CFD的高维气动优化设计问题(30-100维左右)时，面临CFD分析次数过多、计算量过大的困难。为了有效解决这一问题，本项目从代理模型理论与自适应寻优算法入手，在新型代理模型理论与算法、高效优化加点准则、子优化算法、约束处理方法方面，取得了一系列创新的研究成果。研究提出了适用于高维问题的优化算法及框架，并开展了大量的数值算例验证研究，成功将代理优化方法适用的维数拓展至100维，为飞行器气动设计及其他学科相关研究提供了一种更有效的优化设计技术。主要研究成果包括：.1）研究提出了两种新型Cokriging代理模型理论与算法——分层Kriging模型（HK）和梯度增强型分层Kriging模型（GEHK）。HK模型是一种采用低可信度数据辅助预测的变可信度代理模型，相比现有变可信度代理模型方法，模型更加通用，鲁棒，建模效率更高；GEHK模型是一种同时将低可信度分析和梯度信息引入的代理模型，能够进一步提高代理模型的预测精度。.2）发展了基于新型cokriging代理模型的无约束优化方法。对于变可信度代理模型，通过将其与统计学优化加点准则相结合，并对子优化算法展开研究，发展了一种高效的变可信度自适应全局优化算法框架。在此基础上，还发展了一种变可信度优化加点准则，能够自适应地判断新增加样本点的位置和可信度层，实现了对变可信度优化方法效率的进一步提高。此外，还开展了基于梯度增强型代理模型的高维气动优化设计算例研究（108维）。研究结果表明，相比基于Kriging模型的优化方法，GEK方法能够节省10倍以上的三维流场计算量。.3）发展了基于新型cokriging代理模型的约束优化方法。对约束处理方法展开深入研究，通过对约束函数建立代理模型，计算每一样本点处满足约束的概率，获得了考虑约束的自适应优化加点准则，进而发展了基于新型代理模型的约束优化算法框架。此外，还针对所发展的变可信度自适应优化加点准则，发展了一种有效的约束处理方法。通过气动优化算例研究表明，优化结果能够精确收敛于优化问题的最优解，并严格满足约束条件。