Research On Nash-EGO Algorithm For High-dimensional Optimizations And Its Applications In Aerodynamics

Benefited from the Kriging surrogate model and EI(Expected Improvement)updating strategy,the Efficient Global Optimization(EGO)algorithm is more and more popular in aerodynamic shape optimizations for its high accuracy and efficiency.However,subjected to the existence of“Curse-of-dimensionality”,the traditional EGO algorithm is not very suitable in dealing with high-dimensional optimization problems.In order to extend the EGO algorithm to more realistic engineering large-scale optimizations,a novel Nash-EGO algorithm is presented by coupling with the parameter-splitting technology and Nash game strategy.An improved traditional EGO algorithm is first proposed by improving the initial sample points set and enhancing the searching ability of the sub-optimizers.The improved algorithm is tested by several numerical test functions and 2-D bump shape optimization.The results show that,the improved EGO algorithm is more accurate and efficient than the traditional EGO and the Genetic Algorithms(GAs).Mandate to refine the methodology of EGO and extend its usage to a more realistic high-dimensional aerodynamic design problem,a novel Nash-EGO algorithm is presented based on the parameter-splitting technology and Nash game strategy.In the present Nash-EGO,the large-scale design variables are split into several subsets by adopting Nash variable territory splitting,and the EGO optimizer acts as a player of specific Nash game.All the EGO players are coupled with each other and assigned to optimize their own subsets synchronously in parallel to produce the corresponding approximate optimal subsets.Doing in this way,the performance of EGO players could be expected to keep at a high level due to the fact that EGO players now take care of only their own small-scale subsets instead of facing the large-scale problem directly.A set of typical cases with a small number of variables are firstly selected to validate the performance of each EGO player mentioned.Then,the Nash-EGO proposed is tested by representative function cases with a scale of 30 and 90 design variables.It can be learned from the tests that,as compared with conventional EGO,the present algorithm can always find near optimal solutions,which are closer to the theoretical values,and moreover,less CPU time-consuming,up to 584.69 times faster,are achieved.All cases whether30 or 90 design variables have similar efficient performances,which indicates the present algorithm has the potential ability to cope with real large-scale optimizations.The algorithm is first applied to a regular low-dimensional bump shape optimization.The results show that,the proposed Nash-EGO algorithm can also perform very well in regular low-dimensional optimizations with property configurations.Then,the proposed Nash-EGO algorithm is applied to real high-dimensional airfoil/wing optimizations.A finite wing is represented by mounting control airfoils parameterized with a set of design variables,which are manipulated to be increased gradually in purpose of enriching the searching space to accommodate possible more optimal solutions.The enriched design territory is technically split into small subsets to be assigned to the players of Nash-EGO.The algorithm is applied to a constrained drag minimization of RAE2822 airfoil with 14 to 54 design variables and 2 to 8 Nash players to investigate the performance of Nash-EGO,particularly for having an understanding of the influences of the numbers of design variables and Nash players.The results show that,different from traditional EGO algorithm,the Nash-EGO algorithm can archive more accurate optimum value with enriched design space to a certain extent.Meanwhile,more number of Nash players are usually needed when the number of design variables increased.At last,more challenging cases,sectional shape optimizations of DLR-F4 wing with up to 84 design variables,are conducted.The results show that,in comparison with the baselines,up to 6.80%drag reductions can be achieved by Nash-EGO optimizer,meanwhile,the CPU costs are greatly reduced(up to 60.80% speedup)as compared with the counterpart of the traditional EGO.The successful applications presented show the capability of Nash-EGO algorithm for solving real high-dimensional engineering optimizations.