Tối ưu hóa Hệ thống Đa ngành Hiệu quả bằng Phân tích Biến đổi và Lấy mẫu Động - Jangho Park (Virginia Tech)

Efficient Global Optimization of Multidisciplinary System using Variable Fidelity Analysis and Dynamic Sampling Method Jangho Park Dissertation submitted to the Faculty of the Virginia Polytechnic Institute and State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Aerospace Engineering Seongim S Choi, Chair Pradeep Raj, Co-chair William J Devenport Xi Chen April 29, 2019 Blacksburg, Virginia Keywords: Efficient Global Optimization(EGO), Variable-Fidelity(VF) Analysis, Data mining, Gaussian Process Regression(GPR) modeling, Design of Experiment(DoE) Copyright 2019, Jangho Park Efficient Global Optimization of Multidisciplinary System using Variable Fidelity Analysis and Dynamic Sampling Method Jangho Park (ABSTRACT) Work in this dissertation is motivated by reducing the design cost at the early design stage while maintaining high design accuracy throughout all design stages. It presents four key design methods to improve the performance of Efficient Global Optimization for multidis- ciplinary problems. First, a fidelity-calibration method is developed and applied to lower- fidelity samples. Function values analyzed by lower fidelity analysis methods are updated to have equivalent accuracy to that of the highest fidelity samples, and these calibrated data sets are used to construct a variable-fidelity Kriging model.

For the design of exper- iment (DOE), a dynamic sampling method is developed and includes filtering and infilling data based on mathematical criteria on the model accuracy. In the sample infilling process, multi-objective optimization for exploitation and exploration of design space is carried out. To indicate the fidelity of function analysis for additional samples in the variable-fidelity Kriging model, a dynamic fidelity indicator with the overlapping coefficient is proposed. For the multidisciplinary design problems, where multiple physics are tightly coupled with different coupling strengths, multi-response Kriging model is introduced and utilizes the method of iterative Maximum Likelihood Estimation (iMLE).

Through the iMLE process, a large number of hyper-parameters in multi-response Kriging can be calculated with great accuracy and improved numerical stability. The optimization methods developed in the study are validated with analytic functions and showed considerable performance improve- ment. Consequentially, three practical design optimization problems of NACA0012 airfoil, Multi-element NLR 7301 airfoil, and all-moving-wingtip control surface of tailless aircraft are performed, respectively. The results are compared with those of existing methods, and it is concluded that these methods guarantee the equivalent design accuracy at computational cost reduced significantly.

Efficient Global Optimization of Multidisciplinary System using Variable Fidelity Analysis and Dynamic Sampling Method Jangho Park (GENERAL AUDIENCE ABSTRACT) In recent years, as the cost of aircraft design is growing rapidly, and aviation industry is interested in saving time and cost for the design, an accurate design result during the early design stages is particularly important to reduce overall life cycle cost. The purpose of the work to reducing the design cost at the early design stage with design accuracy as high as that of the detailed design. The method of an efficient global optimization (EGO) with variable-fidelity analysis and multidisciplinary design is proposed. Using the variable-fidelity analysis for the function evaluation, high fidelity function evaluations can be replaced by low-fidelity analyses of equivalent accuracy, which leads to considerable cost reduction.

As the aircraft system has sub-disciplines coupled by multiple physics, including aerodynamics, structures, and ther- modynamics, the accuracy of an individual discipline affects that of all others, and thus the design accuracy during in the early design states. Four distinctive design methods are developed and implemented into the standard Efficient Global Optimization (EGO) framework: 1) the variable-fidelity analysis based on error ap- proximation and calibration of low-fidelity samples, 2) dynamic sampling criteria for both filtering and infilling samples, 3) a dynamic fidelity indicator (DFI) for the selection of analysis fidelity for infilled samples, and 4) Multi-response Kriging model with an iterative Maximum Likelihood estimation (iMLE). The methods are validated with analytic functions, and the improvement in cost efficiency through the overall design process is observed, while maintaining the design accuracy, by a comparison with existing design methods. For the practical applications, the methods are applied to the design optimization of airfoil and complete aircraft configuration, respectively.

The design results are compared with those by existing methods, and it is found the method results design results of accuracies equivalent to or higher than high-fidelity analysis-alone design at cost reduced by orders of magnitude. Contents List of Figures vii List of Tables x 1 Introduction 1 1.1 Motivations and Objectives .1 Optimization methods for design problems .2 Efficient Global Optimization(EGO) .3 Variable fidelity analysis .4 Kriging surrogate model .3 Scope of Research Work .1 Variable fidelity analysis and adaptive sampling .2 Sample infilling and dynamic fidelity indicator .3 Multi-Response Kriging model for multidisciplinary problems. 24 2 Single Discipline Optimization 26 2.1 Kriging Surrogate Model .2 Variable Fidelity Analysis .3 Dynamic and Adaptive Sampling Method .1 Filtering low-accuracy samples .2 Infilling samples for single fidelity optimization .3 Infilling samples for variable fidelity optimization .4 Dynamic Fidelity Indicator .5 Variable Fidelity EGO Framework. 50 3 Extension to Multidisciplinary Optimization 52 3.1 Multi-response Kriging (MRK) .1 Reduced-Covariance Multi-Response Kriging (RC-MRK) .2 Fully-Expanded-Covariance Multi-Response Kriging (FEC-MRK) .3 Comparisons between RC-MRK and FEC-MRK .4 Iterative method for Maximum Likelihood (iMLE) Estimation .2 MRK for Variable Fidelity Optimization .1 Infill sampling criteria .1 Parametric study on dynamic switching approach .2 EGO Framework with Variable Fidelity Analysis .2 Sensitivity to filtering criteria .3 Multi-Response Kriging model .1 Aeroelastic Shape Optimization Airfoil .2 Aerodynamic shape optimization of high-lift multi-element airfoil .2 High-fidelity aerodynamic analysis .3 Aerodynamic shape optimization of control surface of tailless aircraft .1 Innovative control effectors .3 Variable-fidelity aerodynamic analysis.

109 6 Conclusion 112 Bibliography 115 vi List of Figures 1.1 Life cycle cost and impact on cost for each design phrase .2 Schematics of EGO process .3 Example of different fidelity analysis methods in Aerodynamics .4 Comparison of lift coefficient of NACA 0015 airfoil .5 Example of Gaussian process with 5 samples .6 Probability density function of estimated function value .7 Developed design methods for Efficient Global Optimization .1 OV L coefficient between two probability functions .2 Pareto-front for ISC .3 Comparison of probability density function of calibrated lower fidelity Krig- ing, the highest fidelity Kriging, and variable fidelity Kriging at an arbitrary design point .4 Pareto-front for ISC in variable-fidelity Kriging .5 The schematics of variable fidelity EGO framework .1 The comparison of the number of hyper-parameter for different number of input and output variate .2 The schematics of iterative Maximum Likelihood Estimation (k = iteration number) .3 Example of Kriging in variable fidelity optimization .4 Example of the use of multi-response Kriging in variable fidelity multidisci- plinary design problem .1 Configurations and iso-contours of six-hump camel back function .2 Configurations and iso-contours of Matlab peak function .3 Parameter study results for six hump camel back function .4 Parameter study results for Matlab peak function .5 2D contour and 3D plot of six hump camel back function with different fidelities 77 4.6 Design history of 3D plots of VF optimization framework for six hump camel back function .7 Comparison of variable fidelity and high fidelity optimization for six hump camel back function .8 2D contour and 3D plot of simplified Rastrigin function with different fidelities 80 4.9 Design history of VF optimization framework with the simplified Rastrigin function .10 Comparison of variable fidelity and high fidelity optimization for simplified Rastrigin function .11 Design iterations and the CCF w.t the varying filtering criteria (six hump camel back function) .12 Design iterations and the CCF w.t the varying filtering criteria (simplified Rastrigin function) .13 The contour of weakly coupled functions .14 The result of global accuracy test for weakly coupled function .15 The sliced contour of strongly coupled functions (z1 = 0) .16 The result of global accuracy test for strongly coupled function .17 The contour of objective function for weakly coupled case .18 The comparison of f1 and f2 contour at the last iteration (Weakly coupled function) .19 The result of EGO test for weakly coupled functions .20 The contour of objective function for strongly coupled case(z1 = 0) .21 The result of EGO test for strongly coupled functions .1 NACA 0012 diagram with design variables .2 Aerodynamic analysis result of NACA0012 baseline airfoil .3 Aerodynamic force and aeroelastic torsion of the wing .4 Schematic of airfoil optimization process .5 Configuration of design variables in NLR 7301 multi-element airfoil .6 Validation of CFD analysis for baseline airfoil .7 Comparison of objective value variations w.t the number of sample points .8 Comparison of baseline and design results in pressure coefficient contour .9 Comparison of baseline and design results in geometry .10 ICE 101 configuration and the models for wind tunnel test .11 Control surfaces of ICE 101 configuration .12 Design variables of AMT .13 Surface mesh for computational methods .14 Pressure coefficient contour comparison between each method (M=0.15 The comparison of pressure coefficient along the cross section between each method (M=0.16 Design history for the optimum AMT shape between VF optimization and HF optimization .17 The comparison of AMT shape .18 Pressure contour of the optimum AMT design result by variable fidelity op- timization. 111 ix List of Tables 2.1 The algorithms of the approximated q-EI .1 Comparison of the number of hyper-parameters in various GPR models .1 A total required number of sample points (Six hump camel back function) .2 A total required number of sample points (Matlab Peak Function) .1 NACA 0012 airfoil optimization results .2 Design variable bound for NLR 7301 multi-element airfoil optimization .3 Aerodynamic performance of the minimum point .4 Design variable of the minimum point .5 Comparison of Design result for AMT shape. 111 x List of Abbreviations Symbols α multiplicative error β additive error Ia Identity matrix whose size is (a × a) w or W a vector of regression parameter for linear regression ϵ additive Gaussian noise with zero mean and non-zero variance κ switching parameter for MPMO-ISC, dynamic switching approach I+ the set of index of the infilling sample points I− the set of index of the filtered sample points I VF the set of index of the sample points for variable fidelity Kriging I the set of index for sample points prior to the adaptive sampling µ the mean function value of random variate Φ(·) cumulative probability density function ϕ(·) probability density function σ2 variance of Gaussian distribution θG global clustering parameter θL local clustering parameter B covariance matrix between responses C covariance matrix xi c covariance function Cov[·, ·] covariance between two random variates det(·) determinant of a matrix E[·] expected value of random variate EI expected improvement G(·, ·) Gaussian process with mean and variance K the number of total design iteration KRG Kriging interpolation function n the number of observed data points n∗ the number of additional samples npop the total number of optimal candidates on the Pareto-front in MPMO-ISC pro- cess OV L overlapping coefficient p the number of design variables (the dimension of design space) pdf probability distribution function q the number of responses, disciplines or physics (the dimension of response space) Q and R spatial covariance matrix r the number of different fidelity for the sample analysis T Threshold for MPMO-ISC, dynamic switching approach V ar[·] variance of random variate x or x sample location y or y sample response xii Superscripts (k) iteration number Subscripts aug augmented vector g, h index for different responses i, j index for different sample locations m index for function fidelity VF variable fidelity Letter modifiers ¯ mean function value of Gaussian distribution ˆ trial location ˜ location of adaptive infill sampling ′ calibrated function value ∗ Optimized value Abbreviations CCF Computational Cost Factor CFD Computational Fluid Dynamics DFO Derivative Free Optimization EGO Efficient Global Optimization FDM Finite Difference Method FEC-MRK Fully Expanded Covariance Multi-Response Kriging GBO Gradient Based Optimization xiii GPR Gaussian Process Regression iMLE iterative Maximum Likelihood Estimation IOC Impact On Cost ISC Infill Sampling Criteria LCC Life Cycle Cost MDO Multidisciplinary Design Optimization PDF Probability Distribution Function RANS Reynolds-Averaged Navier-Stokes RC-MRK Reduced Covariance Multi-Response Kriging RMSE Root Mean Square Error xiv Chapter 1 Introduction 1.