Derivative-Enhanced Variable Fidelity Kriging Approach Dept. of Mechanical Engineering, University of Wyoming, USA Wataru YAMAZAKI 23 rd, September, 2010 Yamazaki, W., Dept. of Aero. Eng., Tohoku Univ. Wataru YAMAZAKI, Univ. of Wyoming -2- Motivation *Surrogate models for - Efficient Design Optimization - Efficient Aerodynamic Data Modeling - Inexpensive Uncertainty Quantification *For more accurate surrogate models - Gradient/Hessian Information Efficient adjoint approaches - Variable Fidelity Function Information Combination of absolute values of high-fid model and trends of low-fid models High-Fidelity ModelLow-Fidelity Model Experimental dataCFD result RANSInviscid Finer mesh CFD resultCoarser mesh CFD result Converged solutionLoose converged solution Yamazaki, W., Dept. of Aero. Eng., Tohoku Univ. Wataru YAMAZAKI, Univ. of Wyoming -3- Variable Fidelity Kriging Model Consider a random process model estimating a function value by a linear combination of variable fidelity function values Minimizing Mean-Squared-Error (MSE) between exact/estimated function with unbiasedness constraints Solving by using the Lagrange multiplier approach Yamazaki, W., Dept. of Aero. Eng., Tohoku Univ. Wataru YAMAZAKI, Univ. of Wyoming -4- Variable Fidelity Kriging Model Introducing correlation function for covariance terms Correlation is estimated by distance between two pts with radial basis function Unknown parameters are determined by the following system of equations Final form of the variable fidelity Kriging model is Yamazaki, W., Dept. of Aero. Eng., Tohoku Univ. Wataru YAMAZAKI, Univ. of Wyoming -5- Variable Fidelity Kriging Model In matrix form expression Correlation parameters in R and r, factors are estimated by a likelihood maximization approach Yamazaki, W., Dept. of Aero. Eng., Tohoku Univ. Wataru YAMAZAKI, Univ. of Wyoming -6- Variable Fidelity Kriging Model Correlations between all sample points combinations by a RBF Yamazaki, W., Dept. of Aero. Eng., Tohoku Univ. Wataru YAMAZAKI, Univ. of Wyoming -7- Derivative-enhanced Kriging Extension of direct approach of gradient-enhanced Kriging Correlations between F-F, F-G, G-G, F-H, G-H and H-H Up to 4 th order derivatives of correlation function Automatic Differentiation by TAPENADE No sensitive parameter Better matrix conditioning than indirect approach Yamazaki, W., Dept. of Aero. Eng., Tohoku Univ. Wataru YAMAZAKI, Univ. of Wyoming -8- Derivative-enhanced Variable Fidelity Kriging High Fidelity Function Gradient Hessian Hessian Vector 1 st Low Fidelity Function Gradient Hessian Hessian Vector 2 nd Low Fidelity Function Gradient Hessian Hessian Vector A Kriging surrogate model by absolute function values of high-fidelity level and function trends of low-fidelity levels Results & Discussion Yamazaki, W., Dept. of Aero. Eng., Tohoku Univ. Wataru YAMAZAKI, Univ. of Wyoming D Analytical Function Case 2 high-fidelity samples 5 low-fidelity samples (+0.5) 5 another low-fidelity samples (-0.5) Yamazaki, W., Dept. of Aero. Eng., Tohoku Univ. Wataru YAMAZAKI, Univ. of Wyoming D Analytical Function Case 2 high-fidelity samples 5 low-fidelity samples (+0.5) 5 another low-fidelity samples (-0.5) Yamazaki, W., Dept. of Aero. Eng., Tohoku Univ. Wataru YAMAZAKI, Univ. of Wyoming D Analytical Function Case 2 high-fidelity samples 5 low-fidelity samples (+0.5) 5 another low-fidelity samples (-0.5) Yamazaki, W., Dept. of Aero. Eng., Tohoku Univ. Wataru YAMAZAKI, Univ. of Wyoming D Analytical Function Case 2 high-fidelity samples 5 low-fidelity samples (+0.5) 5 another low-fidelity samples (-0.5) Yamazaki, W., Dept. of Aero. Eng., Tohoku Univ. Wataru YAMAZAKI, Univ. of Wyoming D Analytical Function Case 2D Cosine function Analytical gradient/Hessian Latin hypercube sampling for high and low-fidelity samples Comparison by RMSE Yamazaki, W., Dept. of Aero. Eng., Tohoku Univ. Wataru YAMAZAKI, Univ. of Wyoming D Analytical Function Case Only 5 high-fidelity samples Derivative information is useful to construct accurate model Exact functionFuncFunc/GradFunc/Grad/Hess Yamazaki, W., Dept. of Aero. Eng., Tohoku Univ. Wataru YAMAZAKI, Univ. of Wyoming D Analytical Function Case Exact function Only function information for both high/low-fidelity samples 5 high-fidelity samples with low-fidelity samples Low-fidelity information is useful to construct accurate model Yamazaki, W., Dept. of Aero. Eng., Tohoku Univ. Wataru YAMAZAKI, Univ. of Wyoming D Analytical Function Case Yamazaki, W., Dept. of Aero. Eng., Tohoku Univ. Wataru YAMAZAKI, Univ. of Wyoming D Analytical Function Case 50 low-fidelity sample points Best performance in FGH for both high/low-fid samples Yamazaki, W., Dept. of Aero. Eng., Tohoku Univ. Wataru YAMAZAKI, Univ. of Wyoming D Analytical Function Case Only function information for both high/low-fid samples Accuracy of VF model depends on trends of low-fidelity model But anyway helpful at smaller numbers of high-fid samples Yamazaki, W., Dept. of Aero. Eng., Tohoku Univ. Wataru YAMAZAKI, Univ. of Wyoming -20- Mach-AoA Hypersurfaces 2D aerodynamic data modeling of Cl, Cd, Cm Mach= [0.5; 1.5] AoA= [0.0; 5.0] Inviscid steady flow computations around NACA0012 Only function information because of noisy design space High fidelity model by a fine mesh 20,000 elements Computational time factor = 1 Low fidelity model by a coarse mesh 1,700 elements Computational time factor = 1/30 Yamazaki, W., Dept. of Aero. Eng., Tohoku Univ. Wataru YAMAZAKI, Univ. of Wyoming -21- Mach-AoA Hypersurfaces Yamazaki, W., Dept. of Aero. Eng., Tohoku Univ. Wataru YAMAZAKI, Univ. of Wyoming -22- Mach-AoA Hypersurfaces Mean error comparison in drag coefficient Improvements at smaller numbers of high fidelity samples Yamazaki, W., Dept. of Aero. Eng., Tohoku Univ. Wataru YAMAZAKI, Univ. of Wyoming -23- Mach-AoA Hypersurfaces Uncertainty analysis at M=0.8, AoA=2.5 for both Mach/AoA 1000 CFD evaluations for a specified value In total 7000 CFD evaluations (= 1000 x 7) for full-MC Full-MC results for =0.1 Yamazaki, W., Dept. of Aero. Eng., Tohoku Univ. Wataru YAMAZAKI, Univ. of Wyoming -24- Mach-AoA Hypersurfaces Mean of C m Variance of C m More accurate uncertainty analysis by Inexpensive MC with variable fidelity Kriging model Yamazaki, W., Dept. of Aero. Eng., Tohoku Univ. Wataru YAMAZAKI, Univ. of Wyoming D Airfoil Shape Optimization Unstructured mesh CFD Steady inviscid flow, M=0.755 NACA0012, 9 DVs by PARSEC Objective function as lift-constrained drag minimization Adjoint gradient available Geometrical constraint for sectional area Fidelity levels by finer/coarser meshes (1.0 : 0.1) Yamazaki, W., Dept. of Aero. Eng., Tohoku Univ. Wataru YAMAZAKI, Univ. of Wyoming -26- Infill Sampling Criteria for Optimization How to find promising location on surrogate model ? Maximization of Expected Improvement (EI) value Potential of being smaller than current minimum (optimal) Consider both estimated function and uncertainty (RMSE) Yamazaki, W., Dept. of Aero. Eng., Tohoku Univ. Wataru YAMAZAKI, Univ. of Wyoming D Airfoil Shape Optimization HFonly: Start from 16 HF initials, new samples by HF evaluations LFonly: Start from 128 LF initials, new samples by LF evaluations VFM: Start from 128 LF initials, new samples by HF evaluations Adj: Adjoint gradient evaluations only for new optimal designs Yamazaki, W., Dept. of Aero. Eng., Tohoku Univ. Wataru YAMAZAKI, Univ. of Wyoming D Airfoil Shape Optimization To include low-fidelity / derivative information is promising Yamazaki, W., Dept. of Aero. Eng., Tohoku Univ. Wataru YAMAZAKI, Univ. of Wyoming D Airfoil Shape Optimization Shock reduction on upper surface Towards supercritical airfoils in HFonly Additional adjustment of problem definition ? NACA0012, Obj = Optimal by HFonly, Obj = 6.66e-4 Optimal by VFM_Adj, Obj = 1.66e-4 Pressure Distributions Yamazaki, W., Dept. of Aero. Eng., Tohoku Univ. Wataru YAMAZAKI, Univ. of Wyoming -30- Concluding Remarks / Future Works Development of derivative-enhanced variable fidelity Kriging model Combination of absolute function values of high-fidelity samples and function trends of low-fidelity samples More accurate fitting on exact function Efficient inexpensive Monte-Carlo simulation at much lower cost Faster convergence towards global optimum Application to Euler/NS/WTT cases and so on Thank you for your attention !! Appendix Yamazaki, W., Dept. of Aero. Eng., Tohoku Univ. Wataru YAMAZAKI, Univ. of Wyoming -32- Gradient/Hessian-enhanced Kriging Implementation Details Correlation function of a RBF Estimation of hyper parameters by maximizing likelihood function with GA Correlation matrix inversion by Cholesky decomposition Search of new sample point location by maximizing Expected Improvement (EI) value with GA Yamazaki, W., Dept. of Aero. Eng., Tohoku Univ. Wataru YAMAZAKI, Univ. of Wyoming D Analytical Function Case Distribution of estimated Yamazaki, W., Dept. of Aero. Eng., Tohoku Univ. Wataru YAMAZAKI, Univ. of Wyoming -34- Aerodynamic Data Modeling ClCl CdCd NACA0012 M=1.4 AoA=3.5[deg] Noisy in Mach number direction Yamazaki, W., Dept. of Aero. Eng., Tohoku Univ. Wataru YAMAZAKI, Univ. of Wyoming -35- Infill Sampling Criteria for Optimization How to find promising location on surrogate model ? Expected Improvement (EI) value Potential of being smaller than current minimum (optimal) Consider both estimated function and uncertainty (RMSE) EI-based criteria have good balance between global/local searching