1 / 35

Derivative-Enhanced Variable Fidelity Kriging Approach

23 rd , September, 2010. Derivative-Enhanced Variable Fidelity Kriging Approach. Wataru YAMAZAKI. Dept. of Mechanical Engineering, University of Wyoming, USA. Motivation. *Surrogate models for - Efficient Design Optimization - Efficient Aerodynamic Data Modeling

foy
Download Presentation

Derivative-Enhanced Variable Fidelity Kriging Approach

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. 23rd, September, 2010 Derivative-Enhanced Variable FidelityKriging Approach Wataru YAMAZAKI Dept. of Mechanical Engineering, University of Wyoming, USA

  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

  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

  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

  5. Variable Fidelity Kriging Model In matrix form expression Correlation parameters in R and r, factors are estimated by a likelihood maximization approach

  6. Variable Fidelity Kriging Model Correlations between all sample points combinations by a RBF

  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 4th order derivatives of correlation function Automatic Differentiation by TAPENADE No sensitive parameter Better matrix conditioning than indirect approach

  8. Derivative-enhanced Variable Fidelity Kriging High Fidelity Function Gradient Hessian Hessian Vector 1st Low Fidelity Function Gradient Hessian Hessian Vector 2nd 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

  9. Results & Discussion

  10. 1D Analytical Function Case • 2 high-fidelity samples • 5 low-fidelity samples (+0.5) • 5 another low-fidelity samples (-0.5)

  11. 1D Analytical Function Case • 2 high-fidelity samples • 5 low-fidelity samples (+0.5) • 5 another low-fidelity samples (-0.5)

  12. 1D Analytical Function Case • 2 high-fidelity samples • 5 low-fidelity samples (+0.5) • 5 another low-fidelity samples (-0.5)

  13. 1D Analytical Function Case • 2 high-fidelity samples • 5 low-fidelity samples (+0.5) • 5 another low-fidelity samples (-0.5)

  14. 2D Analytical Function Case • 2D Cosine function • Analytical gradient/Hessian • Latin hypercube sampling for high and low-fidelity samples • Comparison by RMSE

  15. 2D Analytical Function Case Exact function Func/Grad Func Func/Grad/Hess • Only 5 high-fidelity samples • Derivative information is useful to construct accurate model

  16. 2D Analytical Function Case Exact function • Only function information for both high/low-fidelity samples • 5 high-fidelity samples with 0-200 low-fidelity samples • Low-fidelity information is useful to construct accurate model

  17. 2D Analytical Function Case

  18. 2D Analytical Function Case • 50 low-fidelity sample points • Best performance in FGH for both high/low-fid samples

  19. 2D 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

  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

  21. Mach-AoA Hypersurfaces

  22. Mach-AoA Hypersurfaces • Mean error comparison in drag coefficient • Improvements at smaller numbers of high fidelity samples

  23. Mach-AoA Hypersurfaces Full-MC results for σ=0.1 • Uncertainty analysis at M=0.8, AoA=2.5 for both Mach/AoA • 1000 CFD evaluations for a specified σ value • In total 7000 CFD evaluations (= 1000 x 7) for full-MC

  24. Mach-AoA Hypersurfaces Mean of Cm Variance of Cm • More accurate uncertainty analysis by Inexpensive MC with variable fidelity Kriging model

  25. 2D 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)

  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)

  27. 2D 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

  28. 2D Airfoil Shape Optimization • To include low-fidelity / derivative information is promising

  29. 2D Airfoil Shape Optimization Pressure Distributions NACA0012, Obj = 0.121 Optimal by HFonly, Obj = 6.66e-4 Optimal by VFM_Adj, Obj = 1.66e-4 Shock reduction on upper surface Towards supercritical airfoils in HFonly Additional adjustment of problem definition ?

  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 !!

  31. Appendix

  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

  33. 2D Analytical Function Case Distribution of estimated

  34. Aerodynamic Data Modeling • NACA0012 • M=1.4 • AoA=3.5[deg] • Noisy in Mach number direction Cl Cd

  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

More Related