Metamodeling-Based Optimization

Transcription

1 Metamodeling-Based Optimization Erdem Acar Postdoctoral Research Associate Mississippi State University

2 Outline What is a metamodel? How do we construct a metamodel? Motivation for using metamodels in optimization Different metamodel types Polynomial response surface Radial basis functions Kriging Ensemble of metamodels for better accuracy and less volatility Application problems Concluding remarks 2

3 What is a Metamodel? The design of advanced vehicular systems such as aircraft and automobiles requires performing high fidelity analyses to ensure a high level of accuracy. Numerically expensive FE or CFD simulations Even though computer processing power, memory and storage space have increased drastically throughout the last 30 years, Venkataraman and Haftka* (2004) reported that analysis models of acceptable accuracy have required at least six to eight hours of computer time (an overnight run). Because the fidelity and complexity of analysis models required by the designers have increased. There is a growing interest in replacing these slow, expensive and mostly noisy simulations with smooth approximate models that produce fast results. These approximate models are referred to as metamodels or surrogate models. * S. Venkataraman and R.T. Haftka, 2004, Structural optimization complexity: what has Moore s law done for us?, Structural and Multidisciplinary Optimization, Vol. 28, pp

4 Metamodel construction Metamodel mathematical relationship between design variables and response y f(x) constructed with few simulations Advantages computationally inexpensive easy coupling with optimization software can filter out numerical noise and smooth over slope discontinuities provide differentiable functions 400 Design of experiments Run simulations at selected points Construct metamodel (i.e., find parameters of mathematical function) 200 Assess quality of fit Metamodel construction 0 0 4

5 Design of experiments Design of experiments (DOE) is a sampling plan in design space (i.e., locations where we conduct simulations) Number of points: thumb rule, times the number of coefficients of fitting polynomial, adaptive sampling Considerations reduce the influence of noise, bias errors (insufficient model), extrapolation Types of DOEs: face-centered central composite design (FCCD), Latin hypercube sampling (LHS), orthogonal array (OA), D-optimal design, combinations (LHS+FCCD), CCD LHS 5

6 Curse of dimensionality Order of polynomial Nv Number of coefficients grows rapidly as order of polynomial increases 6

7 Motivation for the use in optimization Issues with simulations discretization and convergence problems lead to numerical noise high computational cost because of high fidelity required by an analyst Design optimization requires evaluating many designs very high cost! Identify design variables x, objectives f, and constraints g Select models to evaluate (f, g) for designs (x) Optimization Optimization process Use metamodeling based methods for optimization 7

8 Metamodeling based optimization concept Define optimization problem Min f(x1, x2 xn) s.t. g(x) 0 h(x) = 0 Select DOE and evaluate f(x), g(x), h(x) x1 x2 xn Metamodel Numerical simulations f g h Refine design space No Perform optimization Validation? Yes Stop 8

9 Polynomial response surface approximation y ( x) 0 Nv i 1 i xi yβ Xε,y Nv i 1 Nv ii xi2 i j 2 ij xi x j K ˆ b X b= XX Xy -1 Assumption: Normally distributed uncorrelated noise In presence of noise, response surface may be more accurate than observed data Questions to ask: Does the data have noise, and/or can the chosen order polynomial approximate the behavior? y Sampling data points Response surface x 9

10 Kriging (KR) y (x) yˆ (x) Nv i 1 C Z (x), Z (sθ ), Systematic departure i i ( x) Z ( x) Trend model Nv i 1 exp i ( xi si )2 Named after a South African mining engineer D. G. Krige Assumption: Systematic departures Z(x) are correlated, noise is small Gaussian correlation function C(x, s,θ) is most popular Computationally expensive for large problems (N>200) y Sampling data points Systematic Departure Linear Trend Model Kriging x 10

11 Radial basis functions (RBF) fˆ (x) N i 1 i x xi λ are found from φ r : radially symmetric functions [ A ] { λ } ={ f } Aij =φ x j x i =φ r where 3 2 φ r =r log cr φ r =exp cr Gaussian φ r =1/ r c 1.5 φ(r) φ r = r 2 c 2 2 Thin-plate spline Thin plate Gaussian Multiquadric Inverse multiquadric 2.5 Multiquadric 2 Inverse multiquadric where 0 c r Found to be good for modeling fast-changing responses Successful application in crashworthiness responses 11

12 Some issues with metamodel selection Many types of metamodels difficult to know which is the best for a given problem Performance of a metamodel depends on number of samples choice of sampling scheme (DOE) GP nature of the problem RBF Main issue best metamodel is problem and DOE dependent uncertainty in predictions since metamodels are inexpensive to construct, can we do better by using several types of metamodels at once? Advantages PRS: Polynomial response surface KRG: Kriging RBF: Radial basis functions protect from choosing the wrong metamodel GP: estimate uncertainty in predictions Gaussian process 12

13 Weighted average model Weighted sum of responses from NM metamodels NM y WA x = w i y i x i =1 Choice of weights wi should reflect the confidence in metamodels accurate metamodel high weight 13

14 Goel et al. s parametric weighting strategy Parametric weighting strategy based on error Ei w * i Ei Eavg Eavg NSM i 1 1, wi wi* i Ei N SM ; 0 1 wi* Protects against data modeling surrogate Here 0.05 Ei PRESS Use PRESS-based weighting strategy Example of PRESS-based weighted surrogate (PWS) with yˆ pws ( x) w yˆ prs (x) wkrg yˆ krg (x) wrbnn yˆ rbnn (x) PRS, kriging, and prs RBNN 14

15 Acar et al. s weighting strategy Using a parametric weighting strategy is not optimal The choice of parameters are experience based and can change one problem to another We determine the weights by solving min f =Err { y WA x, y act x } NM s.t. wi =1 i=1 NM where y WA x = w i y i x i =1 Will provide example problems later on. 15

16 Accuracy of metamodels How to determine if surrogate models are accurate enough to be used for analysis and optimization? Error on test points should be small compared to the response limitation due to the cost of simulations Cross-validation error: Use the data to estimate errors, e.g., leave-one-out-error or PRESS For polynomials, we have adjusted coefficient of determination R2adj, or root mean square error at data points Estimate of approximation errors is given by pointwise measures like, prediction variance and mean square error 16

17 Major issues in MBO Success of MBO depends on the accuracy of surrogate models f (x) fˆ (x) Sources of uncertainties in surrogate model predictions noise in data accuracy of numerical models and simulations sampling strategies choice of surrogate model (e.g. kriging vs. polynomial response surface) error prediction capability 17

18 Application problems (1) Automotive design problem Design for crashworthiness Design of side rails of an automobile (2) Redesign of Al 356 Cast Control Arm Baseline design is based on experience Redesign using optimization techniques (3) Ensemble of metamodels Application to benchmark functions Application to problem (1) 18

19 (1) Automotive design problem 19

20 Optimization problem and design variables Minimize Mass (Y) Such that Intrusion distances (Y) distance allowables Accelerations (Y) acceleration allowables Y = {DV1-4,thickness) 20

21 Metamodels 21

22 Optimization results Single objective and multiobjective optimization problems considered 6 f c X,Y = [ k=1 wk Ak X, Y Atk X,Y A wk X, Y Atk X,Y 2 ] 22

23 (2) Optimization of Al 356 Cast Control Arm Sizing and shape optimization Control arm of Corvette made of Al 356 Design for panic brake and pothole conditions Total 13 design variables (Y1-13) controlling size and shape GENESIS is used to generate perturbed mesh 23

24 Optimization problem Min W(Y) s.t. g(y) 0 YL Y YU Design for minimum weight The choice of material model determines the constraints If a multiscale material model used If a plasticity model used g Y =D Y Dcr 0 g Y =s vm Y s cr 0 Calculation of D(Y) or σvm(y) is through expensive FEA The use of metamodels for W(Y) and g(y) alleviates the computational burden 24

25 Metamodels PRS RBF GP KR FFNN SVR 25

26 Optimization results Min W(Y) s.t. g(y) 0 YL Y YU The optimization problem is solved with MATLAB fmincon function that uses Sequential quadratic programming (SQP) With multiscale material model Without multiscale material model Normalized weight = Max von Mises = MPa Damage (pred.) = 0.01 Damage (actual) = Normalized weight = Max von Mises (pred.) = MPa Max von Mises (actual) = MPa Damage =

27 (3) Ensemble of metamodels In complex engineering problems, the number of responses of interest is more than one. Example: crashworthiness problems cost (or structural weight) intrusion distances and accelerations at different locations loads transmitted to the passengers Different types of metamodels best for different responses PRS is best for mass RBF is best for floor pan displacement at FFI SVR is best for floor pan displacement at OFI As new information comes, efficiency of metamodels can change Hence, the use of a single metamodel is risky! 27

28 Evaluation of accuracy of metamodels Individual Metamodel errors PRS RBF KR Branin-Hoo Hartman 1.63 CRASH GP Ensemble errors SVR E1 E2 E CRASH: 1.06 ACC_DS_FFI Normalized errors E1: ensemble based on simple averaging 1.25 proposed by E2: parametric ensemble Goel et al. (2006) 1.00 E3: our proposed ensemble PRS RBF KR GP SVR E1 E2 E3 28

29 Ensemble of metamodels: more results CRASH: ACC_DS_FFI Normalized errors Normalized error Branin-Hoo function PRS RBF KR GP SVR E1 E2 E3 PRS RBF KR GP SVR E1 E2 E3 29

30 Concluding remarks Motivation for the use of metamodels in a design optimization framework was given Different metamodeling types and construction procedures were discussed The concept of ensemble of metamodels was presented to reduce the volatility of metamodel predictions while increasing accuracy Several application problems were provided to illustrate the MBO approach The accuracy and efficiency of MBO was demonstrated 30

31 Thank you! 31

32 Gaussian Process (GP) Main assumption: The joint probability distribution of the response follows Gaussian distribution P f N C N, X N = f x =k T C N 1 1 2π N C N fn 1 [ 2 l=1 ] f N = f n x 1, x 2,, x { n n nl } N n=1 CN: covariance matrix with elements Cij x x 1 C =θ exp ij 1 T f N μ C 1 f N μ N 2 Prediction at the N+1 the point k = [ C 1, N 1,,C N, N 1 ] L [ exp i l j r l2 Interpolation mode l 2 ] θ2 x x 1 C =θ exp ij 1 [ L 2 l=1 i l j r l l2 Regression mode 2 ] θ 2 δ ij θ 3 eliminates noise Hyperparameters (θi, ri) are obtained via optimization - e.g., maximizing the marginal likelihood 1 1 N L= log C N f T C 1 f N log 2π 2 2 N N 2 Good for modeling nonlinear responses Good for eliminating noise in regression mode 32

33 Support Vector Regression (SVR) f x = w x b When linear regression is used, prediction is performed via < > : dot product l We want to make the prediction as flat as possible 1 2 w 2 s. t. y i w x i b ε w x i b y i ε Min 1 2 w C ξ i ξ i 2 i=1 s. t. y i w x i b ε ξ i w x i b y i ε ξ Min soft formulation i ξ i,ξ 0 i - Write Laplacian - Write KKT conditions - Substitute from KKT to Lagrangian to get dual form l l l 1 Max α i α α j α x i x j ε α i α y i α i α i j i i 2 i, j=1 i=1 i=1 l s. t. αi αi =0, αi αi [ 0,C ] i=1 l f x = α α x x b i i i=1 i In nonlinear regression, replace the dot product with Kernel functions (e.g., Gaussian kernel) Based on generalized portrait algorithm (Vapnik and Lerner, 1963, Vapnik and Chervonekis (1963, 1974). Russia. 33

34 Metamodeling of Nonlinear and Noisy Functions Metamodeling of nonlinear and noisy responses When the critical response is noisy, optimization using the exact simulation results can not be pursued Metamodeling techniques helps to wipe out the noise (e.g., polynomial response surface approximations) However, when the response is also nonlinear, then more advanced metamodeling techniques needs to be utilized. Challenges and Good news stress damage DV1 DV3 DV2 RBF GP KR FFNN SVR % Error Metamodel RS RBF GP KR FFNN SVR % Error Summary Damage DV2 RS The best metamodel has 29% error. However, at the optimum, damage prediction is more accurate and also conservative damage(pred.) = 0.01 damage(actual) = Nonlinear and noisy behavior of damage Damage is function of many nonlinear and interacting terms Calculation of damage via FEA makes it amenable to numerical noise due to effect of mesh distortion In the control arm problem, some design variables are found to have a nonlinear and noisy influence on the damage. Metamodel Six different metamodels are investigated to model nonlinear and noisy functions Polynomial response surface Radial basis functions Kriging Gaussian process Feed Forward neural networks Support vector regression Gaussian process is found to be the best GP has a regression mode along with interpolation mode that filters the noise DV5 34

35 Journal papers submitted/on the line We used metamodeling based optimization concept in the following papers. 1. Rais-Rohani, M., Solanki, K., Acar, E., Eamon, C., Reliability-Based Design Optimization of Automotive Structures under Crash Loads 2. Solanki, K., Acar, E., Rais-Rohani, M., Eamon, C., and Horstemeyer, M.F., Reliability-based Structural Optimization using a Multiscale Material Model 3. Acar, E., Solanki, K., Rais-Rohani, M., Metamodeling of Nonlinear and Noisy Functions 4. Acar, E., and Rais-Rohani, M., Enhanced Surrogate Modeling via Optimum Ensemble of Metamodels 5. Acar, E., and Solanki, K., Improving accuracy of vehicle crashworthiness response predictions using ensemble of metamodels" Acknowledgements Dr. Rais-Rohani (Aerospace Engineering Department, Mississippi State University) Kiran Solanki (Center for Advanced Vehicular Systems, Mississippi State University) Dr. Eamon, Ali, Mohammad, Bulakorn (CE, AE, Mississippi State University) 35

36 (3) Probabilistic Product-Process Design Optimization Objective Solution procedure The efficiency of products depends on the design at macrostructural design variables as well as microstructural features Combine shape and sizing optimization with multiscale analysis reduced weight increased energy absorption, reduced acceleration manufacturability (reduced rejection rate) Example problem: design of stamped side rail Define DVs, DVm, RV Probabilistic Optimization Design of Experiments (DVs, DVm, RV) ABAQUS stamping Extract manuf. res. (e.g., springback, localized thinning) Metamodel Fm (DVs,DVm,RV) Update element damage indices ABAQUS crash Metamodel Fs (DVs,DVm,RV) W (DVs,DVm) Accomplishments and Future Work Multiobjective optimization Accomplishments Product along with the process to be optimized structural responses (e.g., weight, crashworthiness) manufacturability (e.g., springback, localized thinning) Pareto Frontier ABAQUS finite element models are generated I/O Collaboration of different softwares established e.g., NESSUS-ABAQUS interaction Future work 1 f2(x) Manufacturing objective Extract structural resp. (e.g., weight, energy absorption) Perform stamping and crash analysis for given design of experiments Construct metamodels for structural and manufacturing responses Solve probabilistic optimization problem Structural objective f1(x)