Multidisciplinary System Design Optimization (MSDO)

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1 oday s opics Multidisciplinary System Design Optimization (MSDO) Approimation Methods Lecture 9 6 April 004 Design variable lining Reduced-Basis Methods Response Surface Approimations Kriging Variable-Fidelity Models Karen Willco Massachusetts Institute of echnology - Prof. de Wec and Prof. Willco Massachusetts Institute of echnology - Prof. de Wec and Prof. Willco Why Approimation Methods? We have seen throughout the course the constant trade-off between computational cost and fidelity. high fidelity (e.g. CFD,FEM) intermediate fidelity (e.g. vorte lattice, beam theory) empirical models Fidelity Level can we do better? trade studies increasing difficulty Level of MSDO limited optimization/iteration how to implement? can the results be believed? full MDO from Giesing, 998 Approimation methods provide a way to get high-fidelity model information throughout the optimization without the computational epense. 3 Massachusetts Institute of echnology - Prof. de Wec and Prof. Willco Approimation Methods Recall that the analysis or simcode must be invoed each time the optimizer selects a new design vector to try ypically, hundreds (thousands) of design vectors will be analyzed throughout an optimization run Can use Approimation Models (Surrogate Models) for objective functions and constraints If approimate models are inepensive to evaluate, can analyze many more design vector options without worrying about computational resources Concept first introduced in structural optimization by Barthelemy and Hafta, Massachusetts Institute of echnology - Prof. de Wec and Prof. Willco

2 Approimation Methods Overview Design variable lining Reduced-basis methods Response surface methods Kriging Variable-fidelity methods reduce the number of design variables in the optimization code simcode analysis full order same number of design variables simcode analysis simplified combine high-fidelity and approimation models 5 Massachusetts Institute of echnology - Prof. de Wec and Prof. Willco Design Variable Lining Not all design variables may be independent For eample, symmetry may eist in the problem Define: y = + y C y = + y C r r n n r optimizer uses y, but provides to simcode for analysis 6 Massachusetts Institute of echnology - Prof. de Wec and Prof. Willco Reduced-Basis Methods Consider r feasible design vectors:,,..., r We could consider the desired design to be a linear combination of these basis vectors: Reduced-Basis Methods We can now optimize J() by finding the optimal values for the coefficients α i. dimension n dimension r r * α ι i = i C = + scalar coefficient basis vector added for generality do one full-order evaluation of resulting answer approach is efficient if r << n will give the true optimum only if * lies in the span of { i } basis vectors could be previous designs solutions over a particular range (DoE) derived in some other way (POD) 7 Massachusetts Institute of echnology - Prof. de Wec and Prof. Willco 8 Massachusetts Institute of echnology - Prof. de Wec and Prof. Willco

3 Reduced-Basis Eample Eample using a reduced-basis approach (van der Plaats Fig 7-): airfoil design for a unique application. many airfoil shapes with nown performance are available design variables are (,y) coordinates at chordwise locations (n~00) use four basis airfoil shapes (low-speed airfoils) which contain the n geometry points plus two basis shapes which allow trailing edge thicness to vary r=6 (r<<n) optimize for high speed, maimum lift with a constraint on drag 9 Massachusetts Institute of echnology - Prof. de Wec and Prof. Willco Reduced-Basis Eample Vanderplaats, G. N. Numerical Optimization echniques for Engineering Design. Vanderplaats R&D, 999. Figure 7-. Vanderplaats, G. N. Numerical Optimization echniques for Engineering Design. Vanderplaats R&D, 999. Figure Massachusetts Institute of echnology - Prof. de Wec and Prof. Willco Proper Orthogonal Decomposition Also nown as principal components analysis, Kahunen Loève decomposition, singular value decomposition r = * ι i = i α ϕ he r basis vectors ϕ i are orthogonal hey are computed from M empirical solutions {,,... M } hey are optimal in the sense that they minimize the error between the original and the projected data ma ϕ Massachusetts Institute of echnology - Prof. de Wec and Prof. Willco (, ϕ ) ϕ Proper Orthogonal Decomposition hese optimal basis functions can be calculated by:. Evaluating the correlation matri: i j R ij =. Solving the M M eigenvalue problem: R v = λ v i i i 3. Constructing the basis vectors: j M ϕ = v i = i j i Use components of the jth eigenvector to calculate the jth POD basis vector. he jth eigenvalue tells us how important is the jth basis vector. Massachusetts Institute of echnology - Prof. de Wec and Prof. Willco

4 Approimation Methods Overview Design variable lining Reduced-basis methods Response surface methods Kriging Variable-fidelity methods reduce the number of design variables in the optimization code simcode analysis full order same number of design variables simcode analysis simplified combine high-fidelity and approimation models 3 Massachusetts Institute of echnology - Prof. de Wec and Prof. Willco Response Surface Methodology Keep the same number of design variables, but simplify the simcode analysis Create approimating functions to objective and constraints Optimize using the approimations Update approimations using current optimal solution guess and repeat Response surfaces are smooth even if design space is noisy Polynomial-based modeling technique Provide compact, eplicit functional relationship between response and design variables Least squares is computationally inepensive and easy to implement 4 Massachusetts Institute of echnology - Prof. de Wec and Prof. Willco Local Approimations Most common are aylor series epansions: J ( ) = J ( ) + J ( ) δ + δ H ( ) δ + " 0 δ = Could use the first two terms linear approimation Solve, reanalyze and repeat = sequential linear programming Could also include quadratic term: update requires: J ( ) = J ( ) + J ( ) δ + δ H ( ) δ + " function evaluation n function evaluations n(n+)/ function evaluations 5 Massachusetts Institute of echnology - Prof. de Wec and Prof. Willco Local Approimations It is epensive to update the gradient vector and Hessian matri One approach: perform several approimation cycles updating only the constant term then update the linear term and repeat finally, update the Hessian only when no other progress can be made If the design space is highly nonlinear, there is no guarantee that this approach will wor 6 Massachusetts Institute of echnology - Prof. de Wec and Prof. Willco

5 Response Surface Methodology Another approach: use what information is available to create the approimation Use this approimation to mae a small move in the design variables Analyze the result precisely new function evaluation Use the new function evaluation to improve the approimation to the design space Fit a response surface Can use a quadratic or higher order surface Might choose to use only some of the function evaluations (e.g. those in a local neighborhood) 7 Massachusetts Institute of echnology - Prof. de Wec and Prof. Willco RSM 0 eg.. define J = J ( ) J( ) quadratic approimation: J = J δ + δ δ H J J J = δ + δ δ n n + ( H δ + H δ nn H δ n ) + H δ δ + H 3 δ δ n H δ δ n + H δ δ H δ δ 3 3 n, n n n (all derivatives and entries of H are evaluated at 0 ) ( ) 8 Massachusetts Institute of echnology - Prof. de Wec and Prof. Willco RSM Assume we have evaluated the baseline plus q designs: 0 q 0,,..., J, J,... J We could write q equations of the form ( ) using 0 (, 0 ), 0 (, ),, q (, q J J J J " J J ) here is a total of N=n+n(n+)/ unnowns: J J J,,...,, H, H,..., H nn n q RSM q equations, N unnowns If q<n, only some coefficients can be calculated If q>n, use weighted least squares Weight designs closer to current q more heavily In general, use q n+ initial designs so an initial linear approimation can be provided If have baseline plus n designs, can fit a linear approimation in each direction (i.e. a hyperplane) If have more solutions, can fit a quadratic or higher order surface 9 Massachusetts Institute of echnology - Prof. de Wec and Prof. Willco 0 Massachusetts Institute of echnology - Prof. de Wec and Prof. Willco

6 RSM In other words, fit objective function with a polynomial e.g. quadratic approimation: J b c c ( ) = a 0 + i i + ii i + ij i j i i i, j < i Update model by including a new function evaluation then doing least squares fit to compute the new coefficients Massachusetts Institute of echnology - Prof. de Wec and Prof. Willco Estimation problem: RSM J Xc M = J J J J=vector containing " " M responses # # X = M M M M J " Least squares solution: ( ) c = X X X J c 0 c c = c p c=vector containing p coefficients =M p matri containing M design vector inputs as rows. Columns depend on approimation. Massachusetts Institute of echnology - Prof. de Wec and Prof. Willco Kriging RSM tends to create a global model of the design space (especially if all points weighted equally in the LS) May be of limited accuracy when multiple etrema eist (especially quadratic polynomial models) Kriging : combine global model of design space with a model for local deviations which interpolates sample points Developed in fields of spatial statistics and geostatistics Original statistical techniques developed by mining engineer D.G. Krige Interpolation-based modeling technique Computationally more epensive and not as simple to implement as RSM 3 Massachusetts Institute of echnology - Prof. de Wec and Prof. Willco Kriging Unnown function to be modeled epressed as: J ( ) = f ( ) + Z( ) nown function global model for design space e.g. use RSM to fit f() using M observations Gaussian random function zero mean variance σ localized deviation from global model cov Z ( i ), Z ( j ) = σ R R( i, j ) covariance matri of Z() correlation matri correlation function 4 Massachusetts Institute of echnology - Prof. de Wec and Prof. Willco

7 Approimation Methods Overview Design variable lining Reduced-basis methods Response surface methods Kriging Variable-fidelity methods reduce the number of design variables in the optimization code simcode analysis full order same number of design variables simcode analysis simplified combine high-fidelity and approimation models 5 Massachusetts Institute of echnology - Prof. de Wec and Prof. Willco Initialization High-fidelity model Approimation model Variable Fidelity Models Use information from high-fidelity model to chec approimate designs Optimize using approimate model Recalibrate using high-fidelity model recourse to detailed model J(), g() Optimizer From: Fig., Aleandrov et al. 6 Massachusetts Institute of echnology - Prof. de Wec and Prof. Willco optimization on a simplified model Variable Fidelity Models Questions: What do we do when the design derived from the lowfidelity optimization does not provide an improvement in the true objective? How can we use information about the predictive capability of the approimation model to decide when to go bac to the high-fidelity model? How do we decide when to update the approimate model? Variable Fidelity Models When optimization with approimation model is unsuccessful, there are two possible approaches:. Improve the fidelity of the approimate model. Do less optimization One option: use a trust region approach trust region current iterate 7 Massachusetts Institute of echnology - Prof. de Wec and Prof. Willco design space 8 Massachusetts Institute of echnology - Prof. de Wec and Prof. Willco

8 rust Region Approach Classic approach: Regulate the length of the steps taen by the iterative optimization algorithm Regulate based on quality of current approimation model e.g. Quadratic aylor series model: J ( + s ) q ( + s ) = J ( ) + J( ) s + s B s where s is the prospective step in the design variables,, and B is a model of the Hessian matri at. 9 Massachusetts Institute of echnology - Prof. de Wec and Prof. Willco rust Region Approach Restrict the step size to a region in which we trust the quadratic model to approimate J well Done by adding a constraint on the step trust region subproblem min q ( + s ) s.t. s δ In practice, variables are scaled to improve performance (step size can vary in different directions) hen decide whether to accept the step: + + s if J ( + s ) < J( ) = otherwise 30 Massachusetts Institute of echnology - Prof. de Wec and Prof. Willco rust Region Update rust radius, δ, is updated adaptively Update depends on predictive quality of quadratic model: if model did a good job of predicting J, or if there was more improvement than predicted, then increase δ if model did a bad job of predicting J (J increased or decrease was much lower than predicted), then decrease δ if model did an acceptable job of predicting J, then do not change δ rust Region Update Numerically, compare actual and predicted decrease: J ( ) J ( + s ) r = J ( ) q ( + s ) Define constants r, r (r <r ) and apply the rules: if r< r, decrease the trust radius if r>r, increase the trust radius ypical values are r =0., r =0.75 Note that prediction of ascent/descent is more important than prediction of actual value 3 Massachusetts Institute of echnology - Prof. de Wec and Prof. Willco 3 Massachusetts Institute of echnology - Prof. de Wec and Prof. Willco

9 Classic rust Region Algorithm 0 0 Choose \ n and δ > 0 For =0,,... until convergence do { Find an approimate solution s to the subproblem: min q ( + s ) s.t. s δ Compare the actual and predicted decrease: J ( ) J ( + s ) r = J ( ) q ( + s ) Update and δ } From Fig., Aleandrov et al. 33 Massachusetts Institute of echnology - Prof. de Wec and Prof. Willco β-correlation Method Basic idea is to tae low-fidelity approimate model, J a, and correct it by scaling Define the scale factor: J ( ) β = J ( ) a Given the current design, build a first order model of β about : β ( ) = β ( ) + β( )( ) Optimize with approimate model and use local model of β to scale result : J ( ) β J a ( ) 34 Massachusetts Institute of echnology - Prof. de Wec and Prof. Willco Variable Compleity Model VCM isight uses a Variable Compleity Model (VCM) Use two simcodes for the same physical process: () more accurate, longer running (eact) J() () less accurate, shorter running (appro.) J a () Compute a multiplicative or additive correction factor, σ : 0 J ( ) σ = 0 J a ( ) J ( ) = σ J ( ) a or 0 0 σ = J ( ) J a ( ) J ( ) = σ + J ( ) 35 Massachusetts Institute of echnology - Prof. de Wec and Prof. Willco a Optimization Optimization Simcode J() Conventional optimization with simcode Appro. J a () Conventional optimization with approimate model Optimization Appro. J a () Simcode J() Optimize with approimate model, update with simcode:. Run both models, calculate σ 0. Optimize using approimate model and σ 0 3. Update correction factor using simcode and repeat 36 Massachusetts Institute of echnology - Prof. de Wec and Prof. Willco

10 Lecture Summary Approimation methods are one way to capture high-fidelity responses without the computational cost Design variable lining if physical problem allows Reduced-basis methods use low-order representation of the design vector use previous designs, DoE or POD Response surface methodology use polynomial models weighted least squares Kriging interpolation models global + local behavior Variable-fidelity models rust region approach β-correlation 37 Massachusetts Institute of echnology - Prof. de Wec and Prof. Willco References Barthelemy, J-F. M. and Hafta, R.., Approimation concepts for optimum structural design a review, Structural Optimization, 5:9-44, 993. Giunta, A.A. and Watson, L.., A comparison of approimation modeling techniques: polynomial versus interpolating models, AIAA Paper , 998. LeGresley, P.A. and Alonso, J.J., Airfoil design optimization using reduced order models based on proper orthogonal decomposition, AIAA Paper Aleandrov, N., Dennis, J.E., Lewis, R.M. and orczon, V., A trust region framewor for managing the use of approimation models in optimization, NASA CR-0745, ICASE Report No , October 997. Practical Optimization, P.E. Gill, W. Murray and M.H. Wright, Academic Press, 986. Numerical Optimization echniques for Engineering Design, G.N. Vanderplaats, Vanderplaats R&D, Massachusetts Institute of echnology - Prof. de Wec and Prof. Willco