Simultaneous Robust Design and Tolerancing of Compressor Blades

Transcription

1 Simultaneous Robust Design and Tolerancing of Compressor Blades Eric Dow Aerospace Computational Design Laboratory Department of Aeronautics and Astronautics Massachusetts Institute of Technology GTL Seminar October 1, 2013

2 Motivation Geometric variability is unavoidable and undesirable Illustration of manufacturing variability (from [Garzon, 2003]) Impact of geometric variability can be reduced Robust design: change the nominal design Tolerancing: change the level of variability Current design and tolerancing methods are sequential A Design B C Minimum Cost Tolerance 2

3 Performance Impacts of Geometric Variability Geometric variability introduces both variability and mean shift into compressor performance Illustration of performance mean shift and variability (from [Lavainne, 2003]) Mean adiabatic eciency of a ank-milled integrally bladed rotor (IBR) reduced by approximately 1% [Garzon and Darmofal, 2003] 3

4 Research Objectives 1 Develop a framework for simultaneous robust design and tolerance optimization that incorporates manufacturing and operating costs 2 Develop an approach for probabilistic sensitivity analysis of performance with respect to the level of variability 3 Demonstrate framework eectiveness for design and tolerancing of turbomachinery compressor blades 4

5 The Pitfalls of Single-point Optimization Single-point optimized designs often perform poorly away from design point Polars for baseline and optimized DAE-11 airfoil (from [Drela, 1998]) Optimizer exploits ow features to improve performance at design point Small changes in ow features away from design point may degrade performance Adding additional design parameters does not improve o-design performance (it actually gets worse!) 5

6 Robust Design Optimization Robust optimization: determine design whose performance is relatively unchanged when the system is perturbed as a result of variability Cost Single-point Robust Design Variable Probabilistic robust multi-objective: minimize some combination of the mean and variance of the cost Solution approaches can be gradient-based or derivative-free Computationally expensive: typically involves evaluating performance at a large number of conditions 6

7 Modelling Variability: Random Fields Random eld: a collection of random variables that are indexed by a spatial variable Well-suited for modelling spatially distributed variability Parameters can be chosen to model observed behavior (correlation length, non-stationarity, smoothness) Gaussian random elds: uniquely characterized by mean and covariance e(s) 0 e(s) s s (a) Smooth random eld (b) Non-smooth random eld 7

8 The Karhunen-Loève Expansion Simulate e(s, ω) using the Karhunen-Loève (K-L) expansion: e(s, ω) = ē(s) + λi φ i (s)ξ i (ω) i=1 C(s 1, s 2)φ i (s 2)ds 2 = λ i φ i (s 1) D ξ i (ω) N (0, 1), i.i.d λ φ(s) s (a) K-L eigenvalues (b) K-L eigenfunctions Truncate K-L expansion according to the decay of the eigenvalues: N KL e(s, ω) = ē(s) + λi φ i (s)ξ i (ω) i=1 8

9 Non-stationary Random Fields Covariance kernel consists of a xed correlation function spatially variance standard deviation (, C s1 s2 ρ(s1, s2 ) and σ(s ): ) = σ(s1 )σ(s2 )ρ(s1, s2 ) Stationary e(s) σ(s) s s s Non-stationary e(s) σ(s) s

10 Random Field Model of Manufacturing Variability Manufacturing variability is modeled using a non-stationary Gaussian random eld, characterized by its mean ē(s) and covariance C(s 1, s 2) Map random eld to blade surface through arclength Manufactured blade surface constructed by perturbing design intent geometry in the normal direction: x(s, ω) = x d (s) + e(s, ω)ˆn(s) Baseline Manufactured e(s) s (a) Error eld realization (b) Error eld mapped to blade 10

11 Relating Tolerances to Variance Simulate the eect of spatially varying manufacturing tolerances using a spatially varying variance σ 2 (s) Small σ 2 strict tolerances (a) Loose tolerances (b) Strict tolerances (): design intent geometry (- - -): design intent geometry +/- 2σ(s) 11

12 Performance and Manufacturing Costs Choose blade design and manufacturing tolerances that minimize overall cost C perf (d, σ): economic value of aerodynamic performance per blade Proportional to the moments of the performance of the system: C perf (d, σ) = k me[η(ω; d, σ)] + k v Var[η(ω; d, σ)] C man(σ): manufacturing cost per blade Monotonically decreasing function of allowed geometric variability 1 C man(σ) = k man Ω s σ(s) ds 12

13 Optimization Statement Optimal design (d ) and manufacturing tolerances (σ ) determined by minimizing sum of manufacturing and operating costs Mean pressure ratio Π is constrained above minimum allowable value Π (d, σ ) = argmin C perf (d, σ) + C man(σ) s.t. E[Π(d, σ, ω)] Π Gradient-based optimization: use sensitivity information to choose search directions Sequential Quadratic Programming (SQP) 13

14 Model Problem of Interest In the absence of variability, nominal design variables d determines the PDE solution u Nominal Design Variables d PDE Solution u(y; d) 14

15 Model Problem of Interest In the absence of variability, nominal design variables d determines the PDE solution u Random eld e(s, ω) introduces random noise to the solution Correlation Structure ρ Nominal Design Variables d Random Field e(s, ω; σ) PDE Solution u(y, u(y; ω; d, d) σ) Standard Deviation σ 14

16 Model Problem of Interest In the absence of variability, nominal design variables d determines the PDE solution u Random eld e(s, ω) introduces random noise to the solution System performance characterized by the moments of functionals F (u) of the solution, e.g. E[F ] and Var[F (u)] Correlation Structure ρ Nominal Design Variables d Random Field e(s, ω; σ) PDE Solution u(y, u(y; ω; d, d) σ) Performance Moments E[F (u(y, ω; d, σ))] Standard Deviation σ 14

17 Monte Carlo Method Monte Carlo method: approximate moments using sample average Sample i.i.d. Gaussian and construct error eld through K-L N KL e(s, ω) = ē(s) + λi φ i (s)ξ i (ω) i=1 Perturb geometry according to error eld x(s, ω) = x d (s) + e(s, ω)ˆn(s) Compute ow solution for perturbed geometry Compute performance quantities of interest for each sample geometry E[F ] 1 N MC F n N MC n=1 Requires a large number of samples as error is O(N 1/2 MC ) 15

18 From Design and Tolerances to Mean Performance Design Intent Geometry x d (s; d) Random Field 1 e 1 (s, ω 1 ; σ) Geometry Realization 1 x 1 (s, ω 1 ; d, σ) CFD Geometry 1 Performance η 1 (ω 1, d, σ) Correlation Structure ρ ω 1 Karhunen-Loève Expansion φ i (s), λ i ω 2 Random Field 2 e 2 (s, ω 2 ; σ) Geometry Realization 2 x 2 (s, ω 2 ; d, σ) CFD Geometry 2 Performance η 2 (ω 2, d, σ) Performance Moments E[η(d, σ)] Standard Deviation σ ω NMC.. Random Field N MC e NMC (s, ω NMC ; σ) Geometry Realization N MC x NMC (s, ω NMC ; d, σ) CFD Geometry N MC Performance η NMC (ω NMC, d, σ) 16

19 Sensitivity Analysis Overview Gradient based optimization: need to compute E[F ] (Sensitivity of mean performance to nominal design) d i E[F ] (Sensitivity of mean performance to tolerances) σ i Pathwise sensitivity: exchange dierentiation and integration [ ] [ ] [ ] E[F ] F E[F ] F F e = E = E = E d i d i σ i σ i e σ i Shape sensitivities F / d i and F / e can be computed with Adjoint method Direct sensitivity method Finite dierence/complex step method Sample path sensitivity e/ σ i derived from the K-L expansion 17

20 Pathwise Sensitivity Analysis Objective function is some moment of a performance quantity of interest J = E[F (e(σ i, ξ))] = F (e(σ i, ξ)) p Ξ (ξ) dξ Exchange dierentiation and integration: [ ] J F e F e = p Ξ (ξ) dξ = E σ i e σ i e σ i Key idea: x random numbers and perturb sample realizations Pros: well-suited to handle spatially distributed uncertainty Cons: requires continuous F (excludes failure probability sensitivities) 18

21 Monte Carlo and Pathwise Sensitivities Recall Monte Carlo approximation: E[F ] 1 N MC F n N MC n=1 Pathwise sensitivity analysis: exchange dierentiation and summation E[F ] 1 N MC F n E[F ] 1 d i N MC d i σ i N MC n=1 N MC n=1 F n σ i F n/ σ i is computed for xed realization, i.e. xed ξ in the K-L expansion N KL e(s, ω; σ) = ē(s) + λi φ i (s)ξ i (ω) i=1 19

22 Sample Path Sensitivity For each realization of the random eld, compute sensitivity of random eld e with respect to each σ j Karhunen-Loève eigenvalues/eigenvectors are dierentiable functions of σ: N KL e(s, ω; σ) = ē(s) + λi φ i (s)ξ i (ω) e(s, ω; σ) σ j = N KL i=1 φ i σ j i=1 ( 1 2 λ i φ i + ) φ i λ i ξ i (ω) λ i σ j σ j λ i σ j = φ T i C σ j φ i = (C λ i I ) + C σ j φ i These derivatives exist if the eigenvalues have algebraic multiplicity of one 20

23 Sample Path Sensitivity Holding xed ξ in K-L expansion ensures that δσ(s) small e(s) and e(s) + δe δσ(s) are close δσ e e(s) + e σ δσ(s) σ(s) + δσ(s) 0.8 σ (s ) e (s ) e(s) σ(s) s Initial and perturbed standard deviation s Initial and perturbed error eld x d x d + e x d + e + δe Initial and perturbed manufactured blade 21

24 Subsonic Cascade Example MISES: coupled inviscid/viscous ow solver Blade shape parameterized with Chebyshev polynomials Shape sensitivities computed using nite dierences Baseline (no geometric variability): Π 0 = 1.089, θ 0 = Manufacturing variability modeled using a squared exponential kernel with σ(s) = ) s1 s2 2 C(s 1, s 2) = σ(s 1)σ(s 2) exp ( 2L 2 L = 2/20 E[θ] = E[Π] = Mean pressure ratio constrained to be above Π = Performance cost function only includes mean eciency 22

25 Subsonic Cascade Results Optimal design attains lower loss coecient and allows more variability Performance cost function (Cperf ) is reduced by 6% Tolerance cost function (Cman) is reduced by 47% (a) Baseline and optimized blade (b) Optimal σ(s) distribution 23

26 Summary and Future Work New framework for simultaneous robust design and tolerancing Manufacturing and operating costs incorporated into optimization Create a feedback loop between designers and manufacturers Novel probabilistic sensitivity analysis of performance to level of variability Optimal blade performs better and is cheaper to manufacture Future Work More accurate/ecient shape sensitivities: direct sensitivity method Transonic compressor optimization Investigate solution quality 24

27 Questions? 25

28 Pathwise Sensitivity Sucient Conditions for Unbiasedness (Following Glasserman[Glasserman, 2004]) Assume output Y is a function of m random variables: Pathwise estimate is unbiased if [ ] Y (θ + h) Y (θ) E lim h 0 h Y (θ) = f (X 1(θ),..., X m(θ)) = f (X (θ)) = lim h 0 E [ ] Y (θ + h) Y (θ) h (A1) X i (θ) exists w.p. 1, i = 1,..., m (A2) Denote D f R m as the set where f is dierentiable and require P(X (θ) D F ) = 1 Then Y (θ) exists w.p. 1 and is given by Y (θ) = m i=1 θ Θ f X i (X (θ))x i (θ) 26

29 Pathwise Sensitivity Sucient Conditions for Unbiasedness (A3) The function f is Lipschitz continuous, i.e. k f s.t. x, y R m f (x) f (y) k f x y (A4) There exist random variables k i, i = 1,..., m, s.t. θ 1, θ 2 Θ, and E[k i ] < X i (θ 2) X i (θ 1) k i θ 2 θ 1 Conditions (A3) and (A4) imply that Y is almost surely Lipschitz continuous in θ: Y (θ 2) Y (θ 1) k Y θ 2 θ 1 Thus, we have Y (θ + h) Y (θ) h k Y The interchange of expectation and dierentiation are then justied by the DCT 27

30 Adjoint Method Consider an objective function F that depends on the solution u of some PDE, which in turn depends on some parameter p: which we linearize F = F (u; p) δf = F T δu + F T δp u p The PDE solution satises a residual equation R(u; p) = 0 which can be linearized to give [ ] [ ] R R δu + δp = 0 u p 28

31 Adjoint Method Introduce the adjoint state ψ (Lagrange multiplier), and treat the linearized residual equation as a constraint T δf = F u ( F = u δu + F T δp ψ T p ]) T ψ T [ R u If the adjoint state ψ is chosen to satisfy [ ] T R ψ = F =0 [ R δu + p ([ { ] }} ] ){ R δp u ( [ ]) F T δu + ψ T R δp p p u u then the sensitivity gradient can be computed as ( [ ]) F F T p = ψ T R p p Cost is 2N MC ow solutions (vs 2N MC [N d + N σ] ow solutions for FD) 29

32 Eigenvalue Level Repulsion: von Neumann-Wigner Theorem The codimension of the set of positive denite matrices with repeated eigenvalues is greater than one The space of all SPD n n matrices forms a linear space of dimension N = n(n + 1)/2. Two ways to count this: sum of diagonal and elements above diagonal, or... n dimensions corresponding to the eigenvalues (n 1) dimensions corresponding to rst eigenvector subject to φ 1 = 1 (n 2) dimensions corresponding to second eigenvector subject to φ 2 = 1 and φ T 1 φ 2 = 0... Single dimension corresponding to the second to last eigenvector Final eigenvector is uniquely determined by all others n + n (n i) = n + n(n 1)/2 = n(n + 1)/2 = N i=1 30

33 Eigenvalue Level Repulsion: von Neumann-Wigner Theorem Now consider the space of all SPD n n matrices with exactly two eigenvalues that are equal The space of real SPD n n matrices with more than two equal eigenalues is certainly no larger than this space Similar counting approach n 1 dimensions corresponding to the eigenvalues (n 1) dimensions corresponding to rst simple eigenvector subject to φ 1 = 1 (n 2) dimensions corresponding to second simple eigenvector subject to φ 2 = 1 and φ T 1 φ 2 = 0... Two dimensions corresponding to the last simple eigenvector Eigenspace corresponding to equal eigenvalues is uniquely determined n 2 (n 1) + (n i) = N 2 Starting from a random matrix and moving in a random direction will almost surely result in simple eigenvalues i=1 31

34 Antithetic Variates Consider estimating the mean M = E[F (ξ)], ξ = (ξ 1,..., ξ NKL ) with two samples: ˆM = F (ξ 1) + F (ξ 2 ) F1 + F2 = 2 2 with estimator variance If F (ξ) is monotone, choose Var( ˆM) = Var(F 1) + Var(F 2) + 2Cov(F 1, F 2) 4 ξ 1 = ξ 2 then Cov(F 1, F 2) < 0 and variance is reduced Quantities of interest and their sensitivities are nearly linear when level of uncertainty is small 32

35 References I [Drela, 1998] Drela, M. (1998). Frontiers of Computational Fluid Dynamics 1998, chapter 19, Pros and cons of airfoil optimization, pages World Scientic Publishing. [Garzon, 2003] Garzon, V. E. (2003). Probabilistic Aerothermal Design of Compressor Airfoils. PhD dissertation, Massachusetts Institute of Technology, Department of Aeronautics and Astronautics. [Garzon and Darmofal, 2003] Garzon, V. E. and Darmofal, D. (2003). Impact of geometric variability on axial compressor performance. Journal of Turbomachinery, 125(4): [Glasserman, 2004] Glasserman, P. (2004). Monte Carlo Methods in Financial Engineering, chapter 7, Estimating Sensitivities, pages Springer Verlag, New York. [Lavainne, 2003] Lavainne, J. (2003). Sensitivity of a Compressor Repeating-Stage to Geometry Variation. Master's dissertation, Massachusetts Institute of Technology, Department of Aeronautics and Astronautics. 33