High-Fidelity Multidisciplinary Design Using an Integrated Design Environment

Transcription

1 High-Fidelity Multidisciplinary Design Using an Integrated Design Environment Antony Jameson and Juan J. Alonso Department of Aeronautics and Astronautics Stanford University, Stanford CA AFOSR Joint Contractors Meeting for Applied Analysis and Computational Mathematics Programs June

2 Outline Aero-Structural Wing Shape and Planform Optimization Fast Time Integration Methods for Unsteady Problems : application to shape optimization of a pitching airfoil Filtering the Navier Stokes Equations with an Invertible Filter: implication for LES subgrid modeling 2

3 Aero-Structural Wing Shape and Planform Optimization Shape Optimization Baseline profile (red) Redesigned profile (blue) Weakened Shock Boeing 747 -Planform Optimization Baseline planform (green) Redesigned planform (blue) Unstructured Mesh Optimization 3

4 Background: Levels of CFD HIGHEST Automatic Design Integrate the predictive capability into an automatic design method that incorporates computer optimization. Interactive Calculation Attainable when flow calculation can be performed fast enough But does NOT provide any guidance on how to change the shape if performance is unsatisfactory. LOWEST Flow Prediction Predict the flow past an airplane or its important components in different flight regimes such as take-off or cruise and off-design conditions such as flutter. Substantial progress has been made during the last decade. 4

5 Optimization and Design using Sensitivities Calculated by the Finite Difference Method The simplest approach is to define the geometry as f (x) = Âa b i i (x) where a i = weight, b i (x) = set of shape functions Then using the finite difference method, a cost function has sensitivities If the shape changes is I = I(w,a) (such as C D at constant C L ) I ª I(a + da ) - I(a ) i i i a i da i f(x) a n +1 = a n - l I a i (with small positive l) The resulting improvements is I + di = I - IT IT I da = I - l a a a < I More sophisticated search may be used, such as quasi - Newton. 5

6 Disadvantage of the Finite Difference Method The need for a number of flow calculations proportional to the number of design variables Using 4224 mesh points on the wing as design variables 4225 flow calculations ~ 30 minutes each (RANS) Boeing 747 Too Expensive 6

7 Application of Control Theory GOAL : Drastic Reduction of the Computational Costs Drag Minimization Optimal Control of Flow Equations subject to Shape(wing) Variations Define the cost function I = I(w,F) (for example C D at fixed C L ) and a change in F results in a change È di = I Î Í w T È dw + I Î Í F Suppose that the governing equation R which expresses the dependencd of w and F as R(w,F) = 0 and È dr = R È R Î Í w dw + Î Í F df = 0 T df 7

8 4224 design variables Application of Control Theory Since the variation dr is zero, it can be multiplied by a Lagrange Multiplier y and subtracted from the variation di without changing the result. di = IT w Ï dw + IT F = Ì IT w -yt Ó Ê È R È R df -yt Á Î Í w dw + Î Í F Ë df ˆ È R Ï IT Î Í w dw + Ì Ó Choosing y to satisfy the adjoint equation È R Î Í w T y = I w the first term is eliminated, and we find that where Ï G T = Ì I Ó F di = G T df T F -yt È R -y T Î Í F È R Î Í F df (Adjoint Equation) (Gradient) One Flow Solution + One Adjoint Solution 8

9 Advantages of the Adjoint Method: Gradient for N design variables with cost equivalent to two flow solutions Minimal memory requirement in comparison with automatic differentiation Enables shapes to be designed as free surface No need for user defined shape function No restriction on the design space 4224 design variables 9

10 Outline of the Design Process Flow solution Adjoint solution Gradient calculation Sobolev gradient Shape & Grid Modification Repeated until Convergence to Optimum Shape 10

11 Discrete versus Continuous Adjoint Methods The discrete adjoint method evaluates the adjoint and gradient equations algebraically from the discretized flow equations. The continuous adjoint method evaluates the costate solution from the partial differential adjoint equation. The continuous adjoint method leads to no inconsistency as long as it is combined with a compatible search method In the limit of grid convergence the two approaches yield identical gradients. Numerical tests of a model problem verify slightly superior accuracy with the continuous formulation ( Jameson and Vassberg 2000) 11

12 Summary of the Continuous Flow and Adjoint Equations With computational coordinates x i Euler equations for the flow : (1) x i S ij f j (w) = 0 where S ij are metrices, f j (w) the fluxes. Adjoint equation y f (2) C i = 0, C i = S j ij x i w Boundary condition for the Inverse problem (3) I = 1 2 Ú (p - p t ) 2 ds y 2 n x +y 3 n y +y 3 n z = p - p t Gradient y T (4) di = - Ú ds ij f j dd - Ú Ú ( ds D 21 y 2 + ds 22 y 3 + ds 23 y 4 )pdx 1 dx x b 3 w i 12

13 Point-wise Gradient and Shape Parameterizations If the shape is parameterized as then di = f (x) = Ú Â = da k k   k a k b k (x) g(x)df (x)dx Ú = G k da k where G k = k Ú g(x)b k (x)dx g(x)b k (x)dx 13

14 Sobolev Gradient Key issue for successful implementation of the Continuous adjoint method. Define the gradient with respect to the Sobolev inner product di = < g,df > = Ú ( gdf + eg'df ')dx Set df = - lg, di = - l < g,g > This approximates a continuous descent process df dt = -g The Sobolev gradient g is obtained from the simple gradient g by the smoothing equation g - x e g x = g. Continuous descent path 14

15 Computational Costs with N Design Variables Cost of Search Algorithm Steepest Descent Quasi-Newton Sobolev Gradient (Note: K is independent of N) Total Computational Cost of Design Finite Difference Gradients + Steepest Descent Finite Difference Gradients + Quasi-Newton Search or Response surface Adjoint Gradient + Quasi-Newton Search Adjoint Gradient + Sobolev Gradient (Note: K is independent of N) O(N 2 ) O(N ) O(K ) O(N 3 ) O(N 2 ) O(N ) O(K ) - N~ Big Savings - Enables Calculations on a Laptop 15

16 Trailing Edge Crossover Leading term in gradient for drag reduction may lead to trailing edge crossover crossover This corresponds to a sink in a free stream and hence negative drag. This is prevented by Sobolev gradient or by shape parameterizations which don t allow crossover. 16

17 Example: Viscous RAE Drag Minimization Initial Shape and Non-Smoothed Gradient Initial Shape and Smoothed Gradient 17

18 Example: Viscous RAE Drag Minimization Final Shape and Non-Smoothed Gradient Final Shape and Smoothed Gradient 18

19 Good Shape Parameterization of an Airfoil via Conformal Mapping C Define the mapping by On C z log ds dq + i(a -q - p 2 ) = n  C ke -ikq k= 0 n log dz ds = C  k s k =  ( a k cos(kq) + b k sin(kq) ) + iâ b k cos(kq) + a k sin(kq) k= 0 a(s) n k= 0 ( ) Fourier coefficients define the mapping Spectral convergence with n Trialing edge gap is 2piC 1, so prevent crossover by setting C 1 = 0. s 19

20 Sobolev Gradient for Partial Redesign One may wish to freeze some fo the profile, e.g. the structure box Fixed (Structure box) In this case shape changes needed to be smoothly blended into the frozen geometry. This is accomplished by adding more derivatives to the Sobolev gradient ( ) Then the corresponding smoothed gradient satisfies g - Ê x e g ˆ Á Ê 2 g ˆ e Ë x x 2 Á 4 Ë x 2 = g, Hence a patch can be smoothly blended. Ú < u,v >= uv + e 2 u v + e 4 u v dx g = g x = 0 at both end points. 20

21 Potential Modification of F8 Bearcat Racer to win the RENO Air Racer 21

22 Bearcat Shock Free Result Initial C p Final C p 22

23 RANS Calculations Redesign of the Boeing 747 Wing at its Cruise Mach Number Constraints : Fixed C L = 0.42 : Fixed span-load distribution : Fixed thickness 10% wing drag saving (3 hrs cpu time - 16proc.) ~5% aircraft drag saving baseline redesign 23

24 Redesign of the Boeing 747 Wing at Mach 0.9 Sonic Cruiser Constraints : Fixed C L = 0.42 : Fixed span-load distribution RANS Calculations : Fixed thickness Same C We can fly faster at the same drag. 24

25 Redesign of the Boeing 747: Drag Rise ( Three-Point Design ) Constraints : Fixed C L = 0.42 : Fixed span-load distribution : Fixed thickness RANS Calculations Improved Wing L/D Improved M DD benefit benefit Lower drag at the same Mach Number Fly faster with the same drag 25

26 Planform and Aero-Structural Optimization Design tradeoffs suggest an multidisciplinary design and optimization Range = VL D 1 sfc logw o + W f W o Maximize Minimize Planform variations can further maximize VL/D but affects W O 26

27 Aerodynamic Design Tradeoffs The drag coefficient can be split into C D = C DO + C 2 L pear L is maximized if the two terms are equal. D Induced drag is half of the total drag. If we want to have large drag reduction, we should target the induced drag. Design dilemma Increase b D i = 2L 2 perv 2 b 2 D i decreases W O increases Change span by changing planform 27

28 Break Down of Drag Boeing 747 at C L ~.47 (including fuselage lift ~ 15%) Item Wing Pressure Wing friction Fuselage Tail Nacelles Other Total C D 120 counts (15 shock, 105 induced) Cumulative C D 120 counts Induced Drag is the largest component 28

29 Wing Planform Optimization Simplified Planform Model Wing planform modification can yield larger improvements BUT affects structural weight. 1 I = a 1 C D + a 2 2 where C W = Ú (p - p d ) 2 ds Structural Weight q S ref + a 3 C W Can be thought of as constraints 29

30 Additional Features Needed Structural Weight Estimation Large scale gradient : span, sweep, etc Adjoint gradient formulation for dc w /dx Choice of a 1, a 2, and a 3 Use fully-stressed wing box to estimate the structural weight. Large scale gradient Use summation of mapped gradients to be large scale gradient 30

31 Choice of Weighting Constants Breguet range equation R = VL 1 D sfc logw O + W f W O With fixed V, L, sfc, and (W O + W f W TO ), the variation of R can be stated as Ê dr Á R = - Á dc D + C Á D Ë 1 log W TO W O dw O W O ˆ Ê Á = -Á dc D + Á C D Á Ë 1 log C W TO C WO dc WO C WO ˆ Maximizing Range Minimizing I = C D + a 3 a 1 C W using a 3 a 1 = C D C WO log C W TO C W0 31

32 Planform Optimization of Boeing 747 Constraints : Fixed CL=0.42 Redesign Baseline 1) Longer span reduces the induced drag 2) Less sweep and thicker wing sections reduces structure weight 3) Section modification keeps shock drag minimum Overall: Drag and Weight Savings C D C W Baseline Optimize Section at Fixed planform Optimize both section and planform

33 Planform Optimization of MD11 Constraints : Fixed CL=0.45 Baseline Redesign C D C W Baseline Optimize Section at Fixed planform Optimize both section and planform

34 Pareto Front: Expanding the Range of Designs Use multiple a 3 /a 1 ==> Multiple Optimal Shapes Boundary of realizable designs Pareto front of Boeing

35 Super 747 Design a new wing for the Boeing 747 Strategy Use the same fuselage of Boeing 747 Use a new planform (from the Planform Optimization result) Use new airfoil sections (AJ airfoils) Optimized for fixed lift coefficient 35

36 Baseline Boeing

37 Super

38 Comparison C L C D counts C W counts Boeing (112 pressure, 34.3 viscous) (82,550 lbs) Super (83.8 pressure, 37.2 viscous) (70,620 lbs) Save ~10% of airplane drag Save ~ 7% of airplane structural weight 38

39 Automatic design for the Complete Aircraft Geometry on an Unstructured Mesh (SYNPLANE) Key step: reduce the gradient to a surface integral independent of the mesh perturbation (Jameson, A., and Kim, S., "Reduction of the Adjoint Gradient Formula in the Continuous Limit", 41 st AIAA Aerospace Sciences Meeting & Exhibit, AIAA Paper , Reno, NV, January 6-9, ) di = Ú Ú y T ( ds 2 j f j + C 2 dw * )dx 1 dx 3 - Ú Ú ( ds 21 y 2 + ds 22 y 3 + ds 23 y 4 )pdx 1 dx 3 b w Compared to the previous formulation y T di = - Ú ds ij f j dd - Ú Ú ( ds D 21 y 2 + ds 22 y 3 + ds 23 y 4 )pdx 1 dx x b 3 w i b w This field integral is converted to boundary integral 39

40 Question : If the entire profile or wing is translated as a rigid body, the flow is unchanged. Therefore if one calculates the point-wise gradients g xi for movement in the x i direction, they should satisfy the relation Ú g xi ds = 0 Is this true for the computed gradients? 40

41 Actually it can be verified for some implementations of the field integral formula for the gradient. But the boundary integral formula presents a difficulty in the neighborhood of the front and rear stagnation points where w is not well defined. 41

42 Validation of the Reduced Adjoint RAE 2822 airfoil,grid 128x32, Euler calculation Translate the mesh as a rigid body (Far-field boundary is not changed.) Expect summed gradient around the airfoil to be zero. Initial Grid x Translated Grid (1 chord length in x direction) 42

43 Validation of the Reduced Adjoint Gradient Comparison of Original Adjoint, Reduced Adjoint, and Complex-Step methods Comparison of Original Adjoint and Reduced Adjoint methods 43

44 Gradient Comparison Point-wise gradient Integral of the point-wise gradient over the airfoil surface FD 1E-6 Grad 1E-7 Theoretically there is no change in the flow, the integral of the gradient should become zero. 44

45 Deformation of Unstructured Meshes Movement of an individual surface point influences entire mesh Method 1) Spring method: treat edges as springs. Computationally inexpensive but doesn t guarantee to prevent crossovers. 2) Treat the mesh as a pseudo elastic body. Solve x j s ij = 0, s ij = u i x j + u j x i s = stress, u = displacement for a given boundary displacement. 3) Traction method : treat the mesh as an elastic body, but subject to force inputs proportional to the gradient, instead of displacements - alternative to the Sobolev-gradient. 45

46 Redesign of Falcon Complete aircraft calculation on Unstructured Mesh Shock C D = 234 counts 46

47 Redesign of Falcon Using SYNPLANE Drag reduction 18 counts at fixed C L = 0.4 Weakened Shock C D = 216 counts 47

48 Fast Time Integration Methods for Unsteady Problems 48

49 Potential Applications Flutter Analysis, Flow past Helicopter blades, Rotor-Stator Combinations in Turbomachinery, Zero-Mass Synthetic Jets for Flow Control 49

50 Dual Time Stepping BDF The kth-order accurate backward difference formula (BDF) is of the form D t = 1 Dt The non-linear BDF is solved by inner iterations which advance in pseudo-time t* The second-order BDF solves Implementation via Dt * k 1 Â (D - ) q where D - w n +1 = w n +1 - w n q q=1 dw dt + È 3w - 4wn + w n-1 + R(w) * Í = 0 Î 2Dt RK dual time stepping scheme with variable local (RK-BDF) Nonlinear SGS dual time stepping scheme (SGS-BDF) with Multigrid 50

51 Test Case: NACA64A010 pitching airfoil (CT6 Case) Mach Number Pitching amplitude Reduced Freq. Reynolds Number /- 1.01deg million Cycling to limit cycle Pressure Contours at Various Time Instances (AGARD 702) Results of SGS-BDF Scheme (36 time steps per pitching cycle, 3 iterations per time step ) 51

52 Payoff of Dual-time Stepping BDF Schemes Accurate simulations with an order of magnitude reduction in time steps. order For the pitching airfoil: from ~ 1000 to 36 time steps per pitching cycle with three sub-iterations in each step. 52

53 Frequency Domain and Global Space-Time Multigrid Spectral Methods Application : Time-periodic flows Using a Fourier representation in time, the time period T is divided into N steps. ˆ w k = 1 N N-1 w n e -ikndt  n= 0 Then, the discretization operator is given by D t w n = N 2-1  ikw ˆ k e ikndt k= -N 2 53

54 Method 1 (McMullen et.al.) : Transform the equations into frequency domain and solve them in pseudo-time t* d ˆ w k dt * + ik w ˆ k + R ˆ k = 0 Method 2 (Gopinath et.al.) : Solve the equations in the timedomain. The space-time spectral discretization operator is N 2-1 D t w n = Â d m w n +m, d m = 1 2 (-1)m +1 cot( pm N ),m 0 m=- N 2 +1 This is a central difference operator connecting all time levels, yielding an integrated space-time formulation which requires simultaneous solution of the equations at all time levels. 54

55 Comparison with Experimental Data - C L vs. a RANS Time-Spectral Solution with 4 and 8 intervals per pitching cycle Computed Results - Time spectral 4 time intervals - Time spectral 8 time intervals * AGARD-702:Davis Experimental Data 55

56 Payoff of Time Spectral Schemes Engineering accuracy with very small number of time intervals and same rate of convergence as the BDF. Spectral accuracy for sufficiently smooth solutions. Periodic solutions directly without the need to evolve through 5-10 cycles, yielding an order of magnitude reduction in computing cost beyond the reduction already achieved with the BDF, for a total of two orders of magnitude. 56

57 Example of Shape Optimization in Periodic Unsteady Flow (Nadarajah, McMullen, and Jameson AIAA ) Find the (fixed) shape which minimizes the averaged drag coefficient of a pitching airfoil with the constraint that the average lift coefficient is maintained. 57

58 Optimization Results Non-Linear Frequency Domain result Matched Time Accurate result Initial Shape Final Shape Comparison of the Final Airfoil Geometry between the Time Accurate and NLFD method with Lift Constraints. Comparison of the Initial and Final Airfoil Geometry using the NLFD method with Lift Constraints. 58

59 Pressure Distributions Initial Shape Final Shape Comparison of the Final Pressure Distribution between the Time Accurate and NLFD method with Lift Constraints. Comparison of the Initial and Final Pressure Distribution using the NLFD method with Lift Constraints. 59

60 Convergence History Time Accurate Vs. Non-Linear Frequency Domain method 60

61 Flow Past Helicopter Blades 61

62 Challenges Blade runs into its own wake. Resolve blade-vortex interaction Simulate fully articulated hub that is capable of lead-lag, flapping motions coupled with aeroelastic solver Reduce simulation time through convergence acceleration techniques such as space-time multigrid spectral methods 62

63 Test Case: Non-Lifting Hover 2-bladed NACA0012 Mesh size: Tip Mach number 0º Collective pitch Only one section of the blade needs to be calculated because of periodicity Euler Calculation using BDF scheme 63

64 Preliminary Results using the Backward Different Formula Plots of -Cp at different span stations Results look reasonable but needs refinement Subtle modification to the mesh should yield better agreement with experiments 64

65 Future Work Apply space-time multigrid spectral methods This will enable solving Navier-Stokes calculations in reasonable time Couple the flow solution with an aeroelastic model Simulation of forward flight that will allow blade flapping, lagging and change of cyclic pitch Eventually, introduce aero-structural shape optimization to improve rotor performance 65

66 Filtering the Navier Stokes Equations with an Invertible Filter 66

67 Consider the incompressible Navier--Stokes equations where r u i t + ru j u i + r p = m 2 u i (1) x j x i x i x j u i x i = 0 In large eddy simulation (LES) the solution is filtered to remove the small scales. Typically one sets u i (x) = Ú G(x - x )u( x )dx (2) where the kernel G is concentrated in a band defined by the filter width. Then the filtered equations contain the extra virtual stress t ij = u i u j - ui u j (3) because the filtered value of a product is not equal to the product of the filtered values. This stress has to be modeled. 67

68 A filter which completely cuts off the small scales or the high frequency components is not invertible. The use, on the other hand, of an invertible filter would allow equation (1) to be directly expressed in terms of the filtered quantities. Thus one can identify desirable properties of a filter as 1. Attenuation of small scales 2. Commutativity with the differential operator 3. Invertibility Suppose the filter has the form ui = Pu i (4) which can be inverted as Qui = u i (5) where Q = P -1. Moreover Q should be coercieve, so that Qu > c u (6) for some positive constant c. 68

69 Note that if Q commutes with x i then so does Q -1, since for any quantity f which is sufficiently differentiable Also since Q commutes with (Q -1 f ) = Q -1 Q (Q -1 f ) x i x i x i, = Q -1 x i (QQ -1 f ) = Q -1 x i ( f ) u x i = 0 (7) As an example P can be the inverse Helmholtz operator, so that one can write Ê Qui = 1-a 2 2 ˆ Á ui = u i (8) Ë x k x k where a is a length scale proportional to the largest scales to be retained. One may also introduce a filtered pressure p, satisfying the equation Ê Qp = 1-a 2 2 ˆ Á p = p (9) Ë x k x k 69

70 Now one can substitute equation (8) and(9) for u i and p in equation (1) to get r Ê 2 ˆ Ê Á t 1-a2 ui + r 1-a 2 2 ˆ Á u j Ë x k x k Ë x k x k = m 2 Ê 1-a 2 2 ˆ Á x j x j Ë x k x k Ê 1-a 2 2 ˆ Á ui + Ê 1-a 2 2 ˆ Á p x j Ë x l x l x i Ë x k x k Because the order of the differentiations can be interchanged and the Helmholtz operator satisfies condition(6), it can be removed. The product term can be written as r Ï Ê 1-a 2 2 ˆ Ê Á ui 1-a 2 2 ˆ Ì Á u j x j Ó Ë x k x k Ë x l x l = r x j Ï Ì uiu j -a 2 ui Ó = r Ï 2 Ì uiu j -a 2 x j Ó x k x k = rq x j 2 u j x k x k -a 2 u j ui 2 ui x k x k + a 4 2 ui x k x k 2 u j x l x l ( ui u j ) + 2a 2 ui u j 2 + a 4 ui x k x k x k x k Ï Ê uiu j + a 2 Q -1 2 ui u j 2 + a 2 ui 2 u ˆ j Ì Á Ó Ë x k x k x k x k x l x l 2 u j x l x l According to condition (6), if Qf = 0 for any sufficiently differentiable quantity f, then f = 0. 70

71 Thus the filtered equation finally reduces to with the virtual stress r u i t + r x j ( ui u j ) + p = m 2 ui - r t ij (10) x i x k x k x j Ê t ij = a 2 Q -1 2 ui u j + a 2 2 ui 2 u ˆ j Á (11) Ë x k x k x k x k x l x l The virtual stress may be calculated by solving Ê 1-a 2 2 ˆ Ê Á t ij = a 2 2 ui u j + a 2 2 ui Á Ë x k x k Ë x k x k x k x k 2 u ˆ j (12) x l x l Taking the divergence of equation (10), it also follows that p satisfies the Poisson equation 2 p + r x i x i x i x j ui u j ( ) + r 2 x i x j t ij = 0 (13) 71

72 In a discrete solution scales smaller than the mesh width would not be resolved, amounting to an implicit cut off. There is the possibility of introducing an explicit cut off off in t ij. Also one could use equation (8) to restore an estimate of the unfiltered velocity. In order to avoid solving the Helmholtz equation (12), the inverse Helmholtz operator could be expanded formally as ( 1- a 2 D) -1 = 1+a 2 D +a 4 D where D denotes the Laplacian the approximate virtual stress tensor assumes the form 2 x k x k.now retaining terms up to the fourth power of a, u t ij = 2a 2 i u È Ê j +a 4 2D u i u j Í Á x k x k Î Í Ë x k x k ˆ + Du idu j One ay regard the forms (11) or (14) as prototypes for subgrid scale (SGS) models. (14) 72

73 The inverse Helmholtz operator cuts off the smaller scales quite gradually. One could design filters with a sharper cut off by shaping their frequency response. Denote the Fourier transform of f as where (in one space dimension) ˆ f (k) = f (k) = ˆ f = Ff 1 2p 1 2p Then the general form of an invertible filter is F Pf Ÿ f (x)e -ikx dx Ú - f ˆ (k)e -ikx dk Ú - = S(k) ˆ f (k) Ÿ 1 F Qf = S(k) ˆ f (k) where S(k) should decrease rapidly beyond a cut off wave number inversely proportional to a length scale a. 73

74 In the case of a general filter with inverse Q, the virtual stress follows from the relation Then Qu i u j = u i u j = Qu i Qu j t ij = u i u j - u i u j = Q -1 (Qu i Qu j - Q(u i u j )) This formula provides the form for a family of subgrid-scale models. 74

75 Quotation From John Vassberg You are better off with a design method which optimizes 5000 design variables with 10 function evaluations than one which requires 5000 function evaluations to optimize 10 design variables. 75