COMPLETE CONFIGURATION AERO-STRUCTURAL OPTIMIZATION USING A COUPLED SENSITIVITY ANALYSIS METHOD Joaquim R. R. A. Martins Juan J. Alonso James J. Reuther Department of Aeronautics and Astronautics Stanford University 9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization Atlanta, GA, September 2002 1
Outline • Introduction – High-fidelity aircraft design optimization – The need for aero-structural sensitivities – Sensitivity analysis methods • Theory – Adjoint sensitivity equations – Lagged aero-structural adjoint equations • Results – Optimization problem statement – Aero-structural sensitivity validation – Optimization results • Conclusions 9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization Atlanta, GA, September 2002 2
High-Fidelity Aircraft Design Optimization • Start from a baseline geometry provided by a conceptual design tool. • Required for transonic configurations where shocks are present. • Necessary for supersonic, complex geometry design. • High-fidelity analysis needs high-fidelity parameterization, e.g. to smooth shocks, favorable interference. • Large number of design variables and complex models inccur a large cost. 9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization Atlanta, GA, September 2002 3
Aero-Structural Aircraft Design Optimization • Aerodynamics and structures are core disciplines in aircraft design and are very tightly coupled. • For traditional designs, aerodynamicists know the spanload distributions that lead to the true optimum from experience and accumulated data. What about unusual designs? • Want to simultaneously optimize the aerodynamic shape and structure, since there is a trade-off between aerodynamic performance and structural weight, e.g., � W i � Range ∝ L D ln W f 9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization Atlanta, GA, September 2002 4
The Need for Aero-Structural Sensitivities Aerodynamic Optimization • Sequential optimization does not lead to the true optimum. Structural Optimization • Aero-structural optimization requires 1.6 1.4 coupled sensitivities. 1.2 1 Lift 0.8 Aerodynamic optimum • Add structural element sizes to the design 0.6 (elliptical distribution) 0.4 Aero−structural optimum variables. 0.2 (maximum range) 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Spanwise coordinate, y/b • Including structures in the high-fidelity Optimizer wing optimization will allow larger changes in the design. Aerodynamic Structural Analysis Analysis 9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization Atlanta, GA, September 2002 5
Methods for Sensitivity Analysis • Finite-Difference: very popular; easy, but lacks robustness and accuracy; run solver N x times. d f ≈ f ( x n + h ) − f ( x ) + O ( h ) d x n h • Complex-Step Method: relatively new; accurate and robust; easy to implement and maintain; run solver N x times. d f ≈ Im [ f ( x n + ih )] + O ( h 2 ) d x n h • (Semi)-Analytic Methods: efficient and accurate; long development time; cost can be independent of N x . 9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization Atlanta, GA, September 2002 6
Objective Function and Governing Equations Want to minimize scalar objective function, I = I ( x n , y i ) , which depends on: • x n : vector of design variables, e.g. structural plate thickness. • y i : state vector, e.g. flow variables. Physical system is modeled by a set of governing equations: R k ( x n , y i ( x n )) = 0 , where: • Same number of state and governing equations, i, k = 1 , . . . , N R • N x design variables. 9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization Atlanta, GA, September 2002 7
✁ ✂ ✄ ☎ ✆ � Sensitivity Equations Total sensitivity of the objective function: d I = ∂I + ∂I d y i . d x n ∂x n ∂y i d x n Total sensitivity of the governing equations: d R k = ∂ R k + ∂ R k d y i = 0 . d x n ∂x n ∂y i d x n 9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization Atlanta, GA, September 2002 8
Solving the Sensitivity Equations Solve the total sensitivity of the governing equations ∂ R k d y i = − ∂ R k . ∂y i d x n ∂x n Substitute this result into the total sensitivity equation − d y i / d x n � �� � � − 1 ∂ R k � ∂ R k d I = ∂I − ∂I , d x n ∂x n ∂y i ∂y i ∂x n � �� � − Ψ k where Ψ k is the adjoint vector . 9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization Atlanta, GA, September 2002 9
Adjoint Sensitivity Equations Solve the adjoint equations ∂ R k Ψ k = − ∂I . ∂y i ∂y i Adjoint vector is valid for all design variables. Now the total sensitivity of the the function of interest I is: d I = ∂I ∂ R k + Ψ k d x n ∂x n ∂x n The partial derivatives are inexpensive, since they don’t require the solution of the governing equations. 9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization Atlanta, GA, September 2002 10
✄ ☎ ✆ ✝ ☞ ✌ ✍ ✒ Aero-Structural Adjoint Equations �✂✁ ✎✑✏ ✞✠✟☛✡ Two coupled disciplines: Aerodynamics ( A k ) and Structures ( S l ). � w i � ψ k � A k � � � R k ′ = , y i ′ = , Ψ k ′ = . S l u j φ l Flow variables, w i , five for each grid point. Structural displacements, u j , three for each structural node. 9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization Atlanta, GA, September 2002 11
Aero-Structural Adjoint Equations T � ∂ A k ∂ A k ∂I � ψ k ∂w i ∂u j ∂w i . = − ∂I ∂ S l ∂ S l φ l ∂u j ∂w i ∂u j • ∂ A k /∂w i : a change in one of the flow variables affects only the residuals of its cell and the neighboring ones. • ∂ A k /∂u j : wing deflections cause the mesh to warp, affecting the residuals. • ∂ S l /∂w i : since S l = K lj u j − f l , this is equal to − ∂f l /∂w i . • ∂ S l /∂u j : equal to the stiffness matrix, K lj . • ∂I/∂w i : for C D , obtained from the integration of pressures; for stresses, its zero. • ∂I/∂u j : for C D , wing displacement changes the surface boundary over which drag is integrated; for stresses, related to σ m = S mj u j . 9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization Atlanta, GA, September 2002 12
Lagged Aero-Structural Adjoint Equations Since the factorization of the complete residual sensitivity matrix is impractical, decouple the system and lag the adjoint variables, ∂ A k ψ k = − ∂I − ∂ S l ˜ φ l , ∂w i ∂w i ∂w i � �� � Aerodynamic adjoint ∂ S l φ l = − ∂I − ∂ A k ˜ ψ k , ∂u j ∂u j ∂u j � �� � Structural adjoint Lagged adjoint equations are the single discipline ones with an added forcing term that takes the coupling into account. System is solved iteratively, much like the aero-structural analysis. 9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization Atlanta, GA, September 2002 13
Total Sensitivity The aero-structural sensitivities of the drag coefficient with respect to wing shape perturbations are, d I = ∂I ∂ A k ∂ S l + ψ k + φ l . d x n ∂x n ∂x n ∂x n • ∂I/∂x n : C D changes when the boundary over which the pressures are integrated is perturbed; stresses change when nodes are moved. • ∂ A k /∂x n : the shape perturbations affect the grid, which in turn changes the residuals; structural variables have no effect on this term. • S l /∂x n : shape perturbations affect the structural equations, so this term is equal to ∂K lj /∂x n u j − ∂f l /∂x n . 9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization Atlanta, GA, September 2002 14
3D Aero-Structural Design Optimization Framework • Aerodynamics: FLO107-MB, a parallel, multiblock Navier-Stokes flow solver. • Structures: detailed finite element model with plates and trusses. • Coupling: high-fidelity, consistent and conservative. • Geometry: centralized database for exchanges (jig shape, pressure distributions, displacements.) • Coupled-adjoint sensitivity analysis: aerodynamic and structural design variables. 9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization Atlanta, GA, September 2002 15
Sensitivity of C D wrt Shape 0.016 Coupled adjoint 0.014 Complex step 0.012 Coupled adjoint, fixed displacements Complex step, fixed displacements 0.01 0.008 d C D / d x n 0.006 0.004 0.002 0 -0.002 -0.004 -0.006 1 2 3 4 5 6 7 8 9 10 Design variable, n 9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization Atlanta, GA, September 2002 16
Sensitivity of C D wrt Structural Thickness 0.08 0.06 Coupled adjoint 0.04 Complex step 0.02 d C D / d x n 0 -0.02 -0.04 -0.06 -0.08 11 12 13 14 15 16 17 18 19 20 Design variable, n 9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization Atlanta, GA, September 2002 17
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