COMPLETE CONFIGURATION AERO-STRUCTURAL OPTIMIZATION USING A COUPLED SENSITIVITY ANALYSIS METHOD

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1 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

2 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

High-fidelity analysis needs high-fidelity parameterization, e.g. to smooth shocks, favorable interference.

3 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

Aero-Structural Aircraft Design Optimization Aerodynamics and structures are core disciplines in aircraft design and are very tightly coupled.

4 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., Range L ( ) D ln Wi W f 9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization Atlanta, GA, September

5 The Need for Aero-Structural Sensitivities Aerodynamic Optimization Structural Optimization Sequential optimization does not lead to the true optimum Aero-structural optimization requires coupled sensitivities. 1 Lift Aerodynamic optimum (elliptical distribution) Aero structural optimum (maximum range) Aerodynamic Analysis Spanwise coordinate, y/b Optimizer Structural Analysis Student Version of MATLAB Add structural element sizes to the design variables. Including structures in the high-fidelity wing optimization will allow larger changes in the design. 9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization Atlanta, GA, September

6 Methods for Sensitivity Analysis Finite-Difference: very popular; easy, but lacks robustness and accuracy; run solver N x times. df f(x n + h) f(x) dx n h + O(h) Complex-Step Method: relatively new; accurate and robust; easy to implement and maintain; run solver N x times. df Im [f(x n + ih)] dx n h + O(h 2 ) (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

7 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

8 Sensitivity Equations Total sensitivity of the objective function: di = I + I dy i. dx n x n y i dx n Total sensitivity of the governing equations: dr k dx n = R k x n + R k y i dy i dx n = 0. 9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization Atlanta, GA, September

9 Solving the Sensitivity Equations Solve the total sensitivity of the governing equations R k y i dy i dx n = R k x n. Substitute this result into the total sensitivity equation where Ψ k is the adjoint vector. di = I I dx n x n y i dy i / dx n { [ ] }}{ 1 Rk R k, x n } y i {{ } Ψ k 9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization Atlanta, GA, September

10 Adjoint Sensitivity Equations Solve the adjoint equations R k y i Ψ k = I y i. Adjoint vector is valid for all design variables. Now the total sensitivity of the the function of interest I is: di dx n = I x n + Ψ k R k 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

11 Aero-Structural Adjoint Equations Two coupled disciplines: Aerodynamics (A k ) and Structures (S l ). R k = [ Ak ], y S i = l [ wi ], Ψ u k = j [ ψk ]. 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

12 Aero-Structural Adjoint Equations A k w i S l w i T A k [ψk ] u j S l φ l u j I = w i 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 : residuals. wing deflections cause the mesh to warp, affecting the 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

13 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 } w i {{ w } i Aerodynamic adjoint S l w i φl, S l φ l = I A k ψk, 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

14 Total Sensitivity The aero-structural sensitivities of the drag coefficient with respect to wing shape perturbations are, di = I A k S l + ψ k + φ l. dx 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

Geometry: centralized database for exchanges (jig shape, pressure distributions, displacements.

15 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

16 Sensitivity of C D wrt Shape Coupled adjoint Complex step Coupled adjoint, fixed displacements Complex step, fixed displacements dc D / dx n Design variable, n 9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization Atlanta, GA, September

17 Sensitivity of C D wrt Structural Thickness Coupled adjoint Complex step dc D / dx n Design variable, n 9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization Atlanta, GA, September

18 Structural Stress Constraint Lumping To perform structural optimization, we need the sensitivities of all the stresses in the finite-element model with respect to many design variables. There is no method to calculate this matrix of sensitivities efficiently. Therefore, lump stress constraints g m = 1 σ m σ yield 0, using the Kreisselmeier Steinhauser function KS (g m ) = 1 ρ ln ( m e ρg m ), where ρ controls how close the function is to the minimum of the stress constraints. 9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization Atlanta, GA, September

19 Sensitivity of KS wrt Shape Coupled adjoint Complex step Coupled adjoint, fixed loads Complex, fixed loads dks / dx n Design variable, n 9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization Atlanta, GA, September

20 Sensitivity of KS wrt Structural Thickness Coupled adjoint Complex step Coupled adjoint, fixed loads Complex, fixed loads 50 dks / dx n Design variable, n 9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization Atlanta, GA, September

21 Computational Cost vs. Number of Variables 1200 Coupled adjoint Finite difference Complex step Normalized time N x N x N x Number of design variables (N x ) 9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization Atlanta, GA, September

Supersonic Business Jet Optimization Problem Minimize: I = αc D + βw where C D is that of the cruise condition. Natural laminar flow supersonic business jet Mach = 1.

22 Supersonic Business Jet Optimization Problem Minimize: I = αc D + βw where C D is that of the cruise condition. Natural laminar flow supersonic business jet Mach = 1.5, Range = 5,300nm 1 count of drag = 310 lbs of weight Subject to: KS(σ m ) 0 where KS is taken from a maneuver condition. With respect to: external shape and internal structural sizes. 9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization Atlanta, GA, September

23 Baseline Design 9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization Atlanta, GA, September

24 Aero-Structural Optimization Convergence History Drag (counts) Weight (lbs) KS Drag Weight KS Major iteration number th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization Atlanta, GA, September

25 Aero-Structural Optimization Results 9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization Atlanta, GA, September

26 Comparison with Sequential Optimization C D (counts) σ max σ yield Weight (lbs) Objective All-at-once approach Baseline , 285 Optimized , Sequential approach Aerodynamic optimization Baseline Optimized Structural optimization Baseline , 285 Optimized , 567 Aero-structural analysis , th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization Atlanta, GA, September

27 Conclusions Presented a framework for high-fidelity aero-structural design. The computation of sensitivities using the aero-structural adjoint is extremely accurate and efficient. Demonstrated the usefulness of the coupled adjoint by optimizing a supersonic business jet configuration with external shape and internal structural variables. 9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization Atlanta, GA, September

A COUPLED-ADJOINT METHOD FOR HIGH-FIDELITY AERO-STRUCTURAL OPTIMIZATION

A COUPLED-ADJOINT METHOD FOR HIGH-FIDELITY AERO-STRUCTURAL OPTIMIZATION Joaquim Rafael Rost Ávila Martins Department of Aeronautics and Astronautics Stanford University Ph.D. Oral Examination, Stanford