HIGH-FIDELITY AERO-STRUCTURAL DESIGN OPTIMIZATION OF A SUPERSONIC BUSINESS JET Joaquim R. R. A. Martins Juan J. Alonso James J. Reuther Department of Aeronautics and Astronautics Stanford University AIAA Structures, Structural Dynamics and Materials Conference, Denver, CO, April 2002 1
Outline • Introduction – High-fidelity aircraft design optimization – The need for aero-structural sensitivities – Sensitivity analysis methods – Optimization problem statement • Theory – Adjoint sensitivity equations – Lagged aero-structural adjoint equations • Results – Aero-structural sensitivity validation – Optimization results • Conclusions and Future Work AIAA Structures, Structural Dynamics and Materials Conference, Denver, CO, April 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. • Gradient-based optimization is the most efficient and requires accurate sensitivity information. AIAA Structures, Structural Dynamics and Materials Conference, Denver, CO, April 2002 3
Aero-Structural Aircraft Design Optimization • Aerodynamics traditionally has used a shape corresponding to the flying shape of the wing, assuming that shape can be reproduced. • Wing shape depends on aerodynamic solution, so need to couple aerodynamic and structural analyses to obtain the solution, specially for unusual designs. • Want to optimize the structure as well, since there is a trade-off between aerodynamic performance and structural weight: � W i � Range ∝ L D ln W f AIAA Structures, Structural Dynamics and Materials Conference, Denver, CO, April 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 AIAA Structures, Structural Dynamics and Materials Conference, Denver, CO, April 2002 5
Supersonic Business Jet Optimization Problem minimize I = αC D + βW subject to C L = C L T KS ≥ 0 x S ≥ x S min Lump stress constraints g i = 1 − σ i V M ≥ 0 , σ yield using the Kreisselmeier-Steinhauser Natural laminar flow function supersonic business jet Mach = 1.5, Range = 5,300nm �� � KS ( g i ( x )) = − 1 1 count of drag = 310 lbs of weight e − ρg i ( x ) ρ ln . i AIAA Structures, Structural Dynamics and Materials Conference, Denver, CO, April 2002 6
Methods for Sensitivity Analysis • Finite-Difference: very popular; easy, but lacks robustness and accuracy; run solver n times. f ′ ( x ) ≈ f ( x + h ) − f ( x ) + O ( h ) h • Complex-Step Method: relatively new; accurate and robust; easy to implement and maintain; run solver n times. f ′ ( x ) ≈ Im [ f ( x + ih )] + O ( h 2 ) h • (Semi)-Analytic Methods: efficient and accurate; long development time; cost can be independent of n . AIAA Structures, Structural Dynamics and Materials Conference, Denver, CO, April 2002 7
Objective Function and Governing Equations Want to minimize scalar objective function, I = I ( x, y ) , which depends on: • x : vector of design variables, e.g. structural plate thickness. • y : state vector, e.g. structural displacements. Physical system is modeled by a set of governing equations: R ( x, y ( x )) = 0 , where: • Same number of state and governing equations, n R • n x design variables. AIAA Structures, Structural Dynamics and Materials Conference, Denver, CO, April 2002 8
Variational Equations x y R =0 I Total variation of the objective function: δI = ∂I ∂xδx + ∂I ∂yδy. Variation of the governing equations, δR = ∂R ∂x δx + ∂R ∂y δy = 0 . AIAA Structures, Structural Dynamics and Materials Conference, Denver, CO, April 2002 9
Adjoint Sensitivity Equations Since the variation of the governing equations must be zero, we can add it to the total variation of the objective, � ∂R � δI = ∂I ∂xδx + ∂I ∂x δx + ∂R ∂yδy + ψ T ∂y δy , where ψ is a vector of arbitrary components known as adjoint variables . Re-arrange terms, � ∂I � � ∂I � ∂x + ψ T ∂R ∂y + ψ T ∂R δI = δx + δy. ∂x ∂y If term in blue were zero, term in red would represent the total variation of the objective with respect to the design variables. AIAA Structures, Structural Dynamics and Materials Conference, Denver, CO, April 2002 10
Adjoint Sensitivity Equations Since the adjoint variables are arbitrary, we can find a set such that, � ∂R � T T = − ∂I ψ ∂y ∂y ���� ( n R × 1) � �� � � �� � ( n R × 1) ( n R × n R ) Adjoint valid for all design variables. Now the total sensitivity of the objective is: d I ∂I ∂R ψ T = + d x ∂x ∂x ���� ���� ���� ���� (1 × n R ) (1 × n R ) (1 × n x ) ( n R × n x ) The partial derivatives are inexpensive, since they don’t require the solution of the governing equations. AIAA Structures, Structural Dynamics and Materials Conference, Denver, CO, April 2002 11
Aero-Structural Adjoint Equations x w R =0 R =0 A S u I Two coupled disciplines: Aerodynamics ( R A ) and Structures ( R S ). � R A � w � ψ A � � � R = , y = and ψ = R S u ψ S Flow variables, w , five for each grid point. Structural displacements, u , three for each structural node. AIAA Structures, Structural Dynamics and Materials Conference, Denver, CO, April 2002 12
Aero-Structural Adjoint Equations T ∂R A ∂R A ∂I � � ψ A ∂w ∂u ∂w = − ψ S ∂R S ∂R S ∂I ∂w ∂u ∂u • ∂R A /∂w : a change in one of the flow variables affects only the residuals of its cell and the neighboring ones. • ∂R A /∂u : wing deflections cause the mesh to warp, affecting the residuals. • ∂R S /∂w : since R S = Ku − f , this is equal to − ∂f/∂w . • ∂R S /∂u : equal to the stiffness matrix, K . • ∂I/∂w : for C D , obtained from the integration of pressures; for KS, its zero. • ∂I/∂u : for C D wing displacement changes the surface boundary over which drag is integrated; for KS, related to σ = Su . AIAA Structures, Structural Dynamics and Materials Conference, Denver, CO, April 2002 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, � ∂R A � T � ∂R S � T ψ A = − ∂I ˜ ψ S − ∂w ∂w ∂w � �� � Aerodynamic adjoint equations � ∂R S � T � ∂R A � T ψ S = − ∂I ˜ ψ A . − ∂u ∂u ∂u � �� � Structural adjoint equations 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. AIAA Structures, Structural Dynamics and Materials Conference, Denver, CO, April 2002 14
Total Sensitivity The aero-structural sensitivities of the drag coefficient with respect to wing shape perturbations are, d x = ∂I d I ∂R A ∂R S ∂x + ψ T ∂x + ψ T ∂x . A S • ∂I/∂x : C D changes when the boundary over which the pressures are integrated is perturbed; stresses change when nodes are moved. • ∂R A /∂x : the shape perturbations affect the grid, which in turn changes the residuals; structural variables have no effect on this. • ∂R S /∂x : shape perturbations affect the structural equations, so this is ∂K/∂x · u − ∂f/∂x . AIAA Structures, Structural Dynamics and Materials Conference, Denver, CO, April 2002 15
3D Aero-Structural Design Optimization Framework • Aerodynamics: SYN107-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. AIAA Structures, Structural Dynamics and Materials Conference, Denver, CO, April 2002 16
Sensitivity of C D wrt Shape −3 10 x 10 8 6 4 d C D / d x A 2 0 −2 −4 Coupled adjoint Complex step −6 Avg. rel. error = 3.5% −8 1 2 3 4 5 6 7 8 9 10 Shape variable, x j AIAA Structures, Structural Dynamics and Materials Conference, Denver, CO, April 2002 17
Sensitivity of C D wrt Structural Thickness 0.08 0.06 0.04 0.02 d C D / d x S 0 −0.02 −0.04 −0.06 Coupled adjoint Complex step −0.08 Avg. rel. error = 1.6% −0.1 11 12 13 14 15 16 17 18 19 20 Structural variable, x j AIAA Structures, Structural Dynamics and Materials Conference, Denver, CO, April 2002 18
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