HIGH-FIDELITY AERO-STRUCTURAL DESIGN OF COMPLETE AIRCRAFT CONFIGURATIONS Juan J. Alonso Department of Aeronautics & Astronautics Stanford University [email protected]DLR - Institute for Aerodynamics and Flow Technology Braunschweig, December 16, 2002 DLR Lecture, December 2002 1
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HIGH-FIDELITY AERO-STRUCTURAL DESIGN OF COMPLETE …aero-comlab.stanford.edu/aa200b/lect_notes/dlr2002.pdfHigh-Fidelity Aerodynamic Shape Optimization † Start from a baseline geometry
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• Start from a baseline geometry providedby a conceptual design tool.
• High-fidelity models required for transonicconfigurations where shocks are present,high-dimensionality required to smooththese shocks.
• Accurate models also required forcomplex supersonic configurations, subtleshape variations required to takeadvantage of favorable shock interference.
• Large numbers of design variables andhigh-fidelity models incur a large cost.
• Is this combination of requirements achievable? Can we actually do this?This is a set of conflicting requirements: the airplane may not “close”.
• Classical sonic boom minimization theory says that ∆p ≡ W
L32.
• What is the necessary aircraft length? Can we achieve this with ourtarget TOGW?
• At Stanford, we have decided to focus on
– Using aerodynamic shape optimization to take advantage of thenonlinear interactions between shock waves and expansions andproduce shaped booms.
– Using Multidisciplinary Design Optimization (MDO) methods tominimize the weight of the airframe.
DLR Lecture, December 2002 5
Quiet Supersonic Platform (QSP) Program
Ground Plane
Mid Field
CFD Far Field
Near Field
Method for Computing Ground Boom
Signatures
Symmetry Plane View of CFD Computation
DLR Lecture, December 2002 6
Aero-Structural Aircraft Design Optimization
• Aerodynamics and structures are coredisciplines in aircraft design and are verytightly coupled.
• For traditional designs, aerodynamicistsknow the spanload distributions that leadto the true optimum from experience andaccumulated data. What about unusualdesigns?
• Want to simultaneously optimize theaerodynamic shape and structure, sincethere is a trade-off between aerodynamicperformance and structural weight, e.g.,
Range ∝ L
Dln
(Wi
Wf
)
DLR Lecture, December 2002 7
The Need for Aero-Structural Sensitivities
OptimizationStructural
OptimizationAerodynamic
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Spanwise coordinate, y/b
Lift
Aerodynamic optimum (elliptical distribution)
Aero−structural optimum (maximum range)
Student Version of MATLAB
Aerodynamic Analysis
Optimizer
Structural Analysis
• Sequential optimization does not lead tothe true optimum.
• xn: vector of design variables, inputs (e.g. aerodynamic shape); boundscan be set on these variables.
• gm: vector of constraints (e.g. element von Mises stresses); in generalthese are nonlinear functions of the design variables.
DLR Lecture, December 2002 9
Optimization Methods
• Intuition: decreases with increasing dimensionality.
• Grid or random search: the cost of searching the designspace increases rapidly with the number of design variables.
• Evolutionary/Genetic algorithms: good for discretedesign variables and very robust; are they feasible whenusing a large number of design variables?
• Nonlinear simplex: simple and robust but inefficient formore than a few design variables.
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• Gradient-based: the most efficient for a large numberof design variables; assumes the objective function is“well-behaved”. Convergence only guaranteed to a localminimum.
DLR Lecture, December 2002 10
Gradient-Based Optimization: Design Cycle
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• Analysis computes objective function andconstraints (e.g. aero-structural solver.)
• Optimizer uses the sensitivity informationto search for the optimum solution(e.g. sequential quadratic programming.)
• Sensitivity calculation is usually thebottleneck in the design cycle, particularlyfor large dimensional design spaces.
• Accuracy of the sensitivities is important,specially near the optimum.
DLR Lecture, December 2002 11
Sensitivity Analysis Methods
• Finite Differences: very popular; easy, but lacks robustness andaccuracy; run solver Nx times.
df
dxn≈ f(xn + h)− f(x)
h+O(h).
• Complex-Step Method: relatively new; accurate and robust; easy toimplement and maintain; run solver Nx times.
df
dxn≈ Im [f(xn + ih)]
h+O(h2).
• Algorithmic/Automatic/Computational Differentiation: accurate;ease of implementation and cost varies.
• (Semi)-Analytic Methods: efficient and accurate; long developmenttime; cost can be independent of Nx.
DLR Lecture, December 2002 12
Finite-Difference Derivative Approximations
From Taylor series expansion,
f(x + h) = f(x) + hf ′(x) + h2f′′(x)2!
+ h3f′′′(x)3!
+ . . . .
Forward-difference approximation:
⇒ df(x)dx
=f(x + h)− f(x)
h+O(h).
f(x) 1.234567890123484
f(x + h) 1.234567890123456
∆f 0.000000000000028
x x+h
f(x)
f(x+h)
DLR Lecture, December 2002 13
Complex-Step Derivative Approximation
Can also be derived from a Taylor series expansion about x with a complexstep ih:
f(x + ih) = f(x) + ihf ′(x)− h2f′′(x)2!
− ih3f′′′(x)3!
+ . . .
⇒ f ′(x) =Im [f(x + ih)]
h+ h2f
′′′(x)3!
+ . . .
⇒ f ′(x) ≈ Im [f(x + ih)]h
.
No subtraction! Second order approximation.
DLR Lecture, December 2002 14
Simple Numerical Example
Step Size, h
Norm
aliz
ed E
rror,
e
Complex-StepForward-DifferenceCentral-Difference
Estimate derivative atx = 1.5 of the function,
f(x) =ex
√sin3x + cos3x
.
Relative error defined as:
ε =
∣∣∣f ′ − f ′ref
∣∣∣∣∣∣f ′ref
∣∣∣.
DLR Lecture, December 2002 15
Implementation Procedure
• Cookbook procedure for any programming language:
– Substitute all real type variable declarations with complexdeclarations.
– Define all functions and operators that are not defined for complexarguments.
– A complex-step can then be added to the desired variable and thederivative can be estimated by f ′ ≈ Im[f(x + ih)]/h.
• Fortran 77: write new subroutines, substitute some of the intrinsicfunction calls by the subroutine names, e.g. abs by c abs. But ... needto know variable types in original code.
• Fortran 90: can overload intrinsic functions and operators, includingcomparison operators. Compiler knows variable types and chooses correctversion of the function or operator.
• C/C++: also uses function and operator overloading.