Robust Optimization of Aircraft Conceptual Designsupported by MATLAB AD

M. S. Campobasso, P. Fantini, M. GuenovSchool of Engineering, Cranfield University

Cranfield, United Kingdom

3rd European Workshop on Automatic DifferentiationOxford University Computing Laboratory

1st June 2006

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Outline

� Aircraft Conceptual Design

� Robust Design Optimization� Test Problem

� Summary

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Aircraft conceptual design

First stage of aircraft life cycle.

Aircraft layout determined on the basis of fundamental specifications ( � � � �

range, number of passengers, take-off and landing field length, � � � )

Other parameters ( � � � � approach speed, maximum take-off weight, acqui-sition and operating cost, noise emission) treated either as inequality con-straints or objectives to be optimized.

Subsequent preliminary and detail design may yield variations of the orig-inal layout. Robustness of conceptual design optimization is requiredto minimize possible specs and constraint violations induced by suchchanges.

Main difficulty is dealing with a large number � ��� � � � of design variablesand relatively simple functions, each modeling a particular discipline.

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Baseline for aerodynamic cruise analysis

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Four perturbations of design variables

INNER SWEEP

OUTER SWEEP

INNER DIHEDRAL

OUTER DIHEDRAL

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Robust Design Optimization (RDO)

� Optimize objectives and their variance at the same time� Consider probabilistic satisfaction of constraints

DETERMINISTIC

Optimize �� � ��� � � ��� �subject to

� � � � � � �

� ��� � � ��� �� �

ROBUST

Optimize �� � ��� ��� � � � � ��� � �

subject to

� � � ��� � �

� ��� � � � � �� � � � � �

0 2 4 6−10

10

30

50

70

Do

det. func.det. opt.

0 2 4 6−10

10

30

50

70

Do

Ro

det. func.det. opt.rob. func.rob. opt.

Robust design

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Uncertainty Analysis based on Sensitivity Derivatives (1)

Direct evaluation of � and � � (e.g. Monte Carlo-like approach) compu-tationally unaffordable when the analysis is computationally expensive orsophisticated multiobjective optimization is performed.

Alternative strategy is use of sensitivity derivatives. Consider truncated Tay-lor series expansion of� about� yields

� �� � � � � and � �

��� �

� ��� �

� � ��

Starting point: object-oriented MATLAB framework for multi-disciplinarymulti-objective optimization (comprehensive search of design space yield-ing sets of Pareto-fronts)

Objective: use AD to support robust optimization, enhance accuracy andcomputational efficiency

Uncertainty analysis

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AD and multidisciplinary robust optimization� MAD: MATLAB AD package by Dr. Shaun Forth.

– based on operator overloading.– delivers first derivatives in forward mode.– backward mode and second derivatives under development.

� deployment of MAD in our optimization framework:– derivatives of deterministic objectives and constraints needed

for variances (above analysis).– derivatives needed by MATLAB direct solvers and optimizers

such as ��� �� � � and ��� �� � (inside analysis).

� implementation issue 1: how to propagate derivatives through so-lution procedure of strongly coupled sets of disciplines– fixed-point iterations: piggy-back AD.– ��� �� � � : ?

� implementation issue 2: how to propagate derivatives through op-timization subprocesses such as � � �� � : ??

Uncertainty analysis

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Differentiating ��� �� � �

Let us consider the system of� equations

� ��� � � � �

to be solved for� � ��� with given � � � � .

� is a subset of, or depends on the design (active) variables.

� � �� � �� � is a wrapper of � � �� � � with two functions:

� provide ��� �� � � with the Jacobian � � � � at each step of sol. proc.

� provide the user with the Jacobian � � � � at the end of � � �� � � .

Latter step achieved by solving the system

� �� �

� �� � �

� �� �

�

where � �� and � � � are obtained by declaring either� or � of class � � �� ,

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Differentiating ��� �� �

Minimize � � ��� � � � with respect to� � � � and with given � � � � .

� is a subset of, or depends on the design (active) variables.

� � �� � � � is a wrapper of ��� �� � with two functions:

� provide ��� �� � with the Jacobian � � � � at each step of sol. proc.

� provide the user with the Jacobian � � � � at the end of � � �� � .

Latter step achieved achieved as follows.At unconstrained min, � � ��� � � � � � � � � � ,Differentiating with respect to � ,

� � �� � � � � � � �� � �

� �� �

�Sought derivative � � � � is obtained by solving linear system above. Secondderivatives are instead obtained by using forward and backward AD.

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Robust optimization test case

Objective: optimize Maximum Take-off weight (MTOW)Specs: 1) � of passengers, 2) � of engines, 3) cruise Mach and altitude ,

� � �

Independent variables: Wing area � � � �� � � , Wing span �� � � � � , WingSweep �� � , thickness to chord ratio ��� � , Engine thrust ��� � � � � � �

Constraints :1. range ��� � � � �� � �

2. take-off field length ��� � � � � � � �3. wing fuel / fuselage fuel �� � � � � � � �4. cruise thrust coefficient � � � � ��� � �5. climb steed � � � �� � � � � � � � �� �

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Robust objective

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Robust constraints

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Next stages� larger test cases

� second order variances

� robust Pareto front

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