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Structural and Multidisciplinary Optimization

, Volume 51, Issue 5, pp 1113–1132 | Cite as

Coupled aerostructural topology optimization using a level set method for 3D aircraft wings

  • Peter D. DunningEmail author
  • Bret K. Stanford
  • H. Alicia Kim
RESEARCH PAPER

Abstract

The purpose of this work is to develop a level set topology optimization method for an unstructured three-dimensional mesh and apply it to wing box design for coupled aerostructural considerations. The paper develops fast marching and upwind schemes suitable for unstructured meshes, which make the level set method robust and efficient. The method is applied to optimize a representative wing box internal structure for the NASA Common Research Model. The objective is to minimize the total compliance of the wing box. The trim condition that aerodynamic lift must balance the total weight of the aircraft is enforced by allowing the root angle of attack to change. The adjoint method is used to obtain the coupled shape sensitivities required to perform aerostructural optimization of the wing box. Optimum solutions for several aerodynamic and body force load cases, as well as a ground load case, are presented.

Keywords

Level set method 3D unstructured mesh Topology optimization Multi-disciplinary optimization 

Nomenclature

A

Sensitivity factor for angle of attack

a

Vector of Doublet Lattice Method (DLM) box areas

b

Vector defining influence of wing deformed shape on lift

C

Compliance of the structure

cp

Pressure coefficient vector

D

Aerodynamic influence coefficient matrix

E

Material property tensor

e

Number of elements attached to a node

fa

Aerodynamic load vector

fg

Body force load vector

ft

Total load vector

g

Acceleration due to gravity

h

Element edge length

i, j

Indices

K

Global structural stiffness matrix

Kc

Stiffness matrix of an element cut by the boundary

KE

Stiffness matrix of a finite element

k

Iteration number

L

Total lift force

Lc

Lift force from built-in twist and camber

Lα

Lift force from unit angle of attack

N

Load factor

n

Unit normal vector

p

Adjoint state vector

q

Dynamic pressure

Q

Aerodynamic stiffness matrix

S

Force transfer matrix

T

Displacement transfer matrix

t

Fictitious time variable

u

Displacement field or vector

Vn

Velocity function

v

Virtual displacement

Wb

Wing box weight

Wc

Fixed aircraft weight

w

Downwash dependent on deformed wing shape

wc

Constant downwash from built-in camber

x

Point in the design domain

z

Column vector of 1’s

α

Angle of attack

βc

Volume of a cut element that lies inside the structure

βE

Volume of an element

γ

Small number

Γ

Structural boundary

ΓD

Part of boundary subject to displacement boundary conditions

ΓN

Part of boundary subject to aerodynamic loads

Γ0

Part of boundary free from boundary conditions and aerodynamic loads

Δt

Time step

ε

Strain tensor

θ

Arbitrary vector (shape derivative auxiliary variable)

ρ

Material density

ϕ

Implicit function

Ωd

Design domain

Ω

Structural domain

Notes

Acknowledgments

This work is funded by the Fixed Wing project under the National Aeronautics and Space Administration’s (NASA) Fundamental Aeronautics Program. The authors would like to thank Dr. Maxwell Blair for his example DLM code and the Numerical Analysis Group at the Rutherford Appleton Laboratory for their FORTRAN HSL packages (HSL, a collection of Fortran codes for large-scale scientific computation. See http://www.hsl.rl.ac.uk/).

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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Peter D. Dunning
    • 1
    Email author
  • Bret K. Stanford
    • 2
  • H. Alicia Kim
    • 3
  1. 1.National Institute of AerospaceHamptonUSA
  2. 2.NASA Langley Research CenterHamptonUSA
  3. 3.University of BathBathUK

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