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


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.


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



Sensitivity factor for angle of attack


Vector of Doublet Lattice Method (DLM) box areas


Vector defining influence of wing deformed shape on lift


Compliance of the structure


Pressure coefficient vector


Aerodynamic influence coefficient matrix


Material property tensor


Number of elements attached to a node


Aerodynamic load vector


Body force load vector


Total load vector


Acceleration due to gravity


Element edge length

i, j



Global structural stiffness matrix


Stiffness matrix of an element cut by the boundary


Stiffness matrix of a finite element


Iteration number


Total lift force


Lift force from built-in twist and camber


Lift force from unit angle of attack


Load factor


Unit normal vector


Adjoint state vector


Dynamic pressure


Aerodynamic stiffness matrix


Force transfer matrix


Displacement transfer matrix


Fictitious time variable


Displacement field or vector


Velocity function


Virtual displacement


Wing box weight


Fixed aircraft weight


Downwash dependent on deformed wing shape


Constant downwash from built-in camber


Point in the design domain


Column vector of 1’s


Angle of attack


Volume of a cut element that lies inside the structure


Volume of an element


Small number


Structural boundary


Part of boundary subject to displacement boundary conditions


Part of boundary subject to aerodynamic loads


Part of boundary free from boundary conditions and aerodynamic loads


Time step


Strain tensor


Arbitrary vector (shape derivative auxiliary variable)


Material density


Implicit function


Design domain


Structural domain



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


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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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