Structural and Multidisciplinary Optimization

, Volume 58, Issue 3, pp 1205–1220 | Cite as

Toplogical optimization of structures using Fourier representations

  • Daniel A. White
  • Mark L. Stowell
  • Daniel A. Tortorelli


The minimization of compliance subject to a mass constraint is the topology optimization design problem of interest. The goal is to determine the optimal configuration of material within an allowed volume. Our approach builds upon the well-known density method in which the decision variable is the material density in every cell in a mesh. In it’s most basic form the density method consists of three steps: 1) the problem is convexified by replacing the integer material indicator function with a volume fraction, 2) the problem is regularized by filtering the volume fraction field to impose a minimum length scale; 3) the filtered volume fraction is penalized to steer the material distribution toward binary designs. The filtering step is used to yield a mesh-independent solution and to eliminate checkerboard instabilities. In image processing terms this is a low-pass filter, and a consequence is that the decision variables are not independent and a change of basis could significantly reduce the dimension of the nonlinear programming problem. Based on this observation, we represent the volume fraction field with a truncated Fourier representation. This imposes a minimal length scale on the problem, eliminates checkerboard instabilities, and also reduces the number of decision variables by over 100 × (two dimensions) or 1000 × (three dimensions).


Topology optimization Fourier analysis 



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Copyright information

© This is a U.S. Government work and not under copyright protection in the US; foreign copyright protection may apply 2018

Authors and Affiliations

  • Daniel A. White
    • 1
  • Mark L. Stowell
    • 1
  • Daniel A. Tortorelli
    • 1
  1. 1.Lawrence Livermore National LaboratoryLivermoreUSA

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