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Toplogical optimization of structures using Fourier representations

Abstract

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).

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Correspondence to Daniel A. White.

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White, D.A., Stowell, M.L. & Tortorelli, D.A. Toplogical optimization of structures using Fourier representations. Struct Multidisc Optim 58, 1205–1220 (2018). https://doi.org/10.1007/s00158-018-1962-y

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Keywords

  • Topology optimization
  • Fourier analysis