Toplogical optimization of structures using Fourier representations
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 wellknown 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 meshindependent solution and to eliminate checkerboard instabilities. In image processing terms this is a lowpass 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. Furthermore, this imposes a minimal length scale on the problem, eliminates checkerboard instabilities, and also reduces the number of decision variablesmore »
 Authors:

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 Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
 Publication Date:
 Report Number(s):
 LLNLJRNL736779
Journal ID: ISSN 1615147X; 889675
 Grant/Contract Number:
 AC5207NA27344
 Type:
 Accepted Manuscript
 Journal Name:
 Structural and Multidisciplinary Optimization
 Additional Journal Information:
 Journal Volume: 58; Journal Issue: 3; Journal ID: ISSN 1615147X
 Publisher:
 Springer
 Research Org:
 Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
 Sponsoring Org:
 USDOE National Nuclear Security Administration (NNSA)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 97 MATHEMATICS AND COMPUTING; Topology optimization; Fourier analysis
 OSTI Identifier:
 1479078
White, Daniel A., Stowell, Mark L., and Tortorelli, Daniel A.. Toplogical optimization of structures using Fourier representations. United States: N. p.,
Web. doi:10.1007/s001580181962y.
White, Daniel A., Stowell, Mark L., & Tortorelli, Daniel A.. Toplogical optimization of structures using Fourier representations. United States. doi:10.1007/s001580181962y.
White, Daniel A., Stowell, Mark L., and Tortorelli, Daniel A.. 2018.
"Toplogical optimization of structures using Fourier representations". United States.
doi:10.1007/s001580181962y. https://www.osti.gov/servlets/purl/1479078.
@article{osti_1479078,
title = {Toplogical optimization of structures using Fourier representations},
author = {White, Daniel A. and Stowell, Mark L. and Tortorelli, Daniel A.},
abstractNote = {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 wellknown 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 meshindependent solution and to eliminate checkerboard instabilities. In image processing terms this is a lowpass 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. Furthermore, 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).},
doi = {10.1007/s001580181962y},
journal = {Structural and Multidisciplinary Optimization},
number = 3,
volume = 58,
place = {United States},
year = {2018},
month = {4}
}