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Title: 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 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. Furthermore, this imposes a minimal length scale on the problem, eliminates checkerboard instabilities, and also reduces the number of decision variablesmore » by over 100 × (two dimensions) or 1000 × (three dimensions).« less
Authors:
 [1] ;  [1] ;  [1]
  1. Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
Publication Date:
Report Number(s):
LLNL-JRNL-736779
Journal ID: ISSN 1615-147X; 889675
Grant/Contract Number:
AC52-07NA27344
Type:
Accepted Manuscript
Journal Name:
Structural and Multidisciplinary Optimization
Additional Journal Information:
Journal Volume: 58; Journal Issue: 3; Journal ID: ISSN 1615-147X
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/s00158-018-1962-y.
White, Daniel A., Stowell, Mark L., & Tortorelli, Daniel A.. Toplogical optimization of structures using Fourier representations. United States. doi:10.1007/s00158-018-1962-y.
White, Daniel A., Stowell, Mark L., and Tortorelli, Daniel A.. 2018. "Toplogical optimization of structures using Fourier representations". United States. doi:10.1007/s00158-018-1962-y. 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 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. 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/s00158-018-1962-y},
journal = {Structural and Multidisciplinary Optimization},
number = 3,
volume = 58,
place = {United States},
year = {2018},
month = {4}
}