An n material thresholding method for improving integerness of solutions in topology optimization
It is common in solving topology optimization problems to replace an integervalued characteristic function design field with the material volume fraction field, a realvalued approximation of the design field that permits "fictitious" mixtures of materials during intermediate iterations in the optimization process. This is reasonable so long as one can interpolate properties for such materials and so long as the final design is integer valued. For this purpose, we present a method for smoothly thresholding the volume fractions of an arbitrary number of material phases which specify the design. This method is trivial for twomaterial design problems, for example, the canonical topology design problem of specifying the presence or absence of a single material within a domain, but it becomes more complex when three or more materials are used, as often occurs in material design problems. We take advantage of the similarity in properties between the volume fractions and the barycentric coordinates on a simplex to derive a thresholding, method which is applicable to an arbitrary number of materials. As we show in a sensitivity analysis, this method has smooth derivatives, allowing it to be used in gradientbased optimization algorithms. Finally, we present results, which show synergistic effects when usedmore »
 Authors:

^{[1]}
;
^{[1]}
 Univ. of Illinois, UrbanaChampaign, IL (United States). Dept. of Mechanical Science & Engineering
 Publication Date:
 Report Number(s):
 LLNLJRNL683462
Journal ID: ISSN 00295981; TRN: US1702980
 Grant/Contract Number:
 AC5207NA27344
 Type:
 Accepted Manuscript
 Journal Name:
 International Journal for Numerical Methods in Engineering
 Additional Journal Information:
 Journal Volume: 108; Journal Issue: 12; Journal ID: ISSN 00295981
 Publisher:
 Wiley
 Research Org:
 Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
 Sponsoring Org:
 USDOE; Defense Advanced Research Projects Agency (DARPA)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 42 ENGINEERING; 97 MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE; topology design; elasticity; finite element methods; partition of unity
 OSTI Identifier:
 1400087
Watts, Seth, and Tortorelli, Daniel A. An n material thresholding method for improving integerness of solutions in topology optimization. United States: N. p.,
Web. doi:10.1002/nme.5265.
Watts, Seth, & Tortorelli, Daniel A. An n material thresholding method for improving integerness of solutions in topology optimization. United States. doi:10.1002/nme.5265.
Watts, Seth, and Tortorelli, Daniel A. 2016.
"An n material thresholding method for improving integerness of solutions in topology optimization". United States.
doi:10.1002/nme.5265. https://www.osti.gov/servlets/purl/1400087.
@article{osti_1400087,
title = {An n material thresholding method for improving integerness of solutions in topology optimization},
author = {Watts, Seth and Tortorelli, Daniel A.},
abstractNote = {It is common in solving topology optimization problems to replace an integervalued characteristic function design field with the material volume fraction field, a realvalued approximation of the design field that permits "fictitious" mixtures of materials during intermediate iterations in the optimization process. This is reasonable so long as one can interpolate properties for such materials and so long as the final design is integer valued. For this purpose, we present a method for smoothly thresholding the volume fractions of an arbitrary number of material phases which specify the design. This method is trivial for twomaterial design problems, for example, the canonical topology design problem of specifying the presence or absence of a single material within a domain, but it becomes more complex when three or more materials are used, as often occurs in material design problems. We take advantage of the similarity in properties between the volume fractions and the barycentric coordinates on a simplex to derive a thresholding, method which is applicable to an arbitrary number of materials. As we show in a sensitivity analysis, this method has smooth derivatives, allowing it to be used in gradientbased optimization algorithms. Finally, we present results, which show synergistic effects when used with Solid Isotropic Material with Penalty and Rational Approximation of Material Properties material interpolation functions, popular methods of ensuring integerness of solutions.},
doi = {10.1002/nme.5265},
journal = {International Journal for Numerical Methods in Engineering},
number = 12,
volume = 108,
place = {United States},
year = {2016},
month = {4}
}