skip to main content


Title: An n -material thresholding method for improving integerness of solutions in topology optimization

It is common in solving topology optimization problems to replace an integer-valued characteristic function design field with the material volume fraction field, a real-valued 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 two-material 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 gradient-based optimization algorithms. Finally, we present results, which show synergistic effects when usedmore » with Solid Isotropic Material with Penalty and Rational Approximation of Material Properties material interpolation functions, popular methods of ensuring integerness of solutions.« less
ORCiD logo [1] ;  [1]
  1. Univ. of Illinois, Urbana-Champaign, IL (United States). Dept. of Mechanical Science & Engineering
Publication Date:
Report Number(s):
Journal ID: ISSN 0029-5981; TRN: US1702980
Grant/Contract Number:
Accepted Manuscript
Journal Name:
International Journal for Numerical Methods in Engineering
Additional Journal Information:
Journal Volume: 108; Journal Issue: 12; Journal ID: ISSN 0029-5981
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
42 ENGINEERING; 97 MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE; topology design; elasticity; finite element methods; partition of unity
OSTI Identifier: