Premium
An n ‐material thresholding method for improving integerness of solutions in topology optimization
Author(s) -
Watts Seth,
Tortorelli Daniel A.
Publication year - 2016
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.5265
Subject(s) - topology optimization , interpolation (computer graphics) , mathematics , barycentric coordinate system , isotropy , topology (electrical circuits) , mathematical optimization , field (mathematics) , simplex , thresholding , algorithm , computer science , geometry , image (mathematics) , physics , quantum mechanics , combinatorics , artificial intelligence , finite element method , pure mathematics , thermodynamics
Summary 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. 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. Copyright © 2016 John Wiley & Sons, Ltd.