On the comparison of material interpolation schemes and optimal composite properties in plane shape optimization
 Grzegorz Dzierżanowski
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Abstract
This paper deals with the classical problem of material distribution for minimal compliance of twodimensional structures loaded inplane. The main objective of the research consists in investigating the properties of the exact solution to the minimal compliance problem and incorporating them into an approximate solidvoid interpolation model. Consequently, a proposition of a constitutive relation for a porous material arise. The nonsmoothness of stress energy functional known from the approach based on homogenization may be thus avoided which is beneficial for the numerical implementation of the scheme. It is next shown that the simplified variant of the proposed formula justifies and generalizes the RAMP (Rational Approximation of Material Properties) interpolation model of Stolpe and Svanberg (Struct Multidisc Optim 22:116–124). In the second part of the paper, explicit formulae for function θ: Ω → [0, 1] describing the distribution of basic isotropic material in the design space Ω ∈ ℝ^{2} are derived for various interpolation schemes by the requirement of optimality imposed at each x ∈ Ω. Theoretical considerations are illustrated by a code written in MATLAB for typical optimization problems of a cantilever and MBB beam.
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 Title
 On the comparison of material interpolation schemes and optimal composite properties in plane shape optimization
 Open Access
 Available under Open Access This content is freely available online to anyone, anywhere at any time.
 Journal

Structural and Multidisciplinary Optimization
Volume 46, Issue 5 , pp 693710
 Cover Date
 20121101
 DOI
 10.1007/s0015801207882
 Print ISSN
 1615147X
 Online ISSN
 16151488
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 Topology optimization
 Minimal compliance
 Twodimensional elasticity
 Interpolation scheme
 Effective properties
 Industry Sectors
 Authors

 Grzegorz Dzierżanowski ^{(1)}
 Author Affiliations

 1. Department of Structural Mechanics and Computer Aided Engineering, Warsaw University of Technology, al. Armii Ludowej 16, 00637, Warsaw, Poland