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Mathematical Problems in Engineering
Volume 2006 (2006), Article ID 68695, 14 pages

Sequential laminates in multiple-state optimal design problems

Department of Mathematics, University of Zagreb, Bijenička cesta 30, Zagreb 10 000, Croatia

Received 23 November 2004; Revised 2 March 2005; Accepted 4 April 2005

Copyright © 2006 Nenad Antonić and Marko Vrdoljak. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


In the study of optimal design related to stationary diffusion problems with multiple-state equations, the description of the set H={(Aa1,...,Aam):AK(θ)} for given vectors a1,...,amd (m<d) is crucial. K(θ) denotes all composite materials (in the sense of homogenisation theory) with given local proportion θ of the first material. We prove that the boundary of H is attained by sequential laminates of rank at most m with matrix phase αI and core βI (α<β). Examples showing that the information on the rank of optimal laminate cannot be improved, as well as the fact that sequential laminates with matrix phase αI are preferred to those with matrix phase βI, are presented. This result can significantly reduce the complexity of optimality conditions, with obvious impact on numerical treatment, as demonstrated in a simple numerical example.