二値の熱伝導率を持つ領域の第一固有値に対する最適配置(<特集>離散幾何学から提案する新物質創成・物性発現の解明)

概要

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Eigenvalue optimization of the heat equation with spatially inhomogeneous conduction coefficient is concerned. Under the volume constraint, justification of heat conducting domain minimizing and maximizing eigenvalues can be achieved via well-known topology optimization method, the level set method. We consider such optimization problem on various shaped domains with plural boundary conditions, not only for spatially inhomogeneous heat diffusion operator but also the generalized eigenvalue problem of Laplacian which corresponds to the optimization of fundamental frequency on the composite membrane. Our numerical observations show that, in both cases, there are several correspondences between the optimizing shape of conducting domain and eigenfunctions, all of which are independent of the dimension, shape of whole domain and boundary conditions of eigenvalue problem. These numerical observations will be mathematically justified in near future.
2013-12-24