Discrete Geometric Homogenisation and Inverse Homogenisation of an Elliptic Operator

Author: Donaldson, Roger David

Year: 2008

Degree: Dissertation (Ph.D.)

Advisors: Owhadi, Houman; Desbrun, Mathieu

Committee Members: Owhadi, Houman; Marsden, Jerrold E.; Desbrun, Mathieu; Schroeder, Peter; Hou, Thomas Y.

Option: Applied And Computational Mathematics

Abstract

We show how to parameterise a homogenised conductivity in R² by a scalar function s(x), despite the fact that the conductivity parameter in the related up-scaled elliptic operator is typically tensor valued. Ellipticity of the operator is equivalent to strict convexity of s(x), and with consideration to mesh connectivity, this equivalence extends to discrete parameterisations over triangulated domains. We apply the parameterisation in three contexts: (i) sampling s(x) produces a family of stiffness matrices representing the elliptic operator over a hierarchy of scales; (ii) the curvature of s(x) directs the construction of meshes well-adapted to the anisotropy of the operator, improving the conditioning of the stiffness matrix and interpolation properties of the mesh; and (iii) using electric impedance tomography to reconstruct s(x) recovers the up-scaled conductivity, which while anisotropic, is unique. Extensions of the parameterisation to R³ are introduced.

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