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Discrete geometric homogenisation and inverse homogenisation of an elliptic operator

Citation

Donaldson, Roger David (2008) Discrete geometric homogenisation and inverse homogenisation of an elliptic operator. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechETD:etd-05212008-164705

Abstract

We show how to parameterise a homogenised conductivity in $R^2$ 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 $R^3$ are introduced.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:anisotropic; anisotropy; conductivity; convexity; Dirichlet Neumann map; discrete differential geometry; electric impedance tomography; FEM; finite element method; homogenization; inverse problems; metric; metric-based up-scaling; multi-scale; multiscale; upscaling; variational mesh generation; weighted Delaunay triangulation
Degree Grantor:California Institute of Technology
Division:Engineering and Applied Science
Major Option:Applied And Computational Mathematics
Thesis Availability:Mixed availability, specified at file level
Research Advisor(s):
  • Owhadi, Houman (advisor)
  • Desbrun, Mathieu (advisor)
Thesis Committee:
  • Owhadi, Houman (chair)
  • Marsden, Jerrold E.
  • Desbrun, Mathieu
  • Schroeder, Peter
  • Hou, Thomas Y.
Defense Date:2 May 2008
Author Email:rdonald (AT) acm.caltech.edu
Record Number:CaltechETD:etd-05212008-164705
Persistent URL:http://resolver.caltech.edu/CaltechETD:etd-05212008-164705
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:1928
Collection:CaltechTHESIS
Deposited By: Imported from ETD-db
Deposited On:30 May 2008
Last Modified:26 Dec 2012 02:44

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