The construction of periodic unfolding operators on some compact Riemannian manifolds

Sören Dobberschütz; Michael Böhm

doi:10.1515/apam-2013-0013

The construction of periodic unfolding operators on some compact Riemannian manifolds

Sören Dobberschütz, Michael Böhm

Kemisk Institut

1 Citationer (Scopus)

Abstract

The notion of periodic unfolding has become a standard tool in the theory of periodic homogenization. However, all the results obtained so far are only applicable to the "flat" Euclidean space R ⁿ. In this paper, we present a generalization of the method of periodic unfolding applicable to structures defined on certain compact Riemannian manifolds. While many results known from unfolding in domains of R ⁿ can be recovered, for the unfolding of gradients a transport operator has to be defined. This operator connects vector fields on the manifold and in the reference cell, which allows for the formulation of general two-scale problems. We illustrate the use of the new unfolding technique with a simple elliptic model-problem.

Originalsprog	Engelsk
Tidsskrift	Advances in Pure and Applied Mathematics
Vol/bind	5
Udgave nummer	1
Sider (fra-til)	31-45
Antal sider	15
ISSN	1867-1152
DOI	https://doi.org/10.1515/apam-2013-0013
Status	Udgivet - 2014

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10.1515/apam-2013-0013

Citationsformater

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AU - Dobberschütz, Sören

AU - Böhm, Michael

PY - 2014

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N2 - The notion of periodic unfolding has become a standard tool in the theory of periodic homogenization. However, all the results obtained so far are only applicable to the "flat" Euclidean space R n. In this paper, we present a generalization of the method of periodic unfolding applicable to structures defined on certain compact Riemannian manifolds. While many results known from unfolding in domains of R n can be recovered, for the unfolding of gradients a transport operator has to be defined. This operator connects vector fields on the manifold and in the reference cell, which allows for the formulation of general two-scale problems. We illustrate the use of the new unfolding technique with a simple elliptic model-problem.

AB - The notion of periodic unfolding has become a standard tool in the theory of periodic homogenization. However, all the results obtained so far are only applicable to the "flat" Euclidean space R n. In this paper, we present a generalization of the method of periodic unfolding applicable to structures defined on certain compact Riemannian manifolds. While many results known from unfolding in domains of R n can be recovered, for the unfolding of gradients a transport operator has to be defined. This operator connects vector fields on the manifold and in the reference cell, which allows for the formulation of general two-scale problems. We illustrate the use of the new unfolding technique with a simple elliptic model-problem.

KW - homogenization

KW - Periodic unfolding

KW - Riemannian manifolds

U2 - 10.1515/apam-2013-0013

DO - 10.1515/apam-2013-0013

M3 - Journal article

AN - SCOPUS:84896379968

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VL - 5

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EP - 45

JO - Advances in Pure and Applied Mathematics

JF - Advances in Pure and Applied Mathematics

IS - 1

ER -

The construction of periodic unfolding operators on some compact Riemannian manifolds

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