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A conservation law formulation of nonlinear elasticity in general relativity

A conservation law formulation of nonlinear elasticity in general relativity
A conservation law formulation of nonlinear elasticity in general relativity
We present a practical framework for ideal hyperelasticity in numerical relativity. For this purpose, we recast the formalism of Carter and Quintana as a set of Eulerian conservation laws in an arbitrary 3+1 split of spacetime. The resulting equations are presented as an extension of the standard Valencia formalism for a perfect fluid, with additional terms in the stress–energy tensor, plus a set of kinematic conservation laws that evolve a configuration gradient ?Ai. We prove that the equations can be made symmetric hyperbolic by suitable constraint additions, at least in a neighbourhood of the unsheared state. We discuss the Newtonian limit of our formalism and its relation to a second formalism also used in Newtonian elasticity. We validate our framework by numerically solving a set of Riemann problems in Minkowski spacetime, as well as Newtonian ones from the literature.
0264-9381
015005-[53pp]
Gundlach, Carsten
586f1eb5-3185-4b2b-8656-c29c436040fc
Hawke, Ian
fc964672-c794-4260-a972-eaf818e7c9f4
Erickson, Stephanie E.
b0675078-ef6c-482a-a843-51dd820a4fa6
Gundlach, Carsten
586f1eb5-3185-4b2b-8656-c29c436040fc
Hawke, Ian
fc964672-c794-4260-a972-eaf818e7c9f4
Erickson, Stephanie E.
b0675078-ef6c-482a-a843-51dd820a4fa6

Gundlach, Carsten, Hawke, Ian and Erickson, Stephanie E. (2011) A conservation law formulation of nonlinear elasticity in general relativity. Classical and Quantum Gravity, 29 (1), 015005-[53pp]. (doi:10.1088/0264-9381/29/1/015005).

Record type: Article

Abstract

We present a practical framework for ideal hyperelasticity in numerical relativity. For this purpose, we recast the formalism of Carter and Quintana as a set of Eulerian conservation laws in an arbitrary 3+1 split of spacetime. The resulting equations are presented as an extension of the standard Valencia formalism for a perfect fluid, with additional terms in the stress–energy tensor, plus a set of kinematic conservation laws that evolve a configuration gradient ?Ai. We prove that the equations can be made symmetric hyperbolic by suitable constraint additions, at least in a neighbourhood of the unsheared state. We discuss the Newtonian limit of our formalism and its relation to a second formalism also used in Newtonian elasticity. We validate our framework by numerically solving a set of Riemann problems in Minkowski spacetime, as well as Newtonian ones from the literature.

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More information

Published date: 12 December 2011
Organisations: Applied Mathematics

Identifiers

Local EPrints ID: 202587
URI: https://eprints.soton.ac.uk/id/eprint/202587
ISSN: 0264-9381
PURE UUID: 01b5f6f7-0655-409e-9cf3-92bdbf82b535
ORCID for Carsten Gundlach: ORCID iD orcid.org/0000-0001-9585-5375
ORCID for Ian Hawke: ORCID iD orcid.org/0000-0003-4805-0309

Catalogue record

Date deposited: 08 Nov 2011 14:11
Last modified: 20 Jul 2018 00:34

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