Discrete formulation for the dynamics of rods deforming in space

Ana Casimiro, César Rodrigo

Research output: Contribution to journalArticle

Abstract

The movement of rods in an Euclidean space can be described as a field theory on a principal bundle. The dynamics of a rod is governed by partial differential equations that may have a variational origin. If the corresponding smooth Lagrangian density is invariant by some group of transformations, there exist the corresponding conserved Noether currents. Generally, numerical schemes dealing with PDEs fail to reflect these conservation properties. We describe the main ingredients needed to create, from the smooth Lagrangian density, a variational principle for discrete motions of a discrete rod, with the corresponding conserved Noether currents. We describe all geometrical objects in terms of elements on the linear Atiyah bundle using a reduced forward difference operator. We show how this introduces a discrete Lagrangian density that models the discrete dynamics of a discrete rod. The presented tools are general enough to represent a discretization of any variational theory in principal bundles, and its simplicity allows us to perform an iterative integration algorithm to compute the discrete rod evolution in time, starting from any predefined configurations of all discrete rod elements at initial times.

Original languageEnglish
Article number092901
JournalJournal Of Mathematical Physics
Volume60
Issue number9
DOIs
Publication statusPublished - 1 Sep 2019

Fingerprint

rods
formulations
Formulation
bundles
Principal Bundle
Noether
Discrete Dynamics
Michael Francis Atiyah
Euclidean geometry
pulse detonation engines
Difference Operator
variational principles
Variational Principle
ingredients
partial differential equations
Numerical Scheme
Field Theory
Conservation
Euclidean space
conservation

Cite this

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title = "Discrete formulation for the dynamics of rods deforming in space",
abstract = "The movement of rods in an Euclidean space can be described as a field theory on a principal bundle. The dynamics of a rod is governed by partial differential equations that may have a variational origin. If the corresponding smooth Lagrangian density is invariant by some group of transformations, there exist the corresponding conserved Noether currents. Generally, numerical schemes dealing with PDEs fail to reflect these conservation properties. We describe the main ingredients needed to create, from the smooth Lagrangian density, a variational principle for discrete motions of a discrete rod, with the corresponding conserved Noether currents. We describe all geometrical objects in terms of elements on the linear Atiyah bundle using a reduced forward difference operator. We show how this introduces a discrete Lagrangian density that models the discrete dynamics of a discrete rod. The presented tools are general enough to represent a discretization of any variational theory in principal bundles, and its simplicity allows us to perform an iterative integration algorithm to compute the discrete rod evolution in time, starting from any predefined configurations of all discrete rod elements at initial times.",
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Discrete formulation for the dynamics of rods deforming in space. / Casimiro, Ana; Rodrigo, César.

In: Journal Of Mathematical Physics, Vol. 60, No. 9, 092901, 01.09.2019.

Research output: Contribution to journalArticle

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