Evolution equation of Lie-type for finite deformations, and its time-discrete integration

Kód výsledku v IS VaVaI

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Výsledek na webu

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DOI - Digital Object Identifier

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Jazyk výsledku

angličtina

Název v původním jazyce

Evolution equation of Lie-type for finite deformations, and its time-discrete integration

Popis výsledku v původním jazyce

The theory of evolution equations of Lie-type analyses a class of systems of timedependent first-order ordinary differential equations on a Lie group (resp. homogeneous space), which are generated by vector fields related to a corresponding finite dimensional Lie algebra. Their interesting geometric features give rise to important tools, and have originated new mathematical techniques and notions used for investigating differential equations. Here, we shall identify such a type of evolution equation within solid mechanics wherein it describes the evolution of finite deformations on the space of all symmetric positive-definite matrices resp. on the general linear group. In fact, while the position and shape of a deformed body take place in the usual three-dimensional Euclidean space R^3, a corresponding progress of the deformation tensor C makes up a trajectory in the space of all symmetric positive-definite matrices – a negatively curved Riemannian symmetric manifold (a specific homogeneous space). We prove that a well-known relation between deformation rate delta C and symmetric velocity gradient d, via deformation gradient F, can actually be interpreted as an equation of Lie-type describing evolution of the deformation tensor C on the configuration space. The same applies to deformation gradient F, which evolves on the general linear group. As a consequence, this identification leads to geometrically consistent time-discrete integration schemes for finite deformation processes, such as the Runge-Kutta-Munthe-Kaas method, or also briefly mentioned the semi-discrete Magnus and Fer expansion methods.

Název v anglickém jazyce

Evolution equation of Lie-type for finite deformations, and its time-discrete integration

Popis výsledku anglicky

The theory of evolution equations of Lie-type analyses a class of systems of timedependent first-order ordinary differential equations on a Lie group (resp. homogeneous space), which are generated by vector fields related to a corresponding finite dimensional Lie algebra. Their interesting geometric features give rise to important tools, and have originated new mathematical techniques and notions used for investigating differential equations. Here, we shall identify such a type of evolution equation within solid mechanics wherein it describes the evolution of finite deformations on the space of all symmetric positive-definite matrices resp. on the general linear group. In fact, while the position and shape of a deformed body take place in the usual three-dimensional Euclidean space R^3, a corresponding progress of the deformation tensor C makes up a trajectory in the space of all symmetric positive-definite matrices – a negatively curved Riemannian symmetric manifold (a specific homogeneous space). We prove that a well-known relation between deformation rate delta C and symmetric velocity gradient d, via deformation gradient F, can actually be interpreted as an equation of Lie-type describing evolution of the deformation tensor C on the configuration space. The same applies to deformation gradient F, which evolves on the general linear group. As a consequence, this identification leads to geometrically consistent time-discrete integration schemes for finite deformation processes, such as the Runge-Kutta-Munthe-Kaas method, or also briefly mentioned the semi-discrete Magnus and Fer expansion methods.

Druh

C - Kapitola v odborné knize

CEP obor

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OECD FORD obor

10301 - Atomic, molecular and chemical physics (physics of atoms and molecules including collision, interaction with radiation, magnetic resonances, Mössbauer effect)

Projekt

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Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2017

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název knihy nebo sborníku

Emerging Concepts in Evolution Equations

ISBN

978-1-53610-861-3

Počet stran výsledku

30

Strana od-do

1-30

Počet stran knihy

80

Název nakladatele

Nova Science

Místo vydání

Hauppauge (NY)

Kód UT WoS kapitoly

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