Discrete variational Lie group formulation of geometrically exact beam dynamics

Abstract Recently, the authors proposed a geometrically exact beam finite element formulation on the Lie group SE(3). Some important numerical and theoretical aspects leading to a computationally efficient strategy were obtained. For instance, the formulation leads to invariant equilibrium equations under rigid body motions and a locking free element. In this paper we discuss some important aspects of this formulation. The invariance property of the equilibrium equations under rigid body motions is discussed and brought out in simple analytical examples. The discretization method based on the exponential map is recalled and a geometric interpretation is given. Special attention is also dedicated to the consistent interpolation of the velocities.

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Multisymplectic variational integrators and space/time symplecticity

Analysis and Applications ◽

10.1142/s0219530515500025 ◽

2016 ◽

Vol 14 (03) ◽

pp. 341-391 ◽

Cited By ~ 7

Author(s):

François Demoures ◽

François Gay-Balmaz ◽

Tudor S. Ratiu

Keyword(s):

Lie Group ◽

Numerical Scheme ◽

Variational Integrators ◽

Beam Model ◽

Structure Preserving ◽

Dimensional Spacetime ◽

Geometrically Exact Beam ◽

Temporal And Spatial ◽

Group Symmetries

Multisymplectic variational integrators are structure-preserving numerical schemes especially designed for PDEs derived from covariant spacetime Hamilton principles. The goal of this paper is to study the properties of the temporal and spatial discrete evolution maps obtained from a multisymplectic numerical scheme. Our study focuses on a (1+1)-dimensional spacetime discretized by triangles, but our approach carries over naturally to more general cases. In the case of Lie group symmetries, we explore the links between the discrete Noether theorems associated to the multisymplectic spacetime discretization and to the temporal and spatial discrete evolution maps, and emphasize the role of boundary conditions. We also consider in detail the case of multisymplectic integrators on Lie groups. Our results are illustrated with the numerical example of a geometrically exact beam model.

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EFFECTS OF CHEMICAL REACTION AND SLIP IN THE BOUNDARY LAYER OF MHD NANOFLUID FLOW THROUGH A SEMI-INFINITE STRETCHING SHEET WITH THERMOPHORESIS AND BROWNIAN MOTION: THE LIE GROUP ANALYSIS

Nanoscience and Technology An International Journal ◽

10.1615/nanoscitechnolintj.2018025363 ◽

2018 ◽

Vol 9 (1) ◽

pp. 47-68 ◽

Cited By ~ 1

Author(s):

Sawan Kumar Rawat ◽

Alok Kumar Pandey ◽

Manoj Kumar

Keyword(s):

Boundary Layer ◽

Brownian Motion ◽

Chemical Reaction ◽

Lie Group ◽

Stretching Sheet ◽

Group Analysis ◽

Lie Group Analysis ◽

Nanofluid Flow ◽

And Brownian Motion ◽

Flow Through

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Symmetry properties of fractional order transport equations

Proceedings of the Mavlyutov Institute of Mechanics ◽

10.21662/uim2012.1.010 ◽

2012 ◽

Vol 9 (1) ◽

pp. 59-64

Author(s):

R.K. Gazizov ◽

A.A. Kasatkin ◽

S.Yu. Lukashchuk

Keyword(s):

Differential Equations ◽

Fractional Differential Equations ◽

Lie Group ◽

Initial Conditions ◽

Group Analysis ◽

Equivalence Transformation ◽

Point Change ◽

Infinitesimal Operator ◽

Symmetry Properties ◽

Fractional Differential

In the paper some features of applying Lie group analysis methods to fractional differential equations are considered. The problem related to point change of variables in the fractional differentiation operator is discussed and some general form of transformation that conserves the form of Riemann-Liouville fractional operator is obtained. The prolongation formula for extending an infinitesimal operator of a group to fractional derivative with respect to arbitrary function is presented. Provided simple example illustrates the necessity of considering both local and non-local symmetries for fractional differential equations in particular cases including the initial conditions. The equivalence transformation forms for some fractional differential equations are discussed and results of group classification of the wave-diffusion equation are presented. Some examples of constructing particular exact solutions of fractional transport equation are given, based on the Lie group methods and the method of invariant subspaces.

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