Simo-Vu Quoc rods using Clifford algebra

Abstract

SUMMARYWe present an alternative derivation of Simo and Vu Quoc's numerical algorithm 1 for modelling the non-linear dynamic behaviour of rods. The original derivation uses di erential topology, describing large rotations using the Lie group SO(3) and Lie algebra so(3), but resorting to quaternions for the numerical implementation. The new derivation uses Cli ord or geometric algebra as developed by Hestenes 2; 3 for both formulation and implementation. We contend that the new approach is considerably simpler to follow, and thereby allows alternative modelling strategies to be more readily investigated. The new description is also novel in that all formulae for rotational kinematics are applicable in a Euclidean space of any dimension. Copyright ? 1999 John Wiley & Sons, Ltd.KEY WORDS: Cli ord algebra; geometric algebra; rods; large rotations BACKGROUNDOver the past few decades the foundations of classical mechanics have been rigorously reformulated in terms of di erential topology. The major contributors to this process have been Abraham and Marsden in America, with their deÿnitive tome 'Foundations of Classical Mechanics' 4 and Arnol'd in Russia, whose work in this area is typiÿed by his book 'Mathematical Methods of Classical Mechanics'. 5 The early contributions focused on classical particle mechanics and rigid body mechanics. A subsequent advance was the publication of Marsden and Hughes 'Mathematical Foundations of Elasticity' 6 where the di erential topology reformulation was extended to elastic continuum mechanics, providing a fundamental alternative to the earlier description based upon classical tensor analysis as exempliÿed by the works of Truesdell, Noll, Gurtin, Naghdi and others in Vols. III, IV and VI of 'Handbuch Der Physik'. 7 This reformulation has continued apace into the realms of computational continuum mechanics most notably through the work of Simo and coworkers, in particular, on the ÿnite element analysis of the large displacement behaviour of rods and shells (e.g. References 1,[8][9][10][11]. This presents a particular problem in that di erential topology rarely forms part of an engineering education, and thus the foundations of applied mechanics are now written in a language that is impenetrable to most engineers. * Correspondence to: F. A. McRobie, Department of Engineering, Cambridge University, Trumpington Street, Cambridge CB2 lPZ, U.K. E-mail: fam@eng.cam.ac.uk Over the last ten years an alternative reformulation has been pioneered by Hestenes and co-workers who, building upon Cli ord algebra and Grassman calculus, have developed a mathematical framework they refer to as Geometric Algebra. In the introductory 'New Foundations of Classical Mechanics' 2 and in the more formal 'Cli ord Algebra to Geometric Calculus' 3 the foundations of this alternative approach are presented. Again any physical emphasis is on particle mechanics. The initial extension of the approach to continuum mechanics has been commenced by Gull et al. 12 In this paper we provide the initial extensions to co...

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