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Article Dans Une Revue Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik Année : 2005

Direct determination of the rotation in the polar decomposition of the deformation gradient by maximizing a Rayleigh quotient

Céline Bouby
Danielle Fortune
  • Fonction : Auteur
W. Pietraszkiewicz
  • Fonction : Auteur
Claude Vallee
  • Fonction : Auteur

Résumé

We develop a new effective method of determining the rotation R and the stretches U and V in the polar decomposition F = RU = VR of the deformation gradient. The method is based on a minimum property of R to have the smallest distance from F in the Euclidean norm. The proposed method does not require to perform any square root and/or inverse operations. With each F having nine independent components we associate a 4 × 4 symmetric traceless matrix Q. The rotation is described by quaternion parameters from which a quadrivector X is formed. It is shown that X corresponding to R maximizes XT QX over all X satisfying XTX = 1. We prefer to equivalently maximize the Rayleigh quotient YTQY/YTY over all non-vanishing Y and to deduce X by subsequent normalization . The maximization of the Rayleigh quotient is performed by a conjugate gradient algorithm with all iterative steps carried out by explicit closed formulae. Efficiency and accuracy of the method is illustrated by a numerical example.

Dates et versions

hal-00137132 , version 1 (16-03-2007)

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Citer

Céline Bouby, Danielle Fortune, W. Pietraszkiewicz, Claude Vallee. Direct determination of the rotation in the polar decomposition of the deformation gradient by maximizing a Rayleigh quotient. Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, 2005, 85 (3), pp.155-162. ⟨10.1002/zamm.200310167⟩. ⟨hal-00137132⟩

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