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

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 x 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 X(T)QX over all X satisfying (XX)-X-T = 1. We prefer to equivalently maximize the Rayleigh quotient Y(T)QY/(YY)-Y-T over all non-vanishing Y and to deduce X by subsequent normalization X = Y/root(YY)-Y-T. 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.