Direct determination of the rotation in the polar decomposition of the deformation gradient by maximizing a Rayleigh quotient
Direct determination of the rotation in the polar decomposition of the deformation gradient by maximizing a Rayleigh quotient
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通过最大化瑞利商直接确定变形梯度极分解中的旋转
DOI：
10.1002/zamm.200310167
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发表时间：
2005
期刊：
影响因子：
--
通讯作者：
Claude Vallée
中科院分区：
文献类型：
--
作者：
Céline Bouby;D. Fortuné;W. Pietraszkiewicz;Claude Vallée
本文提出了一种新的有效的方法来确定变形梯度的极分解F = RU = VR中的旋转R和伸展U和V。该方法基于R的最小属性，以在欧几里得范数中具有与F的最小“距离”。所提出的方法不需要执行任何平方根和/或逆运算。每个F都有9个独立分量，我们将其与一个4 × 4对称无迹矩阵Q相关联。旋转由四元数参数描述，从四元数参数形成四向量X。它表明，X对应于R最大化XT QX在所有X满足XTX = 1。我们倾向于等价地最大化所有非零Y上的瑞利商YTQY/YTY，并通过随后的归一化$X = Y/\sqrt{Y^{T}Y}$来推导X。瑞利商的最大化是由共轭梯度算法进行的，所有的迭代步骤进行明确的封闭公式。数值算例表明了该方法的有效性和准确性。
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 $X = Y/\sqrt{Y^{T}Y}$. 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.