Positive definiteness of paired symmetric tensors and elasticity tensors
Positive definiteness of paired symmetric tensors and elasticity tensors
复制标题
配对对称张量和弹性张量的正定性
DOI：
10.1016/j.cam.2018.01.025
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发表时间：
2017-05
影响因子：
2.4
通讯作者：
Liqun Qi
中科院分区：
文献类型：
--
作者：
Zhenghai Huang;Liqun Qi
本文考虑高阶成对对称张量和强成对对称张量。固体力学中的弹性张量和高阶弹性张量都是强成对对称张量。一个（强）成对对称张量称为正定的，如果由它定义的齐次多项式是正定的。弹性力学和高阶弹性张量的正定性是固体力学中的强椭圆性，它在非线性弹性理论中占有重要地位。本文主要研究四阶三维和六阶三维（强）成对对称张量的正定性。我们首先证明了所涉及的（强）成对对称张量是正定的当且仅当它的最小M-特征值是正的。其次，我们提出了几个充分必要条件下，有关的（强）成对对称张量是正定的。第三，研究了由四阶三维或六阶三维（强）成对对称张量定义的齐次多项式可以写成多项式平方和的条件，并进一步提出了判断相关（强）成对对称张量是否正定的几个必要和/或充分条件。第四，利用半定松弛方法，提出了一种计算四阶三维（强）对称配对张量的最小M-特征值的序列半定规划方法，并利用该方法检验了该张量的正定性。初步的数值结果证实了我们的理论研究结果。
In this paper, we consider higher order paired symmetric tensors and strongly paired symmetric tensors. Elasticity tensors and higher order elasticity tensors in solid mechanics are strongly paired symmetric tensors. A (strongly) paired symmetric tensor is said to be positive definite if the homogeneous polynomial defined by it is positive definite. Positive definiteness of elasticity and higher order elasticity tensors is strong ellipticity in solid mechanics, which plays an important role in nonlinear elasticity theory. We mainly investigate positive definiteness of fourth order three dimensional and sixth order three dimensional (strongly) paired symmetric tensors. We first show that the concerned (strongly) paired symmetric tensor is positive definite if and only if its smallest M-eigenvalue is positive. Second, we propose several necessary and sufficient conditions under which the concerned (strongly) paired symmetric tensor is positive definite. Third, we study the conditions under which the homogeneous polynomial defined by a fourth order three dimensional or sixth order three dimensional (strongly) paired symmetric tensor can be written as a sum of squares of polynomials, and further, propose several necessary and/or sufficient conditions to judge whether the concerned (strongly) paired symmetric tensors are positive definite or not. Fourth, by using semidefinite relaxation we propose a sequential semidefinite programming method to compute the smallest M-eigenvalue of a fourth order three dimensional (strongly) paired symmetric tensor, by which we can check positive definiteness of the concerned tensor. The preliminary numerical results confirm our theoretical findings.
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DOI：
10.1002/zamm.201500167
发表时间：
2016-09
期刊：
ZAMM ‐ Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik
影响因子：
--
作者：
L. Zubov;A. Rudev
通讯作者：
L. Zubov;A. Rudev
DOI：
10.1016/j.jsc.2012.10.001
发表时间：
2011-09
期刊：
--
影响因子：
--
作者：
Shenglong Hu;Zhenghai Huang;C. Ling;L. Qi
通讯作者：
Shenglong Hu;Zhenghai Huang;C. Ling;L. Qi
影响因子：
0.8
作者：
B. Reznick
通讯作者：
B. Reznick
影响因子：
9.8
作者：
de Jong, Maarten;Chen, Wei;Angsten, Thomas;Jain, Anubhav;Notestine, Randy;Gamst, Anthony;Sluiter, Marcel;Ande, Chaitanya Krishna;van der Zwaag, Sybrand;Plata, Jose J.;Toher, Cormac;Curtarolo, Stefano;Ceder, Gerbrand;Persson, Kristin A.;Asta, Mark
通讯作者：
Asta, Mark
影响因子：
2
作者：
Y. Wang;M. Aron
通讯作者：
Y. Wang;M. Aron