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December  2014, 9(4): 617-634. doi: 10.3934/nhm.2014.9.617

Symmetry and positive definiteness of the tensor-valued spring constant derived from P1-FEM for the equations of linear elasticity

1. 

Waseda Institute for Advanced Study, Waseda University, 3-4-1, Okubo, Shinjuku, Tokyo 169-8555, Japan

2. 

Institute of Science and Engineering, Kanazawa University, Kakuma, Kanazawa 920-1192, Japan

Received  July 2014 Revised  September 2014 Published  December 2014

We study spring-block systems which are equivalent to the P1-finite element methods for the linear elliptic partial differential equation of second order and for the equations of linear elasticity. Each derived spring-block system is consistent with the original partial differential equation, since it is discretized by P1-FEM. Symmetry and positive definiteness of the scalar and tensor-valued spring constants are studied in two dimensions. Under the acuteness condition of the triangular mesh, positive definiteness of the scalar spring constant is obtained. In case of homogeneous linear elasticity, we show the symmetry of the tensor-valued spring constant in the two dimensional case. For isotropic elastic materials, we give a necessary and sufficient condition for the positive definiteness of the tensor-valued spring constant. Consequently, if Poisson's ratio of the elastic material is small enough, like concrete, we can construct a consistent spring-block system with positive definite tensor-valued spring constant.
Citation: Hirofumi Notsu, Masato Kimura. Symmetry and positive definiteness of the tensor-valued spring constant derived from P1-FEM for the equations of linear elasticity. Networks & Heterogeneous Media, 2014, 9 (4) : 617-634. doi: 10.3934/nhm.2014.9.617
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show all references

References:
[1]

International Journal for Numerical Methods in Engineering, 45 (1999), 601-620. doi: 10.1002/(SICI)1097-0207(19990620)45:5<601::AID-NME598>3.0.CO;2-S.  Google Scholar

[2]

Springer, New York, 2002. doi: 10.1007/978-1-4757-3658-8.  Google Scholar

[3]

Computers and Geotechnics, 27 (2000), 225-247. doi: 10.1016/S0266-352X(00)00013-6.  Google Scholar

[4]

Journal of Japan Society of Civil Engineers, Division A: Structural Engineering/Earthquake Engineering & Applied Mechanics, 68 (2012), 10-17. doi: 10.2208/jscejam.68.10.  Google Scholar

[5]

North-Holland, Amsterdam, 1978.  Google Scholar

[6]

Brooks/Cole-Thomson Learning, Belmont, CA, 2004. Google Scholar

[7]

Journal of the Mechanics and Physics of Solids, 53 (2005), 681-703. doi: 10.1016/j.jmps.2004.08.005.  Google Scholar

[8]

Kyoto University RIMS Kokyuroku, 1848 (2013), 171-186. Google Scholar

[9]

SIAM Journal on Numerical Analysis, 47 (2009), 2518-2549. doi: 10.1137/080729566.  Google Scholar

[10]

John Wiley & Sons, Chichester, 2004. doi: 10.1002/0470020180.  Google Scholar

[11]

Journal of Scientific Computing, 38 (2009), 1-14. doi: 10.1007/s10915-008-9217-5.  Google Scholar

[12]

John Wiley and Sons, Chichester, 1992.  Google Scholar

[13]

Journal für die Reine und Angewandte Mathematik, 134 (1908), 198-287. Google Scholar

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