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

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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

Hirofumi Notsu ^1, and Masato Kimura ^2,

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

Abstract
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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.

Keywords: finite element method, linear elasticity, Spring-block system, spring constant..

Mathematics Subject Classification: Primary: 74B05, 74G15, 74S05; Secondary: 74A4.

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

References:

[1]	T. Belytschko and T. Black, Elastic crack growth in finite elements with minimal remeshing, 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.
[2]	S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, Springer, New York, 2002. doi: 10.1007/978-1-4757-3658-8.
[3]	F. Camborde, C. Mariotti and F. V. Donzé, Numerical study of rock and concrete behaviour by discrete element modelling, Computers and Geotechnics, 27 (2000), 225-247. doi: 10.1016/S0266-352X(00)00013-6.
[4]	H. Chen, L. Wijerathne, M. Hori and T. Ichimura, Stability of dynamic growth of two anti-symmetric cracks using PDS-FEM, 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.
[5]	P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978.
[6]	J. M. Gere, Mechanics of Materials, Brooks/Cole-Thomson Learning, Belmont, CA, 2004.
[7]	M. Hori, K. Oguni and H. Sakaguchi, Proposal of FEM implemented with particle discretization for analysis of failure phenomena, Journal of the Mechanics and Physics of Solids, 53 (2005), 681-703. doi: 10.1016/j.jmps.2004.08.005.
[8]	M. Kimura and H. Notsu, A mathematical model of fracture phenomena on a spring-block system, Kyoto University RIMS Kokyuroku, 1848 (2013), 171-186.
[9]	J. Karátson and S. Korotov, An algebraic discrete maximum principle in Hilbert space with applications to nonlinear cooperative elliptic systems, SIAM Journal on Numerical Analysis, 47 (2009), 2518-2549. doi: 10.1137/080729566.
[10]	A. Munjiza, The Combined Finite-Discrete Element Method, John Wiley & Sons, Chichester, 2004. doi: 10.1002/0470020180.
[11]	H. Notsu and M. Tabata, A single-step characteristic-curve finite element scheme of second order in time for the incompressible Navier-Stokes equations, Journal of Scientific Computing, 38 (2009), 1-14. doi: 10.1007/s10915-008-9217-5.
[12]	A. Okabe, B. Boots, K. Sugihara and S.-N. Choi, Spatial Tessellation: Concepts and Applications of Voronoi Diagrams, John Wiley and Sons, Chichester, 1992.
[13]	G. Voronoi, Nouvelles applications des paramètres continus à la théorie des formes quadratiques, deuxième memoire, recherche sur les parallelloèdres primitifs, Journal für die Reine und Angewandte Mathematik, 134 (1908), 198-287.

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References:

[1]	T. Belytschko and T. Black, Elastic crack growth in finite elements with minimal remeshing, 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.
[2]	S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, Springer, New York, 2002. doi: 10.1007/978-1-4757-3658-8.
[3]	F. Camborde, C. Mariotti and F. V. Donzé, Numerical study of rock and concrete behaviour by discrete element modelling, Computers and Geotechnics, 27 (2000), 225-247. doi: 10.1016/S0266-352X(00)00013-6.
[4]	H. Chen, L. Wijerathne, M. Hori and T. Ichimura, Stability of dynamic growth of two anti-symmetric cracks using PDS-FEM, 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.
[5]	P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978.
[6]	J. M. Gere, Mechanics of Materials, Brooks/Cole-Thomson Learning, Belmont, CA, 2004.
[7]	M. Hori, K. Oguni and H. Sakaguchi, Proposal of FEM implemented with particle discretization for analysis of failure phenomena, Journal of the Mechanics and Physics of Solids, 53 (2005), 681-703. doi: 10.1016/j.jmps.2004.08.005.
[8]	M. Kimura and H. Notsu, A mathematical model of fracture phenomena on a spring-block system, Kyoto University RIMS Kokyuroku, 1848 (2013), 171-186.
[9]	J. Karátson and S. Korotov, An algebraic discrete maximum principle in Hilbert space with applications to nonlinear cooperative elliptic systems, SIAM Journal on Numerical Analysis, 47 (2009), 2518-2549. doi: 10.1137/080729566.
[10]	A. Munjiza, The Combined Finite-Discrete Element Method, John Wiley & Sons, Chichester, 2004. doi: 10.1002/0470020180.
[11]	H. Notsu and M. Tabata, A single-step characteristic-curve finite element scheme of second order in time for the incompressible Navier-Stokes equations, Journal of Scientific Computing, 38 (2009), 1-14. doi: 10.1007/s10915-008-9217-5.
[12]	A. Okabe, B. Boots, K. Sugihara and S.-N. Choi, Spatial Tessellation: Concepts and Applications of Voronoi Diagrams, John Wiley and Sons, Chichester, 1992.
[13]	G. Voronoi, Nouvelles applications des paramètres continus à la théorie des formes quadratiques, deuxième memoire, recherche sur les parallelloèdres primitifs, Journal für die Reine und Angewandte Mathematik, 134 (1908), 198-287.

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