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

Previous Article
Homogenization of a poro-elasticity model coupled with diffusive transport and a first order reaction for concrete
NHM Home
This Issue
Next Article
Mathematical modelling of a mushy region formation during sulphation of calcium carbonate

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
Related Papers

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 & 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. 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, (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. 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, 68 (2012), 10. doi: 10.2208/jscejam.68.10.
[5]	P. G. Ciarlet, The Finite Element Method for Elliptic Problems,, North-Holland, (1978).
[6]	J. M. Gere, Mechanics of Materials,, Brooks/Cole-Thomson Learning, (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. 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.
[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. doi: 10.1137/080729566.
[10]	A. Munjiza, The Combined Finite-Discrete Element Method,, John Wiley & Sons, (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. 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, (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.

show all references

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. 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, (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. 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, 68 (2012), 10. doi: 10.2208/jscejam.68.10.
[5]	P. G. Ciarlet, The Finite Element Method for Elliptic Problems,, North-Holland, (1978).
[6]	J. M. Gere, Mechanics of Materials,, Brooks/Cole-Thomson Learning, (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. 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.
[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. doi: 10.1137/080729566.
[10]	A. Munjiza, The Combined Finite-Discrete Element Method,, John Wiley & Sons, (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. 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, (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.

[1]	M. Pellicer, J. Solà-Morales. Spectral analysis and limit behaviours in a spring-mass system. Communications on Pure & Applied Analysis, 2008, 7 (3) : 563-577. doi: 10.3934/cpaa.2008.7.563
[2]	J. Samuel Jiang, Hans G. Kaper, Gary K Leaf. Hysteresis in layered spring magnets. Discrete & Continuous Dynamical Systems - B, 2001, 1 (2) : 219-232. doi: 10.3934/dcdsb.2001.1.219
[3]	So-Hsiang Chou. An immersed linear finite element method with interface flux capturing recovery. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2343-2357. doi: 10.3934/dcdsb.2012.17.2343
[4]	Zhong-Ci Shi, Xuejun Xu, Zhimin Zhang. The patch recovery for finite element approximation of elasticity problems under quadrilateral meshes. Discrete & Continuous Dynamical Systems - B, 2008, 9 (1) : 163-182. doi: 10.3934/dcdsb.2008.9.163
[5]	Cornel M. Murea, H. G. E. Hentschel. A finite element method for growth in biological development. Mathematical Biosciences & Engineering, 2007, 4 (2) : 339-353. doi: 10.3934/mbe.2007.4.339
[6]	Martin Burger, José A. Carrillo, Marie-Therese Wolfram. A mixed finite element method for nonlinear diffusion equations. Kinetic & Related Models, 2010, 3 (1) : 59-83. doi: 10.3934/krm.2010.3.59
[7]	Gero Friesecke, Karsten Matthies. Geometric solitary waves in a 2D mass-spring lattice. Discrete & Continuous Dynamical Systems - B, 2003, 3 (1) : 105-144. doi: 10.3934/dcdsb.2003.3.105
[8]	Maja Miletić, Dominik Stürzer, Anton Arnold. An Euler-Bernoulli beam with nonlinear damping and a nonlinear spring at the tip. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 3029-3055. doi: 10.3934/dcdsb.2015.20.3029
[9]	Zhangxin Chen, Qiaoyuan Jiang, Yanli Cui. Locking-free nonconforming finite elements for planar linear elasticity. Conference Publications, 2005, 2005 (Special) : 181-189. doi: 10.3934/proc.2005.2005.181
[10]	Fang Chen, Ning Gao, Yao- Lin Jiang. On product-type generalized block AOR method for augmented linear systems. Numerical Algebra, Control & Optimization, 2012, 2 (4) : 797-809. doi: 10.3934/naco.2012.2.797
[11]	Nora Aïssiouene, Marie-Odile Bristeau, Edwige Godlewski, Jacques Sainte-Marie. A combined finite volume - finite element scheme for a dispersive shallow water system. Networks & Heterogeneous Media, 2016, 11 (1) : 1-27. doi: 10.3934/nhm.2016.11.1
[12]	Binjie Li, Xiaoping Xie, Shiquan Zhang. New convergence analysis for assumed stress hybrid quadrilateral finite element method. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2831-2856. doi: 10.3934/dcdsb.2017153
[13]	Kun Wang, Yinnian He, Yueqiang Shang. Fully discrete finite element method for the viscoelastic fluid motion equations. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 665-684. doi: 10.3934/dcdsb.2010.13.665
[14]	Junjiang Lai, Jianguo Huang. A finite element method for vibration analysis of elastic plate-plate structures. Discrete & Continuous Dynamical Systems - B, 2009, 11 (2) : 387-419. doi: 10.3934/dcdsb.2009.11.387
[15]	Donald L. Brown, Vasilena Taralova. A multiscale finite element method for Neumann problems in porous microstructures. Discrete & Continuous Dynamical Systems - S, 2016, 9 (5) : 1299-1326. doi: 10.3934/dcdss.2016052
[16]	Qingping Deng. A nonoverlapping domain decomposition method for nonconforming finite element problems. Communications on Pure & Applied Analysis, 2003, 2 (3) : 297-310. doi: 10.3934/cpaa.2003.2.297
[17]	Runchang Lin. A robust finite element method for singularly perturbed convection-diffusion problems. Conference Publications, 2009, 2009 (Special) : 496-505. doi: 10.3934/proc.2009.2009.496
[18]	Caterina Calgaro, Meriem Ezzoug, Ezzeddine Zahrouni. Stability and convergence of an hybrid finite volume-finite element method for a multiphasic incompressible fluid model. Communications on Pure & Applied Analysis, 2018, 17 (2) : 429-448. doi: 10.3934/cpaa.2018024
[19]	Christos V. Nikolopoulos, Georgios E. Zouraris. Numerical solution of a non-local elliptic problem modeling a thermistor with a finite element and a finite volume method. Conference Publications, 2007, 2007 (Special) : 768-778. doi: 10.3934/proc.2007.2007.768
[20]	Toyohiko Aiki. On the existence of a weak solution to a free boundary problem for a model of a shape memory alloy spring. Discrete & Continuous Dynamical Systems - S, 2012, 5 (1) : 1-13. doi: 10.3934/dcdss.2012.5.1

2017 Impact Factor: 1.187

American Institute of Mathematical Sciences