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

Hirofumi Notsu; Masato Kimura; Hirofumi Notsu; Masato Kimura

doi:10.3934/nhm.2014.9.617

Networks and Heterogeneous Media

2014, Volume 9, Issue 4: 617-634. doi: 10.3934/nhm.2014.9.617

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

Hirofumi Notsu ^{1
,},
Masato Kimura ^{2
,}

1.
Waseda Institute for Advanced Study, Waseda University, 3-4-1, Okubo, Shinjuku, Tokyo 169-8555
2.
Institute of Science and Engineering, Kanazawa University, Kakuma, Kanazawa 920-1192

Received: 01 July 2014 Revised: 01 September 2014
Primary: 74B05, 74G15, 74S05; Secondary: 74A45.

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.
- Spring-block system,
- linear elasticity,
- finite element method,
- 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[J]. Networks and Heterogeneous Media, 2014, 9(4): 617-634. doi: 10.3934/nhm.2014.9.617

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Abstract

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.

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