Symmetry and positive definiteness of the tensor-valued spring constant derived from P1-FEM for the equations of linear elasticity
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1.
Waseda Institute for Advanced Study, Waseda University, 3-4-1, Okubo, Shinjuku, Tokyo 169-8555
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2.
Institute of Science and Engineering, Kanazawa University, Kakuma, Kanazawa 920-1192
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Received:
01 July 2014
Revised:
01 September 2014
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Primary: 74B05, 74G15, 74S05; Secondary: 74A45.
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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.
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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