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

Hirofumi Notsu, Masato Kimura

Research output: Contribution to journalArticle

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.

Original languageEnglish
Pages (from-to)617-634
Number of pages18
JournalNetworks and Heterogeneous Media
Volume9
Issue number4
DOIs
Publication statusPublished - 2014
Externally publishedYes

Fingerprint

Positive Definiteness
Linear Elasticity
Tensors
Elasticity
Tensor
Finite element method
Symmetry
Partial differential equations
Elastic Material
Poisson ratio
Scalar
Concretes
Poisson's Ratio
Triangular Mesh
Linear partial differential equation
Elliptic Partial Differential Equations
Positive definite
Two Dimensions
Partial differential equation
Finite Element Method

Keywords

  • Finite element method
  • Linear elasticity
  • Spring constant
  • Spring-block system

ASJC Scopus subject areas

  • Applied Mathematics
  • Statistics and Probability
  • Engineering(all)
  • Computer Science Applications

Cite this

Symmetry and positive definiteness of the tensor-valued spring constant derived from P1-FEM for the equations of linear elasticity. / Notsu, Hirofumi; Kimura, Masato.

In: Networks and Heterogeneous Media, Vol. 9, No. 4, 2014, p. 617-634.

Research output: Contribution to journalArticle

@article{b213364bf38146aa8090816c12a96746,
title = "Symmetry and positive definiteness of the tensor-valued spring constant derived from P1-FEM for the equations of linear elasticity",
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.",
keywords = "Finite element method, Linear elasticity, Spring constant, Spring-block system",
author = "Hirofumi Notsu and Masato Kimura",
year = "2014",
doi = "10.3934/nhm.2014.9.617",
language = "English",
volume = "9",
pages = "617--634",
journal = "Networks and Heterogeneous Media",
issn = "1556-1801",
publisher = "American Institute of Mathematical Sciences",
number = "4",

}

TY - JOUR

T1 - Symmetry and positive definiteness of the tensor-valued spring constant derived from P1-FEM for the equations of linear elasticity

AU - Notsu, Hirofumi

AU - Kimura, Masato

PY - 2014

Y1 - 2014

N2 - 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.

AB - 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.

KW - Finite element method

KW - Linear elasticity

KW - Spring constant

KW - Spring-block system

UR - http://www.scopus.com/inward/record.url?scp=84915822171&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84915822171&partnerID=8YFLogxK

U2 - 10.3934/nhm.2014.9.617

DO - 10.3934/nhm.2014.9.617

M3 - Article

AN - SCOPUS:84915822171

VL - 9

SP - 617

EP - 634

JO - Networks and Heterogeneous Media

JF - Networks and Heterogeneous Media

SN - 1556-1801

IS - 4

ER -