Higher-order and higher floating-point precision numerical approximations of finite strain elasticity moduli

Research output: Contribution to journalArticle

Abstract

Two real‐domain numerical approximation methods for accurate computation of finite strain elasticity moduli are developed and their accuracy and computational efficiency investigated, with reference to hyperelastic constitutive models with known analytical solutions. The methods are higher‐order and higher floating‐point precision numerical approximation, the latter being novel in this context. A general formula for higher‐order approximation finite difference schemes is derived and a new procedure is proposed to implement increased floating‐point precision. The accuracy of the approximated elasticity moduli is investigated numerically using higher‐order approximations in standard double precision and increased quadruple precision. It is found that as the order of the approximation increases, the elasticity moduli tend towards the analytical solution. Using higher floating‐point precision, the approximated elasticity moduli for all orders of approximation are found to be more accurate than the standard double precision evaluation of the analytical moduli. Application of the techniques to a finite element problem shows that the numerically approximated methods obtain convergence equivalent to the analytical method but require greater computational effort. It is concluded that numerical approximation of elasticity moduli is a powerful and effective means of implementing advanced constitutive models in the finite element method without prior derivation of difficult analytical solutions.
LanguageEnglish
Number of pages19
JournalInternational Journal for Numerical Methods in Engineering
Early online date25 Jul 2019
DOIs
Publication statusE-pub ahead of print - 25 Jul 2019

Fingerprint

Finite Strain
Floating point
Numerical Approximation
Elasticity
Modulus
Higher Order
Analytical Solution
Higher Order Approximation
Constitutive models
Constitutive Model
Modulus of elasticity
Computational efficiency
Quadruple
Order of Approximation
Analytical Methods
Finite Difference Scheme
Approximation Methods
Computational Efficiency
Finite element method
Finite Element Method

Keywords

  • elasticity moduli
  • numerical differentiation
  • higher-order approximation
  • higher floating-point precision
  • hyperelasticity
  • nonlinear finite element method

Cite this

@article{57925adc1edc48c7bcec2357f0e265e3,
title = "Higher-order and higher floating-point precision numerical approximations of finite strain elasticity moduli",
abstract = "Two real‐domain numerical approximation methods for accurate computation of finite strain elasticity moduli are developed and their accuracy and computational efficiency investigated, with reference to hyperelastic constitutive models with known analytical solutions. The methods are higher‐order and higher floating‐point precision numerical approximation, the latter being novel in this context. A general formula for higher‐order approximation finite difference schemes is derived and a new procedure is proposed to implement increased floating‐point precision. The accuracy of the approximated elasticity moduli is investigated numerically using higher‐order approximations in standard double precision and increased quadruple precision. It is found that as the order of the approximation increases, the elasticity moduli tend towards the analytical solution. Using higher floating‐point precision, the approximated elasticity moduli for all orders of approximation are found to be more accurate than the standard double precision evaluation of the analytical moduli. Application of the techniques to a finite element problem shows that the numerically approximated methods obtain convergence equivalent to the analytical method but require greater computational effort. It is concluded that numerical approximation of elasticity moduli is a powerful and effective means of implementing advanced constitutive models in the finite element method without prior derivation of difficult analytical solutions.",
keywords = "elasticity moduli, numerical differentiation, higher-order approximation, higher floating-point precision, hyperelasticity, nonlinear finite element method",
author = "Connolly, {Stephen John} and Donald MacKenzie and Yevgen Gorash",
note = "This is the peer reviewed version of the following article: Connolly, SJ, MacKenzie, D & Gorash, Y 2019, 'Higher-order and higher floating-point precision numerical approximations of finite strain elasticity moduli' International Journal for Numerical Methods in Engineering, pp. 1-19., which has been published in final form at https://doi.org/10.1002/nme.6176. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Use of Self-Archived Versions.",
year = "2019",
month = "7",
day = "25",
doi = "10.1002/nme.6176",
language = "English",
journal = "International Journal for Numerical Methods in Engineering",
issn = "0029-5981",

}

TY - JOUR

T1 - Higher-order and higher floating-point precision numerical approximations of finite strain elasticity moduli

AU - Connolly, Stephen John

AU - MacKenzie, Donald

AU - Gorash, Yevgen

N1 - This is the peer reviewed version of the following article: Connolly, SJ, MacKenzie, D & Gorash, Y 2019, 'Higher-order and higher floating-point precision numerical approximations of finite strain elasticity moduli' International Journal for Numerical Methods in Engineering, pp. 1-19., which has been published in final form at https://doi.org/10.1002/nme.6176. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Use of Self-Archived Versions.

PY - 2019/7/25

Y1 - 2019/7/25

N2 - Two real‐domain numerical approximation methods for accurate computation of finite strain elasticity moduli are developed and their accuracy and computational efficiency investigated, with reference to hyperelastic constitutive models with known analytical solutions. The methods are higher‐order and higher floating‐point precision numerical approximation, the latter being novel in this context. A general formula for higher‐order approximation finite difference schemes is derived and a new procedure is proposed to implement increased floating‐point precision. The accuracy of the approximated elasticity moduli is investigated numerically using higher‐order approximations in standard double precision and increased quadruple precision. It is found that as the order of the approximation increases, the elasticity moduli tend towards the analytical solution. Using higher floating‐point precision, the approximated elasticity moduli for all orders of approximation are found to be more accurate than the standard double precision evaluation of the analytical moduli. Application of the techniques to a finite element problem shows that the numerically approximated methods obtain convergence equivalent to the analytical method but require greater computational effort. It is concluded that numerical approximation of elasticity moduli is a powerful and effective means of implementing advanced constitutive models in the finite element method without prior derivation of difficult analytical solutions.

AB - Two real‐domain numerical approximation methods for accurate computation of finite strain elasticity moduli are developed and their accuracy and computational efficiency investigated, with reference to hyperelastic constitutive models with known analytical solutions. The methods are higher‐order and higher floating‐point precision numerical approximation, the latter being novel in this context. A general formula for higher‐order approximation finite difference schemes is derived and a new procedure is proposed to implement increased floating‐point precision. The accuracy of the approximated elasticity moduli is investigated numerically using higher‐order approximations in standard double precision and increased quadruple precision. It is found that as the order of the approximation increases, the elasticity moduli tend towards the analytical solution. Using higher floating‐point precision, the approximated elasticity moduli for all orders of approximation are found to be more accurate than the standard double precision evaluation of the analytical moduli. Application of the techniques to a finite element problem shows that the numerically approximated methods obtain convergence equivalent to the analytical method but require greater computational effort. It is concluded that numerical approximation of elasticity moduli is a powerful and effective means of implementing advanced constitutive models in the finite element method without prior derivation of difficult analytical solutions.

KW - elasticity moduli

KW - numerical differentiation

KW - higher-order approximation

KW - higher floating-point precision

KW - hyperelasticity

KW - nonlinear finite element method

U2 - 10.1002/nme.6176

DO - 10.1002/nme.6176

M3 - Article

JO - International Journal for Numerical Methods in Engineering

T2 - International Journal for Numerical Methods in Engineering

JF - International Journal for Numerical Methods in Engineering

SN - 0029-5981

ER -