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

• Published in: International journal for numerical methods in engineering 2019-12, Vol.120 (10), p.1184-1201
• Main Authors: Connolly, Stephen John, Mackenzie, Donald, Gorash, Yevgen
• Format: Article
• Language: English
• Subjects: Accuracy; Approximation; Computational efficiency; Constitutive models; Elasticity; elasticity moduli; Exact solutions; Finite difference method; Finite element method; higher floating‐point precision; higher‐order approximation; hyperelasticity; Mathematical models; nonlinear finite element method; numerical differentiation; Numerical methods; Strain
• ISSN: 0029-5981; 1097-0207
• DOI: 10.1002/nme.6176

Summary Two real‐domain numerical approximation methods for accurate computation of finite strain elasticity moduli are developed and their accuracy and computational efficiency are 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 toward 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.