Higher‐order and higher floating‐point precision numerical approximations of finite strain elasticity moduli
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 high...
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| Published in: | International journal for numerical methods in engineering Vol. 120; no. 10; pp. 1184 - 1201 |
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| Format: | Journal Article |
| Language: | English |
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07.12.2019
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| ISSN: | 0029-5981, 1097-0207 |
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| Abstract | 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. |
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| AbstractList | 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. 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. |
| Author | Connolly, Stephen John Mackenzie, Donald Gorash, Yevgen |
| Author_xml | – sequence: 1 givenname: Stephen John orcidid: 0000-0001-6286-0469 surname: Connolly fullname: Connolly, Stephen John email: stephen.connolly@strath.ac.uk organization: University of Strathclyde – sequence: 2 givenname: Donald orcidid: 0000-0002-1824-1684 surname: Mackenzie fullname: Mackenzie, Donald organization: University of Strathclyde – sequence: 3 givenname: Yevgen orcidid: 0000-0003-2802-7814 surname: Gorash fullname: Gorash, Yevgen organization: University of Strathclyde |
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| Cites_doi | 10.1002/(SICI)1097-0207(20000520)48:2<159::AID-NME871>3.0.CO;2-Y 10.1016/j.cma.2013.11.005 10.1016/S0022-5096(99)00017-4 10.1007/s00419-014-0939-6 10.1016/j.commatsci.2012.02.027 10.1016/0045-7825(96)01019-5 10.2514/6.2011-886 10.1016/j.ijsolstr.2017.12.010 10.1016/0045-7825(85)90070-2 10.1515/jmbm-2012-0007 10.1007/s00466-009-0395-2 10.5254/1.3538357 10.1098/rspa.1999.0431 10.2514/1.J052184 10.1090/S0025-5718-1988-0935077-0 10.1016/j.cam.2004.12.026 10.1007/s00419-017-1259-4 10.1016/j.jmps.2005.04.010 10.1108/02644409710166190 10.1016/j.finel.2014.05.016 10.1016/j.cma.2014.08.020 10.1016/S0377-0427(99)00358-1 10.1016/j.compstruc.2014.04.009 10.1115/1.4002375 10.1007/s00419-012-0610-z 10.1137/1.9781611971200 10.1080/10255842.2015.1118467 10.1016/S0045-7825(99)00296-0 10.1098/rspa.2012.0167 10.1098/rsta.1948.0002 10.1007/s003660200028 10.1016/j.jmps.2005.04.006 10.1016/j.ijimpeng.2019.03.005 10.1115/1.2979872 |
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Two real‐domain numerical approximation methods for accurate computation of finite strain elasticity moduli are developed and their accuracy and... Two real‐domain numerical approximation methods for accurate computation of finite strain elasticity moduli are developed and their accuracy and computational... |
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| SubjectTerms | 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 |
| Title | Higher‐order and higher floating‐point precision numerical approximations of finite strain elasticity moduli |
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