Top

Computational Mechanics

Published in:

01-11-2015 | Original Paper

Stabilized plane and axisymmetric Lobatto finite element models

Authors: Y. C. Hu, K. Y. Sze, Y. X. Zhou

Published in: Computational Mechanics | Issue 5/2015

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

Abstract

High order elements are renowned for their high accuracy and convergence. Among them, Lobatto spectral finite elements are commonly used in explicit dynamic analyses as their mass matrices when evaluated by the Lobatto integration rule are diagonal. While there are numerous advanced first and second order elements, advanced high order elements are rarely seen. In this paper, generic stabilization schemes are devised for the reduced integrated plane and axisymmetric elements. Static and explicit dynamic tests are considered for evaluating the relatively merits of the stabilized and conventional elements. The displacement errors of the stabilized elements are less than those of the conventional Lobatto elements. When the material is nearly incompressible, the stabilized elements are also more accurate in terms of the energy error norm. This advantage is of practical importance for bio-tissue and hydrated soil analyses.

previous article Computational modeling of the nonlinear stochastic dynamics of horizontal drillstrings

next article A simple way to improved formulation of analysis

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

inform now

Available only for authorised users

Patera AT (1984) A spectral element method for fluid dynamics: laminar flow in a channel expansion. J Comput Phys 54:468–488MATHCrossRef

Maday Y, Meiron D, Patera AT, Rønquist EM (1993) Analysis of iterative methods for the steady and unsteady Stokes problem: application to spectral element discretizations. SIAM J Sci Comput 14:310–337MATHCrossRefMathSciNet

Komatitsch D, Vilotte JP (1998) The spectral element method: an efficient tool to simulate the seismic response of 2D and 3D geological structures. Bull Seismol Soc Am 88:368–392

Komatitsch D, Vilotte JP, Vai R, Castillo-Covarrubias JM, Sanchez-Sesma FJ (1999) The spectral element method for elastic wave equations-application to 2-D and 3-D seismic problems. Int J Numer Methods Eng 45:1139–1164MATHCrossRef

Dauksher W, Emery AF (2000) The solution of elastostatic and elastodynamic problems with Chebyshev spectral finite elements. Comput Methods Appl Mech Eng 188:217–233MATHCrossRef

Canuto C, Hussaini MY, Quarteroni A, Zang TA (2006) Spectral methods: fundamentals in single domains. Springer, Berlin

De Basabe JD, Sen MK (2007) Grid dispersion and stability criteria of some common finite-element methods for acoustic and elastic wave equations. Geophysics 72:T81–T95CrossRef

Kudela P, Żak A, Krawczuk M, Ostachowicz W (2007) Modelling of wave propagation in composite plates using the time domain spectral element method. J Sound Vib 302:728–745MATHCrossRef

Seriani G, Oliveira SP (2008) Dispersion analysis of spectral element methods for elastic wave propagation. Wave Motion 45:729–744MATHCrossRefMathSciNet

10.

Witkowski W, Rucka M, Chróścielewski J, Wilde K (2012) On some properties of 2D spectral finite elements in problems of wave propagation. Finite Elem Anal Des 55:31–41CrossRef

11.

Xing Y, Liu B (2009) High-accuracy differential quadrature finite element method and its application to free vibrations of thin plate with curvilinear domain. Int J Numer Methods Eng 80:1718–1742MATHCrossRefMathSciNet

12.

Zhong H, Yu T (2009) A weak form quadrature element method for plane elasticity problems. Appl Math Model 33:3801–3814MATHCrossRefMathSciNet

13.

Pian THH (1964) Derivation of element stiffness matrices by assumed stress distributions. AIAA J 2:1333–1336CrossRef

14.

Lee SW, Rhiu JJ (1986) A new efficient approach to the formulation of mixed finite element models for structural analysis. Int J Numer Methods Eng 23:1629–1641MATHCrossRef

15.

Rhiu JJ, Lee SW, Russell RM (1990) Two higher-order shell finite elements with stabilization matrix. AIAA J 28:1517–1524CrossRef

16.

Sze KY (1993) A novel approach for devising higher-order hybrid elements. Int J Numer Methods Eng 36:3303–3316MATHCrossRefMathSciNet

17.

Sze KY (1994) Stabilization schemes for 12-node to 21-node brick elements based on orthogonal and consistently assumed stress modes. Comput Methods Appl Mech Eng 119:325–340MATHCrossRef

18.

Sze KY, Yi S, Tay MH (1997) An explicit hybrid stabilized eighteen-node solid element for thin shell analysis. Int J Numer Methods Eng 40:1839–1856CrossRef

19.

Sze KY, Wu D (2011) Transition finite element families for adaptive analysis of axisymmetric elasticity problems. Finite Elem Anal Des 47:360–372CrossRefMathSciNet

20.

Jog CS, Annabattula R (2006) The development of hybrid axisymmetric elements based on the Hellinger-Reissner variational principle. Int J Numer Methods Eng 65:2279–2291MATHCrossRef

21.

Malkus DS, Hughes TJR (1978) Mixed finite element methods–reduced and selective integration techniques: a unification of concepts. Comput Methods Appl Mech Eng 15:63–81MATHCrossRef

22.

Brito KD, Sprague MA (2012) Reissner-Mindlin Legendre spectral finite elements with mixed reduced quadrature. Finite Elem Anal Des 58:74–83CrossRefMathSciNet

23.

Belytschko T, Ong JSJ, Liu WK, Kennedy JM (1984) Hourglass control in linear and nonlinear problems. Comput Methods Appl Mech Eng 43:251–276MATHCrossRef

24.

Belytschko T, Bachrach WE (1986) Efficient implementation of quadrilaterals with high coarse-mesh accuracy. Comput Methods Appl Mech Eng 54:279–301MATHCrossRefMathSciNet

25.

Sze KY, Fan H, Chow CL (1995) Elimination of spurious pressure and kinematic modes in biquadratic nine-node plane element. Int J Numer Methods Eng 38:3911–3932MATHCrossRefMathSciNet

26.

Reese S, Wriggers P (2000) A stabilization technique to avoid hourglassing in finite elasticity. Int J Numer Methods Eng 48:79–109MATHCrossRef

27.

Gruttmann F, Wagner W (2004) A stabilized one-point integrated quadrilateral Reissner-Mindlin plate element. Int J Numer Methods Eng 61:2273–2295MATHCrossRef

28.

Sze KY, Zheng SJ, Lo SH (2004) A stabilized eighteen-node solid element for hyperelastic analysis of shells. Finite Elem Anal Des 40:319–340CrossRef

29.

Liu GH, Sze KY (2010) Axisymmetric quadrilateral elements for large deformation hyperelastic analysis. Int J Mech Mater Des 6:197–207CrossRef

30.

Simo JC, Rifai MS (1990) A class of mixed assumed strain methods and the method of incompatible modes. Int J Numer Methods Eng 29:1595–1638MATHCrossRefMathSciNet

31.

Korelc J, Wriggers P (1996) Consistent gradient formulation for a stable enhanced strain method for large deformations. Eng Comput 13:103–123CrossRef

32.

Glaser S, Armero F (1997) On the formulation of enhanced strain finite elements in finite deformations. Eng Comput 14:759–791MATHCrossRef

33.

Kasper EP, Taylor RL (2000) A mixed-enhanced strain method: part I: geometrically linear problems. Comput Struct 75:237–250CrossRef

34.

Romero I, Bischoff M (2007) Incompatible bubbles: a non-conforming finite element formulation for linear elasticity. Comput Methods Appl Mech Eng 196:1662–1672MATHCrossRefMathSciNet

35.

Bert CW, Malik M (1996) Differential quadrature method in computational mechanics: a review. Appl Mech Rev 49:1–27CrossRef

36.

Hughes TJR (2000) The finite element method: linear static and dynamic finite element analysis. Dover Publications, New York

37.

Szabó BA, Babuška I (1991) Finite element analysis. Wiley, New YorkMATH

38.

Timoshenko S, Goodier JN (1970) Theory of elasticity. McGraw-Hill, New YorkMATH

39.

Sani RL, Gresho PM, Lee RL, Griffiths DF (1981) The cause and cure (?) of the spurious pressures generated by certain FEM solutions of the incompressible Navier-Stokes equations: part 1. Int J Numer Methods Fluids 1:17–43MATHCrossRefMathSciNet

40.

Noh G, Bathe KJ (2013) An explicit time integration scheme for the analysis of wave propagations. Comput Struct 129:178–193CrossRef

41.

Achenbach JD (1973) Wave propagation in elastic solids. North-Holland, AmsterdamMATH

Title: Stabilized plane and axisymmetric Lobatto finite element models
Authors: Y. C. Hu
K. Y. Sze
Y. X. Zhou
Publication date: 01-11-2015
Publisher: Springer Berlin Heidelberg
Published in: Computational Mechanics / Issue 5/2015
Print ISSN: 0178-7675
Electronic ISSN: 1432-0924
DOI: https://doi.org/10.1007/s00466-015-1207-5

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Technik"

Other articles of this Issue 5/2015

Computational modeling of the nonlinear stochastic dynamics of horizontal drillstrings

Three-dimensional Green’s function for an anisotropic multi-layered half-space

Strict upper and lower bounds of stress intensity factors at 2D elastic notches based on constitutive relation error estimation

A low frequency elastodynamic fast multipole boundary element method in three dimensions

Orthotropic rotation-free thin shell elements

Heaviside enriched extended stochastic FEM for problems with uncertain material interfaces