Abstract
This chapter discusses transfinite macroelements, which are based on Gordon interpolation formula. The latter extends Coons interpolation formula (see Chap. 3) considering internal nodes as well. It will be shown that the standard tensor-product elements of Lagrange family constitute a subclass of transfinite elements, while one may generally use more or less internal nodes in several configurations. Moreover, true transfinite elements with different pattern in the arrangement of the internal nodes, as well as degenerated triangular macroelements, are discussed. A class of Cij macroelements is introduced, by influencing the trial functions as well as the blending functions. This class is so wide that can include even an assemblage of conventional bilinear elements in a structured \(n_{\xi } \times n_{\eta }\) arrangement. A careful programming of the shape functions and their global partial derivatives resulted in a single subroutine that includes all twelve combinations. The theory is supported by several test cases that refer to potential and elasticity problems in simple domains of primitive shapes where a single macroelement is used. In a couple of cases, somehow more complex domains are successfully treated using two or three Gordon macroelements.
Keywords
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Bathe KJ (1996) Finite element procedures. Prentice-Hall, Upper Saddle River, New Jersey
Bercovier M, Shilat E (1993) Enhancement of Gordon-Coons interpolations by ‘bubble functions’. Comput Aided Geom Des 10(3–4):253–265
Brebbia CA, Ferrante AJ (1986) Computational methods for the solution of engineering problems, 3rd revised edn. Pentech Press, London
Cavendish JC, Gordon WJ, Hall CA (1976) Ritz-Galerkin approximations in blending function spaces. Numer Math 26:155–178
Cavendish JC, Gordon WJ, Hall CA (1977) Substructured macro elements based on locally blending interpolation. Int J Numer Meth Eng 11:1405–1421
Cavendish JC, Hall CA, Zienkiewicz OC (1978) Blended infinite elements for paraboblic boundary value problems. Int J Numer Meth Eng 12(12):1841–1851
Cavendish JC, Hall CA (1984) A new class of transitional blended finite elements for the analysis of solid structures. Int J Numer Meth Eng 28:241–253
Dimitriou V (2004) Adaptive finite elements and related meshes. Ph.D. Dissertation (advisor: Kanarachos AE), National Technical University of Athens, School of Mechanical Engineering, Athens, Aug 2004 (in Greek). Online available from: http://thesis.ekt.gr/thesisBookReader/id/16430#page/132/mode/2up
El-Zafrany A, Cookson RA (1986) Derivation of Lagrangian and Hermitian shape functions for triangular elements. Int J Numer Meth Eng 23(2):275–285
El-Zafrany A, Cookson RA (1986) Derivation of Lagrangian and Hermitian shape functions for quadrilateral elements. Int J Numer Meth Eng 23(10):1939–1958
Gordon WJ, Hall CA (1973) Transfinite element methods: blending-function interpolation over arbitrary curved element domains. Numer Math 21:109–112
Gordon WJ, Thiel LC (1982) Transfinite mappings and their application to grid generation. In: Thompson JP (ed) Numerical grid generation. Elsevier. Appl Math Comput 11–12:171–233
Nardini D, Brebbia CA (1983) A new approach to free vibration analysis using boundary elements. Appl Math Model 7:157–162
Nardini D, Brebbia CA (1985) Boundary integral formulation of mass matrices for dynamic analysis. In: Brebbia CA (ed) Topics in boundary element research, vol 2. Springer, Berlin, pp 191–208
Partridge PW, Brebbia CA, Wrobel LC (1992) The dual reciprocity boundary element method. Computational Mechanics Publications, Southampton; Elsevier, New York, pp. 236–238
Provatidis CG (2004) Solution of two-dimensional Poisson problems in quadrilateral domains using transfinite Coons interpolation. Commun Numer Methods Eng 20(7):521–533
Provatidis CG (2005) Performance of large Gordon-Coons finite elements in 2-D potential problems. In: Georgiou G, Papanastasiou P, Papadrakakis M (eds) Proceedings of the 5th GRACM international congress on computational mechanics, Limassol, Cuprus, 29 June–1 July 2005, vol 2, pp 805–812. Dedicated to the Memory of Professor John H. Argyris. Online available from: http://euclid.mas.ucy.ac.cy/~georgios/bookfiles/GRACM05v2.pdf
Provatidis CG (2006) Coons-patch macroelements in two-dimensional parabolic problems. Appl Math Model 30:319–351
Provatidis CG (2006) Free vibration analysis of two-dimensional structures using Coons-patch macroelements. Finite Elem Anal Des 42(6):18–531
Provatidis CG (2006) Transient elastodynamic analysis of two-dimensional structures using Coons-patch macroelements. Int J Solids Struct 43:6688–6706
Provatidis CG (2009) Eigenanalysis of two-dimensional acoustic cavities using transfinite interpolation. J Algorithms Comput Technol 3(4):477–502
Provatidis CG (2012) Two-dimensional elastostatic analysis using Coons-Gordon interpolation. Meccanica 47(4):951–967
Scott FC (1968) A quartic, two dimensional isoparametric element. Undergraduate project, Univ. of Wales, Swansea
Timoshenko S, Goodier JN (1970) Theory of elasticity, 3rd edn. McGraw-Hill, New York
Zienkiewicz OC (1977) The finite element method. McGraw-Hill, London
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Provatidis, C.G. (2019). GORDON’s Transfinite Macroelements. In: Precursors of Isogeometric Analysis. Solid Mechanics and Its Applications, vol 256. Springer, Cham. https://doi.org/10.1007/978-3-030-03889-2_4
Download citation
DOI: https://doi.org/10.1007/978-3-030-03889-2_4
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-03888-5
Online ISBN: 978-3-030-03889-2
eBook Packages: EngineeringEngineering (R0)