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Erschienen in:
Initial Attempts on CAD/CAE Integration Kapitel lesen Erstes Kapitel lesen

2019 | OriginalPaper | Buchkapitel

3. COONS’ Interpolation as a Vehicle to Derive Large Isoparametric Elements

verfasst von : Christopher G. Provatidis

Erschienen in: Precursors of Isogeometric Analysis

Verlag: Springer International Publishing

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Abstract

This chapter explains that Coons interpolation, which chronologically is the first formula for the mathematical representation of surface patches in Computational Geometry, can be used to derive the closed-form analytical expressions of shape functions that appear in classical isoparametric finite elements of the Serendipity family. Moreover, it is shown that not only Lagrange polynomials but also reduced natural cardinal cubic B-splines as well as most of other interpolations discussed in Chap. 2 can be also used as trial functions along the four edges of a Coons patch macroelement (“C-element”). The latter is generally a large isoparametric finite element with nodal points along the boundary of the patch. In the case of using Curry-Schoenberg (de Boor) B-splines or NURBS along each edge in the Coons patch, the nodal points are merely replaced by the control points. The performance of all these elements is thoroughly investigated through ten examples in two-dimensional and axisymmetric potential and elasticity problems.

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Vorheriges Kapitel Elements of Approximation and Computational Geometry
Nächstes Kapitel GORDON’s Transfinite Macroelements
Literatur
1.
Bathe KJ (1990) finite element procedures. Prentice-Hall, Upper Saddle River, New JerseyMATH
2.
Beissel S, Belytschko T (1996) Nodal integration of the element-free Galerkin method. Comput Methods Appl Mech Eng 139(1–4):49–74MathSciNetCrossRef
3.
Blevins RD (1979) Formulas for natural frequency and mode shape. Van Nostrand Reinhold, New York
4.
Carslaw HS, Jaeger JC (1986) Conduction of heat in solids, 2nd edn. Oxford Science Publications, Oxford
5.
Cavendish JC, Gordon WJ, Hall CA (1977) Substructured macro elements based on locally blended interpolation. Int J Numer Meth Eng 11:1405–1421CrossRef
6.
Cavendish JC, Hall CA (1984) A new class of transitional blended finite elements for the analysis of solid structures. Int J Numer Meth Eng 20:241–253CrossRef
7.
8.
Courant R, Hilbert D (1966) Methods of mathematical physics, vol I, 1st English edn (translated and revised from the German original). InterScience Publishers, New York
9.
De Boor C (1972) On calculating with B-splines. J Appl Theory 6(1):50–62
10.
11.
Gordon WJ, Hall CA (1973) Transfinite element methods: blending-function interpolation over arbitrary curved element domains. Numer Math 21:109–112MathSciNetCrossRef
12.
Kanarachos A, Deriziotis D (1989) On the solution of Laplace and wave propagation problems using ‘C-elements’. Finite Elem Anal Des 5:97–109CrossRef
13.
Kanarachos A, Provatidis C (1987) Performance of mass matrices for the BEM dynamic analysis of wave propagation problems. Comput Methods Appl Mech Eng 63:155–165CrossRef
14.
Kanarachos A, Provatidis C, Deriziotis D, Foteas N (1999) A new approach of the FEM analysis of two-dimensional elastic structures using global (Coons’s) interpolation functions. In: Wunderlich J (ed) CD Proceedings first european conference on computational mechanics (ECCM’99): (Paper 763.pdf: pp 1–19), Munich-Germany, 28 Aug–3 Sept 1999. Available online at: http://​users.​ntua.​gr/​cprovat/​yliko/​763.​pdf
15.
Kreith F (1968) Principles of heat transfer, 2nd edn. International Textbook Company, Pennsylvania
16.
Lamé G (1852) Leçons sur la Théorie Mathématique de l’ Élasticité des Corps Solides. Gautheir-Villars, Paris
17.
Lin F, Hewitt WT (1994) Expressing coons-gordon surfaces as NURBS. Comput Aided Des 26(2):145–155CrossRef
18.
Nardini D, Brebbia CA (1983) A new approach to free vibration analysis using boundary elements. Appl Math Model 7:157–162CrossRef
19.
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
20.
Pipes LA, Harvill LR (1981) Applied mathematics for engineers and physicists, 3rd edn. McGraw-Hill International Book Company, International Student Edition
21.
Provatidis CG (2005) Analysis of box-like structures using 3-D Coons’ interpolation. Commun Numer Methods Eng 21:443–456CrossRef
22.
Provatidis CG (2012) Two-dimensional elastostatic analysis using Coons-Gordon interpolation. Meccanica 47:951–967MathSciNetCrossRef
23.
Provatidis CG, Kanarachos AE (1995) Further research on the performance of consistent mass matrices using BEM for symmetric/nonsymmetric formulations. Comput Mech 16:197–207MathSciNetCrossRef
24.
Provatidis C, Kanarachos A (2001) Performance of a macro-FEM approach using global interpolation (Coons’) functions in axisymmetric potential problems. Comput Struct 79:1769–1779CrossRef
25.
Provatidis CG (2002) A comparative study between Coons-patch macroelements and boundary elements in two-dimensional potential problems. In: Tsahalis D (ed) Proceedings 4th GRACM congress on computational mechanics, Patras, 27–29 June, 2002, pp 43–50
26.
27.
Provatidis CG (2004) On DR/BEM for eigenvalue analysis of 2-D acoustics. Comput Mech 35:41–53CrossRef
28.
Provatidis C, Kanarachos A (2000) On the use of Coons’ interpolation in CAD/CAE systems. In: Mastorakis N (ed.) Systems and control: theory and applications. World Scientific and Engineering Society Press, pp 343–348; Second international conference on mathematics and computers in mechanical engineering (MCME 2000), pp 10–15 July 2000, Vouliagmeni, Greece. (ISBN: 960-8052-11-4). Online available on: http://​www.​wseas.​us/​e-library/​conferences/​athens2000/​Papers2000/​547.​pdf
29.
Provatidis C G (2003) Analysis of axisymmetric structures using Coons’ interpolation. Finite Elem Anal Des 39:535–558CrossRef
30.
Provatidis CG (2004) Coons-patch macroelements in two-dimensional eigenvalue and scalar wave propagation problems. Comput Struct 82(4–5):383–395MathSciNetCrossRef
31.
Provatidis C (2003) Frequency analysis and transient response of two-dimensional structures using Coons-patch macroelements. In: Brennan MJ, Ferman MA, Rizzi SA, Wenz K (eds) Proceedings of the VIII international conference on recent advances in structural dynamics. Organized by the Institute of Sound & Vibration Research University of Southampton, Southampton, UK, 14–16 July 2003
32.
Provatidis CG (2006) Free vibration analysis of two-dimensional structures using Coons-patch macroelements. Finite Elem Anal Des 42(6):18–531CrossRef
33.
34.
Provatidis CG (2006) Coons-patch macroelements in two-dimensional parabolic problems. Appl Math Model 30:319–351CrossRef
35.
Provatidis CG (2006) Transient elastodynamic analysis of two-dimensional structures using Coons-patch macroelements. Int J Solids Struct 43(22–23):6688-6706CrossRef
36.
Timoshenko S, Goodier JN (1970) Theory of elasticity, 3rd edn. McGraw-Hill, New YorkMATH
37.
Wang L, Zhu X, Tang Z (1997) COONS type blended B-spline (CNSBS) surface and its conversion to NURBS surface. Comput Graph 21(3):297–303CrossRef
38.
Weaver W Jr, Timoshenko SP, Young DH (1990) Vibration problems in engineering, 5th edn. Wiley, New York
39.
Zienkiewicz OC (1977) The finite element method. McGraw-Hill, LondonMATH
40.
Zienkiewicz OC, Taylor RL (1988) The finite element method. In: Basic Formulation and Linear Problems, 4th edn, vol 1. McGraw-Hill, London
Metadaten
Titel
COONS’ Interpolation as a Vehicle to Derive Large Isoparametric Elements
verfasst von
Christopher G. Provatidis
Copyright-Jahr
2019
Buch
Precursors of Isogeometric Analysis

Print ISBN: 978-3-030-03888-5

Electronic ISBN: 978-3-030-03889-2

Copyright-Jahr: 2019

DOI
https://doi.org/10.1007/978-3-030-03889-2_3

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