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Two-dimensional elastostatic analysis using Coons-Gordon interpolation

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Abstract

During the last years blending-function (Coons’) interpolation has been utilized for the construction of large 2D and 3D finite elements with degrees of freedom appearing along the boundaries of the domain. In the particular case of elasticity problems, these so-called “boundary-only Coons macroelements” have been applied to the analysis of simple structures in which adequate accuracy was remarked. This paper continues the research investigating, for the first time, the role of internal nodes in the accuracy of the numerical solution using various trial functions along the boundary in conjunction with various blending functions (piecewise-linear, cubic B-splines and Lagrange polynomials). The performance and limits of the proposed Coons-Gordon macroelements are tested in typical 2D elastostatic examples, where they are also compared with conventional four-node bilinear finite elements of the same mesh density. It was definitely found that although the ‘boundary-only formulation’ of the proposed Coons macroelements successfully pass some well-established patch tests and may be very accurate in some simple test cases, in general, it must be substituted by the ‘transfinite formulation’ (Coons-Gordon) where a sufficient number of internal nodes is necessary to ensure convergence.

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References

  1. Zienkiewicz OC, Taylor RL (2000) The finite element method: the basis, vol 1, 5th edn. Butterworth-Heinemann, Oxford

    MATH  Google Scholar 

  2. Strutt JW (Lord Rayleigh) (1870) On the theory of resonance. Trans R Soc (Lond) A161:77–118

    Google Scholar 

  3. Ritz W (1909) Über eine neue Methode zur Lösung gewissen Variations-Probleme der Mathematischen Physik. J Reine Angew Math 135:1–61

    Article  Google Scholar 

  4. Taylor RL (2011) Isogeometric analysis of nearly incompressible solids. Int J Numer Methods Eng 87:273–288

    Article  Google Scholar 

  5. Mote CD Jr (1971) Local-global finite element. Int J Numer Methods Eng 3:565–574

    Article  MathSciNet  MATH  Google Scholar 

  6. Brebbia CA, Dominguez J (1992) Boundary elements: an introductory course, 2nd edn. Computational Mechanics Publications, Southampton

    MATH  Google Scholar 

  7. Tullberg O, Bolteus L (1982) A critical study of different boundary stiffness matrices. In: Brebbia CA (ed) Boundary elements in engineering. Springer, Berlin, pp 621–635

    Google Scholar 

  8. Jirousek J (1978) Basis for development of large elements locally satisfying all field equations. Comput Methods Appl Mech Eng 14:65–92

    Article  ADS  MATH  Google Scholar 

  9. Jirousek J, Teodorescu P (1982) Large finite elements for the solution of problems in the theory of elasticity. Comput Struct 15:575–587

    Article  MathSciNet  MATH  Google Scholar 

  10. Aristodemo M, Leone A, Mazza M (2001) Energy based boundary elements for finite element analysis. Meccanica 36:463–477

    Article  MATH  Google Scholar 

  11. Wachspress E (1975) A rational finite element basis. Academic Press, New York

    MATH  Google Scholar 

  12. Dasgupta G (2003) Integration within polygonal finite elements. J Aerosp Eng 16:9–18

    Article  Google Scholar 

  13. Soleimani S, Qajarjazi A, Bararnia H, Barari AA, Domairry G (2011) Entropy generation due to natural convection in a partially heated cavity by local RBF-DQ method. Meccanica 46:1023–1033

    Article  Google Scholar 

  14. Liu GR (2003) Mesh free methods: moving beyond the finite element method. CRC Press, Boca Raton

    MATH  Google Scholar 

  15. Atluri SN (2004) The meshless method (MLPG) for domain and BIE discretizations. Tech Science Press, Encino

    Google Scholar 

  16. Rajagopal S, Gupta N (2011) Meshfree modelling of fracture—a comparative study of different methods. Meccanica 46:1145–1158

    Article  Google Scholar 

  17. Szabó B, Babuška I (1991) Finite element analysis. Wiley-Interscience, New York

    MATH  Google Scholar 

  18. Ovunc B (1978) In-plane vibration of plates by continuous mass matrix method. Comput Struct 8:723–731

    Article  MATH  Google Scholar 

  19. Zienkiewicz OC, Taylor RL (2000) The finite element method: fluid dynamics, vol 3, 5th edn. Butterworth-Heinemann, Oxford

    Google Scholar 

  20. Chan SK, Tuba IS, Wilson WK (1970) On the finite element method in linear fracture mechanics. Eng Fract Mech 2:1–17

    Article  Google Scholar 

  21. Henshell RD, Shaw KG (1975) Crack tip elements are unnecessary. Int J Numer Methods Eng 9:495–509

    Article  MATH  Google Scholar 

  22. Barsoum RS (1976) On the use of isoparametric finite elements in linear fracture mechanics. Int J Numer Methods Eng 10:25–37

    Article  MATH  Google Scholar 

  23. Pu SL, MA Hussain, Lorensen WE (1978) The collapsed cubic isoparametric element as a singular element for crack problems. Int J Numer Methods Eng 12:1727–1742

    Article  MATH  Google Scholar 

  24. Tracey DM (1971) Finite elements for determination of crack tip elastic stress intensity factors. Eng Fract Mech 3:255–266

    Article  Google Scholar 

  25. Atluri SN (1983) Higher-order, special and singular finite elements. In: Noor AK, Pilkey W (eds) State of the art survey of finite element technology. ASME, New York, pp 87–126

    Google Scholar 

  26. Shen SF (1975) An aerodynamicist looks at the finite element method. In: Gallagher RH et al. (eds) Finite elements in fluids, vol 2. Wiley-Interscience, London, pp 179–204

    Google Scholar 

  27. Bettess P (1992) Infinite elements. Penshaw Press, Sunderland

    Google Scholar 

  28. Franke R (1982) Scattered data interpolation: test of some methods. Math Comput 48:181–200

    MathSciNet  Google Scholar 

  29. Micchelli CA (1986) Interpolation of scattered data: distance matrices and conditionally positive definite functions. Constr Approx 2:11–22

    Article  MathSciNet  MATH  Google Scholar 

  30. Coons SA (1967) Surfaces for computer aided design of space form. Project MAC, MIT (1964), revised for MAC-TR-41 (1967). Springfield, VA, U.S.A.: Available by CFSTI, Sills Building, 5285 Port Royal Road

  31. Gordon WJ (1971) Blending functions methods of bivariate multivariate interpolation and approximation. SIAM J Numer Anal 8:158–177

    Article  MathSciNet  MATH  Google Scholar 

  32. Gordon WJ, Hall CA (1973) Transfinite element methods blending function interpolation over arbitrary curved element domains. Numer Math 21:109–112

    Article  MathSciNet  MATH  Google Scholar 

  33. Beissel S, Belytschko T (1996) Nodal integration of the element-free Galerkin method. Comput Methods Appl Mech Eng 139:49–74

    Article  MathSciNet  ADS  MATH  Google Scholar 

  34. Azadi M, Shariyat M (2010) Nonlinear transient transfinite element thermal analysis of thick-walled FGM cylinders with temperature-dependent material properties. Meccanica 45:305–318

    Article  MathSciNet  Google Scholar 

  35. Kanarachos A, Deriziotis D (1989) On the solution of Laplace and wave propagation problems using C-elements. Finite Elem Anal Des 5:97–109

    Article  MATH  Google Scholar 

  36. Provatidis C, Kanarachos A (2001) Performance of a macro-FEM approach using global interpolation (Coons’) functions in axisymmetric potential problems. Comput Struct 79:1769–1779

    Article  Google Scholar 

  37. Provatidis CG (2006) Coons-patch macroelements in two-dimensional parabolic problems. Appl Math Model 30:319–351

    Article  MATH  Google Scholar 

  38. Provatidis CG (2005) Three-dimensional Coons macroelements in Laplace and acoustic problems. Comput Struct 83:1572–1583

    Article  MathSciNet  Google Scholar 

  39. Kanarachos A, Provatidis Ch, 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. München, Germany

    Google Scholar 

  40. Provatidis CG, Kanarachos AE (2000) On the use of Coons interpolation in CAD/CAE applications. In: Mastorakis N (ed) Systems and control: theory and applications. World Scientific and Engineering Society, Singapore, pp 343–348

    Google Scholar 

  41. Provatidis CG (2003) Analysis of axisymmetric structures using Coons’ interpolation. Finite Elem Anal Des 39:535–558

    Article  Google Scholar 

  42. Provatidis CG (2005) Analysis of box-like structures using 3-D Coons’ interpolation. Commun Numer Methods Eng 21:443–456

    Article  MATH  Google Scholar 

  43. Dimitriou V (2004) Adaptive finite elements and related meshes. Dissertation, National Technical University of Athens, School of Mechanical Engineering, Greece (in Greek)

  44. Franssen M, Veltkamp RC, Wesselink W (2000) Efficient evaluation of triangular B-spline surfaces. Comput Aided Geom Des 17:863–877

    Article  MathSciNet  MATH  Google Scholar 

  45. Kim HJ, Seo YD, Youn SK (2009) Isogeometric analysis for trimmed CAD surfaces. Comput Methods Appl Mech Eng 198:2982–2995

    Article  ADS  MATH  Google Scholar 

  46. Pan XF, Zhang X, Lu MW (2005) Meshless Galerkin least-squares method. Comput Mech 35:182–189

    Article  MathSciNet  MATH  Google Scholar 

  47. Timoshenko S, Goodier JN (1970) Theory of elasticity, 3rd edn. McGraw-Hill, New York

    MATH  Google Scholar 

  48. Provatidis CG (2011) Some issues on CAD/CAE integration: Global interpolation using isoparametric and isogeometric techniques. In: Boudouvis A, Stavroulakis G (eds) Proceedings 7th GRACM international congress on computational mechanics. Athens, 30 June–2 July 2011

    Google Scholar 

  49. Aristodemo M (1985) A high-continuity finite element model for two-dimensional elastic problems. Comput Struct 21:987–993

    Article  MATH  Google Scholar 

  50. Schramm U, Pilkey WD (1993) The coupling of geometric descriptions and finite element using NURBS—A study in shape optimization. Finite Elem Anal Des 15:11–34

    Article  MATH  Google Scholar 

  51. Hughes TJR, Cottrell JA, Bazilevs Y (2005) Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput Methods Appl Mech Eng 194:4135–4195

    Article  MathSciNet  ADS  MATH  Google Scholar 

  52. Inoue K, Kikuchi Y, Masuyama T (2005) A NURBS finite element method for product shape design. J Eng Des 16(2):157–174

    Article  Google Scholar 

  53. DeBoor C (2001) A practical guide to splines. Springer, New York. Revised Edition

    Google Scholar 

  54. Hughes TJR, Reali A, Sangalli G (2010) Efficient quadrature for NURBS-based isogeometric analysis. Comput Methods Appl Mech Eng 199:301–313

    Article  MathSciNet  ADS  MATH  Google Scholar 

  55. Provatidis CG (2008) Global collocation method for 2-D rectangular domains. J Mech Mater Struct 3:185–194

    Article  Google Scholar 

  56. Provatidis CG (2008) Free vibration analysis of elastic rods using global collocation. Arch Appl Mech 78:241–250

    Article  MATH  Google Scholar 

  57. Provatidis CG (2009) Integration-free Coons macroelements for the solution of 2-D Poisson problems. Int J Numer Methods Eng 77:536–557

    Article  MathSciNet  MATH  Google Scholar 

  58. Provatidis CG, Ioannou KS (2010) Static analysis of two-dimensional elastic structures using global collocation. Arch Appl Mech 80:389–400

    Article  Google Scholar 

  59. Auricchio F, Beirão da Veiga L, Hughes TJR, Reali A, Sangalli G (2010) Isogeometric collocation methods. Math Models Methods Appl Sci 20:2075–2107

    Article  MathSciNet  MATH  Google Scholar 

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Provatidis, C.G. Two-dimensional elastostatic analysis using Coons-Gordon interpolation. Meccanica 47, 951–967 (2012). https://doi.org/10.1007/s11012-011-9489-y

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