Skip to main content
SpringerLink
Log in

Three-Dimensional Macroelements

  • Chapter
  • First Online:
Precursors of Isogeometric Analysis

Part of the book series: Solid Mechanics and Its Applications ((SMIA,volume 256))

  • 1145 Accesses

Abstract

This chapter deals with three-dimensional macroelements (large solid bricks) based on several CAD-based interpolation formulas. First, three alternative expressions are derived for the boundary-only Coons interpolation; the first of them is complete, whereas the next two cover particular cases. It will be shown that the classical eight-node trilinear and the twenty-node triquadratic solid elements are the simplest ones of the Coons family. Second, Gordon interpolation in conjunction with internal nodes is fully explained, and it is shown that the classical 27-node tensor-product Lagrangian element is the simplest element of this class. Based on the two aforementioned CAD interpolations (i) simplified edge -only Coons macroelements will be developed, which are hierarchically improved (ii) through enhanced boundary-only formulation (based on the entire faces) as well as (iii) full tensor-product Coons–Gordon macroelements. Nonrational Bézier elements as well as elements based on B-splines will be presented. For the sake of brevity, numerical results reduce to the acoustic analysis of rectangular and spherical cavities, where all formulations are compared to the closed-form exact solution and are thoroughly criticized. The reader is also referred to previously published numerical examples that include potential (Laplace and Poisson’s equations) and elasticity problems in bodies of cuboidal (parallelepiped with six rectangular faces), cylindrical and spherical shape, using a single macroelement.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 149.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 199.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Ainsworth M, Davydov O, Schumaker L (2016) Bernstein-Bézier finite elements on tetrahedral–hexahedral–pyramidal partitions. Comput Methods Appl Mech Eng 304:140–170

    Article  Google Scholar 

  2. Beer G, Watson JO (2002) Introduction to finite and boundary element methods for engineers (Chap. 11). Wiley, Chichester, pp 357–377

    Google Scholar 

  3. Cobb JE (1988) Tilling the sphere with rational Bezier patches. UUCS-88-009, University of Utah

    Google Scholar 

  4. Coons SA (1964) Surfaces for computer aided design of space form. Project MAC, MIT (1964), revised for MAC-TR-41 (1967). Springfield, VA 22161, USA. Available online: http://publications.csail.mit.edu/lcs/pubs/pdf/MIT-LCS-TR-041.pdf

  5. Cook WA (1974) Body oriented (natural) co-ordinates for generating three-dimensional meshes. Int J Numer Meth Eng 8:27–43

    Article  Google Scholar 

  6. Eringen AC, Suhubi ES (1974) Elastodynamics, vols I & II. Academic Press, New York and London

    Google Scholar 

  7. Graff KF (1975) Wave motion in elastic solids, Oxford University Press. Also: Dover (1991)

    Google Scholar 

  8. Kanarachos A, Grekas D, Provatidis C (1995) Generalized formulation of Coons’ interpolation. In: Kaklis P, Sapidis N (eds) Computer aided geometric design from theory to practice, 1995, NTUA, Chap. 7, pp 65–76 (a one-day research seminar to honour the first visit of Prof. G. Farin from the Arizona University to the National Technical University of Ahens, Greece). ISBN 960-254-068-0

    Google Scholar 

  9. Liu B, Xing Y, Wang Z, Lu X, Sun H (2017) Non-uniform rational Lagrange functions and its applications to isogeometric analysis of in-plane and flexural vibration of thin plates. Comput Methods Appl Mech Eng 321:173–208

    Article  MathSciNet  Google Scholar 

  10. Piegl L, Tiller W (1997) The NURBS book. Springer, Berlin

    Book  Google Scholar 

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

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  13. Provatidis CG (2006) Three-dimensional Coons macroelements: application to eigenvalue and scalar wave propagation problems. Int J Numer Meth Eng 65:111–134

    Article  Google Scholar 

  14. Provatidis CG (2014) Bézier versus Lagrange polynomials-based finite element analysis of 2-D potential problems. Adv Eng Softw 73:22–34

    Article  Google Scholar 

  15. Provatidis CG (2018) Engineering analysis with CAD-based macroelements. Arch Appl Mech 88:121–140

    Article  Google Scholar 

  16. Provatidis CG, Vossou CG, Theodorou EG (2006) On the CAD/CAE integration using Coons interpolation. In: Tsahalis D (ed) Proceedings 2nd international conference “from scientific computing to computational engineering”, Athens, Greece, 5–8 July 2006

    Google Scholar 

  17. Schillinger D, Ruthala PK, Nguyen LH (2016) Lagrange extraction and projection for NURBS basis functions: a direct link between isogeometric and standard nodal finite element formulations. Int J Numer Meth Eng 108(6):515–534

    Article  MathSciNet  Google Scholar 

  18. Szabó B, Babuška I (1991) Finite element analysis. Wiley, New York

    MATH  Google Scholar 

  19. Zienkiewicz OC (1977) The finite element method. McGraw-Hill, London

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mechanical Engineering, National Technical University of Athens, Athens, Greece

    Christopher G. Provatidis

Authors
  1. Christopher G. Provatidis
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Christopher G. Provatidis .

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Provatidis, C.G. (2019). Three-Dimensional 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_10

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-03889-2_10

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-03888-5

  • Online ISBN: 978-3-030-03889-2

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation