# Three-Dimensional Macroelements

• Christopher G. Provatidis
Chapter
Part of the Solid Mechanics and Its Applications book series (SMIA, volume 256)

## 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.

## Keywords

3D Coons interpolation C-element p-method Transfinite Bernstein–Bézier element B-spline solid element Rational bézier Cuboid Sphere

## References

1. 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
2. 2.
Beer G, Watson JO (2002) Introduction to finite and boundary element methods for engineers (Chap. 11). Wiley, Chichester, pp 357–377Google Scholar
3. 3.
Cobb JE (1988) Tilling the sphere with rational Bezier patches. UUCS-88-009, University of UtahGoogle Scholar
4. 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. 5.
Cook WA (1974) Body oriented (natural) co-ordinates for generating three-dimensional meshes. Int J Numer Meth Eng 8:27–43
6. 6.
Eringen AC, Suhubi ES (1974) Elastodynamics, vols I & II. Academic Press, New York and LondonGoogle Scholar
7. 7.
Graff KF (1975) Wave motion in elastic solids, Oxford University Press. Also: Dover (1991)Google Scholar
8. 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 960-254-068-0Google Scholar
9. 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
10. 10.
Piegl L, Tiller W (1997) The NURBS book. Springer, Berlin
11. 11.
Provatidis CG (2005) Analysis of box-like structures using 3-D Coons’ interpolation. Commun Numer Methods Eng 21:443–456
12. 12.
Provatidis CG (2005) Three-dimensional Coons macroelements in Laplace and acoustic problems. Comput Struct 83:1572–1583
13. 13.
Provatidis CG (2006) Three-dimensional Coons macroelements: application to eigenvalue and scalar wave propagation problems. Int J Numer Meth Eng 65:111–134
14. 14.
Provatidis CG (2014) Bézier versus Lagrange polynomials-based finite element analysis of 2-D potential problems. Adv Eng Softw 73:22–34
15. 15.
Provatidis CG (2018) Engineering analysis with CAD-based macroelements. Arch Appl Mech 88:121–140
16. 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 2006Google Scholar
17. 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
18. 18.
Szabó B, Babuška I (1991) Finite element analysis. Wiley, New York
19. 19.
Zienkiewicz OC (1977) The finite element method. McGraw-Hill, London