Rational Elements (BEZIER, NURBS)

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


This chapter deals with single macroelements in which the approximation of the variable U is based mostly on rational Bézier and less on nonuniform rational B-splines (NURBS). Since univariate rational Bernstein–Bézier polynomials is a special case of univariate NURBS, it becomes obvious that tensor-product rational Bézier is also a specific case of tensor-product NURBS. The major significance of rational elements is that they accurately represent the geometry of conics (circles, ellipses, parabolas, and hyperbolas). In an instructive way, we focus on the analysis of a circular cavity using a single tensor-product macroelement. It is shown that a single quadratic Bézier macroelement, although is capable of accurately representing the entire circle, it leads to a numerical solution of low quality (slightly worse than the classical nine-node finite element of Lagrangian type). In both cases, this is due to its insufficiency to approximate the eigensolutions (e.g., the eigenvectors in dynamics). Nevertheless, after a sufficient degree elevation which maintains the shape of the circle, it is shown that the higher-order Bézier converges to the exact solution. The presentation continues with a very short summary on the NURBS-based dominating IGA, and the reader is advised for further study.


Rational Bézier Circular macroelement MATLAB Freeware software Degree elevation Circular acoustic cavity NURBS issues 


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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.School of Mechanical EngineeringNational Technical University of AthensAthensGreece

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