Understanding Bézier Extraction, Bézier Elements, and Hermite Elements in NURBS-Based Isogeometric Analysis

Christopher G. Provatidis

doi:10.24423/cames.2025.1756

Christopher G. Provatidis Mechanical Design and Control Systems Division, School of Mechanical Engineering, National Technical University of Athens, Athens

DOI: http://dx.doi.org/10.24423/cames.2025.1756

Abstract

This paper discusses the current need for Bézier extraction in isogeometric analysis (IGA), and proposes a straightforward procedure to improve the accuracy of the numerical solution without increasing the number of B-spline elements. It shows that after knot insertion, where the shape and parameterization of the domain are preserved, the number of degrees of freedom (DOFs) leading to enhanced numerical accuracy of the numerical solution increases as well. Of particular interest is the fact that the control points implicitly introduced during Bézier extraction in IGA can be explicitly used to form Bézier elements with C⁰-continuity in several ways. Similarly, for any inner knot multiplicity less than the polynomial degree p, accuracy increases while maintaining the same number of B-spline elements. In conclusion, the set of the extracted Bézier or Hermite elements eventually leads to superior accuracy and performance compared to C^p−1-continuity. Nevertheless, if we consider a certain fixed number of DOFs for all three competing models (for p = 3), results show that the C²-continuous model is the most accurate, while the C¹-continuous model (Hermite extraction) is more accurate than the C⁰-continuous model (Bézier extraction). The study includes six static and eigenvalue potential problems with known closed-form exact solution.

Keywords

Bézier extraction, isogeometric analysis, NURBS and finite elements and Helmholtz equation, error estimator,

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