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

• Published in: Computer assisted methods in engineering and science Vol. 32; no. 1
• Main Author: Christopher G. Provatidis
• Format: Journal Article
• Language: English
• Published: Institute of Fundamental Technological Research Polish Academy of Sciences 2025
• Subjects: Bézier extraction; error estimator; isogeometric analysis; NURBS and finite elements and Helmholtz equation
• ISSN: 2299-3649; 2956-5839
• DOI: 10.24423/cames.2025.1756

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 Cp−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.