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Volume 42, Issue 5
Degree Elevation and Knot Insertion for Generalized Bézier Surfaces and Their Application to Isogeometric Analysis

Mengyun Wang, Ye Ji & Chungang Zhu

J. Comp. Math., 42 (2024), pp. 1197-1225.

Published online: 2024-07

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  • Abstract

Generalized Bézier surfaces are a multi-sided generalization of classical tensor product Bézier surfaces with a simple control structure and inherit most of the appealing properties from Bézier surfaces. However, the original degree elevation changes the geometry of generalized Bézier surfaces such that it is undesirable in many applications, e.g. isogeometric analysis. In this paper, we propose an improved degree elevation algorithm for generalized Bézier surfaces preserving not only geometric consistency but also parametric consistency. Based on the knot insertion of B-splines, a novel knot insertion algorithm for generalized Bézier surfaces is also proposed. Then the proposed algorithms are employed to increase degrees of freedom for multi-sided computational domains parameterized by generalized Bézier surfaces in isogeometric analysis, corresponding to the traditional $p$-, $h$-, and $k$-refinements. Numerical examples demonstrate the effectiveness and superiority of our method.

  • AMS Subject Headings

65D07, 65D17, 68U07

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-42-1197, author = {Wang , MengyunJi , Ye and Zhu , Chungang}, title = {Degree Elevation and Knot Insertion for Generalized Bézier Surfaces and Their Application to Isogeometric Analysis}, journal = {Journal of Computational Mathematics}, year = {2024}, volume = {42}, number = {5}, pages = {1197--1225}, abstract = {

Generalized Bézier surfaces are a multi-sided generalization of classical tensor product Bézier surfaces with a simple control structure and inherit most of the appealing properties from Bézier surfaces. However, the original degree elevation changes the geometry of generalized Bézier surfaces such that it is undesirable in many applications, e.g. isogeometric analysis. In this paper, we propose an improved degree elevation algorithm for generalized Bézier surfaces preserving not only geometric consistency but also parametric consistency. Based on the knot insertion of B-splines, a novel knot insertion algorithm for generalized Bézier surfaces is also proposed. Then the proposed algorithms are employed to increase degrees of freedom for multi-sided computational domains parameterized by generalized Bézier surfaces in isogeometric analysis, corresponding to the traditional $p$-, $h$-, and $k$-refinements. Numerical examples demonstrate the effectiveness and superiority of our method.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2301-m2022-0116}, url = {http://global-sci.org/intro/article_detail/jcm/23275.html} }
TY - JOUR T1 - Degree Elevation and Knot Insertion for Generalized Bézier Surfaces and Their Application to Isogeometric Analysis AU - Wang , Mengyun AU - Ji , Ye AU - Zhu , Chungang JO - Journal of Computational Mathematics VL - 5 SP - 1197 EP - 1225 PY - 2024 DA - 2024/07 SN - 42 DO - http://doi.org/10.4208/jcm.2301-m2022-0116 UR - https://global-sci.org/intro/article_detail/jcm/23275.html KW - Generalized Bézier surface, Degree elevation, Knot insertion, Isogeometric analysis, Refinement. AB -

Generalized Bézier surfaces are a multi-sided generalization of classical tensor product Bézier surfaces with a simple control structure and inherit most of the appealing properties from Bézier surfaces. However, the original degree elevation changes the geometry of generalized Bézier surfaces such that it is undesirable in many applications, e.g. isogeometric analysis. In this paper, we propose an improved degree elevation algorithm for generalized Bézier surfaces preserving not only geometric consistency but also parametric consistency. Based on the knot insertion of B-splines, a novel knot insertion algorithm for generalized Bézier surfaces is also proposed. Then the proposed algorithms are employed to increase degrees of freedom for multi-sided computational domains parameterized by generalized Bézier surfaces in isogeometric analysis, corresponding to the traditional $p$-, $h$-, and $k$-refinements. Numerical examples demonstrate the effectiveness and superiority of our method.

Wang , MengyunJi , Ye and Zhu , Chungang. (2024). Degree Elevation and Knot Insertion for Generalized Bézier Surfaces and Their Application to Isogeometric Analysis. Journal of Computational Mathematics. 42 (5). 1197-1225. doi:10.4208/jcm.2301-m2022-0116
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