Generalizations of non-uniform rational B-splines via decoupling of the weights: theory, software and applications
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ORIGINAL ARTICLE
Generalizations of non‑uniform rational B‑splines via decoupling of the weights: theory, software and applications Alireza H. Taheri1 · Saeed Abolghasemi2 · Krishnan Suresh1 Received: 24 January 2019 / Accepted: 7 June 2019 © Springer-Verlag London Ltd., part of Springer Nature 2019
Abstract We introduce a new class of curves and surfaces by exploring multiple variations of non-uniform rational B-splines. These variations which are referred to as generalized non-uniform rational B-splines (GNURBS) serve as an alternative interactive shape design tool, and provide improved approximation abilities in certain applications. GNURBS are obtained by decoupling the weights associated with control points along different physical coordinates. This unexplored idea brings the possibility of treating the weights as additional degrees of freedoms. It will be seen that this proposed concept effectively improves the capability of NURBS, and circumvents its deficiencies in special applications. Further, it is proven that these new representations are merely disguised forms of classic NURBS, guaranteeing a strong theoretical foundation, and facilitating their utilization. A few numerical examples are presented which demonstrate superior approximation results of GNURBS compared to NURBS in both cases of smooth and non-smooth fields. Finally, in order to better demonstrate the behavior and abilities of GNURBS in comparison to NURBS, an interactive MATLAB toolbox has been developed and introduced. Keywords NURBS · Isoparametric · Weights · Decoupling · Generalization
1 Introduction Non-uniform rational B-splines (NURBS) are perhaps the most popular curve and surface representation method in computer-aided design/computer-aided manufacturing (CAD/CAM). They were first introduced in 1975 by Versprille [1] as rational extension of B-splines. NURBS form the backbone of CAD, and are considered the dominant technology for engineering design [2]; further, they have also been extensively used in several applications including isogeometric analysis (IGA) [3], NURBS-augmented finite element analysis [4], shape optimization [5, 6], topology optimization [7, 8], material modeling [9, 10], reverse engineering [11], G-code generation [12] etc. Recent generalizations of NURBS-based technology include T-splines [13, 14] which constitute a superset of NURBS, and provide the local refinement properties by * Krishnan Suresh [email protected] 1
Department of Mechanical Engineering, UW-Madison, Madison, WI 53706, USA
School of Mechanical Engineering, Shahrood University of Technology, Shahrood, Iran
2
allowing for some unstructured-ness. An alternative generalization of NURBS, referred to as Generalized Hierarchical NURBS (H-NURBS), were introduced in 2008 by Chen et al. [15] by extending the idea of hierarchical B-splines to NURBS. Similar to T-splines, H-NURBS primarily bring the possibility of local refinement with tensor-product surfaces. A novel shape-adjustable generalized Bézier curve with multiple shape parameters has
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