Point Inversion for triparametric NURBS
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SHORT ORIGINAL PAPER
Point Inversion ior triparametric NURBS Leonardo Orazi1
· Barbara Reggiani1
Received: 21 January 2020 / Accepted: 23 September 2020 © Springer-Verlag France SAS, part of Springer Nature 2020
Abstract In this paper a Point Inversion algorithm for NURBS volumes is presented. This algorithm is an extension to 3D of the classic Newton–Raphson iteration, thus implying the calculation of the NURBS volume partial derivatives. Explicit formulas for the derivatives are deduced and reported, in turn requiring the evaluation of the corresponding derivatives of the rational basis functions. Also these derivatives are inferred and shown together with some examples of applications. The method, applicable to any free-form shape NURBS volume is moreover compared with Fast Mapping, a novel approach presented here that, once applied to simple NURBS boxes give performance 20–50 times higher. Keywords Point Inversion · triparametric NURBS · NURBS volume · NURBS derivatives · Basis functions derivatives
1 Introduction NURBS curves and surfaces are ubiquitously used in modern 3D systems like CAD and CAM because of their well known representational power and flexibility. In contrast, despite having the same characteristics of their less-dimensional relatives, NURBS volumes (or triparametric NURBS) are historically less used: this mainly because of the fact that solid bodies are usually described in 3D software by their boundary surfaces. However, recently, NURBS volumes are increasingly selected because they offer interesting possibilities. As examples, in Free Form Deformation (FFD) [1] and Constrained Free Form Deformation [2,3] NURBS volumes are used to deform a 3D geometry, being it a tessellated surface or a NURBS surface. FFD can be performed by appropriately embedding the geometry in a NURBS volume, and deformation of the geometry is obtained indirectly by deforming the NURBS volume. Another use of triparametric NURBS is in FEA (Finite Element Analysis) when isogeometric-meshless paradigm is employed [4], their smoothness and capability to represent material properties allows for better and more realistic simulation results [5], [6]. Indeed, in [6], it is proved that
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Leonardo Orazi [email protected] Barbara Reggiani [email protected]
1
Department of Sciences and Methods for Engineering, University of Modena and Reggio, Via Amendola 2, 42122 Emilia, Reggio Emilia, Italy
the use of NURBS volumes helps in representing complex materials properties and in performing isogeometric analysis while improving also the convergence rate and the stability of simulations. Another important aspect is the significative reduction of model complexity and associativity issues [7]. To the best of the authors knowledge, there is no explicit formulation of an algorithm for Point Inversion when dealing with triparametric NURBS, and this despite the growing use of NURBS volumes. Indeed, Point Inversion is one of the fundamental operations in the NURBS framework, being the procedure of obtaining the parame
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