From NURBS to NURPS geometries

Abstract

Quadratic Powell-Sabin splines and their rational extension, the socalled NURPS surfaces, are an interesting alternative for classical tensor-product NURBS in the context of isogeometric analysis, because they allow the use of local refinements while retaining a Bspline like representation and exact description of conic sections. In this paper we present a simple and effective strategy to convert a given planar geometry defined by a quadratic NURBS representation into a NURPS representation, suitable for the analysis.Keywords : Powell-Sabin splines, NURBS, NURPS, isogeometric analysis. AbstractQuadratic Powell-Sabin splines and their rational extension, the so-called NURPS surfaces, are an interesting alternative for classical tensor-product NURBS in the context of isogeometric analysis, because they allow the use of local refinements while retaining a B-spline like representation and exact description of conic sections. In this paper we present a simple and effective strategy to convert a given planar geometry defined by a quadratic NURBS representation into a NURPS representation, suitable for the analysis.

Optimizing domain parameterization in isogeometric analysis based on Powell–Sabin splines

“…Therefore, they are a natural choice for the functions in (2.4), where the reference domain Ω 0 is the unit square. On the other hand, splines defined on triangulations are an interesting alternative to tensor-product B-splines/NURBS in IgA [17,34,35], because they inherently support local refinement and they offer more flexibility in choosing the shape of the parameter domain.…”

“…PS splines are C 1 quadratic polynomial splines defined on any given triangulation endowed with a specific macro-structure [26], and they can be described in terms of basis functions possessing all the nice properties of classical Bsplines, the so-called PS B-splines [7]. Bivariate rational piecewise quadratic geometries (like conics and quadrics) can be represented exactly as NURPS surfaces, and planar quadratic NURBS geometries can be easily and efficiently converted into a NURPS form [34]. Thanks to their structure based on triangulations, PS/NURPS splines offer the flexibility of classical triangular finite elements with respect to local refinement, and they are not confined to a quadrilateral parameter domain but allow for any polygonal parameter domain.…”

“…Examples are T-splines [3,5], hierarchical splines [12,38], LR-splines [8,18], and splines on triangulations [17,34,35].…”

“…Powell-Sabin (PS) splines and their rational extension, the so-called Non-Uniform Rational PS (NURPS) surfaces, are an attractive alternative tool for solving PDEs [31], in particular in the context of IgA [4,34,35]. PS splines are C 1 quadratic polynomial splines defined on any given triangulation endowed with a specific macro-structure [26], and they can be described in terms of basis functions possessing all the nice properties of classical Bsplines, the so-called PS B-splines [7].…”

A multi-patch nonsingular isogeometric boundary element method using trimmed elements

“…Among the most popular candidate basis functions are T-splines [66], which have been used successfully in IGA [5,63,64]. Competitors to T-splines for analysis include PHTsplines [53,75], hierarchical B-splines or NURBS [62,73] and Powell-Sabin splines [30,31] among many others.…”