Algorithms for geometrical operations with NURBS surfaces.

2025-04-30 0 0 456.64KB 9 页 10玖币

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Gernot Beera,∗

aInstitute of Structural Analysis, Graz University of Technology, Lessingstraße 25/II, 8010 Graz, Austria

Abstract

The aim of the paper is to show algorithms for geometrical manipulations on NURBS surfaces. These

include generating NURBS surfaces that pass through given points, calculating the minimum distance to

a point and include line to surface and surface to surface intersections.

Keywords: NURBS, Isogeometric analysis

1. Introduction

Nonuniform rational B-splines or NURBS have been used by the Computer Aided Design (CAD)

community for decades. The reason for this is that they are very suitable for deﬁning geometrical

shapes and for geometrical operations. A great number of publications on NURBS exist, here we quote

the NURBS book [8] and a paper [5] that give a good summary. With the publication of the book

Isogeometric Analysis [4] it was suggested that NURBS would also beneﬁt numerical simulation methods

such as the Finite Element (FEM)and Boundary Element method (BEM). This was followed by a number

of papers discussing the implementation, for example ([6, 1, 3, 10, 7, 9]). A book on the implementation

of isogeometric methods into BEM simulation programs was published [2].

A proper implementation of NURBS into FEM and BEM programs, however required revisiting of

some geometrical operations. While the CAD community is mainly interested in the graphical display

of geometry, the simulation community is interested in generating an ’analysis suitable’ geometry, i.e.

one that allows a suitable volume or boundary discretisation. In this paper we concentrate of surfaces,

as they are used in BEM, and present algorithms for some geometrical operations. It should be noted

that CAD programs have very sophisticated algorithms for computing surface to surface intersections.

However, these algorithms produce data that are not ’analysis suitable’.

2. B-splines and NURBS

B-splines are an attractive alternative to Lagrange polynomials and Serendipity functions predom-

inantly used in simulation. The basis for creating the functions is the knot vector. This is a vector

containing a series of non-decreasing values of the local coordinate ξ:

Ξ = ξ0ξ1··· ξN(1)

A B-spline basis function of order p= 0 (constant) is given by:

Bi,0(ξ) = 1if ξi6ξ < ξi+1

0otherwise (2)

Higher order basis functions are deﬁned by referencing lower order functions:

Bi,p(ξ) = ξ−ξi

ξi+p−ξi

Bi,p−1(ξ) + ξi+p+1 −ξ

ξi+p+1 −ξi+1

Bi+1,p−1(ξ)(3)

B-spines are associate with anchors. The location of the i-th anchor in the parameter space can be

computed by:

ξi=ξi+1 +ξi+2 +···+ξi+p

pi= 0,1,...,I−1.(4)

∗Corresponding author. mail: gernot.beer@tugraz.at, web: www.ifb.tugraz.at

Preprint submitted to Advances in Engineering Software October 25, 2022

arXiv:2210.13160v1 [math.NA] 24 Oct 2022

Nonuniform rational B-splines or NURBS are based on B-splines but have improved properties for

the deﬁnition of geometry. NURBS of order pare deﬁned as:

i(ξ) = Bi,p(ξ) wi

j=0 Bj,p(ξ) wj

(5)

where I+1 is the number of basis functions and wiare weights. This can be extended to two dimensions

using a tensor product:

Np,q

ij (ξ,η) = Np

i(ξ)Nq

j(η)(6)

where Nq

j(η)is a NURBS of order qin ηdirection.

The coordinates of a point on a curve with the local coordinate ξcan be computed by:

x(ξ) =

j=1

Rj(ξ)cj(7)

where Rj(ξi)are NURBS basis functions deﬁned in equation 5 except that numbering starts at 1 instead

of 0. cjare control point coordinates. The vector tangential in direction ξcan be computed by:

v1=∂x(ξ)

∂ξ =

j=1

∂Rj(ξ)

∂ξ cj(8)

The coordinates of a point on a surface with the local coordinate ξ,η can be computed by:

x(ξ,η) =

j=1

Rj(ξ,η)cj(9)

where Rj(ξ,η)are NURBS basis functions deﬁned in equation 6, except that numbering starts at 1 instead

of 0. The basis functions are numbered consecutively with a single index jinstead of two indices iand

j. For a surface the vector in direction ηcan be additionally computed by:

v2=∂x(ξ,η)

∂η =

j=1

∂Rj(ξ, η)

∂η cj(10)

As an example we show in Figure 1 a surface created with the knot vectors:

Ξ = 000111(11)

H=0011

i.e. quadratic in ξ-direction and linear in ηdirection. The control point coordinates and weights are

shown in table 1.

i j x y z w

0 0 0 1 0 1

1 0 0 1 1 0.707

2 0 0 0 1 1

0 1 1 1 0 1

1 1 1 1 1 0.707

2 1 1 0 1 1

Table 1: Control point coordinates and weights for deﬁnition of surface.

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AlgorithmsforgeometricaloperationswithNURBSsurfaces.GernotBeera,aInstituteofStructuralAnalysis,GrazUniversityofTechnology,Lessingstraÿe25/II,8010Graz,AustriaAbstractTheaimofthepaperistoshowalgorithmsforgeometricalmanipulationsonNURBSsurfaces.TheseincludegeneratingNURBSsurfacesthatpassthroughgivenpo...

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Algorithms for geometrical operations with NURBS surfaces.

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