NURBS.jl

This package provides functionality to define and evaluate B-spline, Curry-Schoenberg, and NURBS (non-uniform rational B-spline) basis functions, their derivatives, as well as curves and surfaces based on B-spline and NURBS basis functions.

Overview

The following aspects are implemented (✓) and planned (⌛):

B-spline, Curry-Schoenberg & NURBS evaluation

✓ Basis & derivatives
✓ Curves & derivatives
✓ Surfaces & derivatives

Fundamental operations

✓ File I/O (.step)
✓ Knot manipulation
- knot insertion / refinement
- knot removal
- splitting of curves and surfaces
✓ Transformation of curves and surfaces
- scaling
- translating
- rotating
- mirroring
⌛ Degree elevation / reduction
⌛ Construction of common geometries

Connectivity

✓ Determine patch connectivity
- identify interfaces between patches
- introduce per patch local numbering for vertices and edges
✓ Virtual Bezier mesh connectivty (for FEM)
- introduce on each patch a virtual Bezier mesh
- determine adjacency information of mesh cells

Concerning the B-splines itself, chapters 2-5.4 of [1] are implemented so far (adapted to 1-based indexing).

Note

Open knot vectors are assumed everywhere, if not stated otherwise.

Note

The parametric space is assumed to be $[0, 1]$ or ${[0,1]}^2$ everywhere.

Installation

Installing NURBS is done by entering the package manager (enter ] at the julia REPL) and issuing:

pkg> add NURBS

References

The implementation is based on

[1] L. Piegl, The NURBS Book, Berlin Heidelberg, Springer-Verlag, 1997.
[2] R.N. Simpson, et. al, A Two-Dimensional Isogeometric Boundary Element Method for Elastostatic Analysis, Comput. Methods Appl. Mech. Engrg., 2012.
[3] C. de Boor, A Practical Guide to Splines, revised ed., Appl. Math. Sci., vol. 27, Springer-Verlag, New York, 2001.
[4] L. Beirão da Veiga, A. Buffa, G. Sangalli, R. Vázquez, Analysis-suitable T-splines of arbitrary degree: Definition, linear independence and approximation properties, Math. Models Methods Appl. Sci. 23, 2013.