Mathematical utilities¶

B-Splines and NURBS¶

We start this section by recalling some basic properies about B-splines curves and surfaces. We also recall some fundamental algorithms (knot insertion and degree elevation).

For a basic introduction to the subject, we refer to the books [LP95] and [Far02].

A B-Splines family, of order , can be generated using a non-decreasing sequence of knots .

B-Splines series¶

The j-th B-Spline of order is defined by the recurrence relation:

where,

for and .

We note some important properties of a B-splines basis:

• B-splines are piecewise polynomial of degree ,
• Compact support; the support of is contained in ,
• If , then only the B-splines are non vanishing at ,
• Positivity: ,
• Partition of unity ,
• Local linear independence,
• If a knot has a multiplicity then the B-spline is at .

Knots vector families¶

There are two kind of knots vectors, called clamped and unclamped. Both families contains uniform and non-uniform sequences.

The following are examples of such knots vectors

1. Clamped knots (open knots vector)
• uniform

• non-uniform

1. Unclamped knots
• uniform

• non-uniform

B-Spline curve¶

The B-spline curve in associated to knots vector and the control polygon is defined by :

In (Fig. ref{figBSplineCurve}), we give an example of a quadratic B-Spline curve, and its corresponding knot vector and control points.

We have the following properties for a B-spline curve:

• If , then is just a B’ezier-curve,
• is a piecewise polynomial curve,
• The curve interpolates its extremas if the associated multiplicity of the first and the last knot are maximum (i.e. equal to ), i.e. open knot vector,
• Invariance with respect to affine transformations,
• Strong convex-hull property:

if , then is inside the convex-hull associated to the control points ,

• Local modification : moving the control point affects , only in the interval ,
• The control polygon approaches the behavior of the curve.

Note

In order to model a singular curve, we can use multiple control points : .

Multivariate tensor product splines¶

Let us consider knot vectors . For simplicity, we consider that these knot vectors are open, which means that knots on each side are duplicated so that the spline is interpolating on the boundary, and of bounds and . In the sequel we will use the notation . Each knot vector , will generate a basis for a Schoenberg space, . The tensor product of all these spaces is also a Schoenberg space, namely , where . The cube , will be referred to as a patch.

The basis for is defined by a tensor product :

where, .

A typical cell from is a cube of the form : .

Deriving a B-spline curve¶

The derivative of a B-spline curve is obtained as:

where , and are generated using the knot vector , which is obtained from by reducing by one the multiplicity of the first and the last knot (in the case of open knot vector), i.e. by removing the first and the last knot.

More generally, by introducing the B-splines family generated by the knots vector obtained from by removing the first and the last knot times, we have the following result:

proposition¶

The derivative of the curve is given by

where, for

By denoting and the first and second derivative of the B-spline curve , it is easy to show that:

We have,

• ,
• ,
• ,
• .

Example¶

Let us consider the quadratic B-spline curve associated to the knots vector and the control points :

we have,

where

The B-splines are associated to the knot vector .

Fundamental geometric operations

By inserting new knots into the knot vector, we add new control points without changing the shape of the B-Spline curve. This can be done using the DeBoor algorithm [dB01]. We can also elevate the degree of the B-Spline family and keep unchanged the curve [HHM05]. In (Fig. ref{refinement_curve_B_Spline}), we apply these algorithms on a quadratic B-Spline curve and we show the position of the new control points.

Knot insertion¶

After modification, we denote by the new parameters. are the new control points.

One can insert a new knot , where . For this purpose we use the DeBoor algorithm [dB01]:

Many other algorithms exist, like blossoming for fast insertion algorithm. For more details about this topic, we refer to [NT93].

Order elevation¶

We can elevate the order of the basis, without changing the curve. Several algorithms exist for this purpose. We used the one by Huang et al. [PP91], [HHM05].

A quadratic B-spline curve and its control points. The knot vector is .

The curve after a h-refinement by inserting the knots while the degree is kept equal to .

The curve after a p-refinement, the degree was raised by (using cubic B-splines).

The curve after duplicating the multiplicity of the internal knots , this leads to a B’ezier description. We can then, split the curve into pieces (sub-domains), each one will corresponds to a quadratic B’ezier curve.

Rotation¶

Todo

not yet available

Scaling¶

Todo

not yet available

References

 [dB01] (1, 2) C. de Boor. A Practical Guide to Splines. Applied Mathematical Sciences. Springer New York, 2001. ISBN 9780387953663. URL: https://books.google.de/books?id=m0QDJvBI_ecC.
 [Far02] G. Farin. Curves and surfaces for CAGD: a practical guide. Morgan Kaufmann Pub. Inc., San Francisco, CA, USA, 2002. ISBN 1-55860-737-4.
 [HHM05] (1, 2) Qi-Xing Huang, Shi-Min Hu, and Ralph R. Martin. Fast degree elevation and knot insertion for b-spline curves. Computer Aided Geometric Design, 22(2):183 – 197, 2005. URL: http://www.sciencedirect.com/science/article/B6TYN-4DXBTHR-2/2/d5b3eec2f4c230c8051623c1c000beae, doi:DOI: 10.1016/j.cagd.2004.11.001.
 [LP95] W. Tiller L. Piegl. The NURBS Book. Springer-Verlag, Berlin, Heidelberg, 1995. second ed.
 [NT93] Goldman R. N. and Lyche T. Knot Insertion and Deletion Algorithms for B-Spline Curves and Surfaces. SIAM, Philadelphia, USA, 1993. ISBN 9780898713060.
 [PP91] Hartmut Prautzsch and Bruce Piper. A fast algorithm to raise the degree of spline curves. Comput. Aided Geom. Des., 8:253–265, October 1991. URL: http://portal.acm.org/citation.cfm?id=124930.124932, doi:10.1016/0167-8396(91)90015-4.

DeRham sequences¶

Section author: A. Ratnani

here without boundary conditions

Pullbacks¶

In the case where the physical domain is the image of a logical domain by a smooth mapping (at least ), we have the following parallel diagrams

Where the mappings and are called pullbacks and are given by

where is the jacobian matrix of the mapping .

Note

The pullbacks and are isomorphisms between the corresponding spaces.

Discrete Spaces¶

Let us suppose that we have a sequence of finite subspaces for each of the spaces involved in the DeRham sequence. The discrete DeRham sequence stands for the following commutative diagram between continuous and discrete spaces

When using a Finite Elements methods, we often deal with a reference element, and thus we need also to apply the pullbacks on the discrete spaces. In fact, we have again the following parallel diagram

Since, the pullbacks are isomorphisms in the previous diagram, we can define a one-to-one correspondance

We have then, the following results

Projectors¶

In some cases, one may need to define projectors on smooth functions

Discrete DeRham sequence for B-Splines¶

Buffa et al [BSV09] show the construction of a discrete DeRham sequence using B-Splines.

1. DeRham sequence is reduced to

1. The recursion formula for derivative writes

1. we have which is a change of basis as a diagonal matrix
2. Now if $u in S^p$, with and expansion $u = sum_i u_i N_i^p$, we have

1. If we introduce the B-Splines coefficients vector for the derivative), we have

where is the incidence matrix (of entries and )

Let be the identity matrix, we have

in the 2D case:

in the 3D case:

References

 [BSV09] A. Buffa, G. Sangalli, and R. Vazquez. Isogeometric analysis in electromagnetics: b-splines approximation. Comput. Methods Appl. Mech. Engrg, 199:1143–1152, 2009.

GLT¶

Where do the GLTs come from?¶

The main aim of this paragraph is to present a crucial example that highlights the importance of the GLT algebra when dealing with linear systems coming from the discretization of PDEs. Let us start with some preliminaries. In detail, we will recall the notion of symbol of a matrix-sequence and the basic idea behind the GLT theory.

Spectral preliminaries¶

The following one is a rather informal definition of symbol of a matrix-sequence.

example:

When , , , means

References

 [GMP+14] Carlo Garoni, Carla Manni, Francesca Pelosi, Stefano Serra-Capizzano, and Hendrik Speleers. On the spectrum of stiffness matrices arising from isogeometric analysis. Numerische Mathematik, 127(4):751–799, 2014. URL: http://dx.doi.org/10.1007/s00211-013-0600-2, doi:10.1007/s00211-013-0600-2.