B-splines

B-splines can be derived recursively from a piecewise constant function with ${\displaystyle p=0}$:^[1]

${\displaystyle N_{i,0}(\xi )={\mathcal {I}}_{[\xi _{i},\xi _{i+1})}(s)\quad 1\leq i\leq n}$

Using De Boor's algorithm, it is possible to generate B-splines of arbitrary order ${\displaystyle p}$:^[1]

${\displaystyle N_{i,p}(\xi )={\frac {\xi -\xi _{i}}{\xi _{i+p}-\xi _{i}}}N_{i,p-1}(\xi )+{\frac {\xi _{i+p+1}-\xi }{\xi _{i+p+1}-\xi _{i+1}}}N_{i+1,p-1}(\xi )\quad 1\leq i\leq n}$

valid for both uniform and non-uniform knot vectors. For the previous formula to work properly, let the division of two zeros to be equal to zero, i.e. ${\displaystyle {\dfrac {0}{0}}:=0}$.

B-splines that are generated in this way own both the partition of unity and positivity properties, i.e.:^[1]

${\displaystyle \sum _{i=1}^{n}N_{i,p}(\xi )=1\quad \forall \xi }$

${\displaystyle N_{i,p}(\xi )\geq 0\quad \forall \xi }$

So as to calculate derivatives or order ${\displaystyle k}$ of the ${\displaystyle i^{th}}$ B-splines of degree ${\displaystyle p}$, another recursive formula can be employed:^[1]

${\displaystyle {\frac {d^{k}}{d^{k}\xi }}N_{i,p}(\xi )={\frac {p!}{(p-k)!}}\sum _{j=0}^{k}\alpha _{k,j}N_{i+j,p-k}(\xi )}$

where:

${\displaystyle \alpha _{0,0}=1}$

${\displaystyle \alpha _{k,0}={\frac {\alpha _{k-1,0}}{\xi _{i+p-k+1}-\xi _{i}}},}$

${\displaystyle \alpha _{k,j}={\frac {\alpha _{k-1,j}-\alpha _{k-1,j-1}}{\xi _{i+p+j-k+1}-\xi _{i+j}}}\quad j=1,...,k-1,}$

${\displaystyle \alpha _{k,k}={\frac {-\alpha _{k-1,j-1}}{\xi _{i+p+1}-\xi _{i+k}}}.}$

whenever the denominator of an ${\displaystyle \alpha _{j,j}}$ coefficient is zero, the entire coefficient is forced to be zero as well.

A B-spline curve can be written in the following way:^[8]

${\displaystyle {\textbf {C}}(\xi )=\sum _{i=1}^{n}N_{i,p}(\xi ){\textbf {B}}_{i}}$

where ${\displaystyle n}$ is the number of basis functions ${\displaystyle N_{i,p}}$, and ${\displaystyle {\textbf {B}}_{i}\in \mathbb {R} ^{d}}$ is the ${\displaystyle i^{th}}$ control point, with ${\displaystyle d}$ dimension of the space in which the curve is immersed.

An extension to the two dimensional case can be easily obtained from B-splines curves.^[8] In particular B-spline surfaces are introduced as:^[8]

${\displaystyle {\textbf {S}}(\xi ,\eta )=\sum _{i=1}^{n}\sum _{j=1}^{m}N_{i,p}(\xi )M_{j,q}(\eta ){\textbf {B}}_{i,j}}$

where ${\displaystyle n}$ and ${\displaystyle m}$ are the numbers of basis functions ${\displaystyle N_{i,p}}$ and ${\displaystyle M_{j,q}}$ defined on two different knot vectors ${\displaystyle \Xi =\{\xi _{1},\xi _{2},...,\xi _{n+p+1}\}}$, ${\displaystyle {\mathcal {H}}=\{\eta _{1},\eta _{2},...,\eta _{m+q+1}\}}$, ${\displaystyle {\textbf {B}}_{i,j}}$ represents now a matrix of control points (also called control net).

Finally, B-splines solids, that need three sets of B-splines basis functions and a tensor of control points, can be defined as:^[8]

${\displaystyle {\textbf {S}}(\xi ,\eta ,\zeta )=\sum _{i=1}^{n}\sum _{j=1}^{m}\sum _{k=1}^{l}N_{i,p}(\xi )M_{j,q}(\eta )L_{k,r}(\zeta ){\textbf {B}}_{i,j,k}}$

NURBS

In IGA basis functions are also employed to develop the computational domain and not only for representing the numerical solution. For this reason they should have all the properties that permit to represent the geometry in an exact way. B-splines, due to their intrinsic structure, are not able to generate properly circular shapes for instance.^[1] In order to circumvent this issue, non-uniform rational B-splines, also known as NURBS, are introduced in the following way:^[1]

${\displaystyle R_{i}^{p}(s)={\frac {N_{i,p}(s)\omega _{i}}{W(s)}}}$

where ${\displaystyle N_{i,p}}$ is a one dimensional B-spline, ${\displaystyle W(s)=\sum _{i=1}^{n}N_{i,p}(s)\omega _{i}}$ is referred to as weighting function, and finally ${\displaystyle \omega _{i}}$ is the ${\displaystyle i^{th}}$ weight.

Following the idea developed in the subsection about B-splines, NURBS curve are generated as follows:^[1]

${\displaystyle {\textbf {C}}(s)=\sum _{i=1}^{n}R_{i}^{p}(s){\textbf {B}}_{i}}$

with ${\displaystyle {\textbf {B}}_{i}}$ ${\displaystyle i^{th}}$ vector of control points.

The extension of NURBS basis functions to manifolds of higher dimensions (for instance 2 and 3) is given by:^[1]

${\displaystyle R_{i,j}^{p,q}(s,t)={\frac {N_{i,p}(s)M_{j,q}(t)\omega _{i,j}}{\sum _{a=1}^{n}\sum _{b=1}^{m}N_{a,p}(s)M_{b,q}(t)\omega _{a,b}}}}$

${\displaystyle R_{i,j,k}^{p,q,r}(s,t,w)={\frac {N_{i,p}(s)M_{j,q}(t)L_{k,r}(w)\omega _{i,j,k}}{\sum _{a=1}^{n}\sum _{b=1}^{m}\sum _{c=1}^{l}N_{a,p}(s)M_{b,q}(t)L_{c,r}(w)\omega _{a,b,c}}}}$