Welcome to spl’s documentation!¶

SPL is a Python/Fortran 2003 library for B-Splines/NURBS and Computer Aided Design Algorithms.

SPL can be used in three different ways:

Fortran 90/95 subroutines through the file fortran/src/bsplines/bspline.F90
Fortran 2003 objects, mainly the mapping and cad objects
The same objects as in 2. but through Python

First Steps with SPL¶

This document is meant to give a tutorial-like overview of SPL.

The green arrows designate “more info” links leading to advanced sections about the described task.

By reading this tutorial, you’ll be able to:

compile a simple SPL file
get familiar with parallel programing paradigms
create, modify and build a SPL project.

Install SPL¶

Examples¶

In this section, we describe some features of SPL on simple examples.

See script.

Dive into SPL¶

Contents¶

Introduction¶

Input and Output¶

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, $(N_i)_{ 1 \leqslant i \leqslant n}$ of order $k$ , can be generated using a non-decreasing sequence of knots $T=(t_i)_{1\leqslant i \leqslant n + k}$ .

B-Splines series¶

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

$N_j^k = w_j^k N_j^{k-1} + ( 1 - w_{j+1}^k ) N_{j+1}^{k-1}$

where,

$w_j^k (x) = \frac{x-t_j}{t_{j+k-1}-t_{j}} \hspace{2cm} N_j^1(x) = \chi_{ \left[ t_j, t_{j+1} \right[ }(x)$

for $k \geq 1$ and $1 \leq j \leq n$ .

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

B-splines are piecewise polynomial of degree $p=k-1$ ,
Compact support; the support of $N_j^k$ is contained in $\left[ t_j, t_{j+k} \right]$ ,
If $x \in~ ] t_j,t_{j+1} [$ , then only the B-splines $\{ N_{j-k+1}^k,\cdots,N_{j}^k \}$ are non vanishing at $x$ ,
Positivity: $\forall j \in \{1,\cdots,n \}~~N_j(x) >0, ~~\forall x \in ] t_j, t_{j+k} [$ ,
Partition of unity $\sum_{i=1}^n N_i^{k}(x) = 1, \forall x \in \mathbb{R}$ ,
Local linear independence,
If a knot $t_i$ has a multiplicity $m_i$ then the B-spline is $\mathcal{C}^{(p-m_i)}$ at $t_i$ .

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

Clamped knots (open knots vector)

uniform

$T_1 &= \{0, 0, 0, 1, 2, 3, 4, 5, 5, 5 \} \\ T_2 &= \{-0.2, -0.2, 0.0, 0.2, 0.4, 0.6, 0.8, 0.8 \}$

non-uniform

$T_3 &= \{0, 0, 0, 1, 3, 4, 5, 5, 5 \} \\ T_4 &= \{-0.2, -0.2, 0.4, 0.6, 0.8, 0.8 \}$

Unclamped knots

uniform

$T_5 &= \{0, 1, 2, 3, 4, 5, 6, 7 \} \\ T_6 &= \{-0.2, 0.0, 0.2, 0.4, 0.6, 0.8, 1.0 \}$

non-uniform

$T_7 &= \{0, 0, 3, 4, 7, 8, 9 \} \\ T_8 &= \{-0.2, 0.2, 0.4, 0.6, 1.0, 2.0, 2.5 \}$

B-Spline curve¶

The B-spline curve in $\mathbb{R}^d$ associated to knots vector $T=(t_i)_{1\leqslant i \leqslant n + k}$ and the control polygon $(\mathbf{P}_i)_{ 1 \leqslant i \leqslant n}$ is defined by :

$\mathcal{C}(t) = \sum_{i=1}^n N_i^k(t) \textbf{P}_i$

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 $n=k$ , then $\mathcal{C}$ is just a B’ezier-curve,
$\mathcal{C}$ 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 $k$ ), i.e. open knot vector,
Invariance with respect to affine transformations,
Strong convex-hull property:

if $t_i \leq t \leq t_{i+1}$ , then $\mathcal{C}(t)$ is inside the convex-hull associated to the control points $\mathbf{P}_{i-p},\cdots,\mathbf{P}_{i}$ ,

Local modification : moving the $i^{th}$ control point $\mathbf{P}_{i}$ affects $\mathcal{C}(t)$ , only in the interval $[t_i,t_{i+k}]$ ,
The control polygon approaches the behavior of the curve.

Note

In order to model a singular curve, we can use multiple control points : $\mathbf{P}_{i}=\mathbf{P}_{i+1}$ .

Multivariate tensor product splines¶

Let us consider $d$ knot vectors $\mathcal{T} = \{T^1,T^2,\cdots,T^d\}$ . For simplicity, we consider that these knot vectors are open, which means that $k$ knots on each side are duplicated so that the spline is interpolating on the boundary, and of bounds $0$ and $1$ . In the sequel we will use the notation $I=[0,1]$ . Each knot vector $T^i$ , will generate a basis for a Schoenberg space, $\mathcal{S}_{k_{i}}(T^i,I)$ . The tensor product of all these spaces is also a Schoenberg space, namely $\mathcal{S}_{\mathbf{k}}(\mathcal{T})$ , where $\mathbf{k}=\{k_1,\cdots,k_d\}$ . The cube $\mathcal{P}=I^d=[0,1]^d$ , will be referred to as a patch.

The basis for $\mathcal{S}_{\mathbf{k}}(\mathcal{T})$ is defined by a tensor product :

$N_{\mathbf{i}}^{\mathbf{k}} := N_{i_1}^{k_1} \otimes N_{i_2}^{k_2} \otimes \cdots \otimes N_{i_d}^{k_d}$

where, $\mathbf{i}=\{i_1,\cdots , i_d \}$ .

A typical cell from $\mathcal{P}$ is a cube of the form : $Q_{\mathbf{i}}=[\xi_{i_1}, \xi_{i_1+1}] \otimes \cdots \otimes [\xi_{i_d}, \xi_{i_d+1}]$ .

Deriving a B-spline curve¶

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

$\mathcal{C}^{\prime}(t) = \sum_{i=1}^{n} {N_{i}^{k}}^{\prime}(t) \mathbf{P}_i = \sum_{i=1}^{n} \left(\frac{p}{t_{i+p}-t_{i}}N_{i}^{k-1}(t) \mathbf{P}_i - \frac{p}{t_{i+1+p}-t_{i+1}}N_{i+1}^{k-1}(t) \mathbf{P}_i \right) = \sum_{i=1}^{n-1} {N_{i}^{k-1}}^{\ast}(t) \mathbf{Q}_i$

where $\mathbf{Q}_i = p \frac{\mathbf{P}_{i+1} - \mathbf{P}_i}{t_{i+1+p}-t_{i+1}}$ , and $\{{N_{i}^{k-1}}^{\ast},~~1 \leq i \leq n-1\}$ are generated using the knot vector $T^{\ast}$ , which is obtained from $T$ 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 $\{ {N_{i}^{k-j}}^{\ast}, 1 \leq i \leq n-j \}$ generated by the knots vector $T^{j^{\ast}}$ obtained from $T$ by removing the first and the last knot $j$ times, we have the following result:

proposition¶

The $j^{th}$ derivative of the curve $\mathcal{C}$ is given by

$\mathcal{C}^{(j)}(t) = \sum_{i=1}^{n-j} {N_{i}^{k-j}}^{\ast}(t) \mathbf{P}_i^{(j)}`$

where, for $j>0$

$\mathbf{P}_i^{(j)} = \frac{p-j+1}{t_{i+p+1}-t_{i+j}} \left( \mathbf{P}_{i+1}^{(j-1)} - \mathbf{P}_i^{(j-1)} \right) \\ \mbox{and} ~ ~ ~ \mathbf{P}_i^{(0)} = \mathbf{P}_i.$

By denoting $\mathcal{C}^{\prime}$ and $\mathcal{C}^{\prime\prime}$ the first and second derivative of the B-spline curve $\mathcal{C}$ , it is easy to show that:

We have,

$\mathcal{C}^{\prime}(0) = \frac{p}{t_{p+2}} \left(\mathbf{P}_{2} - \mathbf{P}_1\right)$ ,
$\mathcal{C}^{\prime}(1) = \frac{p}{1-t_{n}} \left(\mathbf{P}_{n} - \mathbf{P}_{n-1}\right)$ ,
$\mathcal{C}^{\prime\prime}(0) = \frac{p(p-1)}{t_{p+2}} \left( \frac{1}{t_{p+2}}\mathbf{P}_{1} - \{ \frac{1}{t_{p+2}} + \frac{1}{t_{p+3}} \} \mathbf{P}_2 + \frac{1}{t_{p+3}}\mathbf{P}_{3} \right)$ ,
$\mathcal{C}^{\prime\prime}(1) = \frac{p(p-1)}{1-t_{n}} \left( \frac{1}{1-t_{n}}\mathbf{P}_{n} - \{ \frac{1}{1-t_{n}} + \frac{1}{1-t_{n-1}} \} \mathbf{P}_{n-1} + \frac{1}{1-t_{n-1}}\mathbf{P}_{n-2} \right)$ .

Example¶

Let us consider the quadratic B-spline curve associated to the knots vector $T=\{000~\frac{2}{5}~\frac{3}{5}~111 \}$ and the control points $\{ P_i, 1 \leq i \leq 5 \}$ :

$\mathcal{C}(t) = \sum_{i=1}^{5} {N_{i}^{3}}^{\prime}(t) \mathbf{P}_i$

we have,

$\mathcal{C}^{\prime}(t) = \sum_{i=1}^{4} {N_{i}^{2}}^{\ast}(t) \mathbf{Q}_i$

where

$\mathbf{Q}_1 = 5 \{\mathbf{P}_{2} - \mathbf{P}_1\}, ~~~~\mathbf{Q}_2 = \frac{10}{3} \{ \mathbf{P}_{3} - \mathbf{P}_2\}, \\ \mathbf{Q}_3 = \frac{10}{3} \{ \mathbf{P}_{4} - \mathbf{P}_3\},~~~~\mathbf{Q}_4 = 5 \{\mathbf{P}_{5} - \mathbf{P}_4\}.$

The B-splines $\{ {N_{i}^{2}}^{\ast},~~1 \leq i \leq 4\}$ are associated to the knot vector $T^{\ast}=\{00~\frac{2}{5}~\frac{3}{5}~11 \}$ .

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 $\widetilde{n}, \widetilde{k}, \widetilde{T}$ the new parameters. $(\textbf{Q}_i)$ are the new control points.

One can insert a new knot $t$ , where $t_j \leqslant t < t_{j+1}$ . For this purpose we use the DeBoor algorithm [dB01]:

$\widetilde{n} = n+1 \\ \widetilde{k} = k \\ \widetilde{T} = \{ t_1,.., t_j, t, t_{j+1},.., t_{n+k}\} \\ \alpha_i = \left\{\begin{array}{cc}1 & 1 \leqslant i \leqslant j-k+1 \\\frac{t-t_i}{t_{i+k-1}-t_i} & j-k+2 \leqslant i \leqslant j \\0 & j+1 \leqslant i \end{array}\right. \\ \textbf{Q}_i = \alpha_i \textbf{P}_i + (1-\alpha_i) \textbf{P}_{i-1}$

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 $T = \{ 000, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1 1 1 \}$ .

The curve after a h-refinement by inserting the knots $\{ 0.15, 0.35\}$ while the degree is kept equal to $2$ .

The curve after a p-refinement, the degree was raised by $1$ (using cubic B-splines).

The curve after duplicating the multiplicity of the internal knots $\{ \frac{1}{4}, \frac{1}{2}, \frac{3}{4} \}$ , this leads to a B’ezier description. We can then, split the curve into $4$ pieces (sub-domains), each one will corresponds to a quadratic B’ezier curve.

Translation¶

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.

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 $d_n=n$ , $d=1$ , $D=[0,\pi]$ , $\{A_n\}_n\sim_\lambda f$ means

References

Exterior Algebra¶

Let $V$ be a real vector space of dimension $n$ .

Definition, Alternating algebraic forms:

For each $k$ , we define $\Alt^k V$ as the space of alternating $k$ -linear maps $V \times \cdots \times V \rightarrow \mathbb{R}$ .

Note

$\Alt^0 = \mathbb{R}$ ,
$\Alt^1 = V^{*}$ is the dual space of $V$ (the space of covectors)

Definition, Exterior product:

For $\omega \in \Alt^j$ and $\eta \in \Alt^k$ , their exterior (wedge) product is given by:

$(\omega \wedge \eta ) (v_1, \cdots, v_{j+k}) = \sum_{\sigma} (\mathrm{sign}~ \sigma) \omega (v_{\sigma(1)}, \cdots, v_{\sigma(j)}) \eta (v_{\sigma(j+1)}, \cdots, v_{\sigma(j+k)})$

for all $v_i \in V$ . Where the sum is over all permutations $\sigma$ of $\{ 1,\cdots,j+k \}$ , for which $\sigma(1)< \cdots <\sigma(j)$ and $\sigma(j+1)< \cdots <\sigma(j+k)$ .

Note

The exterior product is bilinear, associative,
anti-commutative: $\eta \wedge \omega = (-1)^{jk} \omega \wedge \eta$ for all $\omega \in \Alt^j$ and $\eta \in \Alt^k$ .

Definition, Grassmann Algebra:

Grassmann Algebra is defined by:

$\Alt V := \bigoplus_k \Alt^k V$

This is a anti-commutative graded algebra. Also called Exterior Algebra of $V^{*}$

In the case of $V=\mathbb{R}^n$ , we have:

$\Alt V^0 \sim \mathbb{R}$ ,
$\Alt V^1 \sim \mathbb{R}^n$ ,
$\Alt V^{n-1} \sim \mathbb{R}^n$ , using Riesz representation theorem,
$\Alt V^n \sim \mathbb{R}$ , using the map $v \longmapsto \det(v,v_1,\cdots,v_{n-1})$ .

Basis¶

Let $v_1,\cdots,v_n$ be a basis of $V$ and $\mu_1,\cdots,\mu_n$ the associated dual basis for $V^*$ ( $\mu_i(v_j) = \delta_{ij}$ ).

For any increasing permutations $\sigma, \rho : \{ 1,\cdots,k \} \longrightarrow \{ 1,\cdots,n \}$ , we have:

$\mu_{\sigma(1)} \wedge \cdots \wedge \mu_{\sigma(k)} (v_{\rho(1)}, \cdots, v_{\rho(k)}) = \chi_{\sigma,\rho}$

thus the $\binom {n}{k}$ algebraic $k$ -forms $\mu_{\sigma(1)} \wedge \cdots \wedge \mu_{\sigma(k)}$ , form a basis for $\Alt^k V$ and $\dim \Alt^k V = \binom {n}{k}$ .

Definition, Interior product:

Let $\omega$ be a $k$ -form, and $v \in V$ . The interior product of $\omega$ and $v$ is the $(k-1)$ -form $\omega \lrcorner v$ defined by:

$\omega \lrcorner v (v_1,\cdots,v_{k-1}) = \omega (v,v_1,\cdots,v_{k-1})$

We have for $\omega \in \Alt^k V$ , $\eta \in \Alt^l V$ and $v \in V$ :

$(\omega \wedge \eta) \lrcorner v = (\omega \lrcorner v)\wedge \eta + (-1)^k \omega \wedge (\eta \lrcorner v)$

Definition, Inner product:

If $V$ is has an inner product, then $\Alt^k V$ is endowed with an inner product given by:

$(\omega , \eta) = \sum_{\rho} \omega (v_{\rho(1)}, \cdots, v_{\rho(k)}) \eta (v_{\rho(1)}, \cdots, v_{\rho(k)}), ~~~\forall \omega, \eta \in \Alt^k V.$

where the sum is over increasing sequences $\rho : \{ 1,\cdots,k \} \longrightarrow \{ 1,\cdots,n \}$ , and $v_1, \cdots,v_n$ is any orthonormal basis.

Orientation and Volume form¶

Todo

add Orientation and Volume form

Definition, Pullback:

A linear transformation of vector spaces $L: V \rightarrow W$ induces a transformation $L^{*}: \Alt W \rightarrow \Alt V$ , called the pullback, and given by:

$L^{*} \omega (v_{1}, \cdots, v_{k}) = \omega (L v_{1}, \cdots, L v_{k}),~~~~~ \forall \omega \in \Alt^k W,~~~ v_{1}, \cdots, v_{k} \in V$

The pullback acts contravariantly: if $U \xrightarrow{~K~} V \xrightarrow{~L~} W$ then,

$\Alt W \xrightarrow{~K^{*}~} \Alt V \xrightarrow{~L^{*}~} \Alt U$
$L^{*} (\omega \wedge \eta) = L^{*} \omega \wedge L^{*} \eta$

Let V be a subspace of W. For the inclusion $\imath_V : V \longrightarrow W$ , we can define its pullback $\imath_V^{*}$ : this is a surjection of $\Alt W$ onto $\Alt V$ .

If W has an inner product and $\pi_V : W \longrightarrow V$ is the orthogonal projection. We can define its pullback $\pi_V^{*}$ : this an injection of $\Alt V$ onto $\Alt W$ .

Let us consider the composition : $W$ shortstack{ $\pi_V$ \ $\longrightarrow$ } $V$ shortstack{ $\imath_V$ \ $\longrightarrow$ } $W$ , and its pullback $\pi_V^* \imath_V^*$ .

Definition, The tangential and normal parts:

$\pi_V^* \imath_V^*$ associates for each $\omega \in \Alt^k$ its tangential part $\omega_{\parallel}$ with respect to $V$ :

$(\pi_V^* \imath_V^* \omega) (v_1,\cdots,v_k) = \omega (\pi_V v_1, \cdots, \pi_V v_k), ~~~~~\forall v_1,\cdots,v_k \in W.$

$\omega - \pi_V^* \imath_V^* \omega$ associates for each $\omega \in \Alt^k$ its normal part $\omega_{\perp}$ with respect to $V$ .

The tangential part of $\omega$ vanishes if and only if the image of $\omega$ in $\Alt^k V$ vanishes.

Let $V$ be an oriented inner product space, with volume form $\mbox{vol}$ . Let $\omega \in \Alt^k V$ . We can define a new linear map $L_{\omega}$ as the composition of $\Alt^{n-k} V \longrightarrow \Alt^n V$ such as:

$\mu \longmapsto \omega \wedge \mu$

and the canonical isomorphism of $\Alt^n V$ onto $\mathbb{R}$ , and using the Riesz representation theorem, there exists an element $\star \omega \in \Alt^{n-k} V$ such that : $L_{\omega} (\mu) = (\star \omega , \mu)$ , i.e.:

$\omega \wedge \mu = (\star \omega , \mu) \mbox{vol}, ~~~\omega \in \Alt^{k}, ~\mu \in \Alt^{n-k}$

Definition, The Hodge star operation:

The linear map which maps $\Alt^k V$ onto $\Alt^{n-k} V$ $\omega \longmapsto \star \omega$ is called the Hodge star operator.

If $e_1,\cdots,e_n$ is any positively oriented orthonormal basis, and $\sigma$ a permutation, we have

$\omega(e_{\sigma(1)}, \cdots, e_{\sigma(k)}) = (\mathrm{sign} \sigma) \star \omega(e_{\sigma(k+1)}, \cdots, e_{\sigma(n)})$

$\star \star \omega = (-1)^{k(n-k)} \omega, ~~~\forall \omega \in \Alt^k V$ , thus the Hodge star is an isometry.
$(\star \omega)_{\parallel} = \star (\omega_{\perp})$ and $(\star \omega)_{\perp} = \star (\omega_{\parallel})$
the image of $\star \omega$ in $\Alt^k V$ vanishes if and only if $\omega_{\perp}$ vanishes.

$\begin{tabular}{|c|l|} \hline $\Alt^0 \mathbb{R}^3 \cong \mathbb{R}$ & $c \leftrightarrow c$ \\ % \hline $\Alt^1 \mathbb{R}^3 \cong \mathbb{R}^3$ & $u_1 \diff x_1 + u_2 \diff x_2 + u_3 \diff x_3 \leftrightarrow u$ \\ % \hline $\Alt^2 \mathbb{R}^3 \cong \mathbb{R}^3$ & $u_3 \diff x_1 \wedge \diff x_2 - u_2 \diff x_1 \wedge \diff x_3 + u_1 \diff x_2 \wedge \diff x_3 + \leftrightarrow u$ \\ % \hline $\Alt^3 \mathbb{R}^3 \cong \mathbb{R}$ & $c \diff x_1 \wedge \diff x_2 \wedge \diff x_3 \leftrightarrow c$ \\ \hline \end{tabular}$

$\begin{tabular}{|c|l|} \hline $ \wedge : \Alt^1 \mathbb{R}^3 \times \Alt^1 \mathbb{R}^3 \longrightarrow \Alt^2 \mathbb{R}^3$ & $\times : \mathbb{R}^3 \times \mathbb{R}^3 \longrightarrow \mathbb{R}^3$ \\ $ \wedge : \Alt^1 \mathbb{R}^3 \times \Alt^2 \mathbb{R}^3 \longrightarrow \Alt^3 \mathbb{R}^3$ & $\cdot : \mathbb{R}^3 \times \mathbb{R}^3 \longrightarrow \mathbb{R}$ \\ \hline \end{tabular}$

$\begin{tabular}{|c|l|} \hline $ L^* : \Alt^0 \mathbb{R}^3 \longrightarrow \Alt^0 \mathbb{R}^3 $ & $\id : \mathbb{R} \longrightarrow \mathbb{R}$ \\ $ L^* : \Alt^1 \mathbb{R}^3 \longrightarrow \Alt^1 \mathbb{R}^3 $ & $L^T : \mathbb{R}^3 \longrightarrow \mathbb{R}^3$ \\ $ L^* : \Alt^2 \mathbb{R}^3 \longrightarrow \Alt^2 \mathbb{R}^3 $ & $(\det L )L^{-1} : \mathbb{R}^3 \longrightarrow \mathbb{R}^3$ \\ $ L^* : \Alt^3 \mathbb{R}^3 \longrightarrow \Alt^3 \mathbb{R}^3 $ & $(\det L) : \mathbb{R} \longrightarrow \mathbb{R}$ ~~~($c \longmapsto c \det L$) \\ \hline \end{tabular}$

$\begin{tabular}{|c|l|} \hline $ \lrcorner v : \Alt^1 \mathbb{R}^3 \longrightarrow \Alt^0 \mathbb{R}^3 $ & $v \cdot : \mathbb{R}^3 \longrightarrow \mathbb{R}$ \\ $ \lrcorner v : \Alt^2 \mathbb{R}^3 \longrightarrow \Alt^1 \mathbb{R}^3 $ & $v \times : \mathbb{R}^3 \longrightarrow \mathbb{R}^3$ \\ $ \lrcorner v : \Alt^3 \mathbb{R}^3 \longrightarrow \Alt^2 \mathbb{R}^3 $ & $v : \mathbb{R} \longrightarrow \mathbb{R}^3$ ~~~($c \longmapsto c v$) \\ \hline \end{tabular}$

$\begin{tabular}{|c|l|} \hline inner product on $\Alt^k \mathbb{R}^3$ induced & dot product on $\mathbb{R}$ and $\mathbb{R}^3$ \\ by dot product on $\mathbb{R}^3$ & \\ $\volume = \diff x_1 \wedge \diff x_2 \wedge \diff x_3$ & $(v_1,v_2,v_3) \longmapsto \det(v_1|v_2|v_3)$ \\ \hline \end{tabular}$

$\begin{tabular}{|c|l|} \hline $ \star : \Alt^0 \mathbb{R}^3 \longrightarrow \Alt^3 \mathbb{R}^3 $ & $\id : \mathbb{R} \longrightarrow \mathbb{R}$ \\ $ \star : \Alt^1 \mathbb{R}^3 \longrightarrow \Alt^2 \mathbb{R}^3 $ & $\id : \mathbb{R}^3 \longrightarrow \mathbb{R}^3$ \\ \hline \end{tabular}$

Exterior Calculus on manifolds and Differential forms¶

Let $\Omega$ be a smooth manifold, of dimension $n$ .

$\forall x \in \Omega$ we denote by $T_x \Omega$ the tangent space. This is a vector space of dimension $n$ ,
tangent bundle $\{ (x,v), ~~ x \in \Omega, v \in T_x \Omega \}$ ,
Applying the exterior algebra to the tangent spaces, we obtain the exterior forms bundle, whose elements are pairs $(x,\mu)$ with $x \in \Omega$ and $\mu \in \Alt^k T_x \Omega$ .
a differential $k$ -form $\omega$ is a section of this bundle. This is a map which associates to each $x \in \Omega$ an element $\omega_x \in \Alt^k T_x \Omega$ ,
if the map $\mathcal{L}_{\omega}^k : x \longmapsto \omega_x (v_1(x), \cdots, v_k(x))$ is smooth (whenever $v_i$ are smooth), we say that $\omega$ is a smooth differential $k$ -form,
we define $\Lambda^k(\Omega)$ the space of all smooth $k$ -forms on $\Omega$ ,
$\Lambda^0(\Omega) = \mathcal{C}^{\infty}(\Omega)$ ,
if the map $\mathcal{L}_{\omega}^k$ is $\mathcal{C}^{m}(\Omega)$ , we define differential $k$ -forms with less smoothness $\mathcal{C}^{m} \Lambda^k (\Omega)$ .

Let $\Omega$ be a smooth manifold, of dimension $n$ .

Exterior product:

if $\omega \in \Lambda^k(\Omega)$ and $\eta \in \Lambda^j(\Omega)$ , we may define $\omega \wedge \eta$ as $(\omega \wedge \eta)_x = \omega_x \wedge \eta_x$ and the Grassmann algebra $\Lambda(\Omega) := \bigoplus_k \Lambda^k(\Omega)$

Differential forms can be differentiated and integrated, without recourse to any additional structure, such as a metric or a measure.

Exterior differentiation:

For each $\omega \in \Lambda^k(\Omega)$ , can define the $(k+1)$ -form $\diff \omega \in \Lambda^{k+1}(\Omega)$ , such as:

$\diff\omega_x(v_1,\cdots,v_{k+1}) = \sum_{j=1}^{k+1} (-1)^j \partial_{v_j} \omega_x(v_1,\cdots,\hat{v_j},\cdots,v_{k+1})$

where the hat is used to indicated a suppressed argument.

This defines a graded linear operator of degree $+1$ , of $\Lambda(\Omega)$ onto $\Lambda(\Omega)$ .

We have the following properties:

$\diff \circ \diff = 0$
$\diff (\omega \wedge \eta) = \diff \omega \wedge \eta + (-1)^k \omega \wedge \diff \eta, ~~\forall \omega \in \Lambda^k(\Omega), \eta \in \Lambda^j(\Omega)$ ,
(Pullback) let $\phi$ be a smooth map of $\Omega$ onto $\Omega^{\prime}$ . Then $\phi^*(\omega \wedge \eta) = \phi^*(\omega) \wedge \phi^*(\eta)$ and $\phi^* (\diff \omega) = \diff (\phi^* \omega)$ ,
(Interior product) the interior product of a differential $k$ -form $\omega$ with a vector field $v$ ,
we obtain a $(k-1)$ -form by : $(\omega \lrcorner v)_x := \omega_x \lrcorner v_x$ ,
(Trace operator) the pullback $i_{\partial \Omega}^*$ of $i_{\partial \Omega}$ is the trace operator $\trace$

Integration:

If is an oriented, piecewise smooth -dimensional submanifold of , and is a continuous -form, then th integral is well defined :
- [0-forms] can be evaluated at points,
- [1-forms] can be integrated over directed curves,
- [2-forms] can be integrated over directed surfaces,
(Inner product) The $L^2$ -inner product of two differential $k$ -forms on an oriented Riemannian manifold $\Omega$ is defined as :

$(\omega,\eta)_{L^2 \Lambda^k} = \int_{\Omega} (\omega_x,\eta_x) \volume = \int \omega \wedge \star \eta$

The completion of $\Lambda^k(\Omega)$ in the corresponding norm defines the Hilbert space $L^2 \Lambda^k(\Omega)$ .

We have the following results:

(Integration) if $\phi$ is an orientation-preserving diffeomorphism, then

$\int_{\Omega} \phi^* \omega = \int_{\Omega^{\prime}} \omega, ~~~ \forall \omega \in \Lambda^n(\Omega^{\prime})$

Theorem, Stokes theorem:

If $\Omega$ is an oriented $n$ -manifold with boundary $\partial \Omega$ , then

$\int_{\Omega} \diff \omega = \int_{\partial \Omega} \trace \omega, ~~~ \forall \omega \in \Lambda^{n-1}(\Omega)$

Theorem, Integration by parts:

If $\Omega$ is an oriented $n$ -manifold with boundary $\partial \Omega$ , then

$\int_{\Omega} \diff \omega \wedge \eta = (-1)^{k-1} \int_{\Omega} \omega \wedge \diff \eta + \int_{\partial \Omega} \trace \omega \wedge \trace \eta, ~~~ \forall \omega \in \Lambda^{k}(\Omega), \eta \in \Lambda^{n-k-1}(\Omega)$

Sobolev spaces of differential forms¶

As for the classical case, we can define the Sobolev spaces as:

$H^s \Lambda^k(\Omega)$ is the space of differential $k$ -forms such that $\mathcal{L}_{\omega}^k \in H^s(\Omega)$ .
$H \Lambda^k(\Omega) = \{ \omega \in L^2 \Lambda^k(\Omega),~~ \diff \omega \in L^2 \Lambda^{k+1}(\Omega) \}$ . The associated norm is :

$\| \omega \|_{H \Lambda^k}^2 = \| \omega \|_{H \Lambda}^2 := \| \omega \|_{L^2 \Lambda^k}^2 + \| \diff \omega \|_{L^2 \Lambda^{k+1}}^2$

$H \Lambda^{0}(\Omega)$ coincides with $H^1 \Lambda^{0}(\Omega)$ ,
$H \Lambda^{n}(\Omega)$ coincides with $L^2 \Lambda^{n}(\Omega)$ ,
for $0 < k < n$ , we have $H^1 \Lambda^k(\Omega) \subset H \Lambda^k(\Omega) \subset L^2 \Lambda^k(\Omega)$ , strictly.

$\begin{tabular}{|c|c c c c c|} \hline $k$ & $\Lambda^k$ & $H \Lambda^k$ & $\diff \omega$ & $\int_f \omega$ & $\kappa \omega$ \\ \hline & & & & & \\ 0 & $\mathcal{C}^{\infty}$ & $H^1$ & $\nabla \omega$ & $\omega(f)$ & $0$ \\ 1 & $\mathcal{C}^{\infty}(\mathbb{R}^3)$ & $H(\rots,\mathbb{R}^3)$ & $\rots \omega$ & $\int_f \omega \cdot t \diff \mathcal{H}_1$ & $x \longmapsto x \cdot \omega(x)$ \\ 2 & $\mathcal{C}^{\infty}(\mathbb{R}^3)$ & $H(\divs, \mathbb{R}^3)$ & $\divs \omega$ & $\int_f \omega \cdot n \diff \mathcal{H}_2$ & $x \longmapsto x \times \omega(x)$ \\ 3 & $\mathcal{C}^{\infty}$ & $L^2$ & $0$ & $\int_f \omega \diff \mathcal{H}_3$ & $x \longmapsto x \omega(x)$ \\ & & & & & \\ \hline \end{tabular}$

Cohomology and De Rham Complex¶

The De Rham complex is the sequence of spaces and mappings

$0 \xrightarrow{\quad} \Lambda^0(\Omega) \xrightarrow{~\diff~} \Lambda^1(\Omega) \xrightarrow{~\diff~} \cdots \xrightarrow{~\diff~} \Lambda^n(\Omega) \xrightarrow{\quad} 0$

Since, $\diff \circ \diff = 0$ , we have

$\mathcal{R}(\diff : \Lambda^{k-1}(\Omega) \longrightarrow \Lambda^k(\Omega)) \subset \mathcal{N}(\diff : \Lambda^{k}(\Omega) \longrightarrow \Lambda^{k+1}(\Omega))$

If $\Omega$ is an oriented Riemannian manifold, we have the following cohomology:

$0 \xrightarrow{\quad} H \Lambda^0(\Omega) \xrightarrow{~\diff~} H \Lambda^1(\Omega) \xrightarrow{~\diff~} \cdots \xrightarrow{~\diff~} H \Lambda^n(\Omega) \xrightarrow{\quad} 0$

The coderivative operator $\delta : \Lambda^{k}(\Omega) \longrightarrow \Lambda^{k-1}(\Omega)$ is defined as:

$\star \delta \omega = (-1)^k \diff \star \omega,~~~ \omega \in \Lambda^k(\Omega)$

we have

$(\diff \omega , \eta ) = (\omega , \delta \eta ) + \int_{\partial \Omega} \trace \omega \wedge \trace \eta, ~~~ \forall \omega \in \Lambda^{k}(\Omega), \eta \in \Lambda^{k+1}(\Omega),$

$\delta$ is a graded linear operator of degree $-1$ .
$\delta$ is the formal adjoint of $\diff$ whenever $\omega$ or $\eta$ vanishes near the boundary.
we define the spaces

$H^* \Lambda^k(\Omega) = \{ \omega \in L^2 \Lambda^k(\Omega),~~ \delta \omega \in L^2 \Lambda^{k-1}(\Omega) \}.$

we have $H^* \Lambda^k(\Omega) = \star H \Lambda^{n-k}(\Omega)$ .

we obtain the dual complex

$0 \xleftarrow{\quad} H^* \Lambda^0(\Omega) \xleftarrow{~\delta~} H^* \Lambda^1(\Omega) \xleftarrow{~\delta~} \cdots \xleftarrow{~\delta~} H^* \Lambda^n(\Omega) \xleftarrow{\quad} 0$

Cohomology with boundary conditions¶

Let $\Lambda_0^k(\Omega)$ be the subspace of $\Lambda^k(\Omega)$ of smooth $k$ -forms with compact support. We have $\diff \Lambda_0^k \subset \Lambda_0^{k+1}$ .

The De Rham complex with the compact support is

$0 \xrightarrow{\quad} \Lambda^0_0(\Omega) \xrightarrow{~\diff~} \Lambda^1_0(\Omega) \xrightarrow{~\diff~} \cdots \xrightarrow{~\diff~} \Lambda^n_0(\Omega) \xrightarrow{\quad} 0$

Recall that the closure of $\Lambda_0^k(\Omega)$ in $H \Lambda^k(\Omega)$ is

$H_0 \Lambda^k(\Omega) = \{ \omega \in H \Lambda^k(\Omega),~~ \trace \omega =0\}.$

The $L^2$ version of the last complex is

$0 \xrightarrow{\quad} H_0 \Lambda^0(\Omega) \xrightarrow{~\diff~} H_0 \Lambda^1(\Omega) \xrightarrow{~\diff~} \cdots \xrightarrow{~\diff~} H_0 \Lambda^n(\Omega) \xrightarrow{\quad} 0$

Definition, Harmonic forms:

The harmonic $k$ -forms are the differential $k$ -forms that verify the differential equations

$\left\{ \begin{aligned} \diff \omega &=& 0,\\ \delta \omega &=& 0,\\ \trace \star \omega &=& 0.\\ \end{aligned} \right.$

this defines the following space,

$\mathfrak{H}^k (\Omega) = \{ \omega \in H \Lambda^k(\Omega) \cap H_0^* \Lambda^k(\Omega),~~\diff \omega = 0, \delta \omega = 0 \}$

We can also define the following space,

$\mathfrak{H}_0^k (\Omega) = \{ \omega \in H_0 \Lambda^k(\Omega) \cap H^* \Lambda^k(\Omega),~~\diff \omega = 0, \delta \omega = 0 \}$

As we can see, $\star \mathfrak{H}^k (\Omega) = \mathfrak{H}_0^{n-k} (\Omega)$ .

Proposition, Poincaré duality:

There is an isomorphism between the $k$ th De Rham cohomology space and the $(n-k)$ th cohomology space with boundary conditions.

Homological Algebra and Hilbert complexes¶

Homological Algebra¶

A cochain complex is a sequence of vector spaces and linear maps

$k$ -cocycles $\mathfrak{Z}^k := \mathcal{N}(d_k)$ ,
$k$ -coboundaries $\mathfrak{B}^k := \mathcal{R}(d_{k-1})$ ,
$k$ -cohomology $\mathcal{H}^k(V) := \mathfrak{Z}^k / \mathfrak{B}^k$ ,
we say that the sequence is exact, if the cohomology vanishes (i.e. $\forall~k,~~ \mathcal{H}^k(V) = \{0\}$ ),
Given two cochain complexes $V,V^{\prime}$ , a cochain map $f =(f_k)$ (such as $\diff^{\prime}_k f_k = f_{k+1} \diff_k$ )

$\begin{array}{ccccccccc} \cdots & \longrightarrow & V_{k-1} & \mbox{\shortstack{$\diff_{k-1}$ \\ $\longrightarrow$}} & V_{k} & \mbox{\shortstack{$\diff_k$ \\ $\longrightarrow$}} & V_{k+1} & \longrightarrow~\cdots \\ & & \downarrow f_{k-1} & & \downarrow f_{k} & & \downarrow f_{k+1} & & \\ \cdots & \longrightarrow & V_{k-1}^{\prime} & \mbox{\shortstack{$\diff_{k-1}^{\prime}$ \\ $\longrightarrow$}} & V_{k}^{\prime} & \mbox{\shortstack{$\diff_k^{\prime}$ \\ $\longrightarrow$}} & V_{k+1}^{\prime} & \longrightarrow~\cdots \end{array}$

$f_k$ maps $k$ -cochains to $k$ -cochains and $k$ -coboundaries to $k$ -coboundaries, thus induces a map $\mathcal{H}^k(f) : \mathcal{H}^k(V) \longrightarrow \mathcal{H}^k(V^{\prime})$ .

Let $V^{\prime} \subset V$ be two cochain complexes,

The inclusion $\imath_V$ is a cochain map and thus induces a map of cohomology $\mathcal{H}^k(V^{\prime}) \longrightarrow \mathcal{H}^k(V)$ ,
If there exists a cochain projection of $V$ onto $V^{\prime}$ , (this leads to $\pi \circ \imath = \id_{V^{\prime}}$ ) so $\mathcal{H}^k(\pi) \circ \mathcal{H}^k(\imath) = \id_{\mathcal{H}^k(V^{\prime})}$ .

$\begin{array}{ccccccc} \cdots & \longrightarrow & V_{k-1} & \mbox{\shortstack{$\diff_{k-1}$ \\ $\longrightarrow$}} & V_{k} & \longrightarrow~\cdots \\ & & \pi_{k-1} \downarrow \uparrow \imath & & \pi_{k} \downarrow \uparrow \imath & & \\ \cdots & \longrightarrow & V_{k-1}^{\prime} & \mbox{\shortstack{$\diff_{k-1}$ \\ $\longrightarrow$}} & V_{k}^{\prime} & \longrightarrow~\cdots \end{array}$

Thus, $\mathcal{H}^k(\imath)$ is injective and $\mathcal{H}^k(\pi)$ is surjective. Hence, if one of the cohomology spaces $\mathcal{H}^k(V)$ vanishes, then so does $\mathcal{H}^k(V^{\prime})$

Cycles and boundaries of the De Rham complex¶

$k$ -cocycles

$\mathfrak{Z}^k = \{ \omega \in H\Lambda^k(\Omega),~~ \diff \omega = 0 \}, ~~~ \mathfrak{Z}^{*k} = \{ \omega \in H^*\Lambda^k(\Omega),~~ \delta \omega = 0 \},$

$\mathfrak{Z}_0^k = \{ \omega \in H_0\Lambda^k(\Omega),~~ \diff \omega = 0 \}, ~~~ \mathfrak{Z}_0^{*k} = \{ \omega \in H_0^*\Lambda^k(\Omega),~~ \delta \omega = 0 \},$

$k$ -coboundaries

$\mathfrak{B}^k = \diff H\Lambda^{k-1}(\Omega), ~~~ \mathfrak{B}^{* k} = \delta \Lambda^{k+1}(\Omega),$

$\mathfrak{B}_0^k = \diff H_0\Lambda^{k-1}(\Omega), ~~~ \mathfrak{B}_0^{* k} = \delta \Lambda_0^{k+1}(\Omega),$

each of the spaces of cycles is closed in $\mathcal{H} \Lambda^k(\Omega)$ ( $\mathcal{H}^* \Lambda^k(\Omega)$ ), as well in $L^2 \Lambda^k(\Omega)$ .
each of the spaces of boundaries is closed in $L^2 \Lambda^k(\Omega)$ .
let $\perp$ denotes the orthogonal complement in $L^2 \Lambda^k(\Omega)$ ,

$\mathfrak{Z}^{k \perp} \subset \mathfrak{B}^{k \perp} = \mathfrak{Z}_0^{* k} , ~~~ \mathfrak{Z}^{* k \perp} \subset \mathfrak{B}^{* k \perp} = \mathfrak{Z}_0^{k}$

$\mathfrak{Z}_0^{k \perp} \subset \mathfrak{B}_0^{k \perp} = \mathfrak{Z}^{* k} , ~~~ \mathfrak{Z}_0^{* k \perp} \subset \mathfrak{B}_0^{* k \perp} = \mathfrak{Z}^{k}$

The Hodge decomposition¶

There are two Hodge decompositions, with different boundary conditions,

$L^2 \Lambda^k(\Omega) = \underbrace{\mathfrak{B}^{k}}_{\mathfrak{Z}_0^{* k\perp}} \oplus \underbrace{\mathfrak{H}^{k} \oplus \mathfrak{B}_0^{* k}}_{\mathfrak{Z}_0^{* k}=\mathfrak{B}^{k\perp}} = \overbrace{\mathfrak{B}^{k} \oplus \mathfrak{H}^{k}}^{\mathfrak{Z}^{k}=\mathfrak{B}_0^{* k\perp}} \oplus \overbrace{\mathfrak{B}_0^{* k}}^{\mathfrak{Z}^{k\perp}}$
$L^2 \Lambda^k(\Omega) = \underbrace{\mathfrak{B}_0^{k}}_{\mathfrak{Z}^{* k\perp}} \oplus \underbrace{\mathfrak{H}_0^{k} \oplus \mathfrak{B}^{* k}}_{\mathfrak{Z}^{* k}=\mathfrak{B}_0^{k\perp}} = \overbrace{\mathfrak{B}_0^{k} \oplus \mathfrak{H}_0^{k}}^{\mathfrak{Z}_0^{k}=\mathfrak{B}^{* k\perp}} \oplus \overbrace{\mathfrak{B}^{* k}}^{\mathfrak{Z}_0^{k\perp}}$

Summary¶

$\begin{tabular}{|c||c|c|c|c|} \hline $\omega^k \in \Lambda^k(\Omega)$ & $k=0$ & $k=1$ & $k=2$ & $k=3$ \\ \hline $\diff \omega^k$ & $\Grad u$ & $\Curl \uu$ & $\Div \uu$ & $-$ \\ $\delta \omega^k$ & $-$ & $-\Div \uu$ & $\Curl \uu$ & $-\Grad u$ \\ $\mathfrak{i}_{\boldsymbol{\beta}} \omega^k$ & $-$ & $\boldsymbol{\beta} \cdot \uu$ & $\uu \times \boldsymbol{\beta}$ & $u \boldsymbol{\beta}$ \\ $\mathfrak{j}_{\boldsymbol{\beta}} \omega^k$ & $u \boldsymbol{\beta}$ & $-\uu \times \boldsymbol{\beta}$ & $\boldsymbol{\beta} \cdot \uu$ & $-$ \\ $L_{\boldsymbol{\beta}} \omega^k$ & $\boldsymbol{\beta} \cdot \Grad u$ & $\Grad \left(\boldsymbol{\beta} \cdot \uu \right) + \left(\Curl \uu \right) \times \boldsymbol{\beta}$ & $\Curl \left(\uu \times \boldsymbol{\beta} \right) + \boldsymbol{\beta} \Div \uu$ & $\Div \left( u \boldsymbol{\beta} \right)$ \\ $\mathcal{L}_{\boldsymbol{\beta}} \omega^k$ & $-\Div \left( u \boldsymbol{\beta} \right)$ & $-\Curl \left(\uu \times \boldsymbol{\beta} \right) - \boldsymbol{\beta} \Div \uu$ & $-\Grad \left(\boldsymbol{\beta} \cdot \uu \right) - \left(\Curl \uu \right) \times \boldsymbol{\beta}$ & $-\boldsymbol{\beta} \cdot \Grad u$ \\ \hline $\tr \omega^k$ & $u(\xx)$ & $\uu(\xx) \times \nn(\xx)$ & $\uu(\xx) \cdot \nn(\xx)$ & $-$ \\ \hline \hline $H \Lambda^k(\Omega)$ & $\Hgrad$ & $\Hcurl$ & $\Hdiv$ & $\Ltwo$ \\ $V_k$ & $\Vgrad$ & $\Vcurl$ & $\Vdiv$ & $\Vltwo$ \\ \hline \end{tabular}$

References

DeRham sequences¶

here without boundary conditions

$\mathbb{R} \hookrightarrow \Hgrad \xrightarrow{\quad \Grad \quad} \Hcurl \xrightarrow{\quad \Curl \quad} \Hdiv \xrightarrow{\quad \Div \quad} \Ltwo \xrightarrow{\quad} 0$

Pullbacks¶

In the case where the physical domain $\Omega := \mathcal{F}(\hat{\Omega})$ is the image of a logical domain $\hat{\Omega}$ by a smooth mapping $\mathcal{F}$ (at least $\mathcal{C}^1$ ), we have the following parallel diagrams

$\begin{array}{ccccccc} \Hgrad & \xrightarrow{\quad \Grad \quad} & \Hcurl & \xrightarrow{\quad \Curl \quad} & \Hdiv & \xrightarrow{\quad \Div \quad} & \Ltwo \\ \igrad \Bigg\uparrow & & \icurl \Bigg\uparrow & & \idiv \Bigg\uparrow & & \iltwo \Bigg\uparrow \\ \HgradLogical & \xrightarrow{\quad \Grad \quad} & \HcurlLogical & \xrightarrow{\quad \Curl \quad} & \HdivLogical & \xrightarrow{\quad \Div \quad} & \LtwoLogical \\ % \end{array}$

Where the mappings $\igrad, \icurl, \idiv$ and $\iltwo$ are called pullbacks and are given by

$\phi (x) :=& \igrad \hat{\phi} (\hat{x}) = \hat{\phi}(\mathcal{F}^{-1}(x)) \\ \Psi (x) :=& \icurl \hat{\Psi} (\hat{x}) = \left( D \mathcal{F} \right)^{-T} \hat{\Psi}(\mathcal{F}^{-1}(x)) \\ \Phi (x) :=& \idiv \hat{\Phi} (\hat{x}) = \frac{1}{J} D \mathcal{F} \hat{\Phi}(\mathcal{F}^{-1}(x)) \\ \rho (x) :=& \iltwo \hat{\rho} (\hat{x}) = \hat{\rho}(\mathcal{F}^{-1}(x))$

where $D \mathcal{F}$ is the jacobian matrix of the mapping $\mathcal{F}$ .

Note

The pullbacks $\igrad, \icurl, \idiv$ and $\iltwo$ 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

$\begin{array}{ccccccc} \Hgrad & \xrightarrow{\quad \Grad \quad} & \Hcurl & \xrightarrow{\quad \Curl \quad} & \Hdiv & \xrightarrow{\quad \Div \quad} & \Ltwo \\ \Pigrad \Bigg\downarrow & & \Picurl \Bigg\downarrow & & \Pidiv \Bigg\downarrow & & \Piltwo \Bigg\downarrow \\ \Vgrad & \xrightarrow{\quad \Grad \quad} & \Vcurl & \xrightarrow{\quad \Curl \quad} & \Vdiv & \xrightarrow{\quad \Div \quad} & \Vltwo \\ % \end{array}$

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

$\begin{array}{ccccccc} \Vgrad & \xrightarrow{\quad \Grad \quad} & \Vcurl & \xrightarrow{\quad \Curl \quad} & \Vdiv & \xrightarrow{\quad \Div \quad} & \Vltwo \\ \igrad \Bigg\uparrow & & \icurl \Bigg\uparrow & & \idiv \Bigg\uparrow & & \iltwo \Bigg\uparrow \\ \VgradLogical & \xrightarrow{\quad \Grad \quad} & \VcurlLogical & \xrightarrow{\quad \Curl \quad} & \VdivLogical & \xrightarrow{\quad \Div \quad} & \VltwoLogical \\ % \end{array}$

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

$\phi :=& \igrad \hat{\phi}, \quad \phi \in \Vgrad, \hat{\phi} \in \VgradLogical \\ \Psi :=& \icurl \hat{\Psi}, \quad \Psi \in \Vcurl, \hat{\Psi} \in \VcurlLogical \\ \Phi :=& \idiv \hat{\Phi}, \quad \Phi \in \Vdiv, \hat{\Phi} \in \VdivLogical \\ \rho :=& \iltwo \hat{\rho}, \quad \rho \in \Vltwo, \hat{\rho} \in \VltwoLogical$

We have then, the following results

$\Grad \phi =& \icurl \Grad \hat{\phi} , \quad \phi \in \Vgrad \\ \Curl \Psi =& \idiv \Curl \hat{\Psi} , \quad \Psi \in \Vcurl \\ \Div \Phi =& \iltwo \Div \hat{\Phi} , \quad \Phi \in \Vdiv$

Projectors¶

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

$\begin{array}{ccccccc} \Cinfinity & \xrightarrow{\quad \Grad \quad} & \Cinfinity & \xrightarrow{\quad \Curl \quad} & \Cinfinity & \xrightarrow{\quad \Div \quad} & \Cinfinity \\ \Pigrad \Bigg\downarrow & & \Picurl \Bigg\downarrow & & \Pidiv \Bigg\downarrow & & \Piltwo \Bigg\downarrow \\ \Vgrad & \xrightarrow{\quad \Grad \quad} & \Vcurl & \xrightarrow{\quad \Curl \quad} & \Vdiv & \xrightarrow{\quad \Div \quad} & \Vltwo \\ \end{array}$

Discrete DeRham sequence for B-Splines¶

Buffa et al [BSV09] show the construction of a discrete DeRham sequence using B-Splines, (here without boundary conditions)

$\begin{array}{ccccccc} \Hgrad & \xrightarrow{\quad \Grad \quad} & \Hcurl & \xrightarrow{\quad \Curl \quad} & \Hdiv & \xrightarrow{\quad \Div \quad} & \Ltwo \\ \Pigrad \Bigg\downarrow & & \Picurl \Bigg\downarrow & & \Pidiv \Bigg\downarrow & & \Piltwo \Bigg\downarrow \\ \Vgradspline & \xrightarrow{\quad \Grad \quad}& \Vcurlspline & \xrightarrow{\quad \Curl \quad} & \Vdivspline& \xrightarrow{\quad \Div \quad} & \Vltwospline \\ \end{array}$

1d case¶

DeRham sequence is reduced to

$\mathbb{R} \hookrightarrow \underbrace{\mathcal{S}^{p}}_{\VgradLogical} \xrightarrow{\quad \Grad \quad} \underbrace{\mathcal{S}^{p-1}}_{\VltwoLogical} \xrightarrow{\quad} 0$

The recursion formula for derivative writes

${N_i^p}'(t)=D_i^{p}(t)-D_{i+1}^{p}(t) \quad \mbox{where} \quad D_{i}^{p}(t) = \frac{p}{t_{i+p+1}-t_i}N_i^{p-1}(t)$

we have $\mathcal{S}^{p-1} = \mathbf{span}\{ N_i^{p-1}, 1 \leq i \leq n-1 \} = \mathbf{span}\{ D_i^p, 1 \leq i \leq n-1 \}$ which is a change of basis as a diagonal matrix
Now if $u \in S^p$ , with and expansion $u = \sum_i u_i N_i^p$ , we have

$\begin{align*} u^{\prime} = \sum_i u_i \left( N_i^p \right)^{\prime} = \sum_i (-u_{i-1} + u_i) D_i^p % \label{} \end{align*}$

If we introduce the B-Splines coefficients vector $\mathbf{u} := \left( u_i \right)_{1 \leq i \leq n}$ (and $\mathbf{u}^{\star}$ for the derivative), we have

$\mathbf{u}^{\star} = D \mathbf{u}$

where $D$ is the incidence matrix (of entries $-1$ and $+1$ )

Discrete derivatives:

$\mathcal{G} = D$

2d case¶

In 2d, the are two De-Rham complexes:

$\begin{array}{ccccc} \Hgrad & \xrightarrow{\quad \Grad \quad} & \Hcurl & \xrightarrow{\quad \Rots \quad} & \Ltwo \\ \Pigrad \Bigg\downarrow & & \Picurl \Bigg\downarrow & & \Piltwo \Bigg\downarrow \\ \Vgrad & \xrightarrow{\quad \Grad \quad} & \Vcurl & \xrightarrow{\quad \Rots \quad} & \Vltwo \\ \end{array}$

and

$\begin{array}{ccccc} \Hgrad & \xrightarrow{\quad \Curl \quad} & \Hdiv & \xrightarrow{\quad \Div \quad} & \Ltwo \\ \Pigrad \Bigg\downarrow & & \Pidiv \Bigg\downarrow & & \Piltwo \Bigg\downarrow \\ \Vgrad & \xrightarrow{\quad \Grad \quad} & \Vdiv & \xrightarrow{\quad \Div \quad} & \Vltwo \\ \end{array}$

Let $I$ be the identity matrix, we have

Discrete derivatives:

$\mathcal{G} = \begin{pmatrix} D \otimes I \\ I \otimes D \end{pmatrix}$

$\mathcal{C} = \begin{pmatrix} I \otimes D \\ - D \otimes I \end{pmatrix} \quad \mbox{[scalar curl],} \quad \mathcal{C} = \begin{pmatrix} - I \otimes D & D \otimes I \end{pmatrix} \quad \mbox{[vectorial curl]}$

$\mathcal{D} = \begin{pmatrix} D \otimes I & I \otimes D \end{pmatrix}$

3d case¶

Discrete derivatives:

$\mathcal{G} = \begin{pmatrix} D \otimes I \otimes I \\ I \otimes D \otimes I \\ I \otimes I \otimes D \end{pmatrix}$

$\mathcal{C} = \begin{pmatrix} 0 & - I \otimes I \otimes D & I \otimes D \otimes I \\ I \otimes I \otimes D & 0 & - D \otimes I \otimes I \\ - I \otimes D \otimes I & D \otimes I \otimes I & 0 \end{pmatrix}$

$\mathcal{D} = \begin{pmatrix} D \otimes I \otimes I & I \otimes D \otimes I & I \otimes I \otimes D \end{pmatrix}$

Note

From now on, we will denote the discrete derivative by $\mathbb{D}_k$ for the one going from $V_k$ to $V_{k+1}$ .

Algebraic identities¶

Let us consider the discretization of the exterior derivative

$\omega^{k+1} = \diff \omega^k$

multiplying by a test function $\eta^{k+1}$ and integrating over the whole computation domain, we get

$\left( \eta^{k+1}, \omega^{k+1} \right)_{k+1} = \left( \eta^{k+1}, \diff \omega^{k} \right)_{k+1}$

let $E^{k+1}$ , $W^{k}$ and $W^{k+1}$ be the vector representation of $\eta^{k+1}$ , $\omega^{k}$ and $\omega^{k+1}$ . We get

${E^{k+1}}^T M_{k+1} W^{k+1} = {E^{k+1}}^T D_{k+1,k} W^{k}$

where

$D_{k+1,k} = \left( \left( \eta^{k+1}_i, \diff \omega^{k}_j \right)_{k+1} \right)_{i,j}$

On the other hand, using the coderivative, we get

$\left( \eta^{k+1}, \omega^{k+1} \right)_{k+1} = \left( \delta \eta^{k+1}, \omega^{k} \right)_{k} + BC$

Let us now introduce the following matrix

$D_{k,k+1} = \left( \left( \delta \eta^{k+1}_i, \omega^{k}_j \right)_{k} \right)_{i,j}$

hence,

${E^{k+1}}^T D_{k,k+1} W^{k} = \left( \mathbb{D}^{\star}_{k+1} E^{k+1} \right)^T M_{k} W^{k}$

Therefor, we have the following important result

Proposition:

$D_{k+1,k} = D_{k,k+1} + BC$
$D_{k+1,k} = M_{k+1} \mathbb{D}^T_k$
$D_{k,k+1} = {\mathbb{D}^{\star}_{k+1}}^T M_{k}$

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.

API¶

You will find here both the Fortran doxygen documentation as well as the Python-API.

Welcome to spl’s documentation!¶

First Steps with SPL¶

Install SPL¶

Examples¶

Dive into SPL¶

Contents¶

Introduction¶

Input and Output¶

B-Splines and NURBS¶

B-Splines series¶

Knots vector families¶

B-Spline curve¶

Multivariate tensor product splines¶

Deriving a B-spline curve¶

proposition¶

Example¶

Knot insertion¶

Order elevation¶

Translation¶

Rotation¶

Scaling¶

GLT¶

Where do the GLTs come from?¶

Spectral preliminaries¶

Exterior Algebra¶

Basis¶

Orientation and Volume form¶

Exterior Calculus on manifolds and Differential forms¶

Sobolev spaces of differential forms¶

Cohomology and De Rham Complex¶

Cohomology with boundary conditions¶

Homological Algebra and Hilbert complexes¶

Homological Algebra¶

Cycles and boundaries of the De Rham complex¶

The Hodge decomposition¶

Summary¶

DeRham sequences¶

Pullbacks¶

Discrete Spaces¶

Projectors¶

Discrete DeRham sequence for B-Splines¶

1d case¶

2d case¶

3d case¶

Algebraic identities¶

API¶

Fortran API¶

Python API¶

spl package¶

Subpackages¶

spl.core package¶

Submodules¶

spl.mapping module¶

spl.utilities module¶

Module contents¶

Indices and tables¶