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Finite Element Embedded Library and Language in C++
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Keywords

Table of Contents

One of Feel++ assets is it finite element embedded language. The language follows the C++ grammar, and provides keywords as well as operations between objects which are, mathematically, tensors of rank 0, 1 or 2.

In all following tables we use the notations :
$f: \mathbb{R}^n \mapsto \mathbb{R}^{m\times p}$ with $n=1,2,3$, $m=1,2,3$, $p=1,2,3$
$\Omega^e$ current mesh element.

The genesis of the language can be found in [14] and an update on Feel++ is available in [13].

Points

Current Point :

Feel++ KeywordMath ObjectDescriptionDimension
P()
$\overrightarrow{P}$$(P_x, P_y, P_z)^T$$d \times 1$
Px()
$P_x$$x$ coordinate of $\overrightarrow{P}$$1 \times 1$
Py()
$P_y$$y$ coordinate of $\overrightarrow{P}$
(value is 0 in 1D)		$1 \times 1$
Pz()
$P_z$$z$ coordinate of $\overrightarrow{P}$
(value is 0 in 1D and 2D)		$1 \times 1$

Element Barycenter Point :

Feel++ KeywordMath ObjectDescriptionDimension
C()
$\overrightarrow{C}$$(C_x, C_y, C_z)^T$$d \times 1$
Cx()
$C_x$$x$ coordinate of $\overrightarrow{C}$$1 \times 1$
Cy()
$C_y$$y$ coordinate of $\overrightarrow{C}$
(value is 0 in 1D)		$1 \times 1$
Cz()
$C_z$$z$ coordinate of $\overrightarrow{C}$
(value is 0 in 1D and 2D)		$1 \times 1$

Normal at Current Point :

Feel++ KeywordMath ObjectDescriptionDimension
N()
$\overrightarrow{N}$$(N_x, N_y, N_z)^T$$d \times 1$
Nx()
$N_x$$x$ coordinate of $\overrightarrow{N}$$1 \times 1$
Ny()
$N_y$$y$ coordinate of $\overrightarrow{N}$
(value is 0 in 1D)		$1 \times 1$
Nz()
$N_z$$z$ coordinate of $\overrightarrow{N}$
(value is 0 in 1D and 2D)		$1 \times 1$

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Vectors and Matrix

Building Vectors

Usual syntax to create vectors :

Feel++ KeywordMath ObjectDescriptionDimension
vec<n>(v_1,v_2,...,v_n)
$\begin{pmatrix} v_1\\v_2\\ \vdots \\v_n \end{pmatrix}$Column Vector with $n$ rows
entries being expressions		$n \times 1$

You can also use expressions and the unit base vectors :

Feel++ KeywordMath ObjectDescription
oneX() $\begin{pmatrix} 1\\0\\0 \end{pmatrix}$Unit vector $\overrightarrow{i}$
oneY() $\begin{pmatrix} 0\\1\\0 \end{pmatrix}$Unit vector $\overrightarrow{j}$
oneZ() $\begin{pmatrix} 0\\0\\1 \end{pmatrix}$Unit vector $\overrightarrow{k}$

Building Matrix

Feel++ KeywordMath ObjectDescriptionDimension
mat<m,n>(m_11,m_12,...,m_mn)
$\begin{pmatrix} m_{11} & m_{12} & ...\\ m_{21} & m_{22} & ...\\ \vdots & & \end{pmatrix}$$m\times n$ Matrix
entries beeing expressions 		$m \times n$
ones<m,n>()
$\begin{pmatrix} 1 & 1 & ...\\ 1 & 1 & ...\\ \vdots & & \end{pmatrix}$$m\times n$ Matrix
Filled with 1 		$m \times n$
zero<m,n>()
$\begin{pmatrix} 0 & 0 & ...\\ 0 & 0 & ...\\ \vdots & & \end{pmatrix}$$m\times n$ Matrix
Filled with 0 		$m \times n$
constant<m,n>(c)
$\begin{pmatrix} c & c & ...\\ c & c & ...\\ \vdots & & \end{pmatrix}$$m\times n$ Matrix
Filled with a constant c 		$m \times n$
eye<n>()
$\begin{pmatrix} 1 & 0 & ...\\ 0 & 1 & ...\\ \vdots & & \end{pmatrix}$Unit diagonal Matrix
of size $n\times n$ 		$n \times n$
Id<n>()
$\begin{pmatrix} 1 & 0 & ...\\ 0 & 1 & ...\\ \vdots & & \end{pmatrix}$Unit diagonal Matrix
of size $n\times n$ 		$n \times n$

Manipulating Vectors and Matrix

Let $A$ be a square matrix of size $n$.

Feel++ KeywordMath ObjectDescriptionDimension
inv(A)
$A^{-1}$Inverse of matrix $A$ $n \times n$
det(A)
$\det (A)$Determinant of matrix $A$ $1 \times 1$
sym(A)
$\text{Sym}(A)$Symmetric part of matrix $A$ : $\frac{1}{2}(A+A^T)$
$n \times n$
antisym(A)
$ \text{Asym}(A)$Antisymmetric part of $A$ : $\frac{1}{2}(A-A^T)$
$n \times n$

Let A and B be two matrix (or two vectors) of same dimension $m \times n$.

Feel++ KeywordMath ObjectDescriptionDimension
trace(A)
$\text{tr}(A)$Trace of matrix $A$
Generalized on non-squared Matrix
Generalized on Vectors 		$1 \times 1$
trans(B)
$B^T$Transpose of matrix $B$
Can be used on non-squared Matrix
Can be used on Vectors 		$n \times m$
inner(A,B)
$ A.B \\ A:B = \text{tr}(A*B^T)$Scalar product of two vectors
Generalized scalar product of two matrix 		$1 \times 1$
cross(A,B)
$ A\times B$Cross product of two vectors$n \times 1$

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Expressions

Following tables present tools to declare and manipulate expressions.

Feel++ KeywordDescription
Px()
Py()
Pz()
cst( c )
Variable $x$
Variable $y$
Variable $z$
Constant function equal to $c$

You can of course use all current operators ( + - / * ) and the usual following functions :

Feel++ KeywordMath ObjectDescription
abs(expr)
$|f(\overrightarrow{x})|$element wise absolute value of $f$
cos(expr)
$\cos(f(\overrightarrow{x}))$element wise cos value of $f$
sin(expr)
$\sin(f(\overrightarrow{x}))$element wise sin value of $f$
tan(expr)
$\tan(f(\overrightarrow{x}))$element wise tan value of $f$
acos(expr)
$\acos(f(\overrightarrow{x}))$element wise acos value of $f$
asin(expr)
$\asin(f(\overrightarrow{x}))$element wise asin value of $f$
atan(expr)
$\atan(f(\overrightarrow{x}))$element wise atan value of $f$
cosh(expr)
$\cosh(f(\overrightarrow{x}))$element wise cosh value of $f$
sinh(expr)
$\sinh(f(\overrightarrow{x}))$element wise sinh value of $f$
tanh(expr)
$\tanh(f(\overrightarrow{x}))$element wise tanh value of $f$
exp(expr)
$\exp(f(\overrightarrow{x}))$element wise exp value of $f$
log(expr)
$\log(f(\overrightarrow{x}))$element wise log value of $f$
sqrt(expr)
$\sqrt{f(\overrightarrow{x})}$element wise sqrt value of $f$
sign(expr)
$\begin{cases} 1 & \text{if}\ f(\overrightarrow{x}) \geq 0\\-1 & \text{if}\ f(\overrightarrow{x}) < 0\end{cases}$element wise sign value of $f$
chi(expr)
$\chi(f(\overrightarrow{x}))=\begin{cases}0 & \text{if}\ f(\overrightarrow{x}) = 0\\1 & \text{if}\ f(\overrightarrow{x}) \neq 0\\\end{cases}$element wise boolean test of $f$

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Operators

Operations

You can use the usual operations and logical operators.

Feel++ KeywordMath ObjectDescription
+
$ f+g$tensor sum
-
$ f-g$tensor substraction
*
$ f*g$tensor product
/
$ f/g$tensor tensor division
( $g$ scalar field)
<
$ f<g$element wise less
<=
$ f<=g$element wise less or equal
>
$ f>g$element wise greater
>=
$ f>=g$element wise greater or equal
==
$ f==g$element wise equal
!=
$ f!=g$element wise not equal
-
$ -g$element wise unary minus
&&
$ f$ and $g$element wise logical and
||
$ f$ or $g$element wise logical or
!
$ !g$element wise logical not

Differential Operators

Feel++ finit element language use test and trial functions. Keywords are different according to the kind of the manipulated function.
Usual operators are for test functions.
t-operators for trial functions.
v-operators to get an evaluation.

Feel++ KeywordMath ObjectDescriptionRankDimension
id(f)
$f$ test function $\mathrm{rank}(f(\overrightarrow{x}))$ $m \times p $
idt(f)
$f$ trial function $\mathrm{rank}(f(\overrightarrow{x}))$ $m \times p $
idv(f)
$f$ evaluation function $\mathrm{rank}(f(\overrightarrow{x}))$ $m \times p $
grad(f)
$\nabla f$ gradient of test function $\mathrm{rank}(f(\overrightarrow{x}))+1$ $m \times n $
$p=1$
gradt(f)
$\nabla f$ grdient of trial function $\mathrm{rank}(f(\overrightarrow{x}))+1$ $m \times n $
$p=1$
gradv(f)
$\nabla f$ evaluation function gradient $\mathrm{rank}(f(\overrightarrow{x}))+1$ $m \times n $
$p=1$
div(f)
$\nabla\cdot f$ divergence of test function $\mathrm{rank}(f(\overrightarrow{x}))-1$ $1 \times 1 $
divt(f)
$\nabla\cdot f$ divergence of trial function $\mathrm{rank}(f(\overrightarrow{x}))-1$ $1 \times 1 $
divv(f)
$\nabla\cdot f$ evaluation function divergence $\mathrm{rank}(f(\overrightarrow{x}))-1$ $1 \times 1 $
curl(f)
$\nabla\times f$ curl of test function 1$n \times 1 $
$m=n$
curlt(f)
$\nabla\times f$ curl of trial function 1 $n \times 1 $
$m=n$
curlv(f)
$\nabla\times f$ evaluation function curl 1 $n \times 1 $
$m=n$
hess(f)
$\nabla^2 f$ hessian of test function 2 $n \times n $
$m=p=1$

Two Valued Operators

Feel++ KeywordMath ObjectDescriptionRankDimension
jump(f)
$[f]=f_0\overrightarrow{N_0}+f_1\overrightarrow{N_1}$ jump of test function 0$n \times 1 $
$m=1$
jump(f)
$[\overrightarrow{f}]=\overrightarrow{f_0}\cdot\overrightarrow{N_0}+\overrightarrow{f_1}\cdot\overrightarrow{N_1}$ jump of test function 0$1 \times 1 $
$m=2$
jumpt(f)
$[f]=f_0\overrightarrow{N_0}+f_1\overrightarrow{N_1}$ jump of trial function 0$n \times 1 $
$m=1$
jumpt(f)
$[\overrightarrow{f}]=\overrightarrow{f_0}\cdot\overrightarrow{N_0}+\overrightarrow{f_1}\cdot\overrightarrow{N_1}$ jump of trial function 0$1 \times 1 $
$m=2$
jumpv(f)
$[f]=f_0\overrightarrow{N_0}+f_1\overrightarrow{N_1}$ jump of function evaluation 0$n \times 1 $
$m=1$
jumpv(f)
$[\overrightarrow{f}]=\overrightarrow{f_0}\cdot\overrightarrow{N_0}+\overrightarrow{f_1}\cdot\overrightarrow{N_1}$ jump of function evaluation0$1 \times 1 $
$m=2$
average(f)
${f}=\frac{1}{2}(f_0+f_1)$ average of test function$\mathrm{rank}( f(\overrightarrow{x}))$$n \times n $
$m=n$
averaget(f)
${f}=\frac{1}{2}(f_0+f_1)$ average of trial function$\mathrm{rank}( f(\overrightarrow{x}))$$n \times n $
$m=n$
averagev(f)
${f}=\frac{1}{2}(f_0+f_1)$ average of function evaluation$\mathrm{rank}( f(\overrightarrow{x}))$$n \times n $
$m=n$
leftface(f)
$f_0$ left test function$\mathrm{rank}( f(\overrightarrow{x}))$$n \times n $
$m=n$
leftfacet(f)
$f_0$ left trial function$\mathrm{rank}( f(\overrightarrow{x}))$$n \times n $
$m=n$
leftfacev(f)
$f_0$ left function evaluation$\mathrm{rank}( f(\overrightarrow{x}))$$n \times n $
$m=n$
rightface(f)
$f_1$ right test function$\mathrm{rank}( f(\overrightarrow{x}))$$n \times n $
$m=n$
rightfacet(f)
$f_1$ right trial function$\mathrm{rank}( f(\overrightarrow{x}))$$n \times n $
$m=n$
rightfacev(f)
$f_1$ right function evaluation$\mathrm{rank}( f(\overrightarrow{x}))$$n \times n $
$m=n$
maxface(f)
$\max(f_0,f_1)$ maximum of right and left
test function		$\mathrm{rank}( f(\overrightarrow{x}))$$n \times p $
maxfacet(f)
$\max(f_0,f_1)$ maximum of right and left
trial function		$\mathrm{rank}( f(\overrightarrow{x}))$$n \times p $
maxfacev(f)
$\max(f_0,f_1)$ maximum of right and left
function evaluation		$\mathrm{rank}( f(\overrightarrow{x}))$$n \times p $
minface(f)
$\min(f_0,f_1)$ minimum of right and left
test function		$\mathrm{rank}( f(\overrightarrow{x}))$$n \times p $
minfacet(f)
$\min(f_0,f_1)$ minimum of right and left
trial function		$\mathrm{rank}( f(\overrightarrow{x}))$$n \times p $
minfacev(f)
$\min(f_0,f_1)$ minimum of right and left
function evaluation		$\mathrm{rank}( f(\overrightarrow{x}))$$n \times p $

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Geometric Transformations

Jacobian Matrix

You can access to the jacobian matrix, $J$, of the geometric transformation, using the keyword :

J()

There are some tools to manipulate this jacobian.

Feel++ KeywordMath ObjectDescription
detJ()
$\det(J)$Determinant of jacobian matrix
invJT()
$(J^{-1})^T$Transposed inverse of jacobian matrix

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