Term Overview

Term Syntax

In general, the syntax of a term call is:

<term name>.<i>.<r>( <arg1>, <arg2>, ... ),

where <i> denotes an integral name (i.e. a name of numerical quadrature to use) and <r> marks a region (domain of the integral).

The following notation is used:

Notation.

symbol

meaning

\Omega

volume (sub)domain

\Gamma

surface (sub)domain

\cal{D}

volume or surface (sub)domain

d

dimension of space

t

time

y

any function

\ul{y}

any vector function

\ul{n}

unit outward normal

q, s

scalar test function

p, r

scalar unknown or parameter function

\bar{p}

scalar parameter function

\ul{v}

vector test function

\ul{w}, \ul{u}

vector unknown or parameter function

\ul{b}

vector parameter function

\ull{e}(\ul{u})

Cauchy strain tensor (\frac{1}{2}((\nabla u) + (\nabla u)^T))

\ull{F}

deformation gradient F_{ij} = \pdiff{x_i}{X_j}

J

\det(F)

\ull{C}

right Cauchy-Green deformation tensor C = F^T F

\ull{E}(\ul{u})

Green strain tensor E_{ij} = \frac{1}{2}(\pdiff{u_i}{X_j} +
\pdiff{u_j}{X_i} + \pdiff{u_m}{X_i}\pdiff{u_m}{X_j})

\ull{S}

second Piola-Kirchhoff stress tensor

\ul{f}

vector volume forces

f

scalar volume force (source)

\rho

density

\nu

kinematic viscosity

c

any constant

\delta_{ij}, \ull{I}

Kronecker delta, identity matrix

\tr{\ull{\bullet}}

trace of a second order tensor (\sum_{i=1}^d \bullet_{ii})

\dev{\ull{\bullet}}

deviator of a second order tensor (\ull{\bullet} - \frac{1}{d}\tr{\ull{\bullet}})

T_K \in \Tcal_h

K-th element of triangulation (= mesh) \Tcal_h of domain \Omega

K \from \Ical_h

K is assigned values from \{0, 1, \dots, N_h-1\}
\equiv \Ical_h in ascending order

The suffix “_0” denotes a quantity related to a previous time step.

Term names are (usually) prefixed according to the following conventions:

Term name prefixes.

prefix

meaning

evaluation modes

meaning

dw

discrete weak

‘weak’

terms having a virtual (test) argument and zero or more unknown arguments, used for FE assembling

ev

evaluate

‘eval’, ‘el_eval’, ‘el_avg’, ‘qp’

terms having all arguments known, modes ‘el_avg’, ‘qp’ are not supported by all ev_ terms

de

discrete einsum

any (work in progress)

multi-linear terms defined using an enriched einsum notation

Term Table

Below we list all the terms available in automatically generated tables. The first column lists the name, the second column the argument lists and the third column the mathematical definition of each term. The terms are devided into the following tables:

The notation <virtual> corresponds to a test function, <state> to a unknown function and <parameter> to a known function. By <material> we denote material (constitutive) parameters, or, in general, any given function of space and time that parameterizes a term, for example a given traction force vector.

Table of basic terms

Table of sensitivity terms

Table of large deformation terms

Table of special terms

Table of multi-linear terms