.. _term_overview: Term Overview ============= Term Syntax ----------- In general, the syntax of a term call is: .. centered:: ..( , , ... ), where  denotes an integral name (i.e. a name of numerical quadrature to use) and  marks a region (domain of the integral). The following notation is used: .. list-table:: Notation. :widths: 20 80 :header-rows: 1 * - symbol - meaning * - :math:\Omega - volume (sub)domain * - :math:\Gamma - surface (sub)domain * - :math:\cal{D} - volume or surface (sub)domain * - :math:d - dimension of space * - :math:t - time * - :math:y - any function * - :math:\ul{y} - any vector function * - :math:\ul{n} - unit outward normal * - :math:q - scalar test or parameter function * - :math:p - scalar unknown or parameter function * - :math:\ul{v} - vector test or parameter function * - :math:\ul{w}, :math:\ul{u} - vector unknown or parameter function * - :math:\ull{e}(\ul{u}) - Cauchy strain tensor (:math:\frac{1}{2}((\nabla u) + (\nabla u)^T)) * - :math:\ull{F} - deformation gradient :math:F_{ij} = \pdiff{x_i}{X_j} * - :math:J - :math:\det(F) * - :math:\ull{C} - right Cauchy-Green deformation tensor :math:C = F^T F * - :math:\ull{E}(\ul{u}) - Green strain tensor :math: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}) * - :math:\ull{S} - second Piola-Kirchhoff stress tensor * - :math:\ul{f} - vector volume forces * - :math:f - scalar volume force (source) * - :math:\rho - density * - :math:\nu - kinematic viscosity * - :math:c, :math:\ul{c}, :math:\ull{c} - any constant * - :math:\delta_{ij}, \ull{I} - Kronecker delta, identity matrix * - :math:\tr{\ull{\bullet}} - trace of a second order tensor (:math:\sum_{i=1}^d \bullet_{ii}) * - :math:\dev{\ull{\bullet}} - deviator of a second order tensor (:math:\ull{\bullet} - \frac{1}{d}\tr{\ull{\bullet}}) * - :math:T_K \in \Tcal_h - :math:K-th element of triangulation (= mesh) :math:\Tcal_h of domain :math:\Omega * - :math:K \from \Ical_h - :math:K is assigned values from :math:\{0, 1, \dots, N_h-1\} \equiv \Ical_h in ascending order The suffix ":math:_0" denotes a quantity related to a previous time step. Term names are (usually) prefixed according to the following conventions: .. list-table:: Term name prefixes. :widths: 5 20 25 50 :header-rows: 1 * - 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 Evaluation modes 'eval', 'el_avg' and 'qp' are defined as follows: .. list-table:: Evaluation modes. :widths: 20 80 :header-rows: 1 * - mode - definition * - 'eval' - :math:\int_{\cal{D}} (\cdot) * - 'el_avg' - vector for :math:K \from \Ical_h: \int_{T_K} (\cdot) / \int_{T_K} 1 * - 'qp' - :math:(\cdot)|_{qp} .. _term_table: 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: * Table of basic terms_ * Table of large deformation terms_ (total/updated Lagrangian formulation) * Table of sensitivity terms_ * Table of special terms_ * Table of multi-linear terms_ The notation  corresponds to a test function,  to a unknown function and  to a known function. By  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. .. include:: term_table.rst