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
$d$	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
d	discrete	‘eval’, ‘el_eval’	terms having all arguments known, the result is the value of the term integral evaluation
ev	evaluate	‘eval’, ‘el_eval’, ‘el_avg’, ‘qp’	terms having all arguments known and supporting all evaluation modes except ‘weak’ (no virtual variables in arguments, no FE assembling)

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

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¶

Basic terms¶
name/class	arguments	definition
dw_advect_div_free `AdvectDivFreeTerm`	`<material>`, `<virtual>`, `<state>`	$\int_{\Omega} \nabla \cdot (\ul{y} p) q = \int_{\Omega} (\underbrace{(\nabla \cdot \ul{y})}_{\equiv 0} + \ul{y} \cdot \nabla) p) q$
dw_bc_newton `BCNewtonTerm`	`<material_1>`, `<material_2>`, `<virtual>`, `<state>`	$\int_{\Gamma} \alpha q (p - p_{\rm outer})$
dw_biot `BiotTerm`	`<material>`, `<virtual>`, `<state>` `<material>`, `<state>`, `<virtual>` `<material>`, `<parameter_v>`, `<parameter_s>`	$\int_{\Omega} p\ \alpha_{ij} e_{ij}(\ul{v}) \mbox{ , } \int_{\Omega} q\ \alpha_{ij} e_{ij}(\ul{u})$
ev_biot_stress `BiotStressTerm`	`<material>`, `<parameter>`	$- \int_{\Omega} \alpha_{ij} \bar{p}$ $\mbox{vector for } K \from \Ical_h: - \int_{T_K} \alpha_{ij} \bar{p} / \int_{T_K} 1$ $- \alpha_{ij} \bar{p}\|_{qp}$
ev_cauchy_strain `CauchyStrainTerm`	`<parameter>`	$\int_{\Omega} \ull{e}(\ul{w})$ $\mbox{vector for } K \from \Ical_h: \int_{T_K} \ull{e}(\ul{w}) / \int_{T_K} 1$ $\ull{e}(\ul{w})\|_{qp}$
ev_cauchy_strain_s `CauchyStrainSTerm`	`<parameter>`	$\int_{\Gamma} \ull{e}(\ul{w})$ $\mbox{vector for } K \from \Ical_h: \int_{T_K} \ull{e}(\ul{w}) / \int_{T_K} 1$ $\ull{e}(\ul{w})\|_{qp}$
ev_cauchy_stress `CauchyStressTerm`	`<material>`, `<parameter>`	$\int_{\Omega} D_{ijkl} e_{kl}(\ul{w})$ $\mbox{vector for } K \from \Ical_h: \int_{T_K} D_{ijkl} e_{kl}(\ul{w}) / \int_{T_K} 1$ $D_{ijkl} e_{kl}(\ul{w})\|_{qp}$
dw_contact `ContactTerm`	`<material>`, `<virtual>`, `<state>`	$\int_{\Gamma_{c}} \varepsilon_N \langle g_N(\ul{u}) \rangle \ul{n} \ul{v}$
dw_contact_plane `ContactPlaneTerm`	`<material_f>`, `<material_n>`, `<material_a>`, `<material_b>`, `<virtual>`, `<state>`	$\int_{\Gamma} \ul{v} \cdot f(d(\ul{u})) \ul{n}$
dw_contact_sphere `ContactSphereTerm`	`<material_f>`, `<material_c>`, `<material_r>`, `<virtual>`, `<state>`	$\int_{\Gamma} \ul{v} \cdot f(d(\ul{u})) \ul{n}(\ul{u})$
dw_convect `ConvectTerm`	`<virtual>`, `<state>`	$\int_{\Omega} ((\ul{u} \cdot \nabla) \ul{u}) \cdot \ul{v}$
dw_convect_v_grad_s `ConvectVGradSTerm`	`<virtual>`, `<state_v>`, `<state_s>`	$\int_{\Omega} q (\ul{u} \cdot \nabla p)$
ev_def_grad `DeformationGradientTerm`	`<parameter>`	$\ull{F} = \pdiff{\ul{x}}{\ul{X}}\|_{qp} = \ull{I} + \pdiff{\ul{u}}{\ul{X}}\|_{qp} \;, \\ \ul{x} = \ul{X} + \ul{u} \;, J = \det{(\ull{F})}$
dw_dg_advect_laxfrie_flux `AdvectionDGFluxTerm`	`<opt_material>`, `<material_advelo>`, `<virtual>`, `<state>`	$\int_{\partial{T_K}} \ul{n} \cdot \ul{f}^{} (p_{in}, p_{out})q$ where $\ul{f}^{}(p_{in}, p_{out}) = \ul{a} \frac{p_{in} + p_{out}}{2} + (1 - \alpha) \ul{n} C \frac{ p_{in} - p_{out}}{2},$
dw_dg_diffusion_flux `DiffusionDGFluxTerm`	`<material_diffusion_tensor>`, `<state>`, `<virtual>` `<material_diffusion_tensor>`, `<virtual>`, `<state>`	$\int_{\partial{T_K}} D \langle \nabla p \rangle [q] \mbox{ , } \int_{\partial{T_K}} D \langle \nabla q \rangle [p]$ where $\langle \nabla \phi \rangle = \frac{\nabla\phi_{in} + \nabla\phi_{out}}{2}$ $[\phi] = \phi_{in} - \phi_{out}$
dw_dg_interior_penalty `DiffusionInteriorPenaltyTerm`	`<material_diffusion_tensor>`, `<material_Cw>`, `<virtual>`, `<state>`	$\int_{\partial{T_K}} \bar{D} C_w \frac{Ord^2}{d(\partial{T_K})}[p][q]$ where $[\phi] = \phi_{in} - \phi_{out}$
dw_dg_nonlinear_laxfrie_flux `NonlinearHyperbolicDGFluxTerm`	`<opt_material>`, `<fun>`, `<fun_d>`, `<virtual>`, `<state>`	$\int_{\partial{T_K}} \ul{n} \cdot f^{} (p_{in}, p_{out})q$ where $\ul{f}^{}(p_{in}, p_{out}) = \frac{\ul{f}(p_{in}) + \ul{f}(p_{out})}{2} + (1 - \alpha) \ul{n} C \frac{ p_{in} - p_{out}}{2},$
dw_diffusion `DiffusionTerm`	`<material>`, `<virtual>`, `<state>` `<material>`, `<parameter_1>`, `<parameter_2>`	$\int_{\Omega} K_{ij} \nabla_i q \nabla_j p \mbox{ , } \int_{\Omega} K_{ij} \nabla_i \bar{p} \nabla_j r$
dw_diffusion_coupling `DiffusionCoupling`	`<material>`, `<virtual>`, `<state>` `<material>`, `<state>`, `<virtual>` `<material>`, `<parameter_1>`, `<parameter_2>`	$\int_{\Omega} p K_{j} \nabla_j q \mbox{ , } \int_{\Omega} q K_{j} \nabla_j p$
dw_diffusion_r `DiffusionRTerm`	`<material>`, `<virtual>`	$\int_{\Omega} K_{j} \nabla_j q$
ev_diffusion_velocity `DiffusionVelocityTerm`	`<material>`, `<parameter>`	$- \int_{\Omega} K_{ij} \nabla_j \bar{p}$ $\mbox{vector for } K \from \Ical_h: - \int_{T_K} K_{ij} \nabla_j \bar{p} / \int_{T_K} 1$ $- K_{ij} \nabla_j \bar{p}$
ev_div `DivTerm`	`<parameter>`	$\int_{\Omega} \nabla \cdot \ul{u}$ $\mbox{vector for } K \from \Ical_h: \int_{T_K} \nabla \cdot \ul{u} / \int_{T_K} 1$ $(\nabla \cdot \ul{u})\|_{qp}$
dw_div `DivOperatorTerm`	`<opt_material>`, `<virtual>`	$\int_{\Omega} \nabla \cdot \ul{v} \mbox { or } \int_{\Omega} c \nabla \cdot \ul{v}$
dw_div_grad `DivGradTerm`	`<opt_material>`, `<virtual>`, `<state>` `<opt_material>`, `<parameter_1>`, `<parameter_2>`	$\int_{\Omega} \nu\ \nabla \ul{v} : \nabla \ul{u} \mbox{ , } \int_{\Omega} \nu\ \nabla \ul{u} : \nabla \ul{w} \\ \int_{\Omega} \nabla \ul{v} : \nabla \ul{u} \mbox{ , } \int_{\Omega} \nabla \ul{u} : \nabla \ul{w}$
dw_elastic_wave `ElasticWaveTerm`	`<material_1>`, `<material_2>`, `<virtual>`, `<state>`	$\int_{\Omega} D_{ijkl}\ g_{ij}(\ul{v}) g_{kl}(\ul{u})$
dw_elastic_wave_cauchy `ElasticWaveCauchyTerm`	`<material_1>`, `<material_2>`, `<virtual>`, `<state>` `<material_1>`, `<material_2>`, `<state>`, `<virtual>`	$\int_{\Omega} D_{ijkl}\ g_{ij}(\ul{v}) e_{kl}(\ul{u}) \;, \int_{\Omega} D_{ijkl}\ g_{ij}(\ul{u}) e_{kl}(\ul{v})$
dw_electric_source `ElectricSourceTerm`	`<material>`, `<virtual>`, `<parameter>`	$\int_{\Omega} c s (\nabla \phi)^2$
ev_grad `GradTerm`	`<parameter>`	$\int_{\Omega} \nabla p \mbox{ or } \int_{\Omega} \nabla \ul{w}$ $\mbox{vector for } K \from \Ical_h: \int_{T_K} \nabla p / \int_{T_K} 1 \mbox{ or } \int_{T_K} \nabla \ul{w} / \int_{T_K} 1$ $(\nabla p)\|_{qp} \mbox{ or } \nabla \ul{w}\|_{qp}$
dw_jump `SurfaceJumpTerm`	`<opt_material>`, `<virtual>`, `<state_1>`, `<state_2>`	$\int_{\Gamma} c\, q (p_1 - p_2)$
dw_laplace `LaplaceTerm`	`<opt_material>`, `<virtual>`, `<state>` `<opt_material>`, `<parameter_1>`, `<parameter_2>`	$\int_{\Omega} c \nabla q \cdot \nabla p \mbox{ , } \int_{\Omega} c \nabla \bar{p} \cdot \nabla r$
dw_lin_convect `LinearConvectTerm`	`<virtual>`, `<parameter>`, `<state>`	$\int_{\Omega} ((\ul{b} \cdot \nabla) \ul{u}) \cdot \ul{v}$ $((\ul{b} \cdot \nabla) \ul{u})\|_{qp}$
dw_lin_convect2 `LinearConvect2Term`	`<material>`, `<virtual>`, `<state>`	$\int_{\Omega} ((\ul{b} \cdot \nabla) \ul{u}) \cdot \ul{v}$ $((\ul{b} \cdot \nabla) \ul{u})\|_{qp}$
dw_lin_elastic `LinearElasticTerm`	`<material>`, `<virtual>`, `<state>` `<material>`, `<parameter_1>`, `<parameter_2>`	$\int_{\Omega} D_{ijkl}\ e_{ij}(\ul{v}) e_{kl}(\ul{u})$
dw_lin_elastic_iso `LinearElasticIsotropicTerm`	`<material_1>`, `<material_2>`, `<virtual>`, `<state>` `<material_1>`, `<material_2>`, `<parameter_1>`, `<parameter_2>`	$\int_{\Omega} D_{ijkl}\ e_{ij}(\ul{v}) e_{kl}(\ul{u}) \mbox{ with } D_{ijkl} = \mu (\delta_{ik} \delta_{jl}+\delta_{il} \delta_{jk}) + \lambda \ \delta_{ij} \delta_{kl}$
dw_lin_prestress `LinearPrestressTerm`	`<material>`, `<virtual>` `<material>`, `<parameter>`	$\int_{\Omega} \sigma_{ij} e_{ij}(\ul{v})$
dw_lin_strain_fib `LinearStrainFiberTerm`	`<material_1>`, `<material_2>`, `<virtual>`	$\int_{\Omega} D_{ijkl} e_{ij}(\ul{v}) \left(d_k d_l\right)$
dw_non_penetration `NonPenetrationTerm`	`<opt_material>`, `<virtual>`, `<state>` `<opt_material>`, `<state>`, `<virtual>`	$\int_{\Gamma} c \lambda \ul{n} \cdot \ul{v} \mbox{ , } \int_{\Gamma} c \hat\lambda \ul{n} \cdot \ul{u} \\ \int_{\Gamma} \lambda \ul{n} \cdot \ul{v} \mbox{ , } \int_{\Gamma} \hat\lambda \ul{n} \cdot \ul{u}$
dw_non_penetration_p `NonPenetrationPenaltyTerm`	`<material>`, `<virtual>`, `<state>`	$\int_{\Gamma} c (\ul{n} \cdot \ul{v}) (\ul{n} \cdot \ul{u})$
dw_nonsym_elastic `NonsymElasticTerm`	`<material>`, `<virtual>`, `<state>` `<material>`, `<parameter_1>`, `<parameter_2>`	$\int_{\Omega} \ull{D} \nabla\ul{u} : \nabla\ul{v}$
dw_ns_dot_grad_s `NonlinearScalarDotGradTerm`	`<fun>`, `<fun_d>`, `<virtual>`, `<state>` `<fun>`, `<fun_d>`, `<state>`, `<virtual>`	$\int_{\Omega} q \cdot \nabla \cdot \ul{f}(p) = \int_{\Omega} q \cdot \text{div} \ul{f}(p) \mbox{ , } \int_{\Omega} \ul{f}(p) \cdot \nabla q$
dw_piezo_coupling `PiezoCouplingTerm`	`<material>`, `<virtual>`, `<state>` `<material>`, `<state>`, `<virtual>` `<material>`, `<parameter_v>`, `<parameter_s>`	$\int_{\Omega} g_{kij}\ e_{ij}(\ul{v}) \nabla_k p \mbox{ , } \int_{\Omega} g_{kij}\ e_{ij}(\ul{u}) \nabla_k q$
ev_piezo_strain `PiezoStrainTerm`	`<material>`, `<parameter>`	$\int_{\Omega} g_{kij} e_{ij}(\ul{u})$
ev_piezo_stress `PiezoStressTerm`	`<material>`, `<parameter>`	$\int_{\Omega} g_{kij} \nabla_k p$
dw_point_load `ConcentratedPointLoadTerm`	`<material>`, `<virtual>`	$\ul{f}^i = \ul{\bar f}^i \quad \forall \mbox{ FE node } i \mbox{ in a region }$
dw_point_lspring `LinearPointSpringTerm`	`<material>`, `<virtual>`, `<state>`	$\ul{f}^i = -k \ul{u}^i \quad \forall \mbox{ FE node } i \mbox{ in a region }$
dw_s_dot_grad_i_s `ScalarDotGradIScalarTerm`	`<material>`, `<virtual>`, `<state>`	$Z^i = \int_{\Omega} q \nabla_i p$
dw_s_dot_mgrad_s `ScalarDotMGradScalarTerm`	`<material>`, `<virtual>`, `<state>` `<material>`, `<state>`, `<virtual>`	$\int_{\Omega} q \ul{y} \cdot \nabla p \mbox{ , } \int_{\Omega} p \ul{y} \cdot \nabla q$
dw_shell10x `Shell10XTerm`	`<material_d>`, `<material_drill>`, `<virtual>`, `<state>`	$\int_{\Omega} D_{ijkl}\ e_{ij}(\ul{v}) e_{kl}(\ul{u})$
dw_stokes `StokesTerm`	`<opt_material>`, `<virtual>`, `<state>` `<opt_material>`, `<state>`, `<virtual>` `<opt_material>`, `<parameter_v>`, `<parameter_s>`	$\int_{\Omega} p\ \nabla \cdot \ul{v} \mbox{ , } \int_{\Omega} q\ \nabla \cdot \ul{u} \mbox{ or } \int_{\Omega} c\ p\ \nabla \cdot \ul{v} \mbox{ , } \int_{\Omega} c\ q\ \nabla \cdot \ul{u}$
dw_stokes_wave `StokesWaveTerm`	`<material>`, `<virtual>`, `<state>`	$\int_{\Omega} (\ul{\kappa} \cdot \ul{v}) (\ul{\kappa} \cdot \ul{u})$
dw_stokes_wave_div `StokesWaveDivTerm`	`<material>`, `<virtual>`, `<state>` `<material>`, `<state>`, `<virtual>`	$\int_{\Omega} (\ul{\kappa} \cdot \ul{v}) (\nabla \cdot \ul{u}) \;, \int_{\Omega} (\ul{\kappa} \cdot \ul{u}) (\nabla \cdot \ul{v})$
d_sum_vals `SumNodalValuesTerm`	`<parameter>`
d_surface `SurfaceTerm`	`<parameter>`	$\int_\Gamma 1$
ev_surface_div `SurfaceDivTerm`	`<parameter>`	$\int_{\Gamma} \nabla \cdot \ul{u}$ $\mbox{vector for } K \from \Ical_h: \int_{T_K} \nabla \cdot \ul{u} / \int_{T_K} 1$ $(\nabla \cdot \ul{u})\|_{qp}$
dw_surface_dot `DotProductSurfaceTerm`	`<opt_material>`, `<virtual>`, `<state>` `<opt_material>`, `<parameter_1>`, `<parameter_2>`	$\int_\Gamma q p \mbox{ , } \int_\Gamma \ul{v} \cdot \ul{u} \mbox{ , } \int_\Gamma \ul{v} \cdot \ul{n} p \mbox{ , } \int_\Gamma q \ul{n} \cdot \ul{u} \mbox{ , } \int_\Gamma p r \mbox{ , } \int_\Gamma \ul{u} \cdot \ul{w} \mbox{ , } \int_\Gamma \ul{w} \cdot \ul{n} p \\ \int_\Gamma c q p \mbox{ , } \int_\Gamma c \ul{v} \cdot \ul{u} \mbox{ , } \int_\Gamma c p r \mbox{ , } \int_\Gamma c \ul{u} \cdot \ul{w} \\ \int_\Gamma \ul{v} \cdot \ull{M} \cdot \ul{u} \mbox{ , } \int_\Gamma \ul{u} \cdot \ull{M} \cdot \ul{w}$
d_surface_flux `SurfaceFluxTerm`	`<material>`, `<parameter>`	$\int_{\Gamma} \ul{n} \cdot K_{ij} \nabla_j \bar{p}$ $\mbox{vector for } K \from \Ical_h: \int_{T_K} \ul{n} \cdot K_{ij} \nabla_j \bar{p}\ / \int_{T_K} 1$ $\mbox{vector for } K \from \Ical_h: \int_{T_K} \ul{n} \cdot K_{ij} \nabla_j \bar{p}$
dw_surface_flux `SurfaceFluxOperatorTerm`	`<opt_material>`, `<virtual>`, `<state>`	$\int_{\Gamma} q \ul{n} \cdot \ull{K} \cdot \nabla p$
ev_surface_grad `SurfaceGradTerm`	`<parameter>`	$\int_{\Gamma} \nabla p \mbox{ or } \int_{\Gamma} \nabla \ul{w}$ $\mbox{vector for } K \from \Ical_h: \int_{T_K} \nabla p / \int_{T_K} 1 \mbox{ or } \int_{T_K} \nabla \ul{w} / \int_{T_K} 1$ $(\nabla p)\|_{qp} \mbox{ or } \nabla \ul{w}\|_{qp}$
ev_surface_integrate `IntegrateSurfaceTerm`	`<opt_material>`, `<parameter>`	$\int_\Gamma y \mbox{ , } \int_\Gamma \ul{y} \mbox{ , } \int_\Gamma \ul{y} \cdot \ul{n} \\ \int_\Gamma c y \mbox{ , } \int_\Gamma c \ul{y} \mbox{ , } \int_\Gamma c \ul{y} \cdot \ul{n} \mbox{ flux }$ $\mbox{vector for } K \from \Ical_h: \int_{T_K} y / \int_{T_K} 1 \mbox{ , } \int_{T_K} \ul{y} / \int_{T_K} 1 \mbox{ , } \int_{T_K} (\ul{y} \cdot \ul{n}) / \int_{T_K} 1 \\ \mbox{vector for } K \from \Ical_h: \int_{T_K} c y / \int_{T_K} 1 \mbox{ , } \int_{T_K} c \ul{y} / \int_{T_K} 1 \mbox{ , } \int_{T_K} (c \ul{y} \cdot \ul{n}) / \int_{T_K} 1$ $y\|_{qp} \mbox{ , } \ul{y}\|_{qp} \mbox{ , } (\ul{y} \cdot \ul{n})\|_{qp} \mbox{ flux } \\ c y\|_{qp} \mbox{ , } c \ul{y}\|_{qp} \mbox{ , } (c \ul{y} \cdot \ul{n})\|_{qp} \mbox{ flux }$
dw_surface_integrate `IntegrateSurfaceOperatorTerm`	`<opt_material>`, `<virtual>`	$\int_{\Gamma} q \mbox{ or } \int_\Gamma c q$
ev_surface_integrate_mat `IntegrateSurfaceMatTerm`	`<material>`, `<parameter>`	$\int_\Gamma m$ $\mbox{vector for } K \from \Ical_h: \int_{T_K} m / \int_{T_K} 1$ $m\|_{qp}$
dw_surface_ltr `LinearTractionTerm`	`<opt_material>`, `<virtual>` `<opt_material>`, `<parameter>`	$\int_{\Gamma} \ul{v} \cdot \ull{\sigma} \cdot \ul{n}, \int_{\Gamma} \ul{v} \cdot \ul{n},$
d_surface_moment `SurfaceMomentTerm`	`<material>`, `<parameter>`	$\int_{\Gamma} \ul{n} (\ul{x} - \ul{x}_0)$
dw_surface_ndot `SufaceNormalDotTerm`	`<material>`, `<virtual>` `<material>`, `<parameter>`	$\int_{\Gamma} q \ul{c} \cdot \ul{n}$
dw_v_dot_grad_s `VectorDotGradScalarTerm`	`<opt_material>`, `<virtual>`, `<state>` `<opt_material>`, `<state>`, `<virtual>` `<opt_material>`, `<parameter_v>`, `<parameter_s>`	$\int_{\Omega} \ul{v} \cdot \nabla p \mbox{ , } \int_{\Omega} \ul{u} \cdot \nabla q \\ \int_{\Omega} c \ul{v} \cdot \nabla p \mbox{ , } \int_{\Omega} c \ul{u} \cdot \nabla q \\ \int_{\Omega} \ul{v} \cdot (\ull{M} \nabla p) \mbox{ , } \int_{\Omega} \ul{u} \cdot (\ull{M} \nabla q)$
dw_vm_dot_s `VectorDotScalarTerm`	`<material>`, `<virtual>`, `<state>` `<material>`, `<state>`, `<virtual>` `<material>`, `<parameter_v>`, `<parameter_s>`	$\int_{\Omega} \ul{v} \cdot \ul{m} p \mbox{ , } \int_{\Omega} \ul{u} \cdot \ul{m} q\\$
d_volume `VolumeTerm`	`<parameter>`	$\int_\Omega 1$
dw_volume_dot `DotProductVolumeTerm`	`<opt_material>`, `<virtual>`, `<state>` `<opt_material>`, `<parameter_1>`, `<parameter_2>`	$\int_\Omega q p \mbox{ , } \int_\Omega \ul{v} \cdot \ul{u} \mbox{ , } \int_\Omega p r \mbox{ , } \int_\Omega \ul{u} \cdot \ul{w} \\ \int_\Omega c q p \mbox{ , } \int_\Omega c \ul{v} \cdot \ul{u} \mbox{ , } \int_\Omega c p r \mbox{ , } \int_\Omega c \ul{u} \cdot \ul{w} \\ \int_\Omega \ul{v} \cdot (\ull{M} \ul{u}) \mbox{ , } \int_\Omega \ul{u} \cdot (\ull{M} \ul{w})$
dw_volume_integrate `IntegrateVolumeOperatorTerm`	`<opt_material>`, `<virtual>`	$\int_\Omega q \mbox{ or } \int_\Omega c q$
ev_volume_integrate `IntegrateVolumeTerm`	`<opt_material>`, `<parameter>`	$\int_\Omega y \mbox{ , } \int_\Omega \ul{y} \\ \int_\Omega c y \mbox{ , } \int_\Omega c \ul{y}$ $\mbox{vector for } K \from \Ical_h: \int_{T_K} y / \int_{T_K} 1 \mbox{ , } \int_{T_K} \ul{y} / \int_{T_K} 1 \\ \mbox{vector for } K \from \Ical_h: \int_{T_K} c y / \int_{T_K} 1 \mbox{ , } \int_{T_K} c \ul{y} / \int_{T_K} 1$ $y\|_{qp} \mbox{ , } \ul{y}\|_{qp} \\ c y\|_{qp} \mbox{ , } c \ul{y}\|_{qp}$
ev_volume_integrate_mat `IntegrateVolumeMatTerm`	`<material>`, `<parameter>`	$\int_\Omega m$ $\mbox{vector for } K \from \Ical_h: \int_{T_K} m / \int_{T_K} 1$ $m\|_{qp}$
dw_volume_lvf `LinearVolumeForceTerm`	`<material>`, `<virtual>`	$\int_{\Omega} \ul{f} \cdot \ul{v} \mbox{ or } \int_{\Omega} f q$
d_volume_surface `VolumeSurfaceTerm`	`<parameter>`	$1 / D \int_\Gamma \ul{x} \cdot \ul{n}$
dw_zero `ZeroTerm`	`<virtual>`, `<state>`	$0$

Table of sensitivity terms¶

Sensitivity terms¶
name/class	arguments	definition
dw_adj_convect1 `AdjConvect1Term`	`<virtual>`, `<state>`, `<parameter>`	$\int_{\Omega} ((\ul{v} \cdot \nabla) \ul{u}) \cdot \ul{w}$
dw_adj_convect2 `AdjConvect2Term`	`<virtual>`, `<state>`, `<parameter>`	$\int_{\Omega} ((\ul{u} \cdot \nabla) \ul{v}) \cdot \ul{w}$
dw_adj_div_grad `AdjDivGradTerm`	`<material_1>`, `<material_2>`, `<virtual>`, `<parameter>`	$w \delta_{u} \Psi(\ul{u}) \circ \ul{v}$
d_sd_convect `SDConvectTerm`	`<parameter_u>`, `<parameter_w>`, `<parameter_mesh_velocity>`	$\int_{\Omega} [ u_k \pdiff{u_i}{x_k} w_i (\nabla \cdot \Vcal) - u_k \pdiff{\Vcal_j}{x_k} \pdiff{u_i}{x_j} w_i ]$
d_sd_diffusion `SDDiffusionTerm`	`<material>`, `<parameter_q>`, `<parameter_p>`, `<parameter_mesh_velocity>`	$\int_{\Omega} \left[ (\dvg \ul{\Vcal}) K_{ij} \nabla_i q\, \nabla_j p - K_{ij} (\nabla_j \ul{\Vcal} \nabla q) \nabla_i p - K_{ij} \nabla_j q (\nabla_i \ul{\Vcal} \nabla p)\right]$
d_sd_div `SDDivTerm`	`<parameter_u>`, `<parameter_p>`, `<parameter_mesh_velocity>`	$\int_{\Omega} p [ (\nabla \cdot \ul{w}) (\nabla \cdot \ul{\Vcal}) - \pdiff{\Vcal_k}{x_i} \pdiff{w_i}{x_k} ]$
d_sd_div_grad `SDDivGradTerm`	`<material_1>`, `<material_2>`, `<parameter_u>`, `<parameter_w>`, `<parameter_mesh_velocity>`	$w \nu \int_{\Omega} [ \pdiff{u_i}{x_k} \pdiff{w_i}{x_k} (\nabla \cdot \ul{\Vcal}) - \pdiff{\Vcal_j}{x_k} \pdiff{u_i}{x_j} \pdiff{w_i}{x_k} - \pdiff{u_i}{x_k} \pdiff{\Vcal_l}{x_k} \pdiff{w_i}{x_k} ]$
d_sd_lin_elastic `SDLinearElasticTerm`	`<material>`, `<parameter_w>`, `<parameter_u>`, `<parameter_mesh_velocity>`	$\int_{\Omega} \hat{D}_{ijkl}\ e_{ij}(\ul{v}) e_{kl}(\ul{u})$ $\hat{D}_{ijkl} = D_{ijkl}(\nabla \cdot \ul{\Vcal}) - D_{ijkq}{\partial \Vcal_l \over \partial x_q} - D_{iqkl}{\partial \Vcal_j \over \partial x_q}$
d_sd_surface_integrate `SDSufaceIntegrateTerm`	`<parameter>`, `<parameter_mesh_velocity>`	$\int_{\Gamma} p \nabla \cdot \ul{\Vcal}$
d_sd_volume_dot `SDDotVolumeTerm`	`<parameter_1>`, `<parameter_2>`, `<parameter_mesh_velocity>`	$\int_{\Omega} p q (\nabla \cdot \ul{\Vcal}) \mbox{ , } \int_{\Omega} (\ul{u} \cdot \ul{w}) (\nabla \cdot \ul{\Vcal})$

Table of large deformation terms¶

Large deformation terms¶
name/class	arguments	definition
dw_tl_bulk_active `BulkActiveTLTerm`	`<material>`, `<virtual>`, `<state>`	$\int_{\Omega} S_{ij}(\ul{u}) \delta E_{ij}(\ul{u};\ul{v})$
dw_tl_bulk_penalty `BulkPenaltyTLTerm`	`<material>`, `<virtual>`, `<state>`	$\int_{\Omega} S_{ij}(\ul{u}) \delta E_{ij}(\ul{u};\ul{v})$
dw_tl_bulk_pressure `BulkPressureTLTerm`	`<virtual>`, `<state>`, `<state_p>`	$\int_{\Omega} S_{ij}(p) \delta E_{ij}(\ul{u};\ul{v})$
dw_tl_diffusion `DiffusionTLTerm`	`<material_1>`, `<material_2>`, `<virtual>`, `<state>`, `<parameter>`	$\int_{\Omega} \ull{K}(\ul{u}^{(n-1)}) : \pdiff{q}{\ul{X}} \pdiff{p}{\ul{X}}$
dw_tl_fib_a `FibresActiveTLTerm`	`<material_1>`, `<material_2>`, `<material_3>`, `<material_4>`, `<material_5>`, `<virtual>`, `<state>`	$\int_{\Omega} S_{ij}(\ul{u}) \delta E_{ij}(\ul{u};\ul{v})$
dw_tl_he_genyeoh `GenYeohTLTerm`	`<material>`, `<virtual>`, `<state>`	$\int_{\Omega} S_{ij}(\ul{u}) \delta E_{ij}(\ul{u};\ul{v})$
dw_tl_he_mooney_rivlin `MooneyRivlinTLTerm`	`<material>`, `<virtual>`, `<state>`	$\int_{\Omega} S_{ij}(\ul{u}) \delta E_{ij}(\ul{u};\ul{v})$
dw_tl_he_neohook `NeoHookeanTLTerm`	`<material>`, `<virtual>`, `<state>`	$\int_{\Omega} S_{ij}(\ul{u}) \delta E_{ij}(\ul{u};\ul{v})$
dw_tl_he_ogden `OgdenTLTerm`	`<material>`, `<virtual>`, `<state>`	$\int_{\Omega} S_{ij}(\ul{u}) \delta E_{ij}(\ul{u};\ul{v})$
dw_tl_membrane `TLMembraneTerm`	`<material_a1>`, `<material_a2>`, `<material_h0>`, `<virtual>`, `<state>`
d_tl_surface_flux `SurfaceFluxTLTerm`	`<material_1>`, `<material_2>`, `<parameter_1>`, `<parameter_2>`	$\int_{\Gamma} \ul{\nu} \cdot \ull{K}(\ul{u}^{(n-1)}) \pdiff{p}{\ul{X}}$
dw_tl_surface_traction `SurfaceTractionTLTerm`	`<opt_material>`, `<virtual>`, `<state>`	$\int_{\Gamma} \ul{\nu} \cdot \ull{F}^{-1} \cdot \ull{\sigma} \cdot \ul{v} J$
dw_tl_volume `VolumeTLTerm`	`<virtual>`, `<state>`	$\begin{array}{l} \int_{\Omega} q J(\ul{u}) \\ \mbox{volume mode: vector for } K \from \Ical_h: \int_{T_K} J(\ul{u}) \\ \mbox{rel\_volume mode: vector for } K \from \Ical_h: \int_{T_K} J(\ul{u}) / \int_{T_K} 1 \end{array}$
d_tl_volume_surface `VolumeSurfaceTLTerm`	`<parameter>`	$1 / D \int_{\Gamma} \ul{\nu} \cdot \ull{F}^{-1} \cdot \ul{x} J$
dw_ul_bulk_penalty `BulkPenaltyULTerm`	`<material>`, `<virtual>`, `<state>`	$\int_{\Omega} \mathcal{L}\tau_{ij}(\ul{u}) e_{ij}(\delta\ul{v})/J$
dw_ul_bulk_pressure `BulkPressureULTerm`	`<virtual>`, `<state>`, `<state_p>`	$\int_{\Omega} \mathcal{L}\tau_{ij}(\ul{u}) e_{ij}(\delta\ul{v})/J$
dw_ul_compressible `CompressibilityULTerm`	`<material>`, `<virtual>`, `<state>`, `<parameter_u>`	$\int_{\Omega} 1\over \gamma p \, q$
dw_ul_he_mooney_rivlin `MooneyRivlinULTerm`	`<material>`, `<virtual>`, `<state>`	$\int_{\Omega} \mathcal{L}\tau_{ij}(\ul{u}) e_{ij}(\delta\ul{v})/J$
dw_ul_he_neohook `NeoHookeanULTerm`	`<material>`, `<virtual>`, `<state>`	$\int_{\Omega} \mathcal{L}\tau_{ij}(\ul{u}) e_{ij}(\delta\ul{v})/J$
dw_ul_volume `VolumeULTerm`	`<virtual>`, `<state>`	$\begin{array}{l} \int_{\Omega} q J(\ul{u}) \\ \mbox{volume mode: vector for } K \from \Ical_h: \int_{T_K} J(\ul{u}) \\ \mbox{rel\_volume mode: vector for } K \from \Ical_h: \int_{T_K} J(\ul{u}) / \int_{T_K} 1 \end{array}$

Table of special terms¶

Special terms¶
name/class	arguments	definition
dw_biot_eth `BiotETHTerm`	`<ts>`, `<material_0>`, `<material_1>`, `<virtual>`, `<state>` `<ts>`, `<material_0>`, `<material_1>`, `<state>`, `<virtual>`	$\begin{array}{l} \int_{\Omega} \left [\int_0^t \alpha_{ij}(t-\tau)\,p(\tau)) \difd{\tau} \right]\,e_{ij}(\ul{v}) \mbox{ ,} \\ \int_{\Omega} \left [\int_0^t \alpha_{ij}(t-\tau) e_{kl}(\ul{u}(\tau)) \difd{\tau} \right] q \end{array}$
dw_biot_th `BiotTHTerm`	`<ts>`, `<material>`, `<virtual>`, `<state>` `<ts>`, `<material>`, `<state>`, `<virtual>`	$\begin{array}{l} \int_{\Omega} \left [\int_0^t \alpha_{ij}(t-\tau)\,p(\tau)) \difd{\tau} \right]\,e_{ij}(\ul{v}) \mbox{ ,} \\ \int_{\Omega} \left [\int_0^t \alpha_{ij}(t-\tau) e_{kl}(\ul{u}(\tau)) \difd{\tau} \right] q \end{array}$
ev_cauchy_stress_eth `CauchyStressETHTerm`	`<ts>`, `<material_0>`, `<material_1>`, `<parameter>`	$\int_{\Omega} \int_0^t \Hcal_{ijkl}(t-\tau)\,e_{kl}(\ul{w}(\tau)) \difd{\tau}$ $\mbox{vector for } K \from \Ical_h: \int_{T_K} \int_0^t \Hcal_{ijkl}(t-\tau)\,e_{kl}(\ul{w}(\tau)) \difd{\tau} / \int_{T_K} 1$ $\int_0^t \Hcal_{ijkl}(t-\tau)\,e_{kl}(\ul{w}(\tau)) \difd{\tau}\|_{qp}$
ev_cauchy_stress_th `CauchyStressTHTerm`	`<ts>`, `<material>`, `<parameter>`	$\int_{\Omega} \int_0^t \Hcal_{ijkl}(t-\tau)\,e_{kl}(\ul{w}(\tau)) \difd{\tau}$ $\mbox{vector for } K \from \Ical_h: \int_{T_K} \int_0^t \Hcal_{ijkl}(t-\tau)\,e_{kl}(\ul{w}(\tau)) \difd{\tau} / \int_{T_K} 1$ $\int_0^t \Hcal_{ijkl}(t-\tau)\,e_{kl}(\ul{w}(\tau)) \difd{\tau}\|_{qp}$
dw_lin_elastic_eth `LinearElasticETHTerm`	`<ts>`, `<material_0>`, `<material_1>`, `<virtual>`, `<state>`	$\int_{\Omega} \left [\int_0^t \Hcal_{ijkl}(t-\tau)\,e_{kl}(\ul{u}(\tau)) \difd{\tau} \right]\,e_{ij}(\ul{v})$
dw_lin_elastic_th `LinearElasticTHTerm`	`<ts>`, `<material>`, `<virtual>`, `<state>`	$\int_{\Omega} \left [\int_0^t \Hcal_{ijkl}(t-\tau)\,e_{kl}(\ul{u}(\tau)) \difd{\tau} \right]\,e_{ij}(\ul{v})$
d_of_ns_surf_min_d_press `NSOFSurfMinDPressTerm`	`<material_1>`, `<material_2>`, `<parameter>`	$\delta \Psi(p) = \delta \left( \int_{\Gamma_{in}}p - \int_{\Gamma_{out}}bpress \right)$
dw_of_ns_surf_min_d_press_diff `NSOFSurfMinDPressDiffTerm`	`<material>`, `<virtual>`	$w \delta_{p} \Psi(p) \circ q$
d_sd_st_grad_div `SDGradDivStabilizationTerm`	`<material>`, `<parameter_u>`, `<parameter_w>`, `<parameter_mesh_velocity>`	$\gamma \int_{\Omega} [ (\nabla \cdot \ul{u}) (\nabla \cdot \ul{w}) (\nabla \cdot \ul{\Vcal}) - \pdiff{u_i}{x_k} \pdiff{\Vcal_k}{x_i} (\nabla \cdot \ul{w}) - (\nabla \cdot \ul{u}) \pdiff{w_i}{x_k} \pdiff{\Vcal_k}{x_i} ]$
d_sd_st_pspg_c `SDPSPGCStabilizationTerm`	`<material>`, `<parameter_b>`, `<parameter_u>`, `<parameter_r>`, `<parameter_mesh_velocity>`	$\sum_{K \in \Ical_h}\int_{T_K} \delta_K\ [ \pdiff{r}{x_i} (\ul{b} \cdot \nabla u_i) (\nabla \cdot \Vcal) - \pdiff{r}{x_k} \pdiff{\Vcal_k}{x_i} (\ul{b} \cdot \nabla u_i) - \pdiff{r}{x_k} (\ul{b} \cdot \nabla \Vcal_k) \pdiff{u_i}{x_k} ]$
d_sd_st_pspg_p `SDPSPGPStabilizationTerm`	`<material>`, `<parameter_r>`, `<parameter_p>`, `<parameter_mesh_velocity>`	$\sum_{K \in \Ical_h}\int_{T_K} \tau_K\ [ (\nabla r \cdot \nabla p) (\nabla \cdot \Vcal) - \pdiff{r}{x_k} (\nabla \Vcal_k \cdot \nabla p) - (\nabla r \cdot \nabla \Vcal_k) \pdiff{p}{x_k} ]$
d_sd_st_supg_c `SDSUPGCStabilizationTerm`	`<material>`, `<parameter_b>`, `<parameter_u>`, `<parameter_w>`, `<parameter_mesh_velocity>`	$\sum_{K \in \Ical_h}\int_{T_K} \delta_K\ [ (\ul{b} \cdot \nabla u_k) (\ul{b} \cdot \nabla w_k) (\nabla \cdot \Vcal) - (\ul{b} \cdot \nabla \Vcal_i) \pdiff{u_k}{x_i} (\ul{b} \cdot \nabla w_k) - (\ul{u} \cdot \nabla u_k) (\ul{b} \cdot \nabla \Vcal_i) \pdiff{w_k}{x_i} ]$
dw_st_adj1_supg_p `SUPGPAdj1StabilizationTerm`	`<material>`, `<virtual>`, `<state>`, `<parameter>`	$\sum_{K \in \Ical_h}\int_{T_K} \delta_K\ \nabla p (\ul{v} \cdot \nabla \ul{w})$
dw_st_adj2_supg_p `SUPGPAdj2StabilizationTerm`	`<material>`, `<virtual>`, `<parameter>`, `<state>`	$\sum_{K \in \Ical_h}\int_{T_K} \tau_K\ \nabla r (\ul{v} \cdot \nabla \ul{u})$
dw_st_adj_supg_c `SUPGCAdjStabilizationTerm`	`<material>`, `<virtual>`, `<parameter>`, `<state>`	$\sum_{K \in \Ical_h}\int_{T_K} \delta_K\ [ ((\ul{v} \cdot \nabla) \ul{u}) ((\ul{u} \cdot \nabla) \ul{w}) + ((\ul{u} \cdot \nabla) \ul{u}) ((\ul{v} \cdot \nabla) \ul{w}) ]$
dw_st_grad_div `GradDivStabilizationTerm`	`<material>`, `<virtual>`, `<state>`	$\gamma \int_{\Omega} (\nabla\cdot\ul{u}) \cdot (\nabla\cdot\ul{v})$
dw_st_pspg_c `PSPGCStabilizationTerm`	`<material>`, `<virtual>`, `<parameter>`, `<state>`	$\sum_{K \in \Ical_h}\int_{T_K} \tau_K\ ((\ul{b} \cdot \nabla) \ul{u}) \cdot \nabla q$
dw_st_pspg_p `PSPGPStabilizationTerm`	`<opt_material>`, `<virtual>`, `<state>` `<opt_material>`, `<parameter_1>`, `<parameter_2>`	$\sum_{K \in \Ical_h}\int_{T_K} \tau_K\ \nabla p \cdot \nabla q$
dw_st_supg_c `SUPGCStabilizationTerm`	`<material>`, `<virtual>`, `<parameter>`, `<state>`	$\sum_{K \in \Ical_h}\int_{T_K} \delta_K\ ((\ul{b} \cdot \nabla) \ul{u})\cdot ((\ul{b} \cdot \nabla) \ul{v})$
dw_st_supg_p `SUPGPStabilizationTerm`	`<material>`, `<virtual>`, `<parameter>`, `<state>`	$\sum_{K \in \Ical_h}\int_{T_K} \delta_K\ \nabla p\cdot ((\ul{b} \cdot \nabla) \ul{v})$
dw_volume_dot_w_scalar_eth `DotSProductVolumeOperatorWETHTerm`	`<ts>`, `<material_0>`, `<material_1>`, `<virtual>`, `<state>`	$\int_\Omega \left [\int_0^t \Gcal(t-\tau) p(\tau) \difd{\tau} \right] q$
dw_volume_dot_w_scalar_th `DotSProductVolumeOperatorWTHTerm`	`<ts>`, `<material>`, `<virtual>`, `<state>`	$\int_\Omega \left [\int_0^t \Gcal(t-\tau) p(\tau) \difd{\tau} \right] q$