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’ terms having all arguments known, the result is the scalar value of the integral
di discrete integrated ‘eval’ like ‘d’ but the result is not a scalar (e.g. a vector)
dq discrete quadrature ‘qp’ terms having all arguments known, the result are the values in quadrature points of elements
ev evaluate ‘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 an automatically generated table. The first column lists the name, the second column the argument lists and the third column the mathematical definition of each term.

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 all terms.
name/class/link arguments definition

dw_adj_convect1

AdjConvect1Term termsAdjointNavierStokes
 <virtual>, <state>, <parameter>

\int_{\Omega} ((\ul{v} \cdot \nabla) \ul{u}) \cdot \ul{w}

dw_adj_convect2

AdjConvect2Term termsAdjointNavierStokes
 <virtual>, <state>, <parameter>

\int_{\Omega} ((\ul{u} \cdot \nabla) \ul{v}) \cdot \ul{w}

dw_adj_div_grad

 <material_1>, <material_2>, <virtual>, <parameter>

w \delta_{u} \Psi(\ul{u}) \circ \ul{v}

dw_bc_newton

BCNewtonTerm terms_dot
 <material_1>, <material_2>, <virtual>, <state>

\int_{\Gamma} \alpha q (p - p_{\rm outer})

dw_biot

BiotTerm termsBiot

<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})

dw_biot_eth

BiotETHTerm termsBiot

<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}

ev_biot_stress

BiotStressTerm termsBiot
 <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}

dw_biot_th

BiotTHTerm termsBiot

<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_strain

CauchyStrainTerm termsLinElasticity
 <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 termsLinElasticity
 <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 sfepy.terms.termsLinElasticity
 <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}

ev_cauchy_stress_eth

CauchyStressETHTerm termsLinElasticity
 <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 termsLinElasticity
 <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_contact_plane

ContactPlaneTerm termsSurface
 <material_f>, <material_n>, <material_a>, <material_b>, <virtual>, <state>

\int_{\Gamma} \ul{v} \cdot f(d(\ul{u})) \ul{n}

dw_convect

ConvectTerm termsNavierStokes
 <virtual>, <state>

\int_{\Omega} ((\ul{u} \cdot \nabla) \ul{u}) \cdot \ul{v}

ev_def_grad

DeformationGradientTerm terms_hyperelastic_base
 <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_diffusion

DiffusionTerm termsLaplace

<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 termsLaplace

<material>, <virtual>, <state>

<material>, <state>, <virtual>

<material>, <parameter_1>, <parameter_2>

\int_{\Omega} p K_{j} \nabla_j q

dw_diffusion_r

DiffusionRTerm termsLaplace
 <material>, <virtual>

\int_{\Omega} K_{j} \nabla_j q

d_diffusion_sa

DiffusionSATerm termsAcoustic
 <material>, <parameter_q>, <parameter_p>, <parameter_v>

\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]

ev_diffusion_velocity

DiffusionVelocityTerm termsLaplace
 <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

 <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 termsNavierStokes
 <opt_material>, <virtual>

\int_{\Omega} \nabla \cdot \ul{v} \mbox { or } \int_{\Omega} c \nabla \cdot \ul{v}

dw_div_grad

DivGradTerm termsNavierStokes

<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_electric_source

ElectricSourceTerm termsElectric
 <material>, <virtual>, <parameter>

\int_{\Omega} c s (\nabla \phi)^2

ev_grad

 <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}

ev_integrate_mat

IntegrateMatTerm termsBasic
 <material>, <parameter>

\int_\Omega m

\mbox{vector for } K \from \Ical_h: \int_{T_K} m / \int_{T_K} 1

m|_{qp}

dw_jump

SurfaceJumpTerm termsSurface
 <opt_material>, <virtual>, <state_1>, <state_2>

\int_{\Gamma} c\, q (p_1 - p_2)

dw_laplace

LaplaceTerm sfepy.terms.termsLaplace

<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 termsNavierStokes
 <virtual>, <parameter>, <state>

\int_{\Omega} ((\ul{b} \cdot \nabla) \ul{u}) \cdot \ul{v}

((\ul{b} \cdot \nabla) \ul{u})|_{qp}

dw_lin_elastic

LinearElasticTerm termsLinElasticity

<material>, <virtual>, <state>

<material>, <parameter_1>, <parameter_2>

\int_{\Omega} D_{ijkl}\ e_{ij}(\ul{v}) e_{kl}(\ul{u})

dw_lin_elastic_eth

LinearElasticETHTerm termsLinElasticity
 <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_iso

LinearElasticIsotropicTerm termsLinElasticity
 <material_1>, <material_2>, <virtual>, <state>

\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_elastic_th

LinearElasticTHTerm termsLinElasticity
 <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})

dw_lin_prestress

LinearPrestressTerm termsLinElasticity

<material>, <virtual>

<material>, <parameter>

\int_{\Omega} \sigma_{ij} e_{ij}(\ul{v})

dw_lin_strain_fib

LinearStrainFiberTerm termsLinElasticity
 <material_1>, <material_2>, <virtual>

\int_{\Omega} D_{ijkl} e_{ij}(\ul{v}) \left(d_k d_l\right)

dw_new_diffusion

NewDiffusionTerm terms_new
 <material>, <virtual>, <state>  

dw_new_lin_elastic

NewLinearElasticTerm terms_new
 <material>, <virtual>, <state>  

dw_new_mass

NewMassTerm terms_new
 <virtual>, <state>  

dw_new_mass_scalar

NewMassScalarTerm terms_new
 <virtual>, <state>  

dw_non_penetration

NonPenetrationTerm terms_constraints

<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}

d_of_ns_surf_min_d_press

NSOFSurfMinDPressTerm termsAdjointNavierStokes
 <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 termsAdjointNavierStokes
 <material>, <virtual>

w \delta_{p} \Psi(p) \circ q

dw_permeability_r

PermeabilityRTerm termsLaplace
 <material>, <virtual>, <index>

\int_{\Omega} K_{ij} \nabla_j q

dw_piezo_coupling

PiezoCouplingTerm termsPiezo

<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

dw_point_load

ConcentratedPointLoadTerm termsPoint
 <material>, <virtual>

\ul{f}^i = \ul{\bar f}^i \quad \forall \mbox{ FE node } i \mbox{ in a region }

dw_point_lspring

LinearPointSpringTerm termsPoint
 <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 terms_dot
 <material>, <virtual>, <state>

Z^i = \int_{\Omega} q \nabla_i p

d_sd_convect

 <parameter_u>, <parameter_w>, <parameter_mesh_velocity>

\int_{\Omega_D} [ 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_div

 <parameter_u>, <parameter_p>, <parameter_mesh_velocity>

\int_{\Omega_D} p [ (\nabla \cdot \ul{w}) (\nabla \cdot \ul{\Vcal}) - \pdiff{\Vcal_k}{x_i} \pdiff{w_i}{x_k} ]

d_sd_div_grad

 <material_1>, <material_2>, <parameter_u>, <parameter_w>, <parameter_mesh_velocity>

w \nu \int_{\Omega_D} [ \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 termsLinElasticity
 <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_st_grad_div

SDGradDivStabilizationTerm termsAdjointNavierStokes
 <material>, <parameter_u>, <parameter_w>, <parameter_mesh_velocity>

\gamma \int_{\Omega_D} [ (\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 termsAdjointNavierStokes
 <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 termsAdjointNavierStokes
 <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 termsAdjointNavierStokes
 <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} ]

d_sd_surface_ndot

SDSufaceNormalDotTerm termsSurface
 <material>, <parameter>, <parameter_mesh_velocity>

\int_{\Gamma} p \ul{c} \cdot \ul{n} \nabla \cdot \ul{\Vcal}

d_sd_volume_dot

SDDotVolumeTerm termsAdjointNavierStokes
 <parameter_1>, <parameter_2>, <parameter_mesh_velocity>

\int_{\Omega_D} p q (\nabla \cdot \ul{\Vcal}) \mbox{ , } \int_{\Omega_D} (\ul{u} \cdot \ul{w}) (\nabla \cdot \ul{\Vcal})

dw_st_adj1_supg_p

SUPGPAdj1StabilizationTerm termsAdjointNavierStokes
 <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 termsAdjointNavierStokes
 <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 termsAdjointNavierStokes
 <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 termsNavierStokes
 <material>, <virtual>, <state>

\gamma \int_{\Omega} (\nabla\cdot\ul{u}) \cdot (\nabla\cdot\ul{v})

dw_st_pspg_c

PSPGCStabilizationTerm termsNavierStokes
 <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 termsNavierStokes

<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 termsNavierStokes
 <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 termsNavierStokes
 <material>, <virtual>, <parameter>, <state>

\sum_{K \in \Ical_h}\int_{T_K} \delta_K\ \nabla p\cdot ((\ul{b} \cdot \nabla) \ul{v})

dw_stokes

<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}

d_sum_vals

SumNodalValuesTerm termsBasic
 <parameter>  

d_surface

SurfaceTerm termsBasic
 <parameter>

\int_\Gamma 1

dw_surface_dot

DotProductSurfaceTerm terms_dot

<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 termsLaplace
 <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}

ev_surface_integrate

IntegrateSurfaceTerm termsBasic
 <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 termsBasic
 <opt_material>, <virtual>

\int_{\Gamma} q \mbox{ or } \int_\Gamma c q

dw_surface_laplace

SurfaceLaplaceLayerTerm termsAcoustic

<material>, <virtual>, <state>

<material>, <parameter_2>, <parameter_1>

\int_{\Gamma} c \partial_\alpha \ul{q}\,\partial_\alpha \ul{p}, \alpha = 1,\dots,N-1

dw_surface_lcouple

SurfaceCoupleLayerTerm termsAcoustic

<material>, <virtual>, <state>

<material>, <state>, <virtual>

<material>, <parameter_1>, <parameter_2>

\int_{\Gamma} c q\,\partial_\alpha p, \int_{\Gamma} c \partial_\alpha p\, q, \int_{\Gamma} c \partial_\alpha r\, s,\alpha = 1,\dots,N-1

dw_surface_ltr

LinearTractionTerm termsSurface
 <material>, <virtual>

\int_{\Gamma} \ul{v} \cdot \ull{\sigma} \cdot \ul{n}

di_surface_moment

SurfaceMomentTerm termsBasic
 <parameter>, <shift>

\int_{\Gamma} \ul{n} (\ul{x} - \ul{x}_0)

dw_surface_ndot

SufaceNormalDotTerm termsSurface

<material>, <virtual>

<material>, <parameter>

\int_{\Gamma} q \ul{c} \cdot \ul{n}

dw_tl_bulk_active

BulkActiveTLTerm terms_hyperelastic_tl
 <material>, <virtual>, <state>

\int_{\Omega} S_{ij}(\ul{u}) \delta E_{ij}(\ul{u};\ul{v})

dw_tl_bulk_penalty

BulkPenaltyTLTerm terms_hyperelastic_tl
 <material>, <virtual>, <state>

\int_{\Omega} S_{ij}(\ul{u}) \delta E_{ij}(\ul{u};\ul{v})

dw_tl_bulk_pressure

BulkPressureTLTerm terms_hyperelastic_tl
 <virtual>, <state>, <state_p>

\int_{\Omega} S_{ij}(p) \delta E_{ij}(\ul{u};\ul{v})

dw_tl_diffusion

DiffusionTLTerm terms_hyperelastic_tl
 <material_1>, <material_2>, <virtual>, <state>, <parameter>

\int_{\Omega} \ull{K}(\ul{u}^{(n-1)}) : \pdiff{q}{X} \pdiff{p}{X}

dw_tl_fib_a

FibresActiveTLTerm terms_fibres
 <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_mooney_rivlin

MooneyRivlinTLTerm terms_hyperelastic_tl
 <material>, <virtual>, <state>

\int_{\Omega} S_{ij}(\ul{u}) \delta E_{ij}(\ul{u};\ul{v})

dw_tl_he_neohook

NeoHookeanTLTerm terms_hyperelastic_tl
 <material>, <virtual>, <state>

\int_{\Omega} S_{ij}(\ul{u}) \delta E_{ij}(\ul{u};\ul{v})

dw_tl_membrane

TLMembraneTerm terms_membrane
 <material_a1>, <material_a2>, <material_h0>, <virtual>, <state>  

dw_tl_surface_traction

SurfaceTractionTLTerm terms_hyperelastic_tl
 <material>, <virtual>, <state>

\int_{\Gamma} \ul{\nu} \cdot \ull{F}^{-1} \cdot \ull{\sigma} \cdot \ul{v} J

dw_tl_volume

 <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}

dw_ul_bulk_penalty

BulkPenaltyULTerm terms_hyperelastic_ul
 <material>, <virtual>, <state>

\int_{\Omega} \mathcal{L}\tau_{ij}(\ul{u}) e_{ij}(\delta\ul{v})/J

dw_ul_bulk_pressure

BulkPressureULTerm terms_hyperelastic_ul
 <virtual>, <state>, <state_p>

\int_{\Omega} \mathcal{L}\tau_{ij}(\ul{u}) e_{ij}(\delta\ul{v})/J

dw_ul_compressible

CompressibilityULTerm terms_hyperelastic_ul
 <material>, <virtual>, <state>, <parameter_u>

\int_{\Omega} 1\over \gamma p \, q

dw_ul_he_mooney_rivlin

MooneyRivlinULTerm terms_hyperelastic_ul
 <material>, <virtual>, <state>

\int_{\Omega} \mathcal{L}\tau_{ij}(\ul{u}) e_{ij}(\delta\ul{v})/J

dw_ul_he_neohook

NeoHookeanULTerm terms_hyperelastic_ul
 <material>, <virtual>, <state>

\int_{\Omega} \mathcal{L}\tau_{ij}(\ul{u}) e_{ij}(\delta\ul{v})/J

dw_ul_volume

 <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}

dw_v_dot_grad_s

VectorDotGradScalarTerm terms_dot

<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} \cdot \nabla p \mbox{ , } \int_{\Omega} \ul{u} \cdot \ull{M} \cdot \nabla q

d_volume

VolumeTerm termsBasic
 <parameter>

\int_\Omega 1

dw_volume_dot

DotProductVolumeTerm terms_dot

<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} \cdot \ul{u} \mbox{ , } \int_\Omega \ul{u} \cdot \ull{M} \cdot \ul{w}

dw_volume_dot_w_scalar_eth

DotSProductVolumeOperatorWETHTerm terms_dot
 <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 terms_dot
 <ts>, <material>, <virtual>, <state>

\int_\Omega \left [\int_0^t \Gcal(t-\tau) p(\tau) \difd{\tau} \right] q

ev_volume_integrate

IntegrateVolumeTerm termsBasic
 <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}

dw_volume_integrate

IntegrateVolumeOperatorTerm termsBasic
 <opt_material>, <virtual>

\int_\Omega q \mbox{ or } \int_\Omega c q

dw_volume_lvf

LinearVolumeForceTerm termsVolume
 <material>, <virtual>

\int_{\Omega} \ul{f} \cdot \ul{v} \mbox{ or } \int_{\Omega} f q

d_volume_surface

VolumeSurfaceTerm termsBasic
 <parameter>

\int_\Gamma \ul{x} \cdot \ul{n}