SfePy NTC

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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:

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}, \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_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_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}

dw_div

DivOperatorTerm

<opt_material>, <virtual>

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

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_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_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}

ev_integrate_mat

IntegrateMatTerm

<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

<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_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_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_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_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}

d_sum_vals

SumNodalValuesTerm

<parameter>  

d_surface

SurfaceTerm

<parameter>

\int_\Gamma 1

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_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

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}

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}

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_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_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