Table of basic terms

Basic terms

name/class

arguments

definition

examples

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

tim.adv.dif

dw_bc_newton

BCNewtonTerm

<material_1>, <material_2>, <virtual>, <state>

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

tim.hea.equ.mul.mat

dw_biot

BiotTerm

<material>, <virtual/param_v>, <state/param_s>

<material>, <state>, <virtual>

\int_{\Omega} p\ \alpha_{ij} e_{ij}(\ul{v}) \mbox{ , } \int_{\Omega} q\ \alpha_{ij} e_{ij}(\ul{u})

the.ela, the.ela.ess, bio.sho.syn, bio.npb, bio, bio.npb.lag

ev_biot_stress

BiotStressTerm

<material>, <parameter>

- \int_{\Omega} \alpha_{ij} p

ev_cauchy_strain

CauchyStrainTerm

<parameter>

\int_{\cal{D}} \ull{e}(\ul{w})

ev_cauchy_stress

CauchyStressTerm

<material>, <parameter>

\int_{\cal{D}} D_{ijkl} e_{kl}(\ul{w})

dw_contact

ContactTerm

<material>, <virtual>, <state>

\int_{\Gamma_{c}} \varepsilon_N \langle g_N(\ul{u}) \rangle \ul{n} \ul{v}

two.bod.con

dw_contact_ipc

ContactIPCTerm

<material_m>, <material_k>, <material_d>, <material_p>, <virtual>, <state>

\int_{\Gamma_{c}} \sum_{k \in C} \nabla b(d_k, \ul{u})

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}

ela.con.pla

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})

ela.con.sph

dw_convect

ConvectTerm

<virtual>, <state>

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

nav.sto, nav.sto.iga, nav.sto

dw_convect_v_grad_s

ConvectVGradSTerm

<virtual>, <state_v>, <state_s>

\int_{\Omega} q (\ul{u} \cdot \nabla p)

poi.fun

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},

adv.2D, adv.dif.2D, adv.1D

dw_dg_diffusion_flux

DiffusionDGFluxTerm

<material>, <state>, <virtual>

<material>, <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}

bur.2D, adv.dif.2D, lap.2D

dw_dg_interior_penalty

DiffusionInteriorPenaltyTerm

<material>, <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}

bur.2D, adv.dif.2D, lap.2D

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},

bur.2D

dw_diffusion

DiffusionTerm

<material>, <virtual/param_1>, <state/param_2>

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

poi.neu, vib.aco, bio.sho.syn, pie.ela, dar.flo.mul, bio.npb, pie.ela, bio, bio.npb.lag

dw_diffusion_coupling

DiffusionCoupling

<material>, <virtual/param_1>, <state/param_2>

<material>, <state>, <virtual>

\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_{\cal{D}} K_{ij} \nabla_j p

ev_div

DivTerm

<opt_material>, <parameter>

\int_{\cal{D}} \nabla \cdot \ul{u} \mbox { , } \int_{\cal{D}} c \nabla \cdot \ul{u}

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/param_1>, <state/param_2>

\int_{\Omega} \nu\ \nabla \ul{v} : \nabla \ul{u} \mbox{ , } \int_{\Omega} \nabla \ul{v} : \nabla \ul{u}

sto, nav.sto, sta.nav.sto, nav.sto, nav.sto.iga, sto.sli.bc

dw_dot

DotProductTerm

<opt_material>, <virtual/param_1>, <state/param_2>

\int_{\cal{D}} q p \mbox{ , } \int_{\cal{D}} \ul{v} \cdot \ul{u}\\ \int_\Gamma \ul{v} \cdot \ul{n} p \mbox{ , } \int_\Gamma q \ul{n} \cdot \ul{u} \mbox{ , }\\ \int_{\cal{D}} c q p \mbox{ , } \int_{\cal{D}} c \ul{v} \cdot \ul{u} \mbox{ , } \int_{\cal{D}} \ul{v} \cdot \ull{c} \cdot \ul{u}

hel.apa, pie.ela, dar.flo.mul, bor, aco, bal, poi.per.bou.con, poi.fun, wel, sto.sli.bc, bur.2D, mod.ana.dec, lin.ela.up, adv.1D, tim.hea.equ.mul.mat, adv.2D, tim.poi.exp, vib.aco, the.ele, osc, tim.adv.dif, hyd, lin.ela.dam, aco, pie.ela, ref.evp, tim.poi

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

the.ele

ev_grad

GradTerm

<opt_material>, <parameter>

\int_{\cal{D}} \nabla p \mbox{ or } \int_{\cal{D}} \nabla \ul{u}\\ \int_{\cal{D}} c \nabla p \mbox{ or } \int_{\cal{D}} c \nabla \ul{u}

dw_integrate

IntegrateOperatorTerm

<opt_material>, <virtual>

\int_{\cal{D}} q \mbox{ or } \int_{\cal{D}} c q

tim.hea.equ.mul.mat, poi.neu, vib.aco, hel.apa, dar.flo.mul, aco, aco, poi.per.bou.con

ev_integrate

IntegrateTerm

<opt_material>, <parameter>

\int_{\cal{D}} y \mbox{ , } \int_{\cal{D}} \ul{y} \mbox{ , } \int_\Gamma \ul{y} \cdot \ul{n}\\ \int_{\cal{D}} c y \mbox{ , } \int_{\cal{D}} c \ul{y} \mbox{ , } \int_\Gamma c \ul{y} \cdot \ul{n} \mbox{ flux }

ev_integrate_mat

IntegrateMatTerm

<material>, <parameter>

\int_{\cal{D}} c

dw_jump

SurfaceJumpTerm

<opt_material>, <virtual>, <state_1>, <state_2>

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

aco

dw_laplace

LaplaceTerm

<opt_material>, <virtual/param_1>, <state/param_2>

\int_{\Omega} c \nabla q \cdot \nabla p

hel.apa, cub, bor, poi.iga, poi.par.stu, aco, poi.per.bou.con, lap.2D, lap.tim.ebc, poi.sho.syn, poi.fun, wel, sto.sli.bc, poi, bur.2D, tim.hea.equ.mul.mat, sin, tim.poi.exp, vib.aco, adv.dif.2D, the.ele, poi.fie.dep.mat, osc, tim.adv.dif, hyd, the.ela.ess, aco, lap.flu.2d, lap.1d, ref.evp, tim.poi, lap.cou.lcb

dw_lin_convect

LinearConvectTerm

<virtual>, <parameter>, <state>

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

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

sta.nav.sto

dw_lin_convect2

LinearConvect2Term

<material>, <virtual>, <state>

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

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

dw_lin_dspring

LinearDSpringTerm

<opt_material>, <material>, <virtual>, <state>

f^{(i)}_k = -f^{(j)}_k = K_{kl} (u^{(j)}_l - u^{(i)}_l)\\ \quad \forall \mbox{ elements } T_K^{i,j}\\ \mbox{ in a region connecting nodes } i, j

dw_lin_dspring_rot

LinearDRotSpringTerm

<opt_material>, <material>, <virtual>, <state>

f^{(i)}_k = -f^{(j)}_k = K_{kl} (u^{(j)}_l - u^{(i)}_l)\\ \quad \forall \mbox{ elements } T_K^{i,j}\\ \mbox{ in a region connecting nodes } i, j

mul.poi.con

dw_lin_elastic

LinearElasticTerm

<material>, <virtual/param_1>, <state/param_2>

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

lin.vis, the.ela, ela, pie.ela, wed.mes, bio.npb, nod.lcb, mix.mes, bio.npb.lag, mat.non, its.4, lin.ela.mM, pre.fib, pie.ela.mac, two.bod.con, bio, lin.ela.iga, lin.ela.tra, mod.ana.dec, lin.ela.up, vib.aco, its.3, ela.con.sph, its.2, lin.ela, mul.nod.lcb, its.1, ela.con.pla, ela.shi.per, mul.poi.con, lin.ela.dam, bio.sho.syn, the.ela.ess, lin.ela.opt, pie.ela, tru.bri, sei.loa, com.ela.mat

dw_lin_elastic_iso

LinearElasticIsotropicTerm

<material_1>, <material_2>, <virtual/param_1>, <state/param_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/param>

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

non.hyp.mM, pre.fib, pie.ela.mac

dw_lin_spring

LinearSpringTerm

<material>, <virtual>, <state>

\ul{f}^{(i)} = - \ul{f}^{(j)} = k (\ul{u}^{(j)} - \ul{u}^{(i)})\\ \quad \forall \mbox{ elements } T_K^{i,j}\\ \mbox{ in a region connecting nodes } i, j

dw_lin_strain_fib

LinearStrainFiberTerm

<material_1>, <material_2>, <virtual>

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

pre.fib

dw_lin_truss

LinearTrussTerm

<material>, <virtual>, <state>

F^{(i)} = -F^{(j)} = EA / l (U^{(j)} - U^{(i)})\\ \quad \forall \mbox{ elements } T_K^{i,j}\\ \mbox{ in a region connecting nodes } i, j

tru.bri, tru.bri

ev_lin_truss_force

LinearTrussInternalForceTerm

<material>, <parameter>

F = EA / l (U^{(j)} - U^{(i)})\\ \quad \forall \mbox{ elements } T_K^{i,j}\\ \mbox{ in a region connecting nodes } i, j

dw_nl_diffusion

NonlinearDiffusionTerm

<fun>, <dfun>, <virtual>, <state>

\int_{\Omega} \nabla q \cdot \nabla p f(p)

poi.non.mat

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}

bio.npb.lag

dw_non_penetration_p

NonPenetrationPenaltyTerm

<material>, <virtual>, <state>

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

bio.sho.syn

dw_nonsym_elastic

NonsymElasticTerm

<material>, <virtual/param_1>, <state/param_2>

\int_{\Omega} \ull{D} \nabla\ul{u} : \nabla\ul{v}

non.hyp.mM

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

bur.2D

dw_piezo_coupling

PiezoCouplingTerm

<material>, <virtual/param_v>, <state/param_s>

<material>, <state>, <virtual>

\int_{\Omega} g_{kij}\ e_{ij}(\ul{v}) \nabla_k p\\ \int_{\Omega} g_{kij}\ e_{ij}(\ul{u}) \nabla_k q

pie.ela, pie.ela

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 }

its.3, she.can, its.2, tru.bri, its.1, its.4

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

adv.2D, adv.dif.2D, adv.1D

dw_shell10x

Shell10XTerm

<material_d>, <material_drill>, <virtual>, <state>

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

she.can

dw_stokes

StokesTerm

<opt_material>, <virtual/param_v>, <state/param_s>

<opt_material>, <state>, <virtual>

\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}

sto, nav.sto, sta.nav.sto, nav.sto, nav.sto.iga, sto.sli.bc, lin.ela.up

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})

ev_sum_vals

SumNodalValuesTerm

<parameter>

dw_surface_flux

SurfaceFluxOperatorTerm

<opt_material>, <virtual>, <state>

\int_{\Gamma} q \ul{n} \cdot \ull{K} \cdot \nabla p

ev_surface_flux

SurfaceFluxTerm

<material>, <parameter>

\int_{\Gamma} \ul{n} \cdot K_{ij} \nabla_j p

dw_surface_ltr

LinearTractionTerm

<opt_material>, <virtual/param>

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

lin.vis, ela.shi.per, lin.ela.opt, wed.mes, tru.bri, com.ela.mat, nod.lcb, lin.ela.tra, mix.mes

ev_surface_moment

SurfaceMomentTerm

<material>, <parameter>

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

dw_surface_ndot

SufaceNormalDotTerm

<material>, <virtual/param>

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

lap.flu.2d

ev_surface_piezo_flux

SurfacePiezoFluxTerm

<material>, <parameter>

\int_{\Gamma} g_{kij} e_{ij}(\ul{u}) n_k

dw_v_dot_grad_s

VectorDotGradScalarTerm

<opt_material>, <virtual/param_v>, <state/param_s>

<opt_material>, <state>, <virtual>

\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{c} \nabla p) \mbox{ , } \int_{\Omega} \ul{u} \cdot (\ull{c} \nabla q)

vib.aco

dw_vm_dot_s

VectorDotScalarTerm

<material>, <virtual/param_v>, <state/param_s>

<material>, <state>, <virtual>

\int_{\Omega} \ul{v} \cdot \ul{c} p \mbox{ , } \int_{\Omega} \ul{u} \cdot \ul{c} q\\

ev_volume

VolumeTerm

<parameter>

\int_{\cal{D}} 1

dw_volume_lvf

LinearVolumeForceTerm

<material>, <virtual>

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

poi.iga, poi.par.stu, bur.2D, adv.dif.2D

dw_volume_nvf

NonlinearVolumeForceTerm

<fun>, <dfun>, <virtual>, <state>

\int_{\Omega} q f(p)

poi.non.mat

ev_volume_surface

VolumeSurfaceTerm

<parameter>

1 / D \int_\Gamma \ul{x} \cdot \ul{n}

dw_zero

ZeroTerm

<virtual>, <state>

0

ela

Table of sensitivity terms

Sensitivity terms

name/class

arguments

definition

examples

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}

ev_sd_convect

SDConvectTerm

<parameter_u>, <parameter_w>, <parameter_mv>

\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 ]

ev_sd_diffusion

SDDiffusionTerm

<material>, <parameter_q>, <parameter_p>, <parameter_mv>

\int_{\Omega} \hat{K}_{ij} \nabla_i q\, \nabla_j p

\hat{K}_{ij} = K_{ij}\left( \delta_{ik}\delta_{jl} \nabla \cdot \ul{\Vcal} - \delta_{ik}{\partial \Vcal_j \over \partial x_l} - \delta_{jl}{\partial \Vcal_i \over \partial x_k}\right)

de_sd_diffusion

ESDDiffusionTerm

<material>, <virtual/param_1>, <state/param_2>, <parameter_mv>

\int_{\Omega} \hat{K}_{ij} \nabla_i q\, \nabla_j p

\hat{K}_{ij} = K_{ij}\left( \delta_{ik}\delta_{jl} \nabla \cdot \ul{\Vcal} - \delta_{ik}{\partial \Vcal_j \over \partial x_l} - \delta_{jl}{\partial \Vcal_i \over \partial x_k}\right)

ev_sd_div

SDDivTerm

<parameter_u>, <parameter_p>, <parameter_mv>

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

ev_sd_div_grad

SDDivGradTerm

<opt_material>, <parameter_u>, <parameter_w>, <parameter_mv>

\int_{\Omega} \hat{I} \nabla \ul{v} : \nabla \ul{u} \mbox{ , } \int_{\Omega} \nu \hat{I} \nabla \ul{v} : \nabla \ul{u}

\hat{I}_{ijkl} = \delta_{ik}\delta_{jl} \nabla \cdot \ul{\Vcal} - \delta_{ik}\delta_{js} {\partial \Vcal_l \over \partial x_s} - \delta_{is}\delta_{jl} {\partial \Vcal_k \over \partial x_s}

de_sd_div_grad

ESDDivGradTerm

<opt_material>, <virtual/param_1>, <state/param_2>, <parameter_mv>

\int_{\Omega} \hat{I} \nabla \ul{v} : \nabla \ul{u} \mbox{ , } \int_{\Omega} \nu \hat{I} \nabla \ul{v} : \nabla \ul{u}

\hat{I}_{ijkl} = \delta_{ik}\delta_{jl} \nabla \cdot \ul{\Vcal} - \delta_{ik}\delta_{js} {\partial \Vcal_l \over \partial x_s} - \delta_{is}\delta_{jl} {\partial \Vcal_k \over \partial x_s}

de_sd_dot

ESDDotTerm

<opt_material>, <virtual/param_1>, <state/param_2>, <parameter_mv>

\int_\Omega q p (\nabla \cdot \ul{\Vcal}) \mbox{ , } \int_\Omega (\ul{v} \cdot \ul{u}) (\nabla \cdot \ul{\Vcal})\\ \int_\Omega c q p (\nabla \cdot \ul{\Vcal}) \mbox{ , } \int_\Omega c (\ul{v} \cdot \ul{u}) (\nabla \cdot \ul{\Vcal})\\ \int_\Omega \ul{v} \cdot (\ull{M}\, \ul{u}) (\nabla \cdot \ul{\Vcal})

ev_sd_dot

SDDotTerm

<parameter_1>, <parameter_2>, <parameter_mv>

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

de_sd_lin_elastic

ESDLinearElasticTerm

<material>, <virtual/param_1>, <state/param_2>, <parameter_mv>

\int_{\Omega} \hat{D}_{ijkl} {\partial v_i \over \partial x_j} {\partial u_k \over \partial x_l}

\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}

ev_sd_lin_elastic

SDLinearElasticTerm

<material>, <parameter_w>, <parameter_u>, <parameter_mv>

\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}

ev_sd_piezo_coupling

SDPiezoCouplingTerm

<material>, <parameter_u>, <parameter_p>, <parameter_mv>

\int_{\Omega} \hat{g}_{kij}\ e_{ij}(\ul{u}) \nabla_k p

\hat{g}_{kij} = g_{kij}(\nabla \cdot \ul{\Vcal}) - g_{kil}{\partial \Vcal_j \over \partial x_l} - g_{lij}{\partial \Vcal_k \over \partial x_l}

de_sd_piezo_coupling

ESDPiezoCouplingTerm

<material>, <virtual/param_v>, <state/param_s>, <parameter_mv>

<material>, <state>, <virtual>, <parameter_mv>

\int_{\Omega} \hat{g}_{kij}\ e_{ij}(\ul{v}) \nabla_k p \mbox{ , } \int_{\Omega} \hat{g}_{kij}\ e_{ij}(\ul{u}) \nabla_k q

\hat{g}_{kij} = g_{kij}(\nabla \cdot \ul{\Vcal}) - g_{kil}{\partial \Vcal_j \over \partial x_l} - g_{lij}{\partial \Vcal_k \over \partial x_l}

de_sd_stokes

ESDStokesTerm

<opt_material>, <virtual/param_v>, <state/param_s>, <parameter_mv>

<opt_material>, <state>, <virtual>, <parameter_mv>

\int_{\Omega} p\, \hat{I}_{ij} {\partial v_i \over \partial x_j} \mbox{ , } \int_{\Omega} q\, \hat{I}_{ij} {\partial u_i \over \partial x_j}

\hat{I}_{ij} = \delta_{ij} \nabla \cdot \Vcal - {\partial \Vcal_j \over \partial x_i}

ev_sd_surface_integrate

SDSufaceIntegrateTerm

<parameter>, <parameter_mv>

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

de_sd_surface_ltr

ESDLinearTractionTerm

<opt_material>, <virtual/param>, <parameter_mv>

\int_{\Gamma} \ul{v} \cdot \left[\left(\ull{\hat{\sigma}}\, \nabla \cdot \ul{\cal{V}} - \ull{{\hat\sigma}}\, \nabla \ul{\cal{V}} \right)\ul{n}\right]

\ull{\hat\sigma} = \ull{I} \mbox{ , } \ull{\hat\sigma} = c\,\ull{I} \mbox{ or } \ull{\hat\sigma} = \ull{\sigma}

ev_sd_surface_ltr

SDLinearTractionTerm

<opt_material>, <parameter>, <parameter_mv>

\int_{\Gamma} \ul{v} \cdot (\ull{\sigma}\, \ul{n}), \int_{\Gamma} \ul{v} \cdot \ul{n},

de_sd_v_dot_grad_s

ESDVectorDotGradScalarTerm

<opt_material>, <virtual/param_v>, <state/param_s>, <parameter_mv>

<opt_material>, <state>, <virtual>, <parameter_mv>

\int_{\Omega} \hat{I}_{ij} {\partial p \over \partial x_j}\, v_i \mbox{ , } \int_{\Omega} \hat{I}_{ij} {\partial q \over \partial x_j}\, u_i

\hat{I}_{ij} = \delta_{ij} \nabla \cdot \Vcal - {\partial \Vcal_j \over \partial x_i}

Table of large deformation terms

Large deformation terms

name/class

arguments

definition

examples

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})

act.fib, com.ela.mat, hyp

dw_tl_bulk_pressure

BulkPressureTLTerm

<virtual>, <state>, <state_p>

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

per.tl, bal

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}}

per.tl

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})

act.fib

dw_tl_fib_e

FibresExponentialTLTerm

<material_1>, <material_2>, <material_3>, <material_4>, <virtual>, <state>

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

dw_tl_fib_spe

FibresSoftPlusExponentialTLTerm

<opt_material_0>, <material_1>, <material_2>, <material_3>, <material_4>, <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})

com.ela.mat, hyp, bal

dw_tl_he_neohook

NeoHookeanTLTerm

<material>, <virtual>, <state>

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

per.tl, act.fib, com.ela.mat, hyp, bal

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>

bal

ev_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

per.tl

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}

per.tl, bal

ev_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

hyp.ul

dw_ul_bulk_pressure

BulkPressureULTerm

<virtual>, <state>, <state_p>

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

hyp.ul.up

dw_ul_compressible

CompressibilityULTerm

<material>, <virtual>, <state>, <parameter_u>

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

hyp.ul.up

dw_ul_he_by_fun

HyperelasticByFunULTerm

<fun>, <virtual>, <state>

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

hyp.ul.by.fun

dw_ul_he_mooney_rivlin

MooneyRivlinULTerm

<material>, <virtual>, <state>

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

hyp.ul.up, hyp.ul

dw_ul_he_neohook

NeoHookeanULTerm

<material>, <virtual>, <state>

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

hyp.ul.up, hyp.ul

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}

hyp.ul.up

Table of special terms

Special terms

name/class

arguments

definition

examples

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}

ev_cauchy_stress_th

CauchyStressTHTerm

<ts>, <material>, <parameter>

\int_{\Omega} \int_0^t \Hcal_{ijkl}(t-\tau)\,e_{kl}(\ul{w}(\tau)) \difd{\tau}

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})

lin.vis

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})

ev_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

ev_sd_st_grad_div

SDGradDivStabilizationTerm

<material>, <parameter_u>, <parameter_w>, <parameter_mv>

\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} ]

ev_sd_st_pspg_c

SDPSPGCStabilizationTerm

<material>, <parameter_b>, <parameter_u>, <parameter_r>, <parameter_mv>

\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} ]

ev_sd_st_pspg_p

SDPSPGPStabilizationTerm

<material>, <parameter_r>, <parameter_p>, <parameter_mv>

\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} ]

ev_sd_st_supg_c

SDSUPGCStabilizationTerm

<material>, <parameter_b>, <parameter_u>, <parameter_w>, <parameter_mv>

\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})

sta.nav.sto

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

sta.nav.sto

dw_st_pspg_p

PSPGPStabilizationTerm

<opt_material>, <virtual/param_1>, <state/param_2>

\sum_{K \in \Ical_h}\int_{T_K} \tau_K\ \nabla p \cdot \nabla q

sta.nav.sto

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})

sta.nav.sto

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})

sta.nav.sto

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

Table of multi-linear terms

Multi-linear terms

name/class

arguments

definition

examples

de_cauchy_stress

ECauchyStressTerm

<material>, <parameter>

\int_{\Omega} D_{ijkl} e_{kl}(\ul{w})

de_convect

EConvectTerm

<opt_material>, <virtual/param_1>, <state/param_2>

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

de_diffusion

EDiffusionTerm

<material>, <virtual/param_1>, <state/param_2>

\int_{\Omega} K_{ij} \nabla_i q\, \nabla_j p

de_div

EDivTerm

<opt_material>, <virtual/param>

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

de_div_grad

EDivGradTerm

<opt_material>, <virtual/param_1>, <state/param_2>

\int_{\Omega} \nabla \ul{v} : \nabla \ul{u} \mbox{ , } \int_{\Omega} \nu\ \nabla \ul{v} : \nabla \ul{u}

de_dot

EDotTerm

<opt_material>, <virtual/param_1>, <state/param_2>

\int_{\cal{D}} q p \mbox{ , } \int_{\cal{D}} \ul{v} \cdot \ul{u}\\ \int_{\cal{D}} c q p \mbox{ , } \int_{\cal{D}} c \ul{v} \cdot \ul{u}\\ \int_{\cal{D}} \ul{v} \cdot (\ull{c}\, \ul{u})

de_grad

EGradTerm

<opt_material>, <parameter>

\int_{\Omega} \nabla \ul{v} \mbox { , } \int_{\Omega} c \nabla \ul{v} \mbox { , } \int_{\Omega} \ul{c} \cdot \nabla \ul{v} \mbox { , } \int_{\Omega} \ull{c} \cdot \nabla \ul{v}

de_integrate

EIntegrateOperatorTerm

<opt_material>, <virtual>

\int_{\cal{D}} q \mbox{ or } \int_{\cal{D}} c q

de_laplace

ELaplaceTerm

<opt_material>, <virtual/param_1>, <state/param_2>

\int_{\Omega} \nabla q \cdot \nabla p \mbox{ , } \int_{\Omega} c \nabla q \cdot \nabla p

de_lin_convect

ELinearConvectTerm

<virtual/param_1>, <parameter>, <state/param_3>

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

de_lin_elastic

ELinearElasticTerm

<material>, <virtual/param_1>, <state/param_2>

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

de_m_flexo

MixedFlexoTerm

<virtual/param_v>, <state/param_t>

<state>, <virtual>

\int_{\Omega} v_{i,j} a_{ij} \\ \int_{\Omega} u_{i,j} \delta a_{ij}

de_m_flexo_coupling

MixedFlexoCouplingTerm

<material>, <virtual/param_t>, <state/param_s>

<material>, <state>, <virtual>

\int_{\Omega} f_{ijkl}\ e_{jk,l}(\ull{\delta w}) \nabla_i p \\ \int_{\Omega} f_{ijkl}\ e_{jk,l}(\ull{w}) \nabla_i q

de_m_sg_elastic

MixedStrainGradElasticTerm

<material>, <virtual/param_1>, <state/param_2>

\int_{\Omega} a_{ijklmn}\ e_{ij,k}(\ull{\delta w}) \ e_{lm,n}(\ull{w})

de_mass

MassTerm

<material_rho>, <material_lumping>, <material_beta>, <virtual>, <state>

M^C = \int_{\cal{D}} \rho \ul{v} \cdot \ul{u} \\ M^L = \mathrm{lumping}(M^C) \\ M^A = (1 - \beta) M^C + \beta M^L \\ A = \sum_e A_e \\ C = \sum_e A_e^T (M_e^A)^{-1} A_e

sei.loa, ela

de_non_penetration_p

ENonPenetrationPenaltyTerm

<material>, <virtual>, <state>

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

de_nonsym_elastic

ENonSymElasticTerm

<material>, <virtual/param_1>, <state/param_2>

\int_{\Omega} \ull{D} \nabla \ul{v} : \nabla \ul{u}

de_s_dot_mgrad_s

EScalarDotMGradScalarTerm

<material>, <virtual/param_1>, <state/param_2>

<material>, <state>, <virtual>

\int_{\Omega} q \ul{y} \cdot \nabla p \mbox{ , } \int_{\Omega} p \ul{y} \cdot \nabla q

de_stokes

EStokesTerm

<opt_material>, <virtual/param_v>, <state/param_s>

<opt_material>, <state>, <virtual>

\int_{\Omega} p\, \nabla \cdot \ul{v} \mbox{ , } \int_{\Omega} q\, \nabla \cdot \ul{u}\\ \int_{\Omega} c\, p\, \nabla \cdot \ul{v} \mbox{ , } \int_{\Omega} c\, q\, \nabla \cdot \ul{u}

de_stokes_traction

StokesTractionTerm

<opt_material>, <virtual/param_1>, <state/param_2>

\int_{\Gamma} \nu \ul{v}\cdot(\nabla \ul{u} \cdot \ul{n})

de_surface_flux

SurfaceFluxOperatorTerm

<material>, <virtual/param_1>, <state/param_2>

<material>, <state>, <virtual>

\int_{\Gamma} q \ul{n} \cdot \ull{K} \cdot \nabla p \mbox{ , } \int_{\Gamma} p \ul{n} \cdot \ull{K} \cdot \nabla q

pie.ela

de_surface_ltr

ELinearTractionTerm

<opt_material>, <virtual/param>

\int_{\Gamma} \ul{v} \cdot \ul{n} \mbox{ , } \int_{\Gamma} c\, \ul{v} \cdot \ul{n}\\ \int_{\Gamma} \ul{v} \cdot (\ull{\sigma}\, \ul{n}) \mbox{ , } \int_{\Gamma} \ul{v} \cdot \ul{f}

de_surface_piezo_flux

SurfacePiezoFluxOperatorTerm

<material>, <virtual/param_1>, <state/param_2>

<material>, <state>, <virtual>

\int_{\Gamma} q g_{kij} e_{ij}(\ul{u}) n_k \mbox{ , } \int_{\Gamma} p g_{kij} e_{ij}(\ul{v}) n_k