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})

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}

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})

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

ev_cauchy_strain

CauchyStrainTerm

<parameter>

\int_{\cal{D}} \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_{\cal{D}} 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}

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_contact

ContactTerm

<material>, <virtual>, <state>

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

two.bod.con

dw_convect_v_grad_s

ConvectVGradSTerm

<virtual>, <state_v>, <state_s>

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

poi.fun

dw_convect

ConvectTerm

<virtual>, <state>

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

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

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

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}

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

adv.dif.2D, lap.2D, bur.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_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_{\cal{D}} 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_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

bio, bio.npb, dar.flo.mul, bio.npb.lag, bio.sho.syn, poi.neu

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}

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

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_{\cal{D}} \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_dot

DotProductTerm

<opt_material>, <virtual>, <state>

<opt_material>, <parameter_1>, <parameter_2>

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

lin.ela.dam, adv.2D, the.ele, tim.poi, tim.adv.dif, poi.fun, ela, bur.2D, adv.1D, poi.per.bou.con, sto.sli.bc, hyd, lin.ela.up, bor, dar.flo.mul, osc, wel, aco, aco, vib.aco, tim.poi.exp

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_elastic_wave

ElasticWaveTerm

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

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

dw_electric_source

ElectricSourceTerm

<material>, <virtual>, <parameter>

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

the.ele

ev_grad

GradTerm

<parameter>

\int_{\cal{D}} \nabla p \mbox{ or } \int_{\cal{D}} \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_{\cal{D}} m

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

m|_{qp}

dw_integrate

IntegrateOperatorTerm

<opt_material>, <virtual>

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

poi.per.bou.con, dar.flo.mul, aco, aco, vib.aco, poi.neu

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 }

\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_jump

SurfaceJumpTerm

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

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

aco

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

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

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_convect

LinearConvectTerm

<virtual>, <parameter>, <state>

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

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

sta.nav.sto

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_elastic

LinearElasticTerm

<material>, <virtual>, <state>

<material>, <parameter_1>, <parameter_2>

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

ela.shi.per, lin.ela.dam, nod.lcb, lin.ela.opt, lin.ela.tra, two.bod.con, bio, com.ela.mat, lin.ela, lin.ela.iga, bio.npb.lag, lin.ela.mM, its.2, its.1, the.ela.ess, ela, bio.sho.syn, the.ela, bio.npb, its.3, lin.ela.up, ela.con.pla, lin.vis, pre.fib, its.4, mat.non, pie.ela.mac, ela.con.sph

dw_lin_prestress

LinearPrestressTerm

<material>, <virtual>

<material>, <parameter>

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

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

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_non_penetration_p

NonPenetrationPenaltyTerm

<material>, <virtual>, <state>

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

bio.sho.syn

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_nonsym_elastic

NonsymElasticTerm

<material>, <virtual>, <state>

<material>, <parameter_1>, <parameter_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>, <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 }

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

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

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_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})

dw_stokes_wave

StokesWaveTerm

<material>, <virtual>, <state>

\int_{\Omega} (\ul{\kappa} \cdot \ul{v}) (\ul{\kappa} \cdot \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}

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

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 \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_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},

ela.shi.per, com.ela.mat, nod.lcb, lin.ela.opt, lin.vis, lin.ela.tra

ev_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}

lap.flu.2d

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\\

dw_volume_lvf

LinearVolumeForceTerm

<material>, <virtual>

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

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

ev_volume_surface

VolumeSurfaceTerm

<parameter>

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

ev_volume

VolumeTerm

<parameter>

\int_{\cal{D}} 1

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_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]

ev_sd_div_grad

SDDivGradTerm

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

\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} ]

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_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})

ev_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}

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}

ev_sd_surface_integrate

SDSufaceIntegrateTerm

<parameter>, <parameter_mesh_velocity>

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

ev_sd_surface_ltr

SDLinearTractionTerm

<opt_material>, <parameter>, <parameter_mv>

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

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

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_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

dw_tl_he_neohook

NeoHookeanTLTerm

<material>, <virtual>, <state>

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

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

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>

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

ev_tl_volume_surface

VolumeSurfaceTLTerm

<parameter>

1 / D \int_{\Gamma} \ul{\nu} \cdot \ull{F}^{-1} \cdot \ul{x} 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}

per.tl

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_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}

\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})

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})

dw_of_ns_surf_min_d_press_diff

NSOFSurfMinDPressDiffTerm

<material>, <virtual>

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

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)

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>, <state>

<opt_material>, <parameter_1>, <parameter_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

<virtual>, <state>

<parameter_1>, <parameter_2>

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

de_div_grad

EDivGradTerm

<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}

de_div

EDivTerm

<opt_material>, <virtual>

<opt_material>, <parameter>

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

de_dot

EDotTerm

<opt_material>, <virtual>, <state>

<opt_material>, <parameter_1>, <parameter_2>

\int_{\cal{D}} q p \mbox{ , } \int_{\cal{D}} \ul{v} \cdot \ul{u} \mbox{ , } \int_{\cal{D}} p r \mbox{ , } \int_{\cal{D}} \ul{u} \cdot \ul{w} \\ \int_{\cal{D}} c q p \mbox{ , } \int_{\cal{D}} c \ul{v} \cdot \ul{u} \mbox{ , } \int_{\cal{D}} c p r \mbox{ , } \int_{\cal{D}} c \ul{u} \cdot \ul{w} \\ \int_{\cal{D}} \ul{v} \cdot \ull{M} \cdot \ul{u} \mbox{ , } \int_{\cal{D}} \ul{u} \cdot \ull{M} \cdot \ul{w}

de_integrate

EIntegrateOperatorTerm

<opt_material>, <virtual>

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

de_laplace

ELaplaceTerm

<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

de_lin_elastic

ELinearElasticTerm

<material>, <virtual>, <state>

<material>, <parameter_1>, <parameter_2>

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

de_non_penetration_p

ENonPenetrationPenaltyTerm

<material>, <virtual>, <state>

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

de_s_dot_mgrad_s

EScalarDotMGradScalarTerm

<material>, <virtual>, <state>

<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>, <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} \\ \int_{\Omega} r\ \nabla \cdot \ul{w} \mbox{ , } \int_{\Omega} c r\ \nabla \cdot \ul{w}