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
$\Omega$	cell (volume) (sub)domain
$\Gamma$	facet (surface) (sub)domain
$\cal{D}$	cell or facet (sub)domain
$d$	dimension of space
$t$	time
$y$	any function
$\ul{y}$	any vector function
$\ul{n}$	unit outward normal
$q$	scalar test or parameter function
$p$	scalar unknown or parameter function
$\ul{v}$	vector test or parameter function
$\ul{w}$ , $\ul{u}$	vector unknown or 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
$c$ , $\ul{c}$ , $\ull{c}$	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
ev	evaluate	‘eval’, ‘el_eval’, ‘el_avg’, ‘qp’	terms having all arguments known, modes ‘el_avg’, ‘qp’ are not supported by all ev_ terms
de	discrete einsum	any (work in progress)	multi-linear terms defined using an enriched einsum notation

Evaluation modes ‘eval’, ‘el_avg’ and ‘qp’ are defined as follows:

Evaluation modes.¶
mode	definition
‘eval’	$\int_{\cal{D}} (\cdot)$
‘el_avg’	vector for $K \from \Ical_h: \int_{T_K} (\cdot) / \int_{T_K} 1$
‘qp’	$(\cdot)\|_{qp}$

Term Table¶

Below we list all the terms available in automatically generated tables. The first column lists the name, the second column the argument lists and the third column the mathematical definition of each term. The terms are devided into the following tables:

Table of basic terms
Table of large deformation terms (total/updated Lagrangian formulation)
Table of sensitivity terms
Table of special terms
Table of multi-linear terms

The notation <virtual> corresponds to a test function, <state> to a unknown function and <parameter> to a known function. By <material> we denote material (constitutive) parameters, or, in general, any given function of space and time that parameterizes a term, for example a given traction force vector.

Table of basic terms¶

Basic terms¶
name/class	arguments	definition	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>`	$\int_{\Omega} p\ \alpha_{ij} e_{ij}(\ul{v}) \mbox{ , } \int_{\Omega} q\ \alpha_{ij} e_{ij}(\ul{u})$	bio.npb.lag, bio, bio.npb, the.ela, bio.sho.syn, the.ela.ess
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.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 $\langle \nabla \phi \rangle = \frac{\nabla\phi_{in} + \nabla\phi_{out}}{2}$ $[\phi] = \phi_{in} - \phi_{out}$	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>`	$\int_{\Omega} K_{ij} \nabla_i q \nabla_j p$	bio.npb.lag, pie.ela, bio, bio.npb, poi.neu, vib.aco, bio.sho.syn, dar.flo.mul, pie.ela
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>`	$- \int_{\cal{D}} K_{ij} \nabla_j p$
dw_div `DivOperatorTerm`	`<opt_material>`, `<virtual>`	$\int_{\Omega} \nabla \cdot \ul{v} \mbox { or } \int_{\Omega} c \nabla \cdot \ul{v}$
ev_div `DivTerm`	`<opt_material>`, `<parameter>`	$\int_{\cal{D}} \nabla \cdot \ul{u} \mbox { , } \int_{\cal{D}} c \nabla \cdot \ul{u}$
dw_div_grad `DivGradTerm`	`<opt_material>`, `<virtual/param_1>`, `<state/param_2>`	$\int_{\Omega} \nu\ \nabla \ul{v} : \nabla \ul{u} \mbox{ , } \int_{\Omega} \nabla \ul{v} : \nabla \ul{u}$	nav.sto, sto.sli.bc, sto, nav.sto.iga, sta.nav.sto, nav.sto
dw_dot `DotProductTerm`	`<opt_material>`, `<virtual/param_1>`, `<state/param_2>`	$\int_{\cal{D}} q p \mbox{ , } \int_{\cal{D}} \ul{v} \cdot \ul{u}\\ \int_\Gamma \ul{v} \cdot \ul{n} p \mbox{ , } \int_\Gamma q \ul{n} \cdot \ul{u} \mbox{ , }\\ \int_{\cal{D}} c q p \mbox{ , } \int_{\cal{D}} c \ul{v} \cdot \ul{u} \mbox{ , } \int_{\cal{D}} \ul{v} \cdot \ull{c} \cdot \ul{u}$	poi.fun, pie.ela, bor, dar.flo.mul, pie.ela, poi.per.bou.con, lin.ela.up, tim.hea.equ.mul.mat, sto.sli.bc, wel, tim.poi, tim.poi.exp, hel.apa, osc, vib.aco, ref.evp, adv.1D, the.ele, mod.ana.dec, bal, hyd, adv.2D, lin.ela.dam, bur.2D, tim.adv.dif, aco, aco
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, dar.flo.mul, aco, poi.per.bou.con, aco, hel.apa
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>`	$\int_{\Omega} c \nabla q \cdot \nabla p$	poi.fun, poi.sho.syn, lap.2D, lap.cou.lcb, bor, poi.per.bou.con, lap.flu.2d, poi.fie.dep.mat, tim.hea.equ.mul.mat, sin, sto.sli.bc, wel, poi.iga, tim.poi, adv.dif.2D, tim.poi.exp, lap.tim.ebc, hel.apa, osc, ref.evp, the.ele, vib.aco, poi.par.stu, the.ela.ess, cub, hyd, lap.1d, bur.2D, poi, tim.adv.dif, aco, aco
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>`	$\int_{\Omega} D_{ijkl}\ e_{ij}(\ul{v}) e_{kl}(\ul{u})$	com.ela.mat, bio, pie.ela, bio.npb, its.4, lin.ela.mM, the.ela, mul.nod.lcb, wed.mes, bio.sho.syn, two.bod.con, lin.ela.tra, pie.ela, ela.shi.per, its.2, lin.ela.opt, sei.loa, lin.ela.up, tru.bri, its.3, lin.ela.iga, ela.con.pla, mul.poi.con, lin.vis, lin.ela, bio.npb.lag, its.1, vib.aco, mat.non, nod.lcb, mod.ana.dec, the.ela.ess, pre.fib, pie.ela.mac, ela, lin.ela.dam, ela.con.sph, mix.mes
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_elastic_l_ad `LinearElasticLADTerm`	`<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_yp_ad `LinearElasticYPADTerm`	`<material_1>`, `<material_2>`, `<virtual>`, `<state>`	$\int_{\Omega} D_{ijkl}\ e_{ij}(\ul{v}) e_{kl}(\ul{u})\\ \mbox{ with } \\ D_{ijkl} = \mu (\delta_{ik} \delta_{jl}+\delta_{il} \delta_{jk}) + \lambda \ \delta_{ij} \delta_{kl}, \\ \mbox{ where } \\ \lambda = E \nu / ((1 + \nu)(1 - 2\nu)), \\ \mu = E / 2(1 + \nu)$	ela.ide
dw_lin_prestress `LinearPrestressTerm`	`<material>`, `<virtual/param>`	$\int_{\Omega} \sigma_{ij} e_{ij}(\ul{v})$	pre.fib, non.hyp.mM, 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_mass_ad `MassADTerm`	`<material>`, `<virtual>`, `<state>`	$\int_{\cal{D}} \rho \ul{v} \cdot \ul{u}$	ela.ide
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.4, its.1, its.3, its.2, tru.bri, she.can
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 `StokesTerm`	`<opt_material>`, `<virtual/param_v>`, `<state/param_s>` `<opt_material>`, `<state>`, `<virtual>`	$\int_{\Omega} p\ \nabla \cdot \ul{v} \mbox{ , } \int_{\Omega} q\ \nabla \cdot \ul{u}\\ \mbox{ or } \int_{\Omega} c\ p\ \nabla \cdot \ul{v} \mbox{ , } \int_{\Omega} c\ q\ \nabla \cdot \ul{u}$	nav.sto, lin.ela.up, sto.sli.bc, sto, nav.sto.iga, sta.nav.sto, nav.sto
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>`
ev_surface_flux `SurfaceFluxTerm`	`<material>`, `<parameter>`	$\int_{\Gamma} \ul{n} \cdot K_{ij} \nabla_j p$
dw_surface_flux `SurfaceFluxOperatorTerm`	`<opt_material>`, `<virtual>`, `<state>`	$\int_{\Gamma} q \ul{n} \cdot \ull{K} \cdot \nabla p$
dw_surface_ltr `LinearTractionTerm`	`<opt_material>`, `<virtual/param>`	$\int_{\Gamma} \ul{v} \cdot \ull{\sigma} \cdot \ul{n}, \int_{\Gamma} \ul{v} \cdot \ul{n},$	com.ela.mat, mix.mes, tru.bri, wed.mes, lin.ela.tra, nod.lcb, ela.shi.per, lin.ela.opt, lin.vis
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>`	$\int_{\Omega} \ul{f} \cdot \ul{v} \mbox{ or } \int_{\Omega} f q$	bur.2D, adv.dif.2D, poi.iga, poi.par.stu
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
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>`	$\int_{\Omega} \hat{I} \nabla \ul{v} : \nabla \ul{u} \mbox{ , } \int_{\Omega} \nu \hat{I} \nabla \ul{v} : \nabla \ul{u}$ $\hat{I}_{ijkl} = \delta_{ik}\delta_{jl} \nabla \cdot \ul{\Vcal} - \delta_{ik}\delta_{js} {\partial \Vcal_l \over \partial x_s} - \delta_{is}\delta_{jl} {\partial \Vcal_k \over \partial x_s}$
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_dot `ESDDotTerm`	`<opt_material>`, `<virtual/param_1>`, `<state/param_2>`, `<parameter_mv>`	$\int_\Omega q p (\nabla \cdot \ul{\Vcal}) \mbox{ , } \int_\Omega (\ul{v} \cdot \ul{u}) (\nabla \cdot \ul{\Vcal})\\ \int_\Omega c q p (\nabla \cdot \ul{\Vcal}) \mbox{ , } \int_\Omega c (\ul{v} \cdot \ul{u}) (\nabla \cdot \ul{\Vcal})\\ \int_\Omega \ul{v} \cdot (\ull{M}\, \ul{u}) (\nabla \cdot \ul{\Vcal})$
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, hyp, com.ela.mat
dw_tl_bulk_pressure `BulkPressureTLTerm`	`<virtual>`, `<state>`, `<state_p>`	$\int_{\Omega} S_{ij}(p) \delta E_{ij}(\ul{u};\ul{v})$	bal, 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_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, bal, 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, hyp, bal, per.tl
dw_tl_he_neohook_ad `NeoHookeanTLADTerm`	`<material>`, `<virtual>`, `<state>`	$\int_{\Omega} S_{ij}(\ul{u}) \delta E_{ij}(\ul{u};\ul{v})$
dw_tl_he_ogden `OgdenTLTerm`	`<material>`, `<virtual>`, `<state>`	$\int_{\Omega} S_{ij}(\ul{u}) \delta E_{ij}(\ul{u};\ul{v})$
dw_tl_he_ogden_ad `OgdenTLADTerm`	`<material_mu>`, `<material_alpha>`, `<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>`	$\begin{array}{l} \int_{\Omega} q J(\ul{u}) \\ \mbox{volume mode: vector for } K \from \Ical_h: \int_{T_K} J(\ul{u}) \\ \mbox{rel\_volume mode: vector for } K \from \Ical_h: \int_{T_K} J(\ul{u}) / \int_{T_K} 1 \end{array}$	bal, per.tl
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>`	$\int_{\Gamma} q g_{kij} e_{ij}(\ul{u}) n_k \mbox{ , } \int_{\Gamma} p g_{kij} e_{ij}(\ul{v}) n_k$