Table of basic terms¶
name/class |
arguments |
definition |
examples |
---|---|---|---|
dw_advect_div_free |
|
\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 |
|
\int_{\Gamma} \alpha q (p - p_{\rm outer}) |
tim.hea.equ.mul.mat |
dw_biot |
|
\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 |
|
- \int_{\Omega} \alpha_{ij} p |
|
ev_cauchy_strain |
|
\int_{\cal{D}} \ull{e}(\ul{w}) |
|
ev_cauchy_stress |
|
\int_{\cal{D}} D_{ijkl} e_{kl}(\ul{w}) |
|
dw_contact |
|
\int_{\Gamma_{c}} \varepsilon_N \langle g_N(\ul{u}) \rangle \ul{n} \ul{v} |
two.bod.con |
dw_contact_ipc |
|
\int_{\Gamma_{c}} \sum_{k \in C} \nabla b(d_k, \ul{u}) |
|
dw_contact_plane |
|
\int_{\Gamma} \ul{v} \cdot f(d(\ul{u})) \ul{n} |
ela.con.pla |
dw_contact_sphere |
|
\int_{\Gamma} \ul{v} \cdot f(d(\ul{u})) \ul{n}(\ul{u}) |
ela.con.sph |
dw_convect |
|
\int_{\Omega} ((\ul{u} \cdot \nabla) \ul{u}) \cdot \ul{v} |
nav.sto, nav.sto.iga, nav.sto |
dw_convect_v_grad_s |
|
\int_{\Omega} q (\ul{u} \cdot \nabla p) |
poi.fun |
ev_def_grad |
|
\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 |
|
\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 |
|
\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 |
|
\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 |
|
\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 |
|
\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 |
|
\int_{\Omega} p K_{j} \nabla_j q \mbox{ , } \int_{\Omega} q K_{j} \nabla_j p |
|
dw_diffusion_r |
|
\int_{\Omega} K_{j} \nabla_j q |
|
ev_diffusion_velocity |
|
- \int_{\cal{D}} K_{ij} \nabla_j p |
|
ev_div |
|
\int_{\cal{D}} \nabla \cdot \ul{u} \mbox { , } \int_{\cal{D}} c \nabla \cdot \ul{u} |
|
dw_div |
|
\int_{\Omega} \nabla \cdot \ul{v} \mbox { or } \int_{\Omega} c \nabla \cdot \ul{v} |
|
dw_div_grad |
|
\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 |
|
\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 |
|
\int_{\Omega} D_{ijkl}\ g_{ij}(\ul{v}) g_{kl}(\ul{u}) |
|
dw_elastic_wave_cauchy |
|
\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 |
|
\int_{\Omega} c s (\nabla \phi)^2 |
the.ele |
ev_grad |
|
\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 |
|
\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 |
|
\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 |
|
\int_{\cal{D}} c |
|
dw_jump |
|
\int_{\Gamma} c\, q (p_1 - p_2) |
aco |
dw_laplace |
|
\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 |
|
\int_{\Omega} ((\ul{w} \cdot \nabla) \ul{u}) \cdot \ul{v} ((\ul{w} \cdot \nabla) \ul{u})|_{qp} |
sta.nav.sto |
dw_lin_convect2 |
|
\int_{\Omega} ((\ul{c} \cdot \nabla) \ul{u}) \cdot \ul{v} ((\ul{c} \cdot \nabla) \ul{u})|_{qp} |
|
dw_lin_dspring |
|
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 |
|
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 |
|
\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 |
|
\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 |
|
\int_{\Omega} \sigma_{ij} e_{ij}(\ul{v}) |
non.hyp.mM, pre.fib, pie.ela.mac |
dw_lin_spring |
|
\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 |
|
\int_{\Omega} D_{ijkl} e_{ij}(\ul{v}) \left(d_k d_l\right) |
pre.fib |
dw_lin_truss |
|
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 |
|
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 |
|
\int_{\Omega} \nabla q \cdot \nabla p f(p) |
poi.non.mat |
dw_non_penetration |
|
\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 |
|
\int_{\Gamma} c (\ul{n} \cdot \ul{v}) (\ul{n} \cdot \ul{u}) |
bio.sho.syn |
dw_nonsym_elastic |
|
\int_{\Omega} \ull{D} \nabla\ul{u} : \nabla\ul{v} |
non.hyp.mM |
dw_ns_dot_grad_s |
|
\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 |
|
\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 |
|
\int_{\Omega} g_{kij} e_{ij}(\ul{u}) |
|
ev_piezo_stress |
|
\int_{\Omega} g_{kij} \nabla_k p |
|
dw_point_load |
|
\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 |
|
\ul{f}^i = -k \ul{u}^i \quad \forall \mbox{ FE node } i \mbox{ in a region } |
|
dw_s_dot_grad_i_s |
|
Z^i = \int_{\Omega} q \nabla_i p |
|
dw_s_dot_mgrad_s |
|
\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 |
|
\int_{\Omega} D_{ijkl}\ e_{ij}(\ul{v}) e_{kl}(\ul{u}) |
she.can |
dw_stokes |
|
\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 |
|
\int_{\Omega} (\ul{\kappa} \cdot \ul{v}) (\ul{\kappa} \cdot \ul{u}) |
|
dw_stokes_wave_div |
|
\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 |
|
||
dw_surface_flux |
|
\int_{\Gamma} q \ul{n} \cdot \ull{K} \cdot \nabla p |
|
ev_surface_flux |
|
\int_{\Gamma} \ul{n} \cdot K_{ij} \nabla_j p |
|
dw_surface_ltr |
|
\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 |
|
\int_{\Gamma} \ul{n} (\ul{x} - \ul{x}_0) |
|
dw_surface_ndot |
|
\int_{\Gamma} q \ul{c} \cdot \ul{n} |
lap.flu.2d |
ev_surface_piezo_flux |
|
\int_{\Gamma} g_{kij} e_{ij}(\ul{u}) n_k |
|
dw_v_dot_grad_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{c} \nabla p) \mbox{ , } \int_{\Omega} \ul{u} \cdot (\ull{c} \nabla q) |
vib.aco |
dw_vm_dot_s |
|
\int_{\Omega} \ul{v} \cdot \ul{c} p \mbox{ , } \int_{\Omega} \ul{u} \cdot \ul{c} q\\ |
|
ev_volume |
|
\int_{\cal{D}} 1 |
|
dw_volume_lvf |
|
\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 |
|
\int_{\Omega} q f(p) |
poi.non.mat |
ev_volume_surface |
|
1 / D \int_\Gamma \ul{x} \cdot \ul{n} |
|
dw_zero |
|
0 |
ela |
Table of sensitivity terms¶
name/class |
arguments |
definition |
examples |
---|---|---|---|
dw_adj_convect1 |
|
\int_{\Omega} ((\ul{v} \cdot \nabla) \ul{u}) \cdot \ul{w} |
|
dw_adj_convect2 |
|
\int_{\Omega} ((\ul{u} \cdot \nabla) \ul{v}) \cdot \ul{w} |
|
dw_adj_div_grad |
|
w \delta_{u} \Psi(\ul{u}) \circ \ul{v} |
|
ev_sd_convect |
|
\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 |
|
\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 |
|
\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 |
|
\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 |
|
\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 |
|
\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 |
|
\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 |
|
\int_{\Omega} p q (\nabla \cdot \ul{\Vcal}) \mbox{ , } \int_{\Omega} (\ul{u} \cdot \ul{w}) (\nabla \cdot \ul{\Vcal}) |
|
de_sd_lin_elastic |
|
\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 |
|
\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 |
|
\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 |
|
\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 |
|
\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 |
|
\int_{\Gamma} p \nabla \cdot \ul{\Vcal} |
|
de_sd_surface_ltr |
|
\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 |
|
\int_{\Gamma} \ul{v} \cdot (\ull{\sigma}\, \ul{n}), \int_{\Gamma} \ul{v} \cdot \ul{n}, |
|
de_sd_v_dot_grad_s |
|
\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¶
name/class |
arguments |
definition |
examples |
---|---|---|---|
dw_tl_bulk_active |
|
\int_{\Omega} S_{ij}(\ul{u}) \delta E_{ij}(\ul{u};\ul{v}) |
|
dw_tl_bulk_penalty |
|
\int_{\Omega} S_{ij}(\ul{u}) \delta E_{ij}(\ul{u};\ul{v}) |
act.fib, com.ela.mat, hyp |
dw_tl_bulk_pressure |
|
\int_{\Omega} S_{ij}(p) \delta E_{ij}(\ul{u};\ul{v}) |
per.tl, bal |
dw_tl_diffusion |
|
\int_{\Omega} \ull{K}(\ul{u}^{(n-1)}) : \pdiff{q}{\ul{X}} \pdiff{p}{\ul{X}} |
per.tl |
dw_tl_fib_a |
|
\int_{\Omega} S_{ij}(\ul{u}) \delta E_{ij}(\ul{u};\ul{v}) |
act.fib |
dw_tl_fib_e |
|
\int_{\Omega} S_{ij}(\ul{u}) \delta E_{ij}(\ul{u};\ul{v}) |
|
dw_tl_fib_spe |
|
\int_{\Omega} S_{ij}(\ul{u}) \delta E_{ij}(\ul{u};\ul{v}) |
|
dw_tl_he_genyeoh |
|
\int_{\Omega} S_{ij}(\ul{u}) \delta E_{ij}(\ul{u};\ul{v}) |
|
dw_tl_he_mooney_rivlin |
|
\int_{\Omega} S_{ij}(\ul{u}) \delta E_{ij}(\ul{u};\ul{v}) |
com.ela.mat, hyp, bal |
dw_tl_he_neohook |
|
\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 |
|
\int_{\Omega} S_{ij}(\ul{u}) \delta E_{ij}(\ul{u};\ul{v}) |
|
dw_tl_membrane |
|
bal |
|
ev_tl_surface_flux |
|
\int_{\Gamma} \ul{\nu} \cdot \ull{K}(\ul{u}^{(n-1)}) \pdiff{p}{\ul{X}} |
|
dw_tl_surface_traction |
|
\int_{\Gamma} \ul{\nu} \cdot \ull{F}^{-1} \cdot \ull{\sigma} \cdot \ul{v} J |
per.tl |
dw_tl_volume |
|
\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 |
|
1 / D \int_{\Gamma} \ul{\nu} \cdot \ull{F}^{-1} \cdot \ul{x} J |
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dw_ul_bulk_penalty |
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\int_{\Omega} \mathcal{L}\tau_{ij}(\ul{u}) e_{ij}(\delta\ul{v})/J |
hyp.ul |
dw_ul_bulk_pressure |
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\int_{\Omega} \mathcal{L}\tau_{ij}(\ul{u}) e_{ij}(\delta\ul{v})/J |
hyp.ul.up |
dw_ul_compressible |
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\int_{\Omega} 1\over \gamma p \, q |
hyp.ul.up |
dw_ul_he_by_fun |
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\int_{\Omega} \mathcal{L}\tau_{ij}(\ul{u}) e_{ij}(\delta\ul{v})/J |
hyp.ul.by.fun |
dw_ul_he_mooney_rivlin |
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\int_{\Omega} \mathcal{L}\tau_{ij}(\ul{u}) e_{ij}(\delta\ul{v})/J |
hyp.ul.up, hyp.ul |
dw_ul_he_neohook |
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\int_{\Omega} \mathcal{L}\tau_{ij}(\ul{u}) e_{ij}(\delta\ul{v})/J |
hyp.ul.up, hyp.ul |
dw_ul_volume |
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\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¶
name/class |
arguments |
definition |
examples |
---|---|---|---|
dw_biot_eth |
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\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} |
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dw_biot_th |
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\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} |
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ev_cauchy_stress_eth |
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\int_{\Omega} \int_0^t \Hcal_{ijkl}(t-\tau)\,e_{kl}(\ul{w}(\tau)) \difd{\tau} |
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ev_cauchy_stress_th |
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\int_{\Omega} \int_0^t \Hcal_{ijkl}(t-\tau)\,e_{kl}(\ul{w}(\tau)) \difd{\tau} |
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dw_lin_elastic_eth |
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\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 |
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\int_{\Omega} \left [\int_0^t \Hcal_{ijkl}(t-\tau)\,e_{kl}(\ul{u}(\tau)) \difd{\tau} \right]\,e_{ij}(\ul{v}) |
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ev_of_ns_surf_min_d_press |
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\delta \Psi(p) = \delta \left( \int_{\Gamma_{in}}p - \int_{\Gamma_{out}}bpress \right) |
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dw_of_ns_surf_min_d_press_diff |
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w \delta_{p} \Psi(p) \circ q |
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ev_sd_st_grad_div |
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\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} ] |
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ev_sd_st_pspg_c |
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\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} ] |
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ev_sd_st_pspg_p |
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\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} ] |
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ev_sd_st_supg_c |
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\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} ] |
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dw_st_adj1_supg_p |
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\sum_{K \in \Ical_h}\int_{T_K} \delta_K\ \nabla p (\ul{v} \cdot \nabla \ul{w}) |
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dw_st_adj2_supg_p |
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\sum_{K \in \Ical_h}\int_{T_K} \tau_K\ \nabla r (\ul{v} \cdot \nabla \ul{u}) |
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dw_st_adj_supg_c |
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\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}) ] |
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dw_st_grad_div |
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\gamma \int_{\Omega} (\nabla\cdot\ul{u}) \cdot (\nabla\cdot\ul{v}) |
sta.nav.sto |
dw_st_pspg_c |
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\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 |
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\sum_{K \in \Ical_h}\int_{T_K} \tau_K\ \nabla p \cdot \nabla q |
sta.nav.sto |
dw_st_supg_c |
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\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 |
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\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 |
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\int_\Omega \left [\int_0^t \Gcal(t-\tau) p(\tau) \difd{\tau} \right] q |
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dw_volume_dot_w_scalar_th |
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\int_\Omega \left [\int_0^t \Gcal(t-\tau) p(\tau) \difd{\tau} \right] q |
Table of multi-linear terms¶
name/class |
arguments |
definition |
examples |
---|---|---|---|
de_cauchy_stress |
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\int_{\Omega} D_{ijkl} e_{kl}(\ul{w}) |
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de_convect |
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\int_{\Omega} c ((\ul{u} \cdot \nabla) \ul{u}) \cdot \ul{v} |
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de_diffusion |
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\int_{\Omega} K_{ij} \nabla_i q\, \nabla_j p |
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de_div |
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\int_{\Omega} \nabla \cdot \ul{v} \mbox { , } \int_{\Omega} c \nabla \cdot \ul{v} |
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de_div_grad |
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\int_{\Omega} \nabla \ul{v} : \nabla \ul{u} \mbox{ , } \int_{\Omega} \nu\ \nabla \ul{v} : \nabla \ul{u} |
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de_dot |
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\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}) |
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de_grad |
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\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} |
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de_integrate |
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\int_{\cal{D}} q \mbox{ or } \int_{\cal{D}} c q |
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de_laplace |
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\int_{\Omega} \nabla q \cdot \nabla p \mbox{ , } \int_{\Omega} c \nabla q \cdot \nabla p |
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de_lin_convect |
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\int_{\Omega} ((\ul{w} \cdot \nabla) \ul{u}) \cdot \ul{v} |
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de_lin_elastic |
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\int_{\Omega} D_{ijkl}\ e_{ij}(\ul{v}) e_{kl}(\ul{u}) |
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de_m_flexo |
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\int_{\Omega} v_{i,j} a_{ij} \\ \int_{\Omega} u_{i,j} \delta a_{ij} |
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de_m_flexo_coupling |
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\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 |
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de_m_sg_elastic |
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\int_{\Omega} a_{ijklmn}\ e_{ij,k}(\ull{\delta w}) \ e_{lm,n}(\ull{w}) |
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de_mass |
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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 |
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\int_{\Gamma} c (\ul{n} \cdot \ul{v}) (\ul{n} \cdot \ul{u}) |
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de_nonsym_elastic |
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\int_{\Omega} \ull{D} \nabla \ul{v} : \nabla \ul{u} |
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de_s_dot_mgrad_s |
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\int_{\Omega} q \ul{y} \cdot \nabla p \mbox{ , } \int_{\Omega} p \ul{y} \cdot \nabla q |
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de_stokes |
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\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} |
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de_stokes_traction |
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\int_{\Gamma} \nu \ul{v}\cdot(\nabla \ul{u} \cdot \ul{n}) |
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de_surface_flux |
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\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 |
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\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} |
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de_surface_piezo_flux |
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\int_{\Gamma} q g_{kij} e_{ij}(\ul{u}) n_k \mbox{ , } \int_{\Gamma} p g_{kij} e_{ij}(\ul{v}) n_k |