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}) |
|
ev_biot_stress |
|
- \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 |
|
\int_{\Omega} p\ \alpha_{ij} e_{ij}(\ul{v}) \mbox{ , } \int_{\Omega} q\ \alpha_{ij} e_{ij}(\ul{u}) |
the.ela.ess, bio.sho.syn, the.ela, bio.npb.lag, bio.npb, bio |
ev_cauchy_strain |
|
\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 |
|
\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 |
|
\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_contact |
|
\int_{\Gamma_{c}} \varepsilon_N \langle g_N(\ul{u}) \rangle \ul{n} \ul{v} |
two.bod.con |
dw_convect_v_grad_s |
|
\int_{\Omega} q (\ul{u} \cdot \nabla p) |
poi.fun |
dw_convect |
|
\int_{\Omega} ((\ul{u} \cdot \nabla) \ul{u}) \cdot \ul{v} |
nav.sto, nav.sto.iga, nav.sto |
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.dif.2D, adv.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_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 \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 |
|
\int_{\Omega} K_{ij} \nabla_i q \nabla_j p \mbox{ , } \int_{\Omega} K_{ij} \nabla_i \bar{p} \nabla_j r |
bio, dar.flo.mul, bio.sho.syn, bio.npb, bio.npb.lag, poi.neu, pie.ela |
dw_div_grad |
|
\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} |
sto.sli.bc, sta.nav.sto, nav.sto, sto, nav.sto.iga, nav.sto |
dw_div |
|
\int_{\Omega} \nabla \cdot \ul{v} \mbox { or } \int_{\Omega} c \nabla \cdot \ul{v} |
|
ev_div |
|
\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 |
|
\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} |
the.ele, lin.ela.dam, pie.ela, aco, aco, sto.sli.bc, bal, tim.poi.exp, bur.2D, poi.fun, tim.poi, dar.flo.mul, adv.2D, hyd, lin.ela.up, adv.1D, ela, poi.per.bou.con, vib.aco, tim.adv.dif, osc, wel, bor |
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_elastic_wave |
|
\int_{\Omega} D_{ijkl}\ g_{ij}(\ul{v}) g_{kl}(\ul{u}) |
|
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{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 |
|
\int_{\cal{D}} m \mbox{vector for } K \from \Ical_h: \int_{T_K} m / \int_{T_K} 1 m|_{qp} |
|
dw_integrate |
|
\int_{\cal{D}} q \mbox{ or } \int_{\cal{D}} c q |
poi.per.bou.con, dar.flo.mul, vib.aco, poi.neu, aco, aco |
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 } \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 |
|
\int_{\Gamma} c\, q (p_1 - p_2) |
aco |
dw_laplace |
|
\int_{\Omega} c \nabla q \cdot \nabla p \mbox{ , } \int_{\Omega} c \nabla \bar{p} \cdot \nabla r |
sin, the.ele, poi.par.stu, aco, lap.1d, aco, lap.flu.2d, lap.tim.ebc, sto.sli.bc, adv.dif.2D, tim.poi.exp, bur.2D, poi.fun, cub, tim.poi, poi, lap.2D, hyd, osc, poi.iga, poi.per.bou.con, the.ela.ess, lap.cou.lcb, poi.fie.dep.mat, vib.aco, tim.adv.dif, poi.sho.syn, wel, bor |
dw_lin_convect2 |
|
\int_{\Omega} ((\ul{b} \cdot \nabla) \ul{u}) \cdot \ul{v} ((\ul{b} \cdot \nabla) \ul{u})|_{qp} |
|
dw_lin_convect |
|
\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 |
|
\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 |
|
\int_{\Omega} D_{ijkl}\ e_{ij}(\ul{v}) e_{kl}(\ul{u}) |
its.2, ela.con.pla, lin.ela.iga, pre.fib, lin.ela.opt, bio.sho.syn, its.3, pie.ela.mac, lin.ela.dam, pie.ela, nod.lcb, mat.non, com.ela.mat, its.4, its.1, bio, two.bod.con, ela.shi.per, ela.con.sph, lin.ela.mM, lin.ela, lin.ela.up, ela, the.ela.ess, lin.vis, the.ela, lin.ela.tra, bio.npb, bio.npb.lag |
dw_lin_prestress |
|
\int_{\Omega} \sigma_{ij} e_{ij}(\ul{v}) |
non.hyp.mM, pre.fib, pie.ela.mac |
dw_lin_strain_fib |
|
\int_{\Omega} D_{ijkl} e_{ij}(\ul{v}) \left(d_k d_l\right) |
pre.fib |
dw_non_penetration_p |
|
\int_{\Gamma} c (\ul{n} \cdot \ul{v}) (\ul{n} \cdot \ul{u}) |
bio.sho.syn |
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_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 \mbox{ , } \int_{\Omega} g_{kij}\ e_{ij}(\ul{u}) \nabla_k q |
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.2, she.can, its.3, its.4, its.1 |
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.dif.2D, adv.2D, adv.1D |
dw_shell10x |
|
\int_{\Omega} D_{ijkl}\ e_{ij}(\ul{v}) e_{kl}(\ul{u}) |
she.can |
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}) |
|
dw_stokes_wave |
|
\int_{\Omega} (\ul{\kappa} \cdot \ul{v}) (\ul{\kappa} \cdot \ul{u}) |
|
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.sli.bc, sta.nav.sto, nav.sto, sto, lin.ela.up, nav.sto.iga, nav.sto |
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 \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 |
|
\int_{\Gamma} \ul{v} \cdot \ull{\sigma} \cdot \ul{n}, \int_{\Gamma} \ul{v} \cdot \ul{n}, |
ela.shi.per, nod.lcb, lin.ela.opt, lin.vis, lin.ela.tra, com.ela.mat |
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 |
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{M} \nabla p) \mbox{ , } \int_{\Omega} \ul{u} \cdot (\ull{M} \nabla q) |
|
dw_vm_dot_s |
|
\int_{\Omega} \ul{v} \cdot \ul{m} p \mbox{ , } \int_{\Omega} \ul{u} \cdot \ul{m} q\\ |
|
dw_volume_lvf |
|
\int_{\Omega} \ul{f} \cdot \ul{v} \mbox{ or } \int_{\Omega} f q |
poi.par.stu, bur.2D, adv.dif.2D, poi.iga |
ev_volume_surface |
|
1 / D \int_\Gamma \ul{x} \cdot \ul{n} |
|
ev_volume |
|
\int_{\cal{D}} 1 |
|
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} \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 |
|
\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 |
|
\int_{\Omega} p [ (\nabla \cdot \ul{w}) (\nabla \cdot \ul{\Vcal}) - \pdiff{\Vcal_k}{x_i} \pdiff{w_i}{x_k} ] |
|
ev_sd_dot |
|
\int_{\Omega} p q (\nabla \cdot \ul{\Vcal}) \mbox{ , } \int_{\Omega} (\ul{u} \cdot \ul{w}) (\nabla \cdot \ul{\Vcal}) |
|
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} |
|
ev_sd_surface_integrate |
|
\int_{\Gamma} p \nabla \cdot \ul{\Vcal} |
|
ev_sd_surface_ltr |
|
\int_{\Gamma} \ul{v} \cdot \ull{\sigma} \cdot \ul{n}, \int_{\Gamma} \ul{v} \cdot \ul{n}, |
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_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}) |
hyp, com.ela.mat, bal |
dw_tl_he_neohook |
|
\int_{\Omega} S_{ij}(\ul{u}) \delta E_{ij}(\ul{u};\ul{v}) |
hyp, per.tl, act.fib, com.ela.mat, 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 |
ev_tl_volume_surface |
|
1 / D \int_{\Gamma} \ul{\nu} \cdot \ull{F}^{-1} \cdot \ul{x} J |
|
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 |
dw_ul_bulk_penalty |
|
\int_{\Omega} \mathcal{L}\tau_{ij}(\ul{u}) e_{ij}(\delta\ul{v})/J |
hyp.ul |
dw_ul_bulk_pressure |
|
\int_{\Omega} \mathcal{L}\tau_{ij}(\ul{u}) e_{ij}(\delta\ul{v})/J |
hyp.ul.up |
dw_ul_compressible |
|
\int_{\Omega} 1\over \gamma p \, q |
hyp.ul.up |
dw_ul_he_mooney_rivlin |
|
\int_{\Omega} \mathcal{L}\tau_{ij}(\ul{u}) e_{ij}(\delta\ul{v})/J |
hyp.ul, hyp.ul.up |
dw_ul_he_neohook |
|
\int_{\Omega} \mathcal{L}\tau_{ij}(\ul{u}) e_{ij}(\delta\ul{v})/J |
hyp.ul, hyp.ul.up |
dw_ul_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} |
hyp.ul.up |
Table of special terms¶
name/class |
arguments |
definition |
examples |
---|---|---|---|
dw_biot_eth |
|
\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 |
|
\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 |
|
\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 |
|
\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} |
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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 |
|
\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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dw_of_ns_surf_min_d_press_diff |
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w \delta_{p} \Psi(p) \circ q |
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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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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} ] |
|
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} ] |
|
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}) ] |
|
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 |
|
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 |
|
\int_{\Omega} D_{ijkl} e_{kl}(\ul{w}) |
|
de_convect |
|
\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 |
|
\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 |
|
\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 |
|
\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 |
|
\int_\Omega q \mbox{ or } \int_\Omega c q |
|
de_laplace |
|
\int_{\Omega} c \nabla q \cdot \nabla p \mbox{ , } \int_{\Omega} c \nabla \bar{p} \cdot \nabla r |
|
de_lin_elastic |
|
\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 |
|
\int_{\Gamma} c (\ul{n} \cdot \ul{v}) (\ul{n} \cdot \ul{u}) |
|
de_s_dot_mgrad_s |
|
\int_{\Omega} q \ul{y} \cdot \nabla p \mbox{ , } \int_{\Omega} p \ul{y} \cdot \nabla q |
|
de_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} \\ \int_{\Omega} r\ \nabla \cdot \ul{w} \mbox{ , } \int_{\Omega} c r\ \nabla \cdot \ul{w} |