sfepy.terms.terms_sensitivity module¶

class sfepy.terms.terms_sensitivity.ESDDiffusionTerm(*args, **kwargs)[source]¶

Diffusion sensitivity analysis term.

Definition:

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

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

Call signature:

de_sd_diffusion	`(material, virtual, state, parameter_mv)`
	`(material, parameter_1, parameter_2, parameter_mv)`

Arguments:

material: $K_{ij}$
virtual/parameter_1: $q$
state/parameter_2: $p$
parameter_mv: $\ul{\Vcal}$

arg_shapes = {'material': 'D, D', 'parameter_1': 1, 'parameter_2': 1, 'parameter_mv': 'D', 'state': 1, 'virtual': (1, 'state')}¶

arg_types = (('material', 'virtual', 'state', 'parameter_mv'), ('material', 'parameter_1', 'parameter_2', 'parameter_mv'))¶

get_function(mat, vvar, svar, par_mv, mode=None, term_mode=None, diff_var=None, **kwargs)[source]¶

modes = ('weak', 'eval')¶

name = 'de_sd_diffusion'¶

class sfepy.terms.terms_sensitivity.ESDDivGradTerm(*args, **kwargs)[source]¶

Sensitivity (shape derivative) of diffusion term de_div_grad.

Definition:

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

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

Call signature:

de_sd_div_grad	`(opt_material, virtual, state, parameter_mv)`
	`(opt_material, parameter_1, parameter_2, parameter_mv)`

Arguments:

material: $\nu$ (viscosity, optional)
virtual/parameter_1: $\ul{v}$
state/parameter_2: $\ul{u}$
parameter_mv: $\ul{\Vcal}$

arg_shapes = [{'opt_material': '1, 1', 'virtual': ('D', 'state'), 'state': 'D', 'parameter_1': 'D', 'parameter_2': 'D', 'parameter_mv': 'D'}, {'opt_material': None}]¶

arg_types = (('opt_material', 'virtual', 'state', 'parameter_mv'), ('opt_material', 'parameter_1', 'parameter_2', 'parameter_mv'))¶

get_function(mat, vvar, svar, par_mv, mode=None, term_mode=None, diff_var=None, **kwargs)[source]¶

modes = ('weak', 'eval')¶

name = 'de_sd_div_grad'¶

class sfepy.terms.terms_sensitivity.ESDDotTerm(*args, **kwargs)[source]¶

Sensitivity (shape derivative) of dot product of scalars or vectors.

Definition:

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

Call signature:

de_sd_dot	`(opt_material, virtual, state, parameter_mv)`
	`(opt_material, parameter_1, parameter_2, parameter_mv)`

Arguments:

material: $c$ or $\ull{M}$ (optional)
virtual/parameter_1: $q$ or $\ul{v}$
state/parameter_2: $p$ or $\ul{u}$
parameter_mv : $\ul{\Vcal}$

arg_shapes = [{'opt_material': '1, 1', 'virtual': (1, 'state'), 'state': 1, 'parameter_1': 1, 'parameter_2': 1, 'parameter_mv': 'D'}, {'opt_material': None}, {'opt_material': '1, 1', 'virtual': ('D', 'state'), 'state': 'D', 'parameter_1': 'D', 'parameter_2': 'D', 'parameter_mv': 'D'}, {'opt_material': 'D, D'}, {'opt_material': None}]¶

arg_types = (('opt_material', 'virtual', 'state', 'parameter_mv'), ('opt_material', 'parameter_1', 'parameter_2', 'parameter_mv'))¶

get_function(mat, vvar, svar, par_mv, mode=None, term_mode=None, diff_var=None, **kwargs)[source]¶

modes = ('weak', 'eval')¶

name = 'de_sd_dot'¶

class sfepy.terms.terms_sensitivity.ESDLinearElasticTerm(*args, **kwargs)[source]¶

Sensitivity analysis of the linear elastic term. $D_{ijkl}$ can be given in symmetric or non-symmetric form.

Definition:

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

$\hat{D}_{ijkl} = D_{ijkl}(\nabla \cdot \ul{\Vcal}) - D_{ijkq}{\partial \Vcal_l \over \partial x_q} - D_{iqkl}{\partial \Vcal_j \over \partial x_q}$

Call signature:

de_sd_lin_elastic	`(material, virtual, state, parameter_mv)`
	`(material, parameter_1, parameter_2, parameter_mv)`

Arguments 1:

material : $\ull{D}$
virtual/parameter_v : $\ul{v}$
state/parameter_s : $\ul{u}$
parameter_mv : $\ul{\Vcal}$

arg_shapes = [{'material': 'S, S', 'virtual': ('D', 'state'), 'state': 'D', 'parameter_1': 'D', 'parameter_2': 'D', 'parameter_mv': 'D'}, {'material': 'D2, D2'}]¶

arg_types = (('material', 'virtual', 'state', 'parameter_mv'), ('material', 'parameter_1', 'parameter_2', 'parameter_mv'))¶

geometries = ['2_3', '2_4', '3_4', '3_8']¶

get_function(mat, vvar, svar, par_mv, mode=None, term_mode=None, diff_var=None, **kwargs)[source]¶

modes = ('weak', 'eval')¶

name = 'de_sd_lin_elastic'¶

class sfepy.terms.terms_sensitivity.ESDLinearTractionTerm(*args, **kwargs)[source]¶

Sensitivity of the linear traction term.

Definition:

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

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

Call signature:

de_sd_surface_ltr	`(opt_material, virtual, parameter_mv)`
	`(opt_material, parameter, parameter_mv)`

Arguments:

material: $c$ , $\ul{\sigma}$ , $\ull{\sigma}$
virtual/parameter: $\ul{v}$
parameter_mv: $\ul{\Vcal}$

arg_shapes = [{'opt_material': 'S, 1', 'virtual': ('D', None), 'parameter_mv': 'D', 'parameter': 'D'}, {'opt_material': None}, {'opt_material': '1, 1'}, {'opt_material': 'D, D'}]¶

arg_types = (('opt_material', 'virtual', 'parameter_mv'), ('opt_material', 'parameter', 'parameter_mv'))¶

get_function(traction, vvar, par_mv, mode=None, term_mode=None, diff_var=None, **kwargs)[source]¶

integration = 'facet'¶

modes = ('weak', 'eval')¶

name = 'de_sd_surface_ltr'¶

class sfepy.terms.terms_sensitivity.ESDPiezoCouplingTerm(*args, **kwargs)[source]¶

Sensitivity (shape derivative) of the piezoelectric coupling term.

Definition:

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

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

Call signature:

de_sd_piezo_coupling	`(material, virtual, state, parameter_mv)`
	`(material, state, virtual, parameter_mv)`
	`(material, parameter_v, parameter_s, parameter_mv)`

Arguments 1:

material : $g_{kij}$
virtual/parameter_v : $\ul{v}$
state/parameter_s : $p$
parameter_mv : $\ul{\Vcal}$

Arguments 2:

material : $g_{kij}$
state : $\ul{u}$
virtual : $q$
parameter_mv : $\ul{\Vcal}$

arg_shapes = {'material': 'D, S', 'parameter_mv': 'D', 'parameter_s': 1, 'parameter_v': 'D', 'state/div': 'D', 'state/grad': 1, 'virtual/div': (1, None), 'virtual/grad': ('D', None)}¶

arg_types = (('material', 'virtual', 'state', 'parameter_mv'), ('material', 'state', 'virtual', 'parameter_mv'), ('material', 'parameter_v', 'parameter_s', 'parameter_mv'))¶

geometries = ['2_3', '2_4', '3_4', '3_8']¶

get_function(mat, vvar, svar, par_mv, mode=None, term_mode=None, diff_var=None, **kwargs)[source]¶

modes = ('grad', 'div', 'eval')¶

name = 'de_sd_piezo_coupling'¶

class sfepy.terms.terms_sensitivity.ESDStokesTerm(*args, **kwargs)[source]¶

Stokes problem coupling term. Corresponds to weak forms of gradient and divergence terms.

Definition:

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

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

Call signature:

de_sd_stokes	`(opt_material, virtual, state, parameter_mv)`
	`(opt_material, state, virtual, parameter_mv)`
	`(opt_material, parameter_v, parameter_s, parameter_mv)`

Arguments 1:

virtual/parameter_v: $\ul{v}$
state/parameter_s: $p$
parameter_mv: $\ul{\Vcal}$

Arguments 2:

state : $\ul{u}$
virtual : $q$
parameter_mv: $\ul{\Vcal}$

arg_shapes = [{'opt_material': '1, 1', 'virtual/grad': ('D', None), 'state/grad': 1, 'virtual/div': (1, None), 'state/div': 'D', 'parameter_v': 'D', 'parameter_s': 1, 'parameter_mv': 'D'}, {'opt_material': None}]¶

arg_types = (('opt_material', 'virtual', 'state', 'parameter_mv'), ('opt_material', 'state', 'virtual', 'parameter_mv'), ('opt_material', 'parameter_v', 'parameter_s', 'parameter_mv'))¶

get_function(coef, vvar, svar, par_mv, mode=None, term_mode=None, diff_var=None, **kwargs)[source]¶

modes = ('grad', 'div', 'eval')¶

name = 'de_sd_stokes'¶

texpr = 'ij,i.j,0'¶

class sfepy.terms.terms_sensitivity.ESDVectorDotGradScalarTerm(*args, **kwargs)[source]¶

Sensitivity of volume dot product of a vector and a gradient of scalar.

Definition:

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

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

Call signature:

de_sd_v_dot_grad_s	`(opt_material, virtual, state, parameter_mv)`
	`(opt_material, state, virtual, parameter_mv)`
	`(opt_material, parameter_v, parameter_s, parameter_mv)`

Arguments 1:

virtual/parameter_v: $\ul{v}$
state/parameter_s: $p$
parameter_mv: $\ul{\Vcal}$

Arguments 2:

state : $\ul{u}$
virtual : $q$
parameter_mv: $\ul{\Vcal}$

name = 'de_sd_v_dot_grad_s'¶

texpr = 'ij,i,0.j'¶

sfepy.terms.terms_sensitivity.get_nonsym_grad_op(sgrad)[source]¶