sfepy.terms.terms_multilinear module¶

class sfepy.terms.terms_multilinear.ECauchyStressTerm(*args, **kwargs)[source]¶

Evaluate Cauchy stress tensor.

It is given in the usual vector form exploiting symmetry: in 3D it has 6 components with the indices ordered as $[11, 22, 33, 12, 13, 23]$ , in 2D it has 3 components with the indices ordered as $[11, 22, 12]$ .

Definition

$\int_{\Omega} D_{ijkl} e_{kl}(\ul{w})$

Call signature

de_cauchy_stress

(material, parameter)

Arguments

material : $D_{ijkl}$
parameter : $\ul{w}$

arg_shapes = {'material': 'S, S', 'parameter': 'D'}¶

arg_types = ('material', 'parameter')¶

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

name = 'de_cauchy_stress'¶

class sfepy.terms.terms_multilinear.EConvectTerm(*args, **kwargs)[source]¶

Nonlinear convective term.

Definition

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

Call signature

de_convect

(virtual, state)

Arguments

virtual : $\ul{v}$
state : $\ul{u}$

arg_shapes = {'state': 'D', 'virtual': ('D', 'state')}¶

arg_types = ('virtual', 'state')¶

get_function(virtual, state, mode=None, term_mode=None, diff_var=None, **kwargs)[source]¶

name = 'de_convect'¶

class sfepy.terms.terms_multilinear.EDivGradTerm(*args, **kwargs)[source]¶

Vector field diffusion term.

Definition

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

Call signature

de_div_grad	`(opt_material, virtual, state)`
	`(opt_material, parameter_1, parameter_2)`

Arguments 1

material : $\nu$ (viscosity, optional)
virtual : $\ul{v}$
state : $\ul{u}$

Arguments 2

material : $\nu$ (viscosity, optional)
parameter_1 : $\ul{u}$
parameter_2 : $\ul{w}$

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

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

get_function(mat, virtual, state, mode=None, term_mode=None, diff_var=None, **kwargs)[source]¶

modes = ('weak', 'eval')¶

name = 'de_div_grad'¶

class sfepy.terms.terms_multilinear.EDivTerm(*args, **kwargs)[source]¶

Weighted divergence term.

Definition

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

Call signature

de_div	`(opt_material, virtual)`
	`(opt_material, parameter)`

Arguments 1

material : $c$ (optional)
virtual : $\ul{v}$

Arguments 2

material : $c$ (optional)
parameter : $\ul{u}$

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

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

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

modes = ('weak', 'eval')¶

name = 'de_div'¶

class sfepy.terms.terms_multilinear.EIntegrateVolumeOperatorTerm(*args, **kwargs)[source]¶

Volume integral of a test function weighted by a scalar function $c$ .

Definition

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

Call signature

de_volume_integrate

(opt_material, virtual)

Arguments

material : $c$ (optional)
virtual : $q$

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

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

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

name = 'de_volume_integrate'¶

class sfepy.terms.terms_multilinear.ELaplaceTerm(*args, **kwargs)[source]¶

Laplace term with $c$ coefficient. Can be evaluated. Can use derivatives.

Definition

$\int_{\Omega} c \nabla q \cdot \nabla p \mbox{ , } \int_{\Omega} c \nabla \bar{p} \cdot \nabla r$

Call signature

de_laplace	`(opt_material, virtual, state)`
	`(opt_material, parameter_1, parameter_2)`

Arguments 1

material : $c$
virtual : $q$
state : $p$

Arguments 2

material : $c$
parameter_1 : $\bar{p}$
parameter_2 : $r$

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

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

get_function(mat, virtual, state, mode=None, term_mode=None, diff_var=None, **kwargs)[source]¶

modes = ('weak', 'eval')¶

name = 'de_laplace'¶

class sfepy.terms.terms_multilinear.ELinearElasticTerm(*args, **kwargs)[source]¶

General linear elasticity term, with $D_{ijkl}$ given in the usual matrix form exploiting symmetry: in 3D it is $6\times6$ with the indices ordered as $[11, 22, 33, 12, 13, 23]$ , in 2D it is $3\times3$ with the indices ordered as $[11, 22, 12]$ .

Definition

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

Call signature

de_lin_elastic	`(material, virtual, state)`
	`(material, parameter_1, parameter_2)`

Arguments 1

material : $D_{ijkl}$
virtual : $\ul{v}$
state : $\ul{u}$

Arguments 2

material : $D_{ijkl}$
parameter_1 : $\ul{w}$
parameter_2 : $\ul{u}$

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

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

get_function(mat, virtual, state, mode=None, term_mode=None, diff_var=None, **kwargs)[source]¶

modes = ('weak', 'eval')¶

name = 'de_lin_elastic'¶

class sfepy.terms.terms_multilinear.ENonPenetrationPenaltyTerm(*args, **kwargs)[source]¶

Non-penetration condition in the weak sense using a penalty.

Definition

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

Call signature

de_non_penetration_p

(material, virtual, state)

Arguments

material : $c$
virtual : $\ul{v}$
state : $\ul{u}$

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

arg_types = ('material', 'virtual', 'state')¶

get_function(mat, virtual, state, mode=None, term_mode=None, diff_var=None, **kwargs)[source]¶

integration = 'surface'¶

name = 'de_non_penetration_p'¶

class sfepy.terms.terms_multilinear.EScalarDotMGradScalarTerm(*args, **kwargs)[source]¶

Volume dot product of a scalar gradient dotted with a material vector with a scalar.

Definition

$\int_{\Omega} q \ul{y} \cdot \nabla p \mbox{ , } \int_{\Omega} p \ul{y} \cdot \nabla q$

Call signature

de_s_dot_mgrad_s	`(material, virtual, state)`
	`(material, state, virtual)`

Arguments 1

material : $\ul{y}$
virtual : $q$
state : $p$

Arguments 2

material : $\ul{y}$
state : $p$
virtual : $q$

arg_shapes = [{'material': 'D, 1', 'virtual/grad_state': (1, None), 'state/grad_state': 1, 'virtual/grad_virtual': (1, None), 'state/grad_virtual': 1}]¶

arg_types = (('material', 'virtual', 'state'), ('material', 'state', 'virtual'))¶

get_function(mat, var1, var2, mode=None, term_mode=None, diff_var=None, **kwargs)[source]¶

modes = ('grad_state', 'grad_virtual')¶

name = 'de_s_dot_mgrad_s'¶

class sfepy.terms.terms_multilinear.EStokesTerm(*args, **kwargs)[source]¶

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

Definition

$\int_{\Omega} p\ \nabla \cdot \ul{v} \mbox{ , } \int_{\Omega} q\ \nabla \cdot \ul{u} \mbox{ or } \int_{\Omega} c\ p\ \nabla \cdot \ul{v} \mbox{ , } \int_{\Omega} c\ q\ \nabla \cdot \ul{u} \\ \int_{\Omega} r\ \nabla \cdot \ul{w} \mbox{ , } \int_{\Omega} c r\ \nabla \cdot \ul{w}$

Call signature

de_stokes	`(opt_material, virtual, state)`
	`(opt_material, state, virtual)`
	`(opt_material, parameter_v, parameter_s)`

Arguments 1

material : $c$ (optional)
virtual : $\ul{v}$
state : $p$

Arguments 2

material : $c$ (optional)
state : $\ul{u}$
virtual : $q$

Arguments 3

material : $c$ (optional)
parameter_v : $\ul{u}$
parameter_s : $p$

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}, {'opt_material': None}]¶

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

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

modes = ('grad', 'div', 'eval')¶

name = 'de_stokes'¶

class sfepy.terms.terms_multilinear.ESurfaceDotTerm(*args, **kwargs)[source]¶

Surface $L^2(\Gamma)$ dot product for both scalar and vector fields.

Definition

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

Call signature

de_surface_dot	`(opt_material, virtual, state)`
	`(opt_material, parameter_1, parameter_2)`

Arguments 1

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

Arguments 2

material : $c$ or $\ull{M}$ (optional)
parameter_1 : $p$ or $\ul{u}$
parameter_2 : $r$ or $\ul{w}$

integration = 'surface'¶

name = 'de_surface_dot'¶

class sfepy.terms.terms_multilinear.ETermBase(*args, **kwargs)[source]¶

Reserved letters:

c .. cells q .. quadrature points d-h .. DOFs axes r-z .. auxiliary axes

Layout specification letters:

c .. cells q .. quadrature points v .. variable component - matrix form (v, d) -> vector v*d g .. gradient component d .. local DOF (basis, node) 0 .. all material axes

build_expression(texpr, *eargs, diff_var=None)[source]¶

can_backend = {'dask_single': <module 'dask.array' from '/home/eldaran/miniconda3/lib/python3.8/site-packages/dask/array/__init__.py'>, 'dask_threads': <module 'dask.array' from '/home/eldaran/miniconda3/lib/python3.8/site-packages/dask/array/__init__.py'>, 'jax': <module 'jax.numpy' from '/home/eldaran/miniconda3/lib/python3.8/site-packages/jax/numpy/__init__.py'>, 'jax_vmap': <module 'jax.numpy' from '/home/eldaran/miniconda3/lib/python3.8/site-packages/jax/numpy/__init__.py'>, 'numpy': <module 'numpy' from '/home/eldaran/miniconda3/lib/python3.8/site-packages/numpy/__init__.py'>, 'numpy_loop': <module 'numpy' from '/home/eldaran/miniconda3/lib/python3.8/site-packages/numpy/__init__.py'>, 'numpy_qloop': <module 'numpy' from '/home/eldaran/miniconda3/lib/python3.8/site-packages/numpy/__init__.py'>, 'opt_einsum': <module 'opt_einsum' from '/home/eldaran/miniconda3/lib/python3.8/site-packages/opt_einsum/__init__.py'>, 'opt_einsum_dask_single': <module 'dask.array' from '/home/eldaran/miniconda3/lib/python3.8/site-packages/dask/array/__init__.py'>, 'opt_einsum_dask_threads': <module 'dask.array' from '/home/eldaran/miniconda3/lib/python3.8/site-packages/dask/array/__init__.py'>, 'opt_einsum_loop': <module 'opt_einsum' from '/home/eldaran/miniconda3/lib/python3.8/site-packages/opt_einsum/__init__.py'>, 'opt_einsum_qloop': <module 'opt_einsum' from '/home/eldaran/miniconda3/lib/python3.8/site-packages/opt_einsum/__init__.py'>}¶

clear_cache()[source]¶

static function_silent(out, eval_einsum, *args)[source]¶

static function_timer(out, eval_einsum, *args)[source]¶

get_eval_shape(*args, **kwargs)[source]¶

get_fargs(*args, **kwargs)[source]¶

get_normals(arg)[source]¶

get_operands(diff_var)[source]¶

get_paths(expressions, operands)[source]¶

layout_letters = 'cqgvd0'¶

make_function(texpr, *args, diff_var=None)[source]¶

set_backend(backend='numpy', optimize=True, layout=None, **kwargs)[source]¶

set_verbosity(verbosity=None)[source]¶

verbosity = 0¶

class sfepy.terms.terms_multilinear.EVolumeDotTerm(*args, **kwargs)[source]¶

Volume $L^2(\Omega)$ weighted dot product for both scalar and vector fields. Can be evaluated. Can use derivatives.

Definition

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

Call signature

de_volume_dot	`(opt_material, virtual, state)`
	`(opt_material, parameter_1, parameter_2)`

Arguments 1

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

Arguments 2

material : $c$ or $\ull{M}$ (optional)
parameter_1 : $p$ or $\ul{u}$
parameter_2 : $r$ or $\ul{w}$

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

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

get_function(mat, virtual, state, mode=None, term_mode=None, diff_var=None, **kwargs)[source]¶

modes = ('weak', 'eval')¶

name = 'de_volume_dot'¶

class sfepy.terms.terms_multilinear.ExpressionArg(**kwargs)[source]¶

static from_term_arg(arg, term, cache)[source]¶

get_dofs(cache, expr_cache, oname)[source]¶

class sfepy.terms.terms_multilinear.ExpressionBuilder(n_add, cache)[source]¶

add_arg_dofs(iin, ein, name, n_components, iia=None)[source]¶

add_bf(iin, ein, name, cell_dependent=False)[source]¶

add_bfg(iin, ein, name)[source]¶

add_constant(name, cname)[source]¶

add_eye(iic, ein, name, iia=None)[source]¶

add_material_arg(arg, ii, ein)[source]¶

add_psg(iic, ein, name, iia=None)[source]¶

add_state_arg(arg, ii, ein, modifier, diff_var)[source]¶

add_virtual_arg(arg, ii, ein, modifier)[source]¶

apply_layout(layout, operands, defaults=None, verbosity=0)[source]¶

build(texpr, *args, diff_var=None)[source]¶

get_expressions(subscripts=None)[source]¶

static join_subscripts(subscripts, out_subscripts)[source]¶

letters = 'defgh'¶

make_eye(size)[source]¶

make_psg(dim)[source]¶

print_shapes(subscripts, operands)[source]¶

transform(subscripts, operands, transformation='loop', **kwargs)[source]¶

sfepy.terms.terms_multilinear.append_all(seqs, item, ii=None)[source]¶

sfepy.terms.terms_multilinear.collect_modifiers(modifiers)[source]¶

sfepy.terms.terms_multilinear.find_free_indices(indices)[source]¶

sfepy.terms.terms_multilinear.get_einsum_ops(eargs, ebuilder, expr_cache)[source]¶

sfepy.terms.terms_multilinear.get_loop_indices(subs, loop_index)[source]¶

sfepy.terms.terms_multilinear.get_output_shape(out_subscripts, subscripts, operands)[source]¶

sfepy.terms.terms_multilinear.get_sizes(indices, operands)[source]¶

sfepy.terms.terms_multilinear.get_slice_ops(subs, ops, loop_index)[source]¶

sfepy.terms.terms_multilinear.parse_term_expression(texpr)[source]¶