sfepy.terms.terms_surface module¶

class sfepy.terms.terms_surface.ContactPlaneTerm(*args, **kwargs)[source]¶

Small deformation elastic contact plane term with penetration penalty.

The plane is given by an anchor point $\ul{A}$ and a normal $\ul{n}$ . The contact occurs in points that orthogonally project onto the plane into a polygon given by orthogonal projections of boundary points $\{\ul{B}_i\}$ , $i = 1, \dots, N_B$ on the plane. In such points, a penetration distance $d(\ul{u}) = (\ul{X} + \ul{u} - \ul{A}, \ul{n})$ is computed, and a force $f(d(\ul{u})) \ul{n}$ is applied. The force depends on the non-negative parameters $k$ (stiffness) and $f_0$ (force at zero penetration):

If $f_0 = 0$ :

$f(d) = 0 \mbox{ for } d \leq 0 \;, \\ f(d) = k d \mbox{ for } d > 0 \;.$
If $f_0 > 0$ :

$f(d) = 0 \mbox{ for } d \leq -\frac{2 r_0}{k} \;, \\ f(d) = \frac{k^2}{4 r_0} d^2 + k d + r_0 \mbox{ for } -\frac{2 r_0}{k} < d \leq 0 \;, \\ f(d) = k d + f_0 \mbox{ for } d > 0 \;.$

In this case the dependence $f(d)$ is smooth, and a (small) force is applied even for (small) negative penetrations: $-\frac{2 r_0}{k} < d \leq 0$ .

Definition:

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

Call signature:

dw_contact_plane

(material_f, material_n, material_a, material_b, virtual, state)

Arguments:

material_f : $[k, f_0]$
material_n : $\ul{n}$ (special)
material_a : $\ul{A}$ (special)
material_b : $\{\ul{B}_i\}$ , $i = 1, \dots, N_B$ (special)
virtual : $\ul{v}$
state : $\ul{u}$

arg_shapes = {'material_a': '.: D', 'material_b': '.: N, D', 'material_f': '1, 2', 'material_n': '.: D', 'state': 'D', 'virtual': ('D', 'state')}¶

arg_types = ('material_f', 'material_n', 'material_a', 'material_b', 'virtual', 'state')¶

static function(out, force, normal, geo, fmode)[source]¶

geometries = ['3_4', '3_8']¶

get_fargs(force_pars, normal, anchor, bounds, virtual, state, mode=None, term_mode=None, diff_var=None, **kwargs)[source]¶

integration = 'facet'¶

name = 'dw_contact_plane'¶

static smooth_f(d, k, f0, a, eps, diff)[source]¶

class sfepy.terms.terms_surface.ContactSphereTerm(*args, **kwargs)[source]¶

Small deformation elastic contact sphere term with penetration penalty.

The sphere is given by a centre point $\ul{C}$ and a radius $R$ . The contact occurs in points that are closer to $\ul{C}$ than $R$ . In such points, a penetration distance $d(\ul{u}) = R - ||\ul{X} + \ul{u} - \ul{C}||$ is computed, and a force $f(d(\ul{u})) \ul{n}(\ul{u})$ is applied, where $\ul{n}(\ul{u}) = (\ul{X} + \ul{u} - \ul{C}) / ||\ul{X} + \ul{u} - \ul{C}||$ . The force depends on the non-negative parameters $k$ (stiffness) and $f_0$ (force at zero penetration):

If $f_0 = 0$ :

$f(d) = 0 \mbox{ for } d \leq 0 \;, \\ f(d) = k d \mbox{ for } d > 0 \;.$
If $f_0 > 0$ :

$f(d) = 0 \mbox{ for } d \leq -\frac{2 r_0}{k} \;, \\ f(d) = \frac{k^2}{4 r_0} d^2 + k d + r_0 \mbox{ for } -\frac{2 r_0}{k} < d \leq 0 \;, \\ f(d) = k d + f_0 \mbox{ for } d > 0 \;.$

In this case the dependence $f(d)$ is smooth, and a (small) force is applied even for (small) negative penetrations: $-\frac{2 r_0}{k} < d \leq 0$ .

Definition:

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

Call signature:

dw_contact_sphere

(material_f, material_c, material_r, virtual, state)

Arguments:

material_f : $[k, f_0]$
material_c : $\ul{C}$ (special)
material_r : $R$ (special)
virtual : $\ul{v}$
state : $\ul{u}$

arg_shapes = {'material_c': '.: D', 'material_f': '1, 2', 'material_r': '.: 1', 'state': 'D', 'virtual': ('D', 'state')}¶

arg_types = ('material_f', 'material_c', 'material_r', 'virtual', 'state')¶

static function(out, force, normals, fd, geo, fmode)[source]¶

geometries = ['3_4', '3_8']¶

get_fargs(force_pars, centre, radius, virtual, state, mode=None, term_mode=None, diff_var=None, **kwargs)[source]¶

integration = 'facet'¶

name = 'dw_contact_sphere'¶

class sfepy.terms.terms_surface.LinearTractionTerm(name, arg_str, integral, region, **kwargs)[source]¶

Linear traction forces, where, depending on dimension of ‘material’ argument, $\ull{\sigma} \cdot \ul{n}$ is $\bar{p} \ull{I} \cdot \ul{n}$ for a given scalar pressure, $\ul{f}$ for a traction vector, and itself for a stress tensor.

The material parameter can have one of the following shapes: 1 or (1, 1), (D, 1), (S, 1) in all modes, or (D, D) in the eval mode only. The symmetric tensor storage (S, 1) is as follows: in 3D S = 6 and the indices ordered as $[11, 22, 33, 12, 13, 23]$ , in 2D S = 3 and the indices ordered as $[11, 22, 12]$ .

Definition:

$\int_{\Gamma} \ul{v} \cdot \ull{\sigma} \cdot \ul{n}, \int_{\Gamma} \ul{v} \cdot \ul{n},$

Call signature:

dw_surface_ltr	`(opt_material, virtual)`
	`(opt_material, parameter)`

Arguments:

material : $\ull{\sigma}$
virtual : $\ul{v}$

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

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

static d_fun(out, traction, val, sg)[source]¶

get_eval_shape(traction, virtual, mode=None, term_mode=None, diff_var=None, **kwargs)[source]¶

get_fargs(traction, virtual, mode=None, term_mode=None, diff_var=None, **kwargs)[source]¶

integration = 'facet'¶

modes = ('weak', 'eval')¶

name = 'dw_surface_ltr'¶

set_arg_types()[source]¶

class sfepy.terms.terms_surface.SDLinearTractionTerm(name, arg_str, integral, region, **kwargs)[source]¶

Sensitivity of the linear traction term.

Definition:

$\int_{\Gamma} \ul{v} \cdot (\ull{\sigma}\, \ul{n}), \int_{\Gamma} \ul{v} \cdot \ul{n},$

Call signature:

ev_sd_surface_ltr

(opt_material, parameter, parameter_mv)

Arguments:

material : $\ull{\sigma}$
parameter : $\ul{v}$

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

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

static d_fun(out, traction, val, grad_mv, div_mv, sg)[source]¶

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

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

integration = 'facet'¶

name = 'ev_sd_surface_ltr'¶

set_arg_types()[source]¶

class sfepy.terms.terms_surface.SDSufaceIntegrateTerm(name, arg_str, integral, region, **kwargs)[source]¶

Sensitivity of scalar traction.

Definition:

$\int_{\Gamma} p \nabla \cdot \ul{\Vcal}$

Call signature:

ev_sd_surface_integrate

(parameter, parameter_mv)

Arguments:

parameter : $p$
parameter_mv : $\ul{\Vcal}$

arg_shapes = {'parameter': 1, 'parameter_mv': 'D'}¶

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

static function(out, val_p, div_v, sg)[source]¶

get_eval_shape(par, par_v, mode=None, term_mode=None, diff_var=None, **kwargs)[source]¶

get_fargs(par, par_v, mode=None, term_mode=None, diff_var=None, **kwargs)[source]¶

integration = 'facet'¶

name = 'ev_sd_surface_integrate'¶

class sfepy.terms.terms_surface.SufaceNormalDotTerm(name, arg_str, integral, region, **kwargs)[source]¶

“Scalar traction” term, (weak form).

Definition:

$\int_{\Gamma} q \ul{c} \cdot \ul{n}$

Call signature:

dw_surface_ndot	`(material, virtual)`
	`(material, parameter)`

Arguments:

material : $\ul{c}$
virtual : $q$

arg_shapes = {'material': 'D, 1', 'parameter': 1, 'virtual': (1, None)}¶

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

static d_fun(out, material, val, sg)[source]¶

static dw_fun(out, material, bf, sg)[source]¶

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

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

integration = 'facet'¶

modes = ('weak', 'eval')¶

name = 'dw_surface_ndot'¶

set_arg_types()[source]¶

class sfepy.terms.terms_surface.SurfaceJumpTerm(name, arg_str, integral, region, **kwargs)[source]¶

Interface jump condition.

Definition:

$\int_{\Gamma} c\, q (p_1 - p_2)$

Call signature:

dw_jump

(opt_material, virtual, state_1, state_2)

Arguments:

material : $c$
virtual : $q$
state_1 : $p_1$
state_2 : $p_2$

arg_shapes = [{'opt_material': '1, 1', 'state_1': 1, 'state_2': 1, 'virtual': (1, None)}, {'opt_material': None}]¶

arg_types = ('opt_material', 'virtual', 'state_1', 'state_2')¶

static function(out, jump, mul, bf1, bf2, sg, fmode)[source]¶

get_fargs(coef, virtual, state1, state2, mode=None, term_mode=None, diff_var=None, **kwargs)[source]¶

integration = 'facet'¶

name = 'dw_jump'¶