def __init__(self, anchor, normal, bounds):
Struct.__init__(self, anchor=nm.array(anchor, dtype=nm.float64),
bounds=nm.asarray(bounds, dtype=nm.float64))
self.normal = nm.asarray(normal, dtype=nm.float64)
norm = nm.linalg.norm
self.normal /= norm(self.normal)
e3 = [0.0, 0.0, 1.0]
dd = nm.dot(e3, self.normal)
rot_angle = nm.arccos(dd)
if nm.abs(rot_angle) < 1e-14:
mtx = nm.eye(3, dtype=nm.float64)
bounds2d = self.bounds[:, :2]
else:
rot_axis = nm.cross([0.0, 0.0, 1.0], self.normal)
mtx = la.make_axis_rotation_matrix(rot_axis, rot_angle)
mm = la.insert_strided_axis(mtx, 0, self.bounds.shape[0])
rbounds = la.dot_sequences(mm, self.bounds)
bounds2d = rbounds[:, :2]
assert_(nm.allclose(nm.dot(mtx, self.normal), e3,
rtol=0.0, atol=1e-12))
self.adotn = nm.dot(self.anchor, self.normal)
self.rot_angle = rot_angle
self.mtx = mtx
self.bounds2d = bounds2d
def get_fargs(self, kappa, virtual, state,
mode=None, term_mode=None, diff_var=None, **kwargs):
from sfepy.discrete.variables import create_adof_conn, expand_basis
geo, _ = self.get_mapping(state)
n_el, n_qp, dim, n_en, n_c = self.get_data_shape(virtual)
ebf = expand_basis(geo.bf, dim)
aux = nm.einsum('i,...ij->...j', kappa, ebf)[0, :, None, :]
kebf = insert_strided_axis(aux, 0, n_el)
if diff_var is None:
econn = state.field.get_econn('volume', self.region)
adc = create_adof_conn(nm.arange(state.n_dof, dtype=nm.int32),
econn, n_c, 0)
vals = state()[adc]
aux = dot_sequences(kebf, vals[:, None, :, None])
out_qp = dot_sequences(kebf, aux, 'ATB')
fmode = 0
else:
out_qp = dot_sequences(kebf, kebf, 'ATB')
fmode = 1
return out_qp, geo, fmode
def get_coors(self, ig=None):
"""
Get the coordinates of vertices of unique facets in group `ig`.
Parameters
----------
ig : int, optional
The element group. If None, the coordinates for all groups
are returned, filled with zeros at places of missing
vertices, i.e. where facets having less then the full number
of vertices (`n_v`) are.
Returns
-------
coors : array
The coordinates in an array of shape `(n_f, n_v, dim)`.
uid : array
The unique ids of facets in the order of `coors`.
"""
cc = self.domain.get_mesh_coors()
if ig is None:
uid, ii = unique(self.uid_i, return_index=True)
facets = self.facets[ii]
aux = insert_strided_axis(facets, 2, cc.shape[1])
coors = nm.where(aux >= 0, cc[facets], 0.0)
else:
uid_i = self.uid_i[self.indx[ig]]
uid, ii = unique(uid_i, return_index=True)
coors = cc[self.facets[ii, :self.n_fps[ig]]]
return coors, uid
def get_fargs(self,
kappa,
virtual,
state,
mode=None,
term_mode=None,
diff_var=None,
**kwargs):
from sfepy.discrete.variables import create_adof_conn, expand_basis
geo, _ = self.get_mapping(state)
n_el, n_qp, dim, n_en, n_c = self.get_data_shape(virtual)
ebf = expand_basis(geo.bf, dim)
aux = nm.einsum('i,...ij->...j', kappa, ebf)[0, :, None, :]
kebf = insert_strided_axis(aux, 0, n_el)
if diff_var is None:
econn = state.field.get_econn('volume', self.region)
adc = create_adof_conn(nm.arange(state.n_dof, dtype=nm.int32),
econn, n_c, 0)
vals = state()[adc]
aux = dot_sequences(kebf, vals[:, None, :, None])
out_qp = dot_sequences(kebf, aux, 'ATB')
fmode = 0
else:
out_qp = dot_sequences(kebf, kebf, 'ATB')
fmode = 1
return out_qp, geo, fmode
def get_coors(self, ig=None):
"""
Get the coordinates of vertices of unique facets in group `ig`.
Parameters
----------
ig : int, optional
The element group. If None, the coordinates for all groups
are returned, filled with zeros at places of missing
vertices, i.e. where facets having less then the full number
of vertices (`n_v`) are.
Returns
-------
coors : array
The coordinates in an array of shape `(n_f, n_v, dim)`.
uid : array
The unique ids of facets in the order of `coors`.
"""
cc = self.domain.get_mesh_coors()
if ig is None:
uid, ii = unique(self.uid_i, return_index=True)
facets = self.facets[ii]
aux = insert_strided_axis(facets, 2, cc.shape[1])
coors = nm.where(aux >= 0, cc[facets], 0.0)
else:
uid_i = self.uid_i[self.indx[ig]]
uid, ii = unique(uid_i, return_index=True)
coors = cc[self.facets[ii,:self.n_fps[ig]]]
return coors, uid
def get_fargs(self,
kappa,
kvar,
dvar,
mode=None,
term_mode=None,
diff_var=None,
**kwargs):
from sfepy.discrete.variables import create_adof_conn, expand_basis
geo, _ = self.get_mapping(dvar)
n_el, n_qp, dim, n_en, n_c = self.get_data_shape(kvar)
ebf = expand_basis(geo.bf, dim)
aux = nm.einsum('i,...ij->...j', kappa, ebf)[0, :, None, :]
kebf = insert_strided_axis(aux, 0, n_el)
div_bf = geo.bfg
div_bf = div_bf.reshape((n_el, n_qp, 1, dim * n_en))
div_bf = nm.ascontiguousarray(div_bf)
if diff_var is None:
avar = dvar if self.mode == 'kd' else kvar
econn = avar.field.get_econn('volume', self.region)
adc = create_adof_conn(nm.arange(avar.n_dof, dtype=nm.int32),
econn, n_c, 0)
vals = avar()[adc]
if self.mode == 'kd':
aux = dot_sequences(div_bf, vals[:, None, :, None])
out_qp = dot_sequences(kebf, aux, 'ATB')
else:
aux = dot_sequences(kebf, vals[:, None, :, None])
out_qp = dot_sequences(div_bf, aux, 'ATB')
fmode = 0
else:
if self.mode == 'kd':
out_qp = dot_sequences(kebf, div_bf, 'ATB')
else:
out_qp = dot_sequences(div_bf, kebf, 'ATB')
fmode = 1
return out_qp, geo, fmode
def test_tensors(self):
import numpy as nm
from sfepy.linalg import dot_sequences, insert_strided_axis
ok = True
a = nm.arange(1, 10).reshape(3, 3)
b = nm.arange(9, 0, -1).reshape(3, 3)
dab = nm.dot(a, b)
dabt = nm.dot(a, b.T)
datb = nm.dot(a.T, b)
datbt = nm.dot(a.T, b.T)
sa = insert_strided_axis(a, 0, 10)
sb = insert_strided_axis(b, 0, 10)
dsab = dot_sequences(sa, sb, mode='AB')
_ok = nm.allclose(dab[None, ...], dsab, rtol=0.0, atol=1e-14)
self.report('dot_sequences AB: %s' % _ok)
ok = ok and _ok
dsabt = dot_sequences(sa, sb, mode='ABT')
_ok = nm.allclose(dabt[None, ...], dsabt, rtol=0.0, atol=1e-14)
self.report('dot_sequences ABT: %s' % _ok)
ok = ok and _ok
dsatb = dot_sequences(sa, sb, mode='ATB')
_ok = nm.allclose(datb[None, ...], dsatb, rtol=0.0, atol=1e-14)
self.report('dot_sequences ATB: %s' % _ok)
ok = ok and _ok
dsatbt = dot_sequences(sa, sb, mode='ATBT')
_ok = nm.allclose(datbt[None, ...], dsatbt, rtol=0.0, atol=1e-14)
self.report('dot_sequences ATBT: %s' % _ok)
ok = ok and _ok
return ok
def test_tensors(self):
import numpy as nm
from sfepy.linalg import dot_sequences, insert_strided_axis
ok = True
a = nm.arange(1, 10).reshape(3, 3)
b = nm.arange(9, 0, -1).reshape(3, 3)
dab = nm.dot(a, b)
dabt = nm.dot(a, b.T)
datb = nm.dot(a.T, b)
datbt = nm.dot(a.T, b.T)
sa = insert_strided_axis(a, 0, 10)
sb = insert_strided_axis(b, 0, 10)
dsab = dot_sequences(sa, sb, mode='AB')
_ok = nm.allclose(dab[None, ...], dsab, rtol=0.0, atol=1e-14)
self.report('dot_sequences AB: %s' % _ok)
ok = ok and _ok
dsabt = dot_sequences(sa, sb, mode='ABT')
_ok = nm.allclose(dabt[None, ...], dsabt, rtol=0.0, atol=1e-14)
self.report('dot_sequences ABT: %s' % _ok)
ok = ok and _ok
dsatb = dot_sequences(sa, sb, mode='ATB')
_ok = nm.allclose(datb[None, ...], dsatb, rtol=0.0, atol=1e-14)
self.report('dot_sequences ATB: %s' % _ok)
ok = ok and _ok
dsatbt = dot_sequences(sa, sb, mode='ATBT')
_ok = nm.allclose(datbt[None, ...], dsatbt, rtol=0.0, atol=1e-14)
self.report('dot_sequences ATBT: %s' % _ok)
ok = ok and _ok
return ok
def _eval_base(self,
coors,
diff=False,
ori=None,
suppress_errors=False,
eps=1e-15):
"""
See PolySpace.eval_base().
"""
from extmods.lobatto_bases import eval_lobatto_tensor_product as ev
c_min, c_max = self.bbox[:, 0]
base = ev(coors, self.nodes, c_min, c_max, self.order, diff)
if ori is not None:
ebase = nm.tile(base, (ori.shape[0], 1, 1, 1))
if self.edge_indx.shape[0]:
# Orient edge functions.
ie, ii = nm.where(ori[:, self.edge_indx] == 1)
ii = self.edge_indx[ii]
ebase[ie, :, :, ii] *= -1.0
if self.face_indx.shape[0]:
# Orient face functions.
fori = ori[:, self.face_indx]
# ... normal axis order
ie, ii = nm.where((fori == 1) | (fori == 2))
ii = self.face_indx[ii]
ebase[ie, :, :, ii] *= -1.0
# ... swapped axis order
sbase = ev(coors, self.sfnodes, c_min, c_max, self.order, diff)
sbase = insert_strided_axis(sbase, 0, ori.shape[0])
# ...overwrite with swapped axes basis.
ie, ii = nm.where(fori >= 4)
ii2 = self.face_indx[ii]
ebase[ie, :, :, ii2] = sbase[ie, :, :, ii]
# ...deal with orientation.
ie, ii = nm.where((fori == 5) | (fori == 6))
ii = self.face_indx[ii]
ebase[ie, :, :, ii] *= -1.0
base = ebase
return base
def _eval_base(self, coors, diff=False, ori=None,
suppress_errors=False, eps=1e-15):
"""
See PolySpace.eval_base().
"""
from .extmods.lobatto_bases import eval_lobatto_tensor_product as ev
c_min, c_max = self.bbox[:, 0]
base = ev(coors, self.nodes, c_min, c_max, self.order, diff)
if ori is not None:
ebase = nm.tile(base, (ori.shape[0], 1, 1, 1))
if self.edge_indx.shape[0]:
# Orient edge functions.
ie, ii = nm.where(ori[:, self.edge_indx] == 1)
ii = self.edge_indx[ii]
ebase[ie, :, :, ii] *= -1.0
if self.face_indx.shape[0]:
# Orient face functions.
fori = ori[:, self.face_indx]
# ... normal axis order
ie, ii = nm.where((fori == 1) | (fori == 2))
ii = self.face_indx[ii]
ebase[ie, :, :, ii] *= -1.0
# ... swapped axis order
sbase = ev(coors, self.sfnodes, c_min, c_max, self.order, diff)
sbase = insert_strided_axis(sbase, 0, ori.shape[0])
# ...overwrite with swapped axes basis.
ie, ii = nm.where(fori >= 4)
ii2 = self.face_indx[ii]
ebase[ie, :, :, ii2] = sbase[ie, :, :, ii]
# ...deal with orientation.
ie, ii = nm.where((fori == 5) | (fori == 6))
ii = self.face_indx[ii]
ebase[ie, :, :, ii] *= -1.0
base = ebase
return base
def get_fargs(self, kappa, kvar, dvar,
mode=None, term_mode=None, diff_var=None, **kwargs):
from sfepy.discrete.variables import create_adof_conn, expand_basis
geo, _ = self.get_mapping(dvar)
n_el, n_qp, dim, n_en, n_c = self.get_data_shape(kvar)
ebf = expand_basis(geo.bf, dim)
aux = nm.einsum('i,...ij->...j', kappa, ebf)[0, :, None, :]
kebf = insert_strided_axis(aux, 0, n_el)
div_bf = geo.bfg
div_bf = div_bf.reshape((n_el, n_qp, 1, dim * n_en))
div_bf = nm.ascontiguousarray(div_bf)
if diff_var is None:
avar = dvar if self.mode == 'kd' else kvar
econn = avar.field.get_econn('volume', self.region)
adc = create_adof_conn(nm.arange(avar.n_dof, dtype=nm.int32),
econn, n_c, 0)
vals = avar()[adc]
if self.mode == 'kd':
aux = dot_sequences(div_bf, vals[:, None, :, None])
out_qp = dot_sequences(kebf, aux, 'ATB')
else:
aux = dot_sequences(kebf, vals[:, None, :, None])
out_qp = dot_sequences(div_bf, aux, 'ATB')
fmode = 0
else:
if self.mode == 'kd':
out_qp = dot_sequences(kebf, div_bf, 'ATB')
else:
out_qp = dot_sequences(div_bf, kebf, 'ATB')
fmode = 1
return out_qp, geo, fmode
def mask_points(self, points):
mm = la.insert_strided_axis(self.mtx, 0, points.shape[0])
points2d = la.dot_sequences(mm, points)[:, :2]
return la.flag_points_in_polygon2d(self.bounds2d, points2d)
def describe_deformation(el_disps, bfg):
"""
Describe deformation of a thin incompressible 2D membrane in 3D
space, composed of flat finite element faces.
The coordinate system of each element (face), i.e. the membrane
mid-surface, should coincide with the `x`, `y` axes of the `x-y`
plane.
Parameters
----------
el_disps : array
The displacements of element nodes, shape `(n_el, n_ep, dim)`.
bfg : array
The in-plane base function gradients, shape `(n_el, n_qp, dim-1,
n_ep)`.
Returns
-------
mtx_c ; array
The in-plane right Cauchy-Green deformation tensor
:math:`C_{ij}`, :math:`i, j = 1, 2`.
c33 : array
The component :math:`C_{33}` computed from the incompressibility
condition.
mtx_b : array
The discrete Green strain variation operator.
"""
sh = bfg.shape
n_ep = sh[3]
dim = el_disps.shape[2]
sym2 = dim2sym(dim-1)
# Repeat el_disps by number of quadrature points.
el_disps_qp = insert_strided_axis(el_disps, 1, bfg.shape[1])
# Transformed (in-plane) displacement gradient with
# shape (n_el, n_qp, 2 (-> a), 3 (-> i)), du_i/dX_a.
du = dot_sequences(bfg, el_disps_qp)
# Deformation gradient F w.r.t. in plane coordinates.
# F_{ia} = dx_i / dX_a,
# a \in {1, 2} (rows), i \in {1, 2, 3} (columns).
mtx_f = du + nm.eye(dim - 1, dim, dtype=du.dtype)
# Right Cauchy-Green deformation tensor C.
# C_{ab} = F_{ka} F_{kb}, a, b \in {1, 2}.
mtx_c = dot_sequences(mtx_f, mtx_f, 'ABT')
# C_33 from incompressibility.
c33 = 1.0 / (mtx_c[..., 0, 0] * mtx_c[..., 1, 1]
- mtx_c[..., 0, 1]**2)
# Discrete Green strain variation operator.
mtx_b = nm.empty((sh[0], sh[1], sym2, dim * n_ep), dtype=nm.float64)
mtx_b[..., 0, 0*n_ep:1*n_ep] = bfg[..., 0, :] * mtx_f[..., 0, 0:1]
mtx_b[..., 0, 1*n_ep:2*n_ep] = bfg[..., 0, :] * mtx_f[..., 0, 1:2]
mtx_b[..., 0, 2*n_ep:3*n_ep] = bfg[..., 0, :] * mtx_f[..., 0, 2:3]
mtx_b[..., 1, 0*n_ep:1*n_ep] = bfg[..., 1, :] * mtx_f[..., 1, 0:1]
mtx_b[..., 1, 1*n_ep:2*n_ep] = bfg[..., 1, :] * mtx_f[..., 1, 1:2]
mtx_b[..., 1, 2*n_ep:3*n_ep] = bfg[..., 1, :] * mtx_f[..., 1, 2:3]
mtx_b[..., 2, 0*n_ep:1*n_ep] = bfg[..., 1, :] * mtx_f[..., 0, 0:1] \
+ bfg[..., 0, :] * mtx_f[..., 1, 0:1]
mtx_b[..., 2, 1*n_ep:2*n_ep] = bfg[..., 0, :] * mtx_f[..., 1, 1:2] \
+ bfg[..., 1, :] * mtx_f[..., 0, 1:2]
mtx_b[..., 2, 2*n_ep:3*n_ep] = bfg[..., 0, :] * mtx_f[..., 1, 2:3] \
+ bfg[..., 1, :] * mtx_f[..., 0, 2:3]
return mtx_c, c33, mtx_b
def describe_deformation(el_disps, bfg):
"""
Describe deformation of a thin incompressible 2D membrane in 3D
space, composed of flat finite element faces.
The coordinate system of each element (face), i.e. the membrane
mid-surface, should coincide with the `x`, `y` axes of the `x-y`
plane.
Parameters
----------
el_disps : array
The displacements of element nodes, shape `(n_el, n_ep, dim)`.
bfg : array
The in-plane base function gradients, shape `(n_el, n_qp, dim-1,
n_ep)`.
Returns
-------
mtx_c ; array
The in-plane right Cauchy-Green deformation tensor
:math:`C_{ij}`, :math:`i, j = 1, 2`.
c33 : array
The component :math:`C_{33}` computed from the incompressibility
condition.
mtx_b : array
The discrete Green strain variation operator.
"""
sh = bfg.shape
n_ep = sh[3]
dim = el_disps.shape[2]
sym2 = dim2sym(dim-1)
# Repeat el_disps by number of quadrature points.
el_disps_qp = insert_strided_axis(el_disps, 1, bfg.shape[1])
# Transformed (in-plane) displacement gradient with
# shape (n_el, n_qp, 2 (-> a), 3 (-> i)), du_i/dX_a.
du = dot_sequences(bfg, el_disps_qp)
# Deformation gradient F w.r.t. in plane coordinates.
# F_{ia} = dx_i / dX_a,
# a \in {1, 2} (rows), i \in {1, 2, 3} (columns).
mtx_f = du + nm.eye(dim - 1, dim, dtype=du.dtype)
# Right Cauchy-Green deformation tensor C.
# C_{ab} = F_{ka} F_{kb}, a, b \in {1, 2}.
mtx_c = dot_sequences(mtx_f, mtx_f, 'ABT')
# C_33 from incompressibility.
c33 = 1.0 / (mtx_c[..., 0, 0] * mtx_c[..., 1, 1]
- mtx_c[..., 0, 1]**2)
# Discrete Green strain variation operator.
mtx_b = nm.empty((sh[0], sh[1], sym2, dim * n_ep), dtype=nm.float64)
mtx_b[..., 0, 0*n_ep:1*n_ep] = bfg[..., 0, :] * mtx_f[..., 0, 0:1]
mtx_b[..., 0, 1*n_ep:2*n_ep] = bfg[..., 0, :] * mtx_f[..., 0, 1:2]
mtx_b[..., 0, 2*n_ep:3*n_ep] = bfg[..., 0, :] * mtx_f[..., 0, 2:3]
mtx_b[..., 1, 0*n_ep:1*n_ep] = bfg[..., 1, :] * mtx_f[..., 1, 0:1]
mtx_b[..., 1, 1*n_ep:2*n_ep] = bfg[..., 1, :] * mtx_f[..., 1, 1:2]
mtx_b[..., 1, 2*n_ep:3*n_ep] = bfg[..., 1, :] * mtx_f[..., 1, 2:3]
mtx_b[..., 2, 0*n_ep:1*n_ep] = bfg[..., 1, :] * mtx_f[..., 0, 0:1] \
+ bfg[..., 0, :] * mtx_f[..., 1, 0:1]
mtx_b[..., 2, 1*n_ep:2*n_ep] = bfg[..., 0, :] * mtx_f[..., 1, 1:2] \
+ bfg[..., 1, :] * mtx_f[..., 0, 1:2]
mtx_b[..., 2, 2*n_ep:3*n_ep] = bfg[..., 0, :] * mtx_f[..., 1, 2:3] \
+ bfg[..., 1, :] * mtx_f[..., 0, 2:3]
return mtx_c, c33, mtx_b
