"""
Isogeometric analysis utilities.
Notes
-----
The functions :func:`compute_bezier_extraction_1d()` and
:func:`eval_nurbs_basis_tp()` implement the algorithms described in [1].
[1] Michael J. Borden, Michael A. Scott, John A. Evans, Thomas J. R. Hughes:
Isogeometric finite element data structures based on Bezier extraction of
NURBS, Institute for Computational Engineering and Sciences, The University
of Texas at Austin, Austin, Texas, March 2010.
"""
from __future__ import absolute_import
import numpy as nm
from sfepy.base.base import assert_
from six.moves import range
def _get_knots_tuple(knots):
if isinstance(knots, nm.ndarray) and (knots.ndim == 1):
knots = (knots,)
elif not isinstance(knots, tuple):
raise ValueError('knots must be 1D array or a tuple of 1D arrays!')
return knots
[docs]def get_raveled_index(indices, shape):
"""
Get a global raveled index corresponding to nD indices into an array of the
given shape.
"""
return nm.ravel_multi_index(indices, shape)
[docs]def get_unraveled_indices(index, shape):
"""
Get nD indices into an array of the given shape corresponding to a global
raveled index.
"""
return nm.unravel_index(index, shape)
[docs]def tensor_product(a, b):
"""
Compute tensor product of two 2D arrays with possibly different shapes. The
result has the form::
c = [[a00 b, a01 b, ...],
[a10 b, a11 b, ...],
...
... ]
"""
c = nm.empty((a.shape[0] * b.shape[0],
a.shape[1] * b.shape[1]), dtype=b.dtype)
n0 = b.shape[0]
n1 = b.shape[1]
for ir in range(a.shape[0]):
for ic in range(a.shape[1]):
c[n1 * ir : n1 * (ir + 1),
n0 * ic : n0 * (ic + 1)] = a[ir, ic] * b
return c
[docs]def create_connectivity_1d(n_el, knots, degree):
"""
Create connectivity arrays of 1D Bezier elements.
Parameters
----------
n_el : int
The number of elements.
knots : array
The knot vector.
degree : int
The basis degree.
Returns
-------
conn : array
The connectivity of the global NURBS basis.
bconn : array
The connectivity of the Bezier basis.
"""
# Get multiplicities of NURBS knots.
n_knots = len(knots)
mul = [0]
ii = degree + 1
while ii < (n_knots - degree - 1):
i0 = ii
while (ii < (n_knots - degree - 2)) and (knots[ii] == knots[ii + 1]):
ii += 1
mul.append(ii - i0 + 1)
ii += 1
mul = nm.array(mul)[:, None]
aux1 = nm.arange(degree + 1)[None, :]
conn = aux1 + nm.cumsum(mul, 0)
# Bezier basis knots have multiplicity equal to degree.
aux2 = nm.arange(n_el)[:, None]
bconn = aux1 + degree * aux2
return conn.astype(nm.int32), bconn.astype(nm.int32)
[docs]def create_connectivity(n_els, knots, degrees):
"""
Create connectivity arrays of nD Bezier elements.
Parameters
----------
n_els : sequence of ints
The number of elements in each parametric dimension.
knots : sequence of array or array
The knot vectors.
degrees : sequence of ints or int
The basis degrees in each parametric dimension.
Returns
-------
conn : array
The connectivity of the global NURBS basis.
bconn : array
The connectivity of the Bezier basis.
"""
if isinstance(degrees, int): degrees = [degrees]
degrees = nm.asarray(degrees)
knots = _get_knots_tuple(knots)
dim = len(n_els)
assert_(dim == len(degrees) == len(knots))
conns = []
bconns = []
n_gfuns = []
n_gbfuns = []
for ii, n_el in enumerate(n_els):
conn1d, bconn1d = create_connectivity_1d(n_el, knots[ii], degrees[ii])
conns.append(conn1d)
bconns.append(bconn1d)
n_gfuns.append(conn1d.max() + 1)
n_gbfuns.append(bconn1d.max() + 1)
n_el = nm.prod(n_els)
n_efuns = degrees + 1
n_efun = nm.prod(n_efuns)
if dim == 3:
def make_conn_3d(conns, n_gfuns):
conn = nm.empty((n_el, n_efun), dtype=nm.int32)
for ie0 in range(n_els[0]):
c0 = conns[0][ie0]
for ie1 in range(n_els[1]):
c1 = conns[1][ie1]
for ie2 in range(n_els[2]):
c2 = conns[2][ie2]
ie = get_raveled_index([ie0, ie1, ie2], n_els)
for il0 in range(n_efuns[0]):
cl0 = c0[il0]
for il1 in range(n_efuns[1]):
cl1 = c1[il1]
for il2 in range(n_efuns[2]):
cl2 = c2[il2]
iloc = get_raveled_index([il0, il1, il2],
n_efuns)
ig = get_raveled_index([cl0, cl1, cl2],
n_gfuns)
conn[ie, iloc] = ig
return conn
conn = make_conn_3d(conns, n_gfuns)
bconn = make_conn_3d(bconns, n_gbfuns)
elif dim == 2:
def make_conn_2d(conns, n_gfuns):
conn = nm.empty((n_el, n_efun), dtype=nm.int32)
for ie0 in range(n_els[0]):
c0 = conns[0][ie0]
for ie1 in range(n_els[1]):
c1 = conns[1][ie1]
ie = get_raveled_index([ie0, ie1], n_els)
for il0 in range(n_efuns[0]):
cl0 = c0[il0]
for il1 in range(n_efuns[1]):
cl1 = c1[il1]
iloc = get_raveled_index([il0, il1], n_efuns)
ig = get_raveled_index([cl0, cl1], n_gfuns)
conn[ie, iloc] = ig
return conn
conn = make_conn_2d(conns, n_gfuns)
bconn = make_conn_2d(bconns, n_gbfuns)
else:
conn = conns[0]
bconn = bconns[0]
return conn, bconn
[docs]def compute_bezier_control(control_points, weights, ccs, conn, bconn):
"""
Compute the control points and weights of the Bezier mesh.
Parameters
----------
control_points : array
The NURBS control points.
weights : array
The NURBS weights.
ccs : list of 2D arrays
The combined element extraction operators.
conn : array
The connectivity of the global NURBS basis.
bconn : array
The connectivity of the Bezier basis.
Returns
-------
bezier_control_points : array
The control points of the Bezier mesh.
bezier_weights : array
The weights of the Bezier mesh.
"""
n_bpoints = bconn.max() + 1
dim = control_points.shape[1]
bezier_control_points = nm.zeros((n_bpoints, dim), dtype=nm.float64)
bezier_weights = nm.zeros(n_bpoints, dtype=nm.float64)
for ie, ec in enumerate(conn):
cc = ccs[ie]
bec = bconn[ie]
ew = weights[ec]
ecp = control_points[ec]
bew = nm.dot(cc.T, ew)
becp = (1.0 / bew[:, None]) * nm.dot(cc.T, ew[:, None] * ecp)
bezier_control_points[bec] = becp
bezier_weights[bec] = bew
return bezier_control_points, bezier_weights
[docs]def get_bezier_topology(bconn, degrees):
"""
Get a topology connectivity corresponding to the Bezier mesh connectivity.
In the referenced Bezier control points the Bezier mesh is interpolatory.
Parameters
----------
bconn : array
The connectivity of the Bezier basis.
degrees : sequence of ints or int
The basis degrees in each parametric dimension.
Returns
-------
tconn : array
The topology connectivity (corner nodes, or vertices, of Bezier
elements) with vertex ordering suitable for a FE mesh.
"""
shape = nm.asarray(degrees) + 1
dim = len(shape)
ii = nm.arange(bconn.shape[1]).reshape(shape)
if dim == 3:
corners = [ii[0, 0, 0], ii[-1, 0, 0], ii[-1, -1, 0], ii[0, -1, 0],
ii[0, 0, -1], ii[-1, 0, -1], ii[-1, -1, -1], ii[0, -1, -1]]
elif dim == 2:
corners = [ii[0, 0], ii[-1, 0], ii[-1, -1], ii[0, -1]]
else:
corners = [ii[0], ii[-1]]
tconn = bconn[:, corners]
return tconn
[docs]def get_patch_box_regions(n_els, degrees):
"""
Get box regions of Bezier topological mesh in terms of element corner
vertices of Bezier mesh.
Parameters
----------
n_els : sequence of ints
The number of elements in each parametric dimension.
degrees : sequence of ints or int
Polynomial degrees in each parametric dimension.
Returns
-------
regions : dict
The Bezier mesh vertices of box regions.
"""
if isinstance(degrees, int): degrees = [degrees]
degrees = nm.asarray(degrees)
n_els = nm.asarray(n_els)
dim = len(n_els)
shape = n_els * degrees + 1
regions = {}
if dim == 3:
aux0 = nm.arange(0, shape[2], degrees[2], dtype=nm.uint32)
aux1 = nm.arange(0, shape[2] * shape[1], shape[2] * degrees[1],
dtype=nm.uint32)
aux2 = nm.arange(0, shape[2] * shape[1] * shape[0],
shape[2] * shape[1] * degrees[0], dtype=nm.uint32)
aux01 = (aux0[None, :] + aux1[:, None]).ravel()
aux02 = (aux0[None, :] + aux2[:, None]).ravel()
aux12 = (aux1[None, :] + aux2[:, None]).ravel()
regions.update({
'xi00' : aux01,
'xi01' : aux01 + shape[2] * shape[1] * (shape[0] - 1),
'xi10' : aux02,
'xi11' : aux02 + shape[2] * (shape[1] - 1),
'xi20' : aux12,
'xi21' : aux12 + shape[2] - 1,
})
elif dim == 2:
aux0 = nm.arange(0, shape[1], degrees[1], dtype=nm.uint32)
aux1 = nm.arange(0, shape[1] * shape[0], shape[1] * degrees[0],
dtype=nm.uint32)
regions.update({
'xi00' : aux0,
'xi01' : aux0 + shape[1] * (shape[0] - 1),
'xi10' : aux1,
'xi11' : aux1 + shape[1] - 1,
})
else:
regions.update({
'xi00' : nm.array([0], dtype=nm.uint32),
'xi01' : nm.array([shape[0] - 1], dtype=nm.uint32),
})
return regions
[docs]def get_facet_axes(dim):
"""
For each reference Bezier element facet return the facet axes followed by
the remaining (perpendicular) axis, as well as the remaining axis
coordinate of the facet.
Parameters
----------
dim : int
The topological dimension.
Returns
-------
axes : array
The axes of the reference element facets.
coors : array
The remaining coordinate of the reference element facets.
"""
if dim == 3:
axes = [[1, 0, 2], [2, 1, 0], [0, 2, 1],
[0, 1, 2], [1, 2, 0], [2, 0, 1]]
coors = [0.0, 0.0, 0.0, 1.0, 1.0, 1.0]
elif dim == 2:
axes = [[0, 1], [1, 0], [0, 1], [1, 0]]
coors = [0.0, 1.0, 1.0, 0.0]
else:
axes = [[0]]
coors = None
return nm.array(axes, dtype=nm.uint32), nm.array(coors, dtype=nm.float64)
[docs]def get_surface_degrees(degrees):
"""
Get degrees of the NURBS patch surfaces.
Parameters
----------
degrees : sequence of ints or int
Polynomial degrees in each parametric dimension.
Returns
-------
sdegrees : list of arrays
The degrees of the patch surfaces, in the order of the reference Bezier
element facets.
"""
if isinstance(degrees, int): degrees = [degrees]
degrees = nm.asarray(degrees)
dim = len(degrees)
if dim == 3:
sdegrees = [(degrees[0], degrees[1]),
(degrees[1], degrees[2]),
(degrees[0], degrees[2]),
(degrees[0], degrees[1]),
(degrees[1], degrees[2]),
(degrees[0], degrees[2])]
sdegrees = nm.array(sdegrees, dtype=nm.uint32)
elif dim == 2:
sdegrees = degrees[[0, 1, 0, 1]]
else:
sdegrees = None
return sdegrees
[docs]def create_boundary_qp(coors, dim):
"""
Create boundary quadrature points from the surface quadrature points.
Uses the Bezier element tensor product structure.
Parameters
----------
coors : array, shape (n_qp, d)
The coordinates of the surface quadrature points.
dim : int
The topological dimension.
Returns
-------
bcoors : array, shape (n_qp, d + 1)
The coordinates of the boundary quadrature points.
"""
# Boundary QP - use tensor product structure.
axes, acoors = get_facet_axes(dim)
n_f = len(axes)
bcoors = nm.empty((n_f, coors.shape[0], coors.shape[1] + 1),
dtype=nm.float64)
ii = nm.arange(bcoors.shape[1], dtype=nm.uint32)
for ik in range(n_f):
for ic in range(bcoors.shape[2] - 1):
bcoors[ik, :, axes[ik, ic]] = coors[:, ic]
bcoors[ik, ii, axes[ik, -1]] = acoors[ik]
return bcoors
[docs]def get_bezier_element_entities(degrees):
"""
Get faces and edges of a Bezier mesh element in terms of indices into the
element's connectivity (reference Bezier element entities).
Parameters
----------
degrees : sequence of ints or int
Polynomial degrees in each parametric dimension.
Returns
-------
faces : list of arrays
The indices for each face or None if not 3D.
edges : list of arrays
The indices for each edge or None if not at least 2D.
vertices : list of arrays
The indices for each vertex.
Notes
-----
The ordering of faces and edges has to be the same as in
:data:`sfepy.discrete.fem.geometry_element.geometry_data`.
"""
if isinstance(degrees, int): degrees = [degrees]
degrees = nm.asarray(degrees)
dim = len(degrees)
shape = degrees + 1
n_dof = nm.prod(shape)
aux = nm.arange(n_dof, dtype=nm.uint32).reshape(shape)
if dim == 3:
faces = [aux[:, :, 0],
aux[0, :, :],
aux[:, 0, :],
aux[:, :, -1],
aux[-1, :, :],
aux[:, -1, :]]
faces = [ii.ravel() for ii in faces]
edges = [aux[:, 0, 0],
aux[-1, :, 0],
aux[:, -1, 0],
aux[0, :, 0],
aux[:, 0, -1],
aux[-1, :, -1],
aux[:, -1, -1],
aux[0, :, -1],
aux[0, 0, :],
aux[0, -1, :],
aux[-1, -1, :],
aux[-1, 0, :]]
vertices = [aux[0, 0, 0],
aux[-1, 0, 0],
aux[-1, -1, 0],
aux[0, -1, 0],
aux[0, 0, -1],
aux[-1, 0, -1],
aux[-1, -1, -1],
aux[0, -1, -1]]
vertices = [ii[None] for ii in vertices]
elif dim == 2:
faces = None
edges = [aux[:, 0],
aux[-1, :],
aux[:, -1],
aux[0, :]]
vertices = [aux[0, 0],
aux[-1, 0],
aux[-1, -1],
aux[0, -1]]
vertices = [ii[None] for ii in vertices]
else:
faces, edges = None, None
vertices = [aux[:1], aux[-1:]]
return faces, edges, vertices
[docs]def eval_bernstein_basis(x, degree):
"""
Evaluate the Bernstein polynomial basis of the given `degree`, and its
derivatives, in a point `x` in [0, 1].
Parameters
----------
x : float
The point in [0, 1].
degree : int
The basis degree.
Returns
-------
funs : array
The `degree + 1` values of the Bernstein polynomial basis.
ders : array
The `degree + 1` values of the Bernstein polynomial basis derivatives.
"""
n_fun = degree + 1
funs = nm.zeros(n_fun, dtype=nm.float64)
ders = nm.zeros(n_fun, dtype=nm.float64)
funs[0] = 1.0
if degree == 0: return funs, ders
for ip in range(1, n_fun - 1):
prev = 0.0
for ifun in range(ip + 1):
tmp = x * funs[ifun]
funs[ifun] = (1.0 - x) * funs[ifun] + prev
prev = tmp
for ifun in range(n_fun):
ders[ifun] = degree * (funs[ifun - 1] - funs[ifun])
prev = 0.0
for ifun in range(n_fun):
tmp = x * funs[ifun]
funs[ifun] = (1.0 - x) * funs[ifun] + prev
prev = tmp
return funs, ders
[docs]def eval_nurbs_basis_tp(qp, ie, control_points, weights, degrees, cs, conn):
"""
Evaluate the tensor-product NURBS shape functions in a quadrature point for
a given Bezier element.
Parameters
----------
qp : array
The quadrature point coordinates with components in [0, 1] reference
element domain.
ie : int
The Bezier element index.
control_points : array
The NURBS control points.
weights : array
The NURBS weights.
degrees : sequence of ints or int
The basis degrees in each parametric dimension.
cs : list of lists of 2D arrays
The element extraction operators in each parametric dimension.
conn : array
The connectivity of the global NURBS basis.
Returns
-------
R : array
The NURBS shape functions.
dR_dx : array
The NURBS shape functions derivatives w.r.t. the physical coordinates.
det : array
The Jacobian of the mapping to the unit reference element.
"""
if isinstance(degrees, int): degrees = [degrees]
degrees = nm.asarray(degrees)
dim = len(degrees)
assert_(dim == len(qp) == len(cs))
n_efuns = degrees + 1
n_efun = nm.prod(n_efuns)
n_efuns_max = n_efuns.max()
assert_(n_efun == conn.shape[1])
# Element connectivity.
ec = conn[ie]
# Element control points and weights.
W = weights[ec]
P = control_points[ec]
# 1D Bernstein basis B, dB/dxi.
B = nm.empty((dim, n_efuns_max), dtype=nm.float64)
dB_dxi = nm.empty((dim, n_efuns_max), dtype=nm.float64)
for ii in range(dim):
(B[ii, :n_efuns[ii]],
dB_dxi[ii, :n_efuns[ii]]) = eval_bernstein_basis(qp[ii], degrees[ii])
# 1D B-spline basis N = CB, dN/dxi = C dB/dxi.
N = nm.empty((dim, n_efuns_max), dtype=nm.float64)
dN_dxi = nm.empty((dim, n_efuns_max), dtype=nm.float64)
n_els = [len(ii) for ii in cs]
ic = get_unraveled_indices(ie, n_els)
for ii in range(dim):
C = cs[ii][ic[ii]]
N[ii, :n_efuns[ii]] = nm.dot(C, B[ii, :n_efuns[ii]])
dN_dxi[ii, :n_efuns[ii]] = nm.dot(C, dB_dxi[ii, :n_efuns[ii]])
# Numerators and denominator for tensor-product NURBS basis R, dR/dxi.
R = nm.empty(n_efun, dtype=nm.float64)
dR_dxi = nm.empty((n_efun, dim), dtype=nm.float64)
w = 0 # w_b
dw_dxi = nm.zeros(dim, dtype=nm.float64) # dw_b/dxi
a = 0 # Basis function index.
if dim == 3:
for i0 in range(n_efuns[0]):
for i1 in range(n_efuns[1]):
for i2 in range(n_efuns[2]):
R[a] = N[0, i0] * N[1, i1] * N[2, i2] * W[a]
w += R[a]
dR_dxi[a, 0] = dN_dxi[0, i0] * N[1, i1] * N[2, i2] * W[a]
dw_dxi[0] += dR_dxi[a, 0]
dR_dxi[a, 1] = N[0, i0] * dN_dxi[1, i1] * N[2, i2] * W[a]
dw_dxi[1] += dR_dxi[a, 1]
dR_dxi[a, 2] = N[0, i0] * N[1, i1] * dN_dxi[2, i2] * W[a]
dw_dxi[2] += dR_dxi[a, 2]
a += 1
elif dim == 2:
for i0 in range(n_efuns[0]):
for i1 in range(n_efuns[1]):
R[a] = N[0, i0] * N[1, i1] * W[a]
w += R[a]
dR_dxi[a, 0] = dN_dxi[0, i0] * N[1, i1] * W[a]
dw_dxi[0] += dR_dxi[a, 0]
dR_dxi[a, 1] = N[0, i0] * dN_dxi[1, i1] * W[a]
dw_dxi[1] += dR_dxi[a, 1]
a += 1
else:
for i0 in range(n_efuns[0]):
R[a] = N[0, i0] * W[a]
w += R[a]
dR_dxi[a, 0] = dN_dxi[0, i0] * W[a]
dw_dxi[0] += dR_dxi[a, 0]
a += 1
# Finish R <- R / w_b.
R /= w
# Finish dR/dxi. D == W C dB/dxi, dR/dxi = (D - R dw_b/dxi) / w_b.
dR_dxi = (dR_dxi - R[:, None] * dw_dxi) / w
# Mapping reference -> physical domain dxi/dx.
# x = sum P_a R_a, dx/dxi = sum P_a dR_a/dxi, invert.
dx_dxi = nm.dot(P.T, dR_dxi)
det = nm.linalg.det(dx_dxi)
dxi_dx = nm.linalg.inv(dx_dxi)
# dR/dx.
dR_dx = nm.dot(dR_dxi, dxi_dx)
return R, dR_dx, det
[docs]def eval_mapping_data_in_qp(qps, control_points, weights, degrees, cs, conn,
cells=None):
"""
Evaluate data required for the isogeometric domain reference mapping in the
given quadrature points. The quadrature points are the same for all Bezier
elements and should correspond to the Bernstein basis degree.
Parameters
----------
qps : array
The quadrature points coordinates with components in [0, 1] reference
element domain.
control_points : array
The NURBS control points.
weights : array
The NURBS weights.
degrees : sequence of ints or int
The basis degrees in each parametric dimension.
cs : list of lists of 2D arrays
The element extraction operators in each parametric dimension.
conn : array
The connectivity of the global NURBS basis.
cells : array, optional
If given, use only the given Bezier elements.
Returns
-------
bfs : array
The NURBS shape functions in the physical quadrature points of all
elements.
bfgs : array
The NURBS shape functions derivatives w.r.t. the physical coordinates
in the physical quadrature points of all elements.
dets : array
The Jacobians of the mapping to the unit reference element in the
physical quadrature points of all elements.
"""
if cells is None:
cells = nm.arange(conn.shape[0])
n_el = len(cells)
n_qp = qps.shape[0]
dim = control_points.shape[1]
n_efuns = degrees + 1
n_efun = nm.prod(n_efuns)
# Output Jacobians.
dets = nm.empty((n_el, n_qp, 1, 1), dtype=nm.float64)
# Output shape functions.
bfs = nm.empty((n_el, n_qp, 1, n_efun), dtype=nm.float64)
# Output gradients of shape functions.
bfgs = nm.empty((n_el, n_qp, dim, n_efun), dtype=nm.float64)
# Loop over elements.
for iseq, ie in enumerate(cells):
# Loop over quadrature points.
for iqp, qp in enumerate(qps):
bf, bfg, det = eval_nurbs_basis_tp(qp, ie,
control_points, weights,
degrees, cs, conn)
bfs[iseq, iqp] = bf
bfgs[iseq, iqp] = bfg.T
dets[iseq, iqp] = det
return bfs, bfgs, dets
[docs]def eval_variable_in_qp(variable, qps,
control_points, weights, degrees, cs, conn,
cells=None):
"""
Evaluate a field variable in the given quadrature points. The quadrature
points are the same for all Bezier elements and should correspond to the
Bernstein basis degree. The field variable is defined by its DOFs - the
coefficients of the NURBS basis.
Parameters
----------
variable : array
The DOF values of the variable with n_c components, shape (:, n_c).
qps : array
The quadrature points coordinates with components in [0, 1] reference
element domain.
control_points : array
The NURBS control points.
weights : array
The NURBS weights.
degrees : sequence of ints or int
The basis degrees in each parametric dimension.
cs : list of lists of 2D arrays
The element extraction operators in each parametric dimension.
conn : array
The connectivity of the global NURBS basis.
cells : array, optional
If given, use only the given Bezier elements.
Returns
-------
coors : array
The physical coordinates of the quadrature points of all elements.
vals : array
The field variable values in the physical quadrature points.
dets : array
The Jacobians of the mapping to the unit reference element in the
physical quadrature points.
"""
if cells is None:
cells = nm.arange(conn.shape[0])
n_el = len(cells)
n_qp = qps.shape[0]
dim = control_points.shape[1]
nc = variable.shape[1]
# Output values of the variable.
vals = nm.empty((n_el * n_qp, nc), dtype=nm.float64)
# Output physical coordinates of QPs.
coors = nm.empty((n_el * n_qp, dim), dtype=nm.float64)
# Output Jacobians.
dets = nm.empty((n_el * n_qp, 1), dtype=nm.float64)
# Loop over elements.
for iseq, ie in enumerate(cells):
ec = conn[ie]
vals_e = variable[ec]
cps_e = control_points[ec]
# Loop over quadrature points.
for iqp, qp in enumerate(qps):
ii = n_qp * iseq + iqp
bf, bfg, det = eval_nurbs_basis_tp(qp, ie,
control_points, weights,
degrees, cs, conn)
vals_qp = nm.dot(bf, vals_e)
vals[ii, :] = vals_qp
coors_qp = nm.dot(bf, cps_e)
coors[ii, :] = coors_qp
dets[ii] = det
return coors, vals, dets