def interp_to_qp(self, dofs):
"""
Interpolate DOFs into quadrature points.
The quadrature order is given by the field approximation order.
Parameters
----------
dofs : array
The array of DOF values of shape `(n_nod, n_component)`.
Returns
-------
data_qp : array
The values interpolated into the quadrature points.
integral : Integral
The corresponding integral defining the quadrature points.
"""
integral = Integral('i', order=self.approx_order)
data_qp = []
for ig, ap in self.aps.iteritems():
bf = ap.get_base('v', False, integral)
bf = bf[:,0,:].copy()
vals = nm.dot(bf, dofs[ap.econn])
vals = nm.swapaxes(vals, 0, 1)
vals.shape = vals.shape + (1,)
data_qp.append(vals)
data_qp = nm.concatenate(data_qp, axis=0)
return data_qp, integral
def interp_to_qp(self, dofs):
"""
Interpolate DOFs into quadrature points.
The quadrature order is given by the field approximation order.
Parameters
----------
dofs : array
The array of DOF values of shape `(n_nod, n_component)`.
Returns
-------
data_qp : array
The values interpolated into the quadrature points.
integral : Integral
The corresponding integral defining the quadrature points.
"""
integral = Integral('i', order=self.approx_order)
bf = self.get_base('v', False, integral)
bf = bf[:, 0, :].copy()
data_qp = nm.dot(bf, dofs[self.econn])
data_qp = nm.swapaxes(data_qp, 0, 1)
data_qp.shape = data_qp.shape + (1, )
return data_qp, integral
def get_surface_basis(self, region):
from sfepy.discrete.integrals import Integral
import sfepy.discrete.iga as iga
from sfepy.discrete.iga.extmods.igac import eval_mapping_data_in_qp
nurbs = self.nurbs
facets = self._get_facets(region.kind_tdim)
# Cell and face(cell) ids for each facet.
fis = region.get_facet_indices()
# Integral given by max. NURBS surface degree.
fdegrees = iga.get_surface_degrees(nurbs.degrees)
order = fdegrees.max()
integral = Integral('i', order=2 * order)
vals, weights = integral.get_qp(self.domain.gel.surface_facet_name)
# Boundary QP - use tensor product structure.
bvals = iga.create_boundary_qp(vals, region.tdim)
# Compute facet basis, jacobians and physical BQP.
all_qp = []
all_fbfs = []
all_dets = []
for ii, (ie, ifa) in enumerate(fis):
qp_coors = bvals[ifa]
bfs, _, dets = eval_mapping_data_in_qp(qp_coors, nurbs.cps,
nurbs.weights,
nurbs.degrees, nurbs.cs,
nurbs.conn, nm.array([ie]))
# Facet basis.
fbfs = bfs[..., facets[ifa]][0, :, 0, :]
# Weight Jacobians by quadrature point weights.
dets = nm.abs(dets) * weights[None, :, None, None]
dets = dets[0, :, 0, :]
# Physical BQP.
fcps = nurbs.cps[nurbs.conn[ie, facets[ifa]]]
qp = nm.dot(fbfs, fcps)
all_qp.append(qp)
all_fbfs.append(fbfs)
all_dets.append(dets)
return all_qp, all_fbfs, all_dets
def compute_nodal_normals(nodes, region, field, return_imap=False):
"""
Nodal normals are computed by simple averaging of element normals of
elements every node is contained in.
"""
dim = region.dim
field.domain.create_surface_group(region)
field._setup_surface_data(region)
# Custom integral with quadrature points very close to facet vertices.
coors = field.gel.surface_facet.coors
centre = coors.sum(axis=0) / coors.shape[0]
qp_coors = coors + 1e-8 * (centre - coors)
# Unit normals -> weights = ones.
qp_weights = nm.ones(qp_coors.shape[0], dtype=nm.float64)
integral = Integral('aux', coors=qp_coors, weights=qp_weights)
normals = nm.zeros((nodes.shape[0], dim), dtype=nm.float64)
mask = nm.zeros((nodes.max() + 1, ), dtype=nm.int32)
imap = nm.empty_like(mask)
imap.fill(nodes.shape[0]) # out-of-range index for normals.
imap[nodes] = nm.arange(nodes.shape[0], dtype=nm.int32)
for ig in region.igs:
cmap, _ = field.get_mapping(ig, region, integral, 'surface')
e_normals = cmap.normal[..., 0]
sd = field.domain.surface_groups[ig][region.name]
econn = sd.get_connectivity()
mask[econn] += 1
# normals[imap[econn]] += e_normals
im = imap[econn]
for ii, en in enumerate(e_normals):
normals[im[ii]] += en
# All nodes must have a normal.
if not nm.all(mask[nodes] > 0):
raise ValueError('region %s has not complete faces!' % region.name)
norm = la.norm_l2_along_axis(normals)[:, nm.newaxis]
if (norm < 1e-15).any():
raise ValueError('zero nodal normal! (a node in volume?)')
normals /= norm
if return_imap:
return normals, imap
else:
return normals
def test_quadratures(self):
"""
Test if the quadratures have orders they claim to have, using
symbolic integration by sympy.
"""
from sfepy.discrete.quadratures import quadrature_tables, tp_geometries
from sfepy.discrete.integrals import Integral
if sm is None:
self.report('cannot import sympy, skipping')
return True
quad = Integral('test_integral')
ok = True
failed = []
for geometry, qps in ordered_iteritems(quadrature_tables):
self.report('geometry:', geometry)
if geometry == '0_1':
continue
elif geometry in tp_geometries:
iter_qp = range(1, 11)
elif geometry == '2_3':
iter_qp = range(1, 25)
elif geometry == '3_4':
iter_qp = range(1, 12)
else:
iter_qp = sorted(qps.keys())
for order in iter_qp:
self.report('order:', order)
aux = self._test_quadratures(quad, geometry, order)
_ok, integral, val = aux
if not _ok:
failed.append((geometry, order, float(integral), val))
ok = ok and _ok
if not ok:
self.report('failed:')
for aux in failed:
self.report(aux, '%.1e' % abs(aux[2] - aux[3]))
return ok
def get_surface_basis(self, region):
"""
Get basis for projections to region's facets.
Notes
-----
Cannot be uses for all fields because IGA does not support surface
mappings.
"""
order = self.approx_order
integral = Integral('i', order=2 * order)
geo, mapping = self.get_mapping(region, integral, 'surface')
pqps = get_physical_qps(region, integral)
qps = pqps.values.reshape(pqps.shape)
bfs = nm.broadcast_to(
geo.bf[..., 0, :],
(qps.shape[0], qps.shape[1], geo.bf.shape[3]),
)
return qps, bfs, geo.det[..., 0]
def set_dofs(self, fun=0.0, region=None, dpn=None, warn=None):
"""
Set the values of DOFs given by the `region` using a function of space
coordinates or value `fun`.
If `fun` is a function, the l2 projection that is global for all region
facets is used to set the DOFs.
If `dpn > 1`, and `fun` is a function, it has to return the values
DOF-by-DOF, i.e. a single one-dimensional vector with all values of the
first component, then of the second one etc. concatenated
together.
Parameters
----------
fun : float or array of length dpn or callable
The DOF values.
region : Region
The region containing the DOFs.
dpn : int, optional
The DOF-per-node count. If not given, the number of field
components is used.
warn : str, optional
The warning message printed when the region selects no DOFs.
Returns
-------
nods : array, shape (n_dof,)
The field DOFs (or node indices) given by the region.
vals : array, shape (dpn, n_dof)
The values of the DOFs, DOF-by-DOF when raveled in C (row-major)
order.
"""
if region is None:
region = self.region
if dpn is None:
dpn = self.n_components
nods = []
vals = []
aux = self.get_dofs_in_region(region)
nods = nm.unique(nm.hstack(aux))
if nm.isscalar(fun):
vals = nm.repeat([fun], nods.shape[0] * dpn)
elif isinstance(fun, nm.ndarray):
assert_(len(fun) == dpn)
vals = nm.repeat(fun, nods.shape[0])
elif callable(fun):
import scipy.sparse as sps
from sfepy.solvers.ls import solve
from sfepy.discrete.integrals import Integral
from sfepy.discrete.fem.utils import prepare_remap
import sfepy.discrete.iga as iga
from sfepy.discrete.iga.extmods.igac import eval_mapping_data_in_qp
nurbs = self.nurbs
facets = self._get_facets(region.kind_tdim)
# Region facet connectivity.
rconn = self.get_econn('surface', region)
# Local connectivity.
remap = prepare_remap(nods, nods.max() + 1)
lconn = [remap[ii] for ii in rconn]
# Cell and face(cell) ids for each facet.
fis = region.get_facet_indices()
# Integral given by max. NURBS surface degree.
fdegrees = iga.get_surface_degrees(nurbs.degrees)
order = fdegrees.max()
integral = Integral('i', order=2 * order)
vals, weights = integral.get_qp(self.domain.gel.surface_facet_name)
# Boundary QP - use tensor product structure.
bvals = iga.create_boundary_qp(vals, region.tdim)
# Compute facet basis, jacobians and physical BQP.
n_dof = len(nods)
rhs = nm.zeros((dpn, n_dof), dtype=nm.float64)
rows, cols, mvals = [], [], []
all_qp = []
all_fbfs = []
all_dets = []
for ii, (ie, ifa) in enumerate(fis):
qp_coors = bvals[ifa]
bfs, _, dets = eval_mapping_data_in_qp(qp_coors, nurbs.cps,
nurbs.weights,
nurbs.degrees, nurbs.cs,
nurbs.conn,
nm.array([ie]))
# Facet basis.
fbfs = bfs[..., facets[ifa]][0, :, 0, :]
# Weight Jacobians by quadrature point weights.
dets = nm.abs(dets) * weights[None, :, None, None]
dets = dets[0, :, 0, :]
# Physical BQP.
fcps = nurbs.cps[nurbs.conn[ie, facets[ifa]]]
qp = nm.dot(fbfs, fcps)
all_qp.append(qp)
all_fbfs.append(fbfs)
all_dets.append(dets)
# DOF values in the physical BQP.
qps = nm.concatenate(all_qp)
vals = nm.asarray(fun(qps))
vals.shape = (dpn, qps.shape[0])
n_qp_face = len(bvals[0])
# Assemble l2 projection system.
for ii, (ie, ifa) in enumerate(fis):
# Assembling indices.
elc = lconn[ii]
fvals = vals[:, n_qp_face * ii:n_qp_face * (ii + 1)]
fbfs = all_fbfs[ii]
dets = all_dets[ii]
# Local projection system.
for idof in range(dpn):
lrhs = (fbfs * (fvals[idof, :, None] * dets)).sum(0)
rhs[idof, elc] += lrhs
lmtx = ((fbfs[..., None] * fbfs[:, None, :]) *
dets[..., None]).sum(0)
er, ec = nm.meshgrid(elc, elc)
rows.append(er.ravel())
cols.append(ec.ravel())
mvals.append(lmtx.ravel())
rows = nm.concatenate(rows)
cols = nm.concatenate(cols)
mvals = nm.concatenate(mvals)
mtx = sps.coo_matrix((mvals, (rows, cols)), shape=(n_dof, n_dof))
vals = nm.zeros((dpn, n_dof), dtype=nm.float64)
# Solve l2 projection system.
for idof in range(dpn):
dofs = solve(mtx, rhs[idof, :])
vals[idof, remap[nods]] = dofs
else:
raise ValueError('unknown function/value type! (%s)' % type(fun))
return nods, vals
def set_dofs(self, fun=0.0, region=None, dpn=None, warn=None):
"""
Set the values of DOFs given by the `region` using a function of space
coordinates or value `fun`.
If `fun` is a function, the l2 projection that is global for all region
facets is used to set the DOFs.
If `dpn > 1`, and `fun` is a function, it has to return the values
DOF-by-DOF, i.e. a single one-dimensional vector with all values of the
first component, then of the second one etc. concatenated
together.
Parameters
----------
fun : float or array of length dpn or callable
The DOF values.
region : Region
The region containing the DOFs.
dpn : int, optional
The DOF-per-node count. If not given, the number of field
components is used.
warn : str, optional
The warning message printed when the region selects no DOFs.
Returns
-------
nods : array, shape (n_dof,)
The field DOFs (or node indices) given by the region.
vals : array, shape (dpn, n_dof)
The values of the DOFs, DOF-by-DOF when raveled in C (row-major)
order.
"""
if region is None:
region = self.region
if dpn is None:
dpn = self.n_components
nods = []
vals = []
aux = self.get_dofs_in_region(region)
nods = nm.unique(nm.hstack(aux))
if nm.isscalar(fun):
vals = nm.repeat([fun], nods.shape[0] * dpn)
elif isinstance(fun, nm.ndarray):
assert_(len(fun) == dpn)
vals = nm.repeat(fun, nods.shape[0])
elif callable(fun):
import scipy.sparse as sps
from sfepy.solvers.ls import solve
from sfepy.discrete.integrals import Integral
from sfepy.discrete.fem.utils import prepare_remap
import sfepy.discrete.iga as iga
from sfepy.discrete.iga.extmods.igac import eval_mapping_data_in_qp
nurbs = self.nurbs
facets = self._get_facets(region.kind_tdim)
# Region facet connectivity.
rconn = self.get_econn('surface', region)
# Local connectivity.
remap = prepare_remap(nods, nods.max() + 1)
lconn = [remap[ii] for ii in rconn]
# Cell and face(cell) ids for each facet.
fis = region.get_facet_indices()
# Integral given by max. NURBS surface degree.
fdegrees = iga.get_surface_degrees(nurbs.degrees)
order = fdegrees.max()
integral = Integral('i', order=2*order)
vals, weights = integral.get_qp(self.domain.gel.surface_facet_name)
# Boundary QP - use tensor product structure.
bvals = iga.create_boundary_qp(vals, region.tdim)
# Compute facet basis, jacobians and physical BQP.
n_dof = len(nods)
rhs = nm.zeros((dpn, n_dof), dtype=nm.float64)
rows, cols, mvals = [], [], []
all_qp = []
all_fbfs = []
all_dets = []
for ii, (ie, ifa) in enumerate(fis):
qp_coors = bvals[ifa]
bfs, _, dets = eval_mapping_data_in_qp(qp_coors, nurbs.cps,
nurbs.weights,
nurbs.degrees,
nurbs.cs,
nurbs.conn,
nm.array([ie]))
# Facet basis.
fbfs = bfs[..., facets[ifa]][0, :, 0, :]
# Weight Jacobians by quadrature point weights.
dets = nm.abs(dets) * weights[None, :, None, None]
dets = dets[0, :, 0, :]
# Physical BQP.
fcps = nurbs.cps[nurbs.conn[ie, facets[ifa]]]
qp = nm.dot(fbfs, fcps)
all_qp.append(qp)
all_fbfs.append(fbfs)
all_dets.append(dets)
# DOF values in the physical BQP.
qps = nm.concatenate(all_qp)
vals = nm.asarray(fun(qps))
vals.shape = (dpn, qps.shape[0])
n_qp_face = len(bvals[0])
# Assemble l2 projection system.
for ii, (ie, ifa) in enumerate(fis):
# Assembling indices.
elc = lconn[ii]
fvals = vals[:, n_qp_face * ii : n_qp_face * (ii + 1)]
fbfs = all_fbfs[ii]
dets = all_dets[ii]
# Local projection system.
for idof in xrange(dpn):
lrhs = (fbfs * (fvals[idof, :, None] * dets)).sum(0)
rhs[idof, elc] += lrhs
lmtx = ((fbfs[..., None] * fbfs[:, None, :])
* dets[..., None]).sum(0)
er, ec = nm.meshgrid(elc, elc)
rows.append(er.ravel())
cols.append(ec.ravel())
mvals.append(lmtx.ravel())
rows = nm.concatenate(rows)
cols = nm.concatenate(cols)
mvals = nm.concatenate(mvals)
mtx = sps.coo_matrix((mvals, (rows, cols)), shape=(n_dof, n_dof))
vals = nm.zeros((dpn, n_dof), dtype=nm.float64)
# Solve l2 projection system.
for idof in xrange(dpn):
dofs = solve(mtx, rhs[idof, :])
vals[idof, remap[nods]] = dofs
else:
raise ValueError('unknown function/value type! (%s)' % type(fun))
return nods, vals
def _eval_basis_transform(self, subs):
"""
"""
from sfepy.discrete import Integral
from sfepy.discrete.fem import Mesh, FEDomain, Field
transform = nm.tile(nm.eye(self.econn.shape[1]),
(self.econn.shape[0], 1, 1))
if subs is None:
return transform
gel = self.gel
ao = self.approx_order
conn = [gel.conn]
mesh = Mesh.from_data('a', gel.coors, None, [conn], [nm.array([0])],
[gel.name])
cdomain = FEDomain('d', mesh)
comega = cdomain.create_region('Omega', 'all')
rcfield = Field.from_args('rc', self.dtype, 1, comega, approx_order=ao)
fdomain = cdomain.refine()
fomega = fdomain.create_region('Omega', 'all')
rffield = Field.from_args('rf', self.dtype, 1, fomega, approx_order=ao)
def assign_transform(transform, bf, subs, ef):
if not len(subs): return
n_sub = (subs.shape[1] - 2) // 2
for ii, sub in enumerate(subs):
for ij in range(n_sub):
ik = 2 * (ij + 1)
fface = ef[sub[ik + 1]]
mtx = transform[sub[ik]]
ix, iy = nm.meshgrid(fface, fface)
cbf = bf[iy, 0, ix]
mtx[ix, iy] = cbf
fcoors = rffield.get_coor()
coors = fcoors[rffield.econn[0]]
integral = Integral('i',
coors=coors,
weights=nm.ones_like(coors[:, 0]))
rcfield.clear_qp_base()
bf = rcfield.get_base('v', False, integral)
if gel.name == '2_4':
fsubs = subs
esubs = None
assign_transform(transform, bf, fsubs, rffield.efaces)
else:
fsubs = subs[0]
esubs = subs[1]
assign_transform(transform, bf, fsubs, rffield.efaces)
if esubs is not None:
assign_transform(transform, bf, esubs, rffield.eedges)
assert_((nm.abs(transform.sum(1) - 1.0) < 1e-15).all())
return transform
def load_and_plot_fun(folder, filename, exact=None):
"""
Parameters
----------
folder : str
folder where to look for files
filename : str
used in {name}.i.vtk, i = 0,1, ... tns - 1
number of time steps
exact : callable
exact solution at the last frame
"""
in_file = head(glob(pjoin(folder, "*.vtk")))
coors, data = load_state_1D_vtk(in_file)
approx_order = data.shape[0] - 1
dmesh = Mesh.from_file(in_file)
domain = FEDomain("", dmesh)
omega = domain.create_region('Omega', 'all')
field = DGField('f',
nm.float64,
'scalar',
omega,
approx_order=approx_order)
# Sufficient quadrature order for the analytical expression.
idiff = Integral('idiff', 20)
u = FieldVariable("u", "unknown", field)
eqs = Equations(
[Equation('balance', SurfaceTerm("s()", "u", idiff, omega, u=u))])
pb = Problem("err_est", equations=eqs)
u.set_data(field.ravel_sol(data.swapaxes(0, 1)))
num_qp = pb.evaluate('ev_volume_integrate.idiff.Omega(u)',
u=u,
integrals=Integrals([idiff]),
mode='qp')
aux = Material('aux', function=sol_fun)
ana_qp = pb.evaluate('ev_volume_integrate_mat.idiff.Omega(aux.u, u)',
aux=aux,
u=u,
integrals=Integrals([idiff]),
mode='qp')
qps = pb.fields["f"].mapping.get_physical_qps(idiff.get_qp("1_2")[0])
fqps = qps.flatten()
plt.figure("Reconstructed solution")
plt.gca().set_ylim(-.5, 3.)
ww_approx, xx = reconstruct_legendre_dofs(coors, None, data)
ww_exact = exact(xx)
XN = xx[-1]
X1 = xx[0]
plt.plot([X1, XN], [2, 2], 'grey', alpha=.6)
plt.plot([X1, XN], [0, 0], 'grey', alpha=.6)
plt.plot(fqps, ana_qp.flatten(), label="$p_{exact}(1, x)$")
plt.plot(fqps, num_qp.flatten(), label="$p_h(1, x)$")
plt.legend()
plt.show()