def test_term_arithmetics(self):
from sfepy.fem import FieldVariable, Integral
from sfepy.terms.terms import Term
integral = Integral('i', order=3)
integral_s = Integral('i', kind='s', order=3)
u = FieldVariable('u', 'parameter', self.field, self.dim,
primary_var_name='(set-to-None)')
t1 = Term.new('d_volume(u)', integral, self.omega, u=u)
t2 = Term.new('d_surface(u)', integral_s, self.gamma1, u=u)
expr = 2.2j * (t1 * 5.5 - 3j * t2) * 0.25
ok = len(expr) == 2
if not ok:
self.report('wrong expression length!')
_ok = nm.allclose(expr[0].sign, 3.025j, rtol=1e-15, atol=0)
if not _ok:
self.report('wrong sign of the first term!')
ok = ok and _ok
_ok = nm.allclose(expr[1].sign, 1.65, rtol=1e-15, atol=0)
if not _ok:
self.report('wrong sign of the second term!')
ok = ok and _ok
return ok
def test_normals(self):
"""
Check orientations of surface normals on the reference elements.
"""
import sfepy
from sfepy.fem import Mesh, Domain, Integral
from sfepy.fem.poly_spaces import PolySpace
from sfepy.fem.mappings import SurfaceMapping
from sfepy.linalg import normalize_vectors
ok = True
for geom in ['2_3', '2_4', '3_4', '3_8']:
mesh = Mesh.from_file('meshes/elements/%s_1.mesh' % geom,
prefix_dir=sfepy.data_dir)
domain = Domain('domain', mesh)
surface = domain.create_region('Surface', 'vertices of surface',
'facet')
domain.create_surface_group(surface)
sd = domain.surface_groups[0][surface.name]
coors = domain.get_mesh_coors()
gel = domain.geom_els[geom].surface_facet
ps = PolySpace.any_from_args('aux', gel, 1)
mapping = SurfaceMapping(coors, sd.get_connectivity(), ps)
integral = Integral('i', order=1)
vals, weights = integral.get_qp(gel.name)
# Evaluate just in the first quadrature point...
geo = mapping.get_mapping(vals[:1], weights[:1])
expected = expected_normals[geom].copy()
normalize_vectors(expected)
_ok = nm.allclose(expected,
geo.normal[:, 0, :, 0],
rtol=0.0,
atol=1e-14)
self.report('%s: %s' % (geom, _ok))
if not _ok:
self.report('expected:')
self.report(expected)
self.report('actual:')
self.report(geo.normal[:, 0, :, 0])
ok = ok and _ok
return ok
def _get_qp(geometry, order):
from sfepy.fem import Integral
from sfepy.fem.geometry_element import GeometryElement
aux = Integral('aux', order=order)
coors, weights = aux.get_qp(geometry)
true_order = aux.qps[geometry].order
output('geometry:', geometry, 'order:', order, 'num. points:',
coors.shape[0], 'true_order:', true_order)
output('min. weight:', weights.min())
output('max. weight:', weights.max())
return GeometryElement(geometry), coors, weights
def _get_qp(geometry, order):
from sfepy.fem import Integral
from sfepy.fem.geometry_element import GeometryElement
aux = Integral('aux', order=order)
coors, weights = aux.get_qp(geometry)
true_order = aux.qps[geometry].order
output('geometry:', geometry, 'order:', order, 'num. points:',
coors.shape[0], 'true_order:', true_order)
output('min. weight:', weights.min())
output('max. weight:', weights.max())
return GeometryElement(geometry), coors, weights
def test_normals(self):
"""
Check orientations of surface normals on the reference elements.
"""
import sfepy
from sfepy.fem import Mesh, Domain, Integral
from sfepy.fem.poly_spaces import PolySpace
from sfepy.fem.mappings import SurfaceMapping
from sfepy.linalg import normalize_vectors
ok = True
for geom in ['2_3', '2_4', '3_4', '3_8']:
mesh = Mesh.from_file('meshes/elements/%s_1.mesh' % geom,
prefix_dir=sfepy.data_dir)
domain = Domain('domain', mesh)
surface = domain.create_region('Surface', 'nodes of surface')
domain.create_surface_group(surface)
sd = domain.surface_groups[0][surface.name]
coors = domain.get_mesh_coors()
gel = domain.geom_els[geom].surface_facet
ps = PolySpace.any_from_args('aux', gel, 1)
mapping = SurfaceMapping(coors, sd.get_connectivity(), ps)
integral = Integral('i', order=1)
vals, weights = integral.get_qp(gel.name)
# Evaluate just in the first quadrature point...
geo = mapping.get_mapping(vals[:1], weights[:1])
expected = expected_normals[geom].copy()
normalize_vectors(expected)
_ok = nm.allclose(expected, geo.normal[:, 0, :, 0],
rtol=0.0, atol=1e-14)
self.report('%s: %s' % (geom, _ok))
if not _ok:
self.report('expected:')
self.report(expected)
self.report('actual:')
self.report(geo.normal[:, 0, :, 0])
ok = ok and _ok
return ok
def test_term_evaluation(self):
from sfepy.fem import Integral, FieldVariable
from sfepy.terms.terms import Term
integral = Integral('i', order=3)
u = FieldVariable('u', 'parameter', self.field, self.dim,
primary_var_name='(set-to-None)')
term = Term.new('d_volume(u)', integral, self.omega, u=u)
term *= 10.0
term.setup()
vol = term.evaluate()
self.report('volume: %.8f == 2000.0' % vol)
ok = nm.allclose(vol, 2000.0, rtol=1e-15, atol=0)
## vec = t1.evaluate() # Returns vector.
## vec = t1.evaluate(u=u_vec) # Returns the same vector.
## mtx = t1.evaluate(diff_var='u') # Returns matrix.
## val = t1.evaluate(v=u_vec, u=u_vec) # Forbidden - virtual variable
## # cannot have value.
return ok
def test_solving(self):
from sfepy.base.base import IndexedStruct
from sfepy.fem \
import FieldVariable, Material, ProblemDefinition, \
Function, Equation, Equations, Integral
from sfepy.fem.conditions import Conditions, EssentialBC
from sfepy.terms import Term
from sfepy.solvers.ls import ScipyDirect
from sfepy.solvers.nls import Newton
u = FieldVariable('u', 'unknown', self.field, self.dim)
v = FieldVariable('v', 'test', self.field, self.dim,
primary_var_name='u')
m = Material('m', lam=1.0, mu=1.0)
f = Material('f', val=[[0.02], [0.01]])
bc_fun = Function('fix_u_fun', fix_u_fun,
extra_args={'extra_arg' : 'hello'})
fix_u = EssentialBC('fix_u', self.gamma1, {'u.all' : bc_fun})
shift_u = EssentialBC('shift_u', self.gamma2, {'u.0' : 0.1})
integral = Integral('i', order=3)
t1 = Term.new('dw_lin_elastic_iso(m.lam, m.mu, v, u)',
integral, self.omega, m=m, v=v, u=u)
t2 = Term.new('dw_volume_lvf(f.val, v)', integral, self.omega, f=f, v=v)
eq = Equation('balance', t1 + t2)
eqs = Equations([eq])
ls = ScipyDirect({})
nls_status = IndexedStruct()
nls = Newton({}, lin_solver=ls, status=nls_status)
pb = ProblemDefinition('elasticity', equations=eqs, nls=nls, ls=ls)
## pb.save_regions_as_groups('regions')
pb.time_update(ebcs=Conditions([fix_u, shift_u]))
state = pb.solve()
name = op.join(self.options.out_dir, 'test_high_level_solving.vtk')
pb.save_state(name, state)
ok = nls_status.condition == 0
if not ok:
self.report('solver did not converge!')
_ok = state.has_ebc()
if not _ok:
self.report('EBCs violated!')
ok = ok and _ok
return ok
def _integrate(var, val_qp):
from sfepy.fem import Integral
from sfepy.fem.mappings import get_jacobian
integral = Integral('i', 2)
det = get_jacobian(var.field, integral)
val = (val_qp * det).sum(axis=1) / det.sum(axis=1)
return val
def make_h1_projection_data(target, eval_data):
"""
Project scalar data given by a material-like `eval_data()` function to a
scalar `target` field variable using the :math:`H^1` dot product.
"""
order = target.field.approx_order * 2
integral = Integral('i', order=order)
un = target.name
v = FieldVariable('v', 'test', target.field, 1, primary_var_name=un)
lhs1 = Term.new('dw_volume_dot(v, %s)' % un,
integral,
target.field.region,
v=v,
**{un: target})
lhs2 = Term.new('dw_laplace(v, %s)' % un,
integral,
target.field.region,
v=v,
**{un: target})
def _eval_data(ts, coors, mode, **kwargs):
if mode == 'qp':
val = eval_data(ts, coors, mode, 'val', **kwargs)
gval = eval_data(ts, coors, mode, 'grad', **kwargs)
return {'val': val, 'gval': gval}
m = Material('m', function=_eval_data)
rhs1 = Term.new('dw_volume_lvf(m.val, v)',
integral,
target.field.region,
m=m,
v=v)
rhs2 = Term.new('dw_diffusion_r(m.gval, v)',
integral,
target.field.region,
m=m,
v=v)
eq = Equation('projection', lhs1 + lhs2 - rhs1 - rhs2)
eqs = Equations([eq])
ls = ScipyDirect({})
nls_status = IndexedStruct()
nls = Newton({}, lin_solver=ls, status=nls_status)
pb = ProblemDefinition('aux', equations=eqs, nls=nls, ls=ls)
pb.time_update()
# This sets the target variable with the projection solution.
pb.solve()
if nls_status.condition != 0:
output('H1 projection: solver did not converge!')
def from_conf(conf, options):
from sfepy.fem import Mesh, Domain, Integral
domains = []
for filename in filename_meshes:
mesh = Mesh.from_file(filename)
domain = Domain('domain_%s' % mesh.name.replace(data_dir, ''),
mesh)
domain.create_region('Omega', 'all')
domain.create_region('Gamma', 'nodes of surface')
domains.append(domain)
integrals = {'Omega' : Integral('iv', kind='v', order=3),
'Gamma' : Integral('is', kind='s', order=3)}
test = Test(domains=domains, integrals=integrals,
conf=conf, options=options)
return test
def test_invariance_qp(self):
from sfepy import data_dir
from sfepy.fem import (Mesh, Domain, H1NodalVolumeField, Variables,
Integral)
from sfepy.terms import Term
from sfepy.fem.mappings import get_physical_qps
mesh = Mesh('source mesh', data_dir + '/meshes/3d/block.mesh')
bbox = mesh.get_bounding_box()
dd = bbox[1, :] - bbox[0, :]
data = nm.sin(4.0 * nm.pi * mesh.coors[:,0:1] / dd[0]) \
* nm.cos(4.0 * nm.pi * mesh.coors[:,1:2] / dd[1])
variables = {
'u': ('unknown field', 'scalar_tp', 0),
'v': ('test field', 'scalar_tp', 'u'),
}
domain = Domain('domain', mesh)
omega = domain.create_region('Omega', 'all')
field = H1NodalVolumeField('scalar_tp',
nm.float64,
1,
omega,
approx_order=1)
ff = {field.name: field}
vv = Variables.from_conf(transform_variables(variables), ff)
u = vv['u']
u.set_from_mesh_vertices(data)
integral = Integral('i', order=2)
term = Term.new('ev_volume_integrate(u)', integral, omega, u=u)
term.setup()
val1, _ = term.evaluate(mode='qp')
val1 = val1.ravel()
qps = get_physical_qps(omega, integral)
coors = qps.get_merged_values()
val2 = u.evaluate_at(coors).ravel()
self.report('max. difference:', nm.abs(val1 - val2).max())
ok = nm.allclose(val1, val2, rtol=0.0, atol=1e-12)
self.report('invariance in qp: %s' % ok)
return ok
def from_conf(conf, options):
from sfepy.fem import Mesh, Domain, Integral
domains = []
for filename in filename_meshes:
mesh = Mesh.from_file(filename)
domain = Domain('domain_%s' % mesh.name.replace(data_dir, ''),
mesh)
domain.create_region('Omega', 'all')
domain.create_region('Gamma', 'vertices of surface', 'facet')
domains.append(domain)
integral = Integral('i', order=3)
test = Test(domains=domains,
integral=integral,
conf=conf,
options=options)
return test
def create_mass_matrix(field):
"""
Create scalar mass matrix corresponding to the given field.
Returns
-------
mtx : csr_matrix
The mass matrix in CSR format.
"""
u = FieldVariable('u', 'unknown', field, 1)
v = FieldVariable('v', 'test', field, 1, primary_var_name='u')
integral = Integral('i', order=field.approx_order * 2)
term = Term.new('dw_volume_dot(v, u)', integral, field.region, v=v, u=u)
eq = Equation('aux', term)
eqs = Equations([eq])
eqs.time_update(None)
dummy = eqs.create_state_vector()
mtx = eqs.create_matrix_graph()
mtx = eqs.eval_tangent_matrices(dummy, mtx)
return mtx
def main():
parser = OptionParser(usage=usage, version="%prog")
parser.add_option(
"-b", "--basis", metavar="name", action="store", dest="basis", default="lagrange", help=help["basis"]
)
parser.add_option(
"-n",
"--max-order",
metavar="order",
type=int,
action="store",
dest="max_order",
default=10,
help=help["max_order"],
)
parser.add_option(
"-m",
"--matrix",
metavar="type",
action="store",
dest="matrix_type",
default="laplace",
help=help["matrix_type"],
)
parser.add_option(
"-g", "--geometry", metavar="name", action="store", dest="geometry", default="2_4", help=help["geometry"]
)
options, args = parser.parse_args()
dim, n_ep = int(options.geometry[0]), int(options.geometry[2])
output("reference element geometry:")
output(" dimension: %d, vertices: %d" % (dim, n_ep))
n_c = {"laplace": 1, "elasticity": dim}[options.matrix_type]
output("matrix type:", options.matrix_type)
output("number of variable components:", n_c)
output("polynomial space:", options.basis)
output("max. order:", options.max_order)
mesh = Mesh.from_file(data_dir + "/meshes/elements/%s_1.mesh" % options.geometry)
domain = Domain("domain", mesh)
omega = domain.create_region("Omega", "all")
orders = nm.arange(1, options.max_order + 1, dtype=nm.int)
conds = []
order_fix = 0 if options.geometry in ["2_4", "3_8"] else 1
for order in orders:
output("order:", order, "...")
field = Field.from_args(
"fu", nm.float64, n_c, omega, approx_order=order, space="H1", poly_space_base=options.basis
)
to = field.approx_order
quad_order = 2 * (max(to - order_fix, 0))
output("quadrature order:", quad_order)
integral = Integral("i", order=quad_order)
qp, _ = integral.get_qp(options.geometry)
output("number of quadrature points:", qp.shape[0])
u = FieldVariable("u", "unknown", field, n_c)
v = FieldVariable("v", "test", field, n_c, primary_var_name="u")
m = Material("m", lam=1.0, mu=1.0)
if options.matrix_type == "laplace":
term = Term.new("dw_laplace(m.mu, v, u)", integral, omega, m=m, v=v, u=u)
n_zero = 1
else:
assert_(options.matrix_type == "elasticity")
term = Term.new("dw_lin_elastic_iso(m.lam, m.mu, v, u)", integral, omega, m=m, v=v, u=u)
n_zero = (dim + 1) * dim / 2
term.setup()
output("assembling...")
tt = time.clock()
mtx, iels = term.evaluate(mode="weak", diff_var="u")
output("...done in %.2f s" % (time.clock() - tt))
mtx = mtx[0][0, 0]
try:
assert_(nm.max(nm.abs(mtx - mtx.T)) < 1e-10)
except:
from sfepy.base.base import debug
debug()
output("matrix shape:", mtx.shape)
eigs = eig(mtx, method="eig.sgscipy", eigenvectors=False)
eigs.sort()
# Zero 'true' zeros.
eigs[:n_zero] = 0.0
ii = nm.where(eigs < 0.0)[0]
if len(ii):
output("matrix is not positive semi-definite!")
ii = nm.where(eigs[n_zero:] < 1e-12)[0]
if len(ii):
output("matrix has more than %d zero eigenvalues!" % n_zero)
output("smallest eigs:\n", eigs[:10])
ii = nm.where(eigs > 0.0)[0]
emin, emax = eigs[ii[[0, -1]]]
output("min:", emin, "max:", emax)
cond = emax / emin
conds.append(cond)
output("condition number:", cond)
output("...done")
plt.figure(1)
plt.semilogy(orders, conds)
plt.xticks(orders, orders)
plt.xlabel("polynomial order")
plt.ylabel("condition number")
plt.grid()
plt.figure(2)
plt.loglog(orders, conds)
plt.xticks(orders, orders)
plt.xlabel("polynomial order")
plt.ylabel("condition number")
plt.grid()
plt.show()
def main():
from sfepy import data_dir
parser = OptionParser(usage=usage, version='%prog')
parser.add_option('-s',
'--show',
action="store_true",
dest='show',
default=False,
help=help['show'])
options, args = parser.parse_args()
mesh = Mesh.from_file(data_dir + '/meshes/2d/rectangle_tri.mesh')
domain = Domain('domain', mesh)
min_x, max_x = domain.get_mesh_bounding_box()[:, 0]
eps = 1e-8 * (max_x - min_x)
omega = domain.create_region('Omega', 'all')
gamma1 = domain.create_region('Gamma1',
'nodes in x < %.10f' % (min_x + eps))
gamma2 = domain.create_region('Gamma2',
'nodes in x > %.10f' % (max_x - eps))
field = H1NodalVolumeField('fu',
nm.float64,
'vector',
omega,
approx_order=2)
u = FieldVariable('u', 'unknown', field, mesh.dim)
v = FieldVariable('v', 'test', field, mesh.dim, primary_var_name='u')
m = Material('m', lam=1.0, mu=1.0)
f = Material('f', val=[[0.02], [0.01]])
integral = Integral('i', order=3)
t1 = Term.new('dw_lin_elastic_iso(m.lam, m.mu, v, u)',
integral,
omega,
m=m,
v=v,
u=u)
t2 = Term.new('dw_volume_lvf(f.val, v)', integral, omega, f=f, v=v)
eq = Equation('balance', t1 + t2)
eqs = Equations([eq])
fix_u = EssentialBC('fix_u', gamma1, {'u.all': 0.0})
bc_fun = Function('shift_u_fun', shift_u_fun, extra_args={'shift': 0.01})
shift_u = EssentialBC('shift_u', gamma2, {'u.0': bc_fun})
ls = ScipyDirect({})
nls_status = IndexedStruct()
nls = Newton({}, lin_solver=ls, status=nls_status)
pb = ProblemDefinition('elasticity', equations=eqs, nls=nls, ls=ls)
pb.save_regions_as_groups('regions')
pb.time_update(ebcs=Conditions([fix_u, shift_u]))
vec = pb.solve()
print nls_status
pb.save_state('linear_elasticity.vtk', vec)
if options.show:
view = Viewer('linear_elasticity.vtk')
view(vector_mode='warp_norm',
rel_scaling=2,
is_scalar_bar=True,
is_wireframe=True)
def _gen_common_data(order, gels, report):
import sfepy
from sfepy.base.base import Struct
from sfepy.linalg import combine
from sfepy.fem import Mesh, Domain, Field, FieldVariable, Integral
from sfepy.fem.global_interp import get_ref_coors
integral = Integral('i', order=order)
for geom, poly_space_base in combine([['2_4', '3_8'],
['lagrange', 'lobatto']]):
report('geometry: %s, base: %s' % (geom, poly_space_base))
mesh0 = Mesh.from_file('meshes/elements/%s_2.mesh' % geom,
prefix_dir=sfepy.data_dir)
gel = gels[geom]
perms = gel.get_conn_permutations()
qps, qp_weights = integral.get_qp(gel.surface_facet.name)
zz = nm.zeros_like(qps[:, :1])
qps = nm.hstack(([qps] + [zz]))
rot = rots[geom]
if rot is not None:
pass
shift = shifts[geom]
rcoors = nm.ascontiguousarray(qps
+ shift[:1, :] - shift[1:, :])
ccoors = nm.ascontiguousarray(qps
+ shift[:1, :] + shift[1:, :])
for ir, pr in enumerate(perms):
for ic, pc in enumerate(perms):
report('ir: %d, ic: %d' % (ir, ic))
mesh = mesh0.copy()
conn = mesh.conns[0]
conn[0, :] = conn[0, pr]
conn[1, :] = conn[1, pc]
cache = Struct(mesh=mesh)
domain = Domain('domain', mesh)
omega = domain.create_region('Omega', 'all')
region = domain.create_region('Facet', rsels[geom])
field = Field.from_args('f', nm.float64, shape=1,
region=omega, approx_order=order,
poly_space_base=poly_space_base)
var = FieldVariable('u', 'unknown', field, 1)
report('# dofs: %d' % var.n_dof)
vec = nm.empty(var.n_dof, dtype=var.dtype)
ap = field.aps[0]
ps = ap.interp.poly_spaces['v']
dofs = field.get_dofs_in_region_group(region, 0,
merge=False)
edofs, fdofs = nm.unique(dofs[1]), nm.unique(dofs[2])
rrc, rcells, rstatus = get_ref_coors(field, rcoors,
cache=cache)
crc, ccells, cstatus = get_ref_coors(field, ccoors,
cache=cache)
yield (geom, poly_space_base, qp_weights, mesh, ir, ic,
ap, ps, rrc, crc, vec, edofs, fdofs)
def main():
parser = OptionParser(usage=usage, version='%prog')
parser.add_option('-b',
'--basis',
metavar='name',
action='store',
dest='basis',
default='lagrange',
help=help['basis'])
parser.add_option('-n',
'--max-order',
metavar='order',
type=int,
action='store',
dest='max_order',
default=10,
help=help['max_order'])
parser.add_option('-m',
'--matrix',
metavar='type',
action='store',
dest='matrix_type',
default='laplace',
help=help['matrix_type'])
parser.add_option('-g',
'--geometry',
metavar='name',
action='store',
dest='geometry',
default='2_4',
help=help['geometry'])
options, args = parser.parse_args()
dim, n_ep = int(options.geometry[0]), int(options.geometry[2])
output('reference element geometry:')
output(' dimension: %d, vertices: %d' % (dim, n_ep))
n_c = {'laplace': 1, 'elasticity': dim}[options.matrix_type]
output('matrix type:', options.matrix_type)
output('number of variable components:', n_c)
output('polynomial space:', options.basis)
output('max. order:', options.max_order)
mesh = Mesh.from_file(data_dir +
'/meshes/elements/%s_1.mesh' % options.geometry)
domain = Domain('domain', mesh)
omega = domain.create_region('Omega', 'all')
orders = nm.arange(1, options.max_order + 1, dtype=nm.int)
conds = []
order_fix = 0 if options.geometry in ['2_4', '3_8'] else 1
for order in orders:
output('order:', order, '...')
field = Field.from_args('fu',
nm.float64,
n_c,
omega,
approx_order=order,
space='H1',
poly_space_base=options.basis)
to = field.approx_order
quad_order = 2 * (max(to - order_fix, 0))
output('quadrature order:', quad_order)
integral = Integral('i', order=quad_order)
qp, _ = integral.get_qp(options.geometry)
output('number of quadrature points:', qp.shape[0])
u = FieldVariable('u', 'unknown', field, n_c)
v = FieldVariable('v', 'test', field, n_c, primary_var_name='u')
m = Material('m', lam=1.0, mu=1.0)
if options.matrix_type == 'laplace':
term = Term.new('dw_laplace(m.mu, v, u)',
integral,
omega,
m=m,
v=v,
u=u)
n_zero = 1
else:
assert_(options.matrix_type == 'elasticity')
term = Term.new('dw_lin_elastic_iso(m.lam, m.mu, v, u)',
integral,
omega,
m=m,
v=v,
u=u)
n_zero = (dim + 1) * dim / 2
term.setup()
output('assembling...')
tt = time.clock()
mtx, iels = term.evaluate(mode='weak', diff_var='u')
output('...done in %.2f s' % (time.clock() - tt))
mtx = mtx[0][0, 0]
try:
assert_(nm.max(nm.abs(mtx - mtx.T)) < 1e-10)
except:
from sfepy.base.base import debug
debug()
output('matrix shape:', mtx.shape)
eigs = eig(mtx, method='eig.sgscipy', eigenvectors=False)
eigs.sort()
# Zero 'true' zeros.
eigs[:n_zero] = 0.0
ii = nm.where(eigs < 0.0)[0]
if len(ii):
output('matrix is not positive semi-definite!')
ii = nm.where(eigs[n_zero:] < 1e-12)[0]
if len(ii):
output('matrix has more than %d zero eigenvalues!' % n_zero)
output('smallest eigs:\n', eigs[:10])
ii = nm.where(eigs > 0.0)[0]
emin, emax = eigs[ii[[0, -1]]]
output('min:', emin, 'max:', emax)
cond = emax / emin
conds.append(cond)
output('condition number:', cond)
output('...done')
plt.figure(1)
plt.semilogy(orders, conds)
plt.xticks(orders, orders)
plt.xlabel('polynomial order')
plt.ylabel('condition number')
plt.grid()
plt.figure(2)
plt.loglog(orders, conds)
plt.xticks(orders, orders)
plt.xlabel('polynomial order')
plt.ylabel('condition number')
plt.grid()
plt.show()
def describe_geometry(self,
field,
gtype,
region,
integral=None,
return_mapping=False):
"""
Compute jacobians, element volumes and base function derivatives
for Volume-type geometries (volume mappings), and jacobians,
normals and base function derivatives for Surface-type
geometries (surface mappings).
Notes
-----
- volume mappings can be defined on a part of an element group,
although the field has to be defined always on the whole group.
- surface mappings are defined on the surface region
- surface mappings require field order to be > 0
"""
domain = field.domain
group = domain.groups[self.ig]
coors = domain.get_mesh_coors(actual=True)
if gtype == 'volume':
if integral is None:
from sfepy.fem import Integral
integral = Integral('i_tmp', 'v', 1)
qp = self.get_qp('v', integral)
iels = region.cells[self.ig]
geo_ps = self.interp.get_geom_poly_space('v')
ps = self.interp.poly_spaces['v']
bf = self.get_base('v', 0, integral, iels=iels)
conn = nm.take(group.conn, iels, axis=0)
mapping = VolumeMapping(coors, conn, poly_space=geo_ps)
vg = mapping.get_mapping(qp.vals,
qp.weights,
poly_space=ps,
ori=self.ori)
out = vg
elif (gtype == 'surface') or (gtype == 'surface_extra'):
assert_(field.approx_order > 0)
if self.ori is not None:
msg = 'surface integrals do not work yet with the' \
' hierarchical basis!'
raise ValueError(msg)
sd = domain.surface_groups[self.ig][region.name]
esd = self.surface_data[region.name]
qp = self.get_qp(sd.face_type, integral)
geo_ps = self.interp.get_geom_poly_space(sd.face_type)
ps = self.interp.poly_spaces[esd.face_type]
bf = self.get_base(esd.face_type, 0, integral)
conn = sd.get_connectivity()
mapping = SurfaceMapping(coors, conn, poly_space=geo_ps)
sg = mapping.get_mapping(qp.vals, qp.weights, poly_space=ps)
if gtype == 'surface_extra':
sg.alloc_extra_data(self.get_v_data_shape()[2])
self.create_bqp(region.name, integral)
qp = self.qp_coors[(integral.name, esd.bkey)]
v_geo_ps = self.interp.get_geom_poly_space('v')
bf_bg = v_geo_ps.eval_base(qp.vals, diff=True)
ebf_bg = self.get_base(esd.bkey, 1, integral)
sg.evaluate_bfbgm(bf_bg, ebf_bg, coors, sd.fis, group.conn)
out = sg
elif gtype == 'point':
out = mapping = None
else:
raise ValueError('unknown geometry type: %s' % gtype)
if out is not None:
# Store the integral used.
out.integral = integral
out.qp = qp
out.ps = ps
# Update base.
out.bf[:] = bf
if return_mapping:
out = (out, mapping)
return out
def _gen_common_data(orders, gels, report):
import sfepy
from sfepy.base.base import Struct
from sfepy.linalg import combine
from sfepy.fem import Mesh, Domain, Field, FieldVariable, Integral
from sfepy.fem.global_interp import get_ref_coors
bases = ([ii for ii in combine([['2_4', '3_8'],
['lagrange', 'lobatto']])]
+ [ii for ii in combine([['2_3', '3_4'],
['lagrange']])])
for geom, poly_space_base in bases:
report('geometry: %s, base: %s' % (geom, poly_space_base))
order = orders[geom]
integral = Integral('i', order=order)
aux = '' if geom in ['2_4', '3_8'] else 'z'
mesh0 = Mesh.from_file('meshes/elements/%s_2%s.mesh' % (geom, aux),
prefix_dir=sfepy.data_dir)
gel = gels[geom]
perms = gel.get_conn_permutations()
qps, qp_weights = integral.get_qp(gel.surface_facet.name)
zz = nm.zeros_like(qps[:, :1])
qps = nm.hstack(([qps] + [zz]))
shift = shifts[geom]
rcoors = nm.ascontiguousarray(qps
+ shift[:1, :] - shift[1:, :])
ccoors = nm.ascontiguousarray(qps
+ shift[:1, :] + shift[1:, :])
for ir, pr in enumerate(perms):
for ic, pc in enumerate(perms):
report('ir: %d, ic: %d' % (ir, ic))
report('pr: %s, pc: %s' % (pr, pc))
mesh = mesh0.copy()
conn = mesh.conns[0]
conn[0, :] = conn[0, pr]
conn[1, :] = conn[1, pc]
cache = Struct(mesh=mesh)
domain = Domain('domain', mesh)
omega = domain.create_region('Omega', 'all')
region = domain.create_region('Facet', rsels[geom], 'facet')
field = Field.from_args('f', nm.float64, shape=1,
region=omega, approx_order=order,
poly_space_base=poly_space_base)
var = FieldVariable('u', 'unknown', field, 1)
report('# dofs: %d' % var.n_dof)
vec = nm.empty(var.n_dof, dtype=var.dtype)
ap = field.aps[0]
ps = ap.interp.poly_spaces['v']
dofs = field.get_dofs_in_region_group(region, 0,
merge=False)
edofs, fdofs = nm.unique(dofs[1]), nm.unique(dofs[2])
rrc, rcells, rstatus = get_ref_coors(field, rcoors,
cache=cache)
crc, ccells, cstatus = get_ref_coors(field, ccoors,
cache=cache)
assert_((rstatus == 0).all() and (cstatus == 0).all())
yield (geom, poly_space_base, qp_weights, mesh, ir, ic,
ap, ps, rrc, rcells[0, 1], crc, ccells[0, 1],
vec, edofs, fdofs)
def eval_in_els_and_qp(expression, ig, iels, coors,
fields, materials, variables,
functions=None, mode='eval', term_mode=None,
extra_args=None, verbose=True, kwargs=None):
"""
Evaluate an expression in given elements and points.
Parameters
----------
expression : str
The expression to evaluate.
fields : dict
The dictionary of fields used in `variables`.
materials : Materials instance
The materials used in the expression.
variables : Variables instance
The variables used in the expression.
functions : Functions instance, optional
The user functions for materials etc.
mode : one of 'eval', 'el_avg', 'qp'
The evaluation mode - 'qp' requests the values in quadrature points,
'el_avg' element averages and 'eval' means integration over
each term region.
term_mode : str
The term call mode - some terms support different call modes
and depending on the call mode different values are
returned.
extra_args : dict, optional
Extra arguments to be passed to terms in the expression.
verbose : bool
If False, reduce verbosity.
kwargs : dict, optional
The variables (dictionary of (variable name) : (Variable
instance)) to be used in the expression.
Returns
-------
out : array
The result of the evaluation.
"""
weights = nm.ones_like(coors[:, 0])
integral = Integral('ie', coors=coors, weights=weights)
domain = fields.values()[0].domain
region = Region('Elements', 'given elements', domain, '')
region.cells = iels + domain.mesh.el_offsets[ig]
region.update_shape()
domain.regions.append(region)
for field in fields.itervalues():
field.clear_mappings(clear_all=True)
for ap in field.aps.itervalues():
ap.clear_qp_base()
aux = create_evaluable(expression, fields, materials,
variables.itervalues(), Integrals([integral]),
functions=functions,
mode=mode, extra_args=extra_args, verbose=verbose,
kwargs=kwargs)
equations, variables = aux
out = eval_equations(equations, variables,
preserve_caches=False,
mode=mode, term_mode=term_mode)
domain.regions.pop()
return out
def main():
parser = OptionParser(usage=usage, version='%prog')
parser.add_option('-b', '--basis', metavar='name',
action='store', dest='basis',
default='lagrange', help=help['basis'])
parser.add_option('-n', '--max-order', metavar='order', type=int,
action='store', dest='max_order',
default=10, help=help['max_order'])
parser.add_option('-m', '--matrix', metavar='type',
action='store', dest='matrix_type',
default='laplace', help=help['matrix_type'])
parser.add_option('-g', '--geometry', metavar='name',
action='store', dest='geometry',
default='2_4', help=help['geometry'])
options, args = parser.parse_args()
dim, n_ep = int(options.geometry[0]), int(options.geometry[2])
output('reference element geometry:')
output(' dimension: %d, vertices: %d' % (dim, n_ep))
n_c = {'laplace' : 1, 'elasticity' : dim}[options.matrix_type]
output('matrix type:', options.matrix_type)
output('number of variable components:', n_c)
output('polynomial space:', options.basis)
output('max. order:', options.max_order)
mesh = Mesh.from_file(data_dir + '/meshes/elements/%s_1.mesh'
% options.geometry)
domain = Domain('domain', mesh)
omega = domain.create_region('Omega', 'all')
orders = nm.arange(1, options.max_order + 1, dtype=nm.int)
conds = []
order_fix = 0 if options.geometry in ['2_4', '3_8'] else 1
for order in orders:
output('order:', order, '...')
field = Field.from_args('fu', nm.float64, n_c, omega,
approx_order=order,
space='H1', poly_space_base=options.basis)
to = field.approx_order
quad_order = 2 * (max(to - order_fix, 0))
output('quadrature order:', quad_order)
integral = Integral('i', order=quad_order)
qp, _ = integral.get_qp(options.geometry)
output('number of quadrature points:', qp.shape[0])
u = FieldVariable('u', 'unknown', field)
v = FieldVariable('v', 'test', field, primary_var_name='u')
m = Material('m', lam=1.0, mu=1.0)
if options.matrix_type == 'laplace':
term = Term.new('dw_laplace(m.mu, v, u)',
integral, omega, m=m, v=v, u=u)
n_zero = 1
else:
assert_(options.matrix_type == 'elasticity')
term = Term.new('dw_lin_elastic_iso(m.lam, m.mu, v, u)',
integral, omega, m=m, v=v, u=u)
n_zero = (dim + 1) * dim / 2
term.setup()
output('assembling...')
tt = time.clock()
mtx, iels = term.evaluate(mode='weak', diff_var='u')
output('...done in %.2f s' % (time.clock() - tt))
mtx = mtx[0][0, 0]
try:
assert_(nm.max(nm.abs(mtx - mtx.T)) < 1e-10)
except:
from sfepy.base.base import debug; debug()
output('matrix shape:', mtx.shape)
eigs = eig(mtx, method='eig.sgscipy', eigenvectors=False)
eigs.sort()
# Zero 'true' zeros.
eigs[:n_zero] = 0.0
ii = nm.where(eigs < 0.0)[0]
if len(ii):
output('matrix is not positive semi-definite!')
ii = nm.where(eigs[n_zero:] < 1e-12)[0]
if len(ii):
output('matrix has more than %d zero eigenvalues!' % n_zero)
output('smallest eigs:\n', eigs[:10])
ii = nm.where(eigs > 0.0)[0]
emin, emax = eigs[ii[[0, -1]]]
output('min:', emin, 'max:', emax)
cond = emax / emin
conds.append(cond)
output('condition number:', cond)
output('...done')
plt.figure(1)
plt.semilogy(orders, conds)
plt.xticks(orders, orders)
plt.xlabel('polynomial order')
plt.ylabel('condition number')
plt.grid()
plt.figure(2)
plt.loglog(orders, conds)
plt.xticks(orders, orders)
plt.xlabel('polynomial order')
plt.ylabel('condition number')
plt.grid()
plt.show()
