def L2error(self, uh, exact, intorder=None):
"""Compute :math:`L^2` error against exact solution.
The computation is based on the following identity:
.. math::
\|u-u_h\|_0^2 = (u,u)+(u_h,u_h)-2(u,u_h).
Parameters
----------
uh : np.array
The discrete solution.
exact : function handle
The exact solution.
intorder : (OPTIONAL) int
The integration order.
Returns
-------
float
The global :math:`L^2` error.
"""
if self.elem_u.maxdeg != self.elem_v.maxdeg:
raise NotImplementedError("elem_u.maxdeg must be elem_v.maxdeg "
"when computing errors!")
if intorder is None:
intorder = 2*self.elem_u.maxdeg
X, W = get_quadrature(self.mesh.refdom, intorder)
# assemble some helper matrices
# the idea is to use the identity: (u-uh,u-uh)=(u,u)+(uh,uh)-2(u,uh)
def uv(u, v):
return u*v
def fv(v, x):
return exact(x)*v
M = self.iasm(uv)
f = self.iasm(fv)
detDF = self.mapping.detDF(X)
x = self.mapping.F(X)
uu = np.sum(np.dot(exact(x)**2*np.abs(detDF), W))
return np.sqrt(uu + np.dot(uh, M.dot(uh)) - 2.*np.dot(uh, f))
def inorm(self, form, interp, intorder=None):
# evaluate norm on all elements
tind = range(self.mesh.t.shape[1])
if not isinstance(interp, dict):
raise Exception("The input solution vector(s) must be in a "
"dictionary! Pass e.g. {0:u} instead of u.")
if intorder is None:
intorder = 2*self.elem_u.maxdeg
# check and fix parameters of form
oldparams = inspect.getargspec(form).args
paramlist = ['u', 'du', 'ddu', 'x', 'h']
fform = self.fillargs(form, paramlist)
X, W = get_quadrature(self.mesh.refdom, intorder)
# mappings
x = self.mapping.F(X, tind) # reference facet to global facet
# jacobian at quadrature points
detDF = self.mapping.detDF(X, tind)
Nbfun_u = self.dofnum_u.t_dof.shape[0]
dim = self.mesh.p.shape[0]
# interpolate the solution vectors at quadrature points
zero = 0.0*x[0]
w, dw, ddw = ({} for i in range(3))
for k in interp:
w[k] = zero
dw[k] = const_cell(zero, dim)
ddw[k] = const_cell(zero, dim, dim)
for j in range(Nbfun_u):
jdofs = self.dofnum_u.t_dof[j, :]
w[k] += interp[k][jdofs][:, None]\
* self.u[j]
for a in range(dim):
dw[k][a] += interp[k][jdofs][:, None]\
* self.du[j][a]
for b in range(dim):
ddw[k][a][b] += interp[k][jdofs][:, None]\
* self.ddu[j][a][b]
# compute the mesh parameter from jacobian determinant
h = np.abs(detDF)**(1.0/self.mesh.dim())
return np.dot(fform(w, dw, ddw, x, h)**2*np.abs(detDF), W)
def __init__(self, mesh, elem_u, elem_v=None, intorder=None):
if not isinstance(mesh, spfem.mesh.Mesh):
raise Exception("First parameter must be an instance of "
"spfem.mesh.Mesh.")
if not isinstance(elem_u, spfem.element.AbstractElement):
raise Exception("Second parameter must be an instance of "
"spfem.element.AbstractElement.")
if elem_v is not None:
if not isinstance(elem_v, spfem.element.AbstractElement):
raise Exception("Third parameter must be an instance of "
"spfem.element.AbstractElement.")
self.mesh = mesh
self.elem_u = elem_u
self.dofnum_u = Dofnum(mesh, elem_u)
self.mapping = mesh.mapping()
# duplicate test function element type if None is given
if elem_v is None:
self.elem_v = elem_u
self.dofnum_v = self.dofnum_u
else:
self.elem_v = elem_v
self.dofnum_v = Dofnum(mesh, elem_v)
if intorder is None:
# compute the maximum polynomial degree from elements
self.intorder = self.elem_u.maxdeg + self.elem_v.maxdeg
else:
self.intorder = intorder
# quadrature points and weights
X, _ = get_quadrature(self.mesh.refdom, self.intorder)
# global quadrature points
x = self.mapping.F(X, range(self.mesh.t.shape[1]))
# pre-compute basis functions at quadrature points
self.u, self.du, self.ddu = self.elem_u.evalbasis(self.mesh, x)
if elem_v is None:
self.v = self.u
self.dv = self.du
self.ddv = self.ddu
else:
self.v, self.dv, self.ddv = self.elem_v.evalbasis(self.mesh, x)
def H1error(self, uh, dexact, intorder=None):
"""Compute :math:`H^1` seminorm error against exact solution.
The computation is based on the following identity:
.. math::
\|\\nabla(u-u_h)\|_0^2 = (\\nabla u,\\nabla u)
+ (\\nabla u_h, \\nabla u_h)
- 2(\\nabla u, \\nabla u_h).
Parameters
----------
uh : np.array
The discrete solution.
dexact : function handle
The derivative of the exact solution.
intorder : (OPTIONAL) int
The integration order.
Returns
-------
float
The global :math:`H^1` error.
"""
if self.elem_u.maxdeg != self.elem_v.maxdeg:
raise NotImplementedError("elem_u.maxdeg must be elem_v.maxdeg "
"when computing errors!")
if intorder is None:
intorder = 2*self.elem_u.maxdeg
X, W = get_quadrature(self.mesh.refdom, intorder)
# assemble some helper matrices
# the idea is to use the identity: (u-uh,u-uh)=(u,u)+(uh,uh)-2(u,uh)
def uv(du, dv):
if not isinstance(du, dict):
return du*dv
elif len(du) == 2:
return du[0]*dv[0] + du[1]*dv[1]
elif len(du) == 3:
return du[0]*dv[0] + du[1]*dv[1] + du[2]*dv[2]
else:
raise NotImplementedError("AssemblerElement.H1error not "
"implemented for current domain "
"dimension!")
def fv(dv, x):
if not isinstance(x, dict):
return dexact(x)*dv
elif len(x) == 2:
return dexact[0](x)*dv[0] + dexact[1](x)*dv[1]
elif len(x) == 3:
return dexact[0](x)*dv[0] + dexact[1](x)*dv[1]\
+ dexact[2](x)*dv[2]
else:
raise NotImplementedError("AssemblerElement.H1error not "
"implemented for current domain "
"dimension!")
M = self.iasm(uv)
f = self.iasm(fv)
detDF = self.mapping.detDF(X)
x = self.mapping.F(X)
if not isinstance(x, dict):
uu = np.sum(np.dot((dexact(x)**2) * np.abs(detDF), W))
elif len(x) == 2:
uu = np.sum(np.dot((dexact[0](x)**2 + dexact[1](x)**2)
* np.abs(detDF), W))
elif len(x) == 3:
uu = np.sum(np.dot((dexact[0](x)**2 + dexact[1](x)**2
+ dexact[2](x)**2) * np.abs(detDF), W))
else:
raise NotImplementedError("AssemblerElement.H1error not "
"implemented for current domain "
"dimension!")
return np.sqrt(uu + np.dot(uh, M.dot(uh)) - 2.*np.dot(uh, f))
def fasm(self, form, find=None, interior=False, intorder=None,
normals=True, interp=None):
"""Facet assembly."""
if find is None:
if interior:
find = self.mesh.interior_facets()
else:
find = self.mesh.boundary_facets()
if intorder is None:
intorder = self.elem_u.maxdeg + self.elem_v.maxdeg
nv = self.mesh.p.shape[1]
nt = self.mesh.t.shape[1]
ne = find.shape[0]
# check and fix parameters of form
oldparams = inspect.getargspec(form).args
if interior:
if 'u1' in oldparams or 'du1' in oldparams:
paramlist = ['u1', 'u2', 'v1', 'v2',
'du1', 'du2', 'dv1', 'dv2',
'x', 'h', 'n', 'w']
bilinear = True
else:
paramlist = ['v1', 'v2', 'dv1', 'dv2',
'x', 'h', 'n', 'w']
bilinear = False
else:
if 'u' in oldparams or 'du' in oldparams:
paramlist = ['u', 'v', 'du', 'dv', 'x', 'h', 'n', 'w']
bilinear = True
else:
paramlist = ['v', 'dv', 'x', 'h', 'n', 'w']
bilinear = False
fform = self.fillargs(form, paramlist)
X, W = get_quadrature(self.mesh.brefdom, intorder)
# boundary element indices
tind1 = self.mesh.f2t[0, find]
tind2 = self.mesh.f2t[1, find]
# mappings
x = self.mapping.G(X, find=find) # reference facet to global facet
Y1 = self.mapping.invF(x, tind=tind1) # global facet to ref element
Y2 = self.mapping.invF(x, tind=tind2) # global facet to ref element
Nbfun_u = self.dofnum_u.t_dof.shape[0]
Nbfun_v = self.dofnum_v.t_dof.shape[0]
detDG = self.mapping.detDG(X, find)
# compute normal vectors
n = {}
if normals:
# normals based on tind1 only
n = self.mapping.normals(Y1, tind1, find, self.mesh.t2f)
# compute the mesh parameter from jacobian determinant
if self.mesh.dim() > 1.0:
h = np.abs(detDG)**(1.0/(self.mesh.dim() - 1.0))
else: # exception for 1D mesh (no boundary h defined)
h = None
# interpolate some previous discrete function at quadrature points
w = {}
if interp is not None:
if not isinstance(interp, dict):
raise Exception("The input solution vector(s) must be in a "
"dictionary! Pass e.g. {0:u} instead of u.")
for k in interp:
w[k] = 0.0*x[0]
for j in range(Nbfun_u):
phi, _ = self.elem_u.gbasis(self.mapping, Y1, j, tind1)
w[k] += interp[k][self.dofnum_u.t_dof[j, tind1], None]*phi
# bilinear form
if bilinear:
# initialize sparse matrix structures
ndata = Nbfun_u*Nbfun_v*ne
if interior:
data = np.zeros(4*ndata)
rows = np.zeros(4*ndata)
cols = np.zeros(4*ndata)
else:
data = np.zeros(ndata)
rows = np.zeros(ndata)
cols = np.zeros(ndata)
for j in range(Nbfun_u):
u1, du1 = self.elem_u.gbasis(self.mapping, Y1, j, tind1)
if interior:
u2, du2 = self.elem_u.gbasis(self.mapping, Y2, j, tind2)
if j == 0:
# these are zeros corresponding to the shapes of u,du
z = const_cell(0, *cell_shape(u2))
dz = const_cell(0, *cell_shape(du2))
for i in range(Nbfun_v):
v1, dv1 = self.elem_v.gbasis(self.mapping, Y1, i, tind1)
if interior:
v2, dv2 = self.elem_v.gbasis(self.mapping, Y2, i, tind2)
# compute entries of local stiffness matrices
if interior:
ixs1 = slice(ne*(Nbfun_v*j + i),
ne*(Nbfun_v*j + i + 1))
ixs2 = slice(ne*(Nbfun_v*j + i) + ndata,
ne*(Nbfun_v*j + i + 1) + ndata)
ixs3 = slice(ne*(Nbfun_v*j + i) + 2*ndata,
ne*(Nbfun_v*j + i + 1) + 2*ndata)
ixs4 = slice(ne*(Nbfun_v*j + i) + 3*ndata,
ne*(Nbfun_v*j + i + 1) + 3*ndata)
data[ixs1] = np.dot(fform(u1, z, v1, z,
du1, dz, dv1, dz,
x, h, n, w)*np.abs(detDG), W)
rows[ixs1] = self.dofnum_v.t_dof[i, tind1]
cols[ixs1] = self.dofnum_u.t_dof[j, tind1]
data[ixs2] = np.dot(fform(z, u2, z, v2,
dz, du2, dz, dv2,
x, h, n, w)*np.abs(detDG), W)
rows[ixs2] = self.dofnum_v.t_dof[i, tind2]
cols[ixs2] = self.dofnum_u.t_dof[j, tind2]
data[ixs3] = np.dot(fform(z, u2, v1, z,
dz, du2, dv1, dz,
x, h, n, w)*np.abs(detDG), W)
rows[ixs3] = self.dofnum_v.t_dof[i, tind1]
cols[ixs3] = self.dofnum_u.t_dof[j, tind2]
data[ixs4] = np.dot(fform(u1, z, z, v2,
du1, dz, dz, dv2,
x, h, n, w)*np.abs(detDG), W)
rows[ixs4] = self.dofnum_v.t_dof[i, tind2]
cols[ixs4] = self.dofnum_u.t_dof[j, tind1]
else:
ixs = slice(ne*(Nbfun_v*j + i), ne*(Nbfun_v*j + i + 1))
data[ixs] = np.dot(fform(u1, v1, du1, dv1,
x, h, n, w)*np.abs(detDG), W)
rows[ixs] = self.dofnum_v.t_dof[i, tind1]
cols[ixs] = self.dofnum_u.t_dof[j, tind1]
return coo_matrix((data, (rows, cols)),
shape=(self.dofnum_v.N, self.dofnum_u.N)).tocsr()
# linear form
else:
if interior:
# could not find any use case
raise Exception("No interior support in linear facet form.")
# initialize sparse matrix structures
data = np.zeros(Nbfun_v*ne)
rows = np.zeros(Nbfun_v*ne)
cols = np.zeros(Nbfun_v*ne)
for i in range(Nbfun_v):
v1, dv1 = self.elem_v.gbasis(self.mapping, Y1, i, tind1)
# find correct location in data,rows,cols
ixs = slice(ne*i, ne*(i + 1))
# compute entries of local stiffness matrices
data[ixs] = np.dot(fform(v1, dv1, x, h, n, w)*np.abs(detDG), W)
rows[ixs] = self.dofnum_v.t_dof[i, tind1]
cols[ixs] = np.zeros(ne)
return coo_matrix((data, (rows, cols)),
shape=(self.dofnum_v.N, 1)).toarray().T[0]
def iasm(self, form, intorder=None, tind=None, interp=None):
"""Return a matrix related to a bilinear or linear form
where the integral is over the interior of the domain.
Parameters
----------
form : function handle
The bilinear or linear form function handle.
The supported parameters can be found in the
following table.
+-----------+----------------------+--------------+
| Parameter | Explanation | Supported in |
+-----------+----------------------+--------------+
| u | solution | bilinear |
+-----------+----------------------+--------------+
| v | test fun | both |
+-----------+----------------------+--------------+
| du | solution derivatives | bilinear |
+-----------+----------------------+--------------+
| dv | test fun derivatives | both |
+-----------+----------------------+--------------+
| x | spatial location | both |
+-----------+----------------------+--------------+
| w | cf. interp | both |
+-----------+----------------------+--------------+
| h | the mesh parameter | both |
+-----------+----------------------+--------------+
The function handle must use these exact names for
the variables. Unused variable names can be omitted.
Examples of valid bilinear forms:
::
def bilin_form1(du, dv):
# Note that the element must be
# defined for two-dimensional
# meshes for this to make sense!
return du[0]*dv[0] + du[1]*dv[1]
def bilin_form2(du, v):
return du[0]*v
bilin_form3 = lambda u, v, x: x[0]**2*u*v
Examples of valid linear forms:
::
def lin_form1(v):
return v
def lin_form2(h, x, v):
import numpy as np
mesh_parameter = h
X = x[0]
Y = x[1]
return mesh_parameter*np.sin(np.pi*X)*np.sin(np.pi*Y)*v
The linear forms are automatically detected to be
non-bilinear through the omission of u or du.
intorder : int
The order of polynomials for which the applied
quadrature rule is exact. By default,
2*Element.maxdeg is used. Reducing this number
can sometimes reduce the computation time.
interp : numpy array
Using this flag, the user can provide
a solution vector that is interpolated
to the quadrature points and included in
the computation of the bilinear form
(the variable w). Useful e.g. when solving
nonlinear problems.
"""
if tind is None:
# assemble on all elements by default
tind = range(self.mesh.t.shape[1])
nt = len(tind)
if intorder is None:
# compute the maximum polynomial degree from elements
intorder = self.elem_u.maxdeg + self.elem_v.maxdeg
# check and fix parameters of form
oldparams = inspect.getargspec(form).args
if 'u' in oldparams or 'du' in oldparams:
paramlist = ['u', 'v', 'du', 'dv', 'x', 'w', 'h']
bilinear = True
else:
paramlist = ['v', 'dv', 'x', 'w', 'h']
bilinear = False
fform = self.fillargs(form, paramlist)
# quadrature points and weights
X, W = get_quadrature(self.mesh.refdom, intorder)
# global quadrature points
x = self.mapping.F(X, tind)
# jacobian at quadrature points
detDF = self.mapping.detDF(X, tind)
Nbfun_u = self.dofnum_u.t_dof.shape[0]
Nbfun_v = self.dofnum_v.t_dof.shape[0]
# interpolate some previous discrete function at quadrature points
w = {}
if interp is not None:
if not isinstance(interp, dict):
raise Exception("The input solution vector(s) must be in a "
"dictionary! Pass e.g. {0:u} instead of u.")
for k in interp:
w[k] = 0.0*x[0]
for j in range(Nbfun_u):
phi, _ = self.elem_u.lbasis(X, j)
w[k] += np.outer(interp[k][self.dofnum_u.t_dof[j, :]], phi)
# compute the mesh parameter from jacobian determinant
h = np.abs(detDF)**(1.0/self.mesh.dim())
# bilinear form
if bilinear:
# initialize sparse matrix structures
data = np.zeros(Nbfun_u*Nbfun_v*nt)
rows = np.zeros(Nbfun_u*Nbfun_v*nt)
cols = np.zeros(Nbfun_u*Nbfun_v*nt)
for j in range(Nbfun_u):
u, du = self.elem_u.gbasis(self.mapping, X, j, tind)
for i in range(Nbfun_v):
v, dv = self.elem_v.gbasis(self.mapping, X, i, tind)
# find correct location in data,rows,cols
ixs = slice(nt*(Nbfun_v*j+i), nt*(Nbfun_v*j+i+1))
# compute entries of local stiffness matrices
data[ixs] = np.dot(fform(u, v, du, dv, x, w, h)
* np.abs(detDF), W)
rows[ixs] = self.dofnum_v.t_dof[i, tind]
cols[ixs] = self.dofnum_u.t_dof[j, tind]
return coo_matrix((data, (rows, cols)),
shape=(self.dofnum_v.N, self.dofnum_u.N)).tocsr()
else:
# initialize sparse matrix structures
data = np.zeros(Nbfun_v*nt)
rows = np.zeros(Nbfun_v*nt)
cols = np.zeros(Nbfun_v*nt)
for i in range(Nbfun_v):
v, dv = self.elem_v.gbasis(self.mapping, X, i, tind)
# find correct location in data,rows,cols
ixs = slice(nt*i, nt*(i+1))
# compute entries of local stiffness matrices
data[ixs] = np.dot(fform(v, dv, x, w, h)*np.abs(detDF), W)
rows[ixs] = self.dofnum_v.t_dof[i, :]
cols[ixs] = np.zeros(nt)
return coo_matrix((data, (rows, cols)),
shape=(self.dofnum_v.N, 1)).toarray().T[0]
def fnorm(self, form, interp, intorder=None, interior=False, normals=True):
if interior:
# evaluate norm on all interior facets
find = self.mesh.interior_facets()
else:
# evaluate norm on all boundary facets
find = self.mesh.boundary_facets()
if not isinstance(interp, dict):
raise Exception("The input solution vector(s) must be in a "
"dictionary! Pass e.g. {0:u} instead of u.")
if intorder is None:
intorder = 2*self.elem_u.maxdeg
# check and fix parameters of form
oldparams = inspect.getargspec(form).args
if 'u1' in oldparams or 'du1' in oldparams or 'ddu1' in oldparams:
if interior is False:
raise Exception("The supplied form contains u1 although "
"no interior=True is given.")
if interior:
paramlist = ['u1', 'u2', 'du1', 'du2', 'ddu1', 'ddu2',
'x', 'n', 't', 'h']
else:
paramlist = ['u', 'du', 'ddu', 'x', 'n', 't', 'h']
fform = self.fillargs(form, paramlist)
X, W = get_quadrature(self.mesh.brefdom, intorder)
# indices of elements at different sides of facets
tind1 = self.mesh.f2t[0, find]
tind2 = self.mesh.f2t[1, find]
# mappings
x = self.mapping.G(X, find=find) # reference facet to global facet
detDG = self.mapping.detDG(X, find)
# compute basis function values at quadrature points
u1, du1, ddu1 = self.elem_u.evalbasis(self.mesh, x, tind=tind1)
if interior:
u2, du2, ddu2 = self.elem_u.evalbasis(self.mesh, x, tind=tind2)
Nbfun_u = self.dofnum_u.t_dof.shape[0]
dim = self.mesh.p.shape[0]
n = {}
t = {}
if normals:
Y = self.mapping.invF(x, tind=tind1) # global facet to ref element
n = self.mapping.normals(Y, tind1, find, self.mesh.t2f)
if len(n) == 2: # TODO fix for 3D and other than triangles?
t[0] = -n[1]
t[1] = n[0]
# interpolate the solution vectors at quadrature points
zero = np.zeros((len(find), len(W)))
w1, dw1, ddw1 = ({} for i in range(3))
if interior:
w2, dw2, ddw2 = ({} for i in range(3))
for k in interp:
w1[k] = zero
dw1[k] = const_cell(zero, dim)
ddw1[k] = const_cell(zero, dim, dim)
if interior:
w2[k] = zero
dw2[k] = const_cell(zero, dim)
ddw2[k] = const_cell(zero, dim, dim)
for j in range(Nbfun_u):
jdofs1 = self.dofnum_u.t_dof[j, tind1]
jdofs2 = self.dofnum_u.t_dof[j, tind2]
w1[k] += interp[k][jdofs1][:, None] * u1[j]
if interior:
w2[k] += interp[k][jdofs2][:, None] * u2[j]
for a in range(dim):
dw1[k][a] += interp[k][jdofs1][:, None]\
* du1[j][a]
if interior:
dw2[k][a] += interp[k][jdofs2][:, None]\
* du2[j][a]
for b in range(dim):
ddw1[k][a][b] += interp[k][jdofs1][:, None]\
* ddu1[j][a][b]
if interior:
ddw2[k][a][b] += interp[k][jdofs2][:, None]\
* ddu2[j][a][b]
h = np.abs(detDG)**(1.0/(self.mesh.dim()-1.0))
if interior:
return np.dot(fform(w1, w2, dw1, dw2, ddw1, ddw2,
x, n, t, h)**2*np.abs(detDG), W), find
else:
return np.dot(fform(w1, dw1, ddw1,
x, n, t, h)**2*np.abs(detDG), W), find
def iasm(self, form, tind=None, interp=None):
if tind is None:
# assemble on all elements by default
tind = range(self.mesh.t.shape[1])
nt = len(tind)
# check and fix parameters of form
oldparams = inspect.getargspec(form).args
if 'u' in oldparams or 'du' in oldparams or 'ddu' in oldparams:
paramlist = ['u', 'v', 'du', 'dv', 'ddu', 'ddv', 'x', 'w', 'h']
bilinear = True
else:
paramlist = ['v', 'dv', 'ddv', 'x', 'w', 'h']
bilinear = False
fform = self.fillargs(form, paramlist)
# quadrature points and weights
X, W = get_quadrature(self.mesh.refdom, self.intorder)
# global quadrature points
x = self.mapping.F(X, tind)
# jacobian at quadrature points
detDF = self.mapping.detDF(X, tind)
Nbfun_u = self.dofnum_u.t_dof.shape[0]
Nbfun_v = self.dofnum_v.t_dof.shape[0]
# interpolate some previous discrete function at quadrature points
w = {}
if interp is not None:
if not isinstance(interp, dict):
raise Exception("The input solution vector(s) must be in a "
"dictionary! Pass e.g. {0:u} instead of u.")
# interpolate the solution vectors at quadrature points
zero = 0.0*x[0]
w = {}
for k in interp:
w[k] = zero
for j in range(self.Nbfun_u):
jdofs = self.dofnum_u.t_dof[j, :]
w[k] += interp[k][jdofs][:, None]*self.u[j]
# compute the mesh parameter from jacobian determinant
h = np.abs(detDF)**(1.0/self.mesh.dim())
# bilinear form
if bilinear:
# initialize sparse matrix structures
data = np.zeros(Nbfun_u*Nbfun_v*nt)
rows = np.zeros(Nbfun_u*Nbfun_v*nt)
cols = np.zeros(Nbfun_u*Nbfun_v*nt)
for j in range(Nbfun_u):
for i in range(Nbfun_v):
# find correct location in data,rows,cols
ixs = slice(nt*(Nbfun_v*j + i), nt*(Nbfun_v*j + i + 1))
# compute entries of local stiffness matrices
data[ixs] = np.dot(fform(self.u[j], self.v[i],
self.du[j], self.dv[i],
self.ddu[j], self.ddv[i],
x, w, h)*np.abs(detDF), W)
rows[ixs] = self.dofnum_v.t_dof[i, tind]
cols[ixs] = self.dofnum_u.t_dof[j, tind]
return coo_matrix((data, (rows, cols)),
shape=(self.dofnum_v.N, self.dofnum_u.N)).tocsr()
else:
# initialize sparse matrix structures
data = np.zeros(Nbfun_v*nt)
rows = np.zeros(Nbfun_v*nt)
cols = np.zeros(Nbfun_v*nt)
for i in range(Nbfun_v):
# find correct location in data,rows,cols
ixs = slice(nt*i, nt*(i + 1))
# compute entries of local stiffness matrices
data[ixs] = np.dot(fform(self.v[i], self.dv[i], self.ddv[i],
x, w, h)*np.abs(detDF), W)
rows[ixs] = self.dofnum_v.t_dof[i, :]
cols[ixs] = np.zeros(nt)
return coo_matrix((data, (rows, cols)),
shape=(self.dofnum_v.N, 1)).toarray().T[0]