def assembly_mass_matrix_D(V):
# ... sizes
[s1, s2] = V.vector_space.starts
[e1, e2] = V.vector_space.ends
[p1, p2] = V.vector_space.pads
# ...
# ... seetings
[k1, k2] = [W.quad_order for W in V.spaces]
[spans_1, spans_2] = [W.spans for W in V.spaces]
[basis_1, basis_2] = [W.quad_basis for W in V.spaces]
[weights_1, weights_2] = [W.quad_weights for W in V.spaces]
[points_1, points_2] = [W.quad_points for W in V.spaces]
[V_1, V_2] = [W for W in V.spaces]
# print(V.spaces[0].quad_order)
# ... data structure
M = StencilMatrix(V.vector_space, V.vector_space)
# ...
# ... build matrices
for ie1 in range(s1, e1 + 1 - p1):
for ie2 in range(s2, e2 + 1 - p2):
i_span_1 = spans_1[ie1]
i_span_2 = spans_2[ie2]
for il_1 in range(0, p1 + 1):
for jl_1 in range(0, p1 + 1):
for il_2 in range(0, p2 + 1):
for jl_2 in range(0, p2 + 1):
i1 = i_span_1 - p1 - 1 + il_1
j1 = i_span_1 - p1 - 1 + jl_1
i2 = i_span_2 - p2 - 1 + il_2
j2 = i_span_2 - p2 - 1 + jl_2
v_s = 0.0
for g1 in range(0, k1):
for g2 in range(0, k2):
di1_0 = coefficients_linear_combinaison(
V_1, i1) * basis_1[ie1, il_1, 0, g1]
di2_0 = coefficients_linear_combinaison(
V_2, i2) * basis_2[ie1, il_2, 0, g1]
bi_0 = di1_0 * di2_0
d_j1 = basis_1[
ie1, jl_1, 0,
g1] * coefficients_linear_combinaison(
V_1, j1)
d_j2 = basis_2[
ie2, jl_2, 0,
g2] * coefficients_linear_combinaison(
V_2, j2)
bj_0 = d_j1 * d_j2
wvol = weights_1[ie1, g1] * weights_2[ie2,
g2]
v_s = v_s + bi_0 * bj_0 * wvol
M[i1, i2, j1 - i1, j2 - i2] += v_s
# ...
return M
def assembly_mass_V(V):
# ... sizes
[s1, s2] = V.vector_space.starts
[e1, e2] = V.vector_space.ends
[p1, p2] = V.vector_space.pads
# ...
# ... seetings
[k1, k2] = [W.quad_order for W in V.spaces]
[spans_1, spans_2] = [W.spans for W in V.spaces]
[basis_1, basis_2] = [W.quad_basis for W in V.spaces]
[weights_1, weights_2] = [W.quad_weights for W in V.spaces]
[points_1, points_2] = [W.quad_points for W in V.spaces]
# print(V.spaces[0].quad_order)
# ... data structure
M = StencilMatrix(V.vector_space, V.vector_space)
# ...
# ... build matrices
for ie1 in range(s1, e1 + 1 - p1):
for ie2 in range(s2, e2 + 1 - p2):
i_span_1 = spans_1[ie1]
i_span_2 = spans_2[ie2]
for il_1 in range(0, p1 + 1):
for jl_1 in range(0, p1 + 1):
for il_2 in range(0, p2 + 1):
for jl_2 in range(0, p2 + 1):
i1 = i_span_1 - p1 - 1 + il_1
j1 = i_span_1 - p1 - 1 + jl_1
i2 = i_span_2 - p2 - 1 + il_2
j2 = i_span_2 - p2 - 1 + jl_2
v_s = 0.0
for g1 in range(0, k1):
for g2 in range(0, k2):
bi_0 = basis_1[ie1, il_1, 0,
g1] * basis_2[ie1, il_2, 0,
g2]
bj_0 = basis_1[ie1, jl_1, 0,
g1] * basis_2[ie2, jl_2, 0,
g2]
# bj_x = basis_1[ie1, jl_1, 1, g1] * basis_2[ie2, jl_2, 0, g2]
# bj_y = basis_1[ie1, jl_1, 0, g1] * basis_2[ie2, jl_2, 1, g2]
wvol = weights_1[ie1, g1] * weights_2[ie2,
g2]
# v_s += (bi_x * bj_x + bi_y * bj_y) * wvol
v_s = v_s + bi_0 * bj_0 * wvol
M[i1, i2, j1 - i1, j2 - i2] += v_s
# ...
return M
def assemblyDSpanMassMatrix(V): """ Assembly the mass matrix of the space V Entries: -------- V: SplineSpace Output: ------- mass : StencilMatrix - the mass matrix """ # ... sizes [s1] = V.vector_space.starts [e1] = V.vector_space.ends [p1] = V.vector_space.pads k1 = V.quad_order spans_1 = V.spans basis_1 = V.quad_basis weights_1 = V.quad_weights knots_1 = V.knots # ... # ... data structure mass = StencilMatrix( V.vector_space, V.vector_space ) # ... # ... build matrices for ie1 in range(s1, e1+1-p1): i_span_1 = spans_1[ie1] for il_1 in range(0, p1+1): for jl_1 in range(0, p1+1): i1 = i_span_1 - p1 + il_1 j1 = i_span_1 - p1 + jl_1 # -- assembly D_j^p B-spline family: D_j^p = alpha * basis_w_0 alpha_j = coeffDspan(V, j1) # -- assembly D_i^p B-spline family: D_i^p = alpha * basis_w_0 alpha_i = coeffDspan(V, i1) v_m = 0.0 for g1 in range(0, k1): bi_0 = basis_1[ie1, il_1, 0, g1] bj_0 = basis_1[ie1, jl_1, 0, g1] di_0 = bi_0 * alpha_i dj_0 = bj_0 * alpha_j wvol = weights_1[ie1, g1] v_m += di_0 * dj_0 * wvol mass[i1, j1 - i1] += v_m return mass
def assemblyMassMatrix(V): """ Assembly the mass matrix of the space V Entries: -------- V: SplineSpace Output: ------- mass : StencilMatrix - the mass matrix """ # ... sizes [s1] = V.vector_space.starts [e1] = V.vector_space.ends [p1] = V.vector_space.pads k1 = V.quad_order spans_1 = V.spans basis_1 = V.quad_basis weights_1 = V.quad_weights # ... # ... data structure mass = StencilMatrix( V.vector_space, V.vector_space ) # ... # ... build matrices for ie1 in range(s1, e1+1-p1): i_span_1 = spans_1[ie1] for il_1 in range(0, p1+1): for jl_1 in range(0, p1+1): i1 = i_span_1 - p1 + il_1 j1 = i_span_1 - p1 + jl_1 v_m = 0.0 for g1 in range(0, k1): bi_0 = basis_1[ie1, il_1, 0, g1] bj_0 = basis_1[ie1, jl_1, 0, g1] wvol = weights_1[ie1, g1] v_m += bi_0 * bj_0 * wvol mass[i1, j1 - i1] += v_m # ... # ... return mass
def __init__(self, V):
"""
Entry:
------
V: SplineSpace used
"""
# ... sizes
[s1] = V.vector_space.starts
[e1] = V.vector_space.ends
[p1] = V.vector_space.pads
k1 = V.quad_order
spans_1 = V.spans
basis_1 = V.quad_basis
weights_1 = V.quad_weights
# ...
# ... data structure
stiffness = StencilMatrix(V.vector_space, V.vector_space)
# ...
# ... build matrices
for ie1 in range(s1, e1 + 1 - p1):
i_span_1 = spans_1[ie1]
for il_1 in range(0, p1 + 1):
for jl_1 in range(0, p1 + 1):
i1 = i_span_1 - p1 + il_1
j1 = i_span_1 - p1 + jl_1
v_s = 0.0
for g1 in range(0, k1):
bi_x = basis_1[ie1, il_1, 1, g1]
bj_x = basis_1[ie1, jl_1, 1, g1]
wvol = weights_1[ie1, g1]
v_s += (bi_x * bj_x) * wvol
stiffness[i1, j1 - i1] += v_s
# ...
# ...
self._stiff = stiffness
def assembly(V, kernel):
# ... sizes
[s1, s2] = V.vector_space.starts
[e1, e2] = V.vector_space.ends
[p1, p2] = V.vector_space.pads
# ...
# ... seetings
[k1, k2] = [W.quad_order for W in V.spaces]
[spans_1, spans_2] = [W.spans for W in V.spaces]
[basis_1, basis_2] = [W.basis for W in V.spaces]
[weights_1, weights_2] = [W.weights for W in V.spaces]
[points_1, points_2] = [W.points for W in V.spaces]
# ...
# ... data structure
M = StencilMatrix(V.vector_space, V.vector_space)
# ...
# ... element matrix
mat = zeros((p1 + 1, p2 + 1, 2 * p1 + 1, 2 * p2 + 1), order='F')
# ...
# ... build matrices
for ie1 in range(s1, e1 + 1 - p1):
for ie2 in range(s2, e2 + 1 - p2):
i_span_1 = spans_1[ie1]
i_span_2 = spans_2[ie2]
bs1 = basis_1[:, :, :, ie1]
bs2 = basis_2[:, :, :, ie2]
w1 = weights_1[:, ie1]
w2 = weights_2[:, ie2]
kernel(p1, p2, k1, k2, bs1, bs2, w1, w2, mat)
i1 = i_span_1 - p1 - 1
i2 = i_span_2 - p2 - 1
M[i1:i1 + p1 + 1, i2:i2 + p2 + 1, :, :] += mat[:, :, :, :]
def assemble_matrix_1d(V, kernel, args=None, M=None):
# ... sizes
[s1] = V.vector_space.starts
[e1] = V.vector_space.ends
[p1] = V.vector_space.pads
k1 = V.quad_order
spans_1 = V.spans
basis_1 = V.basis
weights_1 = V.weights
# ...
# ... data structure if not given
if M is None:
M = StencilMatrix(V.vector_space, V.vector_space)
# ...
# ... element matrix
mat = np.zeros((p1+1,2*p1+1), order='F')
# ...
if args is None:
_kernel = kernel
else:
_kernel = lambda p1, k1, bs, w, mat: kernel(p1, k1, bs, w, mat, *args)
# ... build matrices
for ie1 in range(s1, e1+1-p1):
i_span_1 = spans_1[ie1]
bs = basis_1[:, :, :, ie1]
w = weights_1[:, ie1]
_kernel(p1, k1, bs, w, mat)
s1 = i_span_1 - p1 - 1
M._data[s1:s1+p1+1,:] += mat[:,:]
# ...
return M
# ... numbers of elements and degres
n1 = 8
n2 = 4
p1 = 2
p2 = 1
# ...
# ... Vector Spaces
V = StencilVectorSpace([n1, n2], [p1, p2], [False, False])
V1 = StencilVectorSpace([n1], [p1], [False])
V2 = StencilVectorSpace([n2], [p2], [False])
# ... Inputs
X0 = StencilVector(V)
A = StencilMatrix(V1, V1)
B = StencilMatrix(V2, V2)
# ...
# ... Fill in A, B and X
utils.populate_1d_matrix(A, 5.)
utils.populate_1d_matrix(B, 6.)
utils.populate_2d_vector(X0)
# ...
# ..
Y = kron_dot(B, A, X0)
X = kron_solve(B, A, Y)
print('X0 = \n', X0._data)
def assemblyMassMatrix2D(V):
"""
Require:
--------
@V is a FemTensor V= (V1, V2) with @V1, @V2 are FemSpace
Returns:
--------
@M is a StencilMatrix - the mass matrix for V space
"""
# ... sizes
[s1, s2] = V.vector_space.starts
[e1, e2] = V.vector_space.ends
[p1, p2] = V.vector_space.pads
# ...
# ... seetings
[k1, k2] = [W.quad_order for W in V.spaces]
[spans_1, spans_2] = [W.spans for W in V.spaces]
[basis_1, basis_2] = [W.quad_basis for W in V.spaces]
[weights_1, weights_2] = [W.quad_weights for W in V.spaces]
[points_1, points_2] = [W.quad_points for W in V.spaces]
# ... data structure
M = StencilMatrix(V.vector_space, V.vector_space)
# ...
# ... build matrices
for ie1 in range(s1, e1 + 1 - p1):
for ie2 in range(s2, e2 + 1 - p2):
i_span_1 = spans_1[ie1]
i_span_2 = spans_2[ie2]
for il_1 in range(0, p1 + 1):
for jl_1 in range(0, p1 + 1):
for il_2 in range(0, p2 + 1):
for jl_2 in range(0, p2 + 1):
i1 = i_span_1 - p1 - 1 + il_1
j1 = i_span_1 - p1 - 1 + jl_1
i2 = i_span_2 - p2 - 1 + il_2
j2 = i_span_2 - p2 - 1 + jl_2
v_m = 0.0
for g1 in range(0, k1):
for g2 in range(0, k2):
bi_0 = basis_1[ie1, il_1, 0,
g1] * basis_2[ie1, il_2, 0,
g2]
bj_0 = basis_1[ie1, jl_1, 0,
g1] * basis_2[ie2, jl_2, 0,
g2]
wvol = weights_1[ie1, g1] * weights_2[ie2,
g2]
v_m = v_m + bi_0 * bj_0 * wvol
M[i1, i2, j1 - i1, j2 - i2] += v_m
# ...
return M
def assemblyMassMatrixDspan2D(V):
"""
We compute the mass matrix for the span
$$ D_i^{p} = \frac{p+1}{t_{i+p+1} - t_i} N_i^p $$
where $N_i^p$ are the default splines of V space
Require:
--------
@V is a FemTensor V= (V1, V2) with @V1, @V2 are FemSpace
Returns:
--------
@M is a StencilMatrix - the mass matrix for D_i^p span
"""
# ... sizes
[s1, s2] = V.vector_space.starts
[e1, e2] = V.vector_space.ends
[p1, p2] = V.vector_space.pads
# ...
# ... seetings
[k1, k2] = [W.quad_order for W in V.spaces]
[spans_1, spans_2] = [W.spans for W in V.spaces]
[basis_1, basis_2] = [W.quad_basis for W in V.spaces]
[weights_1, weights_2] = [W.quad_weights for W in V.spaces]
[points_1, points_2] = [W.quad_points for W in V.spaces]
[V_1, V_2] = [W for W in V.spaces]
# ... data structure
M = StencilMatrix(V.vector_space, V.vector_space)
# ...
# ... build matrices
for ie1 in range(s1, e1 + 1 - p1):
for ie2 in range(s2, e2 + 1 - p2):
i_span_1 = spans_1[ie1]
i_span_2 = spans_2[ie2]
for il_1 in range(0, p1 + 1):
for jl_1 in range(0, p1 + 1):
for il_2 in range(0, p2 + 1):
for jl_2 in range(0, p2 + 1):
i1 = i_span_1 - p1 - 1 + il_1
j1 = i_span_1 - p1 - 1 + jl_1
i2 = i_span_2 - p2 - 1 + il_2
j2 = i_span_2 - p2 - 1 + jl_2
v_s = 0.0
for g1 in range(0, k1):
for g2 in range(0, k2):
di1_0 = coefficients_linear_combinaison(
V_1, i1) * basis_1[ie1, il_1, 0, g1]
di2_0 = coefficients_linear_combinaison(
V_2, i2) * basis_2[ie1, il_2, 0, g1]
bi_0 = di1_0 * di2_0
d_j1 = basis_1[
ie1, jl_1, 0,
g1] * coefficients_linear_combinaison(
V_1, j1)
d_j2 = basis_2[
ie2, jl_2, 0,
g2] * coefficients_linear_combinaison(
V_2, j2)
bj_0 = d_j1 * d_j2
wvol = weights_1[ie1, g1] * weights_2[ie2,
g2]
v_s = v_s + bi_0 * bj_0 * wvol
M[i1, i2, j1 - i1, j2 - i2] += v_s
# ...
return M
# ... numbers of elements and degres
n1 = 8
n2 = 4
p1 = 3
p2 = 1
# ...
# ... Vector Spaces
V = StencilVectorSpace([n1, n2], [p1, p2], [False, False])
V1 = StencilVectorSpace([n1], [p1], [False])
V2 = StencilVectorSpace([n2], [p2], [False])
# ... Inputs
Y = StencilVector(V)
A = StencilMatrix(V1, V1)
B = StencilMatrix(V2, V2)
# ...
# ... Fill in A, B and X
utils.populate_1d_matrix(A, 5.)
utils.populate_1d_matrix(B, 6.)
utils.populate_2d_vector(Y)
# ...
# ..
X = kron_solve(B, A, Y)
X_ref = utils.kron_solve_ref(B, A, Y)
print('A = \n', A.toarray())
def assemble_matrix_1d(V, kernel, args=None, M=None, fields={}):
from spl.fem.vector import VectorBasicSobolevSpace
# ...
is_block = False
if isinstance(V, VectorBasicSobolevSpace):
if not (V.is_block):
raise NotImplementedError('Expecting a scalar or vector space.')
is_block = True
# ...
# ... data structure if not given
if M is None:
from spl.linalg.stencil import StencilVector, StencilMatrix
# scalar case
if not is_block:
M = StencilMatrix(V.vector_space, V.vector_space)
else:
n_components = len(V.spaces)
V = V.spaces[0]
M = []
for i in range(0, n_components):
line = []
for j in range(0, n_components):
line.append(StencilMatrix(V.vector_space, V.vector_space))
M.append(line)
# ...
# ... sizes
[s1] = V.vector_space.starts
[e1] = V.vector_space.ends
[p1] = V.vector_space.pads
k1 = V.quad_order
spans_1 = V.spans
basis_1 = V.basis
weights_1 = V.weights
points_1 = V.points
# ...
# ... element matrix
if not is_block:
mat = zeros((p1 + 1, 2 * p1 + 1), order='F')
else:
mats = []
for i in range(0, n_components):
line = []
for j in range(0, n_components):
line.append(zeros((p1 + 1, 2 * p1 + 1), order='F'))
mats.append(line)
# ...
# ... build matrices
# TODO this is only for the parallel case
# for ie1 in range(s1, e1+1-p1):
for ie1 in range(0, V.ncells):
i_span_1 = spans_1[ie1]
bs = basis_1[:, :, :, ie1]
w = weights_1[:, ie1]
u = points_1[:, ie1]
s1 = i_span_1 - p1 - 1
field_coeffs = []
for k, f in list(fields.items()):
field_coeffs.append(f.coeffs[s1:s1 + p1 + 1])
if not is_block:
if args is None:
kernel(p1, k1, bs, u, w, mat, *field_coeffs)
else:
kernel(p1, k1, bs, u, w, mat, *args, *field_coeffs)
else:
_mats = []
for i in range(0, n_components):
for j in range(0, n_components):
_mats.append(mats[i][j])
if args is None:
kernel(p1, k1, bs, u, w, *_mats, *field_coeffs)
else:
kernel(p1, k1, bs, u, w, *_mats, *args, *field_coeffs)
if not is_block:
M._data[s1:s1 + p1 + 1, :] += mat[:, :]
else:
for i in range(0, n_components):
for j in range(0, n_components):
_mat = mats[i][j]
_M = M[i][j]
_M._data[s1:s1 + p1 + 1, :] += _mat[:, :]
# ...
return M
def assemble_matrix_3d(V, kernel, args=None, M=None, fields={}):
from spl.fem.vector import VectorBasicSobolevSpace
# ...
is_block = False
if isinstance(V, VectorBasicSobolevSpace):
if not (V.is_block):
raise NotImplementedError('Expecting a scalar or vector space.')
is_block = True
# ...
# ... data structure if not given
if M is None:
from spl.linalg.stencil import StencilVector, StencilMatrix
# scalar case
if not is_block:
M = StencilMatrix(V.vector_space, V.vector_space)
else:
n_components = len(V.spaces)
V = V.spaces[0]
M = []
for i in range(0, n_components):
line = []
for j in range(0, n_components):
line.append(StencilMatrix(V.vector_space, V.vector_space))
M.append(line)
# ...
# ... sizes
[s1, s2, s3] = V.vector_space.starts
[e1, e2, e3] = V.vector_space.ends
[p1, p2, p3] = V.vector_space.pads
# ...
# ... seetings
[k1, k2, k3] = [W.quad_order for W in V.spaces]
[spans_1, spans_2, spans_3] = [W.spans for W in V.spaces]
[basis_1, basis_2, basis_3] = [W.basis for W in V.spaces]
[weights_1, weights_2, weights_3] = [W.weights for W in V.spaces]
[points_1, points_2, points_3] = [W.points for W in V.spaces]
# ...
# ... element matrix
if not is_block:
mat = zeros(
(p1 + 1, p2 + 1, p3 + 1, 2 * p1 + 1, 2 * p2 + 1, 2 * p3 + 1),
order='F')
else:
mats = []
for i in range(0, n_components):
line = []
for j in range(0, n_components):
line.append(
zeros((p1 + 1, p2 + 1, p3 + 1, 2 * p1 + 1, 2 * p2 + 1,
2 * p3 + 1),
order='F'))
mats.append(line)
# ...
# ... build matrices
# TODO this is only for the parallel case
# for ie1 in range(s1, e1+1-p1):
# for ie2 in range(s2, e2+1-p2):
# for ie3 in range(s3, e3+1-p3):
for ie1 in range(0, V.ncells[0]):
for ie2 in range(0, V.ncells[1]):
for ie3 in range(0, V.ncells[2]):
i_span_1 = spans_1[ie1]
i_span_2 = spans_2[ie2]
i_span_3 = spans_3[ie3]
bs1 = basis_1[:, :, :, ie1]
bs2 = basis_2[:, :, :, ie2]
bs3 = basis_3[:, :, :, ie3]
w1 = weights_1[:, ie1]
w2 = weights_2[:, ie2]
w3 = weights_3[:, ie3]
u1 = points_1[:, ie1]
u2 = points_2[:, ie2]
u3 = points_3[:, ie3]
s1 = i_span_1 - p1 - 1
s2 = i_span_2 - p2 - 1
s3 = i_span_3 - p3 - 1
field_coeffs = []
for k, f in list(fields.items()):
field_coeffs.append(f.coeffs[s1:s1 + p1 + 1,
s2:s2 + p2 + 1,
s3:s3 + p3 + 1])
if not is_block:
if args is None:
kernel(p1, p2, p3, k1, k2, k3, bs1, bs2, bs3, u1, u2,
u3, w1, w2, w3, mat, *field_coeffs)
else:
kernel(p1, p2, p3, k1, k2, k3, bs1, bs2, bs3, u1, u2,
u3, w1, w2, w3, mat, *args, *field_coeffs)
else:
_mats = []
for i in range(0, n_components):
for j in range(0, n_components):
_mats.append(mats[i][j])
if args is None:
kernel(p1, p2, p3, k1, k2, k3, bs1, bs2, bs3, u1, u2,
u3, w1, w2, w3, *_mats, *field_coeffs)
else:
kernel(p1, p2, p3, k1, k2, k3, bs1, bs2, bs3, u1, u2,
u3, w1, w2, w3, *_mats, *args, *field_coeffs)
if not is_block:
M._data[s1:s1 + p1 + 1, s2:s2 + p2 + 1,
s3:s3 + p3 + 1, :, :, :] += mat[:, :, :, :, :, :]
else:
for i in range(0, n_components):
for j in range(0, n_components):
_mat = mats[i][j]
_M = M[i][j]
_M._data[s1:s1 + p1 + 1, s2:s2 + p2 + 1, s3:s3 +
p3 + 1, :, :, :] += _mat[:, :, :, :, :, :]
# ...
return M