1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
|
__authors__ = "Cecile Daversin Catty"
__date__ = "2019-03-26 -- 2019-03-26"
from utils import LagrangeElement
from ufl import (
Coefficient,
Constant,
FunctionSpace,
Measure,
Mesh,
MixedFunctionSpace,
TestFunctions,
TrialFunctions,
action,
conj,
dx,
grad,
inner,
interval,
lhs,
replace,
rhs,
tetrahedron,
triangle,
)
from ufl.algorithms import expand_derivatives, renumbering
from ufl.algorithms.formsplitter import extract_blocks
from ufl.algorithms.formtransformations import compute_form_adjoint
def test_mixed_functionspace(self):
# Domains
domain_3d = Mesh(LagrangeElement(tetrahedron, 1, (3,)))
domain_2d = Mesh(LagrangeElement(triangle, 1, (2,)))
domain_1d = Mesh(LagrangeElement(interval, 1, (1,)))
# Finite elements
f_1d = LagrangeElement(interval, 1)
f_2d = LagrangeElement(triangle, 1)
f_3d = LagrangeElement(tetrahedron, 1)
# Function spaces
V_3d = FunctionSpace(domain_3d, f_3d)
V_2d = FunctionSpace(domain_2d, f_2d)
V_1d = FunctionSpace(domain_1d, f_1d)
# MixedFunctionSpace = V_3d x V_2d x V_1d
V = MixedFunctionSpace(V_3d, V_2d, V_1d)
# Check sub spaces
assert V.num_sub_spaces() == 3
assert V.ufl_sub_space(0) == V_3d
assert V.ufl_sub_space(1) == V_2d
assert V.ufl_sub_space(2) == V_1d
# Arguments from MixedFunctionSpace
(u_3d, u_2d, u_1d) = TrialFunctions(V)
(v_3d, v_2d, v_1d) = TestFunctions(V)
# Measures
dx3 = Measure("dx", domain=domain_3d)
dx2 = Measure("dx", domain=domain_2d)
dx1 = Measure("dx", domain=domain_1d)
# Mixed variational form
# LHS
a_11 = u_1d * v_1d * dx1
a_22 = u_2d * v_2d * dx2
a_33 = u_3d * v_3d * dx3
a_21 = u_2d * v_1d * dx1
a_12 = u_1d * v_2d * dx1
a_32 = u_3d * v_2d * dx2
a_23 = u_2d * v_3d * dx2
a_31 = u_3d * v_1d * dx1
a_13 = u_1d * v_3d * dx1
a = a_11 + a_22 + a_33 + a_21 + a_12 + a_32 + a_23 + a_31 + a_13
# RHS
f_1 = v_1d * dx1
f_2 = v_2d * dx2
f_3 = v_3d * dx3
f = f_1 + f_2 + f_3
# Check extract_block algorithm
# LHS
assert extract_blocks(a, 0, 0) == a_33
assert extract_blocks(a, 0, 1) == a_23
assert extract_blocks(a, 0, 2) == a_13
assert extract_blocks(a, 1, 0) == a_32
assert extract_blocks(a, 1, 1) == a_22
assert extract_blocks(a, 1, 2) == a_12
assert extract_blocks(a, 2, 0) == a_31
assert extract_blocks(a, 2, 1) == a_21
assert extract_blocks(a, 2, 2) == a_11
# RHS
assert extract_blocks(f, 0) == f_3
assert extract_blocks(f, 1) == f_2
assert extract_blocks(f, 2) == f_1
# Test dual space method
V_dual = V.dual()
assert V_dual.num_sub_spaces() == 3
assert V_dual.ufl_sub_space(0) == V_3d.dual()
assert V_dual.ufl_sub_space(1) == V_2d.dual()
assert V_dual.ufl_sub_space(2) == V_1d.dual()
V_dual = V.dual(0, 2)
assert V_dual.num_sub_spaces() == 3
assert V_dual.ufl_sub_space(0) == V_3d.dual()
assert V_dual.ufl_sub_space(1) == V_2d
assert V_dual.ufl_sub_space(2) == V_1d.dual()
def test_lhs_rhs():
V = LagrangeElement(interval, 1)
domain = Mesh(LagrangeElement(interval, 1, (1,)))
space0 = FunctionSpace(domain, V)
space1 = FunctionSpace(domain, V)
mixed_space = MixedFunctionSpace(space0, space1)
def mass(u, v):
return inner(u, v) * dx
def mixed(u, v):
return inner(u.dx(0), v) * dx
def stiffness(u, v):
return inner(grad(u), grad(v)) * dx
def source1(f, v):
return f * v * dx
def source2(f, v):
return f * v.dx(0) * dx
u, p = TrialFunctions(mixed_space)
v, q = TestFunctions(mixed_space)
f = Constant(domain)
g = Constant(domain)
F = mass(u, v) + mixed(u, q) + stiffness(p, q) + source1(f, v) + source2(g, q)
a = lhs(F)
a_blocked = extract_blocks(a)
L = rhs(F)
L_blocked = extract_blocks(L)
assert a_blocked[0][0] == mass(u, v)
assert a_blocked[0][1] is None
assert a_blocked[1][0] == mixed(u, q)
assert renumbering.renumber_indices(a_blocked[1][1]) == renumbering.renumber_indices(
expand_derivatives(stiffness(p, q))
)
assert L_blocked[0] == -source1(f, v)
assert L_blocked[1] == -source2(g, q)
def test_action():
V = LagrangeElement(interval, 1)
domain = Mesh(LagrangeElement(interval, 1, (1,)))
space0 = FunctionSpace(domain, V)
space1 = FunctionSpace(domain, V)
mixed_space = MixedFunctionSpace(space0, space1)
def mass(u, v):
return inner(u, v) * dx
def stiffness(u, v):
return inner(grad(u), grad(v)) * dx
def source1(f, v):
return inner(f, v) * dx
def source2(f, v):
return inner(f, v.dx(0)) * dx
u, _ = TrialFunctions(mixed_space)
v, q = TestFunctions(mixed_space)
f = Constant(domain)
g = Constant(domain)
F = mass(u, v) + stiffness(u, q) + source1(f, v) + source2(g, q)
assert len(F.coefficients()) == 0
F_reduced = action(F)
F_reduced_renumbered = renumbering.renumber_indices(expand_derivatives(F_reduced))
assert len(F_reduced_renumbered.coefficients()) == 1
inserted_coeff = F_reduced_renumbered.coefficients()[0]
# Create reference solution
Fh = mass(inserted_coeff, v) + stiffness(inserted_coeff, q) + source1(f, v) + source2(g, q)
Fr1 = renumbering.renumber_indices(expand_derivatives(Fh))
assert Fr1 == F_reduced_renumbered
# Repeat action, reduce to scalar
coefficients = [Coefficient(space0), Coefficient(space1)]
J = action(F_reduced, coefficients)
J_renumbered = renumbering.renumber_indices(expand_derivatives(J))
# Reference J
J_exp = renumbering.renumber_indices(expand_derivatives(F))
J_ref = replace(J_exp, {u: inserted_coeff, v: coefficients[0], q: coefficients[1]})
# Verify
assert J_renumbered == J_ref
def test_adjoint():
V = LagrangeElement(triangle, 1)
domain = Mesh(LagrangeElement(triangle, 1, (2,)))
space0 = FunctionSpace(domain, V)
space1 = FunctionSpace(domain, V)
mixed_space = MixedFunctionSpace(space0, space1)
u, p = TrialFunctions(mixed_space)
du, dp = TestFunctions(mixed_space)
c = Coefficient(space0)
Jh = (
inner(grad(u), grad(du)) * dx
+ inner(c * dp.dx(0), u) * dx
- inner(du.dx(0), p.dx(1)) * dx
+ inner(p, dp) * dx
)
Jh_adj = compute_form_adjoint(Jh)
blocked_adj = extract_blocks(Jh_adj)
ref_adj = [
[conj(inner(grad(du), grad(u))) * dx, conj(-inner(p.dx(0), du.dx(1))) * dx],
[conj(inner(c * u.dx(0), dp)) * dx, conj(inner(dp, p)) * dx],
]
for i in range(2):
for j in range(2):
assert ref_adj[i][j] == blocked_adj[i][j]
|
