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
|
"""This demo program solves an elastodynamics problem."""
# Copyright (C) 2010 Garth N. Wells
#
# This file is part of DOLFIN.
#
# DOLFIN is free software: you can redistribute it and/or modify
# it under the terms of the GNU Lesser General Public License as published by
# the Free Software Foundation, either version 3 of the License, or
# (at your option) any later version.
#
# DOLFIN is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU Lesser General Public License for more details.
#
# You should have received a copy of the GNU Lesser General Public License
# along with DOLFIN. If not, see <http://www.gnu.org/licenses/>.
#
# Modified by Anders Logg 2008-2011
#
# First added: 2010-04-30
# Last changed: 2012-11-12
from dolfin import *
import matplotlib.pyplot as plt
# Form compiler options
parameters["form_compiler"]["cpp_optimize"] = True
parameters["form_compiler"]["optimize"] = True
def update(u, u0, v0, a0, beta, gamma, dt):
"""Update fields at the end of each time step."""
# Get vectors (references)
u_vec, u0_vec = u.vector(), u0.vector()
v0_vec, a0_vec = v0.vector(), a0.vector()
# Update acceleration and velocity
# a = 1/(2*beta)*((u - u0 - v0*dt)/(0.5*dt*dt) - (1-2*beta)*a0)
a_vec = (1.0/(2.0*beta))*( (u_vec - u0_vec - v0_vec*dt)/(0.5*dt*dt) - (1.0-2.0*beta)*a0_vec )
# v = dt * ((1-gamma)*a0 + gamma*a) + v0
v_vec = dt*((1.0-gamma)*a0_vec + gamma*a_vec) + v0_vec
# Update (u0 <- u0)
v0.vector()[:], a0.vector()[:] = v_vec, a_vec
u0.vector()[:] = u.vector()
# External load
class Traction(UserExpression):
def __init__(self, dt, t, old, **kwargs):
self.t = t
self.dt = dt
self.old = old
super().__init__(**kwargs)
def eval(self, values, x):
# 'Shift' time for n-1 values
t_tmp = self.t
if self.old and t > 0.0:
t_tmp -= self.dt
cutoff_t = 10.0*1.0/32.0;
weight = t_tmp/cutoff_t if t_tmp < cutoff_t else 1.0
values[0] = 1.0*weight
values[1] = 0.0
def value_shape(self):
return (2,)
# Sub domain for clamp at left end
def left(x, on_boundary):
return x[0] < 0.001 and on_boundary
# Sub domain for rotation at right end
def right(x, on_boundary):
return x[0] > 0.99 and on_boundary
# Load mesh and define function space
mesh = Mesh("../dolfin_fine.xml.gz")
# Define function space
V = VectorFunctionSpace(mesh, "CG", 1)
# Test and trial functions
u = TrialFunction(V)
r = TestFunction(V)
E = 1.0
nu = 0.0
mu = E / (2.0*(1.0 + nu))
lmbda = E*nu / ((1.0 + nu)*(1.0 - 2.0*nu))
# Mass density andviscous damping coefficient
rho = 1.0
eta = 0.25
# Time stepping parameters
alpha_m = 0.2
alpha_f = 0.4
beta = 0.36
gamma = 0.7
dt = 1.0/32.0
t = 0.0
T = 10*dt
# Some useful factors
factor_m1 = rho*(1.0-alpha_m)/(beta*dt*dt)
factor_m2 = rho*(1.0-alpha_m)/(beta*dt)
factor_m3 = rho*(1.0-alpha_m-2.0*beta)/(2.0*beta)
factor_d1 = eta*(1.0-alpha_f)*gamma/(beta*dt)
factor_d2 = eta*((1.0-alpha_f)*gamma-beta)/beta
factor_d3 = eta*(gamma-2.0*beta)*(1.0-alpha_f)*dt/(2.0*beta)
# Fields from previous time step (displacement, velocity, acceleration)
u0 = Function(V)
v0 = Function(V)
a0 = Function(V)
# External forces (body and applied tractions
f = Constant((0.0, 0.0))
p = Traction(dt, t, False, degree=1)
p0 = Traction(dt, t, True, degree=1)
# Create mesh function over the cell facets
boundary_subdomains = MeshFunction("size_t", mesh, mesh.topology().dim() - 1)
boundary_subdomains.set_all(0)
force_boundary = AutoSubDomain(right)
force_boundary.mark(boundary_subdomains, 3)
# Define measure for boundary condition integral
dss = ds(subdomain_data=boundary_subdomains)
# Stress tensor
def sigma(r):
return 2.0*mu*sym(grad(r)) + lmbda*tr(sym(grad(r)))*Identity(len(r))
# Forms
a = factor_m1*inner(u, r)*dx + factor_d1*inner(u, r)*dx \
+(1.0-alpha_f)*inner(sigma(u), grad(r))*dx
L = factor_m1*inner(r, u0)*dx + factor_m2*inner(r, v0)*dx \
+ factor_m3*inner(r, a0)*dx \
+ factor_d1*inner(r, u0)*dx + factor_d2*inner(r, v0)*dx \
+ factor_d3*inner(r, a0)*dx \
- alpha_f*inner(grad(r), sigma(u0))*dx \
+ inner(r, f)*dx + (1.0-alpha_f)*inner(r, p)*dss(3) + alpha_f*inner(r, p0)*dss(3)
# Set up boundary condition at left end
zero = Constant((0.0, 0.0))
bc = DirichletBC(V, zero, left)
# FIXME: This demo needs some improved commenting
# Time-stepping
u = Function(V)
vtk_file = File("elasticity.pvd")
while t <= T:
t += dt
print("Time: ", t)
p.t = t
p0.t = t
solve(a == L, u, bc)
update(u, u0, v0, a0, beta, gamma, dt)
# Save solution to VTK format
vtk_file << u
# Plot solution
plot(u, mode="displacement")
plt.show()
|
