linear_elasticity/linear_elastic_damping.py¶

Description

Time-dependent linear elasticity with a simple damping.

Find such that:

where

source code

r"""
Time-dependent linear elasticity with a simple damping.

Find :math:\ul{u} such that:

.. math::
\int_{\Omega} c\ \ul{v} \cdot \pdiff{\ul{u}}{t}
+ \int_{\Omega} D_{ijkl}\ e_{ij}(\ul{v}) e_{kl}(\ul{u})
= 0

where

.. math::
D_{ijkl} = \mu (\delta_{ik} \delta_{jl}+\delta_{il} \delta_{jk}) +
\lambda \ \delta_{ij} \delta_{kl}
\;.
"""
from __future__ import print_function
from __future__ import absolute_import
from copy import deepcopy

import numpy as nm
from sfepy.examples.linear_elasticity.linear_elastic import \
filename_mesh, materials, regions, fields, ebcs, \
integrals, solvers

def print_times(problem, state):
print(nm.array(problem.ts.times))

options = {
'ts' : 'ts',
'save_times' : 'all',
'post_process_hook_final' : print_times,
'output_format' : 'h5',
}

variables = {
'u' : ('unknown field', 'displacement', 0, 1),
'v' : ('test field', 'displacement', 'u'),
}

# Put density to 'solid'.
materials = deepcopy(materials)
materials['solid'][0].update({'c' : 1000.0})

# Moving the PerturbedSurface region.
ebcs = deepcopy(ebcs)
ebcs['PerturbedSurface'][1].update({'u.0' : 'ebc_sin'})

def ebc_sin(ts, coors, **kwargs):
val = 0.01 * nm.sin(2.0*nm.pi*ts.nt)
return nm.tile(val, (coors.shape[0],))

equations = {
'balance_of_forces in time' :
"""dw_dot.i.Omega( solid.c, v, du/dt )
+ dw_lin_elastic.i.Omega( solid.D, v, u ) = 0""",
}

if ts.time > 0.5:
ts.set_time_step(0.1)

return True

solvers = deepcopy(solvers) # Do not spoil linear_elastic.py namespace in tests.
solvers.update({
't0' : 0.0,
't1' : 1.0,
'dt' : None,
'n_step' : 101,