linear_elasticity/linear_elastic_tractions.pyΒΆ
Description
Linear elasticity with pressure traction load on a surface and constrained to one-dimensional motion.
Find
such that:where
and
with given traction pressure .
r"""
Linear elasticity with pressure traction load on a surface and constrained to
one-dimensional motion.
Find :math:`\ul{u}` such that:
.. math::
\int_{\Omega} D_{ijkl}\ e_{ij}(\ul{v}) e_{kl}(\ul{u})
= - \int_{\Gamma_{right}} \ul{v} \cdot \ull{\sigma} \cdot \ul{n}
\;, \quad \forall \ul{v} \;,
where
.. math::
D_{ijkl} = \mu (\delta_{ik} \delta_{jl}+\delta_{il} \delta_{jk}) +
\lambda \ \delta_{ij} \delta_{kl}
\;.
and :math:`\ull{\sigma} \cdot \ul{n} = \bar{p} \ull{I} \cdot \ul{n}`
with given traction pressure :math:`\bar{p}`.
"""
import numpy as nm
def linear_tension(ts, coor, mode=None, **kwargs):
if mode == 'qp':
val = nm.tile(1.0, (coor.shape[0], 1, 1))
return {'val' : val}
def define():
"""Define the problem to solve."""
from sfepy import data_dir
filename_mesh = data_dir + '/meshes/3d/block.mesh'
options = {
'nls' : 'newton',
'ls' : 'ls',
}
functions = {
'linear_tension' : (linear_tension,),
}
fields = {
'displacement': ('real', 3, 'Omega', 1),
}
materials = {
'solid' : ({
'lam' : 5.769,
'mu' : 3.846,
},),
'load' : (None, 'linear_tension')
}
variables = {
'u' : ('unknown field', 'displacement', 0),
'v' : ('test field', 'displacement', 'u'),
}
regions = {
'Omega' : ('all', {}),
'Left' : ('nodes in (x < -4.99)', {}),
'Right' : ('nodes in (x > 4.99)', {}),
}
ebcs = {
'fixb' : ('Left', {'u.all' : 0.0}),
'fixt' : ('Right', {'u.[1,2]' : 0.0}),
}
##
# Balance of forces.
equations = {
'elasticity' :
"""dw_lin_elastic_iso.2.Omega( solid.lam, solid.mu, v, u )
= - dw_surface_ltr.2.Right( load.val, v )""",
}
##
# Solvers etc.
solvers = {
'ls' : ('ls.scipy_direct', {}),
'newton' : ('nls.newton',
{ 'i_max' : 1,
'eps_a' : 1e-10,
'eps_r' : 1.0,
'macheps' : 1e-16,
# Linear system error < (eps_a * lin_red).
'lin_red' : 1e-2,
'ls_red' : 0.1,
'ls_red_warp' : 0.001,
'ls_on' : 1.1,
'ls_min' : 1e-5,
'check' : 0,
'delta' : 1e-6,
'is_plot' : False,
# 'nonlinear' or 'linear' (ignore i_max)
'problem' : 'nonlinear'}),
}
return locals()