linear_elasticity/its2D_1.pyΒΆ

Description

Diametrically point loaded 2-D disk. See Primer.

Find \ul{u} such that:

\int_{\Omega} D_{ijkl}\ e_{ij}(\ul{v}) e_{kl}(\ul{u})
= 0
\;, \quad \forall \ul{v} \;,

where

D_{ijkl} = \mu (\delta_{ik} \delta_{jl}+\delta_{il} \delta_{jk}) +
\lambda \ \delta_{ij} \delta_{kl}
\;.

../../_images/linear_elasticity-its2D_12.png

source code

r"""
Diametrically point loaded 2-D disk. See :ref:`sec-primer`.

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

.. math::
    \int_{\Omega} D_{ijkl}\ e_{ij}(\ul{v}) e_{kl}(\ul{u})
    = 0
    \;, \quad \forall \ul{v} \;,

where

.. math::
    D_{ijkl} = \mu (\delta_{ik} \delta_{jl}+\delta_{il} \delta_{jk}) +
    \lambda \ \delta_{ij} \delta_{kl}
    \;.
"""
from __future__ import absolute_import
from sfepy.mechanics.matcoefs import stiffness_from_youngpoisson
from sfepy.discrete.fem.utils import refine_mesh
from sfepy import data_dir

# Fix the mesh file name if you run this file outside the SfePy directory.
filename_mesh = data_dir + '/meshes/2d/its2D.mesh'

refinement_level = 0
filename_mesh = refine_mesh(filename_mesh, refinement_level)

output_dir = '.' # set this to a valid directory you have write access to

young = 2000.0 # Young's modulus [MPa]
poisson = 0.4  # Poisson's ratio

options = {
    'output_dir' : output_dir,
}

regions = {
    'Omega' : 'all',
    'Left' : ('vertices in (x < 0.001)', 'facet'),
    'Bottom' : ('vertices in (y < 0.001)', 'facet'),
    'Top' : ('vertex 2', 'vertex'),
}

materials = {
    'Asphalt' : ({'D': stiffness_from_youngpoisson(2, young, poisson)},),
    'Load' : ({'.val' : [0.0, -1000.0]},),
}

fields = {
    'displacement': ('real', 'vector', 'Omega', 1),
}

equations = {
   'balance_of_forces' :
   """dw_lin_elastic.2.Omega(Asphalt.D, v, u)
      = dw_point_load.0.Top(Load.val, v)""",
}

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

ebcs = {
    'XSym' : ('Bottom', {'u.1' : 0.0}),
    'YSym' : ('Left', {'u.0' : 0.0}),
}

solvers = {
    'ls' : ('ls.scipy_direct', {}),
    'newton' : ('nls.newton', {
        'i_max' : 1,
        'eps_a' : 1e-6,
    }),
}