linear_elasticity/elastic_shifted_periodic.py

Description

Linear elasticity with linear combination constraints and periodic boundary conditions.

The linear combination constraints are used to apply periodic boundary conditions with a shift in the second axis direction.

Find \ul{u} such that:

\int_{\Omega} D_{ijkl}\ e_{ij}(\ul{v}) e_{kl}(\ul{u})
= - \int_{\Gamma_{bottom}} \ul{v} \cdot \ull{\sigma} \cdot \ul{n}
\;, \quad \forall \ul{v} \;,

\ul{u} = 0 \mbox{ on } \Gamma_{left} \;,

u_1 = u_2 = 0 \mbox{ on } \Gamma_{right} \;,

\ul{u}(\ul{x}) = \ul{u}(\ul{y}) \mbox{ for }
\ul{x} \in \Gamma_{bottom}, \ul{y} \in \Gamma_{top},
\ul{y} = P_1(\ul{x}) \;,

\ul{u}(\ul{x}) = \ul{u}(\ul{y}) + a(\ul{y}) \mbox{ for }
\ul{x} \in \Gamma_{near}, \ul{y} \in \Gamma_{far},
\ul{y} = P_2(\ul{x}) \;,

where

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

and the traction \ull{\sigma} \cdot \ul{n} = \bar{p} \ull{I} \cdot
\ul{n} is given in terms of traction pressure \bar{p}. The function a(\ul{y}) is given (the shift), P_1 and P_2 are the periodic coordinate mappings.

View the results using:

$ ./postproc.py block.vtk --wireframe -b --only-names=u -d'u,plot_displacements,rel_scaling=1,color_kind="scalars",color_name="von_mises_stress"'
../_images/linear_elasticity-elastic_shifted_periodic1.png

source code

r"""
Linear elasticity with linear combination constraints and periodic boundary
conditions.

The linear combination constraints are used to apply periodic boundary
conditions with a shift in the second axis direction.

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

.. math::
    \int_{\Omega} D_{ijkl}\ e_{ij}(\ul{v}) e_{kl}(\ul{u})
    = - \int_{\Gamma_{bottom}} \ul{v} \cdot \ull{\sigma} \cdot \ul{n}
    \;, \quad \forall \ul{v} \;,

    \ul{u} = 0 \mbox{ on } \Gamma_{left} \;,

    u_1 = u_2 = 0 \mbox{ on } \Gamma_{right} \;,

    \ul{u}(\ul{x}) = \ul{u}(\ul{y}) \mbox{ for }
    \ul{x} \in \Gamma_{bottom}, \ul{y} \in \Gamma_{top},
    \ul{y} = P_1(\ul{x}) \;,

    \ul{u}(\ul{x}) = \ul{u}(\ul{y}) + a(\ul{y}) \mbox{ for }
    \ul{x} \in \Gamma_{near}, \ul{y} \in \Gamma_{far},
    \ul{y} = P_2(\ul{x}) \;,

where

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

and the traction :math:`\ull{\sigma} \cdot \ul{n} = \bar{p} \ull{I} \cdot
\ul{n}` is given in terms of traction pressure :math:`\bar{p}`. The function
:math:`a(\ul{y})` is given (the shift), :math:`P_1` and :math:`P_2` are the
periodic coordinate mappings.

View the results using::

  $ ./postproc.py block.vtk --wireframe -b --only-names=u -d'u,plot_displacements,rel_scaling=1,color_kind="scalars",color_name="von_mises_stress"'
"""
from __future__ import absolute_import
import numpy as nm

from sfepy.mechanics.matcoefs import stiffness_from_lame
from sfepy.mechanics.tensors import get_von_mises_stress
import sfepy.discrete.fem.periodic as per
from sfepy import data_dir

filename_mesh = data_dir + '/meshes/3d/block.mesh'

options = {
    'nls' : 'newton',
    'ls' : 'ls',
    'post_process_hook' : 'post_process'
}

def post_process(out, pb, state, extend=False):
    """
    Calculate and output strain and stress for given displacements.
    """
    from sfepy.base.base import Struct

    ev = pb.evaluate
    stress = ev('ev_cauchy_stress.2.Omega(solid.D, u)', mode='el_avg')

    vms = get_von_mises_stress(stress.squeeze())
    vms.shape = (vms.shape[0], 1, 1, 1)
    out['von_mises_stress'] = Struct(name='output_data', mode='cell',
                                     data=vms, dofs=None)

    return out

def linear_tension(ts, coor, mode=None, **kwargs):
    if mode == 'qp':
        val = 0.1 * nm.sin(coor[:, 0] / 10.)

        return {'val' : val.reshape((coor.shape[0], 1, 1))}

def get_shift(ts, coors, region=None):
    val = nm.zeros_like(coors, dtype=nm.float64)

    val[:, 1] = 0.1 * coors[:, 0]
    return val

functions = {
    'get_shift' : (get_shift,),
    'linear_tension' : (linear_tension,),
    'match_y_plane' : (per.match_y_plane,),
    'match_z_plane' : (per.match_z_plane,),
}

fields = {
    'displacement': ('real', 3, 'Omega', 1),
}

materials = {
    'solid' : ({
        'D' : stiffness_from_lame(3, 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' : ('vertices in (x < -4.99)', 'facet'),
    'Right' : ('vertices in (x > 4.99)', 'facet'),
    'Bottom' : ('vertices in (z < -0.99)', 'facet'),
    'Top' : ('vertices in (z > 0.99)', 'facet'),
    'Near' : ('vertices in (y < -0.99)', 'facet'),
    'Far' : ('vertices in (y > 0.99)', 'facet'),
}

ebcs = {
    'fix1' : ('Left', {'u.all' : 0.0}),
    'fix2' : ('Right', {'u.[1,2]' : 0.0}),
}

epbcs = {
    'periodic' : (['Bottom', 'Top'], {'u.all' : 'u.all'}, 'match_z_plane'),
}

lcbcs = {
    'shifted' : (('Near', 'Far'),
                 {'u.all' : 'u.all'},
                 'match_y_plane', 'shifted_periodic',
                 'get_shift'),
}

equations = {
    'elasticity' : """
        dw_lin_elastic.2.Omega(solid.D, v, u)
        = -dw_surface_ltr.2.Bottom(load.val, v)
    """,
}

solvers = {
    'ls' : ('ls.scipy_direct', {}),
    'newton' : ('nls.newton', {
        'i_max'      : 1,
        'eps_a'      : 1e-10,
    }),
}