linear_elasticity/nodal_lcbcs.py

Description

Linear elasticity with nodal linear combination constraints.

Find \ul{u} such that:

\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

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

and \ull{\sigma} \cdot \ul{n} = \bar{p} \ull{I} \cdot \ul{n} with given traction pressure \bar{p}. The constraints are given in terms of coefficient matrices and right-hand sides, see the lcbcs keyword below. For instance, 'nlcbc1' in the 3D mesh case corresponds to

u_0 - u_1 + u_2 = 0 \\
u_0 + 0.5 u_1 + 0.1 u_2 = 0.05

that should hold in the 'Top' region.

This example demonstrates how to pass command line options to a problem description file using --define option of simple.py. Try:

python simple.py examples/linear_elasticity/nodal_lcbcs.py --define='dim: 3'

to use a 3D mesh, instead of the default 2D mesh. The example also shows that the nodal constraints can be used in place of the Dirichlet boundary conditions. Try:

python simple.py examples/linear_elasticity/nodal_lcbcs.py --define='use_ebcs: False'

to replace ebcs with the 'nlcbc4' constraints. The results should be the same for the two cases. Both options can be combined:

python simple.py examples/linear_elasticity/nodal_lcbcs.py --define='dim: 3, use_ebcs: False'

The post_process() function is used both to compute the von Mises stress and to verify the linear combination constraints.

View the 2D results using:

python postproc.py square_quad.vtk --wireframe -b

python postproc.py square_quad.vtk --wireframe -b --only-names=u -d'u,plot_displacements,rel_scaling=1,color_kind="scalars",color_name="von_mises_stress"'

View the 3D results using:

python postproc.py cube_medium_tetra.vtk --wireframe -b

python postproc.py cube_medium_tetra.vtk --wireframe -b --only-names=u -d'u,plot_displacements,rel_scaling=1,color_kind="scalars",color_name="von_mises_stress"'
../_images/linear_elasticity-nodal_lcbcs1.png

source code

r"""
Linear elasticity with nodal linear combination constraints.

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}`. The constraints are given in terms of
coefficient matrices and right-hand sides, see the ``lcbcs`` keyword below. For
instance, ``'nlcbc1'`` in the 3D mesh case corresponds to

.. math::
    u_0 - u_1 + u_2 = 0 \\
    u_0 + 0.5 u_1 + 0.1 u_2 = 0.05

that should hold in the ``'Top'`` region.

This example demonstrates how to pass command line options to a problem
description file using ``--define`` option of ``simple.py``. Try::

  python simple.py examples/linear_elasticity/nodal_lcbcs.py --define='dim: 3'

to use a 3D mesh, instead of the default 2D mesh. The example also shows that
the nodal constraints can be used in place of the Dirichlet boundary
conditions. Try::

  python simple.py examples/linear_elasticity/nodal_lcbcs.py --define='use_ebcs: False'

to replace ``ebcs`` with the ``'nlcbc4'`` constraints. The results should be
the same for the two cases. Both options can be combined::

  python simple.py examples/linear_elasticity/nodal_lcbcs.py --define='dim: 3, use_ebcs: False'

The :func:`post_process()` function is used both to compute the von Mises
stress and to verify the linear combination constraints.

View the 2D results using::

  python postproc.py square_quad.vtk --wireframe -b

  python postproc.py square_quad.vtk --wireframe -b --only-names=u -d'u,plot_displacements,rel_scaling=1,color_kind="scalars",color_name="von_mises_stress"'

View the 3D results using::

  python postproc.py cube_medium_tetra.vtk --wireframe -b

  python postproc.py cube_medium_tetra.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.base.base import output, assert_
from sfepy.mechanics.matcoefs import stiffness_from_lame
from sfepy.mechanics.tensors import get_von_mises_stress
from sfepy import data_dir

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(m.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)

    dim = pb.domain.shape.dim

    us = state().reshape((-1, dim))

    field = pb.fields['displacement']

    if dim == 2:
        ii = field.get_dofs_in_region(pb.domain.regions['Top'])
        output('top LCBC (u.0 - u.1 = 0):')
        output('\n', nm.c_[us[ii], nm.diff(us[ii], 1)])

        ii = field.get_dofs_in_region(pb.domain.regions['Bottom'])
        output('bottom LCBC (u.0 + u.1 = -0.1):')
        output('\n', nm.c_[us[ii], nm.sum(us[ii], 1)])

        ii = field.get_dofs_in_region(pb.domain.regions['Right'])
        output('right LCBC (u.0 + u.1 = linspace(0, 0.1)):')
        output('\n', nm.c_[us[ii], nm.sum(us[ii], 1)])

    else:
        ii = field.get_dofs_in_region(pb.domain.regions['Top'])
        output('top LCBC (u.0 - u.1 + u.2 = 0):')
        output('\n', nm.c_[us[ii], us[ii, 0] - us[ii, 1] + us[ii, 2]])
        output('top LCBC (u.0 + 0.5 u.1 + 0.1 u.2 = 0.05):')
        output('\n', nm.c_[us[ii],
                           us[ii, 0] + 0.5 * us[ii, 1] + 0.1 * us[ii, 2]])

        ii = field.get_dofs_in_region(pb.domain.regions['Bottom'])
        output('bottom LCBC (u.2 - 0.1 u.1 = 0.2):')
        output('\n', nm.c_[us[ii], us[ii, 2] - 0.1 * us[ii, 1]])

        ii = field.get_dofs_in_region(pb.domain.regions['Right'])
        output('right LCBC (u.0 + u.1 + u.2 = linspace(0, 0.1)):')
        output('\n', nm.c_[us[ii], nm.sum(us[ii], 1)])

    return out

def define(dim=2, use_ebcs=True):
    assert_(dim in (2, 3))

    if dim == 2:
        filename_mesh = data_dir + '/meshes/2d/square_quad.mesh'

    else:
        filename_mesh = data_dir + '/meshes/3d/cube_medium_tetra.mesh'

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

    def get_constraints(ts, coors, region=None):
        mtx = nm.ones((coors.shape[0], 1, dim), dtype=nm.float64)

        rhs = nm.arange(coors.shape[0], dtype=nm.float64)[:, None]

        rhs *= 0.1 / (coors.shape[0] - 1)

        return mtx, rhs

    functions = {
        'get_constraints' : (get_constraints,),
    }

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

    materials = {
        'm' : ({
            'D' : stiffness_from_lame(dim, lam=5.769, mu=3.846),
        },),
        'load' : ({'val' : -1.0},),
    }

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

    regions = {
        'Omega' : 'all',
        'Bottom' : ('vertices in (y < -0.499) -v r.Left', 'facet'),
        'Top' : ('vertices in (y > 0.499) -v r.Left', 'facet'),
        'Left' : ('vertices in (x < -0.499)', 'facet'),
        'Right' : ('vertices in (x > 0.499) -v (r.Bottom +v r.Top)', 'facet'),
    }

    if dim == 2:
        lcbcs = {
            'nlcbc1' : ('Top', {'u.all' : None}, None, 'nodal_combination',
                        ([[1.0, -1.0]], [0.0])),
            'nlcbc2' : ('Bottom', {'u.all' : None}, None, 'nodal_combination',
                        ([[1.0, 1.0]], [-0.1])),
            'nlcbc3' : ('Right', {'u.all' : None}, None, 'nodal_combination',
                        'get_constraints'),
        }

    else:
        lcbcs = {
            'nlcbc1' : ('Top', {'u.all' : None}, None, 'nodal_combination',
                        ([[1.0, -1.0, 1.0], [1.0, 0.5, 0.1]], [0.0, 0.05])),
            'nlcbc2' : ('Bottom', {'u.[2,1]' : None}, None, 'nodal_combination',
                        ([[1.0, -0.1]], [0.2])),
            'nlcbc3' : ('Right', {'u.all' : None}, None, 'nodal_combination',
                        'get_constraints'),
        }

    if use_ebcs:
        ebcs = {
            'fix' : ('Left', {'u.all' : 0.0}),
        }

    else:
        ebcs = {}

        lcbcs.update({
            'nlcbc4' : ('Left', {'u.all' : None}, None, 'nodal_combination',
                        (nm.eye(dim), nm.zeros(dim))),
        })

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

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

    return locals()