linear_elasticity/linear_elastic_up.pyΒΆ

Description

Nearly incompressible linear elasticity in mixed displacement-pressure formulation with comments.

Find \ul{u}, p such that:

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

- \int_{\Omega} q\ \nabla \cdot \ul{u}
- \int_{\Omega} \gamma q p
= 0
\;, \quad \forall q \;.

../../_images/linear_elasticity-linear_elastic_up.png

source code

r"""
Nearly incompressible linear elasticity in mixed displacement-pressure
formulation with comments.

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

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

    - \int_{\Omega} q\ \nabla \cdot \ul{u}
    - \int_{\Omega} \gamma q p
    = 0
    \;, \quad \forall q \;.
"""
#!
#! Linear Elasticity
#! =================
#$ \centerline{Example input file, \today}

#! This file models a cylinder that is fixed at one end while the
#! second end has a specified displacement of 0.02 in the x direction
#! (this boundary condition is named PerturbedSurface).
#! The output is the displacement for each node, saved by default to
#! simple_out.vtk. The material is linear elastic.
from __future__ import absolute_import
from sfepy import data_dir

from sfepy.mechanics.matcoefs import stiffness_from_youngpoisson_mixed, bulk_from_youngpoisson

#! Mesh
#! ----

dim = 3
approx_u = '3_4_P1'
approx_p = '3_4_P0'
order = 2
filename_mesh = data_dir + '/meshes/3d/cylinder.mesh'
#! Regions
#! -------
#! Whole domain 'Omega', left and right ends.
regions = {
    'Omega' : 'all',
    'Left' : ('vertices in (x < 0.001)', 'facet'),
    'Right' : ('vertices in (x > 0.099)', 'facet'),
}
#! Materials
#! ---------
#! The linear elastic material model is used.
materials = {
    'solid' : ({'D' : stiffness_from_youngpoisson_mixed(dim, 0.7e9, 0.4),
                'gamma' : 1.0/bulk_from_youngpoisson(0.7e9, 0.4)},),
}
#! Fields
#! ------
#! A field is used to define the approximation on a (sub)domain
fields = {
    'displacement': ('real', 'vector', 'Omega', 1),
    'pressure' : ('real', 'scalar', 'Omega', 0),
}
#! Integrals
#! ---------
#! Define the integral type Volume/Surface and quadrature rule.
integrals = {
    'i' : order,
}
#! Variables
#! ---------
#! Define displacement and pressure fields and corresponding fields
#! for test variables.
variables = {
    'u' : ('unknown field', 'displacement'),
    'v' : ('test field', 'displacement', 'u'),
    'p' : ('unknown field', 'pressure'),
    'q' : ('test field', 'pressure', 'p'),
}
#! Boundary Conditions
#! -------------------
#! The left end of the cylinder is fixed (all DOFs are zero) and
#! the 'right' end has non-zero displacements only in the x direction.
ebcs = {
    'Fixed' : ('Left', {'u.all' : 0.0}),
    'PerturbedSurface' : ('Right', {'u.0' : 0.02, 'u.1' : 0.0}),
}
#! Equations
#! ---------
#! The weak formulation of the linear elastic problem.
equations = {
    'balance_of_forces' :
    """  dw_lin_elastic.i.Omega( solid.D, v, u )
       - dw_stokes.i.Omega( v, p )
       = 0 """,
    'pressure constraint' :
    """- dw_stokes.i.Omega( u, q )
       - dw_volume_dot.i.Omega( solid.gamma, q, p )
       = 0""",
}
#! Solvers
#! -------
#! Define linear and nonlinear solver.
#! Even linear problems are solved by a nonlinear solver - only one
#! iteration is needed and the final residual is obtained for free.
solvers = {
    'ls' : ('ls.schur_complement', {
        'keep': ['p'],
        'eliminate': ['u'],
    }),
    'newton' : ('nls.newton', {
        'i_max'      : 1,
        'eps_a'      : 1e-2,
        'eps_r'      : 1e-10,
    }),
}
#! Options
#! -------
#! Various problem-specific options.
options = {
    'output_dir' : './output',
    'absolute_mesh_path' : True,
}