linear_elasticity/linear_elastic_iga.py

Description

Linear elasticity solved in a single patch NURBS domain using the isogeometric analysis (IGA) approach.

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}
\;.

The domain geometry was created by:

$ ./script/gen_iga_patch.py -d [1,0.5,0.1] -s [11,5,3] --degrees [2,2,2] -o meshes/iga/block3d.iga

View the results using:

$ ./postproc.py block3d.vtk --wireframe -b
$ ./postproc.py block3d.vtk --wireframe -b -d 'u,plot_displacements,rel_scaling=1e0'
../_images/linear_elasticity-linear_elastic_iga1.png

source code

r"""
Linear elasticity solved in a single patch NURBS domain using the isogeometric
analysis (IGA) approach.

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}
    \;.

The domain geometry was created by::

  $ ./script/gen_iga_patch.py -d [1,0.5,0.1] -s [11,5,3] --degrees [2,2,2] -o meshes/iga/block3d.iga

View the results using::

  $ ./postproc.py block3d.vtk --wireframe -b
  $ ./postproc.py block3d.vtk --wireframe -b -d 'u,plot_displacements,rel_scaling=1e0'
"""
from __future__ import absolute_import
from sfepy.mechanics.matcoefs import stiffness_from_lame
from sfepy import data_dir

filename_domain = data_dir + '/meshes/iga/block3d.iga'

regions = {
    'Omega' : 'all',
    'Gamma1' : ('vertices of set xi00', 'facet'),
    'Gamma2' : ('vertices of set xi01', 'facet'),
}

materials = {
    'solid' : ({
        'D' : stiffness_from_lame(3, lam=5.769, mu=3.846),
    },),
}

fields = {
    'displacement': ('real', 'vector', 'Omega', None, 'H1', 'iga'),
}

integrals = {
    'i' : 3,
}

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

ebcs = {
    'u1' : ('Gamma1', {'u.all' : 0.0}),
    'u2' : ('Gamma2', {'u.0' : 0.1, 'u.[1,2]' : 'get_ebcs'}),
}

def get_ebcs(ts, coors, **kwargs):
    import numpy as nm

    aux = nm.empty_like(coors[:, 1:])
    aux[:, 0] = 0.1 * coors[:, 1]
    aux[:, 1] = -0.05 + 0.03 * nm.sin(coors[:, 1] * 5 * nm.pi)

    return aux

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

equations = {
    'balance_of_forces' : """dw_lin_elastic.i.Omega(solid.D, v, u) = 0""",
}

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