linear_elasticity/material_nonlinearity.py

Description

Example demonstrating how a linear elastic term can be used to solve an elasticity problem with a material nonlinearity.

\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-material_nonlinearity.png

source code

# -*- coding: utf-8 -*-
r"""
Example demonstrating how a linear elastic term can be used to solve an
elasticity problem with a material nonlinearity.

.. 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
import numpy as nm

from sfepy.linalg import norm_l2_along_axis
from sfepy import data_dir
from sfepy.mechanics.matcoefs import stiffness_from_lame

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

def post_process(out, pb, state, extend=False):
    from sfepy.base.base import Struct

    mu = pb.evaluate('ev_integrate_mat.2.Omega(nonlinear.mu, u)',
                     mode='el_avg', copy_materials=False, verbose=False)
    out['mu'] = Struct(name='mu', mode='cell', data=mu, dofs=None)

    strain = pb.evaluate('ev_cauchy_strain.2.Omega(u)', mode='el_avg')
    out['strain'] = Struct(name='strain', mode='cell', data=strain, dofs=None)

    return out

strains = [None]

def get_pars(ts, coors, mode='qp',
             equations=None, term=None, problem=None, **kwargs):
    """
    The material nonlinearity function - the Lamé coefficient `mu`
    depends on the strain.
    """
    if mode != 'qp': return

    val = nm.empty(coors.shape[0], dtype=nm.float64)
    val.fill(1e0)

    order = term.integral.order
    uvar = equations.variables['u']

    strain = problem.evaluate('ev_cauchy_strain.%d.Omega(u)' % order,
                              u=uvar, mode='qp')
    if ts.step > 0:
        strain0 = strains[-1]

    else:
        strain0 = strain

    dstrain = (strain - strain0) / ts.dt
    dstrain.shape = (strain.shape[0] * strain.shape[1], strain.shape[2])

    norm = norm_l2_along_axis(dstrain)

    val += norm

    # Store history.
    strains[0] = strain
    return {'D': stiffness_from_lame(dim=3, lam=1e1, mu=val),
            'mu': val.reshape(-1, 1, 1)}

def pull(ts, coors, **kwargs):
    val = nm.empty_like(coors[:,0])
    val.fill(0.01 * ts.step)

    return val

functions = {
    'get_pars' : (get_pars,),
    'pull' : (pull,),
}

options = {
    'ts' : 'ts',
    'output_format' : 'h5',
    'save_times' : 'all',

    'post_process_hook' : 'post_process',
}

regions = {
    'Omega' : 'all',
    'Left' : ('vertices in (x < 0.001)', 'facet'),
    'Right' : ('vertices in (x > 0.099)', 'facet'),
}

materials = {
    'nonlinear' : 'get_pars',
}

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

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

ebcs = {
    'Fixed' : ('Left', {'u.all' : 0.0}),
    'Displaced' : ('Right', {'u.0' : 'pull', 'u.[1,2]' : 0.0}),
}

equations = {
    'balance_of_forces in time' :
    """dw_lin_elastic.2.Omega(nonlinear.D, v, u) = 0""",
}

solvers = {
    'ls' : ('ls.scipy_direct', {}),
    'newton' : ('nls.newton',
                {'i_max'      : 1,
                 'eps_a'      : 1e-10,
                 'eps_r'      : 1.0,
                 }),
    'ts' : ('ts.simple',
            {'t0' : 0.0,
             't1' : 1.0,
             'dt' : None,
             'n_step' : 5,
             'quasistatic' : True,
             'verbose' : 1,
             }),
}