linear_elasticity/linear_elastic_tractions.pyΒΆ

Description

Linear elasticity with pressure traction load on a surface and constrained to one-dimensional motion.

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 function verify_tractions() is called after the solution to verify that the inner surface tractions correspond to the load applied to the external surface. Try running the example with different approximation orders and/or uniform refinement levels:

  • the default options:

    python simple.py examples/linear_elasticity/linear_elastic_tractions.py -O refinement_level=0 -d approx_order=1
    
  • refine once:

    python simple.py examples/linear_elasticity/linear_elastic_tractions.py -O refinement_level=1 -d approx_order=1
    
  • use the tri-quadratic approximation (Q2):

    python simple.py examples/linear_elasticity/linear_elastic_tractions.py -O refinement_level=0 -d approx_order=2
    
../../_images/linear_elasticity-linear_elastic_tractions2.png

source code

r"""
Linear elasticity with pressure traction load on a surface and constrained to
one-dimensional motion.

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 function :func:`verify_tractions()` is called after the solution to verify
that the inner surface tractions correspond to the load applied to the external
surface. Try running the example with different approximation orders and/or uniform refinement levels:

- the default options::

    python simple.py examples/linear_elasticity/linear_elastic_tractions.py -O refinement_level=0 -d approx_order=1

- refine once::

    python simple.py examples/linear_elasticity/linear_elastic_tractions.py -O refinement_level=1 -d approx_order=1

- use the tri-quadratic approximation (Q2)::

    python simple.py examples/linear_elasticity/linear_elastic_tractions.py -O refinement_level=0 -d approx_order=2
"""
from __future__ import absolute_import
import numpy as nm
from sfepy.base.base import output
from sfepy.mechanics.matcoefs import stiffness_from_lame

def linear_tension(ts, coor, mode=None, **kwargs):
    if mode == 'qp':
        val = nm.tile(1.0, (coor.shape[0], 1, 1))

        return {'val' : val}

def verify_tractions(out, problem, state, extend=False):
    """
    Verify that the inner surface tractions correspond to the load applied
    to the external surface.
    """
    from sfepy.mechanics.tensors import get_full_indices
    from sfepy.discrete import Material, Function

    load_force = problem.evaluate(
        'ev_surface_integrate_mat.2.Right(load.val, u)'
    )
    output('surface load force:', load_force)

    def eval_force(region_name):
        strain = problem.evaluate(
            'ev_cauchy_strain_s.i.%s(u)' % region_name, mode='qp',
            verbose=False,
        )
        D = problem.evaluate(
            'ev_surface_integrate_mat.i.%s(solid.D, u)' % region_name,
            mode='qp',
            verbose=False,
        )

        normal = nm.array([1, 0, 0], dtype=nm.float64)

        s2f = get_full_indices(len(normal))
        stress = nm.einsum('cqij,cqjk->cqik', D, strain)
        # Full (matrix) form of stress.
        mstress = stress[..., s2f, 0]

        # Force in normal direction.
        force = nm.einsum('cqij,i,j->cq', mstress, normal, normal)

        def get_force(ts, coors, mode=None, **kwargs):
            if mode == 'qp':
                return {'force' : force.reshape(coors.shape[0], 1, 1)}
        aux = Material('aux', function=Function('get_force', get_force))

        middle_force = - problem.evaluate(
            'ev_surface_integrate_mat.i.%s(aux.force, u)' % region_name,
            aux=aux,
            verbose=False,
        )
        output('%s section axial force:' % region_name, middle_force)

    eval_force('Left')
    eval_force('Middle')
    eval_force('Right')

    return out

def define(approx_order=1):
    """Define the problem to solve."""
    from sfepy import data_dir

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

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

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

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

    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'),
        # Use a parent region to select only facets belonging to cells in the
        # parent region. Otherwise, each facet is in the region two times, with
        # opposite normals.
        'Middle' : ('vertices in (x > -1e-10) & (x < 1e-10)', 'facet', 'Rhalf'),
        'Rhalf' : 'vertices in x > -1e-10',
        'Right' : ('vertices in (x > 4.99)', 'facet'),
    }

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

    integrals = {
        'i' : 2 * approx_order,
    }

    ##
    # Balance of forces.
    equations = {
        'elasticity' :
        """dw_lin_elastic.i.Omega( solid.D, v, u )
         = - dw_surface_ltr.i.Right( load.val, v )""",
    }

    ##
    # Solvers etc.
    solvers = {
        'ls' : ('ls.auto_direct', {}),
        'newton' : ('nls.newton',
                    { 'i_max'      : 1,
                      'eps_a'      : 1e-10,
                      'eps_r'      : 1.0,
                      'macheps'   : 1e-16,
                      # Linear system error < (eps_a * lin_red).
                      'lin_red'    : 1e-2,
                      'ls_red'     : 0.1,
                      'ls_red_warp' : 0.001,
                      'ls_on'      : 1.1,
                      'ls_min'     : 1e-5,
                      'check'     : 0,
                      'delta'     : 1e-6,
                      })
    }

    return locals()