linear_elasticity/its2D_interactive.py

Description

Diametrically point loaded 2-D disk, using commands for interactive use. See Primer.

The script combines the functionality of all the its2D_?.py examples and allows setting various simulation parameters, namely:

  • material parameters
  • displacement field approximation order
  • uniform mesh refinement level

The example shows also how to probe the results as in linear_elasticity/its2D_4.py, and how to display the results using Mayavi. Using sfepy.discrete.probes allows correct probing of fields with the approximation order greater than one.

In the SfePy top-level directory the following command can be used to get usage information:

python examples/linear_elasticity/its2D_interactive.py -h

Notes

The --probe and --show options work simultaneously only if Mayavi and Matplotlib use the same backend type (for example wx).

source code

#!/usr/bin/env python
"""
Diametrically point loaded 2-D disk, using commands for interactive use. See
:ref:`sec-primer`.

The script combines the functionality of all the ``its2D_?.py`` examples and
allows setting various simulation parameters, namely:

- material parameters
- displacement field approximation order
- uniform mesh refinement level

The example shows also how to probe the results as in
:ref:`linear_elasticity-its2D_4`, and how to display the results using Mayavi.
Using :mod:`sfepy.discrete.probes` allows correct probing of fields with the
approximation order greater than one.

In the SfePy top-level directory the following command can be used to get usage
information::

  python examples/linear_elasticity/its2D_interactive.py -h

Notes
-----

The ``--probe`` and ``--show`` options work simultaneously only if Mayavi and
Matplotlib use the same backend type (for example wx).
"""
from __future__ import absolute_import
import sys
from six.moves import range
sys.path.append('.')
from argparse import ArgumentParser, RawDescriptionHelpFormatter

import numpy as nm
import matplotlib.pyplot as plt

from sfepy.base.base import assert_, output, ordered_iteritems, IndexedStruct
from sfepy.discrete import (FieldVariable, Material, Integral, Integrals,
                            Equation, Equations, Problem)
from sfepy.discrete.fem import Mesh, FEDomain, Field
from sfepy.terms import Term
from sfepy.discrete.conditions import Conditions, EssentialBC
from sfepy.mechanics.matcoefs import stiffness_from_youngpoisson
from sfepy.solvers.ls import ScipyDirect
from sfepy.solvers.nls import Newton
from sfepy.discrete.fem.geometry_element import geometry_data
from sfepy.discrete.probes import LineProbe
from sfepy.discrete.projections import project_by_component

from examples.linear_elasticity.its2D_2 import stress_strain
from examples.linear_elasticity.its2D_3 import nodal_stress

def gen_lines(problem):
    """
    Define two line probes.

    Additional probes can be added by appending to `ps0` (start points) and
    `ps1` (end points) lists.
    """
    ps0 = [[0.0,  0.0], [0.0,  0.0]]
    ps1 = [[75.0, 0.0], [0.0, 75.0]]

    # Use enough points for higher order approximations.
    n_point = 1000

    labels = ['%s -> %s' % (p0, p1) for p0, p1 in zip(ps0, ps1)]
    probes = []
    for ip in range(len(ps0)):
        p0, p1 = ps0[ip], ps1[ip]
        probes.append(LineProbe(p0, p1, n_point))

    return probes, labels

def probe_results(u, strain, stress, probe, label):
    """
    Probe the results using the given probe and plot the probed values.
    """
    results = {}

    pars, vals = probe(u)
    results['u'] = (pars, vals)
    pars, vals = probe(strain)
    results['cauchy_strain'] = (pars, vals)
    pars, vals = probe(stress)
    results['cauchy_stress'] = (pars, vals)

    fig = plt.figure()
    plt.clf()
    fig.subplots_adjust(hspace=0.4)
    plt.subplot(311)
    pars, vals = results['u']
    for ic in range(vals.shape[1]):
        plt.plot(pars, vals[:,ic], label=r'$u_{%d}$' % (ic + 1),
                 lw=1, ls='-', marker='+', ms=3)
    plt.ylabel('displacements')
    plt.xlabel('probe %s' % label, fontsize=8)
    plt.legend(loc='best', fontsize=10)

    sym_indices = ['11', '22', '12']

    plt.subplot(312)
    pars, vals = results['cauchy_strain']
    for ic in range(vals.shape[1]):
        plt.plot(pars, vals[:,ic], label=r'$e_{%s}$' % sym_indices[ic],
                 lw=1, ls='-', marker='+', ms=3)
    plt.ylabel('Cauchy strain')
    plt.xlabel('probe %s' % label, fontsize=8)
    plt.legend(loc='best', fontsize=10)

    plt.subplot(313)
    pars, vals = results['cauchy_stress']
    for ic in range(vals.shape[1]):
        plt.plot(pars, vals[:,ic], label=r'$\sigma_{%s}$' % sym_indices[ic],
                 lw=1, ls='-', marker='+', ms=3)
    plt.ylabel('Cauchy stress')
    plt.xlabel('probe %s' % label, fontsize=8)
    plt.legend(loc='best', fontsize=10)

    return fig, results

helps = {
    'young' : "the Young's modulus [default: %(default)s]",
    'poisson' : "the Poisson's ratio [default: %(default)s]",
    'load' : "the vertical load value (negative means compression)"
    " [default: %(default)s]",
    'order' : 'displacement field approximation order [default: %(default)s]',
    'refine' : 'uniform mesh refinement level [default: %(default)s]',
    'probe' : 'probe the results',
    'show' : 'show the results figure',
}

def main():
    from sfepy import data_dir

    parser = ArgumentParser(description=__doc__,
                            formatter_class=RawDescriptionHelpFormatter)
    parser.add_argument('--version', action='version', version='%(prog)s')
    parser.add_argument('--young', metavar='float', type=float,
                      action='store', dest='young',
                      default=2000.0, help=helps['young'])
    parser.add_argument('--poisson', metavar='float', type=float,
                      action='store', dest='poisson',
                      default=0.4, help=helps['poisson'])
    parser.add_argument('--load', metavar='float', type=float,
                      action='store', dest='load',
                      default=-1000.0, help=helps['load'])
    parser.add_argument('--order', metavar='int', type=int,
                      action='store', dest='order',
                      default=1, help=helps['order'])
    parser.add_argument('-r', '--refine', metavar='int', type=int,
                      action='store', dest='refine',
                      default=0, help=helps['refine'])
    parser.add_argument('-s', '--show',
                      action="store_true", dest='show',
                      default=False, help=helps['show'])
    parser.add_argument('-p', '--probe',
                      action="store_true", dest='probe',
                      default=False, help=helps['probe'])
    options = parser.parse_args()

    assert_((0.0 < options.poisson < 0.5),
            "Poisson's ratio must be in ]0, 0.5[!")
    assert_((0 < options.order),
            'displacement approximation order must be at least 1!')

    output('using values:')
    output("  Young's modulus:", options.young)
    output("  Poisson's ratio:", options.poisson)
    output('  vertical load:', options.load)
    output('uniform mesh refinement level:', options.refine)

    # Build the problem definition.
    mesh = Mesh.from_file(data_dir + '/meshes/2d/its2D.mesh')
    domain = FEDomain('domain', mesh)

    if options.refine > 0:
        for ii in range(options.refine):
            output('refine %d...' % ii)
            domain = domain.refine()
            output('... %d nodes %d elements'
                   % (domain.shape.n_nod, domain.shape.n_el))

    omega = domain.create_region('Omega', 'all')
    left = domain.create_region('Left',
                                'vertices in x < 0.001', 'facet')
    bottom = domain.create_region('Bottom',
                                  'vertices in y < 0.001', 'facet')
    top = domain.create_region('Top', 'vertex 2', 'vertex')

    field = Field.from_args('fu', nm.float64, 'vector', omega,
                            approx_order=options.order)

    u = FieldVariable('u', 'unknown', field)
    v = FieldVariable('v', 'test', field, primary_var_name='u')

    D = stiffness_from_youngpoisson(2, options.young, options.poisson)

    asphalt = Material('Asphalt', D=D)
    load = Material('Load', values={'.val' : [0.0, options.load]})

    integral = Integral('i', order=2*options.order)
    integral0 = Integral('i', order=0)

    t1 = Term.new('dw_lin_elastic(Asphalt.D, v, u)',
                  integral, omega, Asphalt=asphalt, v=v, u=u)
    t2 = Term.new('dw_point_load(Load.val, v)',
                  integral0, top, Load=load, v=v)
    eq = Equation('balance', t1 - t2)
    eqs = Equations([eq])

    xsym = EssentialBC('XSym', bottom, {'u.1' : 0.0})
    ysym = EssentialBC('YSym', left, {'u.0' : 0.0})

    ls = ScipyDirect({})

    nls_status = IndexedStruct()
    nls = Newton({}, lin_solver=ls, status=nls_status)

    pb = Problem('elasticity', equations=eqs)

    pb.set_bcs(ebcs=Conditions([xsym, ysym]))

    pb.set_solver(nls)

    # Solve the problem.
    state = pb.solve()
    output(nls_status)

    # Postprocess the solution.
    out = state.create_output_dict()
    out = stress_strain(out, pb, state, extend=True)
    pb.save_state('its2D_interactive.vtk', out=out)

    gdata = geometry_data['2_3']
    nc = len(gdata.coors)

    integral_vn = Integral('ivn', coors=gdata.coors,
                          weights=[gdata.volume / nc] * nc)

    nodal_stress(out, pb, state, integrals=Integrals([integral_vn]))

    if options.probe:
        # Probe the solution.
        probes, labels = gen_lines(pb)

        sfield = Field.from_args('sym_tensor', nm.float64, 3, omega,
                                approx_order=options.order - 1)
        stress = FieldVariable('stress', 'parameter', sfield,
                               primary_var_name='(set-to-None)')
        strain = FieldVariable('strain', 'parameter', sfield,
                               primary_var_name='(set-to-None)')

        cfield = Field.from_args('component', nm.float64, 1, omega,
                                 approx_order=options.order - 1)
        component = FieldVariable('component', 'parameter', cfield,
                                  primary_var_name='(set-to-None)')

        ev = pb.evaluate
        order = 2 * (options.order - 1)
        strain_qp = ev('ev_cauchy_strain.%d.Omega(u)' % order, mode='qp')
        stress_qp = ev('ev_cauchy_stress.%d.Omega(Asphalt.D, u)' % order,
                       mode='qp', copy_materials=False)

        project_by_component(strain, strain_qp, component, order)
        project_by_component(stress, stress_qp, component, order)

        all_results = []
        for ii, probe in enumerate(probes):
            fig, results = probe_results(u, strain, stress, probe, labels[ii])

            fig.savefig('its2D_interactive_probe_%d.png' % ii)
            all_results.append(results)

        for ii, results in enumerate(all_results):
            output('probe %d:' % ii)
            output.level += 2
            for key, res in ordered_iteritems(results):
                output(key + ':')
                val = res[1]
                output('  min: %+.2e, mean: %+.2e, max: %+.2e'
                       % (val.min(), val.mean(), val.max()))
            output.level -= 2

    if options.show:
        # Show the solution. If the approximation order is greater than 1, the
        # extra DOFs are simply thrown away.
        from sfepy.postprocess.viewer import Viewer

        view = Viewer('its2D_interactive.vtk')
        view(vector_mode='warp_norm', rel_scaling=1,
             is_scalar_bar=True, is_wireframe=True)

if __name__ == '__main__':
    main()