linear_elasticity/shell10x_cantilever_interactive.py

Description

Bending of a long thin cantilever beam computed using the dw_shell10x term.

Find displacements of the central plane \ul{u}, and rotations \ul{\alpha} such that:

\int_{\Omega} D_{ijkl}\ e_{ij}(\ul{v}, \ul{\beta})
e_{kl}(\ul{u}, \ul{\alpha})
= - \int_{\Gamma_{right}} \ul{v} \cdot \ul{f}
\;, \quad \forall \ul{v} \;,

where D_{ijkl} is the isotropic elastic tensor, given using the Young’s modulus E and the Poisson’s ratio \nu.

The variable u below holds both \ul{u} and \ul{\alpha} DOFs. For visualization, it is saved as two fields u_disp and u_rot, corresponding to \ul{u} and \ul{\alpha}, respectively.

The material, loading and discretization parameters can be given using command line options.

Besides the default straight beam, two coordinate transformations can be applied (see the --transform option):

  • bend: the beam is bent
  • twist: the beam is twisted

For the straight and bent beam a comparison with the analytical solution coming from the Euler-Bernoulli theory is shown.

See also linear_elasticity/shell10x_cantilever.py example.

Usage Examples

See all options:

python examples/linear_elasticity/shell10x_cantilever_interactive.py -h

Apply the bending transformation to the beam domain coordinates, plot convergence curves w.r.t. number of elements:

python examples/linear_elasticity/shell10x_cantilever_interactive.py output -t bend -p

Apply the twisting transformation to the beam domain coordinates, change number of cells, show the solution:

python examples/linear_elasticity/shell10x_cantilever_interactive.py output -t twist -n 2,51,3 -s

source code

#!/usr/bin/env python
r"""
Bending of a long thin cantilever beam computed using the
:class:`dw_shell10x <sfepy.terms.terms_shells.Shell10XTerm>` term.

Find displacements of the central plane :math:`\ul{u}`, and rotations
:math:`\ul{\alpha}` such that:

.. math::
    \int_{\Omega} D_{ijkl}\ e_{ij}(\ul{v}, \ul{\beta})
    e_{kl}(\ul{u}, \ul{\alpha})
    = - \int_{\Gamma_{right}} \ul{v} \cdot \ul{f}
    \;, \quad \forall \ul{v} \;,

where :math:`D_{ijkl}` is the isotropic elastic tensor, given using the Young's
modulus :math:`E` and the Poisson's ratio :math:`\nu`.

The variable ``u`` below holds both :math:`\ul{u}` and :math:`\ul{\alpha}`
DOFs. For visualization, it is saved as two fields ``u_disp`` and ``u_rot``,
corresponding to :math:`\ul{u}` and :math:`\ul{\alpha}`, respectively.

The material, loading and discretization parameters can be given using command
line options.

Besides the default straight beam, two coordinate transformations can be applied
(see the ``--transform`` option):

- bend: the beam is bent
- twist: the beam is twisted

For the straight and bent beam a comparison with the analytical solution
coming from the Euler-Bernoulli theory is shown.

See also :ref:`linear_elasticity-shell10x_cantilever` example.

Usage Examples
--------------

See all options::

  python examples/linear_elasticity/shell10x_cantilever_interactive.py -h

Apply the bending transformation to the beam domain coordinates, plot
convergence curves w.r.t. number of elements::

  python examples/linear_elasticity/shell10x_cantilever_interactive.py output -t bend -p

Apply the twisting transformation to the beam domain coordinates, change number of cells, show the solution::

  python examples/linear_elasticity/shell10x_cantilever_interactive.py output -t twist -n 2,51,3 -s
"""
from __future__ import absolute_import
from argparse import RawDescriptionHelpFormatter, ArgumentParser
import os
import sys
from six.moves import range
sys.path.append('.')

import numpy as nm

from sfepy.base.base import output, IndexedStruct
from sfepy.base.ioutils import ensure_path
from sfepy.discrete import (FieldVariable, Material, Integral,
                            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.solvers.ls import ScipyDirect
from sfepy.solvers.nls import Newton
from sfepy.linalg import make_axis_rotation_matrix
from sfepy.mechanics.tensors import transform_data
from sfepy.mesh.mesh_generators import gen_block_mesh
import sfepy.mechanics.shell10x as sh

def make_mesh(dims, shape, transform=None):
    """
    Generate a 2D rectangle mesh in 3D space, and optionally apply a coordinate
    transform.
    """
    _mesh = gen_block_mesh(dims, shape, [0, 0], name='shell10x', verbose=False)

    coors = nm.c_[_mesh.coors, nm.zeros(_mesh.n_nod, dtype=nm.float64)]
    coors = nm.ascontiguousarray(coors)

    conns = [_mesh.get_conn(_mesh.descs[0])]

    mesh = Mesh.from_data(_mesh.name, coors, _mesh.cmesh.vertex_groups, conns,
                          [_mesh.cmesh.cell_groups], _mesh.descs)

    if transform == 'bend':
        bbox = mesh.get_bounding_box()
        x0, x1 = bbox[:, 0]

        angles = 0.5 *  nm.pi * (coors[:, 0] - x0) / (x1 - x0)
        mtx = make_axis_rotation_matrix([0, -1, 0], angles[:, None, None])

        coors = mesh.coors.copy()
        coors[:, 0] = 0
        coors[:, 2] = (x1 - x0)

        mesh.coors[:] = transform_data(coors, mtx=mtx)
        mesh.coors[:, 0] -= 0.5 * (x1 - x0)

    elif transform == 'twist':
        bbox = mesh.get_bounding_box()
        x0, x1 = bbox[:, 0]

        angles = 0.5 *  nm.pi * (coors[:, 0] - x0) / (x1 - x0)
        mtx = make_axis_rotation_matrix([-1, 0, 0], angles[:, None, None])

        mesh.coors[:] = transform_data(mesh.coors, mtx=mtx)

    return mesh

def make_domain(dims, shape, transform=None):
    """
    Generate a 2D rectangle domain in 3D space, define regions.
    """
    xmin = (-0.5 + 1e-12) * dims[0]
    xmax = (0.5 - 1e-12) * dims[0]

    mesh = make_mesh(dims, shape, transform=transform)
    domain = FEDomain('domain', mesh)
    domain.create_region('Omega', 'all')
    domain.create_region('Gamma1', 'vertices in (x < %.14f)' % xmin, 'facet')
    domain.create_region('Gamma2', 'vertices in (x > %.14f)' % xmax, 'facet')

    return domain

def solve_problem(shape, dims, young, poisson, force, transform=None):
    domain = make_domain(dims[:2], shape, transform=transform)

    omega = domain.regions['Omega']
    gamma1 = domain.regions['Gamma1']
    gamma2 = domain.regions['Gamma2']

    field = Field.from_args('fu', nm.float64, 6, omega, approx_order=1,
                            poly_space_base='shell10x')
    u = FieldVariable('u', 'unknown', field)
    v = FieldVariable('v', 'test', field, primary_var_name='u')

    thickness = dims[2]
    if transform is None:
        pload = [[0.0, 0.0, force / shape[1], 0.0, 0.0, 0.0]] * shape[1]

    elif transform == 'bend':
        pload = [[force / shape[1], 0.0, 0.0, 0.0, 0.0, 0.0]] * shape[1]

    elif transform == 'twist':
        pload = [[0.0, force / shape[1], 0.0, 0.0, 0.0, 0.0]] * shape[1]

    m = Material('m', D=sh.create_elastic_tensor(young=young, poisson=poisson),
                 values={'.drill' : 1e-7})
    load = Material('load', values={'.val' : pload})

    aux = Integral('i', order=3)
    qp_coors, qp_weights = aux.get_qp('3_8')
    qp_coors[:, 2] = thickness * (qp_coors[:, 2] - 0.5)
    qp_weights *= thickness

    integral = Integral('i', coors=qp_coors, weights=qp_weights, order='custom')

    t1 = Term.new('dw_shell10x(m.D, m.drill, v, u)',
                  integral, omega, m=m, v=v, u=u)
    t2 = Term.new('dw_point_load(load.val, v)',
                  integral, gamma2, load=load, v=v)
    eq = Equation('balance', t1 - t2)
    eqs = Equations([eq])

    fix_u = EssentialBC('fix_u', gamma1, {'u.all' : 0.0})

    ls = ScipyDirect({})

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

    pb = Problem('elasticity with shell10x', equations=eqs)
    pb.set_bcs(ebcs=Conditions([fix_u]))
    pb.set_solver(nls)

    state = pb.solve()

    return pb, state, u, gamma2

def get_analytical_displacement(dims, young, force, transform=None):
    """
    Returns the analytical value of the max. displacement according to
    Euler-Bernoulli theory.
    """
    l, b, h = dims

    if transform is None:
        moment = b * h**3 / 12.0
        u = force * l**3 / (3 * young * moment)

    elif transform == 'bend':
        u = force * 3.0 * nm.pi * l**3 / (young * b * h**3)

    elif transform == 'twist':
        u = None

    return u

helps = {
    'output_dir' : 'output directory',
    'dims' :
    'dimensions of the cantilever [default: %(default)s]',
    'nx' :
    'the range for the numbers of cells in the x direction'
    ' [default: %(default)s]',
    'transform' :
    'the transformation of the domain coordinates [default: %(default)s]',
    'young' : "the Young's modulus [default: %(default)s]",
    'poisson' : "the Poisson's ratio [default: %(default)s]",
    'force' : "the force load [default: %(default)s]",
    'plot' : 'plot the max. displacement w.r.t. number of cells',
    'scaling' : 'the displacement scaling, with --show [default: %(default)s]',
    'show' : 'show the results figure',
    'silent' : 'do not print messages to screen',
}

def main():
    parser = ArgumentParser(description=__doc__.rstrip(),
                            formatter_class=RawDescriptionHelpFormatter)
    parser.add_argument('output_dir', help=helps['output_dir'])
    parser.add_argument('-d', '--dims', metavar='l,w,t',
                        action='store', dest='dims',
                        default='0.2,0.01,0.001', help=helps['dims'])
    parser.add_argument('-n', '--nx', metavar='start,stop,step',
                        action='store', dest='nx',
                        default='2,103,10', help=helps['nx'])
    parser.add_argument('-t', '--transform', choices=['none', 'bend', 'twist'],
                        action='store', dest='transform',
                        default='none', help=helps['transform'])
    parser.add_argument('--young', metavar='float', type=float,
                        action='store', dest='young',
                        default=210e9, help=helps['young'])
    parser.add_argument('--poisson', metavar='float', type=float,
                        action='store', dest='poisson',
                        default=0.3, help=helps['poisson'])
    parser.add_argument('--force', metavar='float', type=float,
                        action='store', dest='force',
                        default=-1.0, help=helps['force'])
    parser.add_argument('-p', '--plot',
                        action="store_true", dest='plot',
                        default=False, help=helps['plot'])
    parser.add_argument('--u-scaling', metavar='float', type=float,
                        action='store', dest='scaling',
                        default=1.0, help=helps['scaling'])
    parser.add_argument('-s', '--show',
                        action="store_true", dest='show',
                        default=False, help=helps['show'])
    parser.add_argument('--silent',
                        action='store_true', dest='silent',
                        default=False, help=helps['silent'])
    options = parser.parse_args()

    dims = nm.array([float(ii) for ii in options.dims.split(',')],
                    dtype=nm.float64)
    nxs = tuple([int(ii) for ii in options.nx.split(',')])
    young = options.young
    poisson = options.poisson
    force = options.force

    output_dir = options.output_dir

    odir = lambda filename: os.path.join(output_dir, filename)

    filename = odir('output_log.txt')
    ensure_path(filename)
    output.set_output(filename=filename, combined=options.silent == False)

    output('output directory:', output_dir)
    output('using values:')
    output("  dimensions:", dims)
    output("  nx range:", nxs)
    output("  Young's modulus:", options.young)
    output("  Poisson's ratio:", options.poisson)
    output('  force:', options.force)
    output('  transform:', options.transform)

    if options.transform == 'none':
        options.transform = None

    u_exact = get_analytical_displacement(dims, young, force,
                                          transform=options.transform)

    if options.transform is None:
        ilog = 2
        labels = ['u_3']

    elif options.transform == 'bend':
        ilog = 0
        labels = ['u_1']

    elif options.transform == 'twist':
        ilog = [0, 1, 2]
        labels = ['u_1', 'u_2', 'u_3']

    label = ', '.join(labels)

    log = []
    for nx in range(*nxs):
        shape = (nx, 2)

        pb, state, u, gamma2 = solve_problem(shape, dims, young, poisson, force,
                                             transform=options.transform)

        dofs = u.get_state_in_region(gamma2)
        output('DOFs along the loaded edge:')
        output('\n%s' % dofs)

        log.append([nx - 1] + nm.array(dofs[0, ilog], ndmin=1).tolist())

    pb.save_state(odir('shell10x_cantilever.vtk'), state)

    log = nm.array(log)

    output('max. %s displacement w.r.t. number of cells:' % label)
    output('\n%s' % log)
    output('analytical value:', u_exact)

    if options.plot:
        import matplotlib.pyplot as plt

        plt.rcParams.update({
            'lines.linewidth' : 3,
            'font.size' : 16,
        })

        fig, ax1 = plt.subplots()
        fig.suptitle('max. $%s$ displacement' % label)

        for ic in range(log.shape[1] - 1):
            ax1.plot(log[:, 0], log[:, ic + 1], label=r'$%s$' % labels[ic])
        ax1.set_xlabel('# of cells')
        ax1.set_ylabel(r'$%s$' % label)
        ax1.grid(which='both')

        lines1, labels1 = ax1.get_legend_handles_labels()

        if u_exact is not None:
            ax1.hlines(u_exact, log[0, 0], log[-1, 0],
                       'r', 'dotted', label=r'$%s^{analytical}$' % label)

            ax2 = ax1.twinx()
            # Assume single log column.
            ax2.semilogy(log[:, 0], nm.abs(log[:, 1] - u_exact), 'g',
                         label=r'$|%s - %s^{analytical}|$' % (label, label))
            ax2.set_ylabel(r'$|%s - %s^{analytical}|$' % (label, label))

            lines2, labels2 = ax2.get_legend_handles_labels()

        else:
            lines2, labels2 = [], []

        ax1.legend(lines1 + lines2, labels1 + labels2, loc='best')

        plt.tight_layout()
        ax1.set_xlim([log[0, 0] - 2, log[-1, 0] + 2])

        suffix = {None: 'straight',
                  'bend' : 'bent', 'twist' : 'twisted'}[options.transform]
        fig.savefig(odir('shell10x_cantilever_convergence_%s.png' % suffix))

        plt.show()

    if options.show:
        from sfepy.postprocess.viewer import Viewer
        from sfepy.postprocess.domain_specific import DomainSpecificPlot

        ds = {'u_disp' :
                  DomainSpecificPlot('plot_displacements',
                                     ['rel_scaling=%f' % options.scaling])}
        view = Viewer(odir('shell10x_cantilever.vtk'))
        view(domain_specific=ds, is_scalar_bar=True, is_wireframe=True,
             opacity={'wireframe' : 0.5})

if __name__ == '__main__':
    main()