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.solvers import use_first_available
from sfepy.solvers.ls import MUMPSSolver, 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 = use_first_available([(MUMPSSolver, {}), (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()