large_deformation/balloon.pyΒΆ

Description

Inflation of a Mooney-Rivlin hyperelastic balloon.

This example serves as a verification of the membrane term (dw_tl_membrane, TLMembraneTerm) implementation.

Following Rivlin 1952 and Dumais, the analytical relation between a relative stretch L = r / r_0 of a thin (membrane) sphere made of the Mooney-Rivlin material of the undeformed radius r_0, membrane thickness h_0 and the inner pressure p is

p = 4 \frac{h_0}{r_0} (\frac{1}{L} - \frac{1}{L^7}) (c_1 + c_2 L^2) \;,

where c_1, c_2 are the Mooney-Rivlin material parameters.

In the equations below, only the surface of the domain is mechanically important - a stiff 2D membrane is embedded in the 3D space and coincides with the balloon surface. The volume is very soft, to simulate a fluid-filled cavity. A similar model could be used to model e.g. plant cells. The balloon surface is loaded by prescribing the inner volume change \omega(t). The fluid pressure in the cavity is a single scalar value, enforced by the 'integral_mean_value' linear combination condition.

Find \ul{u}(\ul{X}) and a constant p such that:

  • balance of forces:

    \intl{\Omega\suz}{} \left( \ull{S}\eff(\ul{u})
- p\; J \ull{C}^{-1} \right) : \delta \ull{E}(\ul{v}; \ul{v}) \difd{V}
+ \intl{\Gamma\suz}{} \ull{S}\eff(\tilde{\ul{u}}) \delta
\ull{E}(\tilde{\ul{u}}; \tilde{\ul{v}}) h_0 \difd{S}
= 0 \;, \quad \forall \ul{v} \in [H^1_0(\Omega)]^3 \;,

  • volume conservation:

    \int\limits_{\Omega_0} \left[\omega(t)-J(u)\right] q\, dx = 0
\qquad \forall q \in L^2(\Omega) \;,

where

\ull{F} deformation gradient F_{ij} = \pdiff{x_i}{X_j}
J \det(F)
\ull{C} right Cauchy-Green deformation tensor C = F^T F
\ull{E}(\ul{u}) Green strain tensor E_{ij} = \frac{1}{2}(\pdiff{u_i}{X_j} +
\pdiff{u_j}{X_i} + \pdiff{u_m}{X_i}\pdiff{u_m}{X_j})
\ull{S}\eff(\ul{u}) effective second Piola-Kirchhoff stress tensor

The effective stress \ull{S}\eff(\ul{u}) is given by:

\ull{S}\eff(\ul{u}) = \mu J^{-\frac{2}{3}}(\ull{I}
- \frac{1}{3}\tr(\ull{C}) \ull{C}^{-1})
+ \kappa J^{-\frac{4}{3}} (\tr(\ull{C}\ull{I} - \ull{C}
- \frac{2}{6}((\tr{\ull{C}})^2 - \tr{(\ull{C}^2)})\ull{C}^{-1})
\;.

The \tilde{\ul{u}} and \tilde{\ul{v}} variables correspond to \ul{u}, \ul{v}, respectively, transformed to the membrane coordinate frame.

Use the following command to show a comparison of the FEM solution with the above analytical relation (notice the nonlinearity of the dependence):

python simple.py examples/large_deformation/balloon.py -d 'plot: True'

The agreement should be very good, even though the mesh is coarse.

View the results using:

python postproc.py unit_ball.h5 --wireframe -b -d'u,plot_displacements,rel_scaling=1' --step=-1

This example uses the adaptive time-stepping solver ('ts.adaptive') with the default adaptivity function adapt_time_step(). Plot the used time steps by:

python script/plot_times.py unit_ball.h5
../../_images/large_deformation-balloon.png

source code

r"""
Inflation of a Mooney-Rivlin hyperelastic balloon.

This example serves as a verification of the membrane term (``dw_tl_membrane``,
:class:`TLMembraneTerm <sfepy.terms.terms_membrane.TLMembraneTerm>`)
implementation.

Following Rivlin 1952 and Dumais, the analytical relation between a
relative stretch :math:`L = r / r_0` of a thin (membrane) sphere made of the
Mooney-Rivlin material of the undeformed radius :math:`r_0`, membrane
thickness :math:`h_0` and the inner pressure :math:`p` is

.. math::

   p = 4 \frac{h_0}{r_0} (\frac{1}{L} - \frac{1}{L^7}) (c_1 + c_2 L^2) \;,

where :math:`c_1`, :math:`c_2` are the Mooney-Rivlin material parameters.

In the equations below, only the surface of the domain is mechanically
important - a stiff 2D membrane is embedded in the 3D space and coincides with
the balloon surface. The volume is very soft, to simulate a fluid-filled
cavity. A similar model could be used to model e.g. plant cells. The balloon
surface is loaded by prescribing the inner volume change :math:`\omega(t)`.
The fluid pressure in the cavity is a single scalar value, enforced by the
``'integral_mean_value'`` linear combination condition.

Find :math:`\ul{u}(\ul{X})` and a constant :math:`p` such that:

- balance of forces:

  .. math::
     \intl{\Omega\suz}{} \left( \ull{S}\eff(\ul{u})
     - p\; J \ull{C}^{-1} \right) : \delta \ull{E}(\ul{v}; \ul{v}) \difd{V}
     + \intl{\Gamma\suz}{} \ull{S}\eff(\tilde{\ul{u}}) \delta
     \ull{E}(\tilde{\ul{u}}; \tilde{\ul{v}}) h_0 \difd{S}
     = 0 \;, \quad \forall \ul{v} \in [H^1_0(\Omega)]^3 \;,

- volume conservation:

  .. math::
     \int\limits_{\Omega_0} \left[\omega(t)-J(u)\right] q\, dx = 0
     \qquad \forall q \in L^2(\Omega) \;,

where

.. list-table::
   :widths: 20 80

   * - :math:`\ull{F}`
     - deformation gradient :math:`F_{ij} = \pdiff{x_i}{X_j}`
   * - :math:`J`
     - :math:`\det(F)`
   * - :math:`\ull{C}`
     -  right Cauchy-Green deformation tensor :math:`C = F^T F`
   * - :math:`\ull{E}(\ul{u})`
     - Green strain tensor :math:`E_{ij} = \frac{1}{2}(\pdiff{u_i}{X_j} +
       \pdiff{u_j}{X_i} + \pdiff{u_m}{X_i}\pdiff{u_m}{X_j})`
   * - :math:`\ull{S}\eff(\ul{u})`
     - effective second Piola-Kirchhoff stress tensor

The effective stress :math:`\ull{S}\eff(\ul{u})` is given by:

.. math::
    \ull{S}\eff(\ul{u}) = \mu J^{-\frac{2}{3}}(\ull{I}
    - \frac{1}{3}\tr(\ull{C}) \ull{C}^{-1})
    + \kappa J^{-\frac{4}{3}} (\tr(\ull{C}\ull{I} - \ull{C}
    - \frac{2}{6}((\tr{\ull{C}})^2 - \tr{(\ull{C}^2)})\ull{C}^{-1})
    \;.

The :math:`\tilde{\ul{u}}` and :math:`\tilde{\ul{v}}` variables correspond to
:math:`\ul{u}`, :math:`\ul{v}`, respectively, transformed to the membrane
coordinate frame.

Use the following command to show a comparison of the FEM solution with the
above analytical relation (notice the nonlinearity of the dependence)::

  python simple.py examples/large_deformation/balloon.py -d 'plot: True'

The agreement should be very good, even though the mesh is coarse.

View the results using::

  python postproc.py unit_ball.h5 --wireframe -b -d'u,plot_displacements,rel_scaling=1' --step=-1

This example uses the adaptive time-stepping solver (``'ts.adaptive'``) with
the default adaptivity function :func:`adapt_time_step()
<sfepy.solvers.ts_solvers.adapt_time_step>`. Plot the used time steps by::

  python script/plot_times.py unit_ball.h5
"""
from __future__ import absolute_import
import os
import numpy as nm

from sfepy.base.base import Output
from sfepy.discrete.fem import MeshIO
from sfepy.linalg import get_coors_in_ball
from sfepy import data_dir

output = Output('balloon:')

def get_nodes(coors, radius, eps, mode):
    if mode == 'ax1':
        centre = nm.array([0.0, 0.0, -radius], dtype=nm.float64)

    elif mode == 'ax2':
        centre = nm.array([0.0, 0.0, radius], dtype=nm.float64)

    elif mode == 'equator':
        centre = nm.array([radius, 0.0, 0.0], dtype=nm.float64)

    else:
        raise ValueError('unknown mode %s!' % mode)

    return get_coors_in_ball(coors, centre, eps)

def get_volume(ts, coors, region=None):
    rs = 1.0 + 1.0 * ts.time

    rv = get_rel_volume(rs)
    output('relative stretch:', rs)
    output('relative volume:', rv)

    out = nm.empty((coors.shape[0],), dtype=nm.float64)
    out.fill(rv)

    return out

def get_rel_volume(rel_stretch):
    """
    Get relative volume V/V0 from relative stretch r/r0  of a ball.
    """
    return nm.power(rel_stretch, 3.0)

def get_rel_stretch(rel_volume):
    """
    Get relative stretch r/r0 from relative volume V/V0 of a ball.
    """
    return nm.power(rel_volume, 1.0/3.0)

def get_balloon_pressure(rel_stretch, h0, r0, c1, c2):
    """
    Rivlin 1952 + Dumais:

    P = 4*h0/r0 * (1/L-1/L^7).*(C1+L^2*C2)
    """
    L = rel_stretch
    p = 4.0 * h0 / r0 * (1.0/L - 1.0/L**7) * (c1 + c2 * L**2)

    return p

def plot_radius(problem, state):
    import matplotlib.pyplot as plt

    from sfepy.postprocess.time_history import extract_time_history

    ths, ts = extract_time_history('unit_ball.h5', 'p e 0')

    p = ths['p'][0]
    L = 1.0 + ts.times[:p.shape[0]]

    L2 = 1.0 + nm.linspace(ts.times[0], ts.times[-1], 1000)
    p2 = get_balloon_pressure(L2, 1e-2, 1, 3e5, 3e4)

    plt.rcParams['lines.linewidth'] = 3
    plt.rcParams['text.fontsize'] = 16

    plt.plot(L2, p2, 'r', label='theory')
    plt.plot(L, p, 'b*', ms=12, label='FEM')

    plt.title('Mooney-Rivlin hyperelastic balloon inflation')
    plt.xlabel(r'relative stretch $r/r_0$')
    plt.ylabel(r'pressure $p$')

    plt.legend(loc='best')

    fig = plt.gcf()
    fig.savefig('balloon_pressure_stretch.pdf')

    plt.show()

def define(plot=False):
    filename_mesh = data_dir + '/meshes/3d/unit_ball.mesh'

    conf_dir = os.path.dirname(__file__)
    io = MeshIO.any_from_filename(filename_mesh, prefix_dir=conf_dir)
    bbox = io.read_bounding_box()
    dd = bbox[1] - bbox[0]

    radius = bbox[1, 0]
    eps = 1e-8 * dd[0]

    options = {
        'nls' : 'newton',
        'ls' : 'ls',
        'ts' : 'ts',
        'save_times' : 'all',
        'output_dir' : '.',
        'output_format' : 'h5',
    }

    if plot:
        options['post_process_hook_final'] = plot_radius

    fields = {
        'displacement': (nm.float64, 3, 'Omega', 1),
        'pressure': (nm.float64, 1, 'Omega', 0),
    }

    materials = {
        'solid' : ({
            'mu' : 50, # shear modulus of neoHookean term
            'kappa' : 0.0, # shear modulus of Mooney-Rivlin term
        },),
        'walls' : ({
            'mu' : 3e5, # shear modulus of neoHookean term
            'kappa' : 3e4, # shear modulus of Mooney-Rivlin term
            'h0' : 1e-2, # initial thickness of wall membrane
        },),
    }

    variables = {
        'u' : ('unknown field', 'displacement', 0),
        'v' : ('test field', 'displacement', 'u'),
        'p' : ('unknown field', 'pressure', 1),
        'q' : ('test field', 'pressure', 'p'),
        'omega' : ('parameter field', 'pressure', {'setter' : 'get_volume'}),
    }

    regions = {
        'Omega'  : 'all',
        'Ax1' : ('vertices by get_ax1', 'vertex'),
        'Ax2' : ('vertices by get_ax2', 'vertex'),
        'Equator' : ('vertices by get_equator', 'vertex'),
        'Surface' : ('vertices of surface', 'facet'),
    }

    ebcs = {
        'fix1' : ('Ax1', {'u.all' : 0.0}),
        'fix2' : ('Ax2', {'u.[0, 1]' : 0.0}),
        'fix3' : ('Equator', {'u.1' : 0.0}),
    }

    lcbcs = {
        'pressure' : ('Omega', {'p.all' : None}, None, 'integral_mean_value'),
    }

    equations = {
        'balance'
            : """dw_tl_he_neohook.2.Omega(solid.mu, v, u)
               + dw_tl_he_mooney_rivlin.2.Omega(solid.kappa, v, u)
               + dw_tl_membrane.2.Surface(walls.mu, walls.kappa, walls.h0, v, u)
               + dw_tl_bulk_pressure.2.Omega(v, u, p)
               = 0""",
        'volume'
            : """dw_tl_volume.2.Omega(q, u)
               = dw_volume_dot.2.Omega(q, omega)""",
    }

    solvers = {
        'ls' : ('ls.scipy_direct', {}),
        'newton' : ('nls.newton', {
            'i_max'      : 6,
            'eps_a'      : 1e-4,
            'eps_r'      : 1e-8,
            'macheps'    : 1e-16,
            'lin_red'    : 1e-2,
            'ls_red'     : 0.5,
            'ls_red_warp': 0.1,
            'ls_on'      : 100.0,
            'ls_min'     : 1e-5,
            'check'      : 0,
            'delta'      : 1e-6,
            'is_plot'    : False,
            'problem'    : 'nonlinear',
        }),
        'ts' : ('ts.adaptive', {
            't0' : 0.0,
            't1' : 5.0,
            'dt' : None,
            'n_step' : 11,

            'dt_red_factor' : 0.8,
            'dt_red_max' : 1e-3,
            'dt_inc_factor' : 1.25,
            'dt_inc_on_iter' : 4,
            'dt_inc_wait' : 3,

            'verbose' : 1,
            'quasistatic' : True,
        }),
    }

    functions = {
        'get_ax1' : (lambda coors, domain:
                     get_nodes(coors, radius, eps, 'ax1'),),
        'get_ax2' : (lambda coors, domain:
                     get_nodes(coors, radius, eps, 'ax2'),),
        'get_equator' : (lambda coors, domain:
                         get_nodes(coors, radius, eps, 'equator'),),
        'get_volume' : (get_volume,),
    }

    return locals()