large_deformation/active_fibres.py¶

Description

Nearly incompressible hyperelastic material model with active fibres.

Large deformation is described using the total Lagrangian formulation. Models of this kind can be used in biomechanics to model biological tissues, e.g. muscles.

Find $\ul{u}$ such that:

$\intl{\Omega\suz}{} \left( \ull{S}\eff(\ul{u}) + K(J-1)\; J \ull{C}^{-1} \right) : \delta \ull{E}(\ul{v}) \difd{V} = 0 \;, \quad \forall \ul{v} \;,$

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 $\ull{S}\eff(\ul{u})$ incorporates also the effects of the active fibres in two preferential directions:

$\ull{S}\eff(\ul{u}) = \mu J^{-\frac{2}{3}}(\ull{I} - \frac{1}{3}\tr(\ull{C}) \ull{C}^{-1}) + \sum_{k=1}^2 \tau^k \ull{\omega}^k \;.$

The first term is the neo-Hookean term and the sum add contributions of the two fibre systems. The tensors $\ull{\omega}^k = \ul{d}^k\ul{d}^k$ are defined by the fibre system direction vectors $\ul{d}^k$ (unit).

For the one-dimensional tensions $\tau^k$ holds simply ( $^k$ omitted):

$\tau = A f_{\rm max} \exp{\left\{-(\frac{\epsilon - \varepsilon_{\rm opt}}{s})^2\right\}} \mbox{ , } \epsilon = \ull{E} : \ull{\omega} \;.$

../_images/large_deformation-active_fibres1.png

source code

# -*- coding: utf-8 -*-
r"""
Nearly incompressible hyperelastic material model with active fibres.

Large deformation is described using the total Lagrangian formulation.
Models of this kind can be used in biomechanics to model biological
tissues, e.g. muscles.

Find :math:`\ul{u}` such that:

.. math::
    \intl{\Omega\suz}{} \left( \ull{S}\eff(\ul{u})
    + K(J-1)\; J \ull{C}^{-1} \right) : \delta \ull{E}(\ul{v}) \difd{V}
    = 0
    \;, \quad \forall \ul{v} \;,

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})` incorporates also the
effects of the active fibres in two preferential directions:

.. math::
    \ull{S}\eff(\ul{u}) = \mu J^{-\frac{2}{3}}(\ull{I}
    - \frac{1}{3}\tr(\ull{C}) \ull{C}^{-1})
    + \sum_{k=1}^2 \tau^k \ull{\omega}^k
    \;.

The first term is the neo-Hookean term and the sum add contributions of
the two fibre systems. The tensors :math:`\ull{\omega}^k =
\ul{d}^k\ul{d}^k` are defined by the fibre system direction vectors
:math:`\ul{d}^k` (unit).

For the one-dimensional tensions :math:`\tau^k` holds simply (:math:`^k`
omitted):

.. math::
    \tau = A f_{\rm max} \exp{\left\{-(\frac{\epsilon - \varepsilon_{\rm
    opt}}{s})^2\right\}} \mbox{ , } \epsilon = \ull{E} : \ull{\omega}
    \;.
"""
from __future__ import print_function
from __future__ import absolute_import
import numpy as nm

from sfepy import data_dir

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

vf_matrix = 0.5
vf_fibres1 = 0.2
vf_fibres2 = 0.3

options = {
    'nls' : 'newton',
    'ls' : 'ls',
    'ts' : 'ts',
    'save_times' : 'all',
    'post_process_hook' : 'stress_strain',
}


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

materials = {
    'solid' : ({
        'K'  : vf_matrix * 1e3, # bulk modulus
        'mu' : vf_matrix * 20e0, # shear modulus of neoHookean term
    },),
    'f1' : 'get_pars_fibres1',
    'f2' : 'get_pars_fibres2',
}

def get_pars_fibres(ts, coors, mode=None, which=0, vf=1.0, **kwargs):
    """
    Parameters
    ----------
    ts : TimeStepper
        Time stepping info.
    coors : array_like
        The physical domain coordinates where the parameters shound be defined.
    mode : 'qp' or 'special'
        Call mode.
    which : int
        Fibre system id.
    vf : float
        Fibre system volume fraction.
    """
    if mode != 'qp': return

    fmax = 10.0
    eps_opt = 0.01
    s = 1.0

    tt = ts.nt * 2.0 * nm.pi

    if which == 0: # system 1
        fdir = nm.array([1.0, 0.0, 0.0], dtype=nm.float64)
        act = 0.5 * (1.0 + nm.sin(tt - (0.5 * nm.pi)))

    elif which == 1: # system 2
        fdir = nm.array([0.0, 1.0, 0.0], dtype=nm.float64)
        act = 0.5 * (1.0 + nm.sin(tt + (0.5 * nm.pi)))

    else:
        raise ValueError('unknown fibre system! (%d)' % which)

    fdir.shape = (3, 1)
    fdir /= nm.linalg.norm(fdir)

    print(act)

    shape = (coors.shape[0], 1, 1)
    out = {
        'fmax' : vf * nm.tile(fmax, shape),
        'eps_opt' : nm.tile(eps_opt, shape),
        's' : nm.tile(s, shape),
        'fdir' : nm.tile(fdir, shape),
        'act' : nm.tile(act, shape),
    }

    return out

functions = {
    'get_pars_fibres1' : (lambda ts, coors, mode=None, **kwargs:
                          get_pars_fibres(ts, coors, mode=mode, which=0,
                                          vf=vf_fibres1, **kwargs),),
    'get_pars_fibres2' : (lambda ts, coors, mode=None, **kwargs:
                          get_pars_fibres(ts, coors, mode=mode, which=1,
                                          vf=vf_fibres2, **kwargs),),
}

variables = {
    'u' : ('unknown field', 'displacement', 0),
    'v' : ('test field', 'displacement', 'u'),
}

regions = {
    'Omega' : 'all',
    'Left' : ('vertices in (x < 0.001)', 'facet'),
    'Right' : ('vertices in (x > 0.099)', 'facet'),
}

##
# Dirichlet BC.
ebcs = {
    'l' : ('Left', {'u.all' : 0.0}),
}

##
# Balance of forces.
integral_1 = {
    'name' : 'i',
    'order' : 1,
}
equations = {
    'balance'
        : """dw_tl_he_neohook.i.Omega( solid.mu, v, u )
           + dw_tl_bulk_penalty.i.Omega( solid.K, v, u )
           + dw_tl_fib_a.i.Omega( f1.fmax, f1.eps_opt, f1.s, f1.fdir, f1.act,
                                  v, u )
           + dw_tl_fib_a.i.Omega( f2.fmax, f2.eps_opt, f2.s, f2.fdir, f2.act,
                                  v, u )
           = 0""",
}

def stress_strain(out, problem, state, extend=False):
    from sfepy.base.base import Struct, debug

    ev = problem.evaluate
    strain = ev('dw_tl_he_neohook.i.Omega( solid.mu, v, u )',
                mode='el_avg', term_mode='strain')
    out['green_strain'] = Struct(name='output_data',
                                 mode='cell', data=strain, dofs=None)

    stress = ev('dw_tl_he_neohook.i.Omega( solid.mu, v, u )',
                mode='el_avg', term_mode='stress')
    out['neohook_stress'] = Struct(name='output_data',
                                   mode='cell', data=stress, dofs=None )

    stress = ev('dw_tl_bulk_penalty.i.Omega( solid.K, v, u )',
                mode='el_avg', term_mode= 'stress')
    out['bulk_stress'] = Struct(name='output_data',
                                mode='cell', data=stress, dofs=None)

    return out

##
# Solvers etc.
solver_0 = {
    'name' : 'ls',
    'kind' : 'ls.scipy_direct',
}

solver_1 = {
    'name' : 'newton',
    'kind' : 'nls.newton',

    'i_max'      : 7,
    'eps_a'      : 1e-10,
    'eps_r'      : 1.0,
    'macheps'    : 1e-16,
    'lin_red'    : 1e-2, # Linear system error < (eps_a * lin_red).
    'ls_red'     : 0.1,
    'ls_red_warp': 0.001,
    'ls_on'      : 1.1,
    'ls_min'     : 1e-5,
    'check'      : 0,
    'delta'      : 1e-6,
}

solver_2 = {
    'name' : 'ts',
    'kind' : 'ts.simple',

    't0'    : 0,
    't1'    : 1,
    'dt'    : None,
    'n_step' : 21, # has precedence over dt!
    'verbose' : 1,
}