multi_physics/biot.py¶

Description

Biot problem - deformable porous medium.

Find $\ul{u}$ , $p$ such that:

$\int_{\Omega} D_{ijkl}\ e_{ij}(\ul{v}) e_{kl}(\ul{u}) - \int_{\Omega} p\ \alpha_{ij} e_{ij}(\ul{v}) = 0 \;, \quad \forall \ul{v} \;, \int_{\Omega} q\ \alpha_{ij} e_{ij}(\ul{u}) + \int_{\Omega} K_{ij} \nabla_i q \nabla_j p = 0 \;, \quad \forall q \;,$

where

$D_{ijkl} = \mu (\delta_{ik} \delta_{jl}+\delta_{il} \delta_{jk}) + \lambda \ \delta_{ij} \delta_{kl} \;.$

source code

r"""
Biot problem - deformable porous medium.

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

.. math::
    \int_{\Omega} D_{ijkl}\ e_{ij}(\ul{v}) e_{kl}(\ul{u})
    - \int_{\Omega}  p\ \alpha_{ij} e_{ij}(\ul{v})
    = 0
    \;, \quad \forall \ul{v} \;,

    \int_{\Omega} q\ \alpha_{ij} e_{ij}(\ul{u})
    + \int_{\Omega} K_{ij} \nabla_i q \nabla_j p
    = 0
    \;, \quad \forall q \;,

where

.. math::
    D_{ijkl} = \mu (\delta_{ik} \delta_{jl}+\delta_{il} \delta_{jk}) +
    \lambda \ \delta_{ij} \delta_{kl}
    \;.
"""
from __future__ import absolute_import
import numpy as nm

from sfepy import data_dir
from sfepy.mechanics.matcoefs import stiffness_from_lame

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

regions = {
    'Omega' : 'all',
    'Bottom' : ('vertices in (z < -0.4999999)', 'facet'),
    'Top' : ('vertices in (z > 0.4999999)', 'facet'),
    'Left' : ('vertices in (x < -0.4999999)', 'facet'),
}

field_1 = {
    'name' : 'displacement',
    'dtype' : nm.float64,
    'shape' : (3,),
    'region' : 'Omega',
    'approx_order' : 1,
}

field_2 = {
    'name' : 'pressure',
    'dtype' : nm.float64,
    'shape' : (1,),
    'region' : 'Omega',
    'approx_order' : 1,
}

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

ebcs = {
    'fix_u' : ('Bottom', {'u.all' : 0.0}),
    'load_u' : ('Top', {'u.2' : 0.2}),
    'load_p' : ('Left', {'p.all' : 1.0}),
}

material_1 = {
    'name' : 'm',
    'values' : {
        'D': stiffness_from_lame(dim=3, lam=1.7, mu=0.3),
        'alpha' : nm.array( [[0.132], [0.132], [0.132],
                             [0.092], [0.092], [0.092]],
                            dtype = nm.float64 ),
        'K' : nm.array( [[2.0, 0.2, 0.0], [0.2, 1.0, 0.0], [0.0, 0.0, 0.5]],
                        dtype = nm.float64 ),
    }
}

integral_1 = {
    'name' : 'i1',
    'order' : 1,
}

integral_2 = {
    'name' : 'i2',
    'order' : 2,
}

equations = {
    'eq_1' :
    """dw_lin_elastic.i2.Omega( m.D, v, u )
     - dw_biot.i1.Omega( m.alpha, v, p )
       = 0""",
    'eq_2' :
    """dw_biot.i1.Omega( m.alpha, u, q ) + dw_diffusion.i1.Omega( m.K, q, p )
       = 0""",
}

solver_0 = {
    'name' : 'ls_d',
    'kind' : 'ls.scipy_direct',
}

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

    'i_max'      : 1,
    '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,
}