homogenization/linear_homogenization_up.pyΒΆ
Description
missing description!
# mixed formulation
# 07.08.2009
#!
#! Homogenization: Linear Elasticity
#! =================================
#$ \centerline{Example input file, \today}
#! Homogenization of heterogeneous linear elastic material - mixed formulation
import numpy as nm
import sfepy.fem.periodic as per
from sfepy.mechanics.matcoefs import stiffness_from_youngpoisson_mixed, bulk_from_youngpoisson
from sfepy.homogenization.utils import define_box_regions, get_box_volume
import sfepy.homogenization.coefs_base as cb
from sfepy import data_dir
from sfepy.base.base import Struct
from sfepy.homogenization.recovery import compute_micro_u, compute_stress_strain_u, compute_mac_stress_part, add_stress_p
def recovery_le( pb, corrs, macro ):
out = {}
dim = corrs['corrs_le']['u_00'].shape[1]
mic_u = - compute_micro_u( corrs['corrs_le'], macro['strain'], 'u', dim )
mic_p = - compute_micro_u( corrs['corrs_le'], macro['strain'], 'p', dim )
out['u_mic'] = Struct( name = 'output_data',
mode = 'vertex', data = mic_u,
var_name = 'u', dofs = None )
out['p_mic'] = Struct( name = 'output_data',
mode = 'cell', data = mic_p[:,nm.newaxis,
:,nm.newaxis],
var_name = 'p', dofs = None )
stress_Y, strain_Y = compute_stress_strain_u( pb, 'i1', 'Y', 'mat.D', 'u', mic_u )
stress_Y += compute_mac_stress_part( pb, 'i1', 'Y', 'mat.D', 'u', macro['strain'] )
add_stress_p( stress_Y, pb, 'i1', 'Y', 'p', mic_p )
strain = macro['strain'] + strain_Y
out['cauchy_strain'] = Struct( name = 'output_data',
mode = 'cell', data = strain,
dofs = None )
out['cauchy_stress'] = Struct( name = 'output_data',
mode = 'cell', data = stress_Y,
dofs = None )
return out
#! Mesh
#! ----
dim = 3
filename_mesh = data_dir + '/meshes/3d/matrix_fiber.mesh'
region_lbn = (0, 0, 0)
region_rtf = (1, 1, 1)
#! Regions
#! -------
#! Regions, edges, ...
regions = {
'Y' : ('all', {}),
'Ym' : ('elements of group 1', {}),
'Yc' : ('elements of group 2', {}),
}
regions.update( define_box_regions( dim, region_lbn, region_rtf ) )
#! Materials
#! ---------
materials = {
'mat' : ({'D' : {'Ym': stiffness_from_youngpoisson_mixed(dim, 7.0e9, 0.4),
'Yc': stiffness_from_youngpoisson_mixed(dim, 70.0e9, 0.2)},
'gamma': {'Ym': 1.0/bulk_from_youngpoisson(7.0e9, 0.4),
'Yc': 1.0/bulk_from_youngpoisson(70.0e9, 0.2)}},),
}
#! Fields
#! ------
#! Scalar field for corrector basis functions.
fields = {
'corrector_u' : ('real', dim, 'Y', 1),
'corrector_p' : ('real', 1, 'Y', 0),
}
#! Variables
#! ---------
#! Unknown and corresponding test variables. Parameter fields
#! used for evaluation of homogenized coefficients.
variables = {
'u' : ('unknown field', 'corrector_u'),
'v' : ('test field', 'corrector_u', 'u'),
'p' : ('unknown field', 'corrector_p'),
'q' : ('test field', 'corrector_p', 'p'),
'Pi' : ('parameter field', 'corrector_u', 'u'),
'Pi1u' : ('parameter field', 'corrector_u', '(set-to-None)'),
'Pi2u' : ('parameter field', 'corrector_u', '(set-to-None)'),
'Pi1p' : ('parameter field', 'corrector_p', '(set-to-None)'),
'Pi2p' : ('parameter field', 'corrector_p', '(set-to-None)'),
}
#! Functions
functions = {
'match_x_plane' : (per.match_x_plane,),
'match_y_plane' : (per.match_y_plane,),
'match_z_plane' : (per.match_z_plane,),
}
#! Boundary Conditions
#! -------------------
#! Fixed nodes.
ebcs = {
'fixed_u' : ('Corners', {'u.all' : 0.0}),
}
if dim == 3:
epbcs = {
'periodic_x' : (['Left', 'Right'], {'u.all' : 'u.all'}, 'match_x_plane'),
'periodic_y' : (['Near', 'Far'], {'u.all' : 'u.all'}, 'match_y_plane'),
'periodic_z' : (['Top', 'Bottom'], {'u.all' : 'u.all'}, 'match_z_plane'),
}
else:
epbcs = {
'periodic_x' : (['Left', 'Right'], {'u.all' : 'u.all'}, 'match_x_plane'),
'periodic_y' : (['Bottom', 'Top'], {'u.all' : 'u.all'}, 'match_y_plane'),
}
all_periodic = ['periodic_%s' % ii for ii in ['x', 'y', 'z'][:dim] ]
#! Integrals
#! ---------
#! Define the integral type Volume/Surface and quadrature rule.
integrals = {
'i1' : ('v', 2),
'i2' : ('s', 2),
}
#! Options
#! -------
#! Various problem-specific options.
options = {
'coefs' : 'coefs',
'requirements' : 'requirements',
'ls' : 'ls', # linear solver to use
'volume' : { #'variables' : ['u'],
#'expression' : 'd_volume.i1.Y( u )',
'value' : get_box_volume( dim, region_lbn, region_rtf ),
},
'output_dir' : 'output',
'coefs_filename' : 'coefs_le_up',
'recovery_hook' : 'recovery_le',
}
#! Equations
#! ---------
#! Equations for corrector functions.
equation_corrs = {
'balance_of_forces' :
""" dw_lin_elastic.i1.Y( mat.D, v, u )
- dw_stokes.i1.Y( v, p ) =
- dw_lin_elastic.i1.Y( mat.D, v, Pi )""",
'pressure constraint' :
"""- dw_stokes.i1.Y( u, q )
- dw_volume_dot.i1.Y( mat.gamma, q, p ) =
+ dw_stokes.i1.Y( Pi, q )""",
}
#! Expressions for homogenized linear elastic coefficients.
expr_coefs = {
'Q1' : """dw_lin_elastic.i1.Y( mat.D, Pi1u, Pi2u )""",
'Q2' : """dw_volume_dot.i1.Y( mat.gamma, Pi1p, Pi2p )""",
}
#! Coefficients
#! ------------
#! Definition of homogenized acoustic coefficients.
def set_elastic_u(variables, ir, ic, mode, pis, corrs_rs):
mode2var = {'row' : 'Pi1u', 'col' : 'Pi2u'}
val = pis.states[ir, ic]['u'] + corrs_rs.states[ir, ic]['u']
variables[mode2var[mode]].set_data(val)
coefs = {
'elastic_u' : {
'requires' : ['pis', 'corrs_rs'],
'expression' : expr_coefs['Q1'],
'set_variables' : set_elastic_u,
'class' : cb.CoefSymSym,
},
'elastic_p' : {
'requires' : ['corrs_rs'],
'expression' : expr_coefs['Q2'],
'set_variables' : [('Pi1p', 'corrs_rs', 'p'), ('Pi2p', 'corrs_rs', 'p')],
'class' : cb.CoefSymSym,
},
'D' : {
'requires' : ['c.elastic_u', 'c.elastic_p'],
'class' : cb.CoefSum,
},
'filenames' : {},
}
requirements = {
'pis' : {
'variables' : ['u'],
'class' : cb.ShapeDimDim,
},
'corrs_rs' : {
'requires' : ['pis'],
'ebcs' : ['fixed_u'],
'epbcs' : all_periodic,
'equations' : equation_corrs,
'set_variables' : [('Pi', 'pis', 'u')],
'class' : cb.CorrDimDim,
'save_name' : 'corrs_le',
'dump_variables' : ['u', 'p'],
'is_linear' : True,
},
}
#! Solvers
#! -------
#! Define linear and nonlinear solver.
solvers = {
'ls' : ('ls.umfpack', {}),
'newton' : ('nls.newton', {'i_max' : 1,
'eps_a' : 1e-4,
'problem' : 'nonlinear', })
}