# diffusion/cube.py¶

Description

Laplace equation (e.g. temperature distribution) on a cube geometry with different boundary condition values on the cube sides. This example was used to create the SfePy logo.

Find such that:

source code

r"""
Laplace equation (e.g. temperature distribution) on a cube geometry with
different boundary condition values on the cube sides. This example was
used to create the SfePy logo.

Find :math:T such that:

.. math::
\int_{\Omega} c \nabla s \cdot \nabla T
= 0
\;, \quad \forall s \;.
"""
from __future__ import absolute_import
from sfepy import data_dir

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

############# Laplace.

material_1 = {
'name' : 'coef',
'values' : {'val' : 1.0},
}

field_1 = {
'name' : 'temperature',
'dtype' : 'real',
'shape' : (1,),
'region' : 'Omega',
'approx_order' : 1,
}

if filename_mesh.find('cube_medium_hexa.mesh') >= 0:
region_1000 = {
'name' : 'Omega',
'select' : 'cells of group 0',
}
integral_1 = {
'name' : 'i',
'order' : 1,
}
solver_0 = {
'name' : 'ls',
'kind' : 'ls.scipy_direct',
}

elif filename_mesh.find('cube_big_tetra.mesh') >= 0:
region_1000 = {
'name' : 'Omega',
'select' : 'cells of group 6',
}
integral_1 = {
'name' : 'i',
'vals'    : [[1./3., 1./3., 1./3.]],
'weights' : [0.5]
}
solver_0 = {
'name' : 'ls',
'kind' : 'ls.scipy_iterative',

'method' : 'cg',
'i_max'   : 1000,
'eps_r'   : 1e-12,
}

variable_1 = {
'name' : 'T',
'kind' : 'unknown field',
'field' : 'temperature',
'order' : 0, # order in the global vector of unknowns
}

variable_2 = {
'name' : 's',
'kind' : 'test field',
'field' : 'temperature',
'dual' : 'T',
}

region_0 = {
'name' : 'Surface',
'select' : 'vertices of surface',
'kind' : 'facet',
}
region_1 = {
'name' : 'Bottom',
'select' : 'vertices in (z < -0.4999999)',
'kind' : 'facet',
}
region_2 = {
'name' : 'Top',
'select' : 'vertices in (z > 0.4999999)',
'kind' : 'facet',
}
region_03 = {
'name' : 'Left',
'select' : 'vertices in (x < -0.4999999)',
'kind' : 'facet',
}

ebc_1 = {
'name' : 'T0',
'region' : 'Surface',
'dofs' : {'T.0' : -3.0},
}
ebc_4 = {
'name' : 'T1',
'region' : 'Top',
'dofs' : {'T.0' : 1.0},
}
ebc_3 = {
'name' : 'T2',
'region' : 'Bottom',
'dofs' : {'T.0' : -1.0},
}
ebc_2 = {
'name' : 'T3',
'region' : 'Left',
'dofs' : {'T.0' : 2.0},
}

equations = {
'nice_equation' : """dw_laplace.i.Omega( coef.val, s, T ) = 0""",
}

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,
}