diffusion/time_poisson.pyΒΆ

Description

Transient Laplace equation with non-constant initial conditions given by a function.

Find T(t) for t \in [0, t_{\rm final}] such that:

\int_{\Omega} s \pdiff{T}{t}
+ \int_{\Omega} c \nabla s \cdot \nabla T
= 0
\;, \quad \forall s \;.

../../_images/diffusion-time_poisson.png

source code

r"""
Transient Laplace equation with non-constant initial conditions given by a
function.

Find :math:`T(t)` for :math:`t \in [0, t_{\rm final}]` such that:

.. math::
    \int_{\Omega} s \pdiff{T}{t}
    + \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/cylinder.mesh'

t0 = 0.0
t1 = 0.1
n_step = 11

material_2 = {
    'name' : 'coef',
    'values' : {'val' : 0.01},
    'kind' : 'stationary', # 'stationary' or 'time-dependent'
}

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

variable_1 = {
    'name' : 'T',
    'kind' : 'unknown field',
    'field' : 'temperature',
    'order' : 0,
    'history' : 1,
}
variable_2 = {
    'name' : 's',
    'kind' : 'test field',
    'field' : 'temperature',
    'dual' : 'T',
}

regions = {
    'Omega' : 'all',
    'Gamma_Left' : ('vertices in (x < 0.00001)', 'facet'),
    'Gamma_Right' : ('vertices in (x > 0.099999)', 'facet'),
}

ebcs = {
    'T1': ('Gamma_Left', {'T.0' : 2.0}),
    'T2': ('Gamma_Right', {'T.0' : -2.0}),
}

def get_ic(coor, ic):
    """Non-constant initial condition."""
    import numpy as nm
    # Normalize x coordinate.
    mi, ma = coor[:,0].min(), coor[:,0].max()
    nx = (coor[:,0] - mi) / (ma - mi)
    return nm.where( (nx > 0.25) & (nx < 0.75 ), 8.0 * (nx - 0.5), 0.0 )

functions = {
    'get_ic' : (get_ic,),
}

ics = {
    'ic' : ('Omega', {'T.0' : 'get_ic'}),
}

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

equations = {
    'Temperature' :
    """dw_volume_dot.i.Omega( s, dT/dt )
     + dw_laplace.i.Omega( coef.val, s, T ) = 0"""
}

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

    'presolve' : True,
}

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,
    'is_linear' : True,
}

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

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

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