diffusion/poisson_functions.pyΒΆ

Description

Poisson equation with source term.

Find u such that:

\int_{\Omega} c \nabla v \cdot \nabla u
= - \int_{\Omega_L} b v = - \int_{\Omega_L} f v p
\;, \quad \forall v \;,

where b(x) = f(x) p(x), p is a given FE field and f is a given general function of space.

This example demonstrates use of functions for defining material parameters, regions, parameter variables or boundary conditions. Notably, it demonstrates the following:

  1. How to define a material parameter by an arbitrary function - see the function get_pars() that evaluates f(x) in quadrature points.
  2. How to define a known function that belongs to a given FE space (field) - this function, p(x), is defined in a FE sense by its nodal values only - see the function get_load_variable().

In order to define the load b(x) directly, the term dw_volume_dot should be replaced by dw_volume_integrate.

../../_images/diffusion-poisson_functions.png

source code

r"""
Poisson equation with source term.

Find :math:`u` such that:

.. math::
    \int_{\Omega} c \nabla v \cdot \nabla u
    = - \int_{\Omega_L} b v = - \int_{\Omega_L} f v p
    \;, \quad \forall v \;,

where :math:`b(x) = f(x) p(x)`, :math:`p` is a given FE field and :math:`f` is
a given general function of space.

This example demonstrates use of functions for defining material parameters,
regions, parameter variables or boundary conditions. Notably, it demonstrates
the following:

1. How to define a material parameter by an arbitrary function - see the
   function :func:`get_pars()` that evaluates :math:`f(x)` in quadrature
   points.
2. How to define a known function that belongs to a given FE space (field) -
   this function, :math:`p(x)`, is defined in a FE sense by its nodal values
   only - see the function :func:`get_load_variable()`.

In order to define the load :math:`b(x)` directly, the term ``dw_volume_dot``
should be replaced by ``dw_volume_integrate``.
"""
from __future__ import absolute_import
import numpy as nm
from sfepy import data_dir

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

options = {
    'nls' : 'newton',
    'ls' : 'ls',
}

materials = {
    'm' : ({'c' : 1.0},),
    'load' : 'get_pars',
}

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

fields = {
    'temperature' : ('real', 1, 'Omega', 1),
    'velocity' : ('real', 'vector', 'Omega', 1),
}

variables = {
    'u' : ('unknown field', 'temperature', 0),
    'v' : ('test field',    'temperature', 'u'),
    'p' : ('parameter field', 'temperature',
           {'setter' : 'get_load_variable'}),
    'w' : ('parameter field', 'velocity',
           {'setter' : 'get_convective_velocity'}),
}

ebcs = {
    'u1' : ('Gamma_Left', {'u.0' : 'get_ebc'}),
    'u2' : ('Gamma_Right', {'u.0' : -2.0}),
}

integrals = {
    'i' : 1,
}

equations = {
    'Laplace equation' :
    """dw_laplace.i.Omega( m.c, v, u )
     - dw_convect_v_grad_s.i.Omega( v, w, u )
     = - dw_volume_dot.i.Omega_L( load.f, v, p )"""
}

solvers = {
    'ls' : ('ls.scipy_direct', {}),
    'newton' : ('nls.newton', {
        'i_max'      : 1,
        'eps_a'      : 1e-10,
    }),
}

def get_pars(ts, coors, mode=None, **kwargs):
    """
    Evaluate the coefficient `load.f` in quadrature points `coors` using a
    function of space.

    For scalar parameters, the shape has to be set to `(coors.shape[0], 1, 1)`.
    """
    if mode == 'qp':
        x = coors[:, 0]

        val = 55.0 * (x - 0.05)

        val.shape = (coors.shape[0], 1, 1)
        return {'f' : val}

def get_middle_ball(coors, domain=None):
    """
    Get the :math:`\Omega_L` region as a function of mesh coordinates.
    """
    x, y, z = coors[:, 0], coors[:, 1], coors[:, 2]

    r1 = nm.sqrt((x - 0.025)**2.0 + y**2.0 + z**2)
    r2 = nm.sqrt((x - 0.075)**2.0 + y**2.0 + z**2)
    flag = nm.where((r1 < 2.3e-2) | (r2 < 2.3e-2))[0]

    return flag

def get_load_variable(ts, coors, region=None):
    """
    Define nodal values of 'p' in the nodal coordinates `coors`.
    """
    y = coors[:,1]

    val = 5e5 * y
    return val

def get_convective_velocity(ts, coors, region=None):
    """
    Define nodal values of 'w' in the nodal coordinates `coors`.
    """
    val = 100.0 * nm.ones_like(coors)

    return val

def get_ebc(coors, amplitude):
    """
    Define the essential boundary conditions as a function of coordinates
    `coors` of region nodes.
    """
    z = coors[:, 2]
    val = amplitude * nm.sin(z * 2.0 * nm.pi)
    return val

functions = {
    'get_pars' : (get_pars,),
    'get_load_variable' : (get_load_variable,),
    'get_convective_velocity' : (get_convective_velocity,),
    'get_middle_ball' : (get_middle_ball,),
    'get_ebc' : (lambda ts, coor, bc, problem, **kwargs: get_ebc(coor, 5.0),),
}