navier_stokes/navier_stokes2d_iga.py¶

Description

Navier-Stokes equations for incompressible fluid flow in 2D solved in a single patch NURBS domain using the isogeometric analysis (IGA) approach.

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

$\int_{\Omega} \nu\ \nabla \ul{v} : \nabla \ul{u} + \int_{\Omega} ((\ul{u} \cdot \nabla) \ul{u}) \cdot \ul{v} - \int_{\Omega} p\ \nabla \cdot \ul{v} = 0 \;, \quad \forall \ul{v} \;, \int_{\Omega} q\ \nabla \cdot \ul{u} = 0 \;, \quad \forall q \;.$

The domain geometry was created by:

sfepy-mesh iga-patch -2 -d 0.1,0.1 -s 10,10 -o meshes/iga/block2d.iga

View the results using:

sfepy-view block2d.vtk -2

../_images/navier_stokes-navier_stokes2d_iga1.png

source code

# -*- coding: utf-8 -*-
r"""
Navier-Stokes equations for incompressible fluid flow in 2D solved in a single
patch NURBS domain using the isogeometric analysis (IGA) approach.

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

.. math::
    \int_{\Omega} \nu\ \nabla \ul{v} : \nabla \ul{u}
    + \int_{\Omega} ((\ul{u} \cdot \nabla) \ul{u}) \cdot \ul{v}
    - \int_{\Omega} p\ \nabla \cdot \ul{v}
    = 0
    \;, \quad \forall \ul{v} \;,

    \int_{\Omega} q\ \nabla \cdot \ul{u}
    = 0
    \;, \quad \forall q \;.

The domain geometry was created by::

  sfepy-mesh iga-patch -2 -d 0.1,0.1 -s 10,10 -o meshes/iga/block2d.iga

View the results using::

  sfepy-view block2d.vtk -2
"""
from __future__ import absolute_import
from sfepy import data_dir

filename_domain = data_dir + '/meshes/iga/block2d.iga'

regions = {
    'Omega' : 'all',
    'Left' : ('vertices of set xi00', 'facet'),
    'Right' : ('vertices of set xi01', 'facet'),
    'Bottom' : ('vertices of set xi10', 'facet'),
    'Top' : ('vertices of set xi11', 'facet'),
    'Walls' : ('r.Left +v r.Right +v r.Bottom', 'facet'),
}

materials = {
    'fluid' : ({'viscosity' : 1.00e-2},),
}

fields = {
    'velocity': ('real', 'vector', 'Omega', 'iga+1', 'H1', 'iga'),
    'pressure': ('real', 'scalar', 'Omega', 'iga', 'H1', 'iga'),
}

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

ebcs = {
    '1_Walls' : ('Walls', {'u.all' : 0.0}),
    '0_Driven' : ('Top', {'u.0' : 1.0, 'u.1' : 0.0}),
    'Pressure' : ('Bottom', {'p.0' : 0.0}),
}

integrals = {
    'i' : 6,
}

equations = {
    'balance' :
    """+ dw_div_grad.i.Omega(fluid.viscosity, v, u)
       + dw_convect.i.Omega(v, u)
       - dw_stokes.i.Omega(v, p) = 0""",

    'incompressibility' :
    """dw_stokes.i.Omega(u, q) = 0""",
}

solvers = {
    'ls' : ('ls.scipy_direct', {}),
    'newton' : ('nls.newton', {
        'i_max'      : 15,
        'eps_a'      : 1e-10,
        'eps_r'      : 1.0,
    }),
}