# acoustics/vibro_acoustic3d_mid.py¶

Description

Vibro-acoustic problem

3D acoustic domain with 2D perforated deforming interface.

Master problem: defined in 3D acoustic domain (vibro_acoustic3d.py)

Slave subproblem: 2D perforated interface (vibro_acoustic3d_mid.py)

Master 3D problem - find (acoustic pressure) and (transversal acoustic velocity) such that:

Slave 2D subproblem - find (plate deflection) and (rotation) such that:

source code

r"""
Vibro-acoustic problem

3D acoustic domain with 2D perforated deforming interface.

*Master problem*: defined in 3D acoustic domain (vibro_acoustic3d.py)

*Slave subproblem*: 2D perforated interface (vibro_acoustic3d_mid.py)

Master 3D problem - find :math:p (acoustic pressure)
and :math:g (transversal acoustic velocity) such that:

.. math::
c^2 \int_{\Omega} \nabla q \cdot \nabla p
- \omega^2 \int_{\Omega} q p
+ i \omega c \int_{\Gamma_{in}} q p
+ i \omega c \int_{\Gamma_{out}} q p
- i \omega c^2 \int_{\Gamma_0} (q^+ - q^-) g
= 2i \omega c \int_{\Gamma_{in}} q \bar{p}
\;, \quad \forall q \;,

- i \omega \int_{\Gamma_0} f (p^+ - p^-)
- \omega^2 \int_{\Gamma_0} F f g
+ \omega^2 \int_{\Gamma_0} C f w
= 0
\;, \quad \forall f \;,

Slave 2D subproblem - find :math:w (plate deflection)
and :math:\ul{\theta} (rotation) such that:

.. math::
\omega^2 \int_{\Gamma_0} C z g
- \omega^2 \int_{\Gamma_0} S z w
+ \int_{\Gamma_0} \nabla z \cdot \ull{G} \cdot \nabla w
- \int_{\Gamma_0} \ul{\theta} \cdot \ull{G} \cdot \nabla z
= 0
\;, \quad \forall z \;,

- \omega^2 \int_{\Gamma_0} R\, \ul{\nu} \cdot \ul{\theta}
+ \int_{\Gamma_0} D_{ijkl} e_{ij}(\ul{\nu}) e_{kl}(\ul{\theta})
- \int_{\Gamma_0} \ul{\nu} \cdot \ull{G} \cdot \nabla w
+ \int_{\Gamma_0} \ul{\nu} \cdot \ull{G} \cdot \ul{\theta}
= 0
\;, \quad \forall \ul{\nu} \;,
"""
from __future__ import absolute_import
import numpy as nm
from sfepy import data_dir
from sfepy.mechanics.matcoefs import stiffness_from_lame

filename_mesh = data_dir + '/meshes/2d/acoustic_wg_mid.vtk'

sound_speed = 343.0
wave_num = 5.5
thickness = 0.01

c = sound_speed
c2 = c**2
w = wave_num * c
w2 = w**2
wc = w * c
wc2 = w * c2

regions = {
'Gamma0_1': 'all',
'Left': ('vertices in (x < 0.001)', 'facet'),
'Right': ('vertices in (x > 0.299)', 'facet'),
}

fields = {
'deflection': ('complex', 'scalar', 'Gamma0_1', 1),
'rotation': ('complex', 'vector', 'Gamma0_1', 1),
'tvelocity': ('complex', 'scalar', 'Gamma0_1', 1),
}

variables = {
'w': ('unknown field', 'deflection'),
'z': ('test field', 'deflection', 'w'),
'theta': ('unknown field', 'rotation'),
'nu': ('test field', 'rotation', 'theta'),
'g0': ('unknown field', 'tvelocity'),
'f0': ('test field', 'tvelocity', 'g0'),
}

ebcs = {
'fixed_l': ('Left', {'w.0': 0.0, 'theta.all': 0.0}),
'fixed_r': ('Right', {'w.0': 0.0, 'theta.all': 0.0}),
}

options = {
}

materials = {
'ac' : ({'c': -1.064e+00, 'T': 9.202e-01,
'hG': thickness * 4.5e10 * nm.eye(2),
'hR': thickness * 0.71,
'h3R': thickness**3 / 3.0 * 0.71,
'h3C': thickness**3 / 3.0 * stiffness_from_lame(2, 1e1, 1e0)}, ),
}

equations = {
'eq_3': """
%e * dw_dot.5.Gamma0_1(ac.c, z, g0)
- %e * dw_dot.5.Gamma0_1(ac.T, z, w)
- %e * dw_dot.5.Gamma0_1(ac.hR, z, w)
+ dw_diffusion.5.Gamma0_1(ac.hG, z, w)
- dw_v_dot_grad_s.5.Gamma0_1(ac.hG, theta, z)
= 0"""\
% (w2, w2, w2),
'eq_4': """
- %e * dw_dot.5.Gamma0_1(ac.h3R, nu, theta)
+ dw_lin_elastic.5.Gamma0_1(ac.h3C, nu, theta)
- dw_v_dot_grad_s.5.Gamma0_1(ac.hG, nu, w)
+ dw_dot.5.Gamma0_1(ac.hG, nu, theta)
= 0"""\
% (w2, ),
}

solvers = {
'ls' : ('ls.scipy_direct', {}),
'newton' : ('nls.newton', {
'i_max' : 1,
'eps_a' : 1e-4,
'eps_r' : 1e-4,
})
}