Homogenization of 2D structures

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Homogenization of 2D structures
y
Ly
x
Zy
Zx
Lx
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Wavenumber Spectrum for 2D structures
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Decomposition in 2D structures
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Coincidence in Pass-Bands
Mean square structural response as function of
forcing wavenumber and frequency
R
?L/c13. Pass-band
?L10. Pairs of resonances
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Subsonic Force Subsonic Plate
Mp cp/c .8 ?L/c 40 ?L 79.
Spectrum
Graphical view
forcing ?L
free plate
?L
radiation
Pressure versus angle
2?
Mp0.7
Mp0.8
Mp0.9
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Sonic Zones
Subsonic forcing
Free plate dispersion curve
Supersonic forcing
Subsonic plate
Supersonic plate
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Dispersion Relation, Plate with Masses
stop-bands
pass-bands
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Approximate vs. ExactHeavy Fluid
Pressure harmonics magnitudes pP -/P versus
frequency. Structural characteristics ?L/?s
80 cs/ c .8 ?0.6 Z/?sL .5 i kL 2.75.
Second harmonic turns on at kL4.01. Solid line
exact solution gray approximation by 1
global interval dash approximation by 3 global
intervals.
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Approximate vs. ExactLight Fluid
Pressure harmonics magnitudes pP -/P versus
frequency. Structural characteristics ?L/?s 7
cs/ c .8 ?0.6 Z/?sL .5 i kL 2.75. Second
harmonic turns on at kL4.01. Solid line
exact solution gray approximation by 1 global
interval dash approximation by 3 global
intervals.
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Approximate vs. ExactRadiating Mode
Displacement magnitude w?c??/P and real part of
the global pressure pP -/P versus frequency.
Structural characteristics ?s/?L.1 cs/ c
.67 ?0.6 m/?sL2.5 (mass impedance case).
Second harmonic turns on at kL4.01. Full
exact dashed approximation.
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Approximate vs. ExactFirst Radiating Mode
Pressure and displacement magnitudes pP -/P ,
w?c??/P of propagating harmonics versus
frequency. Structural characteristics ?s/?L.1
cs/ c .67 ?0.6 m/?sL2.5. Second harmonic
turns on at kL4.01. Solid line exact
solution gray approximation by 1 global
interval dash approximation by 3 global
intervals.
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Summary Fluid-Loaded Infinite Structures

• The wavenumber filtering procedure was applied to
  a fluid loaded structure. A closed system was
  constructed, which provides solution in a given
  wavenumber range.
• The approximate and exact solutions for fluid
  loading were compared for a scattering problem.
  Excellent results were achieved.
• Future goals enhancement of approximation
  method, 3-D realization, beam and plate
  homogenization, numerical implementations.

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Local-Global Relationship
For local solution outside the interval
For global solution in the interval, in
particular, at
The difference gives
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Global Equation
It can be shown that following equation governs
the global part of the solution. In wavenumber
space
On the left hand side global smooth
operator. On the right hand side new modified
forcing, confined in the global wavenumber
interval. Solution to this equation will be
confined in the global interval AND recover exact
displacements at discontinuities.
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Numerical Sequence
Problem A Modified forcing, global operator, no
discontinuities.
Problem B Special forcing, original operator,
no discontinuities.
Problem A provides exact displacements at
discontinuities. Problem B recovers full global
spectrum.
3-Interval global solution with exact
displacements at discontinuities is
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Example 3
Beam with 8 discontinuities excited at A and B
Directivity, dB scale
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Example 1
Beam with 5 masses excited at A and pinned
at B.
B
A
Directivity, dB scale
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Summary Homogenization of Dry Structures

• Global smooth problems were analyzed for simple
  finite configurations. Uncoupled acoustic
  response was compared in global and exact
  problems.
• Degree of discretization is significantly (1-2
  orders) lowered in the global problem.
• Special attention was given to global solutions
  designed to recover discontinuities displacements
  and low wavenumber content. Excellent degree of
  accuracy in model problems was observed for
  relatively high frequencies.
• Future research 2-D homogenization, shell
  homogenization.

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Example Constant Pressure Elements
Directivity, dB scale
uncoupled exact LGH method keff used
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Example constant pressure elements
Directivity, dB scale
uncoupled exact LGH method
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Example constant pressure elements
Directivity, dB scale
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Summary Wavenumber Filtering in BEM

• Homogenization method was applied to the
  numerical solution of a finite-length
  fluid-loaded periodic structure.
• Considerable computational savings were achieved
  roughly an order of magnitude for a fully
  coupled wet structure and even greater reduction
  for a dry structure.
• Future research
• enhanced fluid-loading formulation
• overcoming wavenumber folding error on finite
  structure
• extending the method to 2D structures.