LocalGlobal Homogenization of Periodic Structures

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Local/Global Homogenization
of Periodic Structures Pavel V.
Danilov Duke University, Durham, North Carolina,
USA Ph.D. Thesis Defense December, 9, 2004
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Finite Differences Membrane with Masses
Harmonic forcing
10 points per bay
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Finite Differences Membrane with Masses
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Finite Differences Membrane with Masses
20 points per bay
40 points per bay
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Finite Differences Membrane with Masses
Exact - 80 points per bay
Global
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Finite Differences Membrane with Masses
Exact - 80 points per bay
Global
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What do we want to achieve?
The goal is to solve for the global, long-wave
part separately. Instead of solving the whole
problem may be, too complicated! is it
possible to solve simpler, preferably with no
discontinuities problem in order to recover only
interesting global part? Local part
short-wave harmonics could be recovered later,
if desired.
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Road Map

• Example of the exact solution, fluid-loaded plate
  with discontinuities pass- and stop- bands,
  broad wave number spectrum, dispersion relation,
  radiation patterns.
• Homogenization of dry structures global
  operator, modified forcing, Bloch waves,
  propagation constants, uncoupled acoustic
  radiation.
• Homogenization of fluid-loaded structures
  fluid-loading approximation. Numerical methods
  BEM formulations using filtering and Bloch wave
  harmonics.
• Infinite fixed-fixed arbitrary
  BC

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Structures with Discontinuities
plate or membrane
Impedance model zF/v
y
x
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Example fluid-loaded plate
with discontinuities
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Governing Equations
Acoustic equation in the fluid medium
Structural equation of the plate
Acceleration matching boundary condition
P(x,y,t) pressure field W(x,t) plate
displacement. Plate parameters D plate
flexural rigidity Fluid parameters ?, c
density and speed of sound z discontinuity
impedance.
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Governing Equations in Wavenumber Space
Forcing is assumed to be of the form
Discontinuities couple different wavenumbers.
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Wavenumber Spectrum
?
supersonic
subsonic
-k
k
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Supersonic Force Supersonic Plate
Mp cp/c 1.7 ?L/c 65 ?L 9.
Spectrum
Graphical view
forcing ?L
free plate
?L
radiation
2?
Pressure versus angle
P
Angle
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Bloch waves on dry structure
n-th bay
(n1)-th bay
Global solution
Right propagating Bloch wave whatever happened
in n-th bay will happen in (n1)-th bay after
some time delay
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Bloch Waves on a Beam
Displacement impedance
Rotational impedance
At discontinuities there are jumps in moment
and shear force
Bloch wavenumbers
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Example Semi Infinite String with Masses,
Pass Band
Driven at the left end
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Dispersion Relation Pass- Stop- Bands
Dispersion relation for membrane with masses
Pass-band
Stop-band
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Coincidence in Stop-Bands
Mean square structural response as function of
forcing wavenumber and frequency
?L/c11.5. Stop-band type I
?L/c 15. Stop-band type II
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Discontinuities lead to

• Reflection of structural waves from
  discontinuities leads to wide spectrum response
• Structural eigenmodes are wave groups rather
  than single harmonics
• Dispersion relation shows pass and stop bands,
  in stop bands structural response in bays decays
• Sound radiation occurs even when excited by
  subsonic force.

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Homogenization of Structureswith Discontinuities

• A problem having structural discontinuities is
  replaced with an equivalent self-contained
  smooth problem.
• The goal is to solve problems with high
  complexity by separating local and global effects.

• Potential Benefits
• Not limited to very low frequencies.
• Fine scale local motion is determined after the
  fact.
• Fluid radiation is contained within smooth global
  problem.
• The impact of evanescent fluid modes is
  incorporated in the global structural operator.
• Possible direct numerical implementation of the
  smooth global problem.

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Membrane with spatially periodic discontinuities
Z
L
j
- impedance applied at
, normalized by tension T
F(x) - applied harmonic force
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Global and Local Solutions
when
when
Wavenumber space
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Global Equation
However
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Homogenization of Infinite Structures
Harmonic motion
Global problem has no discontinuities
Wavenumber or Fourier transform space
Physical space
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General Global Solution
In general, if a dynamic system is described by a
linear operator D(?,?), and discontinuities are
given by their impedance Z and spacing L, then
the global operator in the wavenumber space is
The global operator as an infinite differential
operator
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Modified Forcing
Modified forcing can be found either directly by
inverse Fourier transform in wavenumber space or
using forcing Greens function in physical space.
Original forcing
Modified forcing
Then for arbitrary original forcing
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Wavenumber spectrum 1 interval
FT
Global problem recovers first harmonic exactly
Global problem recovers exact displacements at
discontinuities
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Wavenumber spectrum 3 intervals
FT
radiating interval
Global problem recovers first harmonic and exact
displacements at discontinuities. In this case we
recover the global spectrum in the first interval
exactly and adjust wavenumber content in side
intervals to provide exact impedance
displacements
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Homogenization of fixed-fixed problems
Using the method of images the corresponding
problem on an infinite structure is found. Ready
formulas for global response of infinite
structures can be used.
F(x,t)
Forcing is readily given as Fourier series and
modified forcing in the global problem can be
found
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Acoustic radiation
Assuming light fluid loading - acoustic radiation
and structural response are uncoupled - using
Raleigh integral to get radiated pressure field
from the structural motion. In 2-D case
R
?
Directivity functions are further compared for
exact and global solution.
dB scale
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Sample Problem
Derived global problem was solved for a sample
finite problem. Membrane with 9 attached masses
was fixed at both ends and excited by harmonic
forcing. Mesh for global problems was 3-10 points
between masses. Results were compared to exact
solution by finite difference method, 10-100s
points.
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Sample Results Spatial Sine Excitation
Excitation
Directivity, dB scale
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Another example
Excitation (x/L-5)2 sin(12?x/L) ?L/cs 3.56
Z/?scs i 1.5 ksL
Directivity, dB scale
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Finite structures
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Global solution as
a composition of Bloch wave
harmonics
Global solution is composed of Bloch wave
harmonics.

Constant f can be assigned values to get
desirable properties of the global solution,
which becomes less smooth, but may recover, for
example, zeroth moment of displacement in each
bay
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Importance of higher Moments
Global solution can be adjusted to recover these
values exactly. In further examples ONLY the
first one was preserved.
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Example of using Bloch wave harmonics
Beam with 8 discontinuities excited at A and
pinned at B.
B
A
Directivity, dB scale
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Fluid-Loaded Periodic Structure
Physical space
Wavenumber spectrum
Radiating interval
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Example Radiated Wave from a Plate with
Discontinuities.
Sweeping force
x
y
Pressure amplitude in the fluid
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Global and Local Parts in Fluid Loaded Case
Global system governing long wavelength
structural response and far-field acoustic
pressure was constructed for a coupled
fluid-structure problem.
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Fluid Loading Approximation in the Local
Intervals
Higher harmonics of short wavelength are
non-radiating. Then they just add to the
structural inertia, modifying structural
wavenumber
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Original and Global Problems
Dglobal smooth, easily discretized operator for
numerical soln. MF modified forcing,
calculated using original. NOT just
filtering! MP modified pressure, similar to MF,
but contains unknowns.
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Model Problem Sound Reflection from a Membrane
with Discontinuities
See the article Homogenization of fluid-loaded
periodic structures, J.Acoust.Soc.Am., 116(2),
879-890, Sept. 2004.
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Boundary Elements for Fluid Loading
Boundary elements relate pressure and
displacement, using monopole panels. Thus,
pressure at i-th control point is given by
This pressure forcing is then extended following
the method of images, and only lowest harmonics
preserved, properly modified.
After manipulation the modified pressure on the
element was found
Thus, new boundary element formulation was
derived, preserving only lowest wavenumbers in
the solution.
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Coupled Problem
Forcing F(x)ei?t consists of two harmonics
Displacement spectrum
Directivity, dB scale
Re
Im
Pink exact Green homogenized.
Global interval
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Coupled Problem
Localized forcing
Displacement spectrum
Directivity, dB scale
Re
Im
Pink exact Green homogenized.
Global interval
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Coupled Problem Arbitrary BC
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Boundary Elements
Pressure field from boundary element is modeled
as from distribution of equal baffled sources
Volumetric flow Q of the element is found from a
bay problem as a function of pressure and
displacements of the element.
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Building Blocks for any BC Problems
pi Qi
Bloch wave
Bloch wave
wL
wR
The problem of forcing, located in one bay on the
infinite structure can be solved in terms of
Bloch waves. Solution outside the bay yields
income to the displacement
Solution inside the element provides volumetric
flow
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System of Equations
pi
pi1
pi2
wi
wi1
wi2
Instead of original structural equation
homogeneous Bloch wave solution and particular
BEM solution are used. Elements are of length L
just one element per discontinuity. Constants C1,
C2 are to be found from arbitrary boundary
conditions.
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Sample Problem
Directivity function
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Example Constant Pressure Elements
Directivity, dB scale
uncoupled exact LGH method
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Linear vs. Constant Pressure Elements
Directivity, dB scale
exact constant pressure linear pressure
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Summary

• The Local/Global homogenization method is
  presented. Methods for constructing
  self-contained smooth global problems for
  structures with discontinuities are discussed.
• Substantial savings in the discretization level
  are demonstrated for finite-difference methods on
  dry structures and boundary elements methods on
  fluid-loaded structures.
• Accurate results are achieved in global problems
  recovering displacements at discontinuities and
  far-field radiation response.