J
ournal o
f
A
pp
Published Onli
n
http://dx.doi.or
g
Open Access
Mul
t
Proj
3
Depar
t
ABSTRA
C
This paper c
o
layer. The de
p
tained and all
Pareto-frontie
MATLAB.
Keywords: P
o
U
1. Introdu
c
Porous
p
iezo
c
cal imaging,
other
p
iezoel
e
tions are refe
tures for aco
zoelectric ma
t
tivity, extend
e
ing to the aco
u
the developi
n
structures of
t
On the base
o
ous materials
tion problem
structure hav
e
tors and com
b
types of mod
e
transmission.
rating sound
a
In order to
e
acoustic tran
s
including: lo
w
with water,
d
high mechani
c
widely used
d
vices based o
n
their effectiv
e
p
lied Mathemat
i
n
e November 2
0
g
/10.4236/jamp
.
t
iobjec
t
ecto
r
w
A. V.
N
1
Mathem
a
2
Departmen
t
t
ments of Micro
e
C
T
o
ncerns the o
p
p
endences of
t
owe
d
to decr
e
r
calculation
h
o
rous Piezoce
r
nderwate
r
Ap
p
c
tion
c
eramic mater
i
ultrasonic tr
a
e
ctric devices.
r
red to the de
u
stic transduc
e
t
erials due to
t
e
d frequency
u
stic medium.
n
g of methods
t
he transducer
s
o
f developed
n
effective mo
statement a
n
e
been imple
m
b
ined devices
e
rn devices u
s
A projector i
s
a
nd transmitti
n
e
ffectively fu
l
s
ducers have
w
impedance
d
urability; hi
g
c
al load and
h
d
evices in un
d
n
the
p
iezoele
e
applications
c
i
cs and Ph
y
sics
,
0
13 (http://ww
w
.2013.16017
t
ive Op
t
w
ith Po
r
N
asedkin
1
, M
a
tical Modeling
t
of Electric al E
n
e
lectronics &
M
p
timization
p
r
o
t
he effective
m
e
ase the numb
e
h
as been s o lve
d
r
amics; Multi
o
p
lications
i
als are widel
y
a
nsducers, h
y
A large num
b
velopmen
t
of
e
rs based on
t
hei
r
high
p
ie
z
bandwidth an
The aim of
pr
for synthesi
z
s
for underwa
t
n
umerical met
h
duli determin
n
d solving fo
r
m
ented. Hydr
o
transceivers
s
ed for sonar
s
a transduce
r
n
g it to the a
c
l
fill its
p
urp
o
to meet man
y
for better ac
o
g
h sensitivity;
h
ydrostatic
p
re
d
erwate
r
acou
s
ctric effect. T
h
c
aused by
b
ot
h
,
2013, 1, 89-94
w
.scirp.org/jour
n
t
imal
D
r
ous Pi
e
. S. Shevtso
v
Department, S
o
n
gineer ing, Nat
i
M
arine Engineer
i
Email:
m
Receiv
o
ble
m
for mu
l
m
oduli for
p
or
o
er
of design
v
d
using the li
v
o
bjective Opti
m
y
used in me
y
drophones a
n
b
e
r
of investig
a
effective stru
the porous pi
z
oelectric sen
d better matc
h
r
esented work
ing the opti
m
t
e
r
applicatio
n
h
od for the
po
e the optimiz
a
r
whole devi
c
o
phones, proje
are the gene
r
and underwat
r
used for gen
c
oustic mediu
m
o
se, underwat
e
y
requiremen
t
o
ustic matchi
n
ability to ta
k
ssur e. The m
o
s
tics are the d
h
e expansion
o
h
the success
o
4
n
al
/
jamp
)
D
esign
o
e
zocom
v
a
1
, J.-C. Li
u
o
uthern Federal
U
i
onal Taiwan O
c
i
ng, National K
a
m
ariamarcs@bk
.
e
d
October 201
3
l
tilayered ultr
a
o
us
p
iezoelec
t
v
ariables. The
v
e-link of finit
e
m
ization; Pare
d
i-
nd
a-
c-
e-
s
i-
h
-
is
m
al
n
s.
o
r-
a-
c
e
c-
r
al
e
r
e-
m
.
e
r
t
s,
n
g
k
e
o
s
t
e-
o
f
o
f
the
p
ie
z
the mi
c
ogies
t
active
h
for rec
e
the
p
r
o
ance
m
overco
m
eleme
n
of laye
r
of the
w
over t
h
ramics.
diate l
a
transd
u
theoret
i
lectric
p
roper
t
lectric
charac
t
lectric
thickn
e
for the
energy
.
lectric
u
other
pi
In
pr
p
roper
t
of
Und
e
posite
A
u
2
, S.-H. Ch
a
U
niversity, Ros
c
ean Universi ty
,
a
oshiung Marin
e
.
ru
3
a
sonic transdu
t
ric material o
n
multiobjectiv
e
e
-element (FE
)
to Frontier;
M
z
oelectric ma
t
c
ro-electrome
c
t
o create dev
i
h
ydrophones
a
e
iving and ra
d
o
blems at real
m
atching. The
u
m
ing this dif
f
nt
made of de
n
r
s are needed;
w
hole structu
r
h
is difficulty i
s
This entails
a
yers and
b
et
t
u
ce
r
and medi
u
i
cal [4,5] inv
e
ceramics ma
y
t
ies of transd
u
materials. P
o
t
erized by lo
w
modulus
31
d
e
ss
p
iezoelect
r
efficient con
v
.
Therefore, t
h
u
ltrasonic tra
n
i
ezoelectric d
e
r
evious work
s
t
ies of
p
iezoe
l
e
rwate
r
A
ctive
E
a
ng
3
, J.-K.
W
t
ov-on-Don, R
u
,
Zhongzheng,
T
e
University, K
a
ce
r
with acti
v
n
porosity ha
v
e
optimization
)
package Co
M
ultilayered A
c
erials science
c
hanical syste
m
i
ces based on
a
nd
p
rojectors
,
d
iation of dire
c
izing such st
r
u
se of interm
e
f
iculty. How
e
n
se
p
iezoelect
r
and this inv
o
r
e. The effect
i
s
the use of
po
both decreas
e
t
e
r
acoustic a
g
u
m. Various
e
e
stigations sho
w
y
significantl
y
u
cers and exp
a
o
rous
p
iezoco
m
w
e
r
values of
at almost th
r
ic modulus
v
ersion of ele
c
h
e use of
p
oro
u
n
sducers, hyd
r
e
vices is very
p
s
[6,7] it was
l
ectric materi
a
Acous
t
E
leme
n
W
u
3
u
ssia
T
aiwan
a
oshiung, Taiw
a
v
e porous
p
iez
v
e been
p
revi
o
problem
b
as
e
m
sol Multiphy
s
c
oustic Projec
t
and the devel
m
s (MEMS),
nanoscale
m
,
arrays of tra
n
c
te
d
sound [1]
r
uctures is th
e
e
diate layers
a
e
ve
r
in case
o
r
ic ceramics
a
o
lves big ener
g
i
ve method fo
r
o
rous
p
iezoel
e
ed
number of
g
reemen
t
et
e
xperimental [
w
, that
p
orou
s
y
improve th
e
a
n
d
the use o
f
m
posite mate
r
the transvers
e
e same value
33
d
[6,7] re
s
c
trical and m
e
u
s materials i
n
r
ophone struct
u
p
romising tod
a
investigated
a
ls depend on
JAMP
t
ic
n
ts
a
n
oelectric
o
usl
y
ob-
e
d on the
s
ics with
t
o
r
;
oping of
technol-
m
ultilayer
n
sducers
. One of
e
imped-
a
llows to
o
f active
a
number
g
y losses
r
getting
e
ctric ce-
interme-
w
een the
2,3] and
s
piezoe-
e
desired
f
piezoe-
r
ials are
e
piezoe-
e
s of the
s
ponsible
e
chanical
n
piezoe-
u
res and
a
y.
how the
porosity
Open Access
90
for the cera
m
tric hardness.
the full set of
The aim o
f
tion of multi
l
lectric active
expressed thr
o
obtained dep
e
ciently decre
a
underwater a
c
such layers a
s
tric, backing
p
1) we formu
l
electroelastici
optimized ob
j
sure level (S
and the mea
n
frequency ra
n
riables are:
Y
layer,
p
rotect
i
damping
ar
layers;
p
oros
i
investigation
to-frontier ca
l
the 6D space
are feasible.
The coupl
e
the live-link
with MATL
A
about 30% -
4
Figure 1. The
acoustic medi
u
layer, 3—acti
v
5—protective
f
m
ics of differe
n
Such depend
e
material cons
t
f
presented res
e
l
ayere
d
transd
u
element whic
h
o
ugh the valu
e
e
ndencies of e
f
a
se the numbe
r
c
oustic
p
roje
c
s
: an acoustic
w
p
late and
p
rot
e
l
ate the coupl
ty in axial-s
y
j
ectives we in
t
PL), transmit
t
n
-square valu
e
n
ge from 100
Y
oung’s mod
u
i
ve foam lay
e
a
mete
r
and sti
i
ty of an acti
v
we use the
a
l
culation, i.e.
of design vari
e
d problem is
of the FE
pa
A
B. The best
r
4
0% of
p
oros
i
scheme of mu
l
u
m: 1—acoust
v
e porous piezo
e
f
oam layer.
nt
connectivit
y
e
ncies have
be
t
ants.
e
arch is a s tru
c
u
ce
r
with the
h
all material’
e
of porosity.
f
fective modu
l
r
of design v
a
c
to
r
which st
r
w
indow, matc
h
e
ctive foam la
y
e
d
problem o
y
mmetric for
m
t
roduce avera
g
t
ing current
r
e
of the SP
L
to 400 kHz.
u
les of an a
c
e
r, and match
i
ffness dampi
n
v
e
p
iezoelectr
i
a
pproach
b
as
e
building the
s
ables where a
l
numerically i
a
ckage Coms
o
r
esults have
b
i
ty for
p
iezoel
l
tilayered tran
s
ic window lay
e
lectric layer,
4
A. V. NA
S
y
and ferroele
c
e
en obtained
fo
c
tural optimiz
a
porous piezo
s parameter a
r
The
p
revious
l
l
i allow to suf
f
a
riables. For t
h
r
ucture contai
n
h
ing, piezoele
c
y
ers (see Fi
g
u
r
f acoustics a
n
m
ulation. As t
h
g
e
d
sound
p
r
e
r
esponse (TC
R
L
irregularity
i
The design v
a
c
oustic windo
i
ng layer; ma
n
g parameter
o
i
c layer. In t
h
e
d on the Par
s
e
t
of points
i
l
l the objectiv
e
mplemented
b
o
l Multiphysi
c
een obtained
ectric layer a
n
s
ducer placed
er, 2—matchi
n
4
—backing pla
t
S
EDKIN ET
c-
for
a-
e-
r
e
ly
f
i-
h
e
n
s
c-
r
e
n
d
h
e
e
s-
R
)
in
a-
w
ss
o
f
h
is
e-
in
e
s
by
c
s
at
nd
in
n
g
t
e,
mecha
n
ers tha
t
terials.
p
aram
e
than fo
metho
d
timizat
i
2. De
p
Po
r
For th
e
approa
c
effecti
v
ative v
o
use of
t
obtain
compo
s
but an
model
o
cells—
pores [
2
a skele
come t
h
tion th
e
ters ca
n
from
pi
ten-Sa
n
p
resen
t
ten-Sa
n
The
with t
h
dium
f
p
resen
t
thod,
m
by tria
n
into ac
c
corres
p
calcula
t
unifor
m
experi
m
Fi
gu
33
()
S
r
with th
e
the de
p
Sander
of
ol
obtain
e
also ca
n
and n
u
data fr
o
tained
i
and gi
v
and th
e
AL.
n
ical and dam
p
t
may charact
e
Electro-acou
s
e
ter, is signif
i
r
the dense c
e
d
s might be e
f
i
on of enough
p
endencies
r
osit
y
e
effective
p
r
c
h presented
i
v
e moduli met
h
o
lumes for
po
t
he FE techno
l
the structure
s
ite material.
A
adequate mi
c
o
f the
p
iezoce
r
cube s, some
o
2
]. However,
ton at a large
h
is problem t
h
ory can be u
s
n
be built fr
o
i
ezoelectric m
a
n
der’s metho
d
t
e
d
work both
n
der’s method
experimental
h
e obtained re
s
f
erroelectric h
a
t
e
d
below, th
e
m
arked by cir
c
n
gles. The de
p
c
oun
t
the inh
ond to dotte
d
t
ions
p
erfor
m
m
p
olarization
m
ental data.
u
re 2 shows
33 33
()/ (
0
SS
p

e
experimenta
l
p
endencies are
methods. Th
e
a
rization field
ed
without thi
s
n
be seen that
u
merical resul
t
om
[10]. The
i
n [11] strong
l
v
e about 35%
d
e
results f rom
[
p
ing
p
aramete
e
rize a wide r
a
s
tic efficiency
i
cantly highe
r
e
ramics. Obtai
n
f
fectivel
y
use
d
wide ra nge o
f
of the Effe
c
operties dete
r
i
n details in [
6
h
ods [2,6-8],
m
o
rous
p
iezoele
c
l
ogies. There
a
of a two-
p
h
a
At
low
p
erce
n
c
rostructure o
f
r
amic cubic l
a
o
f which are r
a
this model m
a
number of
po
h
e algorithms
s
ed. In case o
f
om
pores, wh
i
a
terial. One o
f
d
[9] has
b
ee
n
methods: ran
d
were used.
data [2,4,1
0
s
ults for a
p
o
r
a
rdness PZT-
4
e
curves, relat
e
c
les, and the
W
p
endences, o
b
o
mogeneity o
f
d
lines; dash-
d
m
ed at taking
i
; and the das
h
the relative
0
)
on
p
orosi
t
l
data [10,11].
linear for
b
o
t
results obtai
n
s inhomogen
e
s
hypothesis
l
all four depen
t
s are in
b
ett
e
values of el
e
l
y decrease w
i
d
iscrepancy
w
[
9].
rs of interme
d
a
nge of
p
oly
m
, defined by
t
r
for porous
n
e
d
data and
p
d
for the struc
t
f
transducers.
c
tive Modu
l
r
mination we
6
,7] and
b
ase
d
m
odeling of r
e
c
tric material
s
a
re several m
e
a
se cubic
p
iez
nt
of porosity
a
f
porous mat
e
a
ttice consistin
a
ndomly decla
r
a
y lose conne
c
o
res. In order
based on the
f
low porosity
t
i
le at high
po
f
such method
n
analyzed i
n
d
o
m
method
a
0
-12] were c
o
r
ous material
w
4
. For all the
e
d to the ran
W
itten-Sander
b
taine
d
witho
u
f
the
p
olarizati
d
o
t
lines corre
s
i
nto account
t
h
e
d
lines ind
i
electric
p
er
m
t
y
p
in co
m
One can conc
t
h random an
d
n
e
d
with the h
y
e
ity differ fr
o
l
ess than 1%
dencies are v
e
er
agreement
w
e
ctric
p
ermitt
i
i
th increasing
w
ith the calcul
a
JAMP
iate lay-
eric ma-
t
he TCR
material
p
ropose
d
t
ural op-
l
i on
used an
d
on the
e
present-
s
and the
e
thods to
oelectric
a simple
e
rial is a
g of unit
r
ed to be
c
tivity of
to over-
percola-
t
he clus-
o
rosity—
s is Wit-
n
[8]. In
a
n
d
Wit-
o
mpare
d
w
ith me-
figures,
d
o
m
me-
method,
ut
taking
on field,
s
pond to
t
he non-
i
cate the
m
ittivity
m
parison
lude that
d
Witten-
y
pothesis
om
those
- 2%. It
e
ry close,
w
ith the
i
vity ob-
porosity
a
ted data
Open Access
Figure 2. The
d
porosity.
After the c
a
()
E
cp

,
(
i
e
characteristic
s
coefficients
d
where
effE
s
a
trix). The co
m
the experime
n
longitudinal (
piezoelectric
m
As can be
related to th
e
creases with i
n
method. It is
o
assuming of
better agree
m
3(a)). At the
s
lectric modul
u
rosity for
b
ot
h
that the max
i
approximatel
y
about 3% - 5
%
At the se
c
during the si
m
ture, the obt
a
polarization
f
Witten-Sande
r
properties for
3. Coupled
Electroe
l
Piezoele
c
In presented
ultrasound
p
i
e
cations in its
surface and t
h
protective la
y
d
ependency of
a
lculation the
f
)
p
,
()
S
ii
p
,
s
of
p
iezoelec
t
31
d
and
33
d
a
re the comp
o
m
parison of th
e
n
tal data [2,
4
Fi
g
ure 3(a))
a
m
oduli.
seen the dep
e
Witten-San
d
n
creasing
p
or
o
o
bvious that t
h
polarization
f
m
en
t
with th
e
s
ame time, th
e
u
s
33
d
are e
s
h
methods (Fi
g
i
mu
m
error is
y
10% - 15%
%
for the data
[
c
on
d
stage of
m
ulation of a
m
a
ined depende
n
f
ield’s inhom
o
r
method ha
v
the active ele
m
Problem o
l
asticit
y
fo
r
c
tric Trans
d
investigation,
e
zoelectric tra
n
axial-symm
e
h
e bottom o f a
y
e
r
made of
f
relative electri
c
f
ull set of the
e
we obtained
t
ric ceramics
(
()
ii
ddp
o
nents of the
c
e
obtained de
p
4
,13] are
p
erf
o
a
nd transvers
a
endency for
d
e
r
method, s
i
o
sity compare
d
h
e relationshi
p
f
ield’s inhom
o
e
experiment
a
e
dependencie
s
s
sentially ind
g
ure 3(b)). It
still quite si
g
for the data
[
2,4].
the
p
resente
d
m
ultilaye
r
tra
n
n
cies at the
a
o
geneity and
v
e been acce
p
m
ent.
f Acoustics
r
a Multila
y
d
ucer
we consider
n
sduce
r
for u
n
e
tric formulat
i
transdu cer ar
e
f
oam. Transd
u
A. V. NA
S
c
permittivity
o
e
ffective mod
u
such importa
n
as
p
iezoelect
r
() (
)
E
i
eps p

c
ompliance m
a
p
endencies wi
t
o
rmed for
b
o
t
a
l (Figure 3(
b
31
d
coefficie
n
i
gnificantly d
d
to the rando
m
p
obtained at t
h
o
geneity is in
a
l data (Fi
g
u
r
s
for the pi ezo
e
penden
t
of p
o
should be no t
e
g
nifican
t
and
from [13], a
n
d
investigatio
n
n
sducer’s stru
a
ssuming of t
h
related to t
h
p
ted as materi
and
y
ered
a model of
a
n
derwate
r
app
i
on. Cylindric
e
covered wit
h
u
ce
r
consists
o
S
EDKIN ET
o
n
u
li
nt
r
ic
)
,
a-
t
h
t
h
b
))
n
t,
e-
m
h
e
a
r
e
e-
o
-
ed
is
nd
n,
c-
h
e
h
e
i
al
a
n
l
i-
al
h
a
o
f
Figure
cients o
four la
y
layer,
m
constr
u
the
bo
match
e
dent u
p
reflect
p
iezoe
l
have
be
the lo
n
acousti
matchi
n
same
f
couple
d
the FE
p
licati
o
monic
quenc
y
the in
h
p
ressu
r
where
is a s
p
freque
n
b
ound
a
b
ound
a
and ax
i
1).
AL.
3. The depen
d
n porosity.
y
ers, such as:
m
atching laye
u
ction is surro
u
o
undar
y
of a
n
e
d layer (PM
L
p
on the PML
at the interfac
l
ectric layer,
p
e
en chosen t
o
n
gitudinal wa
v
c window t
h
n
g layer—1/
4
f
requency of
a
d
problem of
a
package CO
M
o
n modes: P
r
analysis) and
y
response ana
l
h
omogeneous
H
r
e
p
:
0
= 1000 kg/
m
ee
of soun
d
n
cy with
f
a
ry conditions
a
ry between
a
i
al symmetry
d
encies of rela
t
backing
p
lat
r
and acousti
c
u
nde
d
with a
n
n
acoustic
m
L
) is placed s
o
from a non-
P
e (Figure 1).
p
rotective an
d
o
be approxi
m
v
e at thickne
s
h
ickness—ap
p
4
of the long
i
a
bou
t
300 k
H
a
coustics and
M
SOL Multip
h
r
essure Acou
s
Piezo Axial
l
ysis). Sound
w
H
elmholtz eq
u
2
00
1p
c

m
3
is a fluid d
e
d
,
2
f

Hz
denotin
g
are following
a
n acoustic
m
(on the left
b
t
ive piezoelect
r
e,
p
iezoelectr
i
c
window. T
h
n
acoustic me
d
m
ediu
m
the
p
o
that the wa
v
P
ML mediu
m
The thickness
e
d
backing
p
la
t
m
ately a half l
e
s
s vibration
m
p
roximately
3
i
tudinal wave
H
z. We form
u
electroelastic
i
h
ysics and its
s
tics mode (t
Symmetry m
o
w
aves are gov
u
ation for the
2
20
s
p
c
,
e
nsity,
1
5
s
c
rad/s
is the
g
the freque
n
: sound hard
w
m
ediu
m
and t
h
oundary), (se
e
JAMP
91
r
ic coeffi-
i
c active
h
e whole
d
ium. On
p
erfectly
v
es, inci-
m
, do not
e
s of the
t
e layers
ength of
m
ode; an
3
/4, and
e
s at the
u
late the
i
ty using
two ap-
ime-har-
o
de (fre-
e
rned by
acoustic
(1)
5
00 m/s
angular
n
cy. The
w
all (the
h
e PML)
e
Figure
A. V. NASEDKIN ET AL.
Open Access JAMP
92
The constitutive relations for the piezoelectric active
layer are taken in the stress-charge form as follows:
*
E
S


σcεeE
Deεε E, (2)
where ε is the strain tensor, σ is the stress tensor, E
is the electric field vector, D is the electric displace-
ment vector;
E
c is the tensor of elastic stiffness moduli
at constant electric field; e is the tensor of piezoelectric
moduli (stress coefficients); S
ε is the tensor of electric
permittivity moduli at con s tant mechanical stress.
Coupling between solid end acoustic media is pro-
vided by the boundary conditions: the top and the sides
of a transducer undergo both an acoustic pressure and the
inward accelerations. The bottom of a transducer is fixed;
on the left boundary we consider an axial symmetry con-
dition; on the top of piezoelectric layer the constant elec-
tric potential with amplitude 100 V in whole studied fre-
quency band is applied, when the bottom is grounded.
Since the dimensions of the investigated transducer are
quite small, this type of projector cannot be used to gen-
erate directional sound and therefore we will consid er th e
sound pressure level only in a direct ray. When the
transducer is placed into acoustic medium the thickness
vibration mode is excited at frequencies from approx-
imately 100 to 400 kHz. This frequency range was used
during the following optimization of transducer’s para-
meters.
4. Multiobjective Optimization of the
Piezoelectric Transducer
Ther e is a wide ran ge of mat erial s that c an be used as the
constituent layers of a transducer; this proves the possi-
bility to vary their mechanical properties within the wide
scope. It should be noted that we chose tungsten as a
backing plate material to generate the thickness v ibration
mode of a PZT layer because of its large mechanical
stiffness and high acoustic impedance. In order to for-
mulate the optimization problem let us introduce six de-
sign variables: porosity o f an active layer (por), Young’s
modules of an acoustic window layer (aw
E), matching
layer (m
E), and protective foam layer (
f
E
); mass
damping parameter (1
R) and stiffness damping parame-
ter (2
R) of layers. In our investigation we considered
three objectives: sound pressure level (SPL) in direct ray
measured at the 1m distance from the sound source and
transmitting current response (TCR) to be maximized;
the deviation of SPL is to be minimized. SPL is repre-
sented in decibels as follows

1
20lg ref
ppp, (3)
where 1
p is the sound pressure at the measurement
point; and 5
210
ref
p
 Pa is the threshold of sound
pressure. TCR is the ratio of an absolute value of sound
pressure 1
p, to the amplitude of electric current I
through the active element:
1I
SpI. (4)
There is a wide range of approaches to structural opti-
mization. In the framework of multi-criteria optimization
problem (MOO) when several objective functions exist,
there is no unique solution, but a number of optimum
solutions exist. In this situation the most suitable way to
optimization is a calculation of a so-called Pareto opti-
mum or Pareto-frontier. Using the Pareto approach we
suppose the assignment of a set of choices for all objec-
tive components that are Pareto efficient. By confining
the set of choices to only the Pareto-efficients instead of
considering the full range for each parameter, it is possi-
ble to make trade-offs within this set. During the solving
of considered optimization problem the three integrals
were assumed to be optimized:
 
2
1
21
f
f
p
pfdff f
, (5)

2
1
2
21
f
f
ppfdf
pff

, (6)
 
2
1
21
f
f
pf
TCRdff f
If

, (7)
where
p
,
p
and TCR represent an averaged
SPL, deviation of SPL and TCR, respectively; 1
f
and
2
f
are the boundaries of the frequency range.
Obviously the construction of a Pareto-frontier was
complicated for the three-dimensional space of objective
functions. In order to overcome this difficulty the illu-
stration of a Pareto-frontier has been represented using
the level lines. At the numerical problem solving MAT-
LAB varies design parameters for the transducer, calls
the FE model simulated by Comsol Multiphysics, and is
carry out the multiple computations of the objectives.
Then obtained data are being analyzed and illustrated
using the set of complimentary procedures, written in
MATLAB (see Figures 4(a), (b)).
5. Numerical Results and Discussion
At the analysis of the simulation results it was founded
that the influence of both damping designs variables on
the objectives are negligible. So only two obtained pro-
jections of the criteria points set on the spaces of other
design variables are presented in Figure 4. Figure 4(a)
corresponds to the projection on subspace of two design
variables: porosity of active layer and Young’s modulus
Open Access
Figure 4. Pro
j
contour lines
o
variables.
of matching l
side
p
rotecti
v
window. For
ing bounds fo
1500 dB/A
of SPL. On t
h
and dashed li
n
the 2D subs
p
represent the
seen from Fi
g
racterized by
porosity of a
c
maximum va
l
variation of
p
SPL value sh
i
matching lay
e
porosity; on
t
corresponds
t
quency respo
n
j
ections of th
e
o
f objectives le
v
ayer; Figure
4
v
e layer and
Y
the studied o
b
r
the feasible
v
– for TCR, a
n
h
e presented f
i
n
es are the
p
r
o
p
aces of the
d
intersections
g
ure 4(a) that
TCR, reach
e
tive layer is
g
l
ues of SPL
a
orosity. One
c
i
fts with the g
r
er
from about
t
he other han
d
t
o lower
p
or
o
n
se of SPL ar
e
e
criteria set
p
v
els on the su
b
4
(b) for You
n
Y
oung’s mod
u
b
jectives we
u
v
alues:
150
nd
2.5 dB
f
i
gures the ar e
o
jections of P
a
d
esign variabl
e
of optimum
a
the energetic
e
s desirable v
a
g
reate
r
than 0.
3
a
re reached a
t
c
an observe t
h
r
owth of Y ou
n
1.6 GPa till 2.
d
the lower Y
o
o
sity. The mo
e
reached whe
n
A. V. NA
S
p
resented as t
h
b
spaces of desi
g
n
g’s modulus
o
u
lus of acous
t
u
sed the follo
w
dB for the S
P
for
the deviati
o
a
s between sol
i
a
reto frontier
o
e
s. Green are
a
a
reas. It can
b
efficiency, ch
a
lues when t
h
3
. However, t
h
t
a considerab
h
a
t
the optimu
m
n
g’s modulus
o
2 GPa at high
o
ung’s modul
u
s
t
uniform fr
n
the percent
o
S
EDKIN ET
h
e
g
n
o
f
t
ic
w
-
P
L,
o
n
i
d
o
n
a
s
b
e
a-
h
e
h
e
le
m
o
f
e
r
u
s
e-
o
f
p
orosit
y
that th
e
cantly
quenc
y
the val
u
time t
h
windo
w
areas.
For
c
cy res
p
design
For
from t
h
tains
p
last se
t
made
o
(Youn
g
side th
e
the Par
e
For
ponses
the Fi
g
The
g
optima
l
uneve
n
cases
o
p
orosit
y
level i
n
(1), (2
)
The s
o
range i
base o
f
on 10
%
ure 5(
b
utmost
are tak
e
device
taining
the w
h
TCR
d
respon
s
6. Co
n
Ten de
p
Table
1
lated p
r
St
u
de
s
Insi
d
optim
u
Outs
i
optim
u
Den
AL.
y
is greater t
h
e
Young’s m
o
influences o
n
y
response; th
e
u
es greater t
h
h
e Young’s
m
w
layers doe
s
c
larity, we
p
r
e
p
onses obtaine
variables that
the first set
a
h
e obtained o
p
arameters
b
ei
n
t
corresponds
t
o
f dense
p
iezo
e
g
’s modulus
o
e
optimum ar
e
e
to frontier.
each set of
d
for SPL and
ures 5(a) and
g
raphs shown
l
set of desig
n
n
ness of soun
d
o
f active elem
e
y
. The maxi
m
n
the investig
a
)
, and (3) are
o
und pressure
s the best. It
e
f
dense ceram
i
%
. The peak o
b
b
)) shows th
a
resonance
pr
e
n from the o
p
has a sufficie
n
a constant a
m
h
ole frequenc
y
d
oes not wors
e
s
e for the sou
n
n
clusion
p
endencies o
f
1
. Design vari
a
r
ojectors.
u
die
d
s
ign
aw
E
,
GPa
d
e the
u
m area2.5
i
de the
u
m area0.5
s
e PZT 0.5
h
an 0.25. It’s
o
o
dulus of ac
o
n
the uniform
e
optimum q
u
h
an or equal t
o
m
oduli of a
pr
s
not impact
e
sen
t
below th
r
d
for the thre
e
is contained i
n
a
ll the design
p
timu
m
areas
.
n
g outside th
e
t
o a transduc
e
e
lectric ceram
i
o
f acoustic wi
n
e
a, the other f
o
d
esign variabl
TCR are cal
c
5(b), respecti
v
in Figure 5,
a
n
parameters
d
pressure lev
e
e
n
t
with dense
m
u
m
deviation
a
te
d
frequenc
y
8.5 dB/21 d
B
level inside
e
xceeds the S
P
i
cs on 5%, an
d
b
serve
d
in th
e
at
a transduce
r
r
operties whe
n
p
timal area. If
nt
p
erforman
c
m
plitude of t
h
y
range, the
r
e
n the unifor
m
nd
pressure.
f
material cons
a
bles for the t
h
f
E
,
MPa
m
E
,
GPa
30 16
5 20
30 16
o
bvious (Fi
gu
o
ustic windo
w
ity of the S
P
u
antities corre
s
o
2 GPa. At
t
r
otective and
on TCR in
o
r
ee groups of
e
sets of valu
e
n
Table 1.
variables we
.
The second
e
Pareto fron
t
er
with an act
i
i
cs; the first
pa
n
dow) was ta
k
o
u
r
variables
b
es the freque
n
c
ulated and
pl
v
ely.
a
clearly show
s
provides a
m
e
l as compari
n
ceramics (3)
a
of the sound
y
range for th
e
B
/12 dB, resp
the whole f
r
P
L of
p
roject
o
d
SPL for desi
g
e
TCR graph
(
r
(1) demonst
r
n
the design
v
the electronic
c
e that allows
h
e applied
p
o
t
r
esonance fe
a
m
ity of the f
r
tants on
p
oro
s
h
ree examples
17
10
R
2
7
10
R
1.5 0.6
2.3 1.2
1.5 0.6
JAMP
93
u
re 4(b))
w
signifi-
P
L’s fre-
s
pond to
t
he same
acoustic
o
ptimu
m
frequen-
e
s of the
re taken
set con-
t
ier. The
i
ve layer
a
ramete
r
k
en out-
b
elong to
n
cy res-
l
otted on
s
that the
m
uch less
n
g to the
a
nd 20%
pressure
e
designs
ectively.
r
equenc
y
or
on the
gn (2)—
(
see Fig-
r
ates the
v
ariables
exciting
to main-
t
ential in
a
tures of
r
equenc
y
s
ity were
of simu-
7
por
0.4
0.2
0
A. V. NASEDKIN ET AL.
Open Access JAMP
94
Figure 5. The frequency responses of SPL (a) and TCR (b).
successfully obtained for the porous piezocomposite ma-
terials of different connectivity in order to optimize the
hydroacoustic performance of multilayered projector
based on the active PZT layer with varied porosity.
These effective modules were calculated using the FE
method at the assumption of homogeneous and inhomo-
geneous polarization field. The last dependencies were
used at the statement and solving the optimization prob-
le m d ue to t he be st agreement with the experi mental d at a.
Obtained dependencies allowed to reduce the number of
design variables to six (porosity of an active layer;
Young’s modules of an acoustic window layer, protec-
tive and matching layers; mass and stiffness damping
parameters of layers). On the base of the Pareto optimal-
ity the set of feasible designs in a six dimensional design
space was reconstructed using three objectives: averaged
sound pressure level, transmitting current response and
the standard deviation of the SPL in a frequency range
from 100 to 400 kHz. A comparative analysis of three
examples of the simulated designs has been performed. It
showed the best performance of a projector with porosity
near 40% and elastic modules of intermediate layers
tuned to achieve the best acoustic impedances matching
between the structure and acoustic medium.
7. Acknowledgements
This work is partially supported by the Russian Founda-
tion for the Basic Researches (Grant 12-08-31350) and
by National Science Council of Taiwan (Project NSC99-
2923-E-022-001-MY3).
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