Application of the finite element method to two-dimensional radiation problems

Application of the finite element method to two-dimensional radiation problems

Jamal Assaad

IEMN, UMR CNRS 9929, ISEN Department, Institut Sup•rieur d'Electronique du Nord, 41 Boulevard Vauban, 59046 Lille Cedex, France, and IEMN, UMR CNRS 9929, OAE Department, Universitd de Valenciennes et Hainaut-Cambrdsis, le Mont Houy BP 311, 59304- Valenciennes Cedex, France

Jean-No•l Decarpigny IEMN, UMR CNRS 9929, ISEN Department, Institut $up•rieur d'Electronique du Nord, 41 Boulevard Vauban, 59046 Lille Cedex, France

Christian Brunoel

IEMN, UMR CNRS 9929, O•4E Department, Universit• de Valenciennes et Hainaut-Cambr•sis, le Mont Houy BP 311, 59304 - Valenciennes Cedex, France

Rbgis Bossut and Bernard Hamonic IEMN, UMR CNRS 9929, ISEN Department, Institut $up•rieur d'Electronique du Nord, 41 Boulevard Vauban, 59046 Lille Cedex, France

(Received 7 February 1993; accepted for publication 26 March 1993) Acoustic fields radiated by vibrating elastic bodies immersed in an infinite fluid domain are, in general, quite difficult to compute. This paper demonstrates in the two-dimensional (2-D) case that the radiated near field can be easily obtained using the finite element method if dipolar damping elements are attached to the mesh external circular boundary. These elements are specifically designed to absorb completely the first two components of the asymptotic expansion of the radiated field. Then, the paper provides a new extrapolation method to compute far-field pressures from near-field pressures, using the 2-D Helmholtz equation and its solution obeying the Sommerfeld radiation condition. These developments are valid for any radiation problem in 2D. Finally, two test examples are described, the oscillating cylinder of order m and a finite width planar source mounted in a rigid or a soft baffle. This approach is the generalization to 2-D problems of a previously described approach devoted to axisymmetrical and three-dimensional (3-D) problems [R. Bossut et aL, J. Acoust. Soc. Am. 86, 1234-1244 (1989)]. It has been implemented in the ATILA code. It is well suited to the modeling of high-frequency transducers for imaging and nondestructive testing.

PACS numbers: 43.20.Tb, 43.20.Rz, 43.30.Jx, 43.40.Rj

INTRODUCTION

The directivity pattern is a very important character- istic for high-frequency transducers used in acoustic imaging and nondestructive testing (NDT). Generally, such patterns are theoretically determined with the help of quite simplifying assumptions for the displacement field of the transducer radiating surface, via the analytical or numerical computation of classical integrals. • However, if an accurate, self-consistent, displacement field has to be used, the problem becomes very complex and its solution re- quires the simultaneous solving of coupled mechanical, electrical, and acoustical equations.

The finite element method is widely used for the modeling of piezoelectric transducers. With respect to the structure, the mechanical displacement field and the elec- tric field are simultaneously described using their nodal values, and their coupling is introduced by the material constitutive equations. With respect to the radiation load- ing, various approaches are currently used. In the first set, a mechanical impedance matrix is computed which relates, for all the nodes belonging to the transducer radiating face,

the acoustic forces and the normal vibrating velocities. This matrix can be obtained either by a simplified, analytical model or, more accurately, with the help of the inte- gral equation method. In the second set, the fluid is also meshed and a full finite element formulation is preferred. Then, the fluid finite element mesh can be either termi- nated by infinite elements or limited by an external nonre- flecting surface. In the last case, the external surface can be made up with damping elements that absorb quite completely the outgoing acoustic wave. Their use was suggested by several authors, but the first, powerful approach was proposed by A. Bayliss et al., 2 who also provided a general method to build up more and more accurate dampers (monopolar, dipolar,...). This method was implemented in the ATILA code 3-7 for axisymmetrical and 3-D problems by R. Bossut et al., 3 and has been used to model a lot of radiating transducers, 3-7 including interacting ar- rays. Advantages of this last method is the use of a unique, full finite element approach, the attractive numerical char- acteristics associated to the structure of the final system of equations, and the direct computation of the near-field pressure inside the selected external boundary. The main

562 J. Acoust. Sec. Am. 94 (1), July 1993 0001-4966/93/94(1)/562/12/$6.00 @ 1993 Acoustical Society of America 562

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FIG. 1. Schematic description of the structure and fluid domains.

drawback is generally the large size of the numerical problem, though with dipolar dampers the fluid mesh can be quite limited. Moreover, Bossut et al. 3 also proposed a simple extrapolation method to deduce the transducer far field from the computed near field.

The aim of this paper is to generalize the approach proposed by Bossut et al. 3 to 2-D problems. Such problems are frequently encountered for high-frequency piezoelectric transducers because one of the their dimensions is generally quite larger than the two other ones, which allows the use of the plane strain approximation. In this 2-D generalization, the main difficulty is related to the fact that the radiated wave propagation is described by a zeroth-order Hankel function H0(kr), instead of the classical exp(jkr)/ kr term, 8 where r is the field point abscissa and k the wave number.

Following Bayliss et al. 2 and parallel with the work by Pinski et al., 9 the first section of this paper describes the 2-D damping finite elements. Particularly, it gives the whole formulation for the monopolar and dipolar dampers. Respectively, monopolar and dipolar dampers absorb completely the first and the first two contributions of the asymptotic expansion of the radiated field. Then, in the second section, a specific extrapolation method is suggested and its peculiar mathematical properties are analyzed. Fi- nally, in the third section, two test examples are discussed for which analytical results are available: the radiating circular cylinder of order m (Ref. 8) and a planar, finite width source imbedded in a rigid or soft baffle. 1ø A very careful analysis of the accuracy is given, with respect to the damping element as well as to the extrapolation method. Checks concern the near-field and the far-field pressures, that is to say, in this last case, the directivity patterns.

I. FINITE ELEMENT FORMULATION OF THE RADIATION PROBLEM USING DIPOLAR DAMPING

The finite element method is not detailed here. Its description can be found, for example, in Zienkiewicz's book, 11 and only the general equations are set up in this section for a 2-D radiation problem. The harmonic state is assumed and the e jwt term is implicit, where w is the an- gular frequency (w = 2•rf = 2•rc/A, f being the frequency, ,• the wavelength, and c the sound speed equal to 1500 m/s). Notations are defined in Fig. 1. The solid structure •s is in contact with the fluid domain •f through the surface I•i . Here, •f is split into two parts: •fi and •fe,

by a circular boundary I•e of iadius R; ils, fifi, I"e, and I•i are divided into finite elements connected by nodes. The physical quantities of interest are the displacement field u and the pressure field p. Their nodal values are the unknowns and are, respectively, arranged in nodal vectors U and P. The resulting system of equations is 3-7

[K] --o2 [M ] [ --L] •] [--L] T [H]_(.o2[M1] [pU : , (1) p2C2t02

where [K] and [3//] are, respectively, the structure stiffness and consistent mass matrices; [HI and [_M1] are the fluid compressibility and consistent mass matrices; [L] is a con- nectivity matrix that represents the coupling between the structure and the fluid; p is the fluid density; and E contains the nodal values of the applied forces. In the case of a piezoelectric structure, it results from simple operations on the piezoelectric and dielectric stiffness matrices 3-7 and is then proportional to the applied voltage. Here, ß is a vector that contains the nodal values of the normal deriv-

ative of the pressure on the external fluid domain boundary I•e:

r=R

Here, [K], [3//], [HI, [3//1], ILl, F, and ß result, respectively, from the merging of elementary matrices [Ke], [Me], [Hq, [M•], [L e], and vectors F e and •I/e. In particular, •I/e can be written under the form:

lite: ;F [N;I½ dI"ee' (3) e e

where [N•,] is the shape function matrix associated with the finite element Fe e of the external surface I' e.

A solution of the 2-D Helmholtz equation, which obeys the Sommerfeld radiation condition and is regular out of a circle of radius a (r>•a > 0) surrounding the whole radiating structure, is 12

p(kr, O) =H0(2) (kr) • Fn(O Gn(O) n=0 5; n=0

(4)

where k is equal to 2rr/,•, r and 0 are polar coordinates, and F n and G, are multipolar components which only de- pend on 0. This series converges uniformly and absolutely for r > a. Thus, it may be differentiated termwise with respect to r and 0. Here, H0 (2) and H• 2) are Hankel functions of the second kind. The .superscript 2 will be omitted later. Second kind functions are necessary to describe an outgoing pressure field, the case of incoming waves being ex- cluded in the following. The asymptotic expression for the H m Hankel function is

•m(kr) __jm x/2/rrkr exp[--j(kr--rr/4) ], (5) which can be used for kr > 2. Substituting this relation into Eq. (4) provides an asymptotic expression analogous to that used in the 3-D radiation case: 2-3'12

563 d. Acoust. Soc. Am., Vol. 94, No. 1, July 1993 Assaad et aL: Two-dimensional radiation problems 563


.• oe f n( o)

pa(kr, O)=Ho(kr) • •-•, n=O

with

(6)

f n(O) =Fn(0) +jGn(O). (7)

Bayliss et aL 2 have introduced boundary operators that allow one to damp successive contributions in the multipolar expansion of a radiating pressure field. These operators rely upon the asymptotic expansions of the radiated field in 2D or 3D, and using higher-order operators provide better and better accuracy. Such operators have been previously used for 3-D finite element modeling 3-7 and can be also applied to 2-D problems. z9 In this 2-D case, the mth-order operator is defined by 2

Bm ( 3) 1 O 2l--• = II a(kr) +j + ' l= rn kr

o -- O( kr) + j + 3) 2m--i

kr BIn- 1 (8)

and has the fundamental property,

Binpro=O, (9)

where Pm is given by the expansion of Eq. (6) truncated to n (m. Two cases have been considered and are quite use- ful: the monopolar damper (m=l) and the dipolar damper (m = 2).

Assuming that Fe is at a distance R such that kR• 1, the pressure on it is the far-field pressure given by the first term of Eq. (6)'

p(kR,O) =Poe (kR,O) =Ho(kr) f o( O), (10)

where fo(0) is defined as the far-field directivity pattern. Then,

Buv( kr, O) l •=O, (11)

which is the first and simplest boundary condition on F e. It is interesting to note that

lim B•pa(kr, O)Ir=R=O((kR)-S/2), kR-. oo. (12)

This relation gives an error decreasing as (kR)-s/2 rather than (kR) - 3/2, as obtained by using the standard Sommer- feld radiation condition:

lim J q-O(kr) p(kr, O) =0, (13a)

•p remaining constant when r-. oo. (13b) Using Eq. (11) provides a simple expression for the normal derivative of Eq. (2).

•p= -- («+jkR ) [poe (O)/R ]. (14)

Then, the nodal vector ß as defined in Eq. (3) is equal to

•= pc •+jkR [D] R ' (15)

where the [D] matrix results from merging elementary matrices [D •] given by

[De]=pc fF [N;lr[N;ldFee' (16) e e

Relation (15) is the monopolar damping condition. Assuming now that R is decreased such as F e is not in

the far field, the second term in Eq. (6) starts to be important and the pressure on F e becomes

p(kR,O)=Ho(kR) fo(O) q- kR ' (17) In this case, at least the B 2 term has to be used to absorb the outgoing wave completely. If Eq. (8) is developed with m=2, together with the Helmholtz equation such as to eliminate the second-order derivative with respect to r, B 2 is given by

B2=2 •r + j O( kr) - ( kr) 2 002

+ •r+j •r+2J -4(kr)2 . (18) Obviously,

B2 p( kr, O) I R=0. (19)

Moreover,

lim B2Palr=•=O{(kR)-9/4}, kR-. oo. (20) From Eq. (19) the normal derivative of the pressure is given by

I _ ) •rr r=• -- 5 +jkR +2(I+jKR) •-•+ Thus •, as defined by Eq. (3), is equal to

1

-- •+jkR [D]+ l+(kR) 2[D']

with

[it]--•q-T e L c•O ] ON; 5a-) ao.

a o

(21)

P(kR,O) a ,

(22)

(23)

Relation (22) is the dipolar damping condition where [D] and [D'] are frequency-independent sparse symmetric matrices. Their calculation is straightforward using the finite element shape functions. It is interesting to note that, for large kR, Eq. (22) reduces to Eq. (15) obtained by using monopolar dampers.

II. DETERMINATION OF THE FAR-FIELD RADIATION PATTERN: THE EXTRAPOLATION METHOD

This section describes a new approach to calculate the far-field pressure from the near-field one in 2-D cases, which is called hereafter extrapolation algorithm. To obtain the far-field directivity function fo(0) from the near-

564 J. Acoust. Soc. Am., Vol. 94, No. 1, July 1993 Assaad et aL: Two-dimensional radiation problems 564


field pressure p (kr, O) previously defined in Eq. (4), the multipolar components F n (0) and (]n (0) have first to be expressed in terms of F0 (0) and Go (0).

A. Expression of Fn(O) and Gn(O) in terms of the Fou- rier coefficients of fo(O)

Now, f0(O) can be expressed in terms of its Fourier coefficients up to the order l o-- 1 following the relation:

f0(0)= • [alcos(lO)+blsin(lO)]. l=0

(24)

In the following lines and for convenience, a matrix notation will be used that is written under the form:

T = ( 1, cos 0, sin 0 .... , cos((lo- 1 ) 0), sin((lo- 1 ) 0)), (25)

fo = (ao,al ,bl ,a2,b2,...,al o_ 1,bl o_ 1), (26)

f0(0) =Tf0 r . (27)

Karp 12 has demonstrated that, in this case, F0(0) only contains even-numbered Fourier coefficients, and Go(O) only contains odd ones, i.e.,

1

F0(0) =•[ fo(O)+fo(O+rr)] (28a)

and

1

G0(0) =•[ f o( O) - f o( O+ •r) ]. (28b)

Thus using the previous matrix notation:

F0(0) = TefD r (29a) and

jGo(O)=Tøf• r, (29b) where

Te=(1,0,0, cos(20), sin(20),...), (30a)

and

T ø = (0, cos (0), sin ( 0),0,0,... ), (30b)

T=Te+T ø, (31)

f0=fD+f•. (32)

The superscripts e and o correspond to vectors having odd and even terms, respectively, equal to zero.

Substituting Eq. (4) into the Helmholtz equation and following the procedure outlined in the Appendix for ex- pressing F n (O) and 6• n (0) in terms of recurrence formulas, provides

tl 2 + 02/002 Fn(O) =[3n(] n_ 1(0), with /•n-- -- 2n '

(33a)

( n -- 1 ) 2 + 02/002 (]n (O) •- •'nFn_ 1 (0), with ]/n -- 2n '

(33b)

Defining

n2--I 2 (n-- 1 )2_•2 Nn'l---- 2n ' Mn'l-- 2n (34)

and

E[ n/2 ] II (N2k,! M2k-l,!), Xn,l(N'M) -- k=l 1, for n=0,1,

for n>l, (35)

where E[n/2] is the integer part, allows one to obtain the expressions of F n (O) and 6•n (0) in terms of f0 (0) (see the Appendix). For n even:

efeT Fn(O) =Te[Xn,l(N, M) ] -0 , (36a)

Gn( O) = -jTø[Xn,l(M,N) ]ofgr, (36b) where the brackets mean a diagonal matrix whose terms are obtained by varying l from 0 to l o- 1 in Eq. (35). This matrix includes 2l o-- 1 terms, each nonzero value of l corresponding to two identical terms while l=0 gives a single term. For n odd:

Fn( O) = --jTø[Nn,iXn,i(M,N) ]of•r, (37a) efeT Gn( O) = Te[Mn,lXn,l(N,M) ] -o , (35b)

where the brackets have the same meaning as for n even. The following properties are obtained for the multipo-

lar components F n and G n (see the Appendix): (i) Gn--O if n> lo--2 and Fn--O if n> lo--3; (ii) F n and 6•n can be expanded in a Fourier series of

order larger than or equal to n and smaller than lo; (iii) sine (cosine) terms can be omitted out of Eqs.

(24)-(32), in fact, the only operator that is retained in the expressions of F n and 6•n in terms of f0(0) is the second derivative with respect to 0;

(iv) from (i), n must only vary from 0 to lo--2 in Eq. (4).

B. Description of the extrapolation algorithm

Denoting for n even:

{Ho(k?. ) )e [A(n,l) ] = [,. (kr) n Xn,!(N,M)

{Hi(kl. ) --i[ (kr)n Xn,I(M, N) (38a) and for n odd:

(Yo( )o [A(n,l)]----j[ (kr) n Nn,!)(n,!(M,N)

(Hl(kl.) )e + • (kr)n Mn,I•'n,I(N,M) , (38b) the pressure field as defined by Eq. (4) can be expressed as:

p(kr, O)=T[C(l) ]f0 (39)

with

565 J. Acoust. Soc. Am., Vol. 94, No. 1, July 1993 Assaad et al.: Two-dimensional radiation problems 565


M

[C(/)]= • [A(n,/)]. (40) n=0

Here, M is the number of terms retained in the series expansion and is called the extrapolation order. 3 From property (iv) of Sec. II A, M can be a value between 0 and 1 o-- 2. Although the dampers are designed to absorb completely monopolar and dipolar outgoing waves, waves of order larger than 2 can exist on F e. Thus, a M value less than 1 o- 2 is not suitable and M has always been set equal to lo--2. It is also interesting to note that the diagonal matrix in Eq. (39) only depends on kR, and thus should be computed only once for all the nodes belonging to the circular boundary Fe. Eq. (39) can be rewritten for r=R as

p(kR,Oi)= • [alCOs(lOi) +rllsin(lOi) ], 1=0 -

(41)

with

•1 •]1 -C(l), for l>• 1, (42)

al bl

where i varies between 1 and N, where N is the number of nodes on the circular boundary F e. Then, the pressure at these nodes can be evaluated to obtain a system of N equations with 1 o unknowns. This system can be solved either by matrix inversion, if N is equal to 1o, or by the least-squares method if N is greater than 1o. The second choice has always been selected because the first one can lead to a ill-conditioned Vandermond matrix. 13

Demonstration of Eq. (42) is deduced from property (iii) in Sec. II A. Pressure values obtained at the nodes of the circular boundary F e can be interpolated by a Fourier series of order 1 o, thus producing the coefficients (at,rh). Dividing these coefficients by C(I), Eq. (42) gives coefficients (at,bt). Finally, from Eq. (24), the desired fo(O) directivity is obtained.

It is interesting to note that the extrapolation problem then becomes an interpolation problem for the near-field pressure, and that the far-field and the near-field radiation patterns may be expandable in the same Fourier series. The demonstration is trivial, using Eqs. (28a)-(28b) and property (ii) in Sec. II A.

III. TESTS OF THE EXTRAPOLATION ALGORITHM

Examples available to test the extrapolation algorithm are limited. We have selected two of them for which ana-

lytical solutions exist.

A. Oscillating cylinder of order m

In this case, an infinite cylinder of radius a (Fig. 2), is vibrating in such a way that the radial displacement of its surface at the point given by the polar coordinate (a,O) is

llr(O)=Crn COS(m0), (43)

where Cm is a constant which is small compared to a. The velocity potential in the infinite fluid domain outside the cylinder is 8

566 J. Acoust. Soc. Am., Vol. 94, No. 1, July 1993

Y F Radiation nodes

7 luid nødeeSnode s

fi,.(o) = u,.(o) •,. er u,.(O) = cos (mO)

(b) (a)

FIG. 2. (a) Schematic description of the oscillating cylinder of order m. Here, u is the radial displacement. The length of the cylinder is assumed to be infinite. (b) Finite element mesh for the oscillating cylinder section. The radius a is equal to 20 cm. The radius of the circular boundary R is equal to 23 cm.

cp=An•m( kr)cos( mO). (44)

To obtain A m, the kinematic continuity condition on the circular boundary can be used-

J o! lt r = -- '•'• r= a ' (45) Substituting q and Ur as given by Eqs. (43)-(44) in Eq. (45) provides

--jcC m

Am--H•n(ka ) , (46) where H' is the derivative of H with respect to its argu- ment. The near-field pressure Pc radiated by the cylinder may be expressed in terms of the velocity potential for r>a:

pc(r,O) =jpwq. (47)

Substituting Eq. (44) into Eq. (47) and using Eq. (46) yields

Hm(kr)

pc( kr, O) = pcwCm H•( ka--•• cos (m0). (48) Then using the asymptotic expression H m of H m as defined in Eq. (5), the far-field pressure becomes

ß

Hm(kr)

Pc• (kr, O)=pcwCm H•(ka) cos(m0). (49) For this test case, the radius of the cylinder is set equal

to 20 cm. The circular boundary radius is chosen equal to 23 cm (R/a= 1.15)m can take any value from 0 to 3. All finite elements are isoparametric, with 3 nodes per side (second-order interpolation). Only one quarter of the domain is meshed due to symmetry (Fig. 2). A null normal pressure gradient is imposed on the x axis for symmetry reasons. A null normal pressure gradient condition or a null pressure condition are imposed on the y-axis respectively for m even (m =0 and m = 2) and m odd (m = 1 and m=3). The number of nonzero coefficients used to describe the basis of Eq. (25) is set equal to 10.

Assaad et al.: Two-dimensional radiation problems 566


x Nearfield (monopolar) o Nearfield (dlpolar) Farfield (monopolar) Farfield (dlpolar)

......... ,,,,,0,,, .... ,,,,,,,,,,o,,.,.,,,,,,,,.,,o,,,,,,,,,,

lOO

• so ul

z 7o R/a = 1.15 6O ,,=, =

,."' • 4o

o O.Ol o.1 I lO lOO

kR

(.)

lOO

70

50

40

30

R/a = 1.15

m=l

lo o

O.Ol o.1 1 lo lOO

kR

(b)

iJJ

z iJJ

iJJ

lOO

so

ul 8o

z 1'o

. 6o i•. 8o

cl 4o

•_ •o

. 10

0

100 -

so -

80 -

70 -

eo -

50 -

40 -

30 -

20

lO

o o.1 1

kR

(c)

R/a = 1.15

n1=2

lO lOO

0.01

• R/a = 1.15 • m=3

: .:

i

o.1 I lO loo

kR

(d)

FIG. 3. Variations as a function of kR of the relative pressure differences at maximum amplitude (0=0), between analytical solutions (far field and near field) and numerical solutions (with and without extrapolation), using monopolar and dipolar dampers: (a) m=0, (b) m= 1, (c) m=2, (d) m=3.

Variations as a function of kR of the relative differ-

ences between the analytical solution [Eq. (48)] and the numerical solutions for the near-field pressure at the maximum amplitude (0=0) are displayed in Fig. 3 (a)-(d), for m equal 0 to 3. Numerical solutions are obtained by using monopolar dampers (B• operator) and dipolar dampers ( B 2 operator). Similarly, variations of the relative differences between the analytical solution [Eq. (49) ] and the numerical solutions for the farfield pressure at maximum amplitude ( 0 = 0) are also presented [Fig. 3 ( a)-( d ) ]. Numerical solutions are obtained in the last case by extrapolation from the nearfield pressures values.

Relative differences for the near-field and the far-field

pressures are practically the same. Thus, the extrapolation algorithm does not add numerical errors. Monopolar dampers give better results than dipolar dampers for uniform radiation (m=0). This result was recently pub- lished. 9 In fact, the zeroth-order Hankel function Ho is the fundamental solution of the Helmholtz equation and can be approximated by its asymptotic expression [Eq. (5)] for kR>• 1. When kR > m, relative differences decrease as the frequency increases. When kR > 60, relative differences increase because the wavelength becomes smaller than 2«

times the element size in the r-axis direction. For m differ-

ent from zero (non uniform radiation), several remarks

can be proposed. First, for kR > 10, monopolar and dipolar dampers give the same results. In fact, for m smaller than 3, Hankel functions H m can be approximated by their asymptotic values [Eq. (5)]. Then, for kR < 10, dipolar dampers appear to be more adequate. Relative differences present a maximum for kR value around 2 and increase as m increases for kR < 1. Values of these maxima as well as

differences for kR < 1 can be reduced by considering a larger R value. For example, with R = 2a, a 10% reduction has been obtained.

With respect to the directivity, the cos(m0) term generally dominates. For this reason, no directivity pattern has been presented. Figure 4(a) and (b) give the phase variations for numerical far-field pressures computed with dipolar dampers (dipolar with extrapolation) and analytical far-field pressures given by Eq. (49) at 0=0, as well as the difference between them, for m equal to 2 and 3; (kR- re/ 4) is retained as the phase reference. Excellent results are obtained.



lll

ii, I

m

Analytical

+ Numerloml (dipolaf with extrapolation) Dlfferenoe

100 -

00-

80-

70-

60-

50-

40-

30-

20-

10-

0 0.01

•!1 360 315 ' 270

m = 2 •__ 22s R/a = 1.15 180 135 Z

4. z

....... .... ........ ..... ;711,,,' ........ .... ........ , .... .4, 0.1 I 10 100

kR

(a)

uJ

100 -

90-

80- +,,• 70-

6o- m=3

40-

3O

20 - . ............................... %%. 10- 0 , •'v";';":7[,• , ', ?';-;-;-,-• ....... • .... •'4 .....

0.01 0.1 I

kR

360

315 270 225 180

135

4. z ,4

o

-46 10 100

(b)

FIG. 4. Variations as a function of kR of the pressure phase (in degrees) at 0=0, for analytical and numerical far-field solutions and difference between them: (a) m = 2, (b) rn= 3.

B. Radiation from a source of finite width mounted

in a rigid baffle

An acoustic harmonic source of width W mounted in

a rigid wall is first considered (Fig. 5). The displacement of the source is known and the source is immersed in a

FIG. 5. Schematic description of the acoustic harmonic source of width W mounted in a rigid or soft wall.

FIG. 6. Finite element mesh of the acoustic harmonic source: W is equal to 2 mm and the radius R of the circular boundary equal to 1.2 mm.

semi-infinite homogeneous isotropic fluid medium. The external boundary F e is a semi-circle of radius R. Pressure on Fe is classically given by the Rayleigh's formula, •ø

eJrr/4 f W/2 c•P I exp(-jklr-Xl)dx ' p(kr, O)-- 2re •-w/2 •YY y:0 x/klr-xl (50)

where r is the abscissa for a point on-F e and x is the running point on the source. The normal pressure gradient on this source can be expressed in terms of the normal displacement uy:

•yy y=o: •OO)2Uy. (51) As a first example, Uy is assumed to be constant and equal to u0. Then, it can be demonstrated easily that, if

W

and R)-•- (52) Eq. (50) can be rewritten as

p( kR,O) = --j pcrouoHo( kR ) k W sin [ (k W/2) cos 0 ]

2 (kW/2)cosO ' (53)

where H 0 is given by Eq. (5). The far-field analytical directivity pattern [Eq. (10)] is then given by

k W sin [ (k W/2) cos 0 ] fo (0) = -- jpcrouo 2 (k W/2) cos 0 ' (54) This example is modeled (Fig. 6) with R equal to 1.2

mm and W to 2 mm (R/W=0.6). Only one half of the domain is meshed for the sake of symmetry. The fluid mesh is made up with 363 elements, 22 damping elements being used for the boundary circle. On the rigid baffle and along the x-axis direction, a null normal pressure gradient is assumed. The number of non zero coefficients (nc) necessary to describe the T basis of Eq. (25) can be estimated by expanding the analytical farfield pressure [Eq. (53)] in a series of order nc (Ref. 14). Higher-order terms of this series are negligible when the following criterion is valid:

(kW/2 )2 <0.5. (55)

(2n½+ 1 ) (2n½+ 3)

568 d. Acoust. Soc. Am., Vol. 94, No. 1, July 1993 Assaad et al.: Two-dimensional radiation problems 568


Figure 7 describes the variations with respect to kR of the relative far-field pressure differences at maximum amplitude (0=0) between the analytical solution [Eq. (53)] and the numerical solutions. Numerical far-field pressures were obtained by extrapolation [Eq. (10) ] for different values of he, using monopolar [Fig. 7 (a)] and dipolar [Fig. 7 (b) ] dampers, as a function of kR. As obvious, differences decrease as the number of coefficients increases. The criterion

defined by Eq. (55) gives a good estimation of he, which has also been used in any case. Moreover, it can be noted that, when the number of coefficients is twice the number of maxima in the nearfield directivity, the inequality of Eq. (55 ) is verified.

Relative differences between the analytical farfield pressure p(kR,O) given by Eq. (53) and the pressure on the circular boundary ['e without extrapolation are also presented in Fig. 7 (a) and (b). With extrapolation, dipolar dampers give a better result than monopolar dampers. Without extrapolation, both monopolar and dipolar dampers give bad results, because the pressure used to compute the difference is, in fact, the near-field pressure.

The preceding example has been reused setting W equal to 1.2 mm (R = W) and keeping the same element size to wavelength ratio. Relative differences are presented in Fig. 8 for numerical pressures obtained with or without extrapolation and using monopolar and dipolar dampers. When kR < 1, higher-order waves (B m , with m > 2) have to be considered. When 1 <kR < 6, monopolar dampers (without extrapolation) gives immediately the far-field pressure, because the conditions expressed in Eq. (52) are verified. When 2 • kR • 50, monopolar and dipolar dampers, with extrapolation, give good results, but dipolar dampers are the most adequate. When kR • 50, differences increase because the element size is not small enough compared to the wavelength. At the circular boundary, the element size is then greater than or equal to 0.56A.

Finally, relative far-field directivity patterns computed with dipolar dampers are compared with the analytical ones, as expressed by Eq. (54), for some frequencies (Fig. 9) in order to show the effectiveness of the algorithm. In each case, the near-field directivity, which was used to ex- trapolate the far-field one, is also presented. For the maximum frequency, 13 MHz, the element size at the circular boundary r e is equal to 0.74• and the relative difference at maximum amplitude (Fig. 8) is equal to 12%, but the numerical relative directivity obtained is quite accurate.

As a second example, the normal displacement uy is assumed to be harmonic and symmetrical with respect to the transducer normal:

Monopolar Dampere With Extrapolation (R/W=0.6) Number of Coefflclenle

4 10

6 14

8 CRITERION

Mon. opolar D.a.mpere W. Ithout E.xtrapolatl. on..

w

w •L

W

100 -

90 -

80 -

70 -

60 -

60

40

3O

20

lO

o 0.1

!

i://. ./.'/////

ß /- lOO

kR

Dlpolar Dampere With Extrapolation (R/W=0.6) Number of Coefficient.

4 10

0.1 I 10 100 kR

(b)

uy(x)=uo cos[2•'(x/W) ], with W=,•h, (56)

where Ah is the harmonic wavelength. When conditions provided by Eq. (52) are verified, the far-field pressure is given by

FIG. 7. Variations as a function of kR of the relative differences (0--0), between theoretical far-field pressure and numerical pressures obtained by extrapolation, using (a) monopolar dampers and (b) dipolar dampers, for different values of nc. Relative differences between theoretical far-field pressure and numerical pressures without extrapolation are also presented, using (a) monopolar dampers and (b) dipolar dampers.

569 J. Acoust. Soc. Am., Vol. 94, No. 1, July 1993 Assaad et al.' Two-dimensional radiation problems 569


Radiation of a Source In a Rigid Baffle R=W=l.2mm

..... ..M...•..."...•....p...•..!..•..r....?.!.t...h...•..."...t...?...x..t..r...•...p...•..!..•...t.!.•.•.• ..... Monopolar with extrapolation.

I) !p o.•1., r.__?l t h.__o. ut.__e. xt r.__a. p ol__.,t Io__.n•.__ Dlpolar with extrapolation.

100 -

9O

,o i 7o

,o ,\ 40

\\

•o ,% ........ .•...•/ o ............. , 0.1 I 10 100

kR

FIG. 8. Variations as a function of kR of the relative amplitude differences (0=0) between theoretical and numerical pressures, with or without extrapolation, using monopolar and dipolar dampers (R = W= 1.2 mm).

-10

-2o

90' 90'

.... 60' -10 60'

-20 30'

.•o '.,,,•-•o FREQ. =0.2 MHz FREQ. =0.8 MHz

90' 90'

60' 60'

-10 -10

-20 _ . . -, ,• -20 30' -30 ' , , ,•-30

I I ' l! I', I i [00 00 FREQ. - 3 MHz FREQ. - 6.5 MHz

90' $0'

O0* 60'

.10 .10

.20 -20

o' FREQ. - 9 MHz FREQ. = 13 MHz

p(kR,O)=Ho(kR) -- j pccouo -- kW

sin (X+) sin (X_) ) X 2X + 2X_ , (57)

with

X•- 2 cos(0)+ (58)

and H 0 is defined by Eq. (5). The factor between paren- theses in Eq. (57) is the far-field directivity. Positions of main lobes, defined with respect to the y axis are given by

+ arcsin(•/•n). (59)

The model is the same as for the previous example, with W--R--1.2 mm. Relative differences at maximum amplitude between the analytical solution and the numerical solutions are very close to those obtained in the case of uniform radiation (Fig. 8). Thus, previous comments concerning radiation conditions (monopolar and dipolar dampers) can be applied to this example. As Fig. 9, Fig. 10 displays the analytical and the numerical far-field directivities for selected frequencies. The near-field directivity that was used to compute the far-field one is also presented.

FIG. 9. Relative far-field directivity patterns (in dB) for the finite plane source mounted in a rigid baffle (R--W--1.2 mm) and vibrating uniformly. Analytical far-field directivity patterns are drawn in thick lines. Numerical far-field directivity patterns as obtained by extrapolation using dipolar dampers are shown in dotted lines. The near-field directivity patterns, which were used to compute the far-field one, are displayed in thin lines.

C. Radiation of a source of finite width mounted in a soft baffle

In this case, the pressure along the baffle is set equal to zero (soft boundary). An acoustic source of width W, whose displacement is known, is placed in the soft wall (Fig. 5). The pressure on the external surface F e of radius R (Fig. 5) is given by the Sommerfeld's formula: •ø

p(kr, O) =k e jrc/4 f W/2 Ply-o 2• J--w/2

exp(-jklr-xl)

x x/klr-xl cos(x, (r--x))dx, (60)

where r and x have the same definition as in Eq. (50). As for the Rayleigh's formula [Eq. (50)], conditions expressed by Eq. (52) are assumed to be valid. Then, the far-field pressure is equal to

570 J. Acoust. Soc. Am., Vol. 94, No. 1, July 1993 Assaad et al.' Two-dimensional radiation problems 570


GO' GO' GO'

80' SO'

.lO .lO -lO

,.•0 -•0 •0' -20

.•o ', ,,,•.•o .oo O* O* FREQ, -0,2 MHz FREQ, -0,8 MHz

O0*

. 60' 60'

-10

-20 :SO*

O* FREQ, -0,2 MHz FREQ, -0,6 MHz

90' 90' O0* 90'

60' 60' 60' 60'

-10 -10 -10 -10

-20 -20 30' -20 -20 *

-30 -30 -30 ' . ", ..'•-30 O' O' O' O* FREQ, - 3 MHz FREQ, ' 6,5 MHz FREQ, - 3 MHz FREQ, - 6,$ MHz

00'

60'

-10 -10

-20 -20

0 FREQ, m O MHz

I0'

. 60'

0* FREQ, - 13 MHz

00' $0'

60' 60'

-10 -10

-20 -20 *

O' FREQ, - 9 MHz FREQ, -- 13 MHz

FIG. 10. Same as Fig. 9 except that the displacement is assumed to be harmonic.

½--j(kR--w'/4) fW/2 I p(kR,O)=k x/2w.k R sin 0 p e j•øøsø dx. d -- W/2 y=0 (61)

If, the normal displacement uy is constant and equal to u0, the following relation is generally assumed to holds:

øP I = pco2u0 = --jkp[y: O, (62) Y y---0 which means that the pressure at the surface of the source is supposed to be constant. In fact, this equation is valid for plane waves and not for cylindrical waves. However, in this case the farfield directivity can be written as'

sin[ (kW/2)cos O]

p (kR,O) :P0 sin (O) (k W/2) cos 0 ' (63) where P0 is a constant.

With the geometry of Fig. 9, Fig. 11 displays the numerical far-field directivity patterns obtained using extrapolation with dipolar dampers and the analytical far-field directivity patterns [Eq. (63)] for selected frequencies. When the wavelength is smaller than the source width (/• < W), both directivities are similar for the first lobes. But, when/• > W, there is a quite large difference between analytical and numerical solutions. In fact, the equality

FIG. 11. Same as Fig. 9 except that a soft baffle is assumed.

given by Eq. (62) is not verified in that case. Fig. 12 shows the numerical pressure variations along the source surface, computed by assuming uniform displacement, as a function of x and for two different frequencies (0.8 and 6.5

ß f = 0.8 MHz O f = 6.6 MHz

real part ,,, !, .m., .a, .g. !..n..a..r,y.,, .p..a, .r, !,,,

I ?-,ß.,•,.0. ß,, 0 ..... -- '•' "ß ',"' 0 • " ?'o..o._ ,•:.• •.'..:.•., ,•-'

• ,,o,. ø 0 o.s '.o. ø

m -0.6

o.o 0.3 0.6

x-axis (ram)

FIG. 12. Computed pressure variations at the surface of the finite plane source mounted in a soft baffle and vibrating uniformly, as a function of x (0<x < W/2).



MHz). When A > W (f=0.8 MHz, A= 1.88 mm), pressure variations are important, particularly for the imagi- nary part. For A < W (f=6.5 MHz, 2=0.23 mm), variations are oscillating and more negligible, except for points close to the soft baffle. This is the reason why at larger frequency analytical far-field directivity is similar to the numerical directivity for the first lobes.

IV. CONCLUSION

Generally, high-frequency acoustic transducers with one dimension much larger than the two others (bars, cyl- inders) can be numerically modeled within the plane strain approximation. However, when these transducers radiate into an infinite fluid domain, quite large meshes are re- quired to obtain far-field pressures, which result in very expensive computation costs. Using the dipolar dampers, which have been shown to be able to absorb the main part of the outgoing acoustic wave, this limitation has been sup- pressed. However, the resulting pressure at the external circular boundary is not the far-field pressure. Using the Helmholtz equation and its solution, which obeys the Som- merfeld radiation condition, an extrapolation algorithm has been developed to compute the far-field pressure from the pressure nodal values on the boundary circle.

This formulation has been implemented in the ATILA finite element code and successfully tested. Comparisons have been made with analytical results for classical problems, such as the oscillating cylinder and the source imbedded in a half-space with an acoustically hard or soft baffle. It has been shown that monopolar dampers can only provide the far-field pressures under some restrictive conditions. Dipolar damping elements are more reliable and give accurate near-field pressures. Far-field pressures are then, obtained by extrapolation.

With the incorporation of this new radiation formulation, application of the finite element ATILA code can now be extended to the modeling of transducers used in acoustic imaging and nondestructive testing at high frequency. This new prospect will be outlined in a future paper is and can be found in thorough details in one of the authors' thesis. 16

APPENDIX

1. Recurrence formulas for Fnand Gn

Substituting Eq. (4) into the Helmholtz equation, ex- pressing the derivatives of H0(kr) and H i (kr) in terms of Ho(kr) and Hi (kr) by means of the formulas:

H1 S6=--Sl, S•)•=-•-fi--So,

--2H; (H 1 )t k--T-- ana H;' -- -- -- H0 ,

(A1)

and factorizing out the terms multiplying H 0 and Hi, provides

Z I [

H 1 (kr) +

(kr) n

( o2) n2+•-'• Fn(O)--2(n+ 1 )Sn+l(O )

n2+2n+ 1 +•-• Gn(O)

+2(n+l)Fn+l(O) =0. (A2)

The net coefficients of Ho(kr) and Hl(kr) must vanish identically. Thus, the two recurrence relations (33a) and (33b) are immediately obtained.

2. Expression of the multipolar components Fnand Gnin terms of f0(0)

To express F n (0) and G n (0) in terms of F0 (0) and G0(0), two cases have to be considered corresponding to n even and n odd. For n even:

Fn ( O ) =finYn- lfin-- 2 ' ' 'fi2•/1F0(0),

Gn(0): •/n•n--l'}/n--2' ' ' '}/2•lSO(0).

For n odd:

(A3a)

(A3b)

Fn(O) :fin•/n _ lfin_2' ' ' •/2•lGO(0), (A4a)

Gn(O) :•/n•n--lrn-2' ' 'fi2rlFo(0), (A4b)

where •n and •'n are defined in Eqs. ( 3 3a) and (3 3b). Then it can be easily proved that

(cos(10) cos(10) •nksin(lO))=Nn,l(sin(lO)) (A5a) and

(cos(10) cos(10) Yn[sin(lO))=Mn,l(sin(lO)), (A5b) where Nn, t and Mn, • are defined by Eq. (34).

To obtain the expressions given by in Eqs. (36)-(37), only Fn(O) for n even is computed hereafter in terms of f0(0), the demonstration being similar for other expressions. Replacing Fo(O) as given by Eq. (28a) into Eq. (A3a) provides:

E[ n/2 ]

F n (0) = II fi2k'}/2k -- 1 k=l

+ b2l sin (210) ].

•r[ to/2]

• [a2lcos(210) l=0

(A6)

The order of the product and the summation in Eq. (A6) can be changed. Then Eqs. (A5a)-(A5b) can be ac- counted for and Xn,t(N,M) as given by Eq. (35) can be substituted into Eq. (A6). Thus:

•r[ 6/21

Fn(O)= Z l=0

Xn, I(N,M) [a21cos(21O) +b21sin(21O) ]. (A7)

Using matrix notations as defined by Eqs. (25)-(27) and Eqs. (29)-(32), Eq. (A7) can be rewritten as

Fn( O) =Te[xn,•(N,M) ]ef•r. (A8)



In the same way, expressions given by Eqs. (36)-(37) can be found.

3. Properties of the multipolar component functions

Using Eqs. (34)-(35) for n > 1, Xn,• can be expressed as:

E[n/2] (2k)2__ (/)2 (2k_2)2_ (/)2 Xn,,(N,M) = II- 4k X •=• 2(2k- 1) '

(A9a)

E[n/2] [ (2k__ 1)2__ (l)212 X,•,t(M,N) = II -- 8k (2k- 1 ) ' (A9b) k=l

where X,•,t(M,N) is equal to zero if l=2k-1 with k= 1,...,E[n/2] and similarly X,•,t(N,M) is equal to zero if l= 2k with k= 1,...,E[n/2]. Demonstrations of the first and second properties in Sec. II A are easily obtained using the above equations and the expressions of the multipolar components of Fn and G• functions. Demonstration of the third property is obtained using Eqs. (ASa)-(ASb).

•M. C. Junger and D. Feit, Sound, Structures, and Their Interactions (MIT, Cambridge, 1982), 2nd ed.

2 A. Bayliss, M. Gunzburger, and E. Turkel, "Boundary conditions for the numerical solution of elliptic equations in exterior regions," ICASE Report No. 80/1, NASA Langley (1980).

3 R. Bossut and J.-N. Decarpigny, "Finite element modeling of radiating structures using dipolar dampings elements," J. Acoust. Soc. Am. 86, 1234-1244 (1989).

4R. Bossut and J.-N. Decarpigny, "An improvement of the finite radiating element formulation. Application to the modeling of a radiating

free-flooded transducer," J. Acoust. Soc. Am. Suppl. 1 74, S23 (1983). 5 R. Bossut, J.-N. Decarpigny, B. Tocquet, and D. Boucher, "Application of damping elements to the modeling of underwater radiating structures," J. Acoust. Soc. Am. Suppl. 1 79, S51 (1986).

6 B. Hamonic, "Application of the finite element method to the design of power piezoelectric sonar transducers," Proceedings of the International Workshop on Power Sonic and Ultrasonic Transducers Design, edited by B. Hamonic and J.-N. Decarpigny (Springer-Verlag, Berlin, 1988).

7 J.-N. Decarpigny, "Application de la m6thode des 616ments finis/• l'6- tude des transducteurs pi6zo61ectriques," Th•se de Doctorat d'Etat, Universit6 des Sciences et Techniques de Lille, France (1984).

8p. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

9 p.M. Pinski, L. L. Thompson, and N. N. Abboud, "Local high-order radiation boundary conditions for the two-dimensional time-dependent structural acoustics problem," J. Acoust. Soc. Am. 91, 1320-1335 (1992).

løB. Delannoy, H. Lasota, C. Bruneel, R. Torguet, and E. Bridoux, "The infinite planar baffles problem in acoustic radiation and its experimental verification," J. Appl. Phys. 50, 5189-5195 (1979).

• O. C. Zien•kiewicz, The Finite Element Method (McGraw-Hill, New York, 1977), 3rd ed.

•2S. N. Karp, "A convergent 'Farfield' expansion for two-dimensional radiation functions", Comm. Pure Appl. Math XIV, 427-434 (1960).

13 B. Carnahan, H. A. Luther, and J. O. Wilkes, Applied Numerical Meth- ods (Wiley, New York, 1969).

14 G. Arfken, Mathematical Methods for Physicists (Academic, San Di- ego, 1985), 3rd ed.

•5 j. Assaad, C. Bruneel, J.-N. Decarpigny, and B. Nongaillard, "Electro- mechanical coupling coefficients and far-field radiation patterns of lith- ium niobate bars (Y-cut) used in high-frequency acoustical imaging and nondestructive testing," submitted for publication to J. Acoust. Soc. Am. (1993).

16j. Assaad, "Mod61isation des transducteurs haute frequence/t l'aide d e la m6thode des 616ments finis," Th•se de Doctorat, Universit6 de Va- lenciennes et Hainaut-Cambr6sis, France (1992).

573 J. Acoust. Soc. Am., Vol. 94, No. 1, July 1993 Assaad et aL: Two-dimensional radiation problems 573


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Application of the finite element method to two-dimensional radiation problems

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