A Near-to-far-field Transformation Method

for Acoustic Scattering ProblemsTongjing Sun

1, 2, Guijuan Li

2, Kejun Wang

1

1. College of Automation, Harbin Engineering University, Harbin, China 2. Dalian Scientific Test and Control Technology Institute, Dalian, China

stj3042@126.com

Abstract Kirchhoff integral was applied to acoustic scat-

tering problems, and a near-to-far-field transformation method

was developed for finite-difference time-domain (FDTD) method.

Expressions for steady field and transient field were derived re-

spectively. In addition to its efficiency and simplicity of imple-

mentation, it could reduce absorbing boundary condition (ABC)

errors. For absolutely hard and absolutely soft objects, simula-

tions were performed and results were given. A good agreement

between this method and theoretical results was demonstrated,

and accuracy of this method was higher than direct fi-

nite-difference time-domain (FDTD) method clearly. Validity and

accuracy of this method were well proved.

Index terms near-to-far-field transformation, acoustic scat-

tering, Kirchhoff, FDTD

I. INTRODUCTION

Near-to-far-field transformation is a key technology in

finite-difference time-domain (FDTD) method. It is studied

widely for electromagnetic field, and some transformation

methods were developed [1]-[4]. But for acoustic scattering field,

little research has been done. In 1996, Shuozhong Wang ap-

plied finite-difference time-domain (FDTD) method to un-

derwater acoustic scattering problems [5], and he used a Fou-

rier transform method to gain far-field directional patterns.

The Fourier transform method used sound pressure values on

a circle and sphere in near field to derive values in far field in

polar coordinate. Thus shape of extrapolated boundary was confined. In addition, it only could be applied to steady field,

so it was limited in application. Lili Ma used Green Function

method to achieve near-to-far-field extrapolation [6]. This

method was suit for steady field and transient field, but the

form of Green Function is very complex, which makes the

derivation troubling and not easy to understand. It is obvious

that existing methods are all imperfect, and confined the ap-

plication of FDTD method to acoustic scattering problems.

As a mathematics expression, Kirchhoff formulation, viz.

Huygens integral, is usually used to compute radiation of vi-

bration surface [7] [8], and acoustic scattering can be regarded

as reradiation of scatterer as second sound source. Based on this theory, Kirchhoff formulation was applied to FDTD for

acoustic scattering computation, and a near-to-far-field trans-

formation method was developed. Kichhoff expression is a

classical theory in underwater acoustics, therefore it is easy to

understand. In addition, this method is simple to implement,

and can be used not only to steady field but also to transient

field, and it can reduce absorbing boundary condition (ABC)

errors. Simulations are performed in this paper for absolutely

hard and absolutely soft objects, and validity and accuracy of

this method is demonstrated.

II. FDTD FORMULATION IN ACOUSTICS

Consider sound waves propagating in the water. In stead

of the wave equation, we base our work on the basic Euler’s

equation and the equation of continuity. For simplicity, the

discussion is confined to a two-dimensional space. generaliza-tion to three-dimension should be straightforward. In a 2-D

Cartesian coordinate system, the sound pressure p and the

particle velocity u satisfy the following equations:

xup

x tρ ∂∂ = −

∂ ∂, (1)

yup

y tρ

∂∂ = −∂ ∂

, (2)

2

1u u p

x y tcρ∂ ∂ ∂+ = −∂ ∂ ∂

, (3)

where ρ is the density of the medium, and c is the sound

speed.

To derive the finite difference form of these partial dif-

ferential equations, both time and space need to be discretized.

In the space, a square lattice with a grid size of δ is chosen.

Any boundary locations are approximated to the nearest lat-

tice points. The lattice spacing δ should be sufficiently

small, generally one tenth of the wavelength or less, to give an adequate space sampling. Therefore the first order partial de-

rivatives of a field parameter ( ), ;f x y t with respect to x

and y can be approximated as the following central differ-

ences:

( ) ( ) ( ) ( ) ( ), ; 11, 1,

2

n nf x y tf i j f i j

x δ∂

→ + − −∂

, (4)

( ) ( ) ( ) ( ) ( ), ; 1, 1 , 1

2

n nf x y tf i j f i j

y δ∂

→ + − −∂

, (5)

Central differences provide second-order accuracy compared

to biased differences. Similarly, the partial derivative of

( ), ;f x y t with respect to t can be approximated as

( ) ( ) ( ) ( ) ( )1 1, ; 1, ,

2

n nf x y tf i j f i j

t t

+ −∂→ −

∂ Δ. (6)

Here we follow Yee’s notations where i and j are the

spatial indices representing discretized x and y respec-

tively, and n is the temporal index. Substituting these dif-

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Proceedings of the 2007 IEEEInternational Conference on Mechatronics and Automation

August 5 - 8, 2007, Harbin, China

ference expressions into (1) - (3), the following recurrence re-

lations are obtained:

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )2 1 1, , 1, 1,

n n n n

x x

tu i j u i j p i j p i j

δρ− − −Δ= − + − −

(7)

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )2 1 1, , , 1 , 1

n n n n

y y

tu i j u i j p i j p i j

δρ− − −Δ= − + − −

(8)

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )2 1 1, , [ 1, 1,

n n n n

x x

c tp i j p i j u i j u i j

ρδ

− − −Δ= − + − −

( ) ( ) ( ) ( )1 1, 1 , 1 ]

n n

y yu i j u i j− −+ + − − . (9)

The time step tΔ must be small enough to ensure stability of

the algorithm: 1

2

2 2

1 1 1

2t

c x y c

δ−

Δ ≤ + =Δ Δ

, (for two-dimension) (10)

In electromagnetism, a staggered lattice structure is nor-

mally used so that E and H are evaluated at interleaved lattice

points in the space. This is appropriate for the FDTD repre-

sentation of many EM problems. The aim of the present work

is to establish a framework. In these cases, both p and u

will often appear concurrently in boundary condition expres-

sions. Therefore we choose a non-staggered lattice structure

so that both field parameters are evaluated at the same lattice

points. However, staggered lattice structures may also be use-

ful for some acoustic applications.

(7) – (9) provide explicit recurrence relations between

p and u . The present p value can be computed from the

p of the same lattice point two time-steps earlier, together

with the u values of the neighboring points one time-step

earlier. The u values are computed in a similar way. The

computer program can be written to update u and p al-

ternately without solving any system of simultaneous equa-

tions.

III. NEAR-TO-FAR-FIELD TRANSFORMATION METHOD FOR

ACOUSTIC SCATTERING PROBLEMS

FDTD method can’t compute very far field values be-

cause of the limitation of computer resource. Here, we devel-

oped a near-to-far-field transformation method based on

Kirchhoff expression. This method can express the value in

far field using the value in near field.

A. Kirchhoff Principle for Acoustic Scattering If the pressure in a sound field is of the form

( , , ) ( , , ) exp[ ( )]p x y z a x y z i tω ϕ= − − , (11)

and the amplitude is not too large, the pressure satisfies the

scalar wave equation,

022 =+∇ pkp . (12)

As shown in Fig.1, the pressure at any point P exterior

Fig.1 General surface of integral

to S can be determined by the surface integral

∂−∂=S

n

ikrikr

n dSpr

e

r

epPp ])()([

4

1)(

π, (13)

where n∂ denotes differentiation with respect to distance

along the outward normal to S and r is the distance from

P to S , k is the wavenumber, p is the complex pres-

sure on S , and )(Pp is the complex pressure at P . (13) is

valid only for sinusoidal fields, as stated in (11), but it is a

special case of a general theorem which is valid for sound

fields of arbitrary time dependence.

Formulation (13) enables one to choose a closed surface

near the object, to compute the amplitude and phase of p and

pn∂ over the surface, and to determine the pressure at any

point in the far field by numerical evaluation of the integral.

In practice, however, the normal component of the pressure

gradient pn∂ is so difficult to compute or measure that the

quantity ikp has been used as an approximation for it.

Since the point P is in the far field and therefore kr

is very large compared to unity, it can be shown that

rreikre nikrikr

n ∂≈∂ )/()/( , (14)

where second-order terms in )/1( kr have been neglected.

The derivative rn∂ is evaluated at a point on S . The quan-

tity r can be thought of as a scalar field, and rn∂ can be

thought of as the scalar product of the gradient of r and the

unit outward normal vector. From these considerations it can be seen that

θcos−=∂ rn , (15)

Where θ is the angle between the unit normal and a line

connection P and a point on S . Thus we have the ap-

proximation that

θcos)/()/( reikre ikrikrn −≈∂ . (16)

The substitution of these approximations into (3) gives

+−=S

ikr

pdSr

eikPp ))(cos1(

4)( θ

π. (17)

This expression is used to the compute the acoustic scat-tering in far field.

B. Extrapolation Method Based on Kirchhoff Expression We outline the steps needed for simple implementation

of (17) into a standard FDTD code, based on the Yee cell. (17)

is discretized for FDTD implementation about time and space

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( , ) (1 cos )( ) ( , , , )4

ikr

S

ik ep r t p x y z t dS

rθ

π′= − + . (18)

We then employ a change of variable in the time domain; that

is, instead of using retarded time, we express the calculated

field as a function of the advanced time /t t R c′ = − , c is

sound speed, r is the normal vector to observed point P ,

x , y , z is coordinate of point on the closed surface S .

For each observation point, R is a continuous function

whose domain is the integral surface. However, in numerical

evaluation of the integral where R is defined with respect to

each cell, R becomes a discrete function with nonuniform

spacing and, thus, ( , )p r t takes on values at time instances

that are separated by the FDTD time step tΔ . Therefore, for

uniformity and ease of implementation, we present ( , )p r t

as a discrete sequence with a uniform time step equivalent to

tΔ . The time sequence index corresponding to t is denoted

by n and is equal to the nearest integer to /( )n n R c t′ = − Δ ,

which we define as int

/( )n n R c t′ = − Δ . Substituting it in (18),

and we have FDTD extrapolated expression based on

Kirchhoff formulation

( , ) {(1 cos )( )4

ikr

xyz

ik ep r n

rθ

π= − +

xyztcrnzyxp ΔΔ− )})/(,,,(int

. (19)

C. Discussion The extrapolated expression derived the FDTD discrete

relationship between the field at a certain time step at each observation point and the field on the extrapolated boundary

(closed surface or volume that contains the object). Here,

shape of closed surface is not confined, and implement is very

simple. In addition to field on the extrapolated boundary, we

only need to compute the two parameters R and θ which

are calculated once for each of the subsurfaces xyzΔ . When

we compute the transient field in far field, we can iterate near

field on the integral surface to far field that is contributed to, which can be made as the FDTD loop is executed. Thus, the

fields on the extrapolated boundary at each time step don’t all

need to be stored, therefore memory space is decreased. Cor-

responding expression is

int( , /( ) ) {(1 cos )( )

4

ikr

xyz

ik ep r n r c t

rθ

π+ Δ = − +

xyznzyxp Δ)},,,( (20)

IV. NUMERICAL SIMULATION

In order to compare the near-to-far-field transformation

method with exact theoretical solution, infinity length cylin-

der was selected as the object (radius is 1m, center is point

O ), as shown in Fig. 2. Frequency was 1500Hz, sound speed

was 1500m/s, wavelength was 1 meter, 15 grids were sepa-

rated in every wavelength, the extrapolated boundary was a

square containing the object (center is point O , half length of

side is 50 grids), P observation point in far field (distance to

center is 7 meters, viz. 105 grids).

Fig. 2 Sketch map of computed model

Figure 3 showed the scattering field of an ideally hard

circular cylinder insonified by a plane wave, where (a) and (b)

were the distribution of magnitude and phase of the sound

pressure.

(a) Magnitude distribution

(b) Phase distribution

Fig. 3 Sound pressure distribution of scattered field

For absolutely hard and absolutely soft object, acoustic

scattering in far field was simulated using the near-to-far-field

transformation method presented by this paper. Pressure am-

plitudes for points on the surface around the object which was

seven meters distance to center O were computed at the di-

rection of 0° - 360° using near-to-far-field transformation

306

based on Kirchhoff and FDTD direct computing method. Far

field directivity patterns were derived, and compared with

exact theoretical solution, and errors were analyzed

( 0° -180° ), as shown in Fig. 4 and Fig. 5.

(a) Result of direct FDTD computation

(solid is result of direct FDTD computation,

dashed is exact theoretical solution)

(b) Result of extrapolation

(solid is result of extrapolation,

dashed is exact theoretical solution)

c Errors comparison between two methods

(solid is errors of direct FDTD computation, ,

dashed is errors of extrapolation)

Fig. 4 Simulation results of absolutely hard object

(a) Result of direct FDTD computation

(solid is result of direct FDTD computation,

dashed is exact theoretical solution)

(b) Result of extrapolation

(solid is result of extrapolation,

dashed is exact theoretical solution)

c Errors comparison between two methods

(solid is errors of direct FDTD computation, ,

dashed is errors of extrapolation)

Fig. 5 Simulation results of absolutely soft object

In the accuracy of computation aspect , we can see that

the far field direction patterns derived by near-to-far-field

transformation based on Kirchhoff integral is very near to ex-

act theoretical solution no matter what the object is absolutely

hard or absolutely soft. For ideally hard object, the errors are

about 0.1, and near to 0.2 only adjacent 90° and 120° , and

they tend to zero at adjacent 0° . However, errors of direct

FDTD computation are about 0.2, and the most value reaches

0.45. For absolutely soft object, errors are about 0.15, and

close to 0.2 at adjacent 50° , 70° and 120° , and the errors

are quite little. However, errors of direct FDTD computation

are less than absolutely hard object, but they are bigger than

that of near-to-far-field transformation method clearly. In the time of computation aspect, the extrapolation area

of near-to-far-field transformation based on Kirchhoff integral

is 100 grids multiplied by 100 grids, and the area for the ab-

307

sorbed boundary condition area need to be added to (viz. ex-

tend 20 grids outwards), so computing area becomes 120

grids multiplied by 120 grids. At this time, the time of com-

putation is 34 seconds for absolutely hard object, and it is 23

seconds for absolutely soft object. The area of computation of

direct FDTD computation (for observation point the distance

of which to center is 7 meters, viz. 105 grids) is 230 grids

multiplied by 230 grids at least. At this time, the time of

computation is 227 seconds for absolutely hard object, and it

is 151 seconds for absolutely soft object. From the above, we

can see that the time of near-to-far-field transformation based on Kirchhoff integral is only 15% of direct FDTD computa-

tion (for observation point the distance of which to center is 7

meters, viz. 105 grids). If the observation point is more far

from the center of object, and computing and storing abilities

of computer are gone beyond, and computation can’t be done

by direct FDTD method, the near-to-far-field transformation

is the only path.

Results of simulation showed that results of

near-to-far-field transformation method are entirely credible.

In addition, this method can reduce absorbed boundary condi-

tion (ABC) errors, accuracy is higher than direct FDTD method clearly, and the time of computation is saved .

V. CONCLUSION

This paper applied Kirchhoff integral expression to

FDTD for acoustic scattering. The simulation results for ab-

solutely hard and absolutely soft object were given, and com-

pared with exact theoretical solution and direct FDTD com-

putation result. Several conclusions are educed:

(i) Kirchhoff integral expression used to radiation field

can derive the extrapolation formulation of steady field in

FDTD, and it can also deduce that of transient field.

(ii) The extrapolation method is simple to implement and

easy to understand.

(iii) Simulation results of this method are entirely credi-

ble for acoustic scattering, and this method can reduce ab-

sorbed boundary condition (ABC) errors, increase accuracy of

computation, and save the time of computation.

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