#he modal element method has been employed to determine the scattered field from a plane acoustic wave impinging on a two dimensional body. In the modal element method, the scattering body is represented by finite elements, which are coupled to an eigenfunction expansion representing the acoustic pressure in the infinite computational domain surrounding the body. The present paper extends the previous work by developing the algorithm necessary to calculate the acoustics scattering cross section by the modal element method. The scattering cross section is the acoustical equivalent to the Radar Cross Section (RCS) in electromagnetic theory. Since the scattering cross section is evaluated at infinite distance from the body, an asymptotic approximation is used in conjunction with the standard modal element method. For validation, the scattering cross section of the rigid circular cylinder is computed for the frequency range 0.1 is less than or equal to ka is less than or equal to 100. Results show excellent agreement with the analytic solution

Baumeister, Kenneth J.

Kreider, Kevin L.

English

NASA Technical Reports Server

NASA Technical Memorandum 106667
: . ,
I / 2
f.!i /t s ' / I
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Scattering Cross Section of Sound Waves
by the Modal Element Method
(NASA-TM-I06667) SCATTERING CROSS
SECTION OF SOUND WAVES BY THE MODAL
ELEMENT METHOD (NASA. Lewis
Research Center) 7 p
N94-37308
Unclas
Kenneth J. Baumeister
Lewis Research Center
Cleveland, Ohio
G3/7i 0017067
and
Kevin L. Kreider
The University of Akron
Akron, Ohio
Prepared for the
Winter Annual Meeting
sponsored by the American Society of Mechanical Engineers
Chicago, Illinois, November 16-18, 1994
National Aeronautics and
Space Administration
https://ntrs.nasa.gov/search.jsp?R=19940032800 2020-06-16T11:42:46+00:00Z

SCATTERING CROSS SECTION OF SOUND WAVES
BY THE MODAL ELEMENT METHOD
Kenneth J. Baumeister
National Aeronautics and Space Administration
Lewis Research Center
Cleveland, Ohio 44135
Kevin L. Kreider
The University of Akron
Department of Mathematical Sciences
Akron, Ohio 44325-4002
ABSTRACT
The modal element method has been employed to determine the
scattered field from a plane acoustic wave impinging on a two
dimensional body. In the modal element method, the scattering body
is represented by finite elements, which are coupled to an
eigenfunction expansion representing the acoustic pressure in the
infinite computational domain surrounding the body. The present
paper extends the previous work by developing the algorithm
necessary to calculate the acoustic scattering cross section by the
modal element method. The scattering cross section is the acoustical
equivalent to the radar cross section (RCS) in electromagnetic
theory. Since the scattering cross section is evaluated at infinite
distance from the body, an asymptotic approximation is used in
conjunction with the standard modal element method. For valida-
tion, the scattering cross section of the rigid circular cylinder is
computed for the frequency range 0.1 < ka < I00. Results show
excellent agreement with the analytic solution.
electromagnetics, the method has been applied to scattering from
dielectric cylinders (Chang & Mei (1976), Lee & Cendes (1987),
Baumeister & Kreider (1992)) and propagation in ducts (B aumeister
(1991 )). In acoustics, the method has also been applied to scattering
from cylinders (Baumeister & Kreider (1993)) and propagation in
ducts (Astley & Eversman (1981)). In all of these applications, an
eigenfunction expansion is used to represent the acoustic pressure
field in the far field. An asymptotic approximation of this expansion
presents a simple means of determining the scattering cross section.
This paper presents the numerical algorithm for evaluating the
acoustic scattering cross section by the modal element method. The
scattering cross section is the acoustical equivalent to the radar cross
section (RCS) in electromagnetic theory. Since the scattering cross
section is evaluated at infinite distance from the body, asymptotic
approximations are used in conjunction with the standard modal
element method. For validation, the method is applied to scattering
from rigid circular cylinders, for which the analytic solution is
known.
INTRODUCTION
The modal element method, which couples finite element algo-
rithms to eigenfunction expansions, has been employed in calculat-
ing the scattered field from an acoustical plane wave impinging on
a two dimensional body. The primary reasons for employing the
modal element method are (1) to accurately describe the radiation
boundary condition at the computational boundary and (2) to reduce
the size of the numerical grid. In fact, for hard scatterers, the modal
element method can effectively reduce a two dimensional scattering
problem to a one dimensional problem by employing a single line of
elements circumscribing the scattering body.
The modal element method has been given various titles, such as
the unimoment method and the transfinite element method. In
NOMENCLATURE
A* modal amplitude of wave moving radially outwards
a dimensionless circular cylinder radius
H c1_ Hankel function of the first kind
k wave number
m mode number
M f number of modal coefficients used in eigenfunction expansion
p dimensionlessperturbationacoustic pressure
r dimensionless radial coordinate
dimensionless property constant
O angle between radius vector and positive x axis
p. dimensionless property constant
oc acoustic scattering cross section
to dimensionless frequency
Superscript
* complex conjugate
METHOD OF ANALYSIS
The present study is concerned with computing the scattering cross
section of a two dimensional axisymmetrical rigid body due to an
impinging plane wave traveling in the +x direction. In order to
determine the acoustic scattering cross section, the scattered field must
be computed. To accomplish this, the spatial domain is divided into
two subdomains, the homogeneous domain and the finite element
domain, as shown in Fig. 1. The grid system in Fig. 1 is designed to
allow various structures to be imbedded into the grid, as shown by
inserts A and B. For the special case of rigid bodies, the grid system can
be reduced to a single ring of elements, which effectively reduces the
two dimensional problem to a one dimensional problem.
Imbedded structure
(insert B)
Homogeneous .___. _-- Imbedded
domain _ structure
nsert A)
• _-- Polar region
Interracial ..-___p"
boundary S -j" "___
/ t Transition region
/
Rectangular region --J
Figure 1.---Finite element grid system.
In the finite element domain, an approximate solution for the total
(incident + scattered) pressure at the element nodes is calculated by the
Galerkin method. Linear triangular elements are used and the subdomain
interface is approximated by piecewise linear segments. In the homo-
geneous domain, which extends to infinity, the pressure is represented
by an eigenfunction expansion. The modal element method couples
the two solution forms by imposing continuity on the pressure and
velocity at the interface between the two subdomains. This coupling
results in a single matrix equation in which the eigenfunction coeffi-
cients and the pressure at the finite element nodes are calculated
simultaneously, yielding a global representation of the acoustic pres-
sure field.
GOVERNING EQUATION AND BOUNDARY CONDITIONS
The scalar form of the acoustic wave equation can be written as
(Baumeister & Kreider (1993))
0x_.e 0x) 3y_.e 8y.J O02p'P = 0 (1)
The harmonic time dependence e-_ has been factored out. e and LEtare
material property constants and to is the dimensionless frequency. The
wave number is
k = _-kte (2)
In the homogeneous region surrounding the body, e= 1 and it= 1, while
inside the body, e and t.tmay assume other appropriate values depend-
ing on the material. Baumeister and Dahl (1989) present explicit
expressions for e and I.t as a function of medium density, porosity,
viscous loss coefficient and a heat transfer parameter.
At the interface between the finite element region and the analytic
region, continuity is imposed on the pressure and velocity. The
radiation boundary condition at infinity is automatically satisfied by
the eigenfunction expansion introduced in the next section.
ANALYTIC SOLUTION
In the homogeneous domain, an exact expansion for the pressure can
be derived (Morse and Ingard, 1968, p. 401) from eq. (1) by separation
of variables:
Mcoef -1
p=pi +pS =eikX + Eg+mn(ml)(kr)cos(m0). (3)
rn=0
p_ is the incident plane wave; p_ is the scattered wave. This expansion
is valid for symmetric bodies. The modal coefficients A ÷ are un-to
knowns to be determined. Formulas to estimate the number Mr of
modes needed for convergence can be found in Baumeister & Kreider
(1993).
2
FINITE ELEMENT SOLUTION
The solution to eq. (1) will be obtained by linear finite element
theory. In Fig. 1, the finite element region is divided into triangular
elements defined by corner nodal points. It is assumed that all material
properties are constant in each element. In the conventional weighted
residual approach, the unknown pressure field is described in terms of
all the unknown nodal values of the pressure.
The finite element aspects of converting eq. (1) and (3) and bound-
ary conditions into an appropriate set of global difference equations
can be found in Baumeister & Kreider (1993). The resulting set of
global difference equations is solved for the pressure at the nodes and
the modal coefficients A= +. With the modal coefficients determined, a
relatively simple algorithm has been developed to determine the
scattered cross section and is described in the next section.
BISTATIC ACOUSTIC SCATTERING CROSS SECTION
The bistatic acoustic scattering cross section per unit length for a two
dimensional body can be defined as
ipsl2 2=r{pi-pS*)
Oc(0)= lim 2xr_= lim .pi*)
r_o, pl r-_*_ /p I
(4)
The standard superscript c implies a two dimensional problem, where
the cross section is often called the "echoing width" (Ruck, vol. 1,
pp. 23). In three dimensions, the scattered field decays as I/r: while in
two dimensions, considered herein, the scattered field decays as l/r.
Thus, the 2nr is employed in eq. (4) to negate the 2D scattered field
decay and to yield a finite measure of the reflected power at infinitey.
The cross section defined in eq. (4) could be used in range equations
specifically defined for 2D bodies.
For a unit incident plane wave, eq. (4) simplifies to
_c(o)= lim 2rcr(p s .pS*) (5)
r--)oo
Using conventional numerical grids, it is often necessary to extrapo-
late the nodal values of the field on the outer boundary to infinity. For
example, Noack and Anderson (1992) include a boundary element
approximation to determine the scattering cross section at infinity.
Fortunately, no such additional work is required with the modal
element method. Since the scattering cross section is evaluated at
infinity, the asymptotic expansion forthe Hankel functions (Abramowitz
& Stegun, 9.2.3)
i(z-l_) _iron
H(mi)(z)_ 3_._zze L 4 ;e 2
(6)
can be inserted into eq. (3), which is then substituted into eq. (4)
to yield
/ _V"_"" i(kr-/Tt'_ Mc°ef-I -im -_ 1Gc(0) = lim2m" _xkr/z eL 4 J m'_ -0_;'A+em 2c°s(m0)
l _f-'ff'-- -i(kr-lrt) Mc_"_'I-I im--n 1.I _..._._e \ 4 J ___A+e 2cos(toO )
_kl" m=O
M -I . 7t M e -1 . 7t4 ( _ -,m-- "_( :oef , ,m- "]
=--} _)'A+m e 2cos(m0)|| ____A_e 2cos(m0)| (7)
k t m'_--0 )t m=0 )
Once the A= ÷have been calculated, the scattering cross section and the
pressure field can be easily determined.
RESULTS AND COMPARISONS
Consider the following scattering problem: a unit plane wave,
incident from the left, strikes a rigid circular cylinder of dimensionless
radius a = l oriented with its axis normal to the propagation direction.
The rigid body is simulated numerically through an impedance mis-
match induced by setting e = 1- 10 '9 i and I1 = 1 for each internal finite
element and E = 1 and _t = 1 in the homogeneous domain surrounding
the body. This feature allows flexibility in the numerical implementa-
tion of the method. Rigid bodies with absorber coatings may be studied
with only slight modification to the computer code by adding several
finite element rings with the appropriate values of e and It.
This problem is solved for frequencies ranging from ka = 0.1 to
ka = 100 in order to validate the method. This problem was chosen
because it has an exact solution, given by eq. (3) with modal coeffi-
cients (Bowman et al., eq. (2.38))
-(2 - _Sm0)imJ'm(ka) (8)
A+ = H(m1)' (ka)
and because it has been examined using other methods (Ling, 1986,
Fig. 15).
In Fig. 2, the computed cross sections (square symbols) are com-
pared to the corresponding exact solutions (solid line). The results
agree very well with the exact solutions over a wide frequency range.
Results for ka = 50 and ka = I00 show similar agreement but are
omitted for brevity. In addition, the results agree well with the low
frequency calculations (k = 3.2) of Ling (1986, Fig. 15).
Additional validation of the algorithm is obtained by examining
high frequency (ka > 20) asymptotic approximations. In the
backscattered direction (0 = 180), the optical approximation is valid
(Ruck, et al. (1970), eq. (4.1) to (38)) and the backscattering cross
section becomes
o c = xa = _ (a = 1) (9)
c-
o
¢J
o
2
O
¢J
i5
5 B
Theory
Modal element analysis
10--
(b) k = 3.2
i
8
_ .o 6
C3
o 4
nn
2
I I 1 I I I I I I
45 90 135 180 225 270 315 360 0
e
45 90 135 180 225 270 315 360
e
20 _'c) k = 5.0
/
15T-- o
_ 10
2
0
5
0
0
tO
2
0
100 --
(d) k = 25.0
80
60
4O
2O
0 45 90 135 180 225 270 315 360 0
e
Figure 2._Bistatic scattering cross section.
45 90 135 180 225 270 315 360
e
In the forward scattered direction (O = 0), commonly referred to as the
diffracted field, the optical approximation yields (Ruck, et al. (1970),
eq. (4.1) to (41)) the forward cross section
(_c = 4ka 2 = 4k (a = 1) (10)
Figure 3 shows that the asymptotic limits are attained as frequency
increases in both the forward and backscattered directions. Table I
contains the numerical values obtained for various frequencies in these
directions. Even at ka = 100, the relative error in the forward direction
is only about 3 percent.
Some error at high frequencies can be attributed to the approxima-
tion involved with the asymptotic expansion given by eq. (7). How-
ever, the major source of error is in the calculated values of the mode
coefficients Am*in eq. (3). For example, in the k = 100 calculation, over
130 modal coefficients need to be determined. Consequently, some
significant errors are involved in the calculation of the higher order
modes.
1 000 F ----o---- O=O
/ _ 0=180
100 / - - -- 4k limit
"_ 0.1y_ .ol
.OOl I I I
0.1 1 10
ka
Figure 3.---Scattering cross section as a function of ka.
IO0
TABLE I.---COMPARISON OF EXACT AND
0.1
1.0
3.2
5.0
25
50
100
MODAL ELEMENT (ME) BISTATIC
CROSS SECTIONS
Forward (0 = 0)
Exact ME
0.00261 0.00263
1.643 1.659
8.459 8.470
14.77 15.22
90.31 91.00
187.6 189.3
378.7 369.7
Backward (0 = 180)
Exact ME
0.0022 0.00219
3.427 3.437
3.412 3.413
2.795 2.778
3.151 3.153
3.140 3.140
3.106 2.957
CONCLUDING REMARKS
In the modal element method, the scattering body is imbedded in a
finite element grid to calculate the acoustic pressure field. In the far
field, the pressure is represented analytically by a Hankel function
expansion. The asymptotic approximation of the Hankel functions are
used to calculate the scattering cross section. The method is applicable
to problems involving very high or low frequency scattering from
bodies. In the validation examples, the numerical results are in good
agreement with the corresponding exact solutions and asymptotic
analytical limits.
REFERENCES
Abramowitz, M. and Stegun, I.A., 1964, Handbook of Mathematical
Functions With Formulas, Graphs, and Mathematical Tables, Na-
tional Bureau of Standards Applied Mathematics.
Astley, R.J. and Eversman, W., 1981, "Acoustic Transmission in Non-
Uniform Ducts with Mean Flow, Part II: The Finite Element
Method," Journal of Sound and Vibration, Vol. 74, pp. 103-121.
Baumeister, K.J., 1991, "Effect of Surface Deposits on Electromag-
netic Propagation in Uniform Ducts," IEEE Trans. on Magnetics,
Vol. 27, No. 5.
Baumeister, K.J. and Dahl, M.D., "Acoustic Wave Propagation in
Heterogeneous Structures Including Experimental Validation,"
AIAA 89-1044.
Baumeister, K.J. and Kreider, K.L., 1992,"Modal Ring Method for the
Scattering of Electromagnetic Waves," NASA TM-105966.
Baumeister, K.J. and Kreider, K.L., 1993,"Modal Element Method for
Scattering of Sound by Absorbing Bodies," ASME Journal of
Vibration and Acoustics, Vol. ! 15, No. 3, pp. 314-323.
Bowman, J.J., Senior, T.B.A., and Uslenghi, P.L.E., 1969, "Electro-
magnetic and Acoustic Scattering by Simple Shapes," North-Hol-
land, Amsterdam.
Chang, S.K. and Mei, K.K., 1976, "Application of the Unimoment
Method to Electromagnetic Scattering of Dielectric Cylinders,"
IEEE Trans. on AP, pp. 35--42.
Lee, Jin-Fa and Cendes, Z.J., 1987, '"The Transfinite Element Method
for Computing Electromagnetic Scattering From Arbitrary Lossy
Cylinders," AP-S International Symposium, paper AP03-5,
Blacksburg, Virginia.
Ling, R.T., 1986, "Numerical Solution For the Scattering of Sound
Waves by a Circular Cylinder," AIAA 86-1880.
Morse, P.M. and Ingard, K.U., 1968,'_'heoretical Acoustics," McGraw-
Hill, New York.
Noack, R.W. and Anderson, D.A., 1992, '_rime Domain Solutions of
Maxwell's Equations Using a Finite-Volume Formulation," AIAA
Paper 92--045 I.
Ruck, G.T., Barrick, D.E., Smart, W D. and Krichbaum, C.K., 1970,
"Radar CrossSection Handbook," Volume 1 and 2, Plenum Press.
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July 1994 Technical Memorandum
4. TITLE AND SUBTITLE
Scattering Cross Section of Sound Waves by the Modal Element Method
6. AUTHOR(S)
Kenneth J. Baumeister and Kevin L. Kreider
7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)
National Aeronautics and Space Administration
Lewis Research Center
Cleveland, Ohio 44135-3191
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National Aeronautics and Space Administration
Washington, D.C. 20546-0001
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REPORT NUMBER
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NASA TM- 106667
11. SUPPLEMENTARY NOTES
Prepared for the Winter Annual Meeting sponsored by tim Arnedcan Society of Mechanical Engineers, Chicago, Illinois, November 16-18, 1994.
Kenneth J. Baumeister, NASA Lewis Research Center, and Kevin L. Kreider, University of Akron, Department of Mathematical Sciences, Akron, Ohio
44325--4002. Responsible person, Kenneth J. Baumeister, organization code 2660, (216) 433-5886.
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13. ABSTRACT (Maximum 200 words)
The modal element method has been employed to determine the scattered field from a plane acoustic wave impinging on a
two dimensional body. In the model element method, the scattering body is represented by finite dements, which are
coupled to an eigenfunction expansion representing the acoustic pressure in the infinite computational domain surrounding
the body. The present paper extends the previous work by developing the algorithm necessary to calculate the acoustics
scattering cross section by the modal element method. The scattering cross section is the acoustical equivalent to the radar
cross section (RCS) in electromagnetic theory. Since the scattering cross section is evaluated at infinite distance from the
body, an asymptotic approximation is used in conjunction with the standard modal element method. For validation, the
scattering cross section of the rigid circular cylinder is computed for the frequency range 0.1 < ka < 100. Results show
excellent agreement with the analytic solution.
14. SUBJECT TERMS
Acoustic; Scattering; Cross section; Modal element
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