Comparison of an Analytical and Numerical ... - COMSOL.com

Excerpt from the Proceedings **of** the **COMSOL** Conference 2008 Boston

**Comparison** **of** **an** **Analytical** **an**d **Numerical** Solution for the L**an**dmine

Detection Problem

Michelle B. Mattingly 1 , Kevin L. McIlh**an**y, *,2 **an**d Reza Malek-Mad**an**i **,3

1 Applied Research Laboratory, The Pennsylv**an**ia State University, University Park, PA, 2 Physics

Department, United States Naval Academy, Annapolis, MD, 3 Mathematics Department, United States

Naval Academy, Annapolis, MD.

*mcilh**an**y@usna.edu, **research@usna.edu

Abstract: Acoustic l**an**dmine detection is

ac**com**plished using a loud speaker as **an** airborne

source to generate low-frequency waves that

enter the soil at a certain incident **an**gle. At a

specific frequency, the l**an**dmine will “vibrate” at

reson**an**ce, imparting a certain velocity on the

soil particles above it that is detected by a

sc**an**ning Laser Doppler Vibrometer system. The

ability to mathematically predict the soil surface

velocity plots created from these experiments

would enable the technology to be implemented

faster in more challenging environments. An

**an**alytical solution 1 was determined **an**d has been

**com**pared to experimental results. 2 However, the

**an**alytical problem dem**an**ds signific**an**t time **an**d

**com**putational resources. A problem identical to

the **an**alytical solution was implemented with

**COMSOL** TM , **an**d has signific**an**tly reduced the

**com**putational time **an**d resources necessary to

find a solution while remaining accurate to the

**an**alytical result.

Keywords: Acoustic, **Analytical**, L**an**dmine

1. Introduction

The **an**alytical mathematical model 1 **of** the

buried l**an**dmine detection problem involved

solving two Helmholtz equations in a 2-layer

waveguide, subject to boundary conditions

appropriate for mine detection (Fig. 1). In the

atmospheric layer, a point source (delta function)

represented the loudspeaker. The soil was

modeled as a finite porous layer. The top plate

**of** the buried l**an**dmine was represented as a

circular elastic membr**an**e stretched flush over a

cylindrical cavity in a rigid substrate beneath a

porous layer. Mathematically, the homogeneous

Helmholtz equation was used in the porous layer,

with a Green‟s function representation **of** the

membr**an**e response. In the **an**alytical soil

reson**an**ce predictions 1 , pressure was plotted as a

function **of** frequency, **an**d the reson**an**ces appear

as local maximums **an**d minimums.

In **com**paring the experimental 2 **an**d

**an**alytical 1 results, only the frequency at which

the reson**an**ces occur is **of** import**an**ce, since this

Figure 1. A schematic **of** the mathematical model developed to represent the l**an**dmine detection problem.

is what must be predicted mathematically to aid

the experimenter in the field. Similarly, in

**com**paring the **COMSOL** TM model developed in

this paper to the **an**alytical model, the reson**an**t

frequencies are **of** greater import**an**ce th**an** the

predicted amplitude **of** the response. Several

**COMSOL** TM models have been run for

**com**parison with the **an**alytical models that were

previously developed. 1

2. Theory

Derivation **of** the three-dimensional wave

equation in **an** ideal gas was performed for the

**an**alytical **com**putation, 3,4 based on conservation

**of** mass **an**d conservation **of** momentum

arguments. Assuming time-harmonic motion,

the wave equation be**com**es

2 2

P k P 0,

a Helmholtz equation, with k , where is

c

the **an**gular frequency **an**d c is the sound speed

in the medium.

Morse **an**d Ingard 3 argue that modifications

are necessary for sound tr**an**smission in a porous

media. Thus, the wavenumber ( k ), in the soil

(subscripted „s‟) be**com**es

k

2

i

p

s

,

2

cs

where is the flow resistivity **an**d is the

p

effective density in the soil.

Finally, the delta „function‟ is customarily

used to represent a point source radiating

spherically. A pl**an**e wave representation was

also considered but was ultimately rejected due

to the close proximity **of** the source, target, **an**d

area **of** interest.

3. Governing Equations **an**d Methods

The Helmholtz equation described in the

previous section was converted into cylindrical

coordinates for the **an**alytical **an**alysis 1 **an**d

separation **of** variables was applied to the partial

differential equations. Note that the area outside

the cylinder in Fig. 1 retains the mathematics

from a two layer waveguide. 1,5,6,7 The Green‟s

function was derived for a membr**an**e,

representing the top plate **of** the l**an**dmine.

Continuity conditions were imposed between the

cylinder containing the mine **an**d the waveguide

to find the **an**alytical solution to the problem.

3.1 Eigenvalues

A ch**an**ge **of** variables was performed to force

the eigenvalues (ζ mn **an**d τ mn ) to occur at

predictable intervals for ease **of** **com**putation. In

order to perform these **com**putations, parameters

from D. Velea, R. Waxler, **an**d J. Sabatier were

used. 2 The equation defining the eigenvalues is

2

s

1

2

T k J ka

s

mn

2

za

a

k

sin

J

m

cos

s mn

2

mn

cosz

s

k

s

mn

aI

J ka

mn

s

mn

cos

za

a

2

k

s

1

mn

for m 0,1,2 ,..., where

I

I

1

2

a

0

a

0

rJ

mn

rJ

a

m

a

2

m

m

m

2

mn

sinz

s

k

s

mn

I

2

I

2

mn

cosz

s

k

s

mn

2

mn

sin

mn

sinz

s

k

s

mn

krJ

r

k

mn

2

m

mn

mn

J

m

kaJ

m 1 a

mn

kJm

1kaJ

m

mn

a

2 2

a J

m

mn

a

dr

2

2

J

m1

mn

a

aJ

a

r

mJ

.

m1

mn

mn

dr

(See appendix for variable definitions)

3.2 **Analytical** Solution

Let D r

m

2

mn

0

, : r a be the area occupied by

a membr**an**e with a point source at

r , , z a r with **an** atmosphere

0 0 0

,

0

satisfying a pressure release condition at z .

z a

Assume that total pressure is the sum **of** incident

**an**d scattered pressure. The incident pressure in

air-soil waveguide in the absence **of** a membr**an**e

is

i 1 pi

r,

,

z H

m

n

r0

2

p

J

Z

i

m

n

m0 n1

rcos

m

n

0

z

0

z0

Z

n

z,

r r0

i 1 r,

,

z H r

J

m

m0 n1

r cos

m

n 0

0

z

0

z0

Z

n

z,

r r0

m

2

Z

n

where

2

m 0

.

m

1

m 0

Using the eigenvalues **an**d radial continuity

conditions, the final velocity was found to be

1

w

z

p

i

1

i

a

a

m0 n1

J

m

m

n

n

rA

cosm

' z

mn

for r a , where

mn

z sin

a

z

za

' z cos z z .

mn

**an**d

mn

The coefficients,

a

A

mn

a

a

m

n

mn

, are **com**puted from **an**

infinite set **of** algebraic equations that must be

solved numerically from

(1)

lm

J

m1

lm

a H

m

n

a

I

A

(1)

mnl mn

l1

n

J

m

lm

aH

m1

n

a

i

n

2

m

J

m

J

where

1

m

1 z

Z

z

H

r

0

n

1

n

a H

m

n

a

1

n

a H

m1

n

a

0

m

n

0

,

,

I

mnl

0

zs

za

0

z

za

zs

a

s

z

zZ

z

mn,

s

mn,

a

s

a

mn

z

z

Z

Z

n,

s

n,

a

n

z

z

dz

dz

dz

0 z z

a

,

**an**d

Z

Z

n,

a

n,

s

z

s

z 0

2

z B cos k

s

n

z

s

2

sin k

a

n

za

z,

0 z za

2

z B sin k

a

n

za

2

cos k

s

n

z

z

s

,

z

s

z 0

4. **Numerical** Model

The numerical model was created in

**COMSOL** TM using the pressure subsection **of** the

acoustics module. A twenty meter long **an**d ten

meter high rect**an**gle was drawn to represent the

atmosphere, **an**d a twenty meter long **an**d 0.075

meter deep rect**an**gle was drawn to represent the

soil (Fig. 2). A point source was impl**an**ted in

the atmospheric layer, at (0.2, 2), corresponding

to the **an**alytical problem. A “mine” was placed

beneath the soil layer, drawn as a 0.03 by 0.07

rect**an**gle placed at (0, 0.105).

The subdomain settings **of** density, , **an**d

sound speed through the medium, c , were

consistent with the **an**alytical solution, with the

exception **of** the membr**an**e sound speed. This

was approximated from other plastics. 8 The

expression for the wavenumber in the membr**an**e

is **an**alytically given by

2

m

i

m

k

,

T

where

m

from Fig. 1. The boundary

settings were all set to continuity for the internal

boundaries, **an**d “Sound Hard Boundary” for the

exterior boundaries. This was consistent with

the **an**alytical problem. The point at (0.2, 2) was

set as a “Power Source” using the point settings,

with a strength **of** 1 W.

,

.

Figure 2. The **COMSOL** representation **of** the **an**alytical problem represented in Fig. 1. The base model

is exactly as shown in the figure. The extended model increases the depth **of** the soil layer from 0.075 m

to 10 m, extending the soil layer while the mine remains in the same position.

5. Experimental Results

Preliminary results focusing on small

frequency intervals around **an**alytically predicted

reson**an**ces showed expected behavior.

However, once the frequency domain was

exp**an**ded to the full r**an**ge **of** 80 Hz to 300 Hz,

these apparent “reson**an**ces” disappeared,

indicating that they were small fluctuations in

the pressure.

The amplitude **of** the response is signific**an**tly

affected by the power **of** the source. Although

the delta function was used **an**alytically, a point

source with unit power was applied in the

numerical model. This causes variation in the

amplitude **of** the response, but does not affect the

reson**an**t frequencies.

To improve the solution for **com**parison,

certain ch**an**ges were implemented. Damping

was added in the soil **an**d mine subdomains to

increase the frequency sp**an** **of** the reson**an**ces.

Since the **an**alytical model incorporated damping

in these two layers using **com**plex wavenumbers,

this implementation was valid, although little

damping was used to produce the **an**alytical

result. Values **an**d the types **of** damping were

also varied in attempt to reproduce the **an**alytical

solution **an**d ascertain the effects **of** the specific

values on the problem.

Ch**an**ges were also made in the processing

**an**d postprocessing **of** the solution for better

**com**parison. Rather th**an** plotting the pressure,

the velocity was plotted to allow direct

**com**parison with the **an**alytical solution. While

**COMSOL** TM Script could have been used to plot

the parameterized frequency values, the point

selection in the cross sectional plot parameters

menu was utilized to create the plots quickly **an**d

accurately with respect to the point **of** interest.

6. Discussion

**Comparison** **of** the **an**alytical solution (Fig.

3) to the base model (Fig. 4) showed good

agreement on the soil-membr**an**e reson**an**ce that

occurred at f 280 Hz. Velea, Sabatier, **an**d

Figure 3. **Analytical** solution **of** the l**an**dmine

detection problem for the Sabatier 2 parameters.

Waxler 2 note that this is the first acoustical

reson**an**ce, defined as the frequency at which

“the layer **of** soil between the top **of** the mine **an**d

the surface **of** the soil vibrates at maximum

amplitude.” 2 The **COMSOL** TM model also

remained stable at this point, whereas the

**an**alytical model was difficult to keep stable

around this point.

The mech**an**ical reson**an**ces (defined as “the

reson**an**t frequencies **of** **an**y mech**an**ical system

for which the input mech**an**ical react**an**ce goes to

zero” 2 ) did not appear in the **COMSOL** TM model.

Since these reson**an**ces were quite narrow in the

**an**alytical model ( f 100 Hz **an**d f 225 Hz),

it is possible that these reson**an**ces were too

narrow to detect, even with large damping **an**d

small sampling (0.25 Hz).

The wide reson**an**ce at f 240 Hz was

determined to be a soil reson**an**ce, 1 which was

Figure 5. **COMSOL** solution for the extended

l**an**dmine detection problem.

Figure 4. **COMSOL** solution for the l**an**dmine

detection problem for the Sabatier 2 parameters.

captured in the **COMSOL** TM model as well. The

extended model with the same **an**alytic

properties (Fig. 5) also demonstrated strong

reson**an**ces at f 240 Hz **an**d f 280 Hz.

However, **an**other strong reson**an**ce is

observed at f 175Hz, which is not seen in the

base model. This apparent reson**an**ce was

consistent without damping, or with other types

**of** damping. Mathematically, the extended

problem is very difficult to solve **an**d was not

approached **an**alytically. It is possible that this

reson**an**ce is from the column **of** soil beneath the

mine, or **an** interaction between the sides **of** the

mine **an**d the soil. However, the presence **of** the

two expected reson**an**ces reinforces a certain

confidence in the solution.

In Fig. 6, the predicted effects **of** the

l**an**dmine size are considered. In the **an**alytical

solution, the reson**an**t frequency decreases **an**d

the response amplitude increases with **an**

increase in l**an**dmine size. In the pr**of**ile (a

signific**an**t increase in the mine size), the large

reson**an**ce appears to shift to f 220 Hz. This

shift **of** approximately 60 Hz is not proportional

to the **an**alytical shift; however, the **an**alytical

data does not extend to a 0. 40 m, **an**d shows

indications **of** non-linear behavior.

The predicted effects **of** the mine depth are

shown in Fig. 7. **Analytical** predictions indicate

that increasing soil depth increases the reson**an**t

frequency. There were no conclusive predictions

about the amplitude from the soil depth. A

frequency shift has appeared to occur, but a

second reson**an**ce has also appeared that c**an**not

be attributed to the soil. **Analytical**ly, soil

reson**an**ces are predicted at f 140,230, 320 Hz,

Figure 6. Soil surface pr**of**ile after increasing the

l**an**dmine diameter to 0.04 meters.

which do not account for the extra reson**an**ces at

that depth. It is possible that the additional depth

created **an**other acoustical reson**an**ce.

7. Conclusions

In the field **of** physical mine detection,

information speed **an**d clarity is very import**an**t.

The **an**alytical solution **of** the l**an**dmine detection

problem in Fig. 1 took approximately 6-7 months

to derive, underst**an**d, code, **an**d **an**alyze

solutions. **COMSOL** TM reduced this time to

approximately **an** hour per solution, once the

model had been developed.

The solutions produced by **COMSOL** TM were

**com**parable to the **an**alytical solution. Strong

acoustical l**an**dmine reson**an**ces were predicted

by the **an**alytical 1 **an**d **COMSOL** TM models,

which matched the experimental data. 2 Although

there were no mech**an**ical reson**an**ces predicted

by the **COMSOL** TM models, the matching

acoustic **an**d soil reson**an**ces are quite signific**an**t

across the experimental, **an**alytical, **an**d

numerical implementations.

The solutions for **com**parison **of** the l**an**dmine

size **an**d depth produced some interesting results

that will require more **an**alysis **of** the **an**alytical

solution. Due to the **com**putational time, the

**an**alytical **com**parison models were only run for

small frequency r**an**ges. The more detailed

**COMSOL** TM models have indicated more

**com**plicated behaviors th**an** the **an**alytical models

that need to be investigated further.

Overall, the **COMSOL** TM models developed

in this **an**alysis **an**d **com**parison have confirmed

the **an**alytical solution, while raising interesting

questions to further research in this area. The

**com**putational processing time **of** the **an**alytical

Figure 7. Soil surface pr**of**ile after increasing the

soil layer depth to 1 meter.

model restricted the frequencies **of** consideration

in the model **an**alysis, which prevented a full

frequency **an**alysis with regard to the mine

properties. The **COMSOL** TM models have raised

these import**an**t questions, which will help

further research in the field **of** acoustic l**an**dmine

detection.

8. References

1. M. Mattingly, J. Buch**an****an**, M. Korm**an**, **an**d

R. Malek-Mad**an**i, A Mathematical Model for

the Acoustic **an**d Seismic Properties **of** the

L**an**dmine Detection Problem, Trident Scholar

Project Report, USNA 1531-2, report no. 371

(2008)

2. D. Velea, R. Waxler, **an**d J. Sabatier, An

effective fluid model for l**an**dmine detection

using acoustic to seismic coupling, J. Acoust.

Soc. Am., 115(5), 1993-2002 (2004)

3. P. Morse **an**d K. Ingard, Theoretical

Acoustics, 242-253. Princeton University Press,

Princeton, New Jersey (1968)

4. C. Zwikker **an**d C. Kosten, Sound Absorbing

Materials, 18. Elsevier Publishing Corp., New

York (1949)

5. C. Frisk, Oce**an** **an**d Seabed Acoustics.

Prentice Hall, Englewood Cliffs, New Jersey

(1994)

6. C. Boyles, Acoustic Waveguides. John Wiley

**an**d Sons, New York (1984)

7. R. Hackm**an**, Acoustical scattering in **an**

inhomogeneous waveguide, J. Acoust. Soc. Am.,

80, 1447-1458 (1986)

8. S. H**an**lein, W. Hinckley, F. Stecher,

**Comparison** **of** Mech**an**ical **an**d Acoustic

Properties for selected Nonferrous, Ferrous, **an**d

Plastic Materials, National Technical

Information Service, 70-141, 1-113 (1970)

9. Acknowledgements

Special recognition for their attention **an**d

assist**an**ce in **com**pleting this research should be

given to Dr. David Burnett, Dr. James Buch**an****an**,

Dr. Murray S. Korm**an**, Dr. David Bradley,

Michael Zucker, **an**d Timothy Marston. The

work **of** the authors was partially supported by

the Office **of** Naval Research under Gr**an**t

N0001408WR40063.

10. Appendix

Table 2: Properties **of** **COMSOL** Graphics

Figure Notes

Number

4 Damping in the soil - R 1000

Damping in the mine - 100

5 Damping in the soil - R 1000

Damping in the mine - 100

6 a 0. 40 m, damping is consistent

with Fig. 4 **an**d 5

7 z 1m, damping is consistent

s

with Fig. 4 **an**d 5

f

f

Table 1: Definition **of** Variables (In order **of**

appear**an**ce)

Variable Definition

P

k

c

p

mn

mn

, n

T

a

z

a

z

s

r , z

Pressure (Pa)

Wavenumber (1/m), may be

subscripted with „a‟,

atmospheric, or „s‟ soil

(unsubscripted - membr**an**e)

Angular Frequency (Hz)

Sound Speed (m/s), may be

subscripted with „a‟,

atmospheric, or „s‟ soil

Flow Resistivity

Effective Density

, Eigenvalues

Density (kg/m 3 ), may be

subscripted with „a‟,

atmospheric, „m‟ membr**an**e, or

„s‟ soil

Membr**an**e Tension

Diameter **of** the L**an**dmine

Height **of** the atmospheric layer

(m)

Depth **of** soil layer (m)

, Position **of** interest (m)

0 0,

0

r , z Source location (m)

B

m

Arbitrary coefficient

Coefficient from the time

derivative in the differential

equation for the motion **of** a

membr**an**e exposed to pressure