Inverse Characterization of Poro-Elastic Materials Based ...

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Inverse Characterization of Poro-Elastic MaterialsBased on Acoustical Input DataJ Stuart BoltonPurdue University, [email protected]

Kwanwoo Hong

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Inverse Characterization of Poro Elastic Materials based on

Acoustical Input Data

J. Stuart Bolton and Kwanwoo HongRay W. Herrick Laboratories

School of Mechanical EngineeringPurdue University

ASA San Antonio – 29 October 2009ASA San Antonio – 29 October 2009

1

Introduction

Brief history of standing wave tube Brief history of standing wave tube

Four microphone standing wave tube Four-microphone standing wave tube

E ti ti f Bi t t b d ti l Estimation of Biot parameters based on acoustical measurements

2

3

4

Standing Wave Tubeg

Standing wave method for measuring normal Standing wave method for measuring normal incidence absorption coefficients more than 100 years oldy

Method is credited by a number of authors to J Method is credited by a number of authors to J. Tuma (1902)

Subsequent experiments conducted by Weisbach (1910) and Taylor (1913)

5

( ) y ( )

6PHYSICAL REVIEW, 2(4): 270-287 OCT 1913

7J. Acoust. Soc. Am. 19, 420-427, May 1947

8

9

10J. Acoust. Soc. Am. 90(4), 2182-2191, Oct 1991

Four Microphone Methodp

Four-microphone tube for silencer testing Munjal (Duct Acoustics) Two-load method Two-source method

Four-microphone tube for material testing Four microphone tube for material testing Suggested by Joseph Pope Yun and Bolton (1997 SAE) Song and Bolton (2000 JASA) introduce transfer matrix approach Song and Bolton (2000 JASA) introduce transfer matrix approach Many articles since then

11

Transfer Matrix Method

Mic 1 Mic 4Mic 3Mic 2 tjjkxjkx eDeCeP )( 333 tjjkxjkx eBeAeP )( 11

1

AB

CD

Speaker d

tjjkxjkx eDeCeP )( 444

)(sin2)(

21

2112

xxkePePjA

jkxjkx

)(sin2)(

43

4334

xxkePePjC

jkxjkx

tjjkxjkx eBeAeP )( 222

Anechoic x3x2

x4x1

x

)(s 21 xxk

)(sin2)(

21

1221

xxkePePjB

jkxjkx

3

)(sin2)(

43

3443

xxkePePjD

jkxjkx

0011

xxdxdx

VPVPVPVP

T22

012

dxx

VPVPPP

T

termination (not required)

Sound pressuredxx V

PTTTT

VP

2221

1211

0

00

xdxdxxVPVP 00

xdxdxx

VPVP

00

22

021

xdxdxx

dxx

VPVPVV

T00

0022

xdxdxx

xxdxdx

VPVPVPVP

T

2eTjkd

T/1l20TL h

Sound pressureand velocity relationship

Symmetric sample T11=T22, T11T22-T12T21=1

Transmission loss

111cos1 T

dk p

2221001211 )/(2

TcTcTTeT

dkcdkj

dkcjdkTTTT pppp

cos/sinsincos1211

1 T

T/1log20TL 10 where Transmission loss

Transfer matrix

12

dkcdkjTT pppp cos/sin2221

21

121TT

cpp

Limp or rigid porous material

Property of material

Anechoic Transmission LossAviation grade glass fiber (density=9.6 kg/m3, flow resistivity= 31000 Rayls/m)

30

35

40

ExperimentPrediction using FEM (with edge constraint)Prediction without edge constraint

2.9 cm tube

20

25

30

L (d

B)

10

15

TL

Increase in TLdue to

102

103

104

0

5

Frequency (Hz)

due to edge constraint Shearing Resonance

13

q y ( )

Above shearing resonance – finite size sample represents infinite sample

Below shearing resonance – all properties affected by edge-constraint

Estimation of Biot Parameters

Software available to estimate Biot parameters by performing optimal fit to measured acoustical data (flow

i ti it it t t it i h t i ti l th th lresistivity, porosity, tortuosity, viscous characteristic length, thermal characteristic length, bulk density, Young’s modulus, loss factor, Poisson ratio) ESI-FOAM-X (rigid, limp)

COMET/T i ( i id li l ti ) COMET/Trim (rigid, limp, elastic)

Original software based on transversely infinite layered Original software based on transversely infinite layered representation: i.e., edge constraint effects are not included

14

Infinite Panel Model: COMET/TRIM

1515

Infinite Panel Model: Limitation

101

6

8

10

ss [d

B]

0 6

0.8

1

ficie

nt

4

6

Tran

smis

sion

Los

0.4

0.6

Abs

orpt

ion

Coe

ff

1000 2000 3000 4000 5000 60000

2

Frequency [Hz]

ExperimentTrim

1000 2000 3000 4000 5000 60000

0.2

Frequency [Hz]

ExperimentTrim

Note that this model does not simulate the low frequencytransmission loss fluctuation caused by shearing

1616

transmission loss fluctuation caused by shearing resonance of the sample

Finite Element Models: COMET/SAFE

The software COMET/SAFE is used to model and compute the absorption and transmission loss having a finite depth and finite size layer of porous material.A fi it l t b d th t ll f th A finite element based program that allows for the analysis of sound traveling through various media including fluids, solids and foam-like substances.

Finite element implementation is based on u-U and p-Uversions of Biot theory.All d l d i thi k i l d i t i All models used in this work involved axisymmetric elements.

The new version of TRIM supports automated inverse

1717

The new version of TRIM supports automated inverse characterization capability based on SAFE.

Finite Element Model

Note that finite model can simulate the low frequencytransmission loss fluctuation caused by shearing

1818

transmission loss fluctuation caused by shearing resonance of the sample

Inverse Characterization

Questions: Is it possible to determine the Biot parameters from acoustical measurements? Do parameters act independently? How many parametersparameters act independently? How many parameters can be estimated?

To help answer these questions, introduce a procedure b d Si l V l D itibased on Singular Value Decomposition

Singular Value Decomposition is widely used linear algebraic method to identify the principal components inalgebraic method to identify the principal components in the field of image processing and signal processing.

1919

Sensitivity Matrix AnalysisProcedures

1 Li i b i d i i ffi i l

Procedures

1. Linearize absorption and transmission coefficient close to a certain parameter set

2. Use absorption and/or transmission coefficient values for certain number of frequencies to construct a sensitivity matrixsensitivity matrix

3. Perform singular value decomposition on the sensitivity t i d t t i l l t d t i ff timatrix and extract singular values to determine effective

rank (number of independent parameters)

20204. Calculate condition number (the smaller the better)

Sensitivity Matrix Analysis

Linearize the expression for the absorption and

y y

p ptransmission coefficient in the vicinity of a certain parameter setFor 1 frequencyq y

9

10 )()()(

11i

ii

xff dx

xxx i

RealSolution

Calculate by using “central difference scheme”±1 % difference of material properties

ix

2

ix

Solution

ApproximateSolution

2121ix

ix2


For n frequencies the equation can be combined as a matrix

y y

For n frequencies, the equation can be combined as a matrix

19210

...)()(912111

11dxxxxxx

xfxfxf

ff

9

921

921

0

...

...

............)(

...)(

...921

11

dxxxx

xx xfxfxfff

nnnnn

Sensitivity MatrixPerform singular value decomposition: g pM=UΣV*

The rank of the matrix M equals the number of non-zero

2222

singular values which is the same as the number of nonzero elements in the matrix Σ.

Rigid FoamS iti it M t i A l i

Use COMET/TRIM rigid foam type material that has 5


material properties. E.g., Porosity, flow resistivity, tortuosity, viscous and thermal characteristic length.

The nominal values of the material properties are The nominal values of the material properties are

0.8

1

ient

10

15

dB]

0.2

0.4

0.6

Abs

orpt

ion

Coe

ffici

5

10

Tran

smis

sion

Los

s [d

Porosity Flow Tortuosity VCL TCL

1000 2000 3000 4000 5000 60000

0.2

Frequency [Hz]1000 2000 3000 4000 5000 6000

0

Frequency [Hz]

2323

Porosity FlowResistivity

Tortuosity VCL TCL

0.98 50,000 2.0 3.0*10-5 9.0*10-5

Rigid FoamS iti it M t i A l i

Effect of adding frequency data for absorption coefficient


100

Effect of adding frequency

g q y p

104


3

10-2

10-1

Sin

gula

r Val

ue

10

Con

ditio

n N

umbe

r5 10 15 20 25 30 35 40 45 50 55 60

10-4

10-3

1st SV2nd SV3rd SV4th SV5th SV 1163

5 10 15 20 25 30 35 40 45 50 55 60103

C5 10 15 20 25 30 35 40 45 50 55 60

Number of Frequency

Adding additional frequency data reduces the diti b b t th diti b i t bi

5 10 15 20 25 30 35 40 45 50 55 60Number of Frequency

2424

condition number, but the condition number is too big to consider that the sensitivity matrix is well-posed.

Rigid FoamSensitivity Matrix Analysis

Effect of adding frequency data for transmission coefficient


-1

100


g q y

4


10-3

10-2

10 1

Sin

gula

r Val

ue

103

104

Con

ditio

n N

umbe

r5 10 15 20 25 30 35 40 45 50 55 60

10-4

10

1st SV2nd SV3rd SV4th SV5th SV

581

5 10 15 20 25 30 35 40 45 50 55 60102

10C5 10 15 20 25 30 35 40 45 50 55 60

Number of Frequency

Adding additional frequency data reduces the diti b b t th diti b i t bi


2525

condition number, but the condition number is too big to consider that the sensitivity matrix is well-posed.


Sensitivity matrix


Sensitivity matrix

0 2

0.3

0.4a

0.05

T

-0.1

0

0.1

0.2

Diff

eren

ce

-0.1

-0.05

0

Diff

eren

ce500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000

-0.4

-0.3

-0.2

0.1

PorosityFlow resistivityTortuosityViscous CLThermal CL

500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000-0.25

-0.2

-0.15PorosityFlow resistivityTortuosityViscous CLThermal CL

Sensitivities to porosity and flow resistivity are quite close to

500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000Frequency [Hz]

500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000Frequency [Hz]

2626

p y y qeach other for both absorption and transmission coefficients

Singular VectorAb ti C ffi i tAbsorption Coefficient

Note: Effect of material property 1(porosity) and 2(flow resistivity)is almost the same on all five singular vectors.

27


Fixed porosity case result for absorption coefficient


p y p

100

Effect of adding frequency104


10-2

10-1

Sin

gula

r Val

ue

103

ondi

tion

Num

ber

185 10 15 20 25 30 35 40 45 50 55 60

10-4

10-3

S

1st SV2nd SV3rd SV4th SV

5 10 15 20 25 30 35 40 45 50 55 60101

102Co

Fixing porosity reduces the condition number



2828

g p ysignificantly and makes the sensitivity matrix well-posed.


Combine both absorption and transmission coefficient


Combine both absorption and transmission coefficient sensitivity matrix

Singular Value Condition NumberSingular Value

2.11970 87038

Condition Number

110.870380.399760.18173

Adding other acoustical measurements reduces

2929

gthe condition number further


To verify the effect of low and high condition number


To verify the effect of low and high condition number during the automatic inverse characterization in COMET/TRIM, two different cases were studied

Solution Initial value

Found value 1

Found value 2

Porosity 0.98 0.5 0.54 0.98Flow resistivity 50,000 125,000 165,000 51,050

Tortuosity 2.0 6.0 1.47 2.073Viscous C.L 3.0*10-5 9.0*10-5 1.77*10-5 3.08*10-5

3030Thermal C.L 9.0*10-5 2.7*10-4 7.85*10-4 9.06*10-5

O ti l I Ch t i tiOptimal Inverse Characterization

4 parameter search gives near optimal result 4 parameter search gives near optimal result

31

Inverse Characterization based on FEM

Use COMET/SAFE (FEM) elastic foam type material that has 9 material Use COMET/SAFE (FEM) elastic foam type material that has 9 material properties listed below.

Layer thickness = 5 cm, Sample is fixed around circumferential edge

Porosity FlowR i i i

Tortuosity VCL TCL Density Young’sd l

Poisson’si

Lossf

32

Resistivity modulus ratio factor

0.98 50,000 2.0 3.0*10-5 9.0*10-5 9.0 50,000 0.4 0.3

Singular Vectors for Absorption and T i iTransmission

Higher order singular vectors for absorption & transmission coefficient Higher order singular vectors for absorption & transmission coefficient case

E hi h (6th 7th 8th d 9th) d i l t33

Even higher (6th, 7th, 8th, and 9th) order singular vectors have wide range of values – all parameters independent

S l ti I iti l U fi d

Inverse Characterization Results

Solution Initial guess Unfixed

Porosity 0.98 0.75 0.88

Flow resistivity 50,000 45,000 51,203y , , ,

Tortuosity 2.0 1.7 2.04

Viscous C.L 3.0*10‐5 3.5*10‐5 3.16*10‐5

Thermal C.L 9.0*10‐5 1.05*10‐4 8.66*10‐5

Density 9.0 7.5 9.87

Young’s modulus 50,000 45,000 53,445

Poisson’s ratio 0.4 0.35 0.399

Loss factor 0 3 0 25 0 302

34

Loss factor 0.3 0.25 0.302

• 9 parameters estimated with reasonable accuracy

I Ch t i ti R ltInverse Characterization Results

35

I P d b d FEM

The finite element model’s condition number is

Inverse Procedure based on FEM

significantly smaller than the condition number based on the plane wave model.

Absorption coefficient: 2336 49Transmission coefficient: 1309 178

This result is due to the fact that the finite element model can simulate finite sample size effects such as low frequency shearing resonance of the sample insidelow frequency shearing resonance of the sample inside the tube.

Therefore, the inverse characterizations based on the

36

finite element model have better chance to extract correct material properties.

Conclusions

St di t b id b th b ti d• Standing wave tubes can provide both absorption and transmission loss data for estimation of Biot parameters by inverse methods, but edge constraint effects have a significant impact on the resultssignificant impact on the results

• By using a linearization and SVD procedure, the stability of the inverse process can be improved by removing materialthe inverse process can be improved by removing material properties that makes the sensitivity matrix ill-conditioned.

• Inverse procedures based on finite element models of edge-p gconstrained samples may offer improved performance by making the effect of input parameters more independent

3737

Acknowledgments

P E D k

g

P. E. Doak Joe Pope L&L Products L&L Products United Technologies Research Center 3M Corporation (Jon Alexander) Bruel & Kjaer (Oliviero Olivieri, Jason Kunio, Jorgen

Hald) NASA (Richard Silcox) NASA (Richard Silcox) Richard Yun, Heuk Jin (Bryan) Song, Jinho Song,

Jeong-woo Kim, Taewook Yoo, Kwanwoo Hong, Kang Hou

3838

Hou Tanya Wulf

Inverse Characterization of Poro-Elastic Materials Based ...

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