Numerical Simulation of Acoustic Response of Discontinuous

Chapter 14

Numerical Simulation of Acoustic Response of Discontinuous Viscoelastic Fluids B. C. H. Wendlandt DSTO, Materials Research Laboratory, P.O. Box 50, Ascot Vale, 3032, Victoria, Australia Acoustic response of viscoelastic fluids may be simulated by Lagrangian finite element or Eulerian finite difference schemes. Both techniques have been used to predict the behaviour of discontinuous viscoelastic layers which may contain inserts such as air cavities and which are backed by metal plates. The advantages and disadvantages of applying these numerical techniques to the study of the acoustic response of fluid composites are discussed. The acoustic response of resonant viscoelastic fluid structures to a pressure wave may be simulated by a four-dimensional calculation, three dimensions in space and one in time. The Lagrangian, primitive finite element and Eulerian finite difference schemes form the basis for two models presented in this paper which are able to simulate a wide range of fluid structures containing inclusions of arbitrary spacing, shape and composition. The Lagrangian model subdivides the fluid into a large number of small elements whose size and shape is influenced by the passage of an acoustic wave. This model is especially suitable for the simulation of the acoustic response of the fluid when the material properties are discontinuous or significant deformations of sections of the fluid occur. The Lagrangian model i s , however, computationally expensive, particularly when multidimensional problems are considered. The Eulerian model solves the mathematical equations governing the acoustic response of the fluid rather than simulating the viscoelastic responses of the fluid in detail which simplifies the algorithms but reduces the f l e x i b i l i t y of the model. Both models can readily compute the passage of a pressure pulse or sinusoidal wave through a multilayer, multicavity or multi-inclusion fluid such as coatings used to quieten metal structures. For ease of interpretation, the calculation examples are limited to two space dimensions only. The results of 0097-6156/90/0424-0260$06.00/0 © 1990 American Chemical Society

14. WENDLANDT

261

Numerical Simulation ofAcoustic Response

r e p r e s e n t a t i v e c a l c u l a t i o n s show t h e a c o u s t i c r e s p o n s e o f square, a i r f i l l e d columns and e l a s t i c and v i s c o e l a s t i c c y l i n d r i c a l i n s e r t s embedded i n v i s c o e l a s t i c and e l a s t i c f l u i d s . The i n s e r t s a r e i r r a d i a t e d w i t h sound w h i c h i s i n c i d e n t normal t o t h e l o n g a x i s o f t h e columns and c y l i n d e r s . Model Propagation Model. The p r o p a g a t i o n o f a p r e s s u r e o r s t r e s s wave t h r o u g h a c o m p r e s s i b l e medium i s d e s c r i b e d by t h e laws o f c o n s e r v a t i o n o f mass and momentum and t h e e q u a t i o n o f s t a t e w h i c h r e l a t e s t h e p r e s s u r e o r s t r e s s i n t h e medium t o t h e s t r a i n and i t s material properties Q J . N u m e r i c a l d i f f e r e n c e schemes a b l e t o s i m u l a t e d i s c o n t i n u o u s media i n a s i m p l e f a s h i o n r e q u i r e t h e b a s i c laws o f p h y s i c s g o v e r n i n g p r o p a g a t i o n o f a c o u s t i c waves t o be c a r e f u l l y t r a n s f o r m e d i n t o numerical analogues. Hence t h e s e b a s i c p h y s i c a l p r i n c i p l e s ( 1 ) w i l l be b r i e f l y r e c o n s i d e r e d . C o n s e r v a t i o n o f Mass. The law o f c o n s e r v a t i o n o f mass f o r a c o m p r e s s i b l e medium i s u s u a l l y e x p r e s s e d i n an E u l e r i a n framework as, "the time r a t e change o f mass d e n s i t y a t any p o i n t i s e q u a l t o the n e g a t i v e d i v e r g e n c e o f t h e momentum d e n s i t y a t t h a t p o i n t . " The c o n s e r v a t i o n o f mass e q u a t i o n c a n be e x p r e s s e d i n a frame o f r e f e r e n c e w h i c h moves w i t h t h e f l u i d , t h e L a g r a n g i a n frame. I n m a t h e m a t i c a l n o t a t i o n t h e c o n s e r v a t i o n law becomes i n t h i s frame, ( 2 ) ,

S="PV.v

(1)

where p i s t h e d e n s i t y , y t h e v e l o c i t y v e c t o r o f an element o f t h e medium and t i s t h e time v a r i a b l e . C o n s e r v a t i o n o f Momentum. The law o f c o n s e r v a t i o n o f momentum c a n be e x p r e s s e d f o r a f l u i d i n t e n s o r n o t a t i o n and i n terms o f m a t e r i a l s t r e s s e s and i t s v e l o c i t y components as (2_)

dv.

d o . ,

1

11

,

9 >

j where o. . a r e t h e s t r e s s e s i n t h e m a t e r i a l as d e s c r i b e d below. Equations (1) and (2) a r e r e c o g n i s e d as t h e u s u a l e q u a t i o n s o f m o t i o n i n t h e L a g r a n g i a n frame. 1

S t r e s s - S t r a i n R e l a t i o n s as E q u a t i o n s o f S t a t e . Simple t h e o r y o f e l a s t i c i t y assumes t h a t t h e m a t e r i a l i s i s o t r o p i c and t h a t i n d u c e d s t r e s s e s and s t r a i n s a r e l i n e a r l y r e l a t e d t o each o t h e r as l o n g as t h e y a r e s m a l l . The t h e o r y f u r t h e r assumes t h a t t h e s t r e s s and t h e s t r a i n t e n s o r s always have t h e same axes. P o i s s o n ' s r a t i o and

262

SOUND AND VIBRATION DAMPING WITH POLYMERS

Young's modulus d e s c r i b e t h e e x t e n t t o w h i c h an element o f t h e m a t e r i a l responds when s t r e s s o r s t r a i n i s a p p l i e d a l o n g one direction. I f t h e s t r e s s e s and s t r a i n s a l o n g t h r e e o r t h o g o n a l axes a r e c o n s i d e r e d , t h e n t h e g e n e r a l s t r e s s - s t r a i n r e l a t i o n c a n be w r i t t e n as.

a., lj

= Ae mm

5., I J

+ 2ue. . i j

(3)

where t h e e.,'s a r e t h e s t r a i n s and parameter e 6 . , r e p r e s e n t s t h e sum o f t h e o r t h o g o n a l s t r a i n ; A and \i a r e knowrâs-'Lame' s c o n s t a n t s (_3_). The parameter pi i s a l s o c a l l e d t h e modulus o f r i g i d i t y and measures t h e r e s i s t a n c e o f t h e substance t o distortions. These c o n s t a n t s a r e r e l a t e d t o Young's modulus E and P o i s s o n ' s r a t i o v by A = v E / ( l + v ) ( l - 2 v ) and ju = E / 2 ( l + v ) . Stress-Strain Relations f o r a Fluid. The medium i s h y d r o s t a t i c when t h e d i r e c t s t r e s s e s i n t h r e e o r t h o g o n a l d i r e c t i o n s a r e e q u a l and t h e shear s t r e s s e s a r e z e r o ; o., = a . , = (3A + 2u) e.., i i JJ ^ n S t r e s s - S t r a i n R e l a t i o n s f o r V i s c o e l a s t i c M a t e r i a l s . The v i s c o e l a s t i c b e h a v i o u r o f an e l a s t o m e r v a r i e s w i t h t e m p e r a t u r e , p r e s s u r e , and r a t e o f s t r a i n . T h i s e l a s t i c b e h a v i o u r v a r i e s when stresses are repeatedly reversed. Hence any s i n g l e m a t h e m a t i c a l model c a n o n l y be expected t o approximate t h e e l a s t i c b e h a v i o u r o f a c t u a l s u b s t a n c e s under l i m i t e d c o n d i t i o n s ( 2 ) . The r e s p o n s e o f a t y p i c a l e l a s t o m e r t o s m a l l p r e s s u r e e x c i t a t i o n c a n be e x p r e s s e d by a l i n e a r r e l a t i o n s h i p between t h e s t r e s s e s i n t h e m a t e r i a l and t h e s t r a i n s caused by t h e p r e s s u r e excitation ( 2 ) . The s i m p l e s t g e n e r a l l i n e a r r e l a t i o n s h i p between a change i n s t r e s s and a s s o c i a t e d s t r a i n c a n be w r i t t e n i n t e n s o r n o t a t i o n as

do. . a

- • ij

where B,B material.

5 . . + 2/J(1 + y — ) e. . 1 i j ^ at i j (4)

+ £ — ~ ^ — = A(l + £ 'o a t at

e

and y a r e c h a r a c t e r i s t i c

r e l a x a t i o n times

mm

of the

The l i n e a r model o f E q u a t i o n ( 4 ) g i v e s a r e p r e s e n t a t i o n o f damping o f v i b r a t i o n s by i n t e r n a l f r i c t i o n ( 2 ) . When a s t e a d y s i n u s o i d a l e x c i t a t i o n i s i n v o l v e d , t h e i n t e r n a l f r i c t i o n causes a phase d e l a y i n t h e t r a n s m i s s i o n o f s i g n a l s t h r o u g h t h e m a t e r i a l w h i c h c a n be e x p r e s s e d as a l o s s - t a n g e n t , t a n 6 , w h i c h i s r e l a t e d t o £ , £, y and t h e f r e q u e n c y w, o f t h e s i g n a l by t a n 6 ^ = w(j8 - £ ) / ( l + # P &) f o r d i r e c t o r normal s t r e s s e s and t a n 5 = cj(y £ y ) f o r shear s t r e s s e s . q

Q

+

u

0

14. WENDLANDT

263


The a n g l e 6 measures t h e l a g o f s t r a i n b e h i n d s t r e s s and i s known as t h e l o s s a n g l e o f t h e m a t e r i a l and p r o v i d e s a measure o f the i n t e r n a l damping o f s t r e s s waves. A s i m p l e r model o f v i s c o e l a s t i c i t y , t h e K e l v i n - V o i g t model p l a c e s J6 = 0, t a n 6 ^ = w £ and t a n 6 = wy. q

V

Wave E q u a t i o n . The p r o p a g a t i o n o f t h e d i s p l a c e m e n t v e c t o r u o f an a c o u s t i c e x c i t a t i o n can be d i r e c t l y e x p r e s s e d i n terms o f m a t e r i a l s parameters from t h e law o f c o n s e r v a t i o n o f momentum and E q u a t i o n s (3) and (4) as a wave e q u a t i o n (J*_). In vector notation,

p

= V {A V » u + 7>pi*u}

—

+ 7.pi*7

u

(5)

at

Here t h e K e l v i n - V o i g t model i s assumed t o a d e q u a t e l y d e s c r i b e t h e v i s c o e l a s t i c p r o p e r t i e s o f t h e e l a s t o m e r and t h e Lame c o n s t a n t s can be w r i t t e n t o i n c l u d e t h e c h a r a c t e r i s t i c r e l a x a t i o n times o f t h e material. g become t h e o p e r a t o r s A = A ( l + — ) and pi* = pi (1 + y — ) , where 0 and y a r e v i s c o e l a s t i c r e l a x a t i o n at times. T

n

e

The wave e q u a t i o n may be s e p a r a t e d i n t o two e q u a t i o n s w h i c h d e s c r i b e t o r s i o n and d i l a t a t i o n . Dilatation i s defined by 0 = V.u and i s r e l a t e d t o p r e s s u r e by p = \*0 t h r o u g h t h e equation of continuity. The wave e q u a t i o n w h i c h d e s c r i b e s t h e p r o p a g a t i o n o f d i l a t a t i o n a l waves can be e x p r e s s e d as 2 _ _ _ p -^-f- = V { ( A * + ~pi*)G} + ~pi*V e - (r&)-{V(A*+^*)0+/J*V0}+(V/i*) 2

2

at

.76

p

(6) and i s l i n k e d by s i n k and s o u r c e terms, n o t shown, w h i c h a r i s e where m a t e r i a l parameters v a r y w i t h p o s i t i o n , t o a s i m i l a r e q u a t i o n d e s c r i b i n g t h e p r o p a g a t i o n o f t o r s i o n waves. Terms i n E q u a t i o n (6) w h i c h i n v o l v e (Vp) and 7pi* d e s c r i b e t h e m o d i f y i n g i n f l u e n c e d i s c o n t i n u o u s d e n s i t i e s and srtiear m o d u l i have on t h e p r o p a g a t i o n o f the d i l a t a t i o n a l wave. The pi* r e p r e s e n t s t h e average v a l u e of pi* over a c o m p u t a t i o n a l c e l l and i s d e r i v e d from t h e i n t e g r a l d e f i n i t i o n of i t s divergence. The p r e s e n t study assumes as (_5) t h a t t h e t r a n s f o r m a t i o n o f d i l a t a t i o n t o t o r s i o n waves can be n e g l e c t e d f o r t h e m a t e r i a l response o f i n t e r e s t and t h e s o u r c e and s i n k terms a r e z e r o . The a c o u s t i c response o f many e l a s t o m e r i c f l u i d s can be a d e q u a t e l y d e s c r i b e d by assuming t h a t t h e c o n t r i b u t i o n o f pi* t o t h e p r o p a g a t i o n o f a c o u s t i c waves i s n e g l i g i b l e (_5j . Under t h i s assumption

v {A*e> at

2

p

p

2

(7)

264


L a g r a n g i a n N u m e r i c a l Scheme. The L a g r a n g i a n a p p r o a c h d e f i n e s c e l l s of whose c o r n e r s , and hence b o u n d a r i e s , move w i t h the l o c a l velocity. C e l l c o r n e r movement i s used t o update d e n s i t i e s , s t r a i n s and s t r e s s e s i n the f l u i d . The v e l o c i t y i s t h e n updated t o complete one s t e p i n the time e v o l u t i o n o f f l u i d a c o u s t i c r e s p o n s e . C o n s i d e r i n g m o t i o n i n c a r t e s i a n c o o r d i n a t e s , and i l l u s t r a t i n g t h a t i n the x d i r e c t i o n , the p o s i t i o n o f a c e l l c o r n e r a t time t + 6 t , x ^ - ^ ( t + 5t) i s r e l a t e d t o i t s p r e v i o u s p o s i t i o n X j - ^ C t ) time t by a

x.

t + 6t (t) + f v

( t + 6t) = x.

(s) ds

where i j k a r e the c e l l c o r n e r i n d i c e s i n the t h r e e d i r e c t i o n s o f the c a r t e s i a n c o o r d i n a t e system.

t

(8)

orthogonal

T h i s e x p r e s s i o n can be approximated by

n+1 x

i

j

n

-

k

+ i

j

1

,

x

v

k

(n + /o)

p

6 J

t

(

9

)

k

where i n d e x n i n d i c a t e s a p a r t i c u l a r i n s t a n t o f time v i a t = n5t. C o r r e s p o n d i n g e x p r e s s i o n s can be d e r i v e d f o r the y and z c o o r d i n a t e o f the c e l l c o r n e r . The examples t o be d i s c u s s e d a r e l i m i t e d t o x and y c o o r d i n a t e s o n l y . Hence i n the time i n t e r v a l n 1 t o n, the d e n s i t y v a r i a t i o n o f the c e l l d e s i g n a t e d i j and bounded by the c o r n e r s ( x . , , y . , ) , ( x . , y > ,

*

—

-

p

e

'i±ii £ j

]

-

2

6t A

i

i

i j

[

(

i

+

^

0

X

< W

i

a

j

Vii> 6t

+

W

;

e

" 5t

e

ij - 'ii ij

6t C ( 1 +

+

A

T

i j ^

1 ]

( 1

+

[

]2)

+

S

6t e

:

" V i i i ij

]

fit

+

( )2

f

{A

ij i +

[(1+

îj±i

)

fit (17)

14. WENDLANDT

6

and

t

5

t

5

(*§> . v u d +fcL)e } » . ( P ^ - P ^ )

U

J

e

i

9

j

i " îj+i i j i i

+

]

-V

i

+^

6t

t

td +

i j + 1

2

2

5

^

^-j-! -

6t

4p .,(5x)

267

Numerical Simulation of Acoustic Response

5t

t

5

t

(

ccî-ii) ^ - i j - ^ i - i i ^ l i j "

1

8

)

6t 6t where 5 x , 6 y a r e s p a t i a l i n t e r v a l s between nodes. The d i m e n s i o n a l v a r i a b l e s of E q u a t i o n (18) may be t r a n s f o r m e d i n t o the f o l l o w i n g dimensionless v a r i a b l e s , oc., = 0 . , / ^ ,r = t/t g = x/1, rj = y/1, y. . = p . , / p , l f

0

a., = A. . / A. ., b. . = A. ./A. . , C. . = A. . /A. ., ij l+lj i j ' i j i-lj' i j i j ij+l' i j ' 2

2

B. . = t A. . / l p . ., d. . = A. . ,/A. ., ij 1 i j i j ' i j ij-l' i j '

(19)

p

where 1, t and p a r e l e n g t h , time and d e n s i t y c h a r a c t e r i s t i c o f the c a l c u l a t i o n o f i n t e r e s t . Hence, E q u a t i o n (7) can be w r i t t e n , 1

9

n + 1

= 20 . - 0 : + B. . ( 4 V i j i j i j 5g' n

e ~J;.] l+l J n

+

-

n

X

v

ij

oc.

11+1 6r

L

x1

V

n

x 1

A1

n

+ b. . [ ( i + a. , . ) e , . - a . , . e " ^ . ] - 2 (1 + [ | V ) i j l-l 1 l-l J l-l 1 l-l J 6ry 5r 6r J

^ 6r

* i j" - i i ° i 7 6r

1

]

+

6r

1

oc. . , G .'}]}

4 y . .p ij o

1 + 1 J

I J

5n

i j

(

- (y. y

2

Q

)

.-y y . . .)

i+li i - l i 4 y . .p ij o - a . e ? , . .] -b. .[(i+oc . , ) © ? , . 1+1

li-l 6r

îj+l

1-1

1-1J

6r

268


oc.

ij+1

- d. . [(1+oc. .

ij-l

oc.

6r The e x p e c t e d l o c a l e r r o r of c o m p u t a t i o n i s of second o r d e r i n time and m a t e r i a l p r o p e r t i e s . I n p r a c t i c e an e r r o r of about f i v e p e r c e n t was o b s e r v e d i n runs u s i n g v a l u e s of 0.1 f o r the n o n d i m e n s i o n a l s p a t i a l i n t e r v a l s and a f a c t o r of 1000 i n v a r i a t i o n o f m a t e r i a l p a r a m e t e r s a c r o s s one g r i d i n t e r v a l . S t a b i l i t y of the c o m p u t a t i o n i s governed by the von s t a b i l i t y c o n d i t i o n (_7)for a wave e q u a t i o n .

[(6£)

2

+

2

(5T7) ]

6r

Neumann

(21)

equation. Comparison of L a g r a n g i a n and E u l e r i a n Schemes. The L a g r a n g i a n scheme s u b d i v i d e s the a n e c h o i c c o a t i n g i n t o a l a r g e number o f s m a l l c e l l s , the c o r n e r s of w h i c h a r e moved d u r i n g each c y c l e of computation. The a c o u s t i c d i s p l a c e m e n t s of a p a s s i n g wave a r e c a l c u l a t e d from t h e s e movements. The L a g r a n g i a n scheme p e r m i t s , i n t h i s f a s h i o n , the c a l c u l a t i o n of l a r g e as w e l l as s m a l l s i g n a l responses. The L a g r a n g i a n scheme a l s o e n a b l e s m a t e r i a l i n t e r f a c e s t o be p l a c e d a t the i n t e r f a c e s of c o m p u t a t i o n a l c e l l s . This e n a b l e s a c o u s t i c e x c i t a t i o n c r o s s i n g m a t e r i a l d i s c o n t i n u i t i e s t o be c o n s i d e r e d i n a r e l a t i v e l y r i g o r o u s and s e l f - c o n s i s t e n t manner and i s p a r t i c u l a r l y u s e f u l when the m a t e r i a l p r o p e r t i e s v a r y g r e a t l y a c r o s s an i n s e r t i n t e r f a c e and d e f o r m a t i o n of i n s e r t s , such as a i r c a v i t i e s , occurs. However, m u l t i d i m e n s i o n a l a l g o r i t h m s o f the L a g r a n g i a n schemes a r e d i f f i c u l t t o implement and a r e computationally expensive. The E u l e r i a n f i n i t e d i f f e r e n c e scheme aims t o r e p l a c e the wave e q u a t i o n s w h i c h d e s c r i b e the a c o u s t i c r e s p o n s e o f a n e c h o i c s t r u c t u r e s w i t h a numerical analogue. The r e s p o n s e f u n c t i o n s a r e t y p i c a l l y a p p r o x i m a t e d by s e r i e s of p a r a b o l a s . M a t e r i a l d i s c o n t i n u i t i e s a r e s i m i l a r l y t r e a t e d u n l e s s s p e c i a l boundary c o n d i t i o n s are c o n s i d e r e d . T h i s w i l l i n t r o d u c e some smearing of the s o l u t i o n (6). P r o p a g a t i o n of a c o u s t i c e x c i t a t i o n a c r o s s w a t e r a i r , w a t e r - s t e e l and e l a s t o m e r - a i r have been computed t o a c c u r a c i e s b e t t e r t h a n two p e r c e n t e r r o r (8). In two-dimensional c a l c u l a t i o n s , e r r o r s below f i v e p e r c e n t a r e p r a c t i c a b l e . The p o s i t i o n o f the b o u n d a r i e s a r e i n g e n e r a l c o n s i d e r e d t o be f i x e d . These c o n s t r a i n t s l i m i t the E u l e r i a n scheme t o the c a l c u l a t i o n of a c o u s t i c responses of anechoic s t r u c t u r e s without, simultaneously, considering non-acoustic pressure deformations. However, E u l e r i a n schemes may l e a d t o r e l a t i v e l y s i m p l e a l g o r i t h m s , as e v i d e n t from E q u a t i o n ( 2 0 ) , w h i c h enable m u l t i - d i m e n s i o n a l computations t o be c a r r i e d out i n a r e a s o n a b l e t i m e .

14. WENDLANDT


269

Sample Computation R e s u l t s . The L a g r a n g i a n scheme o f E q u a t i o n (9) to (15) was used t o compute the steady s t a t e r e s p o n s e t o s i n u s o i d a l e x c i t a t i o n of an a n e c h o i c c o a t i n g g l u e d on a s t e e l p l a t e shown i n F i g u r e 1. Symmetry c o n s i d e r a t i o n s p e r m i t the c a l c u l a t i o n s t o be l i m i t e d t o the r e g i o n s between the d o t t e d l i n e s , the u n i t c e l l . Steady s t a t e a c o u s t i c r e s p o n s e of the u n i t c e l l o c c u r r e d f o r the composites c o n s i d e r e d a f t e r the passage of some f i v e a c o u s t i c oscillations. P a t t e r n s of d i r e c t s t r e s s , and shear s t r e s s as shown i n F i g u r e s 2 and 3 were o b t a i n e d . As expected, the c o r n e r s of the c a v i t i e s c o n c e n t r a t e d the s t r e s s e s . V i s c o e l a s t i c energy l o s s c a l c u l a t i o n s , not d i s c u s s e d h e r e , a l s o show t h a t the c o r n e r s of the c a v i t i e s a r e c o n c e n t r a t i o n s of energy l o s s e s . The a c o u s t i c r e s p o n s e of the c o a t i n g r e g i o n above the c a v i t i e s v a r i e d from t h a t of the r e g i o n above the i n t e r c a v i t y r e g i o n s . The r e s u l t s showed t h a t the r e g i o n s between the c a v i t i e s and s t e e l b a c k i n g p l a t e c o n t r i b u t e d l i t t l e t o the a c o u s t i c r e s p o n s e of the coating. The o v e r a l l r e s p o n s e p a t t e r n of the c o a t i n g i s o f c o u r s e f r e q u e n c y dependent, but the r e s u l t s are r e p r e s e n t a t i v e f o r f r e q u e n c i e s from 10 to 100 kHz. The m a t e r i a l was assumed t o have a v i s c o e l a s t i c l o s s tangent of about 0.5 f o r Youngs modulus and the f i r s t and second Lame c o n s t a n t s . T h i s assumption l e a d s t o r e l a t i v e l y h i g h l o s s tangents f o r the b u l k modulus w h i c h i s not r e p r e s e n t a t i v e of a l l e l a s t o m e r s . The model i s a b l e t o c o n s i d e r the a c o u s t i c r e s p o n s e of a s i n g l e i n s e r t as w e l l as a number of c l o s e l y i n t e r a c t i n g i n s e r t s or c a v i t i e s w h i c h m o d i f y the a c o u s t i c f i e l d i n the composite. The E u l e r i a n scheme of E q u a t i o n (20) was used t o examine v a r i o u s a s p e c t s of the r e s p o n s e p r e d i c t e d by the L a g r a n g i a n scheme. The t r a n s m i s s i b i l i t y of a s t e e l p l a t e c o a t e d w i t h an e l a s t o m e r and immersed i n water t o a s i n u s o i d a l p u l s e was computed and i s shown i n F i g u r e 4, (8_). The a c o u s t i c p u l s e r e s p o n s e of an a i r f i l l e d row of square columns t o an i n c i d e n t C a u s s i a n e x c i t a t i o n i s shown i n F i g u r e 5. Here a G a u s s i a n p u l s e i n c i d e n t from the l e f t has passed a row of square, a i r f i l l e d c a v i t i e s s e t i n a l o s s - l e s s e l a s t i c f l u i d . The r e f l e c t e d p u l s e i s seen moving to the l e f t . The a i r i n the c a v i t i e s i s r e s p o n d i n g t o the p u l s e even though the p u l s e had a l r e a d y passed the c a v i t y . T h i s i s expected as the v e l o c i t y of sound i n a i r i s o n l y 1/5 t h a t i n the l o s s - l e s s f l u i d c o n s i d e r e d . The s u r f a c e s o f the c a v i t i e s showed o s c i l l a t i o n s a f t e r the p u l s e had passed the c a v i t i e s . T h i s phenomenon can a l s o be e x p l a i n e d by s u r f a c e wave e f f e c t s (_5_). E q u a t i o n (20) was a l s o used t o compute the a c o u s t i c r e s p o n s e of f l u i d c y l i n d e r s immersed i n water and i n s o n i f i e d normal t o t h e i r a x i s w i t h a s i n u s o i d a l wavepacket. The examples shown h e r e can be c o n s i d e r e d by o t h e r t e c h n i q u e s (5) but serve as a p p r o p r i a t e t e s t s f o r the a c c u r a c y of the model which can then be used t o compute the a c o u s t i c r e s p o n s e s of systems w h i c h cannot be r e a d i l y t r e a t e d by o t h e r methods. The m a t e r i a l p r o p e r t i e s of the c y l i n d e r a r e shown i n T a b l e 1 and were chosen t o enable the c a l c u l a t e d echo s t r u c t u r e of the c y l i n d e r s t o be compared w i t h p r e v i o u s l y p u b l i s h e d a n a l y t i c a l work (_5_) .

270


Figure

2.

Amplitude o f d i r e c t

stress.

WENDIANDT


211


Figure

5.

Table

1

Impulse response o f a i r c a v i t i e s .

M a t e r i a l P r o p e r t i e s of F l u i d s

Density 3

(kg/m ) P

Speed o f Sound (m/s)

A

Loss Tangent

(GPa)

VA/p

tan

6=cjfi

Cylinder A

1500

1000

1.5

Cylinder B

2000

1000

2.0

0

Cylinder C

1500

1000

1.5

0.5

Water

1000

1500

2.25

0

0

14. WENDIANDT


273

The a c o u s t i c r e s p o n s e s o f the l o s s l e s s f l u i d c y l i n d e r s a r e shown i n F i g u r e s 6-8, where I denote t h e i n c i d e n t wavepacket, R the wavepackets r e f l e c t e d by t h e c y l i n d e r s and T l a b e l s t h e wavepackets t r a n s m i t t e d by t h e c y l i n d e r a l o n g t h e a x i s o f advance of t h e i n c i d e n t p a c k e t , normal t o t h e c y l i n d e r a x i s . The a c o u s t i c wave p a t t e r n s a t v a r i o u s s t a g e s o f time e v o l u t i o n o f t h e i n t e r a c t i o n between t h e i n c i d e n t wave and c y l i n d e r a r e shown i n t h e same f i g u r e f o r comparison. The s u b s c r i p t s i n d i c a t e r e l a t i v e time. The i n c i d e n t wave p a t t e r n i s d i s p l a y e d f o r r e f e r e n c e purposes a l s o i n the f i g u r e s . The c a l c u l a t i o n s show t h a t the s p e c u l a r r e f l e c t i o n f r o m a f l u i d c y l i n d e r whose a c o u s t i c impedance e q u a l s t h a t o f w a t e r , c y l i n d e r A i n T a b l e 1, i s v e r y s m a l l , o r n e g l i g i b l e , i n agreement w i t h s i m p l e impedance t h e o r y , F i g u r e 6. The s p e c u l a r r e f l e c t i o n i s l a b e l l e d R^ and i t s computed magnitude f a l l s w i t h i n t h e e r r o r o f computation. The wave w h i c h i s t r a n s m i t t e d f i r s t by t h e c y l i n d e r i s l a b e l l e d T^. The dominant echo from such a c y l i n d e r appears t o be caused by waves t r a v e l l i n g o r c r e e p i n g around t h e o u t s i d e o f t h e c y l i n d e r and i s g e n e r a t e d by t h e c y l i n d e r a t a l a t e r time a f t e r t h e s p e c u l a r r e f l e c t i o n has l e f t t h e c y l i n d e r s u r f a c e . I t i s shown as R i n F i g u r e 7. A t t h i s time a second wavepacket i s emmitted from the o t h e r s i d e o f the c y l i n d e r and i t i s l a b e l l e d T » The s p e c u l a r r e f l e c t i o n f r o m an e l a s t o m e r c y l i n d e r whose a c o u s t i c impedance i s d i f f e r e n t from t h a t o f water ( c y l i n d e r B i n T a b l e 1) i s shown i n F i g u r e 8. The a m p l i t u d e c a l c u l a t e d by t h e p r e s e n t n u m e r i c a l scheme a g r e e s t o w i t h i n 10% w i t h p r e v i o u s work (_5_) w h i c h was based on K i r c h h o f f approximation theory. 2

2

The a m p l i t u d e o f a c o u s t i c e x c i t a t i o n w i t h i n one h a l f o f a l o s s y c y l i n d e r C i s shown i n F i g u r e 9. Symmetry c o n s i d e r a t i o n s p e r m i t c a l c u l a t i o n s t o be l i m i t e d t o one h a l f o f t h e c y l i n d e r . The power l o s s e s a r e p r o p o r t i o n a l t o t h e square o f t h e a m p l i t u d e (4_). F i g u r e 9 shows a m p l i t u d e i n f o r m a t i o n t o b r i n g o u t d e t a i l s o f t h e a c o u s t i c r e s p o n s e a t t h e r i m . The l o s s e s a r e c o n c e n t r a t e d a t t h e back o f t h e c y l i n d e r where f o c u s s i n g w i t h i n t h e c y l i n d e r i s e x p e c t e d from r a y t h e o r y . L o s s e s a l s o appear a t t h e r i m o f t h e c y l i n d e r w h i c h a r e due t o complex i n t e r n a l r e f l e c t i o n s , o r due t o c i r c u m f e r e n t i a l waves moving a t and c l o s e t o t h e s u r f a c e o f t h e cylinder. The p r e s e n t s t u d y has n o t i s o l a t e d c i r c u m f e r e n t i a l o r c r e e p i n g waves from m u l t i p l e i n t e r n a l r e f l e c t i o n phenomena but shows t h e t o t a l l o c a l d i s t u r b a n c e moving t h r o u g h and a l o n g t h e r i m of the c y l i n d e r . The o b s e r v e d c a l c u l a t e d p a t t e r n s a r e caused by the f i v e c y c l e i n c i d e n t wavepacket and s h o u l d be enhanced and perhaps be somewhat d i f f e r e n t f o r l o n g w a v e t r a i n s , hence the 0Q s c a l e i s a r b i t r a r y . Conclusion Two models have been p r e s e n t e d w h i c h a r e based on t h e L a g r a n g i a n and E u l e r i a n d i f f e r e n c e schemes r e s p e c t i v e l y . These models were a b l e t o c a l c u l a t e i n d e t a i l t h e i n t e r a c t i o n o f a c o u s t i c waves w i t h inclusions i n discontinuous f l u i d s . The L a g r a n g i a n model was a b l e t o c o n s i d e r t h e t r a n s f e r o f normal t o shear s t r e s s e s a t f l u i d discontinuities. The E u l e r i a n model i s l i m i t e d t o t h e s t u d y o f d i l a t a t i o n waves but i t s c o m p u t a t i o n a l e f f i c i e n c y e n a b l e s i t t o

274


6

-2

-

-

F i g u r e 6. S p e c u l a r e c h o g e n e r a t e d b y f l u i d c y l i n d e r A. T h e echo i s n o r m a l i s e d w i t h r e s p e c t t o t h e a m p l i t u d e o f t h e i n c i d e n t wave p a c k e t . T h e c i r c u l a r c r o s s s e c t i o n i s l o c a t e d a s shown i n F i g . 3, between x = 6 a n d x = 8 . 3 , where t h e u n i t o f x i s t h r e e w a v e l e n g t h s o f t h e i n c i d e n t wave p a c k e t . I - i n c i d e n t wave packet, R - f i r s t echo o r s p e c u l a r echo generated by c y l i n d e r , - f i r sk t r a n s m i t t e d w a v e u a c k e t . - impedance r e l a t i v e t o impedance o f water.

275


14. WENDLANDT

Figure 7. Dominant echo generated by cylinder A. The echo is normalised with respect to the amplitude of the incident wave packet. The distance X is normalised to three wavelengths of the incident wave packet. I is the incident wave packet (shown for reference only); R is the second, or dominant, echo generated by the cylinder; T the second transmitted wave packet; S the ratio of the speed of sound in cylinderfluidto the speed of sound in surrounding water. 2

2

f

276


0.8

- 0.8

-

-1 -6

F i g u r e 8. S p e c u l a r e c h o g e n e r a t e d b y c y l i n d e r B. T h e e c h o i s n o r m a l i s e d w i t h r e s p e c t t o t h e a m p l i t u d e o f t h e i n c i d e n t wave packet. The d i s t a n c e x i s n o r m a l i s e d t o t h r e e w a v e l e n g t h s o f t h e i n c i d e n t wave p a c k e t . f i r s t o r specular i n c i d e n t wave, R e c h o g e n e r a t e d b y c y l i n d e r , T^ - f i r s t t r a n s m i t t e d wave p a c k e t , Z - r a t i o o f a c o u s t i c impedance o f c y l i n d e r f l u i d t o a c o u s t i c impedance o f water.

Internal Focused Waves

()

Circumferential or Creeping Waves

F i g u r e 9. A m p l i t u d e 0 o f a c o u s t i c e x c i t a t i o n i n c y l i n d e r C. The x a n d y s c a l e s a r e n o r m a l i s e d t o t h e r a d i u s o f t h e c y l i n d e r . The 0 i s arbitrary. Because t h e a c o u s t i c e x c i t a t i o n p a t t e r n i s symmetric about the diameter o f t h ec y l i n d e r , only t h e 0 in half of the cross-section of the cylinder i s displayed.

14. WENDLANDT


111

s t u d y the time e v o l u t i o n of m u l t i p l e r e f l e c t i o n and r e f r a c t i o n a t c u r v e d media i n t e r f a c e s i n a r e a s o n a b l e time on desk t o p computer i n s t a l l a t i o n s , such as a VAX 3100. T o g e t h e r , t h e s e models p r o v i d e u s e f u l t o o l s f o r the study of a wide range of a c o u s t i c problems c o v e r i n g a n e c h o i c c o a t i n g s , machinery mounts and f a b r i c / s k i n interaction. The c h o i c e between L a g r a n g i a n and E u l e r i a n n u m e r i c a l schemes f o r the p r e d i c t i o n of the a c o u s t i c response of v i s c o e l a s t i c f l u i d systems depends on the p a r t i c u l a r problem t o be s o l v e d . W h i l e the L a g r a n g i a n scheme p r o v i d e s a more p r e c i s e s i m u l a t i o n o f d i s c o n t i n u o u s o r composite v i s c o e l a s t i c f l u i d s r a t h e r t h a n a n u m e r i c a l a p p r o x i m a t i o n t o the g o v e r n i n g d i f f e r e n t i a l e q u a t i o n , the r e l a t i v e s i m p l i c i t y of the E u l e r i a n a l g o r i t h m s suggests t h e i r use whenever p o s s i b l e . Only when g r e a t p r e c i s i o n or u n c e r t a i n i t i e s about the e x t e n t of d e f o r m a t i o n s o f the c o a t i n g a r e s i g n i f i c a n t a r e L a g r a n g i a n schemes p r e f e r a b l e t o E u l e r i a n schemes.

Literature Cited 1. 2. 3. 4. 5. 6. 7. 8.

Temkin, S. Elements of Acoustics, John Wiley, New York, 1981. Jaeger, J.C. Elasticity, Fracture and Flow, Methuen, London, 1956. Jeffrey, H. and Jeffrey, B.S. Methods of Mathematical Physics, University Press, Cambridge, 1962. Sommerfeld, A. Mechanics of Deformable Bodies, Academic Press, New York, 1964. Davies, C.M. et al. J. Acoust. Soc. Am. 63, pp 1694-1698, 1978. Brown, D.L. Mathematics of Computation, 1984, 42, pp 369-391. McCracken and Dorn, W.S. Numerical Methods and Fortran Programming, Wiley, London, 1964. Wendlandt, B.C.H. "The Acoustic Properties of Layered Coatings", MRL-R-1034, 1988.

RECEIVED January 24, 1990

Numerical Simulation of Acoustic Response of Discontinuous

Recommend Documents