on evaluation of the green functions for harmonic line loads in a viscoelastic half space

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, VOL. 26, 823-841 (1988)

ON EVALUATION OF THE GREEN FUNCTIONS FOR HARMONIC LINE LOADS IN A VISCOELASTIC HALF SPACE

MARIJAN DRAVINSKI AND TOMI K . MOSSESSIAN

Department of Mechanical Engineering, University of Southern California, Los Angeles, California, U.S .A.

SUMMARY

Comparison of different quadratures for evaluation of the improper wavenumber integrals which arise in evaluation of the Green functions for a viscoelastic half space and harmonic line loadings is investigated. The model is assumed to be of the plane strain type. Extensive testing of the numerical accuracy for various quadratures is performed. A measure of numerical efficiency of the quadratures is proposed and compared for different integration formulae. It was determined that among the procedures tested the Clenshaw-Curtis quadrature offers the most efficient way of evaluating the wavenumber integrals numerically.

INTRODUCTION

In recent years, boundary integral equation methods (BIEM) have proved to be very effective for studying various problems of interest in earthquake engineering and seismology.'. * Namely, standard numerical techniques, such as finite elements and finite differences, require discretization of the entire solution domain, thus making them less efficient for problems involving large characteristic dimensions. In addition, the radiation conditions at infinity cannot be satisfied exactly. Boundary integral equation methods on the other hand require only discretization of the boundary of the ~ c a t t e r e r s ~ . ~ and the radiation conditions at infinity can be modelled e ~ a c t l y . ~ However, a major difficulty in applying BIEM to realistic problems arises in numerical evaluation of the corresponding Green functions. In this paper the Green functions for a steady state plain strain model of a viscoelastic half space are considered. Numerical evaluation of these functions requires calculation of the so-called wavenumber integrals. These improper integrals are of the form f r g(y, o, s) cos(xs) ds, where g is a complex function of oscillatory nature.

Accurate evaluation of the wavenumber integrals requires application of special numerical procedures.6. Standard quadratures will not in general produce correct results. This topic has been a subject of research in several papers. Apsel' and Apsel and Luco6 proposed a technique for evaluation of these integrals by replacing the integrand function g with a quartic polynomial, thus resulting in a Filon type quadrature. Kundu' and Kundu and Ma19 proposed an adaptive Gauss quadrature to accomplish the same task. Recently, Xu and Mal' applied the Clenshaw-Curtis approach in which the integrand function is approximated in terms of Chebyshev polynomials and then integrated to produce a Filon type quadrature.

Evaluation of the wavenumber integrals often requires a considerable amount of the CPU time on a digital computer. In addition, the Green functions itself are only an intermediate numerical step in evaluation of the total response of the elastic half space.'O* ' Consequently, it is essential to apply the most efficient quadrature for numerical evaluation of the Green functions in order to make the problem under consideration solvable on a standard digital computer.

0029-598 1/88/04O823-19%O9.50 0 1988 by John Wiley & Sons, Ltd

Received 4 March 1987 Revised 29 July 1987

824 M. DRAVINSKI AND T. K. MOSSESSIAN

The main topic of this paper is a comparison ofdifferent approaches available at the present time for evaluation of the wavenumber integrals. The list of quadratures considered here is non- exhaustive. The purpose of the analysis presented hereinafter is to demonstrate that the efficiency of various quadratures may differ significantly. The extent of this difference may be so large that choice of an inefficient quadrature for evaluation of the wavenumber integrals may preclude entirely the numerical solution of more complex problems which involve the numerical evaluations of the Green functions.

STATEMENT OF THE PROBLEM

The summary of the Green functions for a viscoelastic half space and harmonic line loadings presented in the Appendix indicates that the major problem in numerical evaluation of the Green functions lies in calculation of the improper wavenumber integrals. Analysis of these integrals shows that it is sufficient to consider the wavenumber integrals of the type

where g is a known function which will be referred to as the integrand function. The integrand in equation (1) depends strongly upon frequency o and distance x. This is illustrated by Figures 1-3 which depict the real and imaginary parts of the integrand in equation (1) for several combinations of the parameters w and x. For the sake of definiteness the integrand function g in Figures 1-3 is chosen to be I/R(s), where R(s) is the Rayleigh function defined in the Appendix. Apparently the integrand contains sharp corners and it may oscillate rapidly within the range of integration. Frequently these oscillations are accompanied by slow attenuation of the integrand function as the variable of integration increases. Numerical evaluation of such integrals using standard integration techniques will not, in general, produce correct results. Therefore, special techniques must be applied to evaluate integrals of the type defined by equation (1) accurately.’

It can be shown” that the sharp corners in the integrand are due to the Rayleigh function R(s). The location of the peaks is frequency dependent and it is related to the zeros of the Rayleigh function R(s). For large values of s, the oscillatory nature of the integrand is controlled mainly by the parameter x.

In this paper, four methods for evaluation of integrals of the type defined by equation (1) are examined in detail. The accuracy of the quadratures is tested in detail and the efficiency of the methods is compared.

SOLUTION OF THE PROBLEM

Improper integral I defined by equation (1) can be written in the following form:

where I = I , + I ,

I , = I T g (y,o,s) cos (xs) ds 0

I , = lm y (Y,o,s) cos (XS) ds T

Parameter T is chosen in such a way that all the zeros of the integrand function g must occur for ~ ( 0 , T) . The integrand of I , exhibits sharp corners within its range of integration. Improper

Figu

re 1

. N

orm

aliz

ed re

al p

art (

solid

line)

and

imag

inar

y pa

rt (d

ashe

d lin

e)

of an

inte

gran

d fu

nctio

n g =

1/R

(s) c

os(x

s) a

s a fu

nctio

n of

s. Pa

ram

eter

s: x

= 10

, w=

1 s

-',

QP=

25, Q

,=50

. If

not

stat

ed d

iffer

ently

the

fol

low

ing

para

met

ers

are

assu

med

: cs*

= 1,

cp*

= 2,

p* =

1

Figu

re 2.

Nor

mal

ized

real

par

t (so

lid lin

e) an

d im

agin

ary p

art (

dash

ed lin

e)

of a

n in

tegr

and

func

tion g

= l/

R(s

) cos

(xs)

as a

func

tion

of s

for x

= 1

and

w

=1

s-1


I. S 2.5 5

Figure 3. Normalized real part (solid line) and imaginary part (dashed line) of an integrand function g = l/R(s)cos.(xs) as a funct ionofsforx=l andw=0.314s- '

integral I, incorporates an oscillatory integrand which attenuates with increase of the integration variable s.

EVALUATION OF IMPROPER INTEGRAL I ,

Integral I, is evaluated numerically by using a procedure due to Longman.' This technique is very effective for calculation of improper integrals where the corresponding integrand oscillates about zero. The algorithm for numerical evaluation of integral I, is based on the result

(4)

where the finite forward differences A'V, are based on integrals of g cos(xs) between its zeros.13 The quadrature for evaluation of the integrals in equation (4) is chosen to be the same as the one used for evaluation of the finite integrals I,, which is considered next.

ILg1/2 V0-1/4AV,y+1/8A2V0- . . .

EVALUATION O F FINITE INTEGRAL I,

Integral I, is evaluated numerically by using four different quadratures: (i) polynomial integration, (ii) spline integration, (iii) adaptive Gauss integration and (iv) Clenshaw-Curtis integration. For

HARMONIC LINE LOADS 827

each of the methods the entire interval [0, 7'l is assumed to be subdivided into N segments with a particular quadrature being applied to each of them. For simplicity, the quadratures over a single segment only are discussed. An outline of each method is presented as follows.

Polynomial integration

Along each of the segments, the integrand function g is replaced by a quartic polynomial so that the segment contribution to the integral I, can be expressed in a Filon type formula

where [a, b ] denotes the segment under consideration. Calculation of the coefficients ak requires evaluation of the integrand function g at five different locations sk, k = 04, within the segment. Solving the resulting system of equations in a closed form, the approximating polynomial is then integrated analytically over the segment according to equation (5).

To avoid poor approximation of the integrand function at the ends of each segment it is assumed that the function is well approximated by the polynomial only at the inner nodes sl, s2 and sj . The nodes so and s4 of the current segment are to be incorporated in the analysis of the previous and the subsequent segments, respectively. It should be noted that, for the sake of simplicity, equation (5 ) is written in a way which does not reflect this 'overlapping' of the segments in the algorithm.

Standard accuracy criteria would require re-evaluation of the integral by using an increased number of subdivisions for the segment under consideration. However, it was shown by Apsel and Luco6 that recalculation of the integral can be avoided by considering instead the magnitudes of the expansion coefficients a k of the current calculation. If the coefficients satisfy certain error criteria the value of the integral is accepted as the final one. Otherwise, the quadrature is repeated over the half of the original segment. The procedure is continued until the error criterion for the accuracy of the integration is being met. The algorithm for the quadrature has been developed in such a way that all previously calculated values of the integrand function can be used in the subsequent calculations.

This approach has been successfully employed by Apsel' and Apsel and Luco6 in evaluation of the Green functions for a three dimensional layered viscoelastic half space subjected to harmonic point loadings.

Spline integration

In this quadrature, the integrand function is replaced by a cubic spline over each segment of integration. Initially, the segment is subdivided into n parts. Integration of the product of the cubic spline and the circular function is performed analytically, resulting in a Filon type formula, similar to the one obtained in the polynomial integration procedure,

where [si, si+ 1] denotes the ith subdivision of the segment under consideration. The coefficients aJi) are determined for each subdivision by requiring that (i) the spline approximation is continuous and has continuous first and second derivatives at all interior nodes within the segment, (ii) the approximation should match the integrand function g at all the nodes of the segment and (iii) the second derivative of the spline approximation is set to zero at the first and last


node of the segment. The resulting tridiagonal system of equations for the unknown coefficients is solved numerically to provide an initial estimate of the integral according to equation (6). Next, the number of subdivisions n is increased to 2n and the calculation is repeated. In this way, all the previously calculated values of the integrand function can be reused in the subsequent calculations. If the difference between two subsequent calculations is less than a prescribed tolerance the value of the integral is accepted as the final one. Otherwise, the number of subdivisions is increased. When the number of subdivisions reaches nmax and the error criterion has not been met, the segment is halved and the entire procedure is repeated for each half of the segment. This process is continued until the desired accuracy criterion of the calculation is met.

The advantage in using spline interpolation, as opposed to interpolation by the polynomials of high degree, is that splines tend to have oscillations of smaller amplitude between the interpolating points. l4

Adaptive Gauss integration

resulting in the following formulae: In this procedure, the Gauss-Legendre quadrature of order M is applied to each segment,

M

Sk= 1/2 [ (b - U)Zk -k b -k U ] (7b)

PM(zk)=@ k = 1, 2 , . . . (74

wk=2/(1 -zk2)~pM'(zk)12 (74 where [a, b] denotes the segment under consideration, P&) is the Legendre polynomial of order M, the prime denotes differentiation, and zk is the kth zero of P,(Z).'~ The algorithm for Gauss integration can be summarized as follows. First, the Gauss-Legendre integration rule of order M is applied to the segment. Then, the order of approximation is increased to M, and integration repeated. If the value of the integral for two successive orders differs less than the prescribed tolerance the calculation is terminated. Otherwise, the order is increased. If the order of integration reaches M,,, and the error criterion is not met the segment is halved and the entire procedure is repeated for each half of the segment.

A similar technique has been successfully used by Kundu' and Kundu and Ma19 for evaluating the response in a three dimensional elastic layered half space due to a dislocation source. Dravinski and Mossessian" utilized this approach for study of surface ground motion due to subsurface inclusion of arbitrary shape and harmonic surface line loadings.

Clenshaw-Curtis integration

Clenshaw and CurtisI6 introduced a method for evaluating a definite integral by expanding the integrand into a finite Chebyshev series and integrating the terms in the series one by one. This has been shown in practice to be an efficient method for evaluating In this method the range of integration over each segment is changed to [ - 1,1]. Therefore, the integrand function of I , (recall, I s has been defined as a segment contribution to the total integral I,) can be approximated by a finite Chebyshev series according to

M

F(z)r C" D,T,(z); - 1 < z < l m = O


where T,(z) is an mth order Chebyshev polynomial of the first kind and C" denotes a finite sum whose first and last terms are to be halved. The coefficients are calculated from the equations'6* l 7

M,, D, = 2 / M x fkcos(mkx/M) (9a)

(9b)

k = O

fk = F (COS (kn/ M)) Therefore, the segment contribution to the integral I, can be written as

I , = g(s)cos(xs)ds = ARe s: where

A=(b-a)/2

B = ( a + b)/2

s = A z + B

and [a, b] denotes the segment under consideration. It can be shown that the following recurrence relation for l,(t) is valid:7

Im+,(t)=2i/t(m+ l)l,(t)+(m+ l)/(m- l)~,-,(t)+2i/[t(m-1)] [e''+(-l)"e-"]; m > 1 (12)

Since the integrals I o ( t ) and Z , ( t ) can be evaluated in a closed form, the recurrence relation (12) provides the means of evaluating the integral I, very efficiently. The Clenshaw-Curtis algorithm can be summarized as follows. Over each of the segments the integrand is approximated in terms of Chebyshev polynomials of order M = 4. Integration of the polynomials is done analytically. If the error criterion for the accuracy of the integral is not being met the order of integration is increased to M = 8 and the error criterion is checked again. If the error criterion for the accuracy of the integral is not satisfied the entire procedure is repeated for each subdivision between two adjacent abscissae (for M = 8).

Once the integral is evaluated by using the Clenshaw-Curtis quadrature the accuracy of integration may be established before any further integration by checking the magnitudes of the coefficients in equation (9). OHara and Smith" showed that an index of possible maximum integration error may be given in terms of expansion coefficients D, (see equation (9)). Numerical testing of the accuracy of integration provided the following tolerance of integration:

~=OW05max(D,/20, (2D3-D,)/96} for M = 4 (1 3 4 E =0.001 max{D9/36, (20, - D9)/224, (D, - D7)/640} for M = 8 (1 3b)

where each term in { } is taken in the absolute value sense, and E <

The Clenshaw-Curtis approach has been successfully employed, in a form slightly modified of the one used in this paper, by Xu and Ma17 for evaluating the spectral response of a layered solid subjected to a transient source. Dravinski and Mossessian" utilized the same approach for evaluating the amplification of strong ground motion due to the presence of dipping layers of arbitrary shape when subjected to incident plane harmonic P, SV and Rayleigh waves.


TESTING PROCEDURES

Testing the numerical accuracy of different quadratures used for evaluation of the Green functions is done for surface line loads and for embedded line sources separately.

Surface load tests

For the surface line loads at very low frequency the results are tested against the corresponding static problem known as Flamant’s problem.’’ The results of the testing for the stress field are summarized in Table I. The subscripts H and V in Table I refer to horizontal and vertical surface line load, respectively. It is apparent that at very low frequency for all quadratures there is very good agreement between Flamant’s solution and the corresponding elastodynamic solution.

Embedded sources tests

For embedded line sources the numerical results for different quadratures are tested against the numerical results of Wong2’ who used a modification of Filon’s method14 and contour integration to evaluate numerically the response of a perfectly elastic half space subjected to embedded harmonic P and SV line loads. The results of the comparison are summarized in Table TI. Here, the numbers in parentheses represent the real and imaginary part of a displacement component. It is evident from Table I1 that the results obtained in this investigation match very closely those of

Table I. Comparison of the results for different quadratures ( P = polynomial; S =spline; G = Gauss; CC = Clenshaw-Curtis) for dynamic stress field ( x lo6) at frequency w=O.OOl s - ’ and the corresponding static solution.

Data: Qp=Q,=500, cp*=2, c,*=p*=l, x = l , y = l

Static P S G cc

-gxxH 159155 159151 159183 159164 159148 -oYYH 159155 159176 159122 159130 159152 - o x y H 159155 159185 159151 159185 159153 -oxXv 159155 159825 157869 160362 158005 - 0 Y Y V 159155 159137 159160 159126 159153 - O X Y v 159155 159176 159122 159130 159152

Table 11. Comparison of the displacement field ( x lo6) due to embedded P and SV harmonic line sources calculated by using different quadratures and the results of Wong” (W = Wong; P = Polynomial; S = spline; G=Gauss; and CC=Clenshaw-Curtis). Data: Qp=Qs= 1000, cp*=2, c,*=p*=l, x = y =,f=1, o = l s - ’

W (217791, 611877) (433731, 178638) (176478, 019148) (1127890, 895556) P (217577, 611740) (434568, 177025) (176197, 019134) (11 13360, 884673)

(217591, 611745) (433229, 178587) (176167, 019143) (1126640, 894875) S G (217585, 611755) (433215, 178585) (1 761 59, 019 136) (1 126600, 894992) cc (217492, 61 1941) (433244, 178531) (175663, 019131) (1126840, 895001)

HARMONIC LINE LOADS 83 1

10 -

5 -

Wong.20 Comparative testing of the results has been done for a wide range of parameters involved in the problem. For the sake of brevity only a few results are reproduced here.

This concludes the testing of the algorithms used for evaluation of the Green functions for an elastic half space and harmonic line loadings.

n + + + + A 0

A A A A A 0 0

D o + + + n D A

+ A 0 0 + + A A A o o o o

+ A A 0'

+ + + A A $ $ o o o

G o A A ~ G '

EVALUATION AND COMPARISON OF RESULTS

To reduce the number of figures the numerical results are presented only for a single 'observation point', i.e. (x, y) =( I, 1). The improper integral corresponding to the horizontal component of the displacement field due to an embedded harmonic P line source is considered first. The integral is evaluated at different frequencies by using four different quadratures as displayed by Figure 4. The number of integrand (or integrand function) evaluations is defined as a measure of quadrature efficiency in calculation of the improper integrals. Throughout the paper, this number is normalized with respect to the number of integrand function evaluations of the Clenshaw-Curtis quadrature for the first data entry, i.e. for o= 1 s - l in Figure 4.

It is evident from the results of Figure 4 that the numerical efficiency of the quadratures may vary markedly from one another. The Clenshaw-Curtis quadrature appears to be the most efficient one while the polynomial integration quadrature is the least efficient one. It should be pointed out that, owing to the normalization of the number of function evaluations, the results of Figure 4 may be

25 -

2 0 -

NFE

15.

0 0

O D 0

0

O D 0 t

0' W

0 10 20 30 40

(I/S)

Figure 4. Number of function evaluations (NFE) as a function of the frequency of the input loading for different quadratures used to evaluate the horizontal component of the displacement field due to embedded harmonic P line source. Throughout the paper circles, triangles, plusses and rectangles correspond to ClenshawCurtis, Gauss, spline, and polynomial quadratures, respectively. The results are normalized with respect to the number of function evaluations of ClenshawXurtis quadrature at o=l s-'. Unless stated differently the following parameters are assumed: x= 1, y = 1,

j = l , QP=Qs=lCQ

832 M. DRAVINSKI A N D T. K. MOSSESSIAN

misleading unless interpreted carefully. For example, at a= 1 s - ' the numerical efficiency of the Gauss and Clenshaw-Curtis quadratures appears to be very close (the actual difference in the efficiency is about 50 per cent). This is only because the number of integrand evaluations for the polynomial quadrature is so large that when displayed in the same scale with the corresponding number for the other quadratures the difference between the more efficient quadratures becomes less obvious. It is interesting to observe that the Gauss quadrature offers a reasonably competitive approach in comparison to the Clenshaw-Curtis one. The spline quadrature is considerably less efficient than the Gauss and the Clenshaw-Curtis procedures, while the polynomial integration approach is the least efficient of the four used in this study. It should be mentioned that the code for the polynomial quadrature is considerably more complicated than the corresponding codes for the other three.

To assess the influence of the parameter x (see equation (1)) upon the number of integrand evaluations, the improper integral corresponding to the horizontal component of the displacement field due to a harmonic P line source is considered again. The results shown by Figure 5 display the number of integrand evaluations for different quadratures as a function of x. These calculations serve as an additional check for the algorithms used for evaluation of the improper integrals. Namely, for a Filon type method the number of function evaluations should remain constant for all x's.

As expected, for the Filon type quadratures (polynomial, spline and Clenshaw-Curtis formulae) the number of integrand evaluations remains the same for all the x's. There are, however, small deviations from a constant for the number of integrand evaluations for the spline quadrature. The reason for these deviations lies in the accuracy criterion used for calculation of the improper integral. Namely, for the polynomial and Clenshaw-Curtis procedures this accuracy criterion is based on the size of the expansion coefficients which are independent of the parameter x. The accuracy criterion for the spline method, on the other hand, is based on the numerical comparison of the actual values of the integral (e.g. for two different numbers of subdivisions) which contain the parameter x. This causes a small deviation from a constant in the number of function evaluations for the spline procedure. It is evident from the results of Figure 5 that increase of the parameter x causes the Gauss quadrature to become more and more inefficient. Still, for the range of x considered here the overall efficiency of the method remains better than that of the polynomial and the spline methods.

n n o o o n o n n o o o o o o o o n n o .I NFE 4 1

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 ,x 0 10 20 30 40

Figure 5. Number of function evaluations as a function of the parameter x for different quadratures when applied for evaluation of the horizontal component of the displacement field due to an embedded P line source of frequency w = 1 s c l

HARMONIC LINE LOADS

25 -

o n n + + +

20 I

833

.- 1 0

n o ? 5 0

+ o o D

NFE I ~

n +

0 ‘ - , 1 w 0 10 20 30 40

(11s)

Figure 6. Number of function evaluations as a function of the frequency for the vertical component of the displacement field due to an embedded P line source

The results depicted by Figure 6 correspond to the vertical component of the displacement field due to an embedded P source. The numerical efficiency of various quadratures remains analogous to the one for the horizontal displacement component. The same is true for the results corresponding to the displacement field due to an embedded SV source displayed by Figures 7 and 8. The number of integrand evaluations for the displacement field due to horizontal and vertical surface line loadings is presented in Figures 9-12.

From the results of Figures 4-12 one can see that, for the quadrdtures tested in this paper, regardless of the nature of the loading or the type of the displacement component, the most efficient quadrature is the Clenshaw-Curtis one. The main reason for this lies in the fact that as the number of subdivisions increases no previously calculated data are being wasted and the accuracy of the results may be established before any further integration by checking the magnitudes of the current expansion coefficients. Although the same is true for the polynomial quadrature approach it appears that the latter is far less efficient in evaluation of the improper wavenumber integrals. The efficiency of the adaptive Gauss quadrature is quite good, while the application of the polynomial and the cubic spline quadratures appears to be a less efficient way of evaluating the Green functions.

Finally, for the sake of completeness a sample set of the numerical values for the Green functions is presented next. The results of Figures 13 and 14 correspond to the displacement components due to vertical and horizontal surface line loads, respectively. The results presented in Figures 13 and 14 may serve as a check for any new quadrature developed for evaluation of the Green functions for a viscoelastic half space.

834 M. DRAVINSKI A N D T. K . MOSSESSIAN

u + 0 0

n + a 0 o + a 0 O f a 0

o + a 0 o + a 0

n + a 0 o + 0 0

a + a 0 O+ Q O

o + a 0 o + a 0 o + Q O

o + a 0

3 9

+ a0 D a 0

m a 0 a + a0

O + Q O

0

3

0 + 0 +

0 + 0 + 0 + 0 +

n + 0 +

o + o t

0 0

a 0 a 0 a 0 a 0 0 0

0 0

0 0

a 0

a 0 o + a 0

o + a 0 a + a 0

O f Q O

o + 0 0

o + a 0 n + a o

a + a 0 o+ a 0 0 + 0 0

0 U

c

2 s -

0

0


0 3

1 + -30

C + 0 0

0 + a0 0 + a 0 0 0 0

0 + QO

0 -c 40

0 4. 00

n a0

@ + ao

0 * 00 0 1.43

c 0 4 4 3

0 + a 0 La3

n + a 0 0 t a0 - a + a o

c 4. 0 0 I

(u n s w s P v) 0

LL z f;l

836 M. DRAVINSKI AND T. K . MOSSESSIAN

e 0

0

E

C 0

@

C

0

0

0

0

+ c3 + co

+ CO

i QO

+ a0 + a i a + a 3

+ a f a 3

+ a +a +a

D +a 0 + Q

0 + a 0 +QO

0 f Q O

o + ac + a0

10 0 10 0 N N u ?

b. 2

8

0

0

n 0

0

0 0

CI

0

0

0

0 + Q O

+ a o 0 +43 0 + Q O

0

+ Q O

+ a o

+ a 0

+ a 0

+ Q O

+ Q O

+ Q O

+ a0 + ao + QO

+ a + QO

+ a0

0 +Q3

U +a 0 + QO


-0.25 ' , w

0 4 6 12 16 20 ( I / s l

Figure 13. Horizontal and vertical components of the surface displacement field due to a vertical harmonic unit line load at the origin of a half space as a function of the frequency. Parameters: x = 1, y=O

Re(u)----- Re ( v ) -. - - Im(u1- - - Im(v)-------

0.2

-0-3 ' , w 0 4 a 12 16 20

(I/s)

Figure 14. Horizontal and vertical components of the surface displacement field due to'a horizontal harmonic unit line load at the origin of a half space as a function of the frequency. Parameters: x = 1, y = O

CONCLUSIONS

Four different quadratures (polynomial, spline, adaptive Gauss and Clenshaw-Curtis) have been examined to assess their efficiency in evaluation of the Green functions for a viscoelastic half space and harmonic line loadings. The Green functions involve improper integrals which require special care to guarantee their accurate evaluation on a digital computer.

The algorithm for each quadrature is presented in detail and tested against some known results. The number of integrand evaluations is introduced as a measure of numerical efficiency of the


quadratures under consideration. The numerical results presented indicate that the Clenshaw- Curtis quadrature is the most efficient one.

ACKNOWLEDGEMENT

During the course of research presented in this paper a number of people contributed directly or indirectly to this work. The authors are grateful to Ajit K. Ma1 for numerous discussions on the numerical evaluation of the Green functions. Discussions with J. E. Luco in the early stages of research are greatly appreciated. Special thanks are due to H. L. Wong, for providing the authors with his Green functions program which has been used for validation of the results, and Randy J. Apse1 who made his Ph.D. thesis available. Richard E. Kaplan was instrumental in making the numerical calculations possible.

This research has been supported in part by a contract 14-08-0001-22023 from the United States Geological Survey.

APPENDIX

Green functions for a two-dimensional half space

c, and S wave speed c,. Attenuation of the medium is defined through21 Consider an elastic half space { - co < x < 00; 0 < y < co} with shear modulus p, P wave speed

l/c,= l/c,*(l -i/2QP) (All

l/c, = l/c,*( 1 - i/2Q,) (A21

where cp* and c,* are intrinsic P and S wave velocities in the absence of dissipation, Q, and Q, are the corresponding measure of attenuations and i = ,/ - 1. Introduction of the attenuation makes the model more realistic since it is well known21'22 that as a wave propagates through real materials its amplitude attenuates owing to internal friction. Observations in seismology show" that Q, and Q, are effectively constant over a wide range of frequencies, hence in this paper Q, and Q, are taken to be frequency independent. Furthermore, introduction of complex velocities affects the numerical integration greatly since this removes the Rayleigh pole from the real wavenumber axis.I2 This allows application of numerical quadrature without resorting to contour integration.

Vertical surface line load

For boundary conditions at y = 0 defined by

cryv = - 6(x)eiW'

crXY = 0

the resulting displacement and stress fields23 can be written in the following form:

u = Y {ce-"Y - ] (A51 u = %? (a(ce-"Y - 2s2 e-b'')} ('46)

cr,, = V { ~ d e - " ~ - 4s' abe-by} ('47)

crXy = - 2 9 {ac(e-"Y - e-by)} ('48) cry, = - %' { ~ ~ e - ~ y - 4s2 abe-bY 1 649)


where

b =(s2 - k2)1/2

c=2s2 - k 2

d=2(s2-hZ)+k2 (A 13) and h and k denote the wavenumbers associated with P and S waves, respectively. The components of the stress tensor are presented normalized with respect to the shear modulus p. The operators 9' and % are defined by

Y { } =(np)-' 1; { }/R(s)s sin(xs)ds (A 14)

Po0

59 { ) = (np)- J { } / R ( s ) cos (xs)ds 0

R(s)=c2-4s2ab (A 16) The branch cuts for a and b are chosen such to guarantee outgoing, bounded waves at infinity.

Horizontal surface line load

For boundary conditions at y=O defined by

rJyy = 0

oXy = - b(x)eio'

the resulting displacement and stress fields23 can be written in the form

Embedded P wave source

For a harmonic P line source located at (0,n the corresponding displacement potentials23 can be written as

~=H,(2)(hr)-H,'Z)(hr')-4V,{s2be-"Y} (A24)

+ = 2 ,4Pp { rz = x2 + (y -f)2; rt2 =x2 + (y +fl2

)

where H,") is a Hankel function of the second kind of order zero. The factor exp(iot) is omitted for brevity. Operators 9, and Wp are defined by

P a 9, 5P( ) = 4i/n J { }/R(s)e-"ls sin (xs)ds

0

840 M . DRAVINSKI AND T. K. MOSSESSIAN

W,,{ }=4i/n{r { }/R(s)e-“Scos(xs)ds

The resulting displacement and stress fields are specified through

u = - hx[ 1 jrH l(z)(hr) - 1 /r‘H1(z)(hr’)] + 2 9 , (h(2~’eC“~ - cepby 1) (A29) u = - h[(y -f)/rH1(’)(hr) + (y +f)/r’Hl(’)(hr’)] + %?, {c(ce-”” - 2sZe-’’)}

oxx=2h[l/r- 2(y-f)z/r3]H1(z)(hr)+ [2(h(y-f)/r)’- kZ]H0(’)(hr)

(A30)

- 2h[l/r’- 2(y +.f)z/r’3]~,(Z)(hr’) - [2(h(y +f)/r’)’ - kz]Ho(’)(hr’)

+4%?:,{sZb(de-“~-ce-’~)} (A311

rrXy=4hx(y -f)/r3H1(’)(hr) - 2(h/r)’x(y -.f)H,(’)(hr)

+ 4hx(y +f)/r’3 H ,(’)(hr’) - 2(h/r’)’x(y +f)H,(’)(hr’)

- 2 .Yp { c2 (e -“y - e - ’y ) } (A32) oyp = 2h(1 / r - 2 ~ ’ / r ~ ) k ~ ~ ( ~ ) ( h r ) + [2(hx/r)’ - k’] H,“)(hr)

- 2 h ( l / r ’ - 2 ~ ~ / r ’ ~ ) H ~ ( ~ ) ( h r ’ ) - [2 (h~ / r ‘ )~ - kz]H0(2)(hr’)

-4VpCs2bc(ep“v-e-bv)} (A331

Embedded S V wave source

For a harmonic SV line source in a half space located at (0 , f ) the corresponding displacement potentialsz3 can be written in the form

4= -2Y,(ce-“Y} (A34)

(A33 t,b =H,“)(kr) + H,‘z)(kr’) - %‘s { c2/be-by}

where operators Ys and Vs are defined by

Y s { }=4i/n{I { }/R(s)e-bSssin(xs)ds

W S { }=4i/nj: { }/R(s)e-bfcos(xs)ds

corresponding displacement and stress fields are specified by

u= -k[(y-f)/rH,(Z)(kr)+(y+f)/r’Hl(z)(kr‘)] + Ws{c(ce-by-2sZe-ay)} (A38)

u = kx( 1 /rH ‘”(kr) - 1 /r’H l(2)(kr‘)] + 2 YS { a(ce - a y - 2s’ e - ’J’ )} (A39) oxx =4kx(y -f)/r3HH,(2)(kr) - 2(k/r)’x(y -f)H,(”(kr)

+ 4kx(y +f)/rf3H1(*)(kr’) - 2(k/r’)’x(y +f)H,(z’(kr‘)

+ 2.Ys {c(de-OY - cePby)} (A40) ox,,= [ I - 2(x/r)’] [2k/r~,(”(kr) - kZH0(’)(kr)]

- [l - 2 ( ~ / r ’ ) ~ ] [2k/r ’H,(2) (kr’ ) -kZH0(2)(kr’ ) ] +4~s{sza~(e -”y -e -by)} (A41)

HARMONIC LINE I.OADS 84 I

REFERENCES

I. R. J. Apsel;Dynamic Green’s functions for layered media and applications to boundary value problem’, Ph.D. Thesis, University of California at San Diego, La Jolla, California, 1979.

2. M. Dravinski. ‘Evaluation of strong ground motion using boundary integral equation approach’, in S. K. Datta, (ed.), Earthquake Source Modeling, Ground Motion, and Structural Response, ASME, AMD Vol. 60, 1984, pp. 61-80.

3. T. A. Cruse. ’A direct formulation and numerical solution of the general transient elastodynamic problem. II’, J . Math.

4. D. M. Cole, D. D. Kosloff, and J . B. Minster, ‘A numerical Boundary integral equation method for elastodynamics’, Bull. Seism. Soc. Am.. 68, 1331-1357 (1978).

5. S. Kobayashi, ‘Some problems of the boundary integral equation method in elastodynamics’. in C. A. Brebbia el ul. (eds.), Boundary Elements, Proc. Fijih Inr. Cot$ Hiroshima, Japan, Springer -Verlag. New York, 1983, pp. 775--784.

6. R. J. Apse1 and J. E. Luco, ‘On the Green functions for a layered half space. Part II’, Bull. Seism. Soc. Am., 73,931-951 ( 1983).

7. P. C. Xu and A. K. Mal, ‘An adaptive scheme for irregularly oscillatory functions’, Waoe Motions, 7,235 -243 (1985). 8. T. Kundu, ‘Computation of surface motion in a stratified half space’, Ph.D. Thesis, University of California at Los

9. T. Kundu and A. K. Mal, ‘Elastic waves in multilayered solids due to a dislocation source’, W a w Motions, 7,459471

10. M. Dravinski and T. K. Mossessian, ‘Scattering of plane harmonic P, SV, and Rayleigh waves by dipping layers of

1 1 . M. Dravinski and T. K. Mosscssian, ‘Amplification of surface ground motion by an inclusion of arbitrary shape for

12. A. Ben-Menahem and S. J. Singh, Seismic Waves and Sources, Springer-Verlag, New York, 1981. 13. I . M. Longman, ‘Note on a method for computing infinite integrals of oscillatory functions’, Proc. Cambridge Phil. Soc.,

14. P. J. Davis and P. Rabinowitz, Methods ujNumerical Inlegrution, Academic Press, New York, 1975. 15. M. Abramowitz and I. A. Stegun, Handbook of Mathematical functions, Dover, New York, 1972. 16. C. W. Clenshaw and A. R. Curtis, ‘A method for numerical integration on an automatic computer’, Numer. Math., 2,

17. H. O‘Hara and F. J. Smith. ‘Error estimation in the Clenshaw-Curtis quadrature formula’, Comp. J . , 11,2I 3-219 (1968). 18. H. O H a r a and F. J. Smith, ‘The evaluation of definite integrals by interval subdivision’, Comp. J., 12, 179-182 (1969). 19. Y. C . Fung, Foundation of Solid Mechanics, Prentice-Hall, Englewood Cliffs, New Jersey, 1965. 20. H. L. Wong, ‘Dynamic soil structure interaction’, Report EERL-75-01 Earthquake Engineering Research Laboratory,

21. K. Aki and P. G. Richards, Quantitutice Seismology, Vol. 1 , W. H. Freeman, San Francisco, California, 1980. 22. L. Knopoff. Q. Rev. Geophys. 2,625-660 (1964). 23. H. Lamb. H.. ‘On propagation of tremors over the surface of an elastic solid’, Phil. Trans. Roy. Soc. London, Ser. A 359,

Atlal. Appl., 22, 341-355 (1968).

Angeles. Los Angeles, California, 1983.

( 1985).

arbitrary shape’. Bull. Seism. Soc. Am., 77, 212-235 (1987).

harmonic line loads’, Numer. Methods Part. Di# Eqs.. 3, 9-25 (1987).

52,164 -768 (1956).

197-205 (1960).

California Institute of Technology, Pasadena, California, 1975.

203, 1 4 2 ( 1904).

on evaluation of the green functions for harmonic line loads in a viscoelastic half space

Documents