Comput. Math. Applic. Vol. 16, No. 1/2, pp. 139-152, 1988 0097-4943/88 $3.00+0.00 Printed in Great Britain. All rights reserved Copyright © 1988 Pergamon Press pie

F I N I T E E L E M E N T M E T H O D F O R T S U N A M I W A V E

P R O P A G A T I O N A N A L Y S I S C O N S I D E R I N G T H E

O P E N B O U N D A R Y C O N D I T I O N

M. KAWAHARA, I T. KODAMA 2 and M. KINOSHITA I

IDepartment of Civil Engineering, Chuo University, 13-27 Kasuga l-chome, Bunkyo-Ku, Tokyo 112, Japan

2Engineering Research Institute, Sato Kogyo Co. Ltd, 47-3 Sanda, Atsugi, Kanagawa 243-02, Japan

Abstraet--A method for the treatment of the open boundary condition in the shallow-water equation to analyze the propagation of tsunami waves using a finite element method is proposed. The key feature of this method is that the analytical solutions of the linear shallow-water equation can be linked with the inner values of the velocity and the water elevation on the open boundary. The two-step explicit method is used to discretize the time function, which has an advantage in problems treating large numbers of elements and unsteady state. Two test examples have been carried out. The numerical results are in good agreement with the analytical solutions. The Tokachi-oki Earthquake tsunami has been computed. The computed results have been compared with the observed tide gauge records. From these comparative studies, it can be concluded that the present method is a useful and effective tool in predicting tsunami generation and propagation.

1. I N T R O D U C T I O N

Tsunamis are long-period waves created by an earthquake occurring in a deep sea bottom. The sea surface is raised suddenly due to the deformation of the sea bottom caused by the earthquake. Subsequently, tsunami waves propagates towards the coast. These waves often cause tremendous damage to property in populated regions and take human lives. Therefore, prediction of tsunami generation and propagation is very important to protect human lives and minimize damage.

A number of numerical methods have been presented in the analysis of tsunamis [1]. Aida [2-4] analyzed tsunami wave propagation using a finite difference method. Kawahara et al. [5-7] presented the two-step explicit finite element method to solve the shallow-water equation.

This paper deals with a new finite element technique to treat open boundaries in the analysis of tsunamis. As the wavelength of a tsunami is rather long, a fairly large-scale area needs to be taken into account in the numerical analysis. However, if a boundary condition can be set up on the open boundary, which is comparable to the effect of an infinite area, the size of the area to be analyzed can be reduced. In this paper, the analytical solutions of the linear shallow-water equations have been used to describe the open boundary condition. Two test examples and the Tokachi-oki Earthquake tsunami are computed by the present method.

2. T H E B A S I C E Q U A T I O N S

The definition sketch of the analytical domain is shown in Fig. 1. The infinite flow region f~ is divided into two domains, the inner f~L and outer f~o domains, by the open boundary Fo. The open boundary FL is the boundary between land and F~ is the infinite boundary. Here and henceforth, equations are expressed using indicial notation representing coordinates Xi (i = 1, 2) and time t, and a conventional summation convention with repeated indices is used.

Tsunami wave propagation in a deep-water region can be expressed by the linearized shallow- water equation. The equations of motion and continuity are described in the following form:

~t +g~L~= 0 (1)

on

a t + hU,,, = 0 (2)

139

140 M. K A w ~ et al.

, t '%

i / %

Ix\ ~ Po /11 1",=. \ /

l %

Fig. 1. Definition sketch of the analytical domain.

where Uj and ~/denote mean velocity and water elevation respectively, and h and g are sea depth and gravity acceleration, respectively. The following two types of boundary conditions are considered. On the boundary FL, the velocity is assumed to be known:

Ui = U, on FL (3)

where ^ denotes a prescribed value on the boundary. The continuity conditions for velocity and water elevation on the open boundary Fo are defined as

,4, on Fo

'1 = ( 5 )

where the overbar indicates the general solutions.

3. ANALYTICAL SOLUTIONS OF THE GOVERNING EQUATIONS

The velocity and water elevation of the shallow-water equation can be expressed in the following form:

Ui = Ui exp ( - togt)exp(fk, x,) (6)

and

= r 7 exp(-- togt)exp(tk~x~) (7)

where Ui and ~ represent the amplitudes of velocity and water elevation respectively, co, ki and t denote angular frequency, the components of wavenumber k and the imaginary unit, respectively. The following equations are obtained by substituting equations (6) and (7) into equations (1) and (2):

hki -Og / L C1 .I

Here, the determinant of equation (8) must be zero in order that 0~ and r7 have nontrivial solutions, which is written as

co ((o2 _ ghk ~) = 0. (9)

The angular frequencies which satisfy equation (9), are obtained as follows:

o91 = 0, o92 = Ck, o h = - Ck, (10)

Prediction of tsunami generation and propagation 141

where C = ~ is the wave velocity. The analytical solutions are expressed as the sum of a sinusoidal function, which can be introduced to allow for the angular frequency, and using equations (6)-(9). The analytical solutions are shown as follows:

= n=l rlo~ 1 ; p(-Ront)exp(ikixi) (11)

in which if3 = flnC(kn) ~ are unknown constants. Equation (l l) can be represented as follows:

~, = W~aff0 ~ (12)

where ~b~ = (U~, ~/)r, ~ = ( 0 , ff)r and W~ denote the coefficients. On the other hand, the nodal values on the open boundary of velocity and water elevation which are discretized in the inner domain, are expressed as follows:

Jp~ = T~vwy (13)

where wv = (Up;, r/p) r and Tx~ denote the coefficients of the respective interpolation functions. The continuity equations on the open boundary can be denoted as follows:

ro(¢~ - ~,) d r = 0. (14)

The unknown constants can be obtained as the following equation using equations (12)-(14):

~120 = W ~ 1T~vwv. (15)

4. THE FINITE ELEMENT METHOD

Equations (1) and (2) are transformed into the following weighted residual equations employing the conventional finite element Galerkin method:

fo . ,16, U * - ~ - d f l - g ~,jr/dr) + g = I 1

and

, -~ d f l - h i rl~U, dD+ h n*O,n, dr = 0 (17)

where U? and p/* denote weighting functions for the velocity and water elevation respectively, and nj denotes the components of the unit normals. The finite element equations (16) and (17) can be described in the following form, employing a linear interpolation function based on triangular finite elements:

M~p 0p, - H~,at/p + S~,~rT~ exp( - i a~ t ) = 0 (18)

and

M~p/jp -- I=,p Up, + R~/~ exp(--i~vt) = 0 (19)

where Up, and/1 a denote the velocity at the flth node of the finite element in the ith direction and the water elevation at the flth node, respectively; and

and

M.~ = fat ( ¢ ~ P ) dD,

I,,B = h ;,~, (¢~, ,¢p) dt~,

H~,a = g fal (~ ' i¢~p) dD

S~,~ = g fro { ~ exp[f(k~x,)]n, } dr

c l { ~ exp[f(k~x,)](k~n,)} dr ~ =

Jro

142 M. KAWAHARA et al.

in which O~ is the interpolation function for both velocity and water elevation. Applying equation (15) to equations (18) and (19), the following finite element equations can be derived:

M~a (1#, - H.,atl# + S~i. W~ ' T~wy = 0 (20)

and

M~afla - I~,#U#, + R~.W~'Tarwr = O. (21)

A two-step explicit finite element method is used to discretize time, which can be derived as follows:

and

first step

and

second step

A~ un+1/2__2~,1 n _~ n - W ~ T~rw~) • # #i - ~#U#, (_H~,#rl#+S~,~ --1 o

At [ n ~.#rl~+ ,/2 = IVl.arl# - --~ ( - ~,# U#, + R.. W~' TaywD;

(22)

(23)

~, H / - - I T u ,n + 1/2~ M~#U n+1 = ff'l.#U~#, -- At(--H=,#fl~#+':2 +~.~,... ~ . ~,", . Pi (24)

-- -- ~ W - I T . . . . +I/2~ (25)

In equations (22)-(25), AI, a denotes the lumped coefficient of M,# and j~r# is the mixed coefficient, which is

h~t~# = eMma + (1 - e)M~a (26)

where e is referred to as the lumping parameter.

5. TEST EXAMPLES

In this section several numerical examples are discussed to illustrate the adaptability of the present finite element method.

(1) Simulation o f progressive waves

As the first computation, wave propagation in an open channel with an uneven bottom is computed. Figure 2 shows the finite element idealization of the one-dimensional channel. For the boundary conditions, the sinusoidal wave is prescribed on the boundary A-B (with an amplitude of 0.1 m and a period of 1 s), the normal velocity is assumed to be zero on the boundaries A-D and B-C, and the analytical solutions of the outer domain are connected on the boundary C-D. The computed water elevation for the upward sloping bottom is plotted in Fig. 3. The case of the downward sloping bottom is shown in Fig. 4. The numerical solution of the present method, the standard finite element solution considering progressive wave conditions and the analytical solution are shown by ©, A and - - , respectively. It is seen that the results of the present method are in good agreement with the analytical solution.

A JJJJJJJJJJ fJlPlJJJJi /JJiJiJ///D

ff, O'Z,~fff 4"lrwwrrw~'~C gC/H" "111 lOre

Fig. 2. Finite element idealization.

Prediction of tsunami generation and propagation

AO

0.06 -/ c .o 0.02 0

-oo2 1 2 ; ' 4 ;

O

-o.o6

- O . 1 0

Fig. 3. Computed water elevation for the upward sloping bottom. (--) Analytical solution; (O) present method; (A) progressive wave condition.

0.101 .~AAAAAA

.~ 0.02 o

I I I I ~ I I I I I I I I , L , , 1 ' 1 ' ' '

1 2 3 4 5 s a 6 b -0 .02 -

X27 ~ -0.06

- 0 . 1 0

Fig. 4. Computed water elevation for the downward sloping bottom. (--) Analytical solution; (O) present method; (A) progressive wave condition.

143

(2) Simulation of the tsunami model

The second example illustrates the tsunami wave propagation problem for the simple analytical domain with open boundary conditions. For the open boundary, two types of shape (shown in Fig. 5) are employed, a circular shape and a square one. For the boundary conditions, the normal velocity is assumed to be zero on the boundaries A-B and B-C, and the analytical solutions of the outer domain are connected on the boundary A-C or A-D-C. A water elevation of 1.0 m is assumed at the wave origin, which is 25 km from point B for the initial condition. The computed water elevation contours at various times are shown in Fig. 6. This figure illustrates similar wave propagations for the two open boundary shapes. In Fig. 7, bird's-eye views of the computed water elevation at various times are illustrated. The computed water elevation of the nodal point at the epicenter is plotted in Fig. 8 as compared with the analytical solution [8] using a circular model. The computed results are in good agreement with the analytical solution. From these comparative studies, the adaptability of the present method is confirmed.

6. THE TOKACHI-OKI TSUNAMI ANALYSIS

In 1968, a severe earthquake of magnitude 7.8 occurred about 150 km off the coast of the Tokachi region, which is called the "Tokachi-oki Earthquake". Following the tremor, a tsunami was generated which struck the coastal region of northern Japan. In this section, this tsunami wave propagation problem is analyzed by the present finite element method. Figure 9 shows the finite element idealization of the regions to be analyzed. The total number of nodal points is 16,547 and the total number of finite elements is 32,386. For the boundary conditions, the reflection condition is used for the coastline; namely, the normal velocity to the coastline is assumed to be zero. The analytical solutions are connected on the open boundary. Figure 10 represents the contour lines

144 M. K A w ~ u t ~ et al.

A

~__~L/L~/q~L/V]/] /1/ 'p' ] , / ' I

O p e n boundory

° / 1

///)/; ///////I C B l, 50 km .

Fig. 5. Finite element idealization.

O p e n boundory

\

Step = 3 6 , t ime = 7 2 0 ( S ) Step = 7 2 , t i m e = 1 4 4 0 ( $ ) St'ep = 1 0 8 , t ime ffi 2 1 6 0 [ s )

Fig. 6. Computed water elevation contours.

Step = 144, time = 2 8 8 0 (s ]

of the water depth and the observation points. The seismic center region is assumed to deform suddenly, as shown in Fig. 11. The computing conditions are as follows: lumping parameter e ffi 0.9, time increment At ffi 3.0 s. Figure 12 shows the comparison between the computed and observed amplitude [9] and arrival time of the first wave at various points along the coast. The numerical results are in good agreement with the observed tide gauge records. Bird's-eye views of the initial wave and the computed water elevation at various times after the earthquake are illustrated in Fig. 13. The wave propagation can be seen clearly in this figure. From this computation, the adaptability of the p ~ o n of tsunami propagation is shown. To compute this example a CPU time of 8.0 rain/1000 steps was required on a HITAC M680H computer.

Prediction of tsunami generation and propagation

Step = 1 8 , t i m e = 3 6 0 (s) S t e p = 3 6 , t ime = 7 2 0 (s )

145

Step = 7 2 , t ime = 1 4 4 0 (s ) Step = 9 0 , t ime = 1800 (s)

Step =108 , t ime = 2 1 6 0 ( s ) S t e p f 1 6 2 t t i m e f 3 2 4 0 ( s )

Fig. 7. Bird's-eye views of the computed water elevation.

146 M. K A w ~ p , x et al.

1.00

0.80

0.60

0.40

0.20

._~ 0.o 0 @ e-

®

~ -0 .20

- 0 . 4 0

-0.6 0

0.8 0

1.00 1.20

LU El

~6 t2 1o8 ' 144 ' 180 ' 216 I I ~$~ 1 2 e 8 324 . " ~ 380

¢1 m m m T i m e ( s ) m m ~ . . . ~ - ~ - -

mlE l m rfl El

Fig. 8. Water elevation at the seismic center. ( ) Analytical solution; (O) present method (circular domain); (~ ) present method (square domain).

Prediction of tsunami generation and propagation 147

I j ....... .

"~".ii ~ :. . . ,

j...

I ("

;("

I l i , a l ~ , ........

i...< /

/ . . .,- . . . . . .

• j

/ I Jj ¢

• - " - 7

I j Q <'

Total, number of nodes : 1 6 5 4 7 Total number of elements : 3 2 3 B 6

Fig. 9. Finite element idealization.

148 M. K ~ w ~ et ai.

(7

I

..................... • . . . . . . . . ¢~... e "

1 ':'/ '":.: i

f g

Fig. 10. Water depth and observation points.

Prediction of tsunami generation and propagation 149

I n i t i a l wave

Fig. 11. Initial condition of the water elevation.

(m)

2. 0

1 . 0

0 . 0

- 1 . 01

records~

(m i n : 0 Observed records Computed results

60

45 . ~

3 0 "~"

! 5 '~ /

I f f I I I I I [ I 0 a b c d e f g h i j

Fig. 12. Computed amplitude and arrival time of the first wave.

Ho

kk

aid

o

Init

ial

wav

e

Fig

. 13

a. B

ird'

s-ey

e vi

ews

of

the

init

ial

wav

e an

d co

mpu

ted

wat

er e

leva

tion

.

-""-

T

=6

min

Fig

. 13

b. B

ird'

s-ey

e vi

ews

of

the

initi

al w

ave

and

com

pute

d w

ater

ele

vati

on.

Fig.

13c

. B

ird'

s-ey

e vi

ews

of t

he i

niti

al w

ave

and

com

pute

d w

ater

ele

vati

on.

T=

t2 m

in

3"

0 0

T=

18

min

Fig.

13d

. B

ird'

s-ey

e vi

ews

of t

he i

niti

al w

ave

and

com

pute

d w

ater

ele

vati

on.

152 M. KAWAHAgA et aL

7. C O N C L U S I O N

A new finite element technique for treating an open boundary is presented. The key feature of this method is that both the finite element solution and the analytical solution of the outer domain are connected at the open boundary. The conclusions obtained are as follows:

(1) The two-step explicit finite element equations, which treat the open boundary condition, were successfully formulated.

(2) To show the adaptability of this method, several comparat ive studies were carried out. The results o f the present method are in good agreement with the analytical solution.

(3) Applying the technique to the Tokachi-oki Earthquake tsunami propagat ion problem and comparing with the observed data, the numerical results are in good agreement with the records.

From the above conclusions, it is seen that the present method is a useful and effective tool in predicting the tsunami generation and propagation.

R E F E R E N C E S

1. H. Kardestuncer (Ed.), Finite Element Handbook (1987). 2. I. Aida, Numerical experiments for the tsunami propagation--the 1964 Niigata tsunami and the 1968 Tokachi-oki

tsunan~. Bull. Earth Res. Inst. Univ. Tokyo 47, 673-700 (1969). 3. I. Aida, Numerical experiments for tsunami caused by moving deformations of sea bottom. Bull. Earth ICes. Inst. Univ.

Tokyo 47, 849-862 (1969). 4. I. Aida, Reliability of a tsunami source model derived from fault parameters. J. Phys. Earth 26, 57-73 (1978). 5. K. Kawahara, Tsunami wave propagation analysis by the finite element method. In Proc. 2nd 'Int. Conf. on Finite

Elements in Water Resources (1978). 6. M. Kawahara, N. Takeuchi and T. Yoshida, Two-step explicit finite element method for tsunami wave propagation

analysis. Int. J. numer. Meth. Engng 12, 331-351 (1978). 7. M. Kawahara, H. Hirano, K. Tsubota and K. Inagaki, Selective lumping finite element method for shallow water flow.

Int. J. numer. Meth. Fluids 2, 89-112 (1982). 8. K. Kajiura, ' T ~ a m i ' . Hydraulics Engineering Series, pp. 66-113 (1966). 9. T. Kishi, 'Tsunami Report', Tokachi-oki Earthquake, pp. 207-215 (1968).