A numerical technique for two-dimensional transient wave propagation analyses

COMMUNICATIONS IN APPLIED NUMERICAL METHODS, VOl. 6, 623-632 (1990)

A NUMERICAL TECHNIQUE FOR TWO-DIMENSIONAL TRANSIENT WAVE PROPAGATION ANALYSES

YALCIN MENGl

AND

A . KAMIL TANRIKULU Cukurova University, Adana, Turkey

SUMMARY A numerical technique is proposed for solving two-dimensional transient wave propagation problems. The technique involves the use of the combination of a transform (such as Fourier, sine, cosine) for one space variable and the method of characteristics. Because of its inclusion of the latter method in the analysis, the proposed technique is capable of describing the sharp variations of the disturbances which may occur in the neighbourhood of wave fronts. In the study, an example problem is presented which involves transient waves in an elastic layer generated by out-of-plane shear stresses acting on its top surface. The formulation of the example problem is kept general in the sense that the variations of the applied shear tractions in the horizontal direction of the layer and in time are kept arbitrary. In the analysis, the Fourier transform is used for the horizontal co-ordinate. The numerical results are obtained and presented for the uniform strip loading case. The assessment of the proposed technique is made by obtaining the results for the case in which the width of the strip load is very large compared to the height of the layer, and by comparing them with exact ones. It is found that the proposed technique yields the results which compare very well with the exact ones and that it is capable of describing the sharp variations at the wave fronts.

1 . INTRODUCTION

As is well known, the method of characteristics is being used widely for one-dimensional wave propagation analysis in areas such as solid mechanics, gas dynamics and hydrodynamics, and its convergence and numerical stability are well established. 1,2 In the method of characteristics we reduce one-dimensional hyperbolic equations, which involve time and one space variable, to ordinary differential equations which are valid along characteristic lines defined in the time-space plane. These reduced equations, called canonical equations, can be integrated either analytically or numerically by taking into account initial and boundary conditions. However, if the wave propagation problem is multidimensional, i.e. if it involves more than one space variable, the construction of the solution by the method of characteristics becomes impractical for the following reasons: (a) for multidimensional wave propagation problems the characteristic manifold is composed of hypersurfaces, not lines as for the one-dimensional case. The analysis based on these characteristic surfaces will be complicated; (b) the canonical equations on characteristic surfaces are partial differential equations, not ordinary differential equations as for the one-dimensional case. The integration of these equations containing partial derivatives with respect t o surface co-ordinates is not easy; (c) taking into account the

0748-8025/90/080623- 10$05 .OO 0 1990 by John Wiley & Sons, Ltd.

Received 2 January 1990 Revised 8 April 1990

624 Y. MENGI AND A. K . TANRIKULU

boundary conditions involves the determination of the intersection lines between characteristic and boundary surfaces, which complicates the analysis further.

In the study, with the object of eliminating the aforementioned shortcomings of the method of characteristics for multidimensional wave propagation problems, a numerical technique is presented. The technique combines a transform technique with the one-dimensional method of characteristics. For simplicity, it is presented in this study for two-dimensional wave propagation problems. The proposed technique is applied to an example problem involving transient SH waves propagating in an elastic layer. The results of this problem indicated that the proposed technique has the capability to describe the sharp variations at wave fronts and it can be used conveniently for multidimensional transient wave propagation analyses.

2. PROPOSED NUMERICAL TECHNIQUE

In what follows, we describe a technique which can be used for two-dimensional wave propagation analysis. The technique involves first the use of a transform (e.g. Fourier transform) for one of the two space variables, then integration of the resulting one- dimensional hyperbolic equations by the method of characteristics, and finally inverting the solution back into real space. It should be noted that since the proposed technique includes the use of a transform method, its applicability should be restricted to linear wave propagation problems.

To explain the technique, first we write two-dimensional wave (hyperbolic) equations symbolically as

L u = f (1)

where u is the dependent variable, L is a linear hyperbolic operator involving the partial derivatives with respect to the time variable t and the space co-ordinates xr(r = 1,2), and f is a prescribed function of t and x,'s. Since equation (1) is a hyperbolic system, it permits the solution which might be discontinuous across some singular propagating surfaces.

We now discuss the steps of the proposed numerical procedure.

First, we apply a transform (such as Fourier, sine, cosine) to equation (1) with respect to one of the two co-ordinates, say, for XZ. This eliminates the partial derivatives with respect to XZ, and the resulting transformed equations can be written in the form of a system of first-order partial differential equations as

A u , ~ + B u , ~ + C = 0 (2)

where u is an n-dimensional vector describing the transformed dependent variables, A and B are some (n x n)-dimensional matrices, C is an n-dimensional vector and the comma designates the partial derivative. For simplicity, the co-ordinate XI is denoted by x in equation (2). The matrices A and B in equation (2) might be the functions of x, t and the transform variable k. The vector C may be the function of x , t , k and a linear function of u. It should be noted that the transform variable k corresponds physically to the wave number in the xz direction. The second step of the procedure involves the solution of the one-dimensional hyperbolic system, equation (2), by the method of characteristics for a given value of the wave number k. The canonical equations associated with equation (2) are given by

du d t

1TA -+11,'C=O (3)

TRANSIENT WAVE PROPAGATION ANALYSES 625

which is valid along the members of the ith characteristic family

d x -= Vi ( i = l - n ) dt (4)

describing curves in x-t plane. The superscript ‘T’ in equation (3) designates the transpose. The vector I; and the wave propagation velocity Vi in equations (3) and (4) are the ith eigenvector and eigenvalue of the eigenvalue problem

( 5 )

d/dt in equation (3) describes the total time derivative along a characteristic line. The canonical equations given in equation (3) constitute n ordinary differential equations each of which is valid along the curves of a different characteristic family. The solution can be constructed by integrating these equations along characteristic lines in the x-t plane and by taking into account the initial and boundary conditions.

(c) In the last step, we invert numerically the solution obtained in (b) back into real space by using the inverse transform relation. It may be noted that the numerical inversion requires the computation of the solution in the step (b) at the discrete wave number points (ko, kl, kz, ...) with an increment A k . In numerical analysis we consider the wave number points up to a certain value of k, which we will call the truncation wave number value and designate by kT. To achieve a desired accuracy in the solution, the kT value and the number of wave number points to be considered in the analysis should be chosen properly. The proper value of kT depends on the smoothness of the solution with respect to XZ. On the other hand, the number of wave number points to be included in the analysis depends on how rapidly the solution decays out along the x2 axis.

(BT - VAT)I = 0.

3 . EXAMPLE PROBLEM

The proposed numerical procedure is applied to a wave propagation problem involving an elastic layer having the shear modulus p, the mass density p and the height H (see Figure 1). The layer is resting on a half-space with shear modulus ji and mass density i . The layer-half- space system is referred to a Cartesian co-ordinate system in which the x-z plane coincides with

Y

Figure 1 . Elastic layer subjected to out-of-plane shear stresses

626 Y. MENGI AND A. K. TANRIKULU

the top surface of the layer and the y axis is directed downwards. The transient waves in the layer are generated by the shear stresses s = s(x, t ) acting on the top surface of the layer parallel to the z axis. It is assumed that the applied shear stresses s are uniform and their extent is infinite along the z axis. The time and x-variations of s are kept arbitrary in the formulation.

The equations of elasticity for the layer can be expressed in terms of first-order partial differential equations as

(6 ) A u , ~ + B u , ~ + D u , ~ = 0

where

A = I (identity matrix)

0 0 0 0 0 - 1

0 - c 2 B=[O 0 -a]; D = [ -,9 : :] (7)

u = (E, E , v )

In equations (7),

w is the displacement component in the z direction and c = layer.

is the shear velocity for the

The boundary condition at the upper surface of the layer is

7yz l y=o = pE ly=o = s(x, t ) (8)

where T~~ is the shear stress component.

boundary condition proposed by Lysmer in Reference 3. It takes the form The boundary condition at the base of the layer will be written in view of the viscous

((Y7yz + pcv) l y = H = ((Y@ + p C V ) l y = H == 0 (9) for our problem, where (Y = pc/pC is the impedance ratio, C being the shear wave velocity in the half-space. (Y = 0 corresponds to the rigid lower boundary, i.e. to the rigid half-space. For (Y # 0, the viscous boundary condition in equation (9) describes approximately the influence of the half-space on the dynamic behaviour of the layer. From the definition of (Y it follows that (Y = 1 corresponds to the absorbing boundary, for which the material properties of the layer and half-space become the same, and the waves propagating downwards in the layer will be absorbed by the half-space.

To apply the numerical procedure discussed in the preceding section to the problem described above, we first take the Fourier transform of the governing equation, equation 6 , with respect to x. This yields

(10) A U ~ ; + B U ~ + c = o

c = ( - ikvF, 0, - c2ikEF)

where superscript F stands for the Fourier transform and the vector C is defined by

(1 1)

In equation (1 l), i is the imaginary number and k is the Fourier transform parameter which corresponds to the wave number for the x direction.

TRANSIENT WAVE PROPAGATION ANALYSES 627

The Fourier transform of the boundary conditions, equations (8) and (9), with respect to x gives

1 = - sF(k , t )

F e F l y = o F /.l (12) (apE +pcu ) l y = H = O

The solution of the problem in transform space is governed by equation (10) and the boundary conditions in equation (12) and zero initial conditions at the time t = O . Since equation (10) is hyperbolic, the solution can be constructed by using the method of characteristics. The canonical equations for the present problem can be obtained by using equations (3), (9, (7) and (11). They are:

dY deF along - = c dt c dt dt

deF 1 duF F dY

- d u F + cikEF = 0,

-+- - -c ikE = 0 , along -= - c dt c dt dt

dY dt

0, along -= 0

For a given value of k , these canonical equations can be integrated easily by using the network shown in Figure 2, which is composed of characteristic Iines, and by taking into account the boundary conditions, equations (12), along the boundary lines B I and B2, and zero initial conditions at t = 0.

The numerical inversion of the solution back into the real space requires the construction of the solution in the characteristic plane at the wave number points (ko, k l , k2, ...) with an increment Ak. The inversion can be performed conveniently for the present problem by using the FFT a l g ~ r i t h m . ~ ~ ’ It may be noted that the same algorithm can be used also for computing the Fourier transform sF(k, t ) of the applied shear stresses.

y = o y = H

Figure 2. The characteristic plane for the example problem

628 Y. MENGI AND A. K . TANRIKULU

4. NUMERICAL RESULTS AND DISCUSSIONS

The numerical results are obtained when the time and x variations of the applied tractions are trapezoidal, i.e. s is given by

s = so f (x )g ( t ) (14)

where so is the intensity of the applied shear stresses, and the functions f and g represent the trapezoidal distributions of s with respect to x and t as shown in Figure 3. From this figure we see that s has a finite time duration of the amount (e + 2d). Spacewise, s is a strip load of the width ( a + 2b) acting between the points x = T ( a / 2 + 6 ) .

The e and d values (see Figure 3) are chosen to be

e=0-4T, d=0 .2T

where T is a characteristic time defined by T = H/c, which is the amount of time for a normal incident shear wave to travel across the layer thickness H. It may be noted that for the chosen values of e and d the duration time of the applied shear stress s is smaller than T.

Two different sets of values are assigned to a and b (see Figure 3). The first one is

a = 2H, b = O-SH (15)

for which the width of the strip load becomes comparable with the layer thickness. The second set is

a = 20H, b = 5H (16)

which corresponds to the case in which the width of the strip load becomes very large compared to the layer thickness.

The computations indicated that the solution in transform space decreases rapidly as the distance 1 x I increases. The selection of the truncation wave number value ( k ~ ) and the number of wave number points ( N ) to be considered in the analysis as

N = 100 2 N ( a + 2 b ) ’

kT= ~

is sufficient to obtain the solution with good accuracy. The time variations of stresses and displacement obtained by using the proposed numerical

technique at various points of the layer are shown in Figures 4-9. Figures 4-6 are devoted to

lg

Figure 3. Space and time variations of the applied shear stresses used in numerical results

2

1.5

1

0.5

0

-0.1

-1

-1.5

-2 0

TRANSIENT WAVE PROPAGATION ANALYSES 629

2 4 8 8

NONDIYENSIONAL TIME

10

Figure 4. Time variation of the shear stress T~~ at x = 0, y = H when the width of the strip load is very large compared to the height of the layer (rigid base case, 01 = 0)

2.5

2

1.5

1

0.5

0

-0.5

-1

-1.5

-2

-2.5

0 2 4 6 8

NONDIYENSIONAL RYt

10

Figure 5 . Time variation of the shear stress 7yr at x = 0, y = H for u = 2H, b = 0.5H (rigid base, 01 = 0)

630 Y . MENGI AND A. K . TANRIKULU

1.1 , 1

0.0

0.8

0.7

0.8

0.5

0.4

0.3

0.2

0.1

0

-0.1

-0.2

-0.3

-0.4

0

0.5

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0.3

0.2

0.1

0

-0.1

-0.2

-0.3

-0.4

2 4 e 8

NONDXNENSIONU TIME

10

Figure 6. Time variation of the shear stress T~~ at x = 0, y = H for a = 2H, b = 0.5H (absorbing lower boundar a = 1)

0.7

0.6

0.5

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0.3

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0.1

0

-0.1

-0.2

-0.3

-0.4 0 2 4 e 8

NONDXNENSIONAL TIME

10

Figure 7. Time variation of the shear stress sxr at x = 1.5H, y = 0 for a = 2H, b = 0.5H (rigid base, a = 0)

TRANSIENT WAVE PROPAGATION ANALYSES 63 1

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

-0.1

0 2 4 6 e 10

NONDIYENSIONAL TIYE

Figure 8. Time variation of the shear stress T~~ at x = 1 -5H, y = 0 for a = 2H. b = 0.5H (absorbing lower boundary, a = 1)

Figure 9. Time variation of the displacement w at x = 0, y = 0 for a = 2H, b = 0-5H (rigid base, 01 = 0)

632 Y. MENGI AND A. K. TANRIKULU

the shear stress 7yz. Figures 7-8 belong to the shear stress 7xz. Figure 9 shows the time variation of the displacement w at the point x = y = 0 of the top surface. The results in all of these figures, except those in Figure 4 are obtained by using the (a , b ) values given in equation (15). For Figure 4, the (a , b ) values in equation (16) are used.

The non-dimensional stresses (Tyz , T ~ ~ ) , displacement W and time i appearing in the figures are defined by

where T is the characteristic time defined previously. Figure 4 gives an indication about the accuracy of the proposed numerical technique. This

figure shows the time variation of 7yz at the point x = 0, y = H of the base when the width of the strip load is very large compared to the height of the layer ( a is taken to be 20H). The base of the layer is rigid ( a = O ) . It is known that when a % H the problem becomes approximately one-dimensional, for which the exact solution is available. The results shown in Figure 4 are obtained by using the proposed technique and by treating the problem as two- dimensional with a = 20H, b = 5H. They coincide almost exactly with the one-dimensional exact solution. The form of the solution in Figure 4 shows further that the proposed numerical method is capable of predicting the sharp variations at wave fronts.

In the results presented in Figures (5, 7 and 9), for which the width of the strip load is comparable with the height of the layer ( a = 2 H ) and the base is rigid (a = 0), the scattering effects are apparent, which are due to the inclined waves reflected at the base.

Comparison of the results in Figure 5 with those in Figure 6 , and of the results in Figure 7 with those in Figure 8, shows the influence of the absorbing boundary on the wave profile. From the figures we see that for the absorbing boundary case the waves propagating downwards in the layer are absorbed by the half-space, and accordingly no reflections occur at the base. It should be noted that the results obtained for the absorbing boundary case in Figures 6 and 8 are approximate. This is because the boundary condition, equation (9), with a = 1 used in the analysis to represent the absorbing boundary, describes the complete absorption only for normal incident waves, not for inclined waves.

REFERENCES

1. R. Courant and D. Hilbert, Method of Mathematical Physics, Vol. 11, Interscience Publishers, New

2. G. B. Witham, Linear and Nonlinear Waves, Wiley, New York, 1974. 3. J. Lysmer and R. L. Kuhlemeyer, ‘Finite dynamic model for infinite media’, J. Eng. Mech. Div.,

4. E. 0. Brigham, The Fast Fourier Transform, Prentice-Hall, Englewood Cliffs, N.J., 1974. 5 . J. W. Cooley, P. A. W. Lewis and P. D. Welch, ‘The fast Fourier transform and its applications,’

York, 1966.

ASCE 95, 859-877 (1969).

IEEE Trans., Education, 12, 27-34 (1969).