A Hybrid Numerical Technique, Combining the ... - GeoScienceWorld

Bulletin of the Seismological Society of America, Vol. 88, No. 4, pp. 1036-1050, August 1998

A Hybrid Numerical Technique, Combining the Finite-Element and Boundary-Element Methods, for Modeling the 3D Response of 2D Scatterers by Bin Zhang, Apostolos S. Papageorgiou, and John L. Tassoulas

Abstract

A hybrid formulation of the 2.5D elastodynamic scattering problem combining the finite-element method and boundary integral equation method is presented and validated. The formulation of the 2.5D boundary integral equation method that is used was presented in detail by Papageorgiou and Pei (1998) and is an extension of the discrete wavenumber boundary integral equation method originally proposed by Kawase (1988) for 2D scattering problems. Modeling of the wave field in the domain of the scatterer is based on the variational principle of virtual displacements, and discretization of the domain is accomplished using the finite-element method. The formulation may be used to study the wave field in models of sedimentary deposits (e.g., valleys) or topography (e.g., canyons or ridges) with a 2D variation in structure but obliquely incident plane waves. The hybrid method exploits the versatility of the finite element method for modeling the scatterer and the effectiveness of the boundary integral equation method for taking care of the radiation condition in the half-space. The advantage of the 2.5D formulation is that it provides the means for calculations of 3D wave fields in scattering problems by requiring a storage comparable to that of the corresponding 2D calculations.

Introduction Sedimentary deposits in the form of valleys and topographic features (such as ridges and canyons), by virtue of their formation process, usually exhibit a two-dimensional (i.e., cylindrical) structure, where the out-of-plane variations (i.e., variations along the longitudinal axis of the cylinder) are small enough to be, for all practical purposes, negligible. However, even though the structure of the heterogeneity/scatterer is two-dimensional, propagation of the excitation may be three-dimensional even in the simplest case of incident plane waves (e.g., plane waves incident from an azimuthal direction other than the normal to the axis of the scatterer). In this case, the response of the 2D scatterer is three-dimensional, the out-of-plane motions being coupled with the in-plane motions. A model that may be used to study the 3D response of a 2D scatterer, without requiring the same level of computational resources as a full-blown 3D type of analysis, is referred to in the published literature as a 2.5D model (e.g., Bleistein, 1986). Several investigators have developed numerical methods to address such a 2.5D problem. Luco et al. (1990), Luco and de Barros (1995), de Barros and Luco (1995), and Pedersen et al. (1994a,b, 1995, 1996) used variants of the in-

direct boundary integral equation method (IBIEM) to study the response of geologic scatterers such as canyons, ridges, and valleys to incident plane waves. Zhang and Chopra (1991 a, b) used the direct boundary integral equation method (DBIEM) to determine the impedance matrix for a 3D foundation (such as that of an arch dam) supported on an infinitely long canyon of uniform cross section cut in a homogeneous half-space. Considering the problem of the interaction of a 2D topographic irregularity with the wave field of a point source, Takenaka et al. (1996) extended to the 2.5D case the IBIEM introduced by Bouchon (1985) and Gaffet and Bouchon (1989) to study 2D topography problems. Similarly, Fujiwara (1996) calculated the three-dimensional wave field in a two-dimensional sedimentary basin due to a point source using the DBIEM. More recently, Takenaka and Kennett (1996a) proposed a 2.5D time-domain elastodynamic equation for seismic wave fields in models with a 2D variation in structure but obliquely incident plane waves. This approach was generalized by Takenaka and Kennett (1996b) for the case of nonplanar waves (i.e., for the case when a source is present) and for general anisotropic media. In a parallel effort, Furamura and Takenaka (1996) developed an 1036

1037

A Hybrid Numerical Technique, Combining the Finite-Element and Boundary-Element Methods

efficient 2.5D formulation for the pseudo-spectral method for point-source excitation and applied it successfully to modeling the waveforms recorded in a refraction study. A 2.5D valley model, along with a waveform inversion scheme, was also used by Aoi et al. (1997) to estimate the irregular shape of the sediment-rock interface. In the present study, we present a hybrid formulation of the 2.5D problem combining the finite-element method with the boundary integral equation method. Khair et al. (1989, 1991) and Liu et al. (1991) used a similar hybrid approach to address the same problem. The formulation of the 2.5D boundary integral equation method that we use was presented in detail by Papageorgiou and Pei (1998) and is an extension of the discrete wavenumber boundary integral equation method originally proposed by Kawase (1988) for 2D scattering problems and subsequently extended to 3D problems by Kim and Papageorgiou (1993). To study the wave field in the scatterer, we start with the variational principle of virtual displacements and we discretize the domain using the finite-element method. The coupling of the two domains is accomplished by imposing the conditions of continuity of displacements and tractions along the common boundary of the scatterer with the half-space. The mathematical formulation developed in the present article has been implemented by Zhang and Papageorgiou (1996) and Pei and Papageorgiou (1996), in studies of valley response, and by Pei (1997), in a study of topography effects. We validate the formulation by comparing our results with those obtained by Luco et al. (1990), Luco and de Barros (1995), and de Barros and Luco (1995). Finally, in order to demonstrate the versatility of the hybrid method, we present snapshots of the scattered wave field generated by a plane body wave incident to a ridge.

Mathematical Formulation o f the Problem Statement of the Problem The model consists of an infinitely long viscoelastic inclusion/scatterer of an arbitrary, but uniform cross section embedded in a homogeneous (or horizontally layered) viscoelastic half-space. The geometry of the model used in the present study is shown in Figure 1. The excitation is represented by plane body waves impinging at an oblique angle with respect to the axis of the scatterer or by plane surface waves incident from any azimuthal direction, as shown in Figure 1. Even though the model is two-dimensional, the response is three-dimensional and has the particular feature of repeating itself with a certain delay for different observers along the axis of the scatterer. Stating this differently, the response depends on the coordinate y along the axis of the scatterer and time t only through the combination (y - Vt), where V is the apparent velocity of propagation of the excitation along the axis of the valley; that is, the wavefield is translationally invariant with respect to y. By taking advantage of the translational invariance and by use of an appro-

priate Green's function, the three-dimensional physical problem may be reduced to a two-dimensional mathematical problem and thus lead to a considerably simpler solution. The wave field in the absence of a scatterer is termed the free-field solution and is represented by u(°)(x, t) at a point defined by the position vector x. In the case of an infinite medium, the free-field solution is the incident wave field, whereas in the case of a half-space, the free-field solution may be constructed by superposing on the incident wave field the waves reflected by the traction-free surface of the half-space. (Some authors refer to the free-field solution as the incoming wave.) When the incoming wave encounters the scatterer, some of the wave is deflected from its original course. The difference between the actual wave field u(x, t) and the undisturbed wave u(°)(x, t), which would be present if the scatterer were not there, is termed the scattered wave field u(S)(x, t) (Courant and Hilbert, 1962). That is, u(x, t) = u (°) (x, t) + u (s) (x, t),

(la)

or, for the case described earlier that satisfies the conditions of translational invariance of the wave field with respect to y,

u(x,

y

-

Vt, z )

=

u (°~ (x, y

-

vt, z)

+ u (s)(x,y -

Vt, z).

(lb)

For the case of harmonic excitation of an incident wave, equations (la) and (lb) may be written in terms of the amplitude of the steady-state harmonic oscillations as follows: U(x, co) = U (°) (x, co) + U (s~ (x, 09).

(lc)

In the three subsections that follow, we summarize the boundary integral formulation that is used to model the wave field in the half-space, we develop a finite-element formulation to model the scatterer, and we describe the coupling of the two domains. Modeling of the Half-Space Using the Boundary Integral Method Starting from the representation theorem (also referred to as the dynamic Betti-Rayleigh theorem; Payton, 1964) for two reduced elastodynamic states in the absence of body forces (Eringen and Suhubi, 1975) and exploiting the translational invariance of the field variables with respect to the y axis for the problem described previously (Fig. 1), Papageorgiou and Pei (1998) have demonstrated that the boundary integral formulation of the problem reduces to the following form [for a detailed derivation of equation 2, in addition to Papageorgiou and Pei (1998), one may also want to consult Kawase (1988)]: CoU~ (Go, co) = f G~[ ' (Xo; Go, ~o) • v . TiL j (Xo, co) dr" F

_ p f H{n)ji H, (Xo; Go, co) " V" U* r (xo, co)dF + ~0)* (Go, co),

(2)

1038

B. Zhang, A. S. Papageorgiou, and J. L. Tassoulas

Figure 1. Scatterer with a 2D structure excited by obliquely incident plane body or surface waves• The angles ~0 and i are the azimuthal and incidence angles, respectively. The segment of the scatter used in the application of the principle of virtual displacements is bounded by the two cross sections S~ and Sb at y = a and y = b, respectively. Sr denotes the interface of the scatterer with the half-space contained between the above-mentioned cross sections•

Y

where the asterisk (*) represents a complex conjugate and P f denotes that the line integral is defined in the sense of the r Cauchy principle value (CPV). The boundary curve F is defined as the intersection of the xz plane with the interface Sr of the scatterer with the half-space (Fig. 1). The field variables Gif' (Xo; ~o, co) and HZ~ji (Xo; ~o, o9) are the amplitudes of the steady-state harmonic displacement and traction Green tensors, respectively, at a point Xo of F induced by a point force moving with a constant velocity V on a line that is parallel to the y axis and intersects the xz plane at a point ~o of the boundary curve F [Explicit expressions for GH' (Xo; ~o, co) and Hffn)j i (Xo; ~o, co) may be found in Papageorgiou and Pei (1998) and Pedersen et al. (1994a).] The coefficient tensor C~j of the free term is determined by the boundary shape around {o and is equal to (~2) ~gj (where ~ij is the Kronecker symbol) in case of a smooth boundary (Hartmann, 1980; Rizzo et aL, 1985; Tassoulas, 1988). Equation (2) is referred to as Somigliana identity (Pao and Varatharajulu, 1976; Rizzo et al., 1985). It can be demonstrated, following the same procedure as in Papageorgiou and Pei (1998), that an identical expression to equation (2) is valid but without the asterisks (*) (i.e., without taking the complex conjugates of the respective variables) (see also Eringen and Suhubi, 1975, pp. 432-433). We will be using this latter form (i.e., without the *) of the Somigliana identity in the development that follows.

To solve the Somigliana identity for arbitrary boundary shape and conditions, the discretization of both boundary shape and values of Uj (Xo, co) and T(n)j (Xo, co) should be introduced in the same manner as in the finite-element method (e.g., Banerjee and Batterfield, 1981; Dominguez, 1993). The simplest boundary element is a constant-value line element (a constant-value line element is an element that is a straight line segment and the field variables Uj and T(n)j are assumed to be constant over its length). Thus, by discretizing the boundary curve F into n line segments F1, F2, . . . . F~, assuming that displacements and tractions are uniformly distributed on each element, and varying the source point {o of the Green tensors over F, the discrete form of the Somigliana identity may be cast in the form of a set of linear algebraic equations as follows: 1/2 {/Jr} = where {UF} = {UF1 (Xl) , UF2 (3[2). . . . .

Uvn (x.)} r,

(3b)

{T(n)F} = {T(nl)V, (Xl), T(.=)r2 (x2). . . . . T(..)r. (x.)} r,

(3C)

o (x2). . . . . Ur° o (x,,)}r, {U~-} = {U or, (x0, Ur~

(3d)

dS fF 2 H~/n2) (X2; Xl) d S (hi) (X1; X2) dS f rz I-I~2)(x2; x2) dS frl HU

x0

fF 1 H(.~) H (Xl ; x0 [ H H] =

fF

H (X1;)In) H(.~)

[HHI {Ur} + [GH {T¢n)r} + {U~-}, (3a)

•.

dS fr2 H~z) (x2; x~) dS

fr,, H(n~) H (X.; X2) xn)

dS ] dS

(3e)

dS

frl G~ (Xa; xl) dS fr~ frl GH (xl; x2) dS fr2

dS • . f t . GH (x.; Xl) dS 1 GH (x2; x2) dS • -- f t . GH (xn; x2) dS

fri GH (Xl; x.) dS fr2

G~ (x2; x.)

G u (x2; xl)

dS . . .

f t . G H(x.; x,)

dS

(3f)

A Hybrid Numerical Technique, Combining the Finite-Element and Boundary-Element Methods HH (nj) (xj; xi)

=

V

I H(nj)11 H' (Xj; Xi) H{nj)21 H' (xj; xi) H~nj)31 H' (Xj; Xi)1 / HE' U' (~?~2 (xj; xi) HE' (nj)22 (Xfi Xi) H{nj)32 (Xj; Xi)l, u' (xj; xi) H,/[' (xj; xi) HE' H(nj)13 (nj)23 (nj)33 (xj; Xi)J

G H (x/xi)

(3h)

GT~ (xj; x;) G~; (xj; x3 G~' %; x~)] Gf; (xj; Xi) a~ (Xj; X,) G3~ (Xfi X/)/, | 61~ (xj; x,) G~ (V x0 G~ (x/ x0_]

=V

a(x,

(3g)

"g

co) = ¢~(x, z)e-i~ei%

(5c)

where k = co/V. Similarly, the virtual displacement field at a point x may be written as

co) = Oa(x, z)e-ikye it°t,

(6a)

Of(x, z)e-ikye it°t,

(6b)

09(x, co) =

Modeling of the Scatterer Using the Finite-Element Method

s

0~(X,

o9) = Off(X, z)e-ikYe i°~t.

6ijSgijdV

b b f (f ~" O~ dSy)e-2ikYdy + f (f pfl " Ofl dSy) e 2ikYdy a

Sy

a

~ ~ • 5fie- 2ikYdSa q- f

Sy

~ " Ofle- 2ikYdSb

Sb

+ f t • 5fie-211~YdSr. + fp~OaidV,

(4a)

(6c)

Substituting (5a,b,c) and (6a,b,c) in (4b), we obtain

Sa

"g

(5a) (5b)

rP(x,

with x~ being the centroid of the ith boundary element.

f n(z 'x) hi = (z 'x) V

( oz)

'(~)~(z 'x) R = (z 'x) ,o

(qoz) )cO = [e]

(9Z)

0

0

~/-

0

o

F

:(686[ "l v 1o blooD "~'a) S~OliOJ se (~)*~ pue ' ( a ) A ' ( a ) n sluotuoaeIds!p Iepou imuotu -olo oql Jo sttuol u! IOAOIluotuolo oql le pIog luotuOaelds!p OR1 sso.~dxo OA~pu~ '.IO.lO111~OSOR1JOgS uo.qoos sso:~a OR1 ozt.1 -o:tast.p o,~ 'poqlotu luotuolo-ol!ug oql l(Idde ol .topao u I

o 0 x~

:suotssaJdxa Sut.moiioJ oql £q IIOAI~ '£[aA!loadsoa 'Xl.alvu~ KIl.Ol.lSVla aql pu~ 'Xl.agvm UOl.lounf uol.lolo&alul, aql 'xl.alvu~ aolmado lm.luaaaffl.p oql aye ~t pue ' N '[e] pu~ '(8[ uo!l~nba oas) aolaaA uo!la~al pag! -potu oR1 s! [zl ',~1'~1] = ,t 'suo!ssoadxo OA~IsnOIAosd oql UI I ]701

J

(6I)

+ ,~*A + n~'~) ~ =

"JP ( ~

J

J

spoq19IN luau~alSt-£wpunoff puv iuaulal~t-dll.U1,d aztl gUl.Ul.qu~oD 'anblu~to9~LIVOl.aaumN pl.JqdH V

1042

B. Zhang, A. S. Papageorgiou, and J. L. Tassoulas

SEMI-CIRCULAR

S;r Sb/ UI

41

in which Ur and F r represent the displacement and forces, respectively, of all the nodes lying on the boundary curve F, while Uz represents the displacements of all the interior nodes.

CANYON

SH - WAVE

~0=0

°

i=0

°

4f , cp=0 ° i=45 °

o--ey

omuy

Coupling of the Two Domains In the previous two sections, we have presented equation (3), which governs the motion of the half-space, and derived equation (30), which governs the motion of the scatterer. These two equations are coupled at the interface of the two domains (i.e., along boundary curve F; Fig. 1) where displacements and tractions are assumed to be continuous. In general, the finite elements of the scatterer along the boundary F may not share the same nodes as the boundary elements of the half-space. Furthermore, the degree of interpolation may be different for the finite elements as compared to that of the boundary elements. The coupling of the two domains is accomplished as follows (e.g., Tassoulas, 1988). At any point (x, z) on a boundary element, the traction vector {p(x, z)} may be expressed as

-0.5

0.5

-0.5

0

0.5

1

SV - WAVE 8

q) = 0 ° i=0

g) = 0 °

o--u, Q--

--[Jz

o -

i = 30 °

°

Uz

-

I

x

I q t

-0'.5

{p(x, z)} = ~ (x, z) ~'r(e) , t~(n)F/,

0

0 "'"

t

O.

015

-0.5

0

0.5

1

(31)

where ~ (x, z) is a 3 × m interpolation function matrix having similar structure as that in equation (27) (m = the number of nodes of the boundary element) and {T~e~r } is the nodal traction vector of the boundary element. Thus, the consistent nodal force vector of the finite element that is in contact to the boundary element is given by

P - WAVE

41

4I

cp= 0 ° i=0o

I

o--u;, e- -U~

ND.

, ,er

,q) = 0 °

o--u~

i = 30 °

o-

-u,

!I

f

F(e)

- ] N r (x, z) {p (x, z)} dFi J

Fi =

--

-0.5

f N r (X, Z) 2~ (X, z) dF~ • /~r(e) ~t(n)FJ,

(32)

Fi

where F i is the segment of contact along the boundary F. Therefore, on the boundary F, the relationship between the vectors of finite-element nodal forces and boundary-element nodal forces may be written as F r = R {T(.)r},

0

0.5

-1

(x/a)

-0.5

0

0.5

(x/a)

Figure 2. Scattering of body waves by a semi-circular canyon of radius a. Comparison of the results obtained by the present method (lines) with the results of Trifunac (1973) (SH waves) and Wong (1982) (P and SV waves) for the 2D case (0 = 0°. Nondimensional frequency i/ = 1.0. Amplitude of incident wave is 1.

(33)

where R is constructed by assembling the elemental matrices

- f N T (x, z) ~ (x, z)dFi.

equation 30). If the boundary-element nodes coincide with the finite-element nodes, then U r = { Ur}. In general, it is possible to write

Fi

The next step is to find the relation between the boundary-element nodal displacement vector { Ur} (see equation 3b) and the finite-element nodal displacement vector U r (see

{ Ur} = I / U r ,

(34)

where R is composed of the values of finite-element inter-

A Hybrid Numerical Technique, Combining the Finite-Elementand Boundary-Element Methods SEMI-CIRCULAR

41

RAYLEIGH

condensed out, and the algorithm continues in the same fashion with another degree of freedom. It should be evident that the storage space requirements of a specific problem are determined by the size of the macroelement previously discussed.

CANYON -

WAVE

4,

g~ = o °

1043

qo = o °

Verification o oo OOo

22

-1

0

1

2

-9 -5

(x/a)

(x/a)

Figure 3. Scattering of Rayleigh waves by a semicircular canyon of radius a. Comparison of the results obtained by the present method (lines) with those of Sanchez-Sesma and Campillo (1993). Nondimensional frequency t/ = 0.5.

polation functions N~(x, z) at the boundary-element nodes (see equations 20a, b, c). For the results presented later, we considered 8-node rectangular and/or 6-node triangular isoparametric finite elements and constant-value line boundary elements. To achieve sufficient accuracy, the discretization we used was such as to have three boundary elements corresponding to each finite element. From equations (3a), (30), (33), and (34), we obtain [SbF-

R[GZ-ll-I(½I+[HHIR)s~

SUbFl )] S( ~UIIJ

= ( - R [6"1 1 0

(35)

The coefficient matrix on the left-hand side of equation (35) represents the impedance matrix of the entire (i.e., half-space plus scatterer) system, while the submatrix - R [GH]-1 ( 1 I + [ H/~]) 1~ may be thought of as corresponding to a macroelement that accounts for the impedance of the halfspace. The impedance matrix in equation (35) may be assembled using standard procedures of the finite-element method, and subsequently, the linear system of equation (35) may be solved. However, for many practical problems, such a strategy would require large storage space. In order to circumvent this obstacle, we adopted the frontal solution method (Irons, 1970). According to this solution method, instead of first assembling the complete impedance matrix and then solving the system of equations, the processes of assembling and reducing the equations are performed at the same time. At each cycle of the algorithm, only those equations that are actually required for the elimination of a specific degree of freedom are assembled, the degree of freedom considered is

For the hybrid method developed in the present article, we have developed a computer code, and we have performed extensive comparison tests with the results of a code based solely on the boundary integral formulation (Papageorgiou and Pei, 1998) as well as transparency tests (in which we consider a homogeneous and isotropic elastic half-space, we define a finite region of it to be the scatterer but having the same elastic properties as the rest of the half-space, we compute the response to some kind of excitation--say an incident plane body wave--using the hybrid method, and we compare the results to the analytical ones). All the mentioned comparisons were very satisfactory. For demonstrative purposes, here we present just a sample of comparisons with results in the frequency domain that appeared in the published literature. Also, in order to show the versatility of the hybrid method, we present in the time domain (snapshots) the scattering of a plane wave by an elevated topographic feature (ridge) of a half-space. This latter example allows the visualization of the complex scattered wave field that is generated by seemingly simple scatterer and source of excitation. It should be pointed out that the following results should not be interpreted as an exhaustive investigation of the physics of the problem of scattering by topographic features or inclusions (this aspect of the problem will be addressed in a separate article) but merely as a measure of the validity and accuracy of the hybrid method developed in the present article. Frequency-Domain Response of Semi-Circular Canyon and Valley We start by comparing the two-dimensional responses of a cylindrical canyon with a semi-circular cross section subjected to SH, SV, P, and Rayleigh waves incident from an azimuthal direction normal to the axis of the cylindrical scatterer (Figs. 2 and 3). Following Luco et al. (1990), the results obtained by the present hybrid method are for a slightly dissipative half-space characterized by ~ = 5 f l and ~a = ~fl = 0.01, where in this case, a = c~ (1 + iffa) and fl =/~(1 + i~B) (the parameters ~a and (p represent the small hysteretic damping ratio for P and S waves, respectively). These results are compared against earlier results obtained by Trifunac (1973) (for SH waves), Wong (1982) (for P and SV waves), and Sanchez-Sesma and Campillo (1993) (for the Raleigh wave) for nondissipative elastic halfspace characterized by a = ,/2fl (i.e., Poisson's ratio 1/3). The dimensionless frequency t/ = (coa)/(nfl) is assumed to be equal to 1.0 for body-wave excitation and 0.5 for the

1044

B. Zhang, A. S. Papageorgiou, and J. L. Tassoulas P-WAVE

2il 'f

SV-WAVE

SH-WAVE

31

2:f

1.5 I

O.

0.5

-1

0

1

2

-2

-1

0

1

2

-1

0

1

-1

0

1

2.5

3 2 1.5[

0il -2

0.5 -1

0

1

-1

2

0

1

2

3 2.5

2 1.

~1.8' 1

0.5~

o.51

0.5

-:0

-i

6 x/a

i

z

_0~ -i

6

i

2

_O~

x/a

Rayleigh-wave excitation, where a is the radius of the semicircular canyon. Figure 4 shows the three-dimensional response of the semi-circular canyon to plane body waves ( t / = 0.5) incident from an azimuthal direction ~0 = 40 ° and with an incidence angle i = 45 °. The results of our method (continuous lines) are compared with those of Luco e t al. (1990). Next, we consider the response of a semi-circular valley. First, we consider the two-dimensional response of the valley to SH, SV, and P waves incident from an azimuthal direction normal to the axis of the cylindrical valley. Figure 5 compares the results of our method to those of Luco and de Barros (1995) [which are in very good agreement with the results of Trifunac (1971) (for SH-wave excitation) and those of Dravinski and Mossessian (1987) (for P- and S V wave excitation)]. The valley (medium 2) and the half-space (medium 1) are characterized by ~1 = 2ill, ~2 = 2/~2, y 11/2 ~2 ~1, P2 = (2•3) Pl (where cl = gi[l + 2i sgn (co) ~ciJ , ci = oq, oz2, ill, f12, and Pl, P2 are the densities of the two media). For the SH-wave excitation, {~1 = ~fll = ~g2 = ~2 = 0.001 and J7 = 1, while for the P- and SV-wave excitations, ~Pl = d~l = {~2 = ~/~2 = 0.005 and r/ = 0.005.

-I

0

x/a

I

2

Figure 4. Scattering of body waves by a semi-circular canyon of radius a. Comparison of the results obtained by the present method (lines) with those of Luco et al. (1990) (represented by circles) for P, SV, and S H waves: ~o = 45°; i = 45°; t/ = 0.5. Amplitude of incident wave is 1.

Figure 6 compares the three-dimensional response of the valley to plane body waves (t/ = 0.5) incident from an azimuthal direction ~0 = 45 ° and with an incidence angle of i = 30 °. The results of our method are compared with those of de Barros and Luco (1995). In all the foregoing comparisons, the agreement varies from very good to excellent. Any small discrepancies ( - 10% or less) may be attributed to details of discretization of the scatterer. The more pronounced discrepancies observed in the results of Figure 3 may be related to the fact that in our treatment, we use the half-space Green's functions (and thus satisfy exactly the stress-free boundary condition over the free surface of the half-space), while Sanchez-Sesma and Campillo (1993) use the full-space Green's function (and thus satisfy the stress-free boundary condition only approximately). [See also Yokoi and Takenaka (1995) regarding errors introduced by such approximations.]

=

Time-Domain Response In order to demonstrate the versatility of the proposed hybrid method, we show snapshots of the time-domain re-

A Hybrid Numerical Technique, Combining the Finite-Element and Boundary-Element Methods SEMI-CIRCULAR VALLEY SH - WAVE 6 q~ = 0 °

o--uy

6

~0 = 0 °

i=0 °

i=45

,

,

-2

-1

,

,

,

0

1

2

.

o~uy o

f~

_0;_2_i

6

i

s

SV - WAVE 6

12,

o~u~

~0 = 0 °

q~ = 0 o

o~u~ 0~

•

"~

i

,~

,

-2

-1

0

1

2

-2-1

3

--U z

d

i

2

a

P- WAVE q~ = 0 °

o~ux

i=0o

=-

D

8 i = 30 °

-u~

eeeel

1 2

o- -U z

/

i

5-2-1

os-vl

,

1...-"-" a

-3

-2

f

-1

0

1

2

3

(x/a)

(x/a)

Figure 5. Scattering of body waves by a semi-circular valley of radius a. Comparison of the results of the present method (lines) with the results of Luco and de Barros (1995). Amplitude of incident wave is 1.

sponse of a homogeneous elastic half-space having a topographic anomaly in the form of a ridge. (We selected a 2D example because the interpretation of the results is more obvious.) The shape of the ridge is described by the pseudorealistic mountain model of Sills (1978), defined by

f(x) = h(1 where c = x/1.

-

c 2)

exp (--3c2),

(36)

1045

The domain that is discretized using finite elements is shown in Figure 7. We used quadrilateral and triangular quadratic isoparametric elements, and we made sure that we had two to three elements per minimum wavelength so as to guarantee satisfactory accuracy. The properties of the elastic half-space are a = 6.24 km/sec, fl = 3.6 km/sec (i.e., Poisson's ration 1/4), and p = 2.8 gm/cm 3. The excitation is modeled as a plane SH wave incident with an angle i = 30 ° (~0 = 0°). The time dependence of the excitation is specified to have the waveform of a Ricker wavelet (Ricker, 1945) with a characteristic frequencyfc = 3.0 Hz. We consider the responses of two ridges that differ in height: the high ridge is characterized by h = 1.5 km and l = 2.0 km (h/l = 0.75), while the low ridge has h = 0.5 km and I = 2.0 km (h/l = 0.25). Snapshots, at 0.1-sec intervals, of the responses of the low- and high-ridge models are displayed in Figures 8 and 9, respectively. The scattered wave field generated by the low ridge is relatively simple, consisting of the phases diffracted by the near- and far-source toes of the ridge. On the contrary, the scattered wave field generated by the high ridge is more complex, exhibiting, in addition to the above-mentioned two phases, a distinct third phase that is radiated from the apex of the ridge. This latter phase, as it propagates downward, interferes with the former two phases, generating large differential motions on the apex as well as on the flanks of the hill (observe the rapid interchange--in time and s p a c e - - o f dark and light regions of the wave field along the free surface of the ridge). Finally, notice that this third phase interferes constructively with the diffracted by the far-source toe phase, generating thus a strong-motion front that sweeps the near-source side of the ridge (follow the snapshots from t = 2.5 sec and onward). This is also clear on the surface motion shown in Figure 10. Phases SL, SR, and Sr in Figure 10 represent phases diffracted from the near-source toe (left toe in Fig. 9), the far-source toe (right toe in Fig. 9), and the top of the ridge, respectively. Figures 8 and 9 exhibit the power of domain methods such as the finite-element method in observing a field variable over the entire domain exploiting just intermediate results with no additional computational cost. This power of domain method becomes evident when the domain of the scatterer is heterogeneous, as is often the case in practical applications. In such a case, use of the boundary element alone becomes very cumbersome.

Conclusions We have presented and validated a hybrid numerical method for the 2.5D problem in elastodynamics. The method exploits the versatility of the finite-element method for modeling the scatterer and the effectiveness of the boundary integral equation method for taking care of the radiation condition in the half-space. The hybrid formulation may be used to study the wave fields in models of sedimentary deposits

1046

B. Zhang, A. S. Papageorgiou, and J. L. Tassoulas P-WAVE

SV-WAVE

SH-WAVE

8~

4

1] -2

-1

01

2

-2

-1

1

0

2

-2

2

-1

0

1

2

3

6

:t

1.5

0.5

0

- - 2 -1

0

1

2

3

-2

-1

1

0

2

51

5 o

41 3

o

21

2

11

ol

- 2 - 1 0 1 2 3

-," - 2

~a

-1

0

1

2

3

3

2

1

0

x/a

x/a

Yl

D=3 krn

! L=6 km Figure 7. Domain of the scatterer and ridge and part of the surrounding half-space that is discretized using finite elements.

1

2

3

Figure 6. Scattering of body waves by a semi-circular valley of radius a. Comparison of surface-displacement amplitudes using the present approach (lines) with those obtained by de Barros and Luco (1995) (circles). (p --- 45°; i = 30°; q = 0.5. Amplitude of incident wave is 1.

(e.g., valleys) or topography (e.g., canyon or ridges) with a 2D variation in structure but obliquely incident plane waves. The formulation may also be extended in a straightforward way to accommodate nonplanar waves (i.e., sources) in view of the fact that sources may be represented by a summation of plane waves (The Weyl Integral) (e.g., Bouchon, 1979). The use of the finite-element method together with the "frontal method of solution" of the resulting system of equations allows for realistic and detailed modeling of the scatterer. The advantage of the 2.5D formulation is that it provides the means for calculations of 3D wave fields in scattering problems by requiring a storage comparable to that of the corresponding 2D calculations.

A Hybrid Numerical Technique, Combining the Finite-Element and Boundary-Element Methods

1047

SH-wave Uy Jnc=30 T=t.7

SH-wave Uy inc=30 T=1.8

SH-wave Uy inc=30 T=I.g

SH-wave Uy lnc=50 T=2.0

SH-wave Uy inc=30 T=2.1

SH-wave Uy inc=30 T=2.2

SH-wave Uy inc=30 T=2.3

SH-wave Uy inc=30 T=2.4

SH-wave Uy inc=30 T=2.5

SH-wave Uy inc=30 T=2.6

SH-wave Uy inc=30 T=2.7

SH-wave Uy Jnc=30 T=2.8

SH-wave Uy inc=30 T=2.9

SH-wave Uy inc=30 T=3,O

SH-wave Uy inc=30 T--3.1

Figure 8.

Snapshots at O.l-sec intervals of the response of the low-ridge model to incident plane SH waves ((o = 0°; i = 30°).

1048

B. Zhang, A. S. Papageorgiou, and J. L. Tassoulas

SH-wave Uy inc=30 1"=1,7

SH-wave Uy inc=30 T=1.8

SH-wave Uy inc=30 T= 1.9

SH-wove Uy inc=30 T=2.0

SH-wave Uy inc=30 T:2.1

SH-wave Uy inc:=30 T=2.2

SH-wave Uy inc=30 T=2.3

SH-wove Uy inc=,.30 T-2.4

SH-wave Uy inc=30 T=2.5

SH-wove Uy inc=30 T=2.6

SH-wave Uy inc=30 T:2.7

SH-wave Uy inc=30 T=2.8

SH-wove Uy ine=30 T=2.9

$H-wave Uy inc=30 T=3.0

SH-wave Uy inc=-30 T=3.1

Figure 9. Snapshots at 0.1-sec intervals of the response of the high-ridge model to incident plane SH waves (9 = 0°; i = 30°).

1049

A Hybrid Numerical Technique, Combining the Finite-Element and Boundary-Element Methods

5

E 0 cO

E 0 0 0 I

-5 0

1

2

3

4

5

Time (sec) F i g u r e 10. Free-surface response of the high-ridge model to incident plane SH waves ((p = 0°; i = 30°). Notice the enhanced response amplitudes, at the apex and near-source flank of the ridge, resulting from constructive interference. Phases SL, SR, and Sr in this figure represent phases diffracted from the near-source toe (left toe in Fig. 9), the far-source toe (right toe in Fig. 9), and the top of the ridge respectively.

References Aoi, S., T. Iwata, H. Fujiwara, and K. Irikura (1997). Boundary shape waveform inversion for two-dimensional basin structure using threecomponent array data of plane incident wave with an arbitrary azimuth, Bull. Seism. Soc. Am. 87, 222-233. Banerjee, P. K. and R. Batterfield (1981). Boundary Element Methods in Engineering Science, McGraw-Hill, London, 452 pp. Bleistein, N. (1986). Two-and-one-half dimensional in-plane wave propagation, Geophys. Prospecting 34, 686-703. Bouchon, M. (1979). Discrete wave number representation of elastic wave fields in three-space dimensions, J. Geophys. Res. 84, 3609-3614. Bouchon, M. (1985). A simple, complete numerical solution to the problem of diffraction of SH waves by an irregular surface, J. Acous. Soc. Am. 77, 1-5. Cook, R. D., D. S. Malkus, and M. E. Plesha (1989). Concepts andApplications of Finite Element Analysis, 3rd ed., Wiley, New York, 630 pp. Courant, R. and D. Hilbert (1962). Methods of Mathematical Physics, Vol. 2, lnterscience Publishers Inc., New York, de Barros, F. C. P. and J. E. Luco (1995). Amplification of obliquely incident waves by a cylindrical valley embedded in a layered half-space, Soil Dyn. Earthquake Eng. 14, 163-175. Dominguez, J. (1993). Boundary Elements in Dynamics, Elsevier Applied Science, London-New York, 707 pp. Dravinski, M. and T. K. Mossessian (1987). Scattering of plane harmonic P, SV, and Rayleigh waves by dipping layers of arbitrary shape, Bull. Seism. Soc. Am. 77, 212-235. Eringen, A. and E. S. Suhubi (1975). Elastodynamics, VoL II, Linear Thetry, Academic, New York. Fujiwara, H. (1996). Three-dimensional wavefield in a two-dimensional basin structure due to point source, J. Phys. Earth 44~ 1-22. Furumura, T. and H. Takenaka (1996). 2.5-D modelling of elastic waves using the pseudospectral method, Geophys. J. Int. 124, 820-832. Gaffet, S. and M. Bouchon (1989). Effects of two-dimensional topographies using the discrete wavenumber-boundary integral equation method in P-SV cases, J. Acoust. Soc. Am. 85, 2277-2283. Hartmann, F. (1980). Computing the C-matrix in non-smooth boundary points, in New Developments in Boundary Element Method, C.A. Brehbia (Editor), CML Publications, Southampton, UK, 367-379.

Irons, B. M. (1970). A frontal solution program for finite element analysis, Int. J. Numer. Meth. Eng. 2, 5-32, Kawase, H. (1988). Time-domain response of a semicircular canyon for incident SV, P, and Rayleigh Waves calculated by the discrete wavenumber boundary element method, Ball. Seism. Soc. Am. 78, 14151437. Khair, K. R., S. K. Datta, and A. H. Shah (1989). Amplification of obliquely incident seismic waves by cylindrical alluvial valleys of arbitrary cross-sectional shape. Part I. Incident P and SV waves, Bull. Seism. Soc. Am. 79, 610-630. Khair, K. R., S. K. Datta, and A. H. Shah (1991). Amplification of obliquely incident seismic waves by cylindrical alluvial valleys of arbitrary cross-sectional shape. Part II. Incident SH and Rayleigh waves, Bull. Seism. Soe. Am. 81, 346-357. Kim, J. and A. S. Papageorgiou (1993). Discrete wavenumber boundaryelement method for 3-D scattering problems, J. Eng. Mech. ASCE, 119, 603~24. Liu, S. W., S. K. Datta, and M. Bouden (1991). Scattering of obliquely incident seismic waves by a cylindrical valley in a layered half-space, Earthquake Eng. Struct, Dyn. 20, 859-870. Luco, J. E. and F. C. P. de Barros (1995). Three-dimensional response of a layered cylindrical valley embedded in a layered half-space, Earthquake Eng. Strucr Dyn. 24, 109-125. Luco, J. E., H. L. Wong, and F. C. P. de Barros (1990). Three-dimensional response of a cylindrical canyon in a layered half-space, Earthquake Eng. Strucr Dyn. 19, 799-817. Pat, Y.-H. and V. Varatharajuhi (1976). Huygens' principle, radiation conditions and integral formulas for the scattering of elastic waves, J. Acoust. Soc. Am. 59, 1361-1371. Papagcorgiou, A. S. and D. Pei (1998). A discrete wavenumber boundary element method for 2.5-D elastodynamic scattering problems, Earthquake Eng. Strucr Dyn. 27, 619-638. Payton, R. G. (1964). An application of the dynamic Betti-Rayleigh reciprocal theorem to moving-point loads in elastic media, Q. Appl. Math 21, 299-313. Pedersen, H., F. J. Sanchez-Sesma, and M. Campillo (1994a). Three-dimensional scattering by two-dimensional topographies, Bull. Seism. Soc. Am. 84, 1169-1183. Pedersen, H., B. LeBrun, D. Hatzfeld, M. Campillo, and P.-Y. Bard

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(1994b). Ground-motion amplitude across ridges, Bull. Seism. Soc. Am. 84, 1786-1800. Pedersen, H. A., M. Campillo, and F. J. Sanchez-Sesma (1995). Azimuth dependent wave amplification in alluvial valleys, Soil Dyn. Earthquake Eng. 14, 289-300. Pedersen, H. A., V. Maupin, and M. Campillo (1996). Wave diffraction in multilayered media with the indirect boundary element method: application to 3-D diffraction on long-period surface waves by 2-D lithospheric smactures, Geophys. J. Int. 125, 545-558. Pei, D. (1997). Study of the 3-D response of 2-D scatterers: earthquake engineering applications, Ph.D. Thesis, Rensselaer Polytechnic Institute, Troy, New York. Pei, D. and A. S. Papageorgiou (1996). Locally generated surface waves in Santa Clara Valley: analysis of observations and numerical simulation, Earthquake Eng. Struct. Dyn. 25, 47-63. Ricker, N. (1945). The computation of output disturbances from amplifiers for true wavelet inputs, Geophysics 10, 207-220. Rizzo, F. J., D. J. Shippy, and M. Rezayat (1985). A boundary integral equation method for radiation and scattering of elastic waves in three dimensions, Int. J. Numer. Meth. Eng. 21, 115-129. Sanchez-Sesma, F. J. and M. Campillo (1993). Topographic effects for incident P, SV and Rayleigh waves, Tectonophysics 218, 113-125. Sills, L. B. (1978). Scattering of horizontally polarized shear waves by surface irregularities, Geophys. Z R. Astr. Soc. 54, 319-348. Takenaka, H. and B. L. N. Kennett (1996a). A 2.5-D time-domain elastodynamic equation for plane-wave incidence, Geophys. J, Int. 125, F5-Fg. Takenaka, H. and B. L. N. Keunett (1996b). A 2.5-D time-domain elastodynamic equation for a general anisotropic medium, Geophys. J. Int. 127, F1-F4. Takenaka, H., B. L. N. Kennett, and H. Fujiwara (1996). Effect of 2-D topography on the 3-D seismic wavefield using a 2.5-D discrete wavenumber-boundary integral equation method, Geophys. J. Int. 124, 741-755. Tassoulas, J. L. (1988). Dynamic soil-smacture interaction, in Boundary

B. Zhang, A. S. Papageorgiou, and J. L. Tassoulas

Element Methods in Structural Analysis, D. Beskos (Editor), ASCE, New York, Chap. 10. Trifunae, M. D. (1971). Surface motion of a semi-cylindrical alluvial valley for incident plane SH waves, Bull. Seism. Soc. Am. 61, 1755-1770. Trifunac, M. D. (1973). Scattering of plane SH waves by a semi-cylindrical canyon, Earthquake Eng. Struct. Dyn. 1, 267-281. Wong, H. L. (1982). Effect of surface topography on the diffraction of P, SV and Rayleigh waves, Bull. Seism. Soc. Am. 72, 1167-1183. Yokoi, T. and H. Takenaka (1995). Treatment of an infinitely extended free surface for indirect formulation of the boundary element method, J. Phys. Earth 43, 79-103. Zhang, L. and A. K. Chopra (1991a). Three-dimensional analysis of spatially varying ground motion around a uniform canyon in a homogeneous half-space, Earthquake Eng. Struet. Dyn. 20, 911-926. Zhang, L. and A. K. Chopra (1991b). Impedance functions for three-dimensional foundations supported on an infinitely-long canyon of uniform cross-section in a homogeneous half-space, Earthquake Eng. Struct. Dyn. 20, 1011-1027. Zhang, B. and A. S. Papageorgiou (1996). Simulation of the response of the Marina District Basin, San Francisco, California, to the 1989 Loma Prieta earthquake, Bull. Seism. Soc. Am. 86, 1382-1400. Department of Civil Engineering Rensselaer Polytechnic Institute Troy, New York 12180-3590 Tel.: (518) 276-6331; fax: (518) 276-4833; E-mail: [email protected] (B.Z., A.S.P) Department of Civil Engineering The University of Texas Austin, Texas 78712-1076 (J.L.T.) Manuscript received 11 August 1997.