BOUNDARY ELEMENT ANALYSIS OF THIN STRUCTURAL PROBLEMS IN 2D ELASTOSTATICS

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1 Journal of Marne Scence and Technology, Vol. 19, No. 4, pp (011) 409 BOUNDARY ELEMENT ANALYSIS OF THIN STRUCTURAL PROBLEMS IN D ELASTOSTATICS Yao-Mng Zhang,, Yan Gu, and Jeng-Tzong Chen Key words: boundary element method, nearly sngular ntegrals, thn structural problems, elastcty. ABSTRACT Thn structures, such as thn flms and coatngs, have been wdely desgned and utlzed n many ndustres recently. However, the wdespread expermental research n thn structural problems underles a general lack of modelng efforts whch can accurately and effcently predct ther performance. In ths paper, the boundary element method (BEM) based on elastcty theory s developed for two-dmensonal (D) thn structures wth the thckness to length rato n the mcro (10-6 ) or nano (10-9 ) scales. The BEM-based approach proposed n ths paper s constructed usng a combnaton of the regularzed ndrect boundary ntegral equatons (BIEs) and a general nonlnear transformaton whch can elmnate the nearly sngular propertes of the ntegral kernels. For the test problems studed, very promsng results are obtaned when the thckness to length rato s n the orders between 10-6 and 10-9, whch s suffcent for modelng most thn structures n the mcro- or nano-scales. I. INTRODUCTION Structures made from plate and shell assembles, also known as thn-walled structures, are mportant n many areas of engneerng. The effcent soluton of such structures has wde applcaton, from the desgn of more conventonal cvl engneerng systems to more sophstcated structures employed, for nstance, by the arcraft ndustry. However, the wdespread expermental research n thn structures underles a general lack of modelng efforts whch can accurately and Paper submtted 0/05/10; revsed 05/04/10; accepted 05/08/10. Author for correspondence: Yao-Mng Zhang (e-mal: zymfc@163.com). Insttute of Appled Mathematcs, Shandong Unversty of Technology, Zbo, Shandong Provnce, Chna. Industral Equpment State Key Laboratng of Structures Analyss, Dalan Unversty of Technology, Dalan 11604, Chna. Department of Harbor and Rver Engneerng, Natonal Tawan Ocean Unversty, Keelung 04, Tawan. effcently predct ther performance. For computatonal models of thn structures or thn shapes n structures, two numercal methods can be employed, namely, the fnte element method (FEM) and the boundary element method (BEM). In the last three decades, the FEM based on plate and shell theores has been a successful tool for the analyss of 3-D thn structures such as plates, shells and layered composte structures to study ther deformaton and stress n macro-scale. However, the FEM element count ncreases dramatcally for thn structures due to aspect rato lmtatons, and t was found out that, as shown n Ref. [14], the number of the fnte elements were so large that the task quckly exceed the capacty of the computer used as the thckness decreases. The BEM s a powerful and effcent computatonal method f ntegrals are evaluated accurately, and the BEM based on the elastcty theory s n general more accurate n stress analyss of structures. Ths accuracy wll be mantaned n the analyss of thn structures as well, f the BEM s mplemented correctly to deal wth the dffcultes assocated wth thn structures. Studes show that the conventonal boundary element method (CBEM) usng the standard Gaussan quadrature fals to yeld relable results for thn-walled structures. The major reason for ths falure s that the kernels ntegraton presents varous orders of near sngulartes, owng to the mesh on one sde of the thn-body beng too close to the mesh on the opposte sde. Nearly sngular ntegrals are not sngular n the sense of mathematcs. However, from the pont of vew of numercal ntegratons, these ntegrals can not be calculated accurately by usng the conventonal numercal quadrature snce the ntegrand oscllates very fercely wthn the ntegraton nterval. Other than the nearly sngular ntegral, many drect and ndrect algorthms for sngular ntegral have been developed and used successfully [1-4, 6, 7, 10, 11, 19, 3, 4]. Therefore, the key pont n achevng the requred accuracy and effcency of the BEM s not the sngular ntegral but the nearly sngular ntegral. Although that dffculty can be overcome by usng very fne meshes, the process requres too much preprocessng and CPU tme. In the past decades, tremendous effort s devoted to derve convenent ntegral forms or sophstcated computatonal technques for calculatng the nearly sngular ntegrals. These

2 410 Journal of Marne Scence and Technology, Vol. 19, No. 4 (011) proposed methods can be dvded on the whole nto two categores: ndrect algorthms and drect algorthms. The ndrect algorthms, whch beneft from the regularzaton deas and technques for the sngular ntegrals, are manly to calculate ndrectly or avod calculatng the nearly sngular ntegrals by establshng new regularzed BIE [5, 13, 17, 0,, 7]. The drect algorthms are calculatng the nearly sngular ntegrals drectly, such as nterval subdvson method [9], specal Gaussan quadrature method [15], exact ntegraton method [18, 8], and varous nonlnear transformaton methods [8, 16, 5, 6]. In a recent study, the above methods have been revewed n detal by Zhang and Sun [6]. Wth the development of the numercal technques for calculaton nearly sngular ntegrals, consderable progress has been made n the applcaton of the BEM to the analyss of thn walled structures n the past few years. Sladek et al. have obtaned amount of the orgnal results n ths feld [1]. Nonsngular ntegral equatons for thn-walled structures are proposed based on a subtracton technque and mathematcal regularzaton. Lu et al. [1, 14] have undertaken a lot of researches n ths feld. The nearly sngular surface ntegrals are transformed nto a sum of weakly sngular ntegrals, and a nonlnear coordnate transformaton s developed for nearly weakly sngular ntegrals. The theory s also appled to nterfacal stress analyss for mult-coatng systems, thermal stress analyss of mult-layer thn flms and coatngs, and also thn pezoelectrc solds. However, n Lu s work only some boundary unknowns can be computed, for example, ths approach only gves the results of radal stresses on the boundary nodes. The tangental stresses on the boundary and the physcal quanttes at nteror ponts need further nvestgaton. Zhou et al. proposed sem-analytcal or analytcal ntegral algorthms to solve -D or 3-D thn body problems [30, 31]. Both boundary and nteror unknowns are computed n ther works whch the geometry boundary s depcted by usng lnear shape functons when nearly sngular ntegrals need to be calculated. Although great progresses have been acheved of each of the above works, t should be pont out that most of these earler methods are nether general used nor provdng accurate results when the thckness of the thn structure s smaller than 1.0E-6. The objectve of ths paper s to develop a general BEM-based smulaton for predctng the physcal quanttes n thn structures when the thckness to length rato s n the orders of 1.0E-6 to 1.0E-9. It s well-known that the geometres of many problems of practcal nterest are created from crcular or ellptc arcs. Arc boundary elements can represent crcular and ellptc boundares exactly, and consequently, errors caused by representng such geometres usng polynomal shape functons can be removed usng exact geometrcal representatons. Therefore, the exact geometrcal representaton s expected to gve more accurate results than lower-order boundary element analyss when nearly sngular ntegrals need to be calculated. Another more mportant reason for usng exact geometrcal representaton can be found n the followng condtons. If the boundary geometry s depcted by usng lower order element, the lnear element of the outer surface wll attach or even pass through the nner boundary f the thckness of the consdered structure s very small. Consequently, the actual geometry of consdered doman can not be descrbed lvely, and thus lowerorder geometry approxmaton wll fal to yeld relable results for such problems. In order to avod ths phenomenon, very fne meshes mush be used n ths stuaton, but ths s bound to requre more preprocessng and CPU tme. Most mportantly, a great number of meshes wll produce a lot of artfcal corners whch wll lead to the uncontnuty of the tangent dervatve of the boundary unknowns. Ths s fatal to many engneerng problems. Obvously, the utlzaton of exact geometrcal representaton can be a good choce to avod ths problem. In vew of these reasons, ths paper wll gve an effcent strategy to treat thn body problems wth crcular and ellptc boundares. Ths paper s an extenson of our prevous work [5] where a new nonlnear transformaton method was proposed and appled to treat the thn body effect occurrng n D potental problems. Heren, we derve a general BEM-based approach to treat thn body problems n D elastostatcs. The BEMbased approach proposed n ths paper s constructed usng a combnaton of the regularzed ndrect BIEs and a general nonlnear transformaton whch can remove or damp out the nearly sngular propertes of the ntegral kernels. It s shown that ths combned BIE formulaton can provde stable results for the analyss of thn body problems. For the test problems studed, very promsng results are obtaned when the thckness to length rato of the coatngs s n the orders between 1.0E-1 and 1.0E-9, whch s suffcent for modelng most thn coated cuttng tools n the mcro- or nano- scales. In concluson, the seemng dffcult task of determnng the physcal quanttes n thn structures can be dealt wth effectvely and effcently. II. NON-SINGULAR BOUNDARY INTEGRAL EQUATIONS (BIES) It s well known that the doman varables can be computed by ntegral equatons only after all the boundary quanttes have been obtaned, and the accuracy of boundary quanttes drectly affects the valdty of the nteror quanttes. However, when calculatng the boundary quanttes, we have to deal wth the sngular boundary ntegrals, and usng the regularzed boundary ntegral equatons (BIEs). In ths paper, we always assume that Ω s a bounded doman n R ; Ω c s ts open complement, and denotes the boundary; t(x) and n(x) (or t and n) are the unt tangent and outward normal vectors of to the doman Ω at the pont x, respectvely. For D elastc problems, the non-sngular BIEs wth ndrect varables are gven n Ref. [9]. Regardless of to the rgd body dsplacement and the body forces, the non-sngular BIEs on Ω c can be expressed as

3 Y.-M. Zhang et al.: Boundary Element Analyss of Thn Structural Problems n D Elastostatcs 411 u y = x u y x d y (1) ( ) φk( ) k(, ), u = u d ( y) [ φk( x) φk( y)] k( y, x) φk( y ) u (, ) [( ) ( )] k yx t x t y d+ [ ( ) ( )] n x n y t uk ( yx, ) k0 d+ ny ( ) [ nk( ) nk( )] G x y n ln r ln r d+ nk( y) [ t( ) t( )] d x x y t ln r + nk( y) [ n( x) n( y)] d, y n For the doman Ω, the non-sngular BIEs are gven as ( ) φk( ) k(, ), ( () u y = x u x y d y (3) 1 nk( ) n( ) u ( ) φ k( ) ( ) [ k ] [ k( ) G δ y y y = y n y + (1 v) φ x φ x y k( y)] uk( y, x) d φk( y) [ t( ) t ( )] uk ( yx, ) uk ( yx, ) d+ [ ( ) ( )] d nx ny t n ( k0 ln r + n( y) [ nk( ) nk( )] d+ nk( ) G x y y x ln r [ t( x) t( y)] d+ nk( y) t ln r [ n( x) n( y)] d, y n For the nternal pont y, the ntegral equatons can be wrtten as ( ) ( ) (, ), ˆ φk k { (4) u y = x u y x d y Ω (5) ( ) ( ) (, ), ˆ φk k u y = x u y x d y Ω (6) In Eqs. (1)~(6),, k = 1, ; k 0 = 1/4π(1 v); G s the shear modulus; φ k (x) s the densty functon to be determned; u k (y, x) denotes the Kelvn fundamental soluton. In Eqs. (5) and (6) ˆΩ = Ω or Ω c. Note that, the Gaussan quadrature s drectly used to calculate the ntegrals n dscretzed equatons n the CBEM. However, f the doman of a consdered problem s thn, some boundares wll be very close to each other. Thus, the dstance r between some boundary nodes and boundary ntegral elements probably approaches zero. Ths causes the ntegrals n dscretzed Eqs. (1)~(4) nearly sngular, and the results of the Gaussan quadrature become nvald. Therefore, the densty functons cannot be calculated accurately, needless to say, to calculate the physcal quanttes at nteror ponts. Moreover, almost all the nteror ponts of thn bodes are very close to the ntegral elements. Thus, there also exst nearly sngular ntegrals n Eqs. (5) and (6). These nearly sngular ntegrals can be expressed as I1 = ψ ( x)lnrd e 1 I = ψ ( ) d x e r α where α > 0, ψ(x) s a well-behaved functon ncludng the Jacoban, the shape functons, and a fnte sum of polynomals dvded by r n. Under such a crcumstance, ether a very fne mesh wth massve ntegraton ponts or a specal ntegraton technque needs to be adopted. In the last two decades, numerous research works have been publshed on ths subject n the BEM lterature. Most of the works have been focused on the numercal approaches, such as subdvsons of the element of ntegraton, adaptve ntegraton schemes, exact ntegraton methods and so on. However, most of these earler methods are ether neffcent or can not provde accurate results when the thckness of the thn structure s smaller than 1.0E-6. In ths work, a very effcent transformaton method s proposed to avod the ntegraton dffculty for thn coatngs wth the thckness to length rato n mcro- or nano- scales. III. THE APPROXIMATION OF GEOMETRY BOUNDARIES The quntessence of the BEM s to dscretze the boundary nto a fnte number of segments, not necessarly equal, whch are called boundary elements. Two approxmatons are made over each of these elements. One s about the geometry of the boundary, whle the other has to do wth the varaton of the unknown boundary quantty over the element. 1. Lnear Element Approxmaton In ths secton, the geometry segment s modeled by a contnuous lnear element. Assumng x 1 = ( x 1 1 1, x), x = ( x 1, x) are the two extreme ponts of the lnear element j, then the element j can be expressed as 1 k 1 k k (7) x ( ξ) = N ( ξ) x + N ( ξ) x, ξ [ 1,1], k = 1, (8) where N 1 (ξ) = (1 ξ)/, N (ξ) = (1 + ξ)/. Lettng s = x 1 1 x, w = y ( x x )/, one has

4 41 Journal of Marne Scence and Technology, Vol. 19, No. 4 (011) r y x sξ + w r, = = = (9) r r r = = = ξ + ξ + = ξ η + r x y rr A B E L [( ) d ] (10) where A = s s /4, B = s w, E = w w, ηη = B/A, L = A, d = 4 AE B / A. Wth the ad of the Eq. (10), the nearly sngular ntegrals n Eq. (7) can be rewrtten as η { η } η 1 { η } 1 1 I1 = + ( )ln[( ) ] ln ( ) 1 g ξ ξ η + d dξ + L g ξ dξ (11) g( ξ ) I = + dξ α α L [( ξ η) + d ] where g( ) s a regular functon that conssts of shape functon and Jacoban.. Arc Element Approxmaton It s well-known that the geometres of many problems of practcal nterest are created from crcular or ellptc arcs. Arc boundary elements can represent crcular and ellptc boundares exactly, and consequently, errors caused by geometres usng polynomal shape functons whch can be removed by usng exact geometrcal representatons. Therefore, the exact geometrcal representaton s expected to gve more accurate results than lower-order or even hgh-order boundary element analyss when nearly sngular ntegrals need to be calculated. An exact geometrcal representaton, termed arc element, for crcular and ellptc boundares has been proposed by Zhang [9] n 004. Consder a crcular are element of radus of curvature R wth ts centre of curvature located at ( x 1, x ). Suppose (R, θ 1 ), (R, θ ) are the coordnates of the two extreme ponts of the arc element j, respectvely. Then the exact boundary elements can be expressed as: x1( ξ ) = x 1+ Rcosθ x( ξ ) = x + Rsnθ (1) where θ = 1 ξ θ θ, ( ξ 1). ξ The Jacoban of transformaton from arc element to ntrnsc coordnate ξ reduces to: J( ) R θ ξ θ 1 = (13) For the nteror pont y = (R 0 cos θ 0, R 0 sn θ 0 ), we can suppose θ 1 < θ 0 < θ and ( x 1, x ) = (0, 0). Then 1 η 1+ η θ0 = θ1+ θ ( 1< η < 1) Usng the procedure descrbed above, the dstance between the source pont and the feld pont can be expressed as: { γ } = = 4 0 sn + r x y RR d (14) R R0 where γ = β(ξ η), β = (θ θ 1 )/4, d =. 4 RR0 Substtutng r nto Eq. (7), then the ntegrals I 1 and I can be dvded nto two parts at pont η as follows η { η } η 1 α { η } 1 1 I1 = + ( )ln(sn ) ln ( ) 1 g ξ γ + d dξ + L g ξ dξ 1 g( ξ ) I = + dξ α L (sn γ + d ) (15) where L = RR0 and g( ) s a regular functon that conssts of shape functon, the Jacoban and a fnte sum of polynomals dvded by r n. IV. THE TRANSFORMATION FOR NEARLY SINGULAR INTEGRAL In Eqs. (11) and (15), f d s very small, the above ntegrals would present varous orders of near sngularty. The key to achevng hgh accuracy s to fnd a method to calculate these ntegrals accurately for a small value of d. The ntegrals I 1 and I n Eqs. (11) and (15) can be reduced to the followng ntegrals by smple deducton = + A gx ( ) T dx 0 = ( x + d ) α A T1 g( x)ln( x d ) dx 0 Introducng the followng nonlnear transformaton x d e t (16) k(1 + t) = ( ), [ 1,1] (17) where k = ln 1+ A d. Substtutng (17) nto (16), then the ntegrals I 1 and I can be rewrtten as follows: + dk g t e + e dt 1 k(1 + t) T1 = dk g()ln t d e dt 1 k(1 + t) k(1 + t) ()ln[( 1) 1] k(1 + t) 1 1 α gte () T = d k dt k(1 + t) α [( e 1) + 1] (18) We can observe that (e k(1+t) 1) Thus, the nte-

5 Y.-M. Zhang et al.: Boundary Element Analyss of Thn Structural Problems n D Elastostatcs 413 p A L = m Fg. 1. A thn plate under constant pressure p (Shear modulus μ = , Posson s rato v = 0.). h y x Normal stresses Present CBEM Exact grand s fully regular even f the value of d s very small. By followng the procedures descrbed above, the near sngularty n the BIEs has been fully regularzed. The fnal ntegral formulatons are obtaned as shown n Eq. (18), and be computed straghtforwardly by usng the standard Gaussan quadrature. V. NUMERICAL EXAMPLES To verfy the method developed above, two smple test problems are studed n whch BEM solutons are compared wth the exact solutons. 1. Test Problem 1: A Thn Plate Frst, a thn plate under external pressure p shown n Fg. 1 s studed. We assume the length of the plate n z drecton s large so that ths problem can be smplfed as a plane stran problem. The length L of the plate n x drecton s constant n ths study, whle the thckness h changes from L to 10-9 L. Note that the thckness s changng from macro-scale to mcroscale, and eventually to nano-scale. On the boundary y = 0, dsplacement components n both x and y drectons are constraned. The fnte element analyss of ths problem was attempted n Ref. [14], but t was soon found out that the number of the D fnte element was so large that the task quckly exceed the capacty of the computer used. In the BEM model, the boundary of the plate s dscretzed wth only 0 lnear boundary elements, 16 elements wth length L = m, and 4 other elements wth thckness h. When the thckness h ranges from 1.0E-1 to 1.0E-10, the results of the normal stresses σ xx and σ yy at nteror pont A(0.8, h/ are shown n Fg. and Fg. 3, respectvely. It s obvous that the results calculated by usng the CBEM deterorate quckly as the thckness less than 1.0E-. In contrast, the results calculated by usng the proposed method are very consstent wth the exact solutons even for the thckness as small as 1.0E-10. Ths proves that the developed transformaton work n the BEM procedure s effectve. For h = 1.0E 9, the stress results at the nteror ponts on the lne x = 1.8 are lsted n Table 1 and Table. These results calculated by the present BEM are all n good agreement wth the exact solutons wth the largest relatve error less than.0e-4 (%). However, the CBEM are nvald to calculate both the radal and tangental stresses at these ponts E-0 1E-04 1E-06 1E-08 1E-10 Thckness h Fg.. Normal stresses σ xx at the nteror pont A of Fg. 1. Normal stresses Present CBEM Exact E-0 1E-04 1E-06 1E-08 1E-10 Thckness h Fg. 3. Normal stresses σ yy at the nteror pont A of Fg. 1. The example problem studed here s a very smple one n thn structures. The purpose of usng ths example s to verfy the correctness of the present transformaton method n dealng wth nearly sngular ntegrals.. Test Problem : Thn Coatng on a Shaft In ths secton, the method developed n ths paper wll be used to solve a problem of a shaft wth a thn coatng [14], as shown n Fg. 4. The shaft and coatng have outer rad r a and r b respectvely, wth ther centre of curvature located at the pont o(0, 0). In ths example, the coated system s loaded by a unform pressure p = 1, and the shaft s consdered to be rgd when compared to the coatng, so the boundary condtons are u x = u y = 0 for all nodes at the shaft/coatng nterface. In ths example, δ = (r b r a )/r a s defned as the thckness to length rato. As r a s held constant at 1, the rato reduces as r b decreases. There are totally 0 dscontnuous arc elements dvded along the shaft and coatng surfaces, regardless of the thck-

6 414 Journal of Marne Scence and Technology, Vol. 19, No. 4 (011) Table 1. Normal stresses σ xx at the nteror ponts on the lne x = 1.8, where h = 1.0E 9. y Exact CBEM Relatve error (%) Present Relatve error (%) 1.0E E E E E-03.0E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-03 Table. Normal stresses σ yy at the nteror ponts on the lne x = 1.8, where h = 1.0E 9. y Exact CBEM Relatve error (%) Present Relatve error (%) 1.0E E E E E-04.0E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-04 r a r b p B o Fg. 4. A shaft wth a thn unform coatng. It s obvous that the results calculated by usng the proposed method, shown n Table 3, are very consstent wth the exact solutons even for the thckness to length rato as small as 1.0E-10. We can see from Table 4 that the results of tangental stresses calculated by usng the proposed method are very consstent wth the exact solutons, wth the largest relatve error less than 0.%, even when the thckness-to-length rato as small as 1.0E 9. For δ = 1.0E 9, convergence curves of the computed tangental stresses at nteror ponts B by usng the presented method are shown n Fg. 5, whch we can observe that the convergence speeds are stll fast even when the thcknessto-length rato s as small as 1.0E 9. In Fg. 5, only the errors of the present method are gven snce the errors of the CBEM are relatvely too large. ness of the structure. The elastc shear modulus s G = Pa, Posson s rato s v = 0.. In 1998, Luo et al. [14] have handled ths coatng system. However, n ther work only boundary unknown radal stresses σ r are computed. The physcal quanttes at nteror ponts need further nvestgaton. For dfferent thckness-to-length ratos, the results of the radal and tangental stresses at nteror pont B((r a + r b )/, 0) are lsted n Table 3. For δ = 1.0E 9, the results of tangental stresses on the lne y = 0 are lsted n Table 4. Both the CBEM and the proposed method are employed for the purpose of comparson. VI. CONCLUSION In ths paper, a BEM-based approach s presented and appled to deal wth -D elastc problems of thn bodes. Usng the transformaton technque demonstrated n ths paper, the seemngly dffcult task of evaluatng the nearly sngular ntegrals n the BIE for -D thn structures can be dealt wth effectvely and effcently. Two numercal examples of elastc thn-walled structures wth thckness-to-length ratos rangng between 1.0E 1 and 1.0E 9, whch s suffcent for modelng most thn structures n ndustral applcatons, are presented to test the proposed method. In concluson, the

7 Y.-M. Zhang et al.: Boundary Element Analyss of Thn Structural Problems n D Elastostatcs 415 Table 3. Radal and Tangental stresses at the nteror pont B of Fg. 4. δ Radal stresses σ r Tangental stresses σ θ Exact Present Exact Present 1.0E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E+00 Table 4. Tangental stress σ θ at the nteror pont on the lne y = 0, where δ = 1.0E 9. x Exact CBEM Relatve error (%) Present Relatve error (%) E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E+00 Relatve error (%) tangental stresses Number of elements Fg. 5. Convergence curves of σ θ at the nteror pont B of Fg. 4. thn-body problem has been overcome successfully by usng the proposed strategy, whch ndcates that BEM s especally accurate and effcent for numercal analyss of thn boy problems. The developed method for analyzng -D thn structures can be extended to model layered coatngs, thn flms or other layered structures, and some work along ths lne for thn structures wll be dscussed later. ACKNOWLEDGMENTS The research s supported by the Natonal Natural Scence Foundaton of Chna ( ), the Openng Fund of the State Key Laboratory of Structural Analyss for Industral Equpment (GZ1017) and the Natonal Natural Scence Foundaton of Shangdong Provnce of Chna (ZR010AZ003). REFERENCES 1. Brebba, C. A., Tells, J. C. F., and Wrobel, L. C., Boundary Element Technques, Sprnger-Verlag, Berln, Germany (1984).. Chen, J. T., Chen, P. Y., and Cha, T. C., Surface moton of multple alluval valleys for ncdent plane SH-waves by usng a sem-analytcal approach, Sol Dynamcs and Earthquake Engneerng, Vol. 8, No. 3, pp (008). 3. Chen, J. T. and Hong, H. K., Revew of dual boundary element methods wth emphass on hypersngular ntegrals and dvergent seres, Appled Mechancs Revews, Vol. 5, No. 1, pp (1999). 4. Chen, J. T. and Shen, W. C., Degenerate scale for multply connected laplace problems, Mechancs Research Communcatons, Vol. 34, No. 1, pp (007b). 5. Granados, J. J. and Gallego, G., Regularzaton of nearly hypersngular ntegrals n the boundary element method, Engneerng Analyss wth Boundary Elements, Vol. 5, No. 3, pp (001). 6. Hong, H.-K. and Chen, J. T., Dervatons of ntegral equatons of elastcty, Journal of Engneerng Mechancs, ASCE, Vol. 114, No. 6, pp (1988). 7. Hsao, S. S., Ln, M. C., and Hu, N. C., Drbem analyss of combned wave refracton and dffracton n the presence of current, Journal of Marne Scence and Technology, Vol. 10, No. 1, pp (00). 8. Huang, Q. and Cruse, T. A., Some notes on sngular ntegral technques n boundary element analyss, Internatonal Journal for Numercal

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