Correction of boundary element analysis to scattering of surface waves in an elastic half-space

Nội dung text: Correction of boundary element analysis to scattering of surface waves in an elastic half-space

1. Tuyển tập Hội nghị khoa học toàn quốc lần thứ nhất về Động lực học và Điều khiển Đà Nẵng, ngày 19-20/7/2019, tr. 369-372, DOI 10.15625/vap.2019000304 Correction of boundary element analysis to scattering of surface waves in an elastic half-space Ductho Lea,* and Haidang Phanb,* aHanoi University of Science and Technology bInstitute of Theoretical and Applied Research, Duy Tan University *Email: ducthole24@gmail.com / haidangphan.vn@gmail.com Abstract points, in principle they are infinite. However, those In this paper, the problem of surface wave propagating in an boundaries can be truncated at a relative short distance elastic half-space media is investigated by boundary element from the surface without lost accuracy because of the method. The boundary element method is studied to obtain the attenuation of Rayleigh wave. far-field displacements generated by time-harmonic loads. An In this article, a new elegant and simple approach to elegant treatment for the truncated point on the boundary is provided in which omitted parts of the boundary are substituted the treatment on the truncated point is proposed by using by two fictitious boundaries in the end of two sides of boundary reciprocity theorem. The integral on omitted parts of by using reciprocity theorems. Calculation examples show that boundary are substituted by integral on fictitious the reflection of surface wave in the end of computational mesh boundaries in an integral formulation obtain by consider was eliminated. The comparisons between numerical results suitable quarter-space under reciprocity approach. and the analytical solution of the problem perform an excellent Computer programs are made to obtain both numerical agreement. results and analytical results from (Phan et al., 2013a) then compare each other. Keywords: Boundary element method (BEM), Rayleigh wave, truncated boundary, time-harmonic loads, half-space, 2. Boundary integral formulation reciprocity approach. Consider surface waves generated by time-harmonic loads in an elastic, isotropic and homogeneous 1. Introduction half-space. The boundary in two dimensions is an infinite Propagation of surface wave in an elastic medium is line which is called  in figure 1. The boundary  is a generally and popularity problem in both research field free of traction except the localized religion 1 where and technical application. Waves generated by need to imposed boundary conditions. The boundary is time-harmonic load in an elastic half-space consist two truncated at two points 1 and N , the fictitious main forms namely body waves and guided waves. boundaries − and + are putted on each truncated Rayleigh wave is a particular form of guided waves that 2 2 propagate without geometry decay in wave transfer points, respectively. direction, but its amplitude shown exponentially decrease with depth. It was first described mathematically by (Lord Rayleigh, 1885), after that many characteristics of Rayleigh wave were studied by (Achenbach, 1973; Evernden, 1954; Adler et al., 1977; Phan et al., 2013a; Phan et al., 2014; Phan et al., 2019) and many more scientists. Boundary element method is the most appropriate one of numerical solutions can use for wave propagating and scattering problems. It was shown many advantages Figure 1. Boundary of problem on the computing time, mass of computation and accuracy of results. Furthermore, the BEM can solve the Follow (Dominguez, 1993) the boundary integral problem on the complex geometry while the analytical equation for a point  without body force can be solution only has approximate results on it. This paper i written as based on many previous works on BEM, see for example, ii (Dominguez, 1993; Phan et al., 2013b; Liu et al., 2011; c()() u = ((,)() u  xpxp − (,)())()  xuxdx  (1) lk i k i  lk i k lk i k Arias and Achenbach, 2004). Where lk,1,2= . u* and p* are full-space fundamental For an elastic half-space, the boundary is infinite lk lk therefore we need to impose suitable boundary solution of displacements and tractions given by conditions. The boundary is truncated in two opposite (Dominguez, 1993), while uk and pk are the points which far enough from load source to ensure that displacements and tractions on boundary, respectively. only have the displacement-field of Rayleigh wave on it. ci is jump coefficient depend on the smooth of Two fictitious boundaries are putted on two truncated lk boundary point where load is applied.
2. Ductho Le and Haidang Phan The boundary condition in  and  is free of − 0 ii 0 Al()()()() i u l 1 = c lk  i u k  i traction, apply this into the eq. (1) we have (9) −p*(,)()()(,)()() xuxdx 0  + u * xpxdx 0  ii * ++  lk i k   lk i k 2 0 1 2 1 cupxlkikilkik()()(,)()() ux dx + 01 + − + (2) Two fictitious boundaries 2 and 2 are infinite in +=px (,)()()(,)()() ux dxuxpx dx lkiklkik principle. However, we can truncate those boundaries at a 1 The displacement field is contributed by three short distance from surface without lost accuracy. characters of waves namely longitudinal, transverse and Generally, at three times of wavelength. Thus, we can − Rayleigh waves. However, at the point which locate far evaluate integrals on and instead of  and from source load the body waves are almost completely +  . decayed thus the far-field displacement solution can be approximated by Rayleigh waves component, see (Phan 4. Calculation results et al., 2013a). Hence, if the truncated points  and  are In this section, we consider the problem of surface 1 N waves generated by distribution of loadings on the located far enough from source region eq. (2) can be surface of a half-space. The numerical results from BEM rewritten as computer program are compared with analytical solution −+ii AuAuculillilNlkiki()()()()()()1 ++ of problem obtain by (Achenbach, 2003; Phan et al., (3) 2013a) to verify the accuracy of calculation. Both += px (,)()()(,)()() ux dxuxpx dx  + lkiklkik truncated BEM (Dominguez, 1993) and corrected BEM 011 Where are used to see which one has better results. 1 In figure 2, the region R0 describe the boundary Aupxux− ()()(,)()() dx =*0 (4) lillkik 1 0 − condition area, the rest of boundary has free of traction. u () l 1 The length of computational mesh is chosen approximate 1 Aupx+ ()()(,)()() ux dx =*0 (5) eighty times of wavelength, each side. lilNlkik 0 + ulN() Material in use is Steel with shear modulus 102 0 is  = 7.987210/ Nm , Lame’s constant is In eq. (4) and eq. (5) ul represent the displacements 102 2 corresponding to unit amplitude time-harmonic Rayleigh  = 11.0310/ Nm, and density is = 7800/kgm . surface waves propagating in a half-space, the expression The frequency is fMHz= 0.5 , the traction chosen is 0 of ul can be found in (Achenbach, 1973). T =  /2. A BEM calculation is constructed base on eq. (3) without first two terms is presented by (Dominguez, 1993). It is a simple truncated boundary integral in which omit the infinite parts are represented in eq. (4) and eq. (5). This leads to the reflection of Rayleigh waves at the end of the calculational mesh that make the amplitude disturbed. 3. Integral over the omitted part of the infinite boundary Figure 2. Example of calculation The integrals in eq. (4) and eq. (5) can be calculated by using reciprocity theorem. Following (Achenbach, The programing implementation is done by 2003), in the domain −+   we can write following the block diagram in figure 3. 2 cuuxii( )( p )(( xpx=− , u )( x )( dx ,*0*0 )( ))( ) (6) lkikilkiklkik −+  2 ++ Where = 01  Apply boundary condition we have px*0(,) u x d () x= ()()() cu ii + lk iklk i k i (7) − + pxu*0*0(,) xd () xuxp ()(,) xd x () () −−   lk iklk ik 2 0 12 1 Therefore, + 0 ii Al()()()() i u l  N= c lk  i u k  i (8) −p*(,)()()(,)()() xuxdx 0  + u * xpxdx 0  −−  lk i k   lk i k 2 0 1 2 1 Similarly, in negative direction we can derive Figure 3. Steps of programing implementation
3. Correction of boundary element analysis to scattering of surface waves in an elastic half-space Figure 4 and figure 5 show the comparison between the analytical and BEM results for backscattering and forward scattering amplitudes of vertical displacement corresponding to a distribution of vertical loading with R0 from 0 . 0 0 0 4m to 0 . 0 0 1 8m while figure 6 and figure 7 present the comparison for the case of a distribution of horizontal loading. The comparisons are in excellent agreement. Figure 7. Forward scattering amplitude of vertical displacement corresponding to horizontal loadings 5. Conclusions This paper has proposed a new elegant and simple approach to correct the BEM calculation on surface waves propagating and scattering problem, reach out to the better accuracy. The proposed method using reciprocity theorem to add, substitute and account for the Figure 4. Back-scattering amplitude of vertical omitted parts in the infinite boundary. The efficiency of displacement corresponding to vertical loadings method has been confirmed by results in a test problem. The comparisons between results from BEM program and analytical solution show an excellent agreement. Reference [1] Achenbach, J.: Wave propagation in elastic solids. Elsevier, 1973. [2] Dominguez, J.: Boundary elements in dynamics. Wit Press, 1993. [3] Achenbach, J.: Reciprocity in elastodynamics. Cambridge University Press, 2003. [4] Evernden, J.F., Direction of approach of Rayleigh waves and related problems (Part II), Bulletin of the Seismological Society of America, vol. 44, no. 2A, pp. Figure 5. Forward scattering amplitude of vertical displacement corresponding to vertical loadings 159-184, 1954. [5] Adler, L., Cook, K.V., Dewey, B.R., King, R.T., Relationship between ultrasonic Rayleigh waves and surface residual stress, Materials Evaluation, vol. 35, no. 7, pp. 93-96, 1977. [6] Phan, H., Cho, Y., Achenbach, J.D., Validity of the reciprocity approach for determination of surface wave motion, Ultrasonics, vol. 53, no. 3, pp. 665-671, 2013a. [7] Phan, H., Cho, Y., Achenbach, J.D., Verification of surface wave solutions obtained by the reciprocity theorem, Ultrasonics, vol. 54, no. 7, pp. 1891-1894, 2014. [8] Phan, H., Cho, Y., Le, Q.H., Pham, C.V., Nguyen, H.T.L., Nguyen, P.T., Bui, T.Q., A closed-form solution Figure 6. Back-scattering amplitude of vertical to propagation of guided waves in a layered half-space displacement corresponding to horizontal loadings under a time-harmonic load: An application of
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