Scattering of plane P- and SV-waves by periodic topography: Modeled by a PIBEM

Engineering Analysis with Boundary Elements 106 (2019) 320–333

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Engineering Analysis with Boundary Elements journal homepage: www.elsevier.com/locate/enganabound

Scattering of plane P- and SV-waves by periodic topography: Modeled by a PIBEM B.A. Zhenning a,b, Xu Gao b, Vincent W. Lee c,∗ a

State Key Laboratory of Hydraulic Engineering Simulation and Safety, Tianjin University, Tianjin 300354, China Department of Civil Engineering, Tianjin University, Tianjin 300072, China c Department of Civil Engineering, University of Southern California, Los Angeles, CA 90089-2531, USA b

a r t i c l e

i n f o

Keywords: Periodic topography Green’s functions P-waves and SV-waves Scattering

a b s t r a c t For model of periodic topography subjected to plane waves, the wave fields have feature of repeating themselves with a certain shift of phase in the frequency domain. By fully exploring this particular feature, a periodic indirect boundary element method (PIBEM) is proposed to study the scattering of plane P- and SV-waves by periodic topography. The discretization effort of the PIBEM is reduced to a single topography using Green’s functions of equivalent distributed loads acting on an inclined line as fundamental solutions. Compared to the traditional way of choosing a certain number of topographies to solve the problem with boundary conditions of other topographies being relaxed, the new method has the merits of higher precision and lower memory requirement. By taking periodic hill and canyon as examples, parametric studies are conducted to investigate the complex effects due to the periodic irregularity. Numerical results show that responses of periodic topography are quite different from those of a single and multiple topographies, demonstrating that it is very difficult to obtain the accurate results by choosing a certain number of topographies. In addition, periodic canyon has a stronger shielding effect on P-waves, while periodic hill has a stronger shielding effect on SV-waves.

1. Introduction Investigations of destructive earthquakes have shown that local topographies can significantly affect the seismic ground motion. For example, extensive damage was observed on the ridges of Canal Beagle during the 1985 Chile earthquake [2]. During the 1989 Prieta Loma earthquake, the top of the Robinwood ridge was destroyed [3]. During the 2008 Wenchuan earthquake, most of the disasters were found near the alluvial valleys and canyon sections, particularly in the upper segment of canyon sections (namely, the turning point from the dale to the canyon) [4]. The above examples fully demonstrate that the topographies result in a significant amplification of seismic ground motion during an earthquake. Note that some topographies in nature are continuously (periodically) distributed, such as periodic bedrock ridges in Puna plateau of northwestern Argentina [53], hilly landform and so on. Continuous (periodic) topography also results in a significant amplification of seismic ground motion. Liu [35] investigated the scattering of P-, SV- and Rayleigh waves by three continuous sequential hills. It was found that strong dynamic interaction existed among the hills, which resulted in much higher seismic ground amplitudes. In addition, wave

∗

scattering of periodic structures are wildly existed in other fields, such as industrial engineering field [54], optical field [49,50], electromagnetic field [51], mechanical engineering field [52] and so on. Thus, wave scattering problem of periodic irregularities is a hot topic worthy to be investigated. In the past few decades, researchers have paid much attention on the scattering of elastic waves by topographies. However, most studies focus on the scattering of elastic wave by a single topography. The analytical methods or numerical methods have been widely used to solve this problem. The analytical methods have been proposed for series solutions of simple topographies. The addressed topographies mainly include semi-circular canyon [5,6], semi-elliptical canyon [7], V-shaped canyon [11–14], shallow circular-arc canyon [8–10], semi-circular alluvial valley [39–42], semi-circular hill [15,16] and semi-elliptical hill [17–19]. Numerical methods for the scattering of elastic waves are much more flexible than the analytical methods. With the rapid update of computer technology in recent years, many numerical methods have been developed to solve the scattering of elastic waves by topographies. The numerical methods include the integral equation method [24,25,44], the accurate finite element-indirect boundary integral equation method

Corresponding author. E-mail address: [email protected] (V.W. Lee).

https://doi.org/10.1016/j.enganabound.2019.05.020 Received 27 January 2019; Received in revised form 15 March 2019; Accepted 18 May 2019 Available online 4 June 2019 0955-7997/© 2019 Elsevier Ltd. All rights reserved.

B.A. Zhenning, X. Gao and V.W. Lee

Engineering Analysis with Boundary Elements 106 (2019) 320–333

Table 1 Advances of the computational wave propagation. Published date

Author

Calculation method

Topography (model)

1975 1982 1982 1982 1989 1990 1990 1992 1994 1994 1996 1996 1996 2000 2006 2006 2006 2008 2008 2008 2010 2010 2010 2010 2010 2011 2011 2012 2012 2013 2013 2013 2014 2014 2015 2015 2015 2016 2016 2017 2017 2018 2018

Wong and Jennings Bard Sanchez-Sesma et al. Wong Lee and Cao Cao and Lee Lee Yuan and Men Yuan and Liao Pederaen et al. Takenaka et al. Bouchon et al. Liu Liang et al. Liang et al. Zhao et al. Kamalian et al. Tsaur and Chang Liang and Ba Chen et al. Tsaur et al. Tsaur Smith Ba and Liang Liu et al. Liang and Fu Chen et al. Zhang et al. Chen et al. Gao and Zhang Lee VW and Amornwongpaibun Amornwongpaibun and Lee VW Yang et al. Fu et al. Fu et al. Zhang et al. Liu et al. Liu et al. Ba et al. Zhang et al. Zhang et al. Tang et al. Fu et al.

Boundary element method Discrete wave number method Boundary element method Boundary element method Wave function expansion method Wave function expansion method Wave function expansion method Wave function expansion method Wave function expansion method Boundary element method Discrete wave number method Discrete wave number method Finite element method Wave function expansion method Wave function expansion method Wave function expansion method Boundary element method Wave function expansion method Boundary element method Semi-analytical approach Wave function expansion method Wave function expansion method Finite element method Boundary element method Analytical method Wave function expansion method Integral equation method Wave function expansion method Integral equation method Wave function expansion method Wave function expansion method Wave function expansion method Wave function expansion method Singular boundary method Singular boundary method Wave function expansion method Integral equation method Accurate finite element-indirect Boundary element method Wave function expansion method Wave function expansion method Singular boundary method Singular boundary method

A single canyon A single hill A single surface irregularities hill A single semi-elliptical canyon A single semi-circular canyon A single semi-elliptical canyon A single shallow circular-arc canyon A single semi-circular hill A single shallow circular-arc canyon A single semi-circular canyon A single 2-D irregular topography A single 3-D topography of arbitrary shape Three sequential hills A single shallow circular-arc canyon A single semi-circular alluvial valley A single semi-circular alluvial valley A single homogeneous and non-homogeneous topographic structures A single V-shaped canyon A single hill Multiple alluvial valleys A single V-shaped canyon A single semi-circular hill A single hill A single canyon Two hills of different geometries A single semi-elliptical hill A semi-circular hill A single V-shaped canyon A semi-elliptical hill A single V-shaped canyon A single semi-elliptical hill A single semi-elliptical hill Two scalene triangular hills A soft infinite circular cylinder A rigid infinite circular cylinder A single semi-circular alluvial valley A single canyon Coupled alluvial basin-mountain terrain Periodic canyon and alluvial valley A single semi-circular canyon A single semi-circular alluvial valley Canyon topographic Periodic structures

[43], the finite element method [23], the discrete wave number method [20–22], the boundary element method [26–32] and the singular boundary method [46]. The models of studies above are confined to a single topography. Fewer works have been undertaken for the case of multiple and periodic continuous topographies (models). Chen et al. [36] studied the scattering of SH-waves by multiple continuous alluvial valleys. Liu et al. [34] investigated the scattering of SH-waves by two continuous hills of different geometries on a half-space. Yang et al. [33] obtained the solution of two scalene triangular hills and a semi-cylindrical canyon for incident SH-waves. As for the case of P- and SV-waves, only Liu [35] applied the finite element method to investigate the scattering of P-, SV- and Rayleigh-waves by three continuous sequential hills. Fu et al. [45,55,56] studied the periodic structures by using singular boundary method. From the above literatures (Table 1), it shows that except for Fu et al. [45,55,56], most of the studies on scattering of continuous topographies used traditional way of choosing a certain number of topographies as the calculation cells. This traditional way must discrete each topography and make all topographies satisfy the boundary conditions at the same time. Therefore, the memory requirement and computational cost will be large. For this reason, PIBEM, which only needs choose one topography as the calculation cells, was first proposed by Ba et al. [37] to study the scattering of continuous (periodic) topographies. However, only the scattering of plane SH-waves by periodic topography

had been given in Ba et al. [37]. For the more complex in-plane case i.e., incident plane P- and SV-waves, there are no publications. Hence, a periodic indirect boundary element method (PIBEM) is proposed to study periodic scattering of in-plane waves in this paper. The paper is organized as follows: First, Green’s functions of equivalent uniformly distributed loads acting on an inclined line are derived. Then, the PIBEM is verified by comparing with the published ones. Next, parametric studies are performed in the frequency domain by taking two kinds of periodic topography (periodic canyon and hill) as examples. Finally, the conclusions are given. 2. Model and theoretical formulations The scattering of P- and SV-waves by periodic topography with arbitrary cross-sections in a layered half-space (Fig. 1) is investigated by using PIBEM. The periodic elements are represented by topography “-∞”, … , “−1”, “0”, “1”, … , “∞”. The traction free boundaries of the topographies are represented by S’ -∞ , … , S’ −1 , S’ 0 , S’ 1 , … , S’ ∞ . The interfaces between topographies and half space are represented by S-∞ , … , S-1 , S0 , S2 , … , S∞ . The layered half-space is composed of a number of horizontally layers and the underlying half-space. The material properties of the layered half-space and hills are assumed to be linear elastic, isotropic and homogenous. The incident angle 𝜓 is measured from the horizontal 321

B.A. Zhenning, X. Gao and V.W. Lee

Engineering Analysis with Boundary Elements 106 (2019) 320–333

Fig. 1. Model of periodic topography subjected to incident P- and SV-waves.

Fig. 2. Decompose the model into a closed topography region and an opened layered half-space region.

direction. 𝜔 is the incident circular frequency. L is the period of the topographies. As for the scattering of P- and SV-waves by periodic topography, if only a certain number of topographies is discretized to solve the problem, errors will be introduced due to other topographies’ boundary conditions being relaxed. Moreover, the memory requirement and CPU time will be large. However, note that the wave fields around each of topographies along the x-axis have the characteristics of repeating themselves with a certain shift of time in the time domain (a certain shift of phase in the frequency domain) when plane P- and SV-waves are incident. Therefore, if any one of the topographies’ boundary condition is satisfied, other topographies’ boundaries will be satisfied automatically. By fully exploring this particular feature, the accurate solution of this problem can be obtained by discretizing a single topography. As shown in Fig. 2, the “0th” topography is selected as the calculation cell. For the convenience of calculation, the region matching technique is used to decompose this model into the enclosed topography region ΩH and the opened layered half-space region ΩL . The wave fields in region ΩL include the scattered fields and the free fields which can be obtained by the direct dynamic stiffness method. However, the wave fields in region ΩH only include the scattered fields. Note that the scattered fields in region ΩH are different from those of region ΩL . The scattered fields in region ΩH are simulated by Green’s functions given by Wolf [1]. The scattered fields in region ΩL are represented by Green’s functions of the uniformly distributed inclined loads proposed in this paper. In the following sub-sections, the free filed response obtained by using the exact dynamic stiffness matrix method is briefly introduced; Green’s functions of equivalent uniformly distributed loads acting on an inclined line are derived and the boundary conditions are introduced to solve the problem.

𝐒R [1], the dynamic equation of motion is constructed as P−SV 𝐒P−SV 𝐔 = 𝐐

(1)

where SP-SV denotes the assembled dynamic stiffness matrix; U is the displacement vector; Q is the external load vector. For the plane P- and SV-waves incident from the underlying halfspace, the last two elements in Q can be determined according to Eq. (2), while other elements are zero. { } { } 𝑃𝑏 𝑢0 = 𝐒R (2) − 𝑆𝑉 P i𝑅𝑏 i𝑤0 where Pb and Rb are the component amplitudes of the load vector, u0 and w0 denote the prescribed outcropping motion. The dynamic responses of any points in the layered half-space can be determined by solving Eq. (1). More details on the free field can be found in Ref. [1]. 2.2. Green’s functions and the scattered field As shown in Fig. 3, the equivalent uniformly distributed loads acting on an inclined line, which is defined as the sum of uniformly distributed loads acting on the periodic lines, can be expressed as 𝑝𝑒 (𝑥, 𝑧, 𝑡) =

∞ ∑ 𝑛=−∞

𝑟𝑒 (𝑥, 𝑧, 𝑡) =

∞ ∑ 𝑛=−∞

𝑝𝑛 (𝑥, 𝑧, 𝑡) = 𝑟𝑛 (𝑥, 𝑧, 𝑡) =

∞ ∑ 𝑛=−∞ ∞ ∑ 𝑛=−∞

𝑝𝑛 (𝑡)𝛿[𝑧 − (𝑥 − 𝑛𝐿) tan 𝜃]

(3)

𝑟𝑛 (𝑡)𝛿[𝑧 − (𝑥 − 𝑛𝐿) tan 𝜃]

(4)

where pn (x, z, t) and rn (x, z, t) are the nth uniformly distributed loads in horizontal and vertical directions, respectively. z is the distance from the upper boundary of the loads, L is the distance between the inclined loads, 𝛿 is the Dirac-delta function, 𝜃 is the angle of the line measured from the horizontal plane. Note that the nth and 0th loads can be expressed as { } { } 𝑝𝑛 (𝑥, 𝑧, 𝑡) 𝑝 (𝑡 ) = {𝛿[𝑧 − (𝑥 − 𝑛𝐿) tan 𝜃]} 𝑛 (5) 𝑟𝑛 (𝑥, 𝑧, 𝑡) 𝑟 𝑛 (𝑡 )

2.1. Exact dynamic stiffness matrix and the free field The free fields responses can be obtained by the direct dynamic stiffness method. By assembling the dynamic stiffness matrix of the horizontal layers 𝐒LP−SV and dynamic stiffness coefficient of underlying bed rock 322

B.A. Zhenning, X. Gao and V.W. Lee

Engineering Analysis with Boundary Elements 106 (2019) 320–333

Fig. 3. Periodic distributed loads acting on inclined lines and equivalent uniformly distributed loads acting on an inclined line over part of a layer.

{

𝑝0 (𝑥, 𝑧, 𝑡) 𝑟0 (𝑥, 𝑧, 𝑡)

}

{ } 𝑝 (𝑡 ) = [𝛿(𝑧 − 𝑥 tan 𝜃)] 0 𝑟 0 (𝑡 )

From Eq. (14), the total dynamic response can be reformulated as

(6)

From Fig. 3, Eqs. (5) and (6) can be reformulated as { } { } {( ) } 𝑝0 𝑝𝑛 (𝑥, 𝑧, 𝑡) = 𝑡 − 𝑛𝐿∕𝑐𝑎 𝛿[𝑧 − (𝑥 − 𝑛𝐿) tan 𝜃] 𝑟𝑛 (𝑥, 𝑧, 𝑡) 𝑟0

𝑛=−∞

∞ ∑ 𝑛=−∞ ∞ ∑

𝑟𝑒 (𝑥, 𝑧, 𝑡) =

𝑛=−∞

( ) 𝑝0 𝑡 − 𝑛𝐿∕𝑐𝑎 𝛿[𝑧 − (𝑥 − 𝑛𝐿) tan 𝜃]

(8)

( ) 𝑟0 𝑡 − 𝑛𝐿∕𝑐𝑎 𝛿[𝑧 − (𝑥 − 𝑛𝐿) tan 𝜃]

(9)

𝑓 (𝑥, 𝑧, 𝜔) = =

𝐹 (𝑥, 𝑡) =

∞

∞

∫−∞ ∫−∞

𝑓 (𝑥, 𝑡)𝑒i𝑘𝑥−i𝜔𝑡 𝑑 𝑥𝑑 𝑡

∞

𝑛=−∞

𝑟𝑒 (𝑘, 𝑧, 𝜔) =

∞ ∑ 𝑛=−∞

′

(15)

∞ ∞ ′ 1 ∑ 𝑓 (𝑘, 𝑧, 𝜔)𝑒i(𝑘−𝑘 )𝑛𝐿 𝑒−i𝑘𝑥 𝑑𝑘 2𝜋 𝑛=−∞ ∫−∞ 0 ∞ ∞ 1 ∑ −i𝑘′ 𝑛𝐿 𝑒 𝑓 (𝑘, 𝑧, 𝜔)𝑒i𝑘(𝑛𝐿−𝑥) 𝑑𝑘 ∫−∞ 0 2𝜋 𝑛=−∞

− i𝑘(𝜆∗ + 𝐺∗ )𝑤,𝑧 (𝑘, 𝑧, 𝜔) = −𝜌𝜔2 𝑢(𝑘, 𝑧, 𝜔) − 𝑝0 (𝑘, 𝑧, 𝜔)

(16)

(17)

∞

(11)

−i𝑘(𝜆∗ + 𝐺∗ )𝑢,𝑧 (𝑘, 𝑧, 𝜔) − 𝑘2 𝐺∗ 𝑤(𝑘, 𝑧, 𝜔) + (𝜆∗ + 2𝐺∗ )𝑤,𝑧𝑧 (𝑘, 𝑧, 𝜔) = −𝜌𝜔2 𝑢(𝑘, 𝑧, 𝜔) − 𝑟0 (𝑘, 𝑧, 𝜔)

Performing the Fourier transformation given by Eq. (10) on Eqs. (8) and (9), the equivalent uniformly distributed loads acting on an inclined line can be expressed as ∞ ∑

𝑛=−∞

𝑓0 (𝑘, 𝑧, 𝜔)𝑒i(𝑘−𝑘 )𝑛𝐿

−(𝜆∗ + 2𝐺∗ )𝑘2 𝑢(𝑘, 𝑧, 𝜔) + 𝐺∗ 𝑢,𝑧𝑧 (𝑘, 𝑧, 𝜔)

(10)

1 𝑓 (𝑘, 𝜔)𝑒−i𝑘𝑥+i𝜔𝑡 𝑑 𝑘𝑑 𝜔 (2𝜋)2 ∫−∞ ∫−∞

𝑝𝑒 (𝑘, 𝑧, 𝜔) =

∞ ∑

From Eq. (16), the final Green’s functions of equivalent uniformly distributed loads acting on an inclined line can be obtained once f0 (k, z, 𝜔) is determined. And in this paper, the method proposed by Wolf [1] is used to solve f0 (k, z, 𝜔). The calculation procedures of f0 (k, z, 𝜔) are illustrated as follows. The dynamic-equilibrium equations for harmonic motion are denoted as

In order to obtain Green’s functions of equivalent uniformly distributed loads acting on an inclined line, the Fourier transform and inverse Fourier transform are introduced 𝑓 (𝑘, 𝜔) =

𝑓𝑛 (𝑘, 𝑧, 𝜔) =

Not that the calculations above are carried out in the frequency and the wave number domains. The dynamic response in the frequency and space domains can be obtained by using the inverse Fourier transform

(7)

where nL/ca is the time shift between the 0th and nth loads. ca =𝑐p𝑅 ∕cos 𝜓 (or ca =𝑐s𝑅 ∕cos 𝜓 ) is the apparent velocities of the P-waves (or SV-waves). 𝑐p𝑅 and 𝑐s𝑅 are shear wave velocities of P- and SV-waves in underlying half-space, respectively. Substituting Eq. (7) into Eqs. (3) and (4), the equivalent uniformly distributed loads acting on an inclined line can be expressed as 𝑝𝑒 (𝑥, 𝑧, 𝑡) =

∞ ∑

𝑓 (𝑘, 𝑧, 𝜔) =

𝑝0 (𝜔)𝑒i𝑘𝑧 cot 𝜃 𝑒i(𝑘−𝑘 )𝑛𝐿 = ′

𝑟0 (𝜔)𝑒i𝑘𝑧 cot 𝜃 𝑒i(𝑘−𝑘 )𝑛𝐿 = ′

∞ ∑ 𝑛=−∞ ∞ ∑ 𝑛=−∞

𝑝0 (𝑘, 𝑧, 𝜔)𝑒i(𝑘−𝑘 )𝑛𝐿

(12)

𝑟0 (𝑘, 𝑧, 𝜔)𝑒i(𝑘−𝑘 )𝑛𝐿

(13)

′

′

where 𝜆∗ and G∗ denotes Lame’s constants. A particular solution of Eqs. (17) and (18) can be written as 𝑢p (𝑘, 𝑧, 𝜔) =

′

𝐴1 exp(i𝑘𝑧 cot 𝜃) 2𝜋𝐴0

𝑤p (𝑘, 𝑧, 𝜔) =

where p0 (k, z, 𝜔) and r0 (k, z, 𝜔) are the amplitudes of the 0th distributed loads in the frequency and wave number domains. k is the wave number with respect to horizontal coordinate. k’ =𝜔 /ca . Let f0 (k, z, 𝜔) is the dynamic response induced by the 0th distributed loads in the frequency and wave number domains, from Eqs. (12) and (13), the dynamic response induced by the nth distributed loads can be expressed as 𝑓𝑛 (𝑘, 𝑧, 𝜔)=𝑓0 (𝑘, 𝑧, 𝜔)𝑒i(𝑘−𝑘 )𝑛𝐿

(18)

𝐴2 exp(i𝑘𝑧 cot 𝜃) 2𝜋𝐴0

where 𝐴1 = 𝑟0 cot 𝜃 + 𝑝0 𝑡2 − (𝑟0 + 𝑝0 cot 𝜃)𝐴3 cot 𝜃, 𝐴2 = 𝑝0 (1 − 𝐴3 ) cot 𝜃 + 𝑟0 (𝑠2 𝐴3 − cot 2 𝜃), 𝐴0 = 𝑘2 𝐺∗ [(𝐴3 − 1)2 cot 2 𝜃 − (𝑠2 𝐴3 − cot 2 𝜃)(𝑡2 − 𝐴3 cot 2 𝜃)], 𝐴3 = 𝑐p∗2 ∕𝑐s∗2

(14) 323

(19)

(20)

B.A. Zhenning, X. Gao and V.W. Lee

Engineering Analysis with Boundary Elements 106 (2019) 320–333

{

Where the superscript “p” denotes the particular solution. p0 and r0 are the 0th uniformly distributed loads in vertical and horizontal direction, respectively. 𝑐p∗ and 𝑐s∗ are shear waves velocities of P- and SV-waves, respectively. 𝜌 is mass density. u and w are the displacement amplitudes in horizontal and vertical direction, respectively. Substituting z = 0 and z = d into Eqs. (19) and (20), the displacement amplitudes at the top and bottom surfaces of the loaded layer (Fig. 3) can be expressed as p

𝑢1 (𝑘, 𝜔)= p

𝑢2 (𝑘, 𝜔)=

𝐴1 𝐴 p 𝑤 (𝑘, 𝜔)= 2 2𝜋𝐴0 1 2𝜋𝐴0

(21a)

𝐴1 exp(i𝑘𝑑 cot 𝜃) p 𝐴 exp(i𝑘𝑑 cot 𝜃) 𝑤2 (𝑘, 𝜔) = 2 2𝜋𝐴0 2𝜋𝐴0

(21b)

p

p

𝜎𝑧 (𝑘, 𝑧, 𝜔) =

i𝑘𝐺 ∗ (𝐴 cot 𝜃 − 𝐴2 ) exp(i𝑘𝑧 cot 𝜃) 2𝜋𝐴0 1

(22)

] i𝑘𝐺 ∗ [ 𝐴 𝐴 cot 𝜃 − 𝐴1 (𝐴3 − 2) exp(i𝑘𝑧 cot 𝜃) 2𝜋𝐴0 3 2

(23)

{

p

p

(24)

[ ] p p 𝑅1 (𝑘, 0, 𝜔) = −𝜎𝑧 (𝑘, 0, 𝜔) = −i𝑘𝐺∗ 𝐴3 𝐴2 cot 𝜃 − 𝐴1 (𝐴3 − 2) ∕2𝜋𝐴0

(25)

p

p

𝑃2 (𝑘, 𝑑, 𝜔) = 𝜏𝑥𝑧 (𝑘, 𝑑, 𝜔) = i𝑘𝐺∗ (𝐴1 cot 𝜃 − 𝐴2 ) exp(i𝑘𝑑 cot 𝜃)∕2𝜋𝐴0

p

{

{

p

p

𝑅𝑖 (𝑘, 𝜔) = −𝑅𝑖 (𝑘, 𝜔) − 𝑅h𝑖 (𝑘, 𝜔), (𝑖 = 1, 2)

(32)

{ } ] 𝑝2 , (𝑥, 𝑧) ∈ ΩL 𝑟2

(33)

{ } [ ] 𝑝2 = 𝑔2L,𝑡 (𝑥, 𝑧, 𝜔) , (𝑥, 𝑧) ∈ ΩL 𝑟2

(34)

}

[ =

} 𝑡L𝑔,𝑥 (𝑥, 𝑧, 𝜔)

𝑟1

{ } ] 𝑝1

𝑔1H,𝑡 (𝑥, 𝑧, 𝜔)

𝑟1

𝑔2L,𝑢 (𝑥, 𝑧, 𝜔)

where u(x, z, 𝜔) and w(x, z, 𝜔) are the displacement amplitudes in the horizontal and vertical directions, respectively. 𝑡𝑥 (𝑥, 𝑧, 𝜔) and 𝑡𝑧 (𝑥, 𝑧, 𝜔) are the tractions in the horizontal and vertical directions, respectively. The subscript “g” denotes parameters corresponding to the scattering fields. [𝑔1H,𝑢 (𝑥, 𝑧, 𝜔)] and [𝑔1H,𝑡 (𝑥, 𝑧, 𝜔)] are Green’s functions (non-periodic) of displacement and traction, respectively. [𝑔2L,𝑢 (𝑥, 𝑧, 𝜔)] and [𝑔2L,𝑡 (𝑥, 𝑧, 𝜔)] are Green’s functions of equivalent uniformly distributed loads acting on an inclined line of displacement and traction, respectively. {p1 } and {r1 } are the uniformly distributed loads acting on the boundary S0 and S’ 0 in the region ΩH . {p2 } and {r2 } are the equivalent loads acting on the boundary S0 in the region ΩL . The superscript “H” and “L” denotes the region ΩH and region ΩL , respectively. 2.3. Boundary conditions As discussed above, the PIBEM only a single topography is needed to be discretized. Therefore, the “0th” topography is chosen as the study cell, and boundary conditions on interface S’ 0 and S0 can be expressed as

(26)

(27)

⎧H ⎫ ⎪𝑡g,𝑥 (𝑠)⎪ ′ ⎬d𝑠 = {0}, 𝑠 ∈ 𝑆0 H ∫𝑠0 +𝑠′ 0 ⎨ ⎪𝑡g,𝑧 (𝑠)⎪ ⎩ ⎭

(35)

} ⎧L {H } ⎛{ L ⎫⎞ 𝑡𝑔,𝑥 (𝑠) ⎜ 𝑡𝑔 ,𝑥 (𝑠) ⎪𝑡𝑓 ,𝑥 (𝑠)⎪⎟ +⎨ d𝑠, 𝑠 ∈ 𝑆0 ⎬⎟d𝑠 = ∫ L ∫𝑠0 ⎜⎜ 𝑡L (𝑠) 𝑡H 𝑠0 +𝑠′ 0 ⎪𝑡𝑓 ,𝑧 (𝑠)⎪⎟ 𝑔,𝑧 (𝑠) 𝑔 ,𝑧 ⎝ ⎩ ⎭⎠

(36)

({ ∫𝑠0

(28)

𝑢L𝑔 (𝑠)

}

{ +

𝑤L𝑔 (𝑠)

𝑢L𝑓 (𝑠)

})

𝑤L𝑓 (𝑠)

d𝑠 =

{ ∫𝑠0 +𝑠′ 0

𝑢H 𝑔 (𝑠)

}

𝑤H 𝑔 (𝑠)

d𝑠, 𝑠 ∈ 𝑆0

(37)

where the subscript “f ” denotes parameters corresponding to the free fields. Substituting (32)–(34) into (36) and (37) yields [ ]{𝑝 } H 1 𝑇p1 = {0} (38) 𝑟1

Therefore, the total external loads needed to be applied to the total system are equal to 𝑃𝑖 (𝑘, 𝜔) = −𝑃𝑖 (𝑘, 𝜔) − 𝑃𝑖h (𝑘, 𝜔), (𝑖 = 1, 2)

, (𝑥, 𝑧) ∈ ΩH

=

𝑔1H,𝑢 (𝑥, 𝑧, 𝜔)

{ } 𝑝1

(31)

[

𝑡L𝑔,𝑧 (𝑥, 𝑧, 𝜔)

In order to fix the top and bottom surfaces, the homogeneous solutions (identified with superscript “h”), which corresponding to the negp p p p ative values of 𝑢1 , 𝑤1 , 𝑢2 and 𝑤2 have to be superimposed on particular solutions. The external loads result from ⎧ 𝑃 h (𝑘, 𝜔) ⎫ ⎧ −𝑢p (𝑘, 𝜔) ⎫ ⎪ 1h ⎪ ⎪ 1p ⎪ ⎪i𝑅1 (𝑘, 𝜔)⎪ ⎪−i𝑤1 (𝑘, 𝜔)⎪ 𝐿 ⎨ h ⎬ = 𝐒P−SV ⎨ p ⎬ ⎪ 𝑃2 (𝑘, 𝜔) ⎪ ⎪ −𝑢2 (𝑘, 𝜔) ⎪ ⎪i𝑅h (𝑘, 𝜔)⎪ ⎪−i𝑤p (𝑘, 𝜔)⎪ ⎩ 2 ⎭ ⎩ ⎭ 2

𝑢L𝑔 (𝑥, 𝑧, 𝜔)

𝑤L𝑔 (𝑥, 𝑧, 𝜔)

p

𝑅2 (𝑘, 𝑑, 𝜔) = 𝜎𝑧 (𝑘, 𝑑, 𝜔) [ ] = i𝑘𝐺∗ 𝐴3 𝐴2 cot 𝜃 − 𝐴1 (𝐴3 − 2) exp(i𝑘𝑑 cot 𝜃)∕2𝜋𝐴0

} 𝑡H 𝑔,𝑥 (𝑥, 𝑧, 𝜔)

]

, (𝑥, 𝑧) ∈ ΩH

=

𝑡H 𝑔,𝑧 (𝑥, 𝑧, 𝜔)

The amplitudes of external load (reaction) formulated in the global coordinate system at the loaded layer’s top and bottom surfaces are expressed as 𝑃1 (𝑘, 0, 𝜔) = −𝜏𝑥𝑧 (𝑘, 0, 𝜔) = i𝑘𝐺∗ (𝐴2 − 𝐴1 cot 𝜃)∕2𝜋𝐴0

[

𝑤H 𝑔 (𝑥, 𝑧, 𝜔)

where the subscript “1” and “2” denote displacement amplitudes at the loaded layer’s top and bottom surfaces, respectively. Using 𝜎 z (k, 𝜔) = 𝜆∗ (u,x + w,z ) + 2G∗ w,z and 𝜏 xz (k, 𝜔) = G∗ (u,z + w,x ), the particular solutions of the stress can be expressed as 𝜏𝑥𝑧 (𝑘, 𝑧, 𝜔) =

}

𝑢H 𝑔 (𝑥, 𝑧, 𝜔)

(29) (30)

Substituting Eqs. (29) and (30) into Eq. (1) and solving Eq. (1), the dynamic response induced by the external loads can be obtained. Finally, f0 (k, z, 𝜔) is obtained by adding the responses restricted in the fixed layer to those of external loads being applied to the layered halfspace. The scattered fields in region ΩH are simulated by Green’s functions (non-periodic). As for the region ΩL , the scattered wave fields are represented by Green’s functions of the uniformly distributed inclined loads. Once the these Green’s functions are obtained, the displacement and traction amplitudes corresponding to the scattered wave fields can be formulated as

[ ]{𝑝 } { } [ ]{𝑝 } H 2 1 + 𝑇𝑓 = 𝑇p2 𝑇pL 𝑟2 𝑟1

(39)

[ ]{𝑝 } { } [ ]{𝑝 } H 2 1 + 𝑉𝑓 = 𝑉p2 𝑉pL 𝑟2 𝑟1

(40)

where [ ] H 𝑇p1 = [ ] 𝑇pL = [ ] H 𝑇p2 =

324

[ ∫𝑠0 +𝑠′ 0 ∫𝑠0

] 𝑔1H,𝑡 (𝑠) d𝑠, 𝑠 ∈ 𝑆0′

[ ] 𝑔2L,𝑡 (𝑠) d𝑠, 𝑠 ∈ 𝑆0 [

∫𝑠0 +𝑠′ 0

] 𝑔1H,𝑡 (𝑠) d𝑠, 𝑠 ∈ 𝑆0

(41)

(42a)

(42b)

B.A. Zhenning, X. Gao and V.W. Lee

Engineering Analysis with Boundary Elements 106 (2019) 320–333

Fig. 4. Comparison with the results for a single hill given by Rubio et al. [38].

[ [

] 𝑉pL = ] H 𝑉p2 =

∫𝑠0

[ ] 𝑔2L,𝑢 (𝑠) d𝑠, 𝑠 ∈ 𝑆0 [

∫𝑠0 +𝑠′ 0

] 𝑔1H,𝑢 (𝑠) d𝑠, 𝑠 ∈ 𝑆0

As can be seen form Fig. 5, the increase of shear wave velocity ratio (CH /CL ) leads to a significant decrease of the surface displacement amplitudes in the region (−1.0 < x/a < 1.0), because the shielding effect of harder soil is stronger than that of the softer soil. In addition, because of the dynamic interaction, the vibration becomes stronger with increase of shear wave velocity ratio.

(43)

(44)

Solving, (39) and (40), {p1 }, {p2 }, {r1 } and {r2 } can be obtained. And combining the solutions with Eqs. (31) and (33), the final total surface displacements in the region ΩH and region ΩL can be expressed as { } [ ]{𝑝 } 𝑢(𝑥, 𝑧, 𝜔) 1 = gH (𝑥, 𝑧, 𝜔) , (𝑥, 𝑧) ∈ ΩH (45) 1 , 𝑢 𝑤(𝑥, 𝑧, 𝜔) 𝑟1 } { } { L [ ]{𝑝 } 𝑢𝑓 (𝑥, 𝑧, 𝜔) 𝑢(𝑥, 𝑧, 𝜔) L 2 g = + ( 𝑥, 𝑧, 𝜔 ) , (𝑥, 𝑧) ∈ ΩL (46) 2,𝑢 𝑤(𝑥, 𝑧, 𝜔) 𝑤L𝑓 (𝑥, 𝑧, 𝜔) 𝑟2

4.2. Comparisons between multiple and periodic hill In order to study the difference of results of the periodic semi-circular hill and multiple semi-circular hills, the cases of multiple hills, one, three, five and seven hills are considered. The surface displacement amplitudes around the middle hill are illustrated. In calculation, the normalized frequency is 𝜂 = 𝜔a/𝜋cs = 1.0 and incident angles are 𝜂 = 5° and 90°, Poisson’s ratio is 1/3, damping ratio is 𝜁 = 0.001 and period of the hills is L/a = 8.0. |u/Ap | (|u/ASV |) is the normalized displacement amplitude in the horizontal direction and |w/Ap | (|w/ASV |) is the normalized displacement amplitude in the vertical direction, where Ap and ASV are the amplitude of the incident P- and SV-waves, respectively. First, Fig. 6 clearly shows that the dynamic responses of periodic hill are significantly different from those of multiple hills. The differences between the model of multiple hills and periodic hill become smaller with the increase of hill numbers. However, even for the cases of seven hills, the displacement amplitudes differences between the multiple hills and periodic hill are more than 24% (more detailed numerical results are shown in Tables 2–5). Thus, it is very difficult to obtain the accurate results by traditional way of choosing a finite number of hills to solve the problem of wave scattering by periodic hill. This indicates that the precision of method proposed in this paper is higher than the traditional way. Second, for the grazing incidence (𝜓 = 5°) of SV-waves, due to the shielding effect, the increase of hill numbers leads to a significant decrease of surface amplitudes. For vertical (𝜓 = 90°) incidence of SVwaves, the displacement amplitudes increase with the increase of hill numbers, because of the dynamic interaction among hills. However, for the case of grazing incidence (𝜓 = 5°) of P-waves, the increase of the hill numbers leads to a significant increase of surface displacement amplitudes.

3. Verification of accuracy The degenerate results of this method are compared with results of a single semi-elliptical hill (the height of the semi-elliptical hill is 2a) in a homogenous half-space [38]. Based on numerical experiments, a period L/a = 500.0 enables the periodic hill to be degenerated to a single hill at normalized frequency 𝜂 = 𝜔a/𝜋c = 1.5, 𝜔 is circular frequency, a is the half-width of hill, c is shear wave velocity of half-space. In calculation, the excitation is P-waves with the incident angle 𝜓 p = 50°, Poisson’s ratio is 1/4 and the damping ratio is 𝜁 =0.01. |u/Ap | is the normalized displacement amplitude in the horizontal direction and |w/Ap | is the normalized displacement amplitude in the vertical direction. From the Fig. 4, one can find that our results are very close to those given by Rubio et al. [38]. 4. Numerical results and discussions In this section, three models are considered: the multiple semicircular hills, the periodic hill and the periodic semi-circular canyon in a homogenous half-space. Parametric studies are conducted in detail in order to investigate the influences of frequency, incident angle, material and height of hill on the surface displacements. 4.1. Surface displacement amplitudes of multiple hills versus different hill materials

4.3. Comparisons between periodic hill and canyon To study the effect of hill materials on displacement amplitudes, the three hills in a homogenous half-space are used as an example. The ratios of shear wave velocity of hills and half-space are CH /CL = 0.5, 1.0 and 2.0. The result of homogenous three hills (CH /CL = 1.0) is given by Ba et al. [48]. In calculation, the excitation is P-waves at incident frequency 𝜂 = 1.0 and incident angles 𝜓 = 5° and 90°. The distances of the hills are L/a = 6.0, 8.0 and 10.0. Fig. 5 shows the displacement amplitudes around the middle hill.

The study is conducted on the results of periodic semi-circular hill and canyon in a homogenous half-space for incident P- and SV-waves at normalized frequency 𝜂 = 0.5. The incident angles are 𝜓 = 5° and 90°. Tables 6–9 show the surface displacement amplitudes of periodic hill (canyon) around the 0th hill (canyon) for L/a = 4.0. As can be seen from Tables 6–9, there are significant differences in the surface displacement amplitudes between periodic hill and periodic 325

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Engineering Analysis with Boundary Elements 106 (2019) 320–333

Table 2 Numerical results of precision comparison between multiple and periodic hills for incident P-waves (u). 𝜓

x/a

−4.0

Three hills

5° 90° 5° 90° 5° 90° 5° 90° 5° 90° 5° 90° 5° 90° 5° 90°

−3.0 −2.0 −0.5 0.5 2.0 3.0 4.0

Five hills

Seven hills

Perio-hills

u

Ratio (%)

u

Ratio (%)

u

Ratio (%)

u

0.35297 0.30389 0.46798 1.59083 0.55139 0.65537 0.24433 0.45029 0.18875 0.45029 0.40406 0.65537 0.52313 1.59083 0.40874 0.30389

18.250 504.876 11.773 29.823 12.129 6.234 9.821 27.310 13.078 27.310 9.900 6.234 1.027 29.823 5.338 504.876

0.34831 0.13865 0.54654 1.86605 0.56142 0.72099 0.16552 0.6095 0.21239 0.60950 0.42057 0.72099 0.50161 1.86605 0.35323 0.13865

19.329 175.975 3.037 17.683 10.530 16.871 25.602 1.609 2.192 1.609 6.219 16.871 5.098 17.683 18.190 175.975

0.38085 0.04756 0.55883 1.85655 0.54717 0.64764 0.21335 0.77175 0.21357 0.77175 0.40597 0.64764 0.47309 1.85655 0.39037 0.04756

11.793 5.334 5.354 18.102 12.801 4.981 4.103 24.582 1.648 24.582 9.474 4.981 10.494 18.102 9.588 5.334

0.43177 0.05024 0.53043 2.26691 0.62750 0.61691 0.22248 0.61947 0.21715 0.61947 0.44846 0.61691 0.52856 2.26691 0.43177 0.05024

Note: Ratio = ∣Dis (multiples hills) − Dis (periodic hill)∣/Dis (periodic hill)×100%.

Table 3 Numerical results of precision comparison between multiple and periodic hills for incident P-waves (w). x/a

−4.0 −3.0 −2.0 −0.5 0.5 2.0 3.0 4.0

𝜓

5° 90° 5° 90° 5° 90° 5° 90° 5° 90° 5° 90° 5° 90° 5° 90°

Three hills

Five hills

Seven hills

Perio-hills

w

Ratio (%)

w

Ratio (%)

w

Ratio (%)

w

0.31646 2.92809 0.15559 1.30800 0.24409 2.48318 0.32736 1.20307 0.57476 1.20307 0.17231 2.48318 0.31225 1.30800 0.35404 2.92809

1.090 6.713 34.549 14.370 8.257 19.913 37.991 28.996 21.942 28.996 40.719 19.913 16.192 14.370 10.654 6.7139

0.28672 2.87736 0.17442 1.57422 0.26855 2.32717 0.40699 1.29622 0.62593 1.29622 0.26267 2.32717 0.36241 1.57422 0.34213 2.87736

10.386 8.330 26.627 3.057 0.935 12.379 22.908 23.498 14.993 23.498 9.632 12.379 2.729 3.057 6.932 8.330

0.30349 2.85238 0.20432 1.64800 0.26275 2.00743 0.42290 1.46909 0.64063 1.46909 0.27443 2.00743 0.32896 1.64800 0.30906 2.85238

5.144 9.126 14.050 7.888 1.244 3.060 19.894 13.295 12.996 13.295 5.587 3.060 11.707 7.888 3.403 9.126

0.31995 3.13883 0.23772 1.52751 0.26606 2.07081 0.52793 1.69437 0.73633 1.69437 0.29067 2.07081 0.37258 1.52751 0.31995 3.13883

Note: Ratio = ∣Dis (multiples hills) − Dis (periodic hill)∣/Dis (periodic hill)×100.

Table 4 Numerical results of precision comparison between multiple and periodic hills for incident SV-waves (u). x/a

−4.0 −3.0 −2.0 −0.5 0.5 2.0 3.0 4.0

𝜓

5° 90° 5° 90° 5° 90° 5° 90° 5° 90° 5° 90° 5° 90° 5° 90°

Three hills

Five hills

Seven hills

Perio-hills

u

Ratio (%)

u

Ratio (%)

u

Ratio (%)

u

0.28345 1.67620 0.23103 1.65223 0.17559 3.08083 0.07099 1.65978 0.50588 1.65978 0.15149 3.08083 0.26812 1.65223 0.29676 1.67620

10.204 23.148 1.706 9.728 4.968 4.674 54.696 5.121 0.335 5.121 10.741 4.674 4.013 9.728 5.987 23.148

0.39082 1.52098 0.29500 1.76009 0.20663 3.11243 0.02269 1.66226 0.59637 1.66226 0.17749 3.11243 0.32294 1.76009 0.37465 1.52098

23.810 11.744 25.511 3.834 11.831 3.696 50.555 5.278 18.283 5.278 4.578 3.696 15.612 3.834 18.688 11.744

0.38606 1.33631 0.26372 1.88127 0.15911 3.06299 0.05503 1.52662 0.54863 1.52662 0.13961 3.06299 0.29551 1.88127 0.37205 1.33631

22.302 1.823 12.202 2.786 13.888 5.226 19.917 3.312 8.814 3.312 17.741 5.226 5.792 2.786 17.864 1.823

0.31566 1.36113 0.23504 1.83027 0.18477 3.23188 0.04589 1.57892 0.50419 1.57892 0.16972 3.23188 0.27933 1.83027 0.31566 1.36113

Note: Ratio = ∣Dis (multiples hills) − Dis (periodic hill)∣/Dis (periodic hill)×100%.

326

B.A. Zhenning, X. Gao and V.W. Lee

Engineering Analysis with Boundary Elements 106 (2019) 320–333

Fig. 5. (a) Horizontal displacement amplitudes of multiple hills with different hill materials, (b) vertical displacement amplitudes of multiple hills with different hill materials. Table 5 Numerical results of precision comparison between multiple and periodic hills for incident SV-waves (w). x/a

−4.0 −3.0 −2.0 −0.5 0.5 2.0 3.0 4.0

𝜓

5° 90° 5° 90° 5° 90° 5° 90° 5° 90° 5° 90° 5° 90° 5° 90°

Three hills

Five hills

Seven hills

Perio-hills

w

Ratio (%)

w

Ratio (%)

w

Ratio (%)

w

0.19640 0.19834 0.24460 0.38738 0.35100 0.27466 0.26866 0.83645 0.35869 0.83645 0.28106 0.27466 0.35269 0.38738 0.38604 0.19834

28.413 355.221 23.065 24.565 15.101 23.585 18.201 31.129 24.716 31.129 61.668 23.585 43.773 24.565 40.711 355.221

0.27044 0.11828 0.32424 0.48383 0.43742 0.37540 0.31167 0.87693 0.44549 0.87693 0.22544 0.37540 0.29530 0.48383 0.32227 0.11828

1.425 7.471 1.985 5.783 5.803 4.443 5.106 27.796 6.498 27.796 29.675 4.443 20.378 5.783 17.467 7.471

0.30132 0.04768 0.33135 0.42391 0.42545 0.33540 0.26736 1.01545 0.40631 1.01545 0.16367 0.33540 0.22882 0.42391 0.24024 0.04768

9.831 9.433 4.221 17.452 2.907 6.686 18.597 16.391 14.721 16.391 5.856 6.686 6.722 17.452 12.433 9.433

0.27435 0.04357 0.31793 0.51353 0.41343 0.35943 0.32844 1.21452 0.47645 1.21452 0.17385 0.35943 0.24531 0.51353 0.27435 0.04357

Note: Ratio = ∣Dis (multiples hills) − Dis (periodic hill)∣/Dis (periodic hill)×100%.

canyon. The displacement amplitudes of periodic canyon in the region (1.0 < x/a < 4.0) are smaller than those of periodic hill for the grazing incidence (𝜓 = 5°) of P-waves. This phenomenon indicates that the shielding effect of periodic canyon is stronger than that of the periodic hill for P-waves. For the vertical (𝜓 = 90°) incidence of P-waves, the displacement amplitudes of the periodic canyon in the region (−1.0 < x/a < 1.0)

are larger than those of the periodic hill, which indicates that the dynamic interaction between periodic canyon is more obvious. However, for the grazing incidence (𝜓 = 5°) of SV-waves, the displacement amplitudes of the periodic hill in the region (1.0 < x/a < 4.0) are smaller than those of the periodic canyon. The results show that periodic hill has a stronger shielding effect on SV-waves. Note that the displacement 327

B.A. Zhenning, X. Gao and V.W. Lee

Engineering Analysis with Boundary Elements 106 (2019) 320–333

Table 6 Displacement amplitudes of periodic hill and periodic canyon for incident P-waves (𝜓 = 5º). x/a

−4.0 −3.5 −3.0 −2.5 −2.0 −1.5 −0.5 0.0 0.5 1.5 2.0 2.5 3.0 3.5 4.0

Single canyon

Single hill

Perio-canyons

Perio-hills

u

w

u

w

u

w

u

w

0.42370 0.28235 0.34368 0.50500 0.68605 0.85902 0.60560 0.36340 0.17868 0.21786 0.28662 0.36997 0.44195 0.49363 0.52156

0.46565 0.51673 0.42953 0.23949 0.10685 0.24227 0.19716 0.23692 0.14679 0.44070 0.49157 0.50901 0.49570 0.45513 0.39118

0.65313 0.72651 0.65759 0.53991 0.47813 0.42352 0.37015 0.35210 0.37196 0.37930 0.40064 0.41815 0.42799 0.42921 0.42258

0.40723 0.16166 0.17413 0.44009 0.55935 0.46695 0.07500 0.37410 0.59691 0.41305 0.37639 0.34175 0.30458 0.26652 0.23243

0.32622 0.24199 0.16488 0.16112 0.15851 0.17894 0.27333 0.32622 0.24199 0.16115 0.15852 0.17891 0.19738 0.27333 0.32622

0.12357 0.18228 0.33424 0.26617 0.16968 0.13580 0.17306 0.12357 0.18228 0.26611 0.16960 0.13592 0.19601 0.17306 0.12357

0.23247 0.22188 0.21661 0.34677 0.39606 0.36361 0.24176 0.23247 0.22188 0.34674 0.39602 0.36365 0.17288 0.24176 0.23247

0.25853 0.25497 0.22583 0.24104 0.26444 0.20211 0.23446 0.25853 0.25497 0.24102 0.26447 0.20217 0.18943 0.23966 0.25853

Table 7 Displacement amplitudes of periodic hill and periodic canyon for incident P-waves (𝜓 = 90º). x/a

±4.0 ±3.5 ±3.0 ±2.5 ±2.0 ±1.5 ±0.5 0.0

Single canyon

Single hill

Perio-canyons

Perio-hills

u

w

u

w

u

w

u

w

0.78023 0.76922 0.73342 0.67565 0.59949 0.50941 0.27406 0.00000

1.76869 1.27763 1.43956 2.07932 2.64022 2.87651 1.53594 1.37025

0.79913 0.79411 0.76019 0.69803 0.60905 0.49610 0.16000 0.00000

1.75883 2.24392 2.60371 2.63941 2.27539 1.55380 2.28816 2.68052

0.00000 2.09167 1.90259 1.45827 0.02449 1.46248 2.09167 0.00000

5.62147 4.01960 1.82498 1.22962 0.75354 1.22587 4.01960 5.62147

0.00000 0.14776 1.32963 1.27832 0.00000 1.28000 0.14776 0.00000

1.47531 1.25995 1.22000 3.35907 4.24332 3.36101 1.25995 1.47531

Table 8 Displacement amplitudes of periodic hill and periodic canyon for incident SV-waves (𝜓 = 5º). x/a

−4.0 −3.5 −3.0 −2.5 −2.0 −1.5 −0.5 0.0 0.5 1.5 2.0 2.5 3.0 3.5 4.0

Single canyon

Single hill

Perio-canyons

Perio-hills

u

w

u

w

u

w

u

w

0.20220 0.24028 0.16302 0.06950 0.22417 0.33921 0.08216 0.07808 0.02163 0.16875 0.18149 0.18995 0.19237 0.18924 0.18117

0.32367 0.22919 0.29033 0.34932 0.28777 0.26659 0.34547 0.23224 0.12129 0.18539 0.17162 0.16523 0.16302 0.16449 0.16963

0.18297 0.12600 0.15344 0.15632 0.17939 0.25168 0.29809 0.36941 0.34697 0.04770 0.05320 0.06580 0.08020 0.09440 0.10745

0.39908 0.35428 0.19679 0.28211 0.40285 0.31115 0.26719 0.21900 0.23433 0.17064 0.16766 0.16888 0.17170 0.17528 0.17934

0.03507 0.17114 0.19611 0.15674 0.16513 0.34443 0.17554 0.03507 0.17114 0.15644 0.16510 0.34475 0.41195 0.17554 0.03507

0.61315 0.38592 0.09863 0.23950 0.36863 0.33451 0.43078 0.61315 0.38592 0.23928 0.36864 0.33453 0.29889 0.43078 0.61315

0.27808 0.25548 0.08280 0.12708 0.08500 0.08660 0.23157 0.27808 0.25548 0.12703 0.08490 0.08640 0.08550 0.23157 0.27808

0.26874 0.30398 0.18011 0.20749 0.34417 0.29943 0.18480 0.26874 0.30398 0.20747 0.34403 0.29943 0.05800 0.19544 0.26874

Table 9 Displacement amplitudes of periodic hill and periodic canyon for incident SV-waves (𝜓 = 90º). x/a

±4.0 ±3.5 ±3.0 ±2.5 ±2.0 ±1.5 ±0.5 0.0

Single canyon

Single hill

Perio-canyons

Perio-hills

u

w

u

w

u

w

u

w

2.49303 2.21554 1.89353 1.80371 2.06283 2.46842 1.0513 1.4919

1.02673 1.03293 1.00511 0.93705 0.82610 0.72439 0.86750 0.00000

2.13130 1.71584 1.54771 1.65630 1.78570 1.69468 1.94000 2.24000

0.80545 0.85203 0.88430 0.90633 0.93065 0.99438 1.24516 0.00000

0.87821 1.25423 2.07885 1.73866 1.51957 1.74036 1.25423 0.87821

0.00000 1.10319 1.93759 1.37092 0.01899 1.37364 1.10319 0.00000

0.94700 1.33792 1.46849 1.84334 2.11000 1.84000 1.33792 0.94700

0.00000 1.26404 1.64000 1.34063 0.00000 1.34000 1.26404 0.00000

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Engineering Analysis with Boundary Elements 106 (2019) 320–333

Fig. 6. (a) Displacement amplitudes of periodic hill and multiple hills for incident P-waves, (b) displacement amplitudes of periodic hill and multiple hills for incident SV-waves.

amplitudes of periodic topography are repeated in the region (−4.0≤ x ≤ 0.0 and 0.0≤ x ≤ 4.0) for the period L/a = 4.0, due to the periodic characteristics. Besides, due to the dynamic interaction between adjacent topographies, the surface displacement amplitudes corresponding to the periodic topography are different from those of the single topography. For the grazing incidence (𝜓 = 5°) of P-waves, the displacement amplitudes of the periodic topography are significantly smaller than those of the

single topography, because the shielding effect of periodic topography is stronger than that of the single topography. So the periodic topography can be used to shield elastic waves. In construction, in order to level the ground surface, large amounts of soil are excavated. This soil often needs to be transported away from the construction site, which is costly for the construction project. If this soil is used to construct a periodic hill and valley landscape, which is around the building. It can not only save the cost of transporting soil, but also shield elastic waves [47].

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Engineering Analysis with Boundary Elements 106 (2019) 320–333

Fig. 7. (a) Horizontal displacement amplitudes of periodic hill for incident P-waves under different heights of hill (h/a = 1.0, 1.5 and 2.0), (b) vertical displacement amplitudes of periodic hill for incident P-waves under different heights of hill (h/a = 1.0, 1.5 and 2.0).

For the vertical (𝜓 = 90°) incidence, the displacement amplitudes of periodic topography are larger than those of the single topography, due to dynamic interaction between topographies. It can be found that the periodic topography has significant amplification effects on the seismic ground motion. And some topographies in nature are indeed periodically distributed, such as periodic bedrock ridges, hilly landform and so on. So the exact solution obtained in this paper can be used to determine the seismic ground motion of these periodic topographies, so as to achieve the purpose of earthquake prevention and disaster reduction.

Besides, for the grazing incidence (𝜓 = 5°) of SV-waves, the displacement amplitudes of periodic topography are slightly smaller than those of the single topography. 4.4. Surface displacement amplitudes versus different heights of hill To study the effect of hill height on the displacement amplitudes, the periodic hill of three heights (h/a = 1.0, 1.5 and 2.0) in a homogenous half-space is used as an example. In calculation, the 330

B.A. Zhenning, X. Gao and V.W. Lee

Engineering Analysis with Boundary Elements 106 (2019) 320–333

Fig. 8. (a) Horizontal displacement amplitudes of periodic hill for incident P-waves under different materials of hill (CH /CL = 0.5, 1.0 and 2.0), (b) vertical displacement amplitudes of periodic hill for incident P-waves under different materials of hill (CH /CL = 0.5, 1.0 and 2.0).

excitation is P-waves at normalized frequencies of 𝜂=0.5, 1.0 and 1.5 and incident angles of 𝜓 = 5°, 30° and 90°. The period of the hills is L/a = 8.0. Fig. 7 shows the displacement amplitudes around the 0th hill. In Fig 7, h is the height of hill and a is the half-width of hill. Fig. 7 clearly shows that height of hill has great influence on surface displacement amplitudes, which is dependent on incident angle and frequency. For the grazing incidence (𝜓 = 5°) of P-waves, the increase of the height leads to a significant increase of the sur-

face displacement amplitudes at low incident frequency (𝜂 = 0.5). The surface displacement amplitudes decrease with the increase of height at high incident frequency (𝜂 = 1.5). These indicate that the lower hill has a stronger shielding effect on P-waves at low incident frequency and the higher hill has a stronger shielding effect on Pwaves at high incident frequency. In addition, for oblique incidence (𝜓 = 30°) and vertical (𝜓 = 90°) incidence, due to the dynamic interaction, the displacement amplitudes become larger as the increase of height. 331

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Engineering Analysis with Boundary Elements 106 (2019) 320–333

4.5. Surface displacement amplitudes versus different materials of hill

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To study the effect of material of the hill on displacement amplitudes, the periodic semi-circular hill of three materials in a homogenous halfspace is used as an example. The ratios of shear wave velocity of hills and half-space are CH /CL = 0.5, 1.0 and 2.0. In calculation, the excitation is P-waves at incident frequencies 𝜂 = 0.5, 1.0 and 1.5 and incident angles 𝜓 = 5°, 30° and 90°. The period of the hills is L/a = 8.0. Fig. 8 shows the displacement amplitudes around the 0th hill. As can be seen form Fig. 8, material of hill has a significant influence on surface displacement amplitudes. For the grazing incidence (𝜓 = 5°) of P-waves, because the shielding effect of harder soil is stronger than that of the softer soil, the increase of shear wave velocity ratio (CH /CL ) leads to a significant decrease of the surface displacement amplitudes. Besides, the dynamic interaction between periodic hill becomes weaker with increase of shear wave velocity ratio. In addition, the vibration in the region (−1.0 < x/a < 1.0) becomes stronger with decrease of shear wave velocity ratio. 5. Conclusions In this paper, a periodic indirect boundary element method (PIBEM) using Green’s functions of equivalent uniformly distributed loads acting on an inclined line is proposed to study scattering of in-plane waves (Pand SV-waves) by periodic topography in a half-space. This method, which only requires a single topography to be discretized, has advantages of higher precision and lower memory requirement. By using this method, the complex effects of periodic canyon and hill on the seismic ground motion are studied and main conclusions are as follows. First, the displacement amplitudes of periodic hill are quite different from those of multiple hills. It is very difficult to obtain the accurate results by choosing a finite number of hills to solve the problem of scattering by periodic hill. Second, the dynamic responses of periodic hill are significantly different from those of periodic canyon. The periodic canyon has a stronger shielding effect on P-waves, while periodic hill has a stronger shielding effect on SV-waves. Third, for periodic hill, the lower hill has a stronger shielding effect on P-waves at low incident frequency (𝜂 = 0.5) and the higher hill has a stronger shielding effect on P-waves at high incident frequency (𝜂 = 1.5). Fourth, the dynamic interaction between periodic hill becomes weaker with increase of shear wave velocity ratio. The PIBEM proposed in this paper is also meaningful for periodic scattering in other fields. For example, scattering of light from the periodic structure [49,50], electromagnetic scattering by periodic structure [51], scattering of plane waves from a finite periodic structure of ferrite cylinders [52], and so on. Acknowledgment This study is supported by National Natural Science Foundation of China under grant No. 51778413 and 51578373, which is gratefully acknowledged. Supplementary materials Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.enganabound.2019.05.020. References [1] Wolf JP. Dynamic soil–structure interaction. Englewood Cliffs: Prentice-Hall; 1985. [2] Celebi M. Topographical and geological amplifications determined from strong-motion and aftershock records of the 3 March 1985 Chile earthquake. Bull Seismol Soc Am 1987;77(4):1147–67. [3] Hartzell SH, Carver DL, King KW. Initial investigation of site and topographic effects at Robinwood Ridge, California. Bull Seismol Soc Am 1994;84(5):1336–59.

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