#### Abstract

By using the harmonic response analysis in the finite element analysis method, the seismic dynamic interaction of the three-dimensional urban building group-homogeneous sedimentary basin is studied. The viscoelastic artificial boundary is introduced, which can overcome both the defects of low-frequency drift and high-frequency instability, and the equivalent load in frequency domain is obtained by fast Fourier transform for loading, to explore the influence of the different incident frequencies (0.5–5.0 Hz), different numbers (196, 400, 676), and spacing (55 m, 62.5 m, 70 m) of building groups on the ground motion of homogeneous sedimentary basin under the incidence of SV wave. Numerical results illustrate that at low frequency, the displacement cloud image of the homogeneous sedimentary basin model shows an obvious phenomenon of “central focusing.” With the increase of frequency, the displacement cloud image gradually changes from “central focusing” to “multipoint focusing.” Meanwhile, the displacement peak gradually moves from the surface to the center of the basin. At a certain incident frequency, the existence of dense building groups will change the spatial distribution of displacement amplitude in the basin. Under the action of high-frequency incident waves, denser building groups with more buildings and smaller building spacing have a more pronounced weakening effect on the seismic response of homogeneous sedimentary basins. The displacement response of the center of the basin is generally large. When planning important buildings, the center area should be avoided as much as possible. For existing buildings, structural reinforcement is needed. It is of great significance for the planning and layout of buildings in the soft sedimentary basin and the reasonable spacing of buildings to reduce the risk of urban earthquake disaster.

#### 1. Introduction

Site-City Interaction (SCI) under seismic wave incidence is an active research topic in many fields such as the geotechnical engineering, earthquake engineering, and geophysics. In recent years, with the acceleration of urbanization, buildings become more and dense, making the scale of “Site-Building Groups” larger and larger, which brings new challenges to the seismic safety assessment of the existing buildings on the site and the seismic fortification of the proposed project. In general, the analysis methods of site-city interaction can be divided into the analytical method and numerical method. Analytic solutions usually refer to the wave function expansion method [1–5], the complex variable method [6,7], etc. Numerical solutions mainly include finite element method [8–11], finite difference method [12–14], boundary element method [15–17], discrete wave number method [18], and hybrid method [19,20].

As a special site, the significant amplification effect of the sedimentary basin on ground motion has been confirmed by a large number of examples of earthquake damage [20–23]. In terms of theoretical research, Trifunac [24] and Lee [25], respectively, solved the two-dimensional scattering problem of incident plane SH waves in a semicylindrical alluvial valley and the analytical solution of incident plane waves in a three-dimensional hemispherical depression topography by using the wave function expansion method. Li et al. [26] used the finite difference method to study the site amplification effect of the octave earthquake anomaly area in Hanyuan County during the 2008 Wenchuan earthquake and concluded that the valley topography had a significant impact on the ground motion.

However, the presence of buildings/groups will have a significant impact on the site effect, and many experts and scholars have carried out a lot of research on this. Wirgin and Bard [27] studied the resonance law between the soil layer and a large number of buildings by establishing a two-dimensional “SCI” numerical model and explained some special seismic damage phenomena in the Mexico earthquake (1985). Clouteau and Aubry [28] analyzed the influence of the distribution of building groups on the ground motion characteristics and structural response based on the three-dimensional boundary element method. Tsogka and Wirgin [29] used the finite element method to study the “SCI” problem of soft soil-building groups (homogeneous block model) under SH wave excitation, and the results showed that the seismic response of building groups showed great heterogeneity due to multiple coherent effects of seismic waves. Kham et al. [30] used the two-dimensional boundary element method to analyze the influence of factors such as spatial distribution and natural frequency of urban buildings under the incidence of SH wave and found that the “SCI” effect was most obvious when the frequencies of buildings and foundations were the same. Taborda and Bielak [31] found that the existence of building groups significantly affected the seismic response in and around the city. Using the periodic green function, Volkov and Zheltukhin [32] proposed an efficient method to study the interaction between building groups and the overall resonance effect. He et al. [33] used the wave finite element method to establish a two-dimensional model of buildings and sedimentary valley. By comparing the seismic responses in frequency domain and time domain of individual sedimentary valley and building groups and sedimentary valley, they found that the existence of building groups has a significant impact on the seismic response of the valley.

It is worth noting that the above studies mainly focus on the single two-dimensional or three-dimensional site effect, as well as the dynamic interaction between the site and several buildings, and there is a lack of research on the dynamic interaction between the three-dimensional site-building groups. This paper uses the viscoelastic boundary wave finite element simulation method, build up to 676 buildings 3D building groups-homogeneous sedimentary basin model, and harmonic response analysis is introduced into solving the frequency domain solution, and explores the effects of the number and spacing of urban dense building groups on the ground motion of homogeneous sedimentary basins, so as to provide the theory basis for seismic fortification, etc.

#### 2. Calculation Model

As shown in Figure 1, a three-dimensional dense building groups-homogeneous sedimentary basin model. In which, 1(a) is the homogeneous sedimentary basin with no buildings and 1(b) is building groups-homogeneous sedimentary basin model. Among them, the 8-node solid element and 10-node high-order solid element are selected for the soil, and the 8-node solid element is selected for the foundation. The three-dimensional beam element is used for the simulation of the building part. The building dynamic model adopts the equivalent concentrated mass rod model and by controlling the single particle system model to be the same as the first-order natural vibration period of the building to achieve the purpose of equivalent simplification. The foundation is directly coupled with the soil contacting part. Due to the symmetry of the calculation model, a 1/4 model is taken in the frequency domain to improve the calculation efficiency.

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#### 3. Viscoelastic Boundary and Ground Motion Input Method

##### 3.1. Viscoelastic Boundary

In the simulation of soil boundary, in order to eliminate the reflection of waves on the truncated boundary and reduce the computational freedom, the viscoelastic boundary is used to treat the boundary of local foundation, as shown in Figure 2. Compared with viscous boundary [34] and transmission boundary [35], the viscoelastic boundary is an effective and suitable for solving complex seismic wave problems of artificial boundary [36]; it overcomes the low frequency drift problems caused by the viscous boundary, to simulate the artificial boundary outside the elastic recovery of semi-infinite medium performance, low-frequency and high-frequency stability is good, and easy to use.

In Figure 2, , are, respectively, the radial spring stiffness coefficient and tangential spring stiffness coefficient of viscoelastic boundary and , , respectively, are the radial spring damping coefficient and tangential spring damping coefficient of viscoelastic boundary. Among them,where is the shear modulus of foundation, is the distance between the scattered wave source and the artificial boundary, is the mass density of foundation, is the compression wave velocity of soil, and is the shear wave velocity of soil. In this article, and .

##### 3.2. Ground Motion Input Method

From the stress relationship at the viscoelastic boundary of the truncated soil mass, the equivalent nodal force formula on the viscoelastic artificial boundary can be obtained:where *K* is the stiffness coefficient matrix of the elastic element in the viscoelastic boundary and *C* is the damping coefficient matrix of the damper in the viscoelastic boundary. SV wave is incident from the top of bedrock, and its displacement and acceleration can be calculated according to one-dimensional wave theory, in which is the stress at the corresponding node on the viscoelastic boundary of the truncated soil in the free field andis the influence area of each node on the viscoelastic artificial boundary. The equivalent load expressions (3)–(5) in the time domain can be obtained from the calculation and derivation of geometric equation and constitutive equation:where the superscript of the equivalent load represents the normal direction of the truncated boundary plane where the node is located and the subscript represents the equivalent load direction, is the soil density, is the shear wave velocity, *H* is the distance from the bottom boundary to the surface, and *h*_{d} is the distance from the node to the boundary on the artificial boundary.

As the number of elements of the 3D model in this paper reaches 10^{6}, the harmonic response analysis solution module is used to conduct frequency domain analysis of the site. First, the equivalent load in the time domain is converted to the equivalent load in the frequency domain by fast Fourier transform, and then the load is loaded for solution. The equivalent load on each viscoelastic boundary in the frequency domain is as follows:where is the circular frequency of the incident wave, is the SV wave number, and is the depth of the node (the distance from the surface).

#### 4. Boundary Stability Verification and Accuracy Verification

##### 4.1. Boundary Stability Verification

In order to test the correctness and stability of viscoelastic boundary, the truncation range of soil was selected as 2*Ra*, 3*Ra*, and 4*Ra* for comparative analysis and the sedimentary basin radius . The higher dimensionless frequency are used in the free field verification to reflect the good applicability of viscoelastic boundary. The model parameters are shown in Table 1, and the simple harmonic SV wave is incident vertically from the bottom, and the incident amplitude is 1.

The *x*-direction displacement amplitude of the surface node on the *x* and *y* axes of the free field is shown in Figure 3. When the boundary dimensions are 2*Ra*, 3*Ra*, and 4*Ra*, the maximum displacement in the *x*-direction of the surface node on the *x*-axis is 2.08904, 2.016, and 2.05671, and the minimum value is 1.93878, 1.91517, and 1.99879, respectively. The maximum displacement in the *x*-direction of the surface node on the *y*-axis is 2.08373, 2.00932, and 2.02299, and the minimum value is 1.99496, 1.92641, and 1.96558, respectively.

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Figure 4 is the scatter diagram of the error analysis of the displacement amplitude and theoretical value of the ground node in the *x* direction on the *x* and *y* axes of the free field. Similar to the law of fixed boundary, the error between the simulation result and the theoretical value decreases with the increase of the model size. The three cases all meet the accuracy requirements within 5% of the engineering error. Considering the calculation accuracy and efficiency comprehensively, this paper selects 3*Ra* for the truncated soil area for the next numerical simulation analysis.

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##### 4.2. Accuracy Verification

In order to test the applicability of using viscoelastic boundary to solve the seismic response of sedimentary basins, the results of finite element simulation of viscoelastic boundary wave are compared with the theoretical results in [37]. Take the hemispherical sedimentary basin as the verification object; the model parameters are shown in Table 2; the incident wave is the SV wave with amplitude of 1. The results are shown in Figure 5.

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It can be seen that the method in this paper is suitable for solving the seismic response of sedimentary basins. At the same time, the results in the figure also show that the presence of sedimentary basins has a more obvious amplification effect on seismic waves, and the further the basin center, the more significant the amplification effect, and the amplification effect in the main displacement direction of the basin center reaches 6 times. The focusing effect of sedimentary basin on incident wave is very significant.

#### 5. Numerical Analysis

##### 5.1. Site and Building Models

The simple point dynamic system is used to replace the building to simplify the modeling and reduce the calculation. The period of the dynamic system is the same as that of the building so as to achieve the dynamic equivalence of the structure. The single building is 64 m in height, and *D* is the spacing between buildings. The length and width of the foundation are 30 m, the height is 7.5 m, and the total mass is 2.21 × 10^{7} kg. For the sedimentary basin, the radius *R* is 1000 m and the depth of the basin is 200 m. Other model parameters of the site and buildings are shown in Table 3. Take the dimensionless displacement and the incident wave amplitude is 1.0, and compare the magnification to discuss the amplification effect of the basin.

##### 5.2. The Influence of Building Quantity on the Amplification Effect of Ground Motion in Homogeneous Sedimentary Basin

By comparing the displacement amplitude of the homogeneous sedimentary basin with different number of buildings and no buildings and the position of the larger reaction area, the influence of the number of buildings on the seismic dynamic response of the homogeneous sedimentary basin is obtained.

Figures 6(a)–6(d)–11(a)–11(d) is the displacement reflection cloud image of the homogeneous sedimentary basin with the change of the number of buildings under the incidence of SV wave at different frequencies, and *f* is the frequency of the incident wave. Taking 6(a) as an example, the left side shows the *x*-direction displacement cloud map on the *x*-*y* plane of the homogeneous sedimentary basin, and the right side shows the *x*-direction displacement cloud map on the *x*-*z* plane and *y*-*z* plane of the homogeneous sedimentary basin from top to bottom.

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As shown in Figure 6, at low frequency (*f* = 0.5 Hz), the presence of the basin has a strong “focusing effect” on the site, and the site at the center of the basin is very obvious, which makes the buildings located in the center of the sedimentary basin susceptible to great influence. For *f* = 1 Hz, the buildings act as a “secondary source” and produces a coherent effect with the scattered waves in the sedimentary basin, resulting in spatial redistribution of surface ground motion. With the increase of the number of buildings, the spatial position of the peak displacement gradually shifted from the surface point on the *x*-axis to the surface point on the *y*-axis, and the peak displacement position within the basin gradually moved to the center of the basin. At the same time, the existence of buildings will increase the displacement response of sedimentary basin. If there are no buildings in the basin, the peak displacement on the surface and inside of the basin are 5.5, as illustrated in Figure 7(a), and the peak displacement on the surface and inside of the basin are 6.8 and 7.4 (see Figure 7(d)) when the number of buildings 26 × 26, which increases by 23.6% and 34.5%, respectively.

Different from the 1 Hz case, when the frequency is 2 Hz, the buildings can effectively weaken the seismic response of the basin, the peak of displacement amplitude of no building reaches 7.9 (see Figure 8(a)), while the number of buildings is 26 × 26 and the basin displacement peaks is 5.1 (see Figure 8(d)), reducing 35.4%. Moreover, the buildings still has an obvious “focusing effect” on the scattered waves, and the peak displacement position in the basin is closer to the center of the basin as the number of buildings increases. Due to the coherent effect of the buildings on the waves, the peak displacement position in the *x*-direction of the whole basin shifted from near the surface to near the bottom of the basin.

Figure 9 shows the displacement reflection of the homogeneous sedimentary basin at the frequency is 3 Hz. The presence of the buildings reduces the peak displacement in the basin space, the peak of displacement amplitude of no building is 14.9 (see Figure 9(a)), while the number of buildings is 14 × 14 and the basin displacement peaks are 8.8 (see Figure 9(b)), reduced by 40.9%. However, the peak displacement in the *x*-direction moves from the basin interior to the surface, resulting in a 57.4% increase in the *x*-direction displacement of the basin surface, as shown in Figures 9(a) and 9(d); the peak of displacement amplitude of no building is 6.8 and the number of buildings 26 × 26 is 10.7 because of the scattering effect of the buildings as a “secondary source” on the wave.

When the SV wave is incident at a frequency of 4 Hz, as illustrated in Figure 10, the “coherence effect” of buildings and homogeneous sedimentary basin is very obvious, and the existence of buildings will reduce the displacement response of homogeneous sedimentary basin and the displacement peak in the *x*-direction in the basin decreases with the increase of the number of buildings, such as the surface peak displacement of the no buildings’ basin is 9.1 and the number of buildings 14 × 14, 20 × 20, and 26 × 26 are 8.7, 8.8, and 7.8, respectively. Figure 11 shows the case of the frequency 5 Hz; the buildings have the most significant weakening effect on the seismic response within the basin; when there were no buildings and the number of buildings 20 × 20, the peak displacement of the homogeneous sedimentary basin was 18.1 and 6.2, respectively, with a decrease of 65.7%. From the perspective of the whole basin, the 26 × 26 buildings has the best effect on the reduction of displacement amplitude in most areas of the basin space.

In general, at low frequency (*f* = 0.5 Hz), the “coherence effect” of the buildings on the scattered waves is not obvious, and the seismic response of the homogeneous basin is hardly affected. At a certain frequency (*f* = 1 Hz), the buildings will not only change the position of the peak displacement of the basin but also increase the peak displacement of the basin. At high frequency (*f* ≥ 2 Hz), the existence of buildings weakens the seismic response of the homogeneous basin to different degrees, but the more the buildings, the better the weakening effect.

##### 5.3. The Influence of Building Spacing on the Amplification Effect of Ground Motion in Homogeneous Sedimentary Basin

In order to explore the influence of building spacing on the homogeneous sedimentary basin under the action of earthquake, the displacement amplitude of the homogeneous sedimentary basin under different building spacing and no building was analyzed, as well as the location of the larger reaction area. In order to reduce unnecessary calculations, only 20 × 20 buildings are selected, and the seismic displacement response of homogeneous sedimentary basin under different building spacing is shown in Figures 12(a)–12(d)–17(a)–17(d).

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When *f* = 0.5 Hz, as shown in Figure 12, in the case of different building spacing and no building, the homogeneous sedimentary basins all show obvious “focusing effect,” which can make the buildings in the middle of the basin destroyed seriously. And the peak displacement in the *x*-direction of both the surface and the interior of the homogeneous basin is 5.2, indicating that the building spacing has no obvious influence on the seismic response of the homogeneous sedimentary basin at low-frequency incidence.

Figure 13 shows the displacement response of the basin with the different building spacing and no building at 1 Hz. It can be seen that the ground motion response of the surface of the basin will generate redistribution, changing from the “central focusing” of *f* = 0.5 Hz to the “multipoint focusing” along the *y*-axis, and the sites near *y*/*r* = 2/5 and 4/5 of basin reflected very clearly and the buildings is easily affected. With the increase of building spacing, the displacement response of the basin gradually decreases. For example, in the case that the number of buildings is 20 × 20 and the building spacing *D* = 55 m, 62.5 m, and 70 m, the peak displacements of the basin are 5.8, 5.4, and 5.3, respectively. When *f* = 2 Hz, as shown in Figure 14, the buildings shows obvious characteristics of weakening the seismic response of the basin and with the increase of building spacing, the weakening effect of buildings becomes weaker, such as the surface peak displacement of the no buildings basin is 7.9, and the building spacing *D* = 55 m, 62.5 m and 70 m are 5.1, 6 and 6.3, decreased by 35.4%, 24.1% and 20.3% respectively.

When *f* = 3 Hz, the buildings also weakens the seismic response of the basin and the weakening effect is the best when the building spacing is 70 m, the peak displacement on the surface of the homogeneous basin in the *x*-direction of the no buildings is 6.8 (see Figure 15(a)), and that of *D* = 70 m is 5.5(see Figure 15(d)), decreased by 19.1%. However, when the building spacing is 55 m, the coherence effect of scattered waves in the basin is the strongest, and the peak displacements in the basin of the no buildings and the building when the spacing is 55 m are 6.8 and 9.4 (see Figures 15(a) and 15(b)), increased by 38.9%. At the same time, this also led to a more dramatic “focusing effect,” which further magnified the peak displacement of the basin surface.

Figures 16 and 17 show the seismic displacement response of homogeneous sedimentary basin under different building spacings at 4 Hz and 5 Hz, respectively. The buildings significantly weakens the peak displacement in the homogeneous basin, and the smaller the building spacing, the more significant the weakening effect. For example, the displacement peaks of homogeneous sedimentary basins are 18.1 (see Figure 17(a)), 6.5 (see Figure 17(b)), and 6.9 (see Figure 17(c)), when there are no buildings, the building spacing is 55 m and the building spacing is 62.5 m, and the seismic response of sedimentary basins is reduced by 64.1% and 61.9% due to the presence of buildings.

In summary, at low frequencies, the variation of building spacing has almost no effect on the peak displacement of homogeneous sedimentary basins. When the frequency is between 1 Hz and 3 Hz, the denser buildings will lead to the intensification of the “focusing effect” of the basin and the increase of the peak displacement in the basin. However, the existence of building groups can reduce the displacement amplitude of the whole basin.

#### 6. Conclusions

In this paper, the harmonic response analysis module in the finite element simulation software ANSYS is used to carry out frequency domain analysis on the three-dimensional building group-homogeneous sedimentary basin model. The following conclusions are obtained:(1)At low frequency (*f* = 0.5 Hz), the changes of the number of buildings and building spacing have little impact on the peak displacement in the *x*-direction of the homogeneous sedimentary basin, and the displacement amplitude of basin shows the obvious phenomenon of “central focusing.” So, it is necessary to avoid the central position as far as possible when building site selection in the basin.(2)At a certain low frequency (1.0 Hz in this paper), as the buildings acts as a “secondary source” and generates a dramatic “coherence effect” with the scattered waves, the displacement response of the sedimentary basin is redistributed. The peak displacement of the basin moves to the center, and the position of the peak displacement of the surface also shifts from the node on the *x*-axis to the node on the *y*-axis.(3)With the increase of frequency, the displacement amplitude of the homogeneous sedimentary basin changes from “central focusing” to “multipoint focusing,” and the displacement peak gradually moves from the surface to the center of the basin. In addition, the peak displacement in the *x*-direction of the homogeneous sedimentary basin basically increases with the increase of incident wave frequency.(4)The existence of buildings will weaken the seismic response of homogeneous sedimentary basin to different degrees. The larger the number of buildings, the smaller the space between buildings and the more significant the weakening effect of seismic response to homogeneous sedimentary basin.

#### Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Acknowledgments

This study was financially supported by the National Natural Science Youth Fund of China (Grant no. 51608521).