Summary In this paper, the complete formulation of the attenuated orthotropic time-domain half-space boundary element method was proposed to analyze the transient SH-wave scattering problems. By changing the spatial variable, the time-domain half-space Green's functions were obtained in an isotropic-like analytical process by solving the singular form of the scalar wave equation. The stress-free boundary condition of the ground surface was satisfied using the wave source image theory. To include the materials damping, the Barkan approach was used in the formulation to attenuate the half-space by applying a constant logarithmic reduction into the modified boundary integral equation. Finally, the closed-form attenuated orthotropic half-space scalar kernels were obtained in the time-domain for displacement/traction fields by analytical integration of Green's functions. The method was easily implemented in a time-domain computer code to analyze the seismic homogenous orthotropic medium. To elaborate the model of the heterogeneous problems, a sub-structuring approach was presented to satisfy the continuity conditions at the interface depending on the position of the node and normal. Then, several seismic problems including a surface canyon, an underground cavity, a subsurface inclusion, and an alluvial valley subjected to SH-wave was solved and compared with the literature to validate the surface response in the case of isotropic convergence. To evaluate the accuracy and the time of analysis for the proposed method versus the full-space boundary element models, a comparative test was carried out for two examples. A favorite agreement was testified between the responses to verify the proposed method's capability for simple modeling of the orthotropic topographic features. Finally, the ability of the method to analyze the problems with high degrees of freedom was investigated by solving a half-space model including 25 subsurface orthotropic circular inclusions.

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An advanced direct half-plane time-domain boundary element method (BEM) was applied to obtain the seismic response of a linear elastic irregular surface including periodically distributed semi-sine shaped valleys subjected to propagating obliquely incident plane SH waves. After developing the method for complex multiple surface topographies, some verification examples were solved and compared with those of the published works. Then, the transient response of a rough surface with 2–16 semi-sine shaped valleys was determined as synthetic seismograms. In this regard, the depth ratio of the valleys was sensitized. Finally, amplification patterns of the surface were presented in some cases. The results showed that the method was able to analyze the multipart models in the time-domain. Moreover, the response of the sinusoidal corrugated surfaces was very effective against seismic waves in forming different patterns. The method was recommended to researchers for transient analysis of complex engineering structures and composite materials in nanoscale.

In this paper, the geometrical effects of shallow twin lined tunnels with different cross sections are investigated to obtain the anti-plane seismic ground motion under vertical/horizontal incident plane SH waves. A model of long two-dimensional lined tunnels is established and embedded in a homogeneous linear elastic half-plane by an applied numerical time-domain boundary element approach. In addition to a brief introduction to the formulation of the method, by considering five tunnel sections including circular, elliptical, horseshoe, square and rectangular, the surface response is sensitized to observe the normalized displacement amplitude/amplification ratio. In this regard, the angle of the incident wave and the frequency of the response are also included in changing the response pattern. To illustrate the results in both time and frequency domains, they are presented as blanket charts, snapshots, and three-/two-dimensional diagrams. The results showed that the seismic response of the surface is extremely affected by the geometric parameters of underground tunnels, which can create different conditions on the ground surface with shifting the direction of the wavefront. Article Highlights Geometrical effect of twin horizontally overlapping lined tunnels. Applying a time-domain half-plane boundary element method. Illustrating the response in time and frequency domains. The effect of depth and distance ratios on the seismic ground motion. Propagating vertical and horizontal incident SH-wave type. Graphic Abstract

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