In order to simulate the propagation process of subway vibration of parallel tunnels in semi-infinite rocks or soils, time domain boundary element method (TD-BEM) formulation for analyzing the dynamic response of twin-parallel circular tunnels in an elastic semi-infinite medium is developed in this paper. The time domain boundary integral equations of displacement and stress for the elastodynamic problem are presented based on Betti’s reciprocal work theorem, ignoring contributions from initial conditions and body forces. In the process of establishing time domain boundary integral equations, some virtual boundaries are constructed between finite boundaries and the free boundary to form a boundary to refer to the time domain boundary integral equations for a single circular tunnel under dynamic loads. The numerical treatment and solving process of time domain boundary integral equations are given in detail, including temporal discretization, spatial discretization and the assembly of the influencing coefficients. In the process of the numerical implementation, infinite boundary elements are incorporated in time domain boundary element method formulation to satisfy stress free conditions on the ground surface, in addition, to reduce the discretization of the boundary of the ground surface. The applicability and efficiency of the presented time domain boundary element formulation are verified by a deliberately designed example.

Due to the demanding development and the limited space in urban areas, complex underground structures including the parallel subway tunnels, spaced at a limited distance, are being constructed. Although the subway brings convenience, a moving vehicle on uneven rails produces a dynamic force between wheels and rails and causes vibration, which has potential negative effects on sub-infrastructure beneath the rails, twin-parallel tunnels and even the structures on the ground [

Without loss of generality, taking the unballasted track in a subway project as an example, the whole propagation process of subway vibration could be roughly divided into two phases: First, the coupled vibration between rails and the wheels of the vehicle acts on the track slab, sequentially passes through the CA mortar layer, the railway ballast, the lining of the tunnel and the outer protective layer of the lining. And after that, the vibration propagates in the semi-infinite surrounding rocks or soils. Since the layout of the parallel subway tunnels is generally in the form of long cylinders in semi-infinite rocks or soils, the subway vibration analysis system could be illustrated by a plane model, as shown in

In order to fully understand the propagation law of subway vibration, great efforts have been devoted to developing numerical models for the dynamic track-tunnel-soil system, among which finite element method (FEM) is widely used. However, FEM requires an artificial boundary when dealing with infinite or semi-infinite problems, so it is not the ideal numerical tool. Since boundary element method (BEM) is good at simulating the radiational wave propagation in semi-infinite soils, the FEM–BEM coupled model would be most attractive [

In comparison with FEM, BEM has irreplaceable advantages in dealing with infinite or semi-infinite problems, automatically satisfying Sommerfeld’s radiation conditions at the infinity for infinite and semi-infinite domains, restricting the discretization only on the boundary of the computational domain for elastic analysis [

BEM for elastodynamic problems could be classified according to the nature of the adopted fundamental solutions [

TD-BEM formulation for elastodynamics was originally proposed for the first time in two journal papers by Mansur et al. [

The above mentioned studies are focused mainly on infinite domain problems, where only the discretization on the boundary is required and external boundaries are not required to be artificially truncated. However, many practical geotechnical engineering problems require models associated with the semi-infinite domain. The stress free conditions on the ground surface should be taken into consideration when dealing with semi-infinite problems by TD-BEM [

The idea of the infinite boundary element and the reciprocal decaying shape function were first proposed by Watson [

By removing the FEM sub-model in the ideal FEM–BEM coupled model in

This paper presents the semi-infinite TD-BEM formulation for analyzing the dynamic response of twin-paralleled circular tunnels under dynamic loads. Time domain boundary integral equations for this problem are presented, and then, the numerical implementation by introducing the infinite boundary element is presented. Finally, the presented TD-BEM formulation for twin-parallel tunnels in an elastic semi-infinite medium, in which the infinite boundary element and time domain fundamental solutions are incorporated, is verified by a deliberately designed example.

The boundaries for the semi-infinite domain problem include finite boundaries

where

It is noted that the boundaries AB, BA, CD, and DC are virtual boundaries, and their relations are shown in

Thus, the displacement boundary integral equation can be re-formulated, as:

where

The displacement and traction fundamental solutions in

The corresponding integrals of fundamental solutions are expressed by

In _{s}_{d}

where

The tensors _{ik}_{ik}_{ik}_{ik}_{ik}_{ik}

where

The parameters that are relevant to both space and time in the

where

It should be noted that the subscript

In

In the following context, all the finite parts of the singular integrals are analogously calculated.

The stress integral equation for internal points is obtained by combining Hooke’s law with the derivatives of

where

In references [

where

The tensors _{ijk}_{ijk}_{ijk}_{ijk}_{ijk}_{ijk}

By recalling the expressions,

The displacement is assumed to have the linear variation in time, whereas the traction is assumed to be constant. For the given time interval [_{m −1}, _{m}

where the time interpolation functions are defined, as:

From the expressions of fundamental solutions, ^{2}, respectively. When ^{2}, which could be mathematically expressed, as:

The displacement solution and traction solution can be obtained by the superpositions of displacement and traction fundamental solutions respectively, so decay rates of ^{2}, respectively, when the field point

Therefore, besides the satisfaction of the convergence criterion, the interpolation mode of the infinite boundary element needs to reflect the above mentioned decaying characteristics at the infinity. The requirements about the shape functions of infinite boundary elements are

Delta function property,

Completeness,

When the field point

There are left and right forms for the infinite boundary elements, as shown in

It is obvious that the above shape functions possess delta function property and completeness. Moreover, when

For the right infinite boundary element, these relations are also applicable when

It is also noted that the stress on the ground surface is free, i.e.,

The linear interpolation functions of the displacement and the traction in space can be expressed as:

Therefore, the variables of the displacement and the traction after discretization in both time and space can be rewritten as:

where _{q}

So far, according to the decaying characteristics of fundamental solutions from the finite boundary to the infinity, the shape functions for infinite boundary elements are constructed to mathematically describe the relationships between the infinite boundary and the infinity for the field variables. Moreover, the shape functions satisfy the convergence criterion.

The numerical solutions to

In

The values of influencing coefficients are calculated by using the effective method in [

The influencing coefficients at node

The influencing coefficients of the displacement at node

where the superscripts

To further simplify, the influencing coefficients of linear spatial elements are also assembled. The influencing coefficients of element

where _{ei}

When the source point

where

In

This section is to verify the applicability and the accuracy of the presented TD-BEM formulation for twin-parallel tunnels in an elastic semi-infinite medium, where the infinite boundary elements are incorporated and the time domain fundamental solutions are adopted.

For the purpose of verifying the algorithm, the example with analytical solutions should be the best, because of the uniqueness and the objectivity of the analytical solution. Unfortunately or fortunately, although the analytical solution to the response of the twin-parallel tunnels under dynamic load in an elastic semi-infinite medium is not available, the analytical solution to the response of the single tunnel under dynamic load in an elastic infinite medium is available. Therefore, the numerical example for twin-parallel tunnels is deliberately designed to utilize the analytical solution to the response of the single tunnel under dynamic load in an elastic infinite medium to verify the presented TD-BEM formulation for twin-parallel tunnels under dynamic load in an elastic semi-infinite medium. As shown in _{0} = 3 m) are deliberately assumed to be buried in enough deep depth (^{3}.

In this scenario, theoretically, the response of monitoring point A could be classified into the early phase and the late phase. In the early phase, the wave initiated from the left tunnel has arrived at and passed by point A, while the reflected wave from the ground, due to the wave from the left tunnel, has not yet arrived, let alone the more faraway wave from the right tunnel. In the late phase, the wave at monitoring point A is subsequently disturbed by the reflected wave from the ground and by the propagating wave from the right tunnel. Therefore, the response of monitoring point A for the problem of twin-parallel tunnels in half-plane in the early phase should be the same as the response of the problem of a single circular tunnel in an infinite medium. According to the analytical wave velocity (

_{r}

From

This study is concerned with developing TD-BEM formulation for dynamic analysis for twin-parallel circular tunnels in a semi-infinite medium, with the following conclusions:

Time domain fundamental solutions are adopted in the boundary integral equations in TD-BEM formulation for twin-parallel tunnels under dynamic loads in an elastic semi-infinite medium. The time domain fundamental solutions make it possible to directly and effectively obtain the time history of the response in the presented TD-BEM formulation.

In the process of the numerical implementation, infinite boundary elements are incorporated in TD-BEM formulation to satisfy the stress free conditions on the ground surface, moreover, to reduce the discretization of the boundary of the ground surface.

The applicability and the accuracy of the TD-BEM formulation are verified by comparing the calculated results from the presented TD-BEM with those from the method of characteristics in the deliberately designed example.

The presented TD-BEM formulation could be a step forward in the ideal FEM-BEM coupled model for twin-parallel tunnels under dynamic loads in an elastic semi-infinite medium and could be applicable for blasting projects with twin-parallel blasting holes.