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# Development of TD-BEM Formulation for Dynamic Analysis for Twin-Parallel Circular Tunnels in an Elastic Semi-Innite Medium

Weidong Lei1, Hai Zhou1,*, Hongjun Li2, Rui Chen1

1 School of Civil and Environmental Engineering, Harbin Institute of Technology, Shenzhen, 518055, China
2 Urban and Rural Construction Institute, Hebei Agricultural University, Baoding, 071001, China

* Corresponding Author: Hai Zhou. Email:

Computer Modeling in Engineering & Sciences 2021, 126(2), 577-597. https://doi.org/10.32604/cmes.2021.011857

## Abstract

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.

## Keywords

Lei, W., Zhou, H., Li, H., Chen, R. (2021). Development of TD-BEM Formulation for Dynamic Analysis for Twin-Parallel Circular Tunnels in an Elastic Semi-Innite Medium. CMES-Computer Modeling in Engineering & Sciences, 126(2), 577–597.

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