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In this paper, an advanced formulation of time-domain, two-dimensional Boundary Element Method (BEM) with material damping is presented. Full space two-dimensional visco-elastodynamic time-convoluted kernels are proposed in order to incorporate proportional damping. This approach is applied to carry out site response analysis of viscoelastic topographic structures subjected to SV and P incident waves. Seismic responses of horizontally layered site, semi-circular canyons, slope topography and ridge sections subjected to these incident waves are analyzed in order to demonstrate the accuracy of the kernels and the applicability of the presented viscoelastic boundary element algorithm. The results show an excellent agreement with recent published results obtained in frequency domain. Also, the effects of different material damping ratios on site response are investigated.