A Boundary Element Formulation based on the Convolution Quadrature Method for the Dynamic Behaviour Analysis of the Unsaturated Soils
Résumé
The site effects generated by topographical features are among the sources of amplifications and de-amplifications of the seismic signals, which can be important over large frequency domains. Site response analysis of topographical structures could only be solved accurately, economically and under realistic conditions, with the aid of numerical methods such as the Boundary Element Method (BEM). The BEM is a very effective numerical tool for dynamic analysis of linear elastic bounded and unbounded media. The method is very attractive for wave propagation problems, because the discretization is done only on the boundary, yielding smaller meshes and systems of equations. Another advantage is that this method represents efficiently the outgoing waves through infinite domains, which is very useful when dealing with scattered waves by topographical structures. This paper aims at obtaining an advanced formulation of the time-domain Boundary Element Method (BEM) for two-dimensional dynamic analysis of unsaturated soil. Unlike the usual time-domain BEM the present formulation applies a Convolution Quadrature developed by Lubich (1988) which requires only the Laplace-domain instead of the time-domain fundamental solutions. In this paper first of all, the set of fully coupled governing differential equations of a porous medium saturated by two compressible fluids (water and air) subjected to dynamic loadings is obtained. These phenomenal formulations are presented based on the experimental observations and with respect to the poromechanics theory within the framework of the suction-based mathematical model presented by Gatmiri (1997) and Gatmiri et al. (1998). In this model, the effect of deformations on the suction distribution in the soil skeleton and the inverse effect are included in the formulation via a suction-dependent formulation of state surfaces of void ratio and degree of saturation. In this formulation, the solid skeleton displacements , water pressure and air pressure are presumed to be independent variables. Secondly, the Boundary Integral Equation (BIE) is developed directly from those equations via the use of the weighted residuals method for the first time in a way that permits an easy discretization and implementation in a numerical code. The associated fundamental solution obtained by Maghoul et al. (2009) is used in the BIE. Then, once the BIE is derived, the focus shifts to an overview of the general purpose numerical implementation. Finally, the resulting BEM time domain formulation represents the first of its kind for two-dimensional dynamic problems in unsaturated soils.