Wave propagation in a simplified modelled poroelastic continuum: fundamental solutions and a time domain boundary element formulation

Abstract

SUMMARYIn finite element formulations for poroelastic continua a representation of Biot's theory using the unknowns solid displacement and pore pressure is preferred. Such a formulation is possible either for quasi-static problems or for dynamic problems if the inertia effects of the fluid are neglected. Contrary to these formulations a boundary element method (BEM) for the general case of Biot's theory in time domain has been published (Wave Propagation in Viscoelastic and Poroelastic Continua: A Boundary Element Approach. Lecture Notes in Applied Mechanics. Springer: Berlin, Heidelberg, New York, 2001.). If the advantages of both methods are required it is common practice to couple both methods. However, for such a coupled FE/BE procedure a BEM for the simplified dynamic Biot theory as used in FEM must be developed.Therefore, here, the fundamental solutions as well as a BE time stepping procedure is presented for the simplified dynamic theory where the inertia effects of the fluid are neglected. Further, a semi-analytical one-dimensional solution is presented to check the proposed BE formulation. Finally, wave propagation problems are studied using either the complete Biot theory as well as the simplified theory. These examples show that no significant differences occur for the selected material.

The solution of elastostatic and dynamic problems using the boundary element method based on spherical Hankel element framework

“…For problems such as stress concentration or infinite domains, BEM can be applied to achieve better accuracy in comparison with finite element method. Many usages of BEM in solving problems related to wave propagation, crack, buckling, optimization, etc are reported in the literature …”

An Efficient Boundary Element Method Using a Modal Transformation for Transient Acoustic Problem

“…Laplace transformation has been applied to solve the acoustic problem during the collision between a rigid sphere and a cylinder (2) . A time-domain boundary element solution has been applied to solve the transient acoustic phenomenon in the open space around a structure (3)(4)(5) . Fourier transformation bas been also applied to obtain a transient acoustic solution expressed as a combination of steady-state boundary element solutions at various frequencies (6) .…”

Special Solutions for Dynamic Poroelasticity in 1-D Formulation

“…соответствующим ГИУ. Продемонстрируем применение метода численного обращения преобразования Лапласа для построения интеграла по времени от матрицы фундаментальных решений: статье[4] на основе метода квадратур сверток получен подобный результат для U 11 . На рис.…”

“…1 приведены полученные нами результаты, причем на рис. 1,а результат из[4] представлен штриховой линией. Рассматривался водонасыщенный грунт, характеризующийся параметрами: 2 ; ν = 0,298; ρ = 1884 кг/м 3 ; ρ f = 1000 кг/м 3 ; φ = 0,48; (Н .…”

“…с). Точка пространства, для которой проводились вычисления, выбрана со следующими координатами: (0,1, 0,15, 0,2).Сравнение результата из[4] с проведенными нами численными экспериментами показывает, что примененный метод численного обращения дает более точный и устойчивый результат по сравнению с методом квадратур сверток, описанным в[3].…”