Development of boundary-element time-step scheme in solving 3D poroelastodynamics problems

Igor Vorobtsov,Andrey Petrov,Aleksandr Belov,E Buzaud,E Cadoni,H Couque,A Cosculluela

doi:10.1051/epjconf/201818301042

Igor Vorobtsov, Andrey Petrov + Show 5 more

Open Access

https://doi.org/10.1051/epjconf/201818301042

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Abstract

The development of time-step boundary-element scheme for the three dimensional boundaryvalue problems of poroelastodynamics is presented. The poroelastic continuum is described using Biot’s mathematical model. Poroelastic material is assumed to consist of a solid phase constituting an elastic formdefining skeleton and carrying most of the loading, and two fluid phases filling the pores. Dynamic equations of the poroelastic medium are written for unknown functions of displacement of the elastic skeleton and pore pressures of the filling materials. Green’s matrices and, based on it, boundary integral equations are written in Laplace domain. Discrete analogue are obtained by applying the collocation method to a regularized boundary integral equation. Boundary element scheme is based on time-step method of numerical inversion of Laplace transform. A modification of the time-step scheme on the nodes of Runge-Kutta methods is considered. The Runge-Kutta scheme is exemplified with 2-and 3-stage Radau schemes. The results of comparing the two schemes in analyzing a numerical example are presented.

Highlights

1 Introduction In BEM-modeling of dynamic processes, two main approaches can be conventionally discerned: solving in time, using a time-step scheme [1], and solving in Laplace or Fourier transforms with the subsequent inversion of the transforms [2]
Matrices of fundamental solutions can only be constructed in Fourier and Laplace images
That is why the pioneering formulations of the boundary-element method for the dynamics of poroelastic media of the Biot model were published in Laplace transforms [3, 4]

Summary

Introduction

In BEM-modeling of dynamic processes, two main approaches can be conventionally discerned: solving in time, using a time-step scheme [1], and solving in Laplace or Fourier transforms with the subsequent inversion of the transforms [2]. In [5,6,7,8,9,10], to construct a time-step boundary-element scheme based on fundamental solutions in Laplace transforms, the quadrature convolution method introduced by Lyubich [11, 12] is used. Work [17] reviews a wide range of approaches to using a boundary-element scheme in combination with the convolution quadrature method, based on both Euler scheme and other schemes of the Runge-Kutta family. In [18,19,20,21], dynamic wave processes in elastic media with coupled fields are automatically modeled using an analogous boundary-element scheme based on the timestep method of numerically inverting Laplace transforms. The present study describes a modification of such a time-step method on the nodes of Runge-Kutta schemes and a time-step BEM scheme constructed on its basis

Problem formulation

BEM application

Laplace transform inversion

Numerical example

Conclusion