A combined application of boundary-element and Runge-Kutta methods in three-dimensional elasticity and poroelasticity

Aleksandr Ipatov,Leonid Igumnov,Andrey Petrov,E. Cadoni,Aleksandr Belov

doi:10.1051/epjconf/20159404026

Aleksandr Ipatov, Leonid Igumnov + Show 3 more

Open Access

https://doi.org/10.1051/epjconf/20159404026

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Abstract

The report presents the development of the time-boundary element methodology and a description of the related software based on a stepped method of numerical inversion of the integral Laplace transform in combination with a family of Runge-Kutta methods for analyzing 3-D mixed initial boundary-value problems of the dynamics of inhomogeneous elastic and poro-elastic bodies. The results of the numerical investigation are presented. The investigation methodology is based on direct-approach boundary integral equations of 3-D isotropic linear theories of elasticity and poroelasticity in Laplace transforms. Poroelastic media are described using Biot models with four and five base functions. With the help of the boundary-element method, solutions in time are obtained, using the stepped method of numerically inverting Laplace transform on the nodes of Runge-Kutta methods. The boundary-element method is used in combination with the collocation method, local element-by-element approximation based on the matched interpolation model. The results of analyzing wave problems of the effect of a non-stationary force on elastic and poroelastic finite bodies, a poroelastic half-space (also with a fictitious boundary) and a layered half-space weakened by a cavity, and a half-space with a trench are presented. Excitation of a slow wave in a poroelastic medium is studied, using the stepped BEM-scheme on the nodes of Runge-Kutta methods.

Highlights

The report presents the development of the time-boundary element methodology and a description of the related software based on a stepped method of numerical inversion of the integral Laplace transform in combination with a family of Runge-Kutta methods for analyzing 3-D mixed initial boundary-value problems of the dynamics of inhomogeneous elastic and poro-elastic bodies
The investigation methodology is based on directapproach boundary integral equations of 3-D isotropic linear theories of elasticity and poroelasticity in Laplace transforms
The boundary-value problem can be reduced to the following boundary integral equation [1, 3, 4]: Tik(x, y, s)ui (y, s) − Ti0k(x, y, s)ui (x, s)

Summary

Problem formulation

The governing equations for the elasticity problem in Laplace domain are as follows. + (K + G/3)grad div, u denotes Dirichlet boundary and σ – Neumann boundary, G, K are elastic moduli, Fi is bulk body force, ρ is material density, s is the Laplace transform parameter. The governing equations for the elasticity problem in Laplace domain are as follows. + (K + G/3)grad div, u denotes Dirichlet boundary and σ – Neumann boundary, G, K are elastic moduli, Fi is bulk body force, ρ is material density, s is the Laplace transform parameter. The governing equations of partially saturated poroelasticity in the Laplace domain with five unknowns – solid displacements ui , the pore wetting fluid pressure pw, and the pore non-wetting fluid pressure pa – given by [2]. Φ Sw + κwρws φ Sa + κaρas where κw and κa the phase permeability of the wetting and the non-wetting fluid are given by κw = Krwk/ηw and κa = Krak/ηa respectively.

Boundary integral equations and discretization

Laplace transform inversion

Poroelastic column

Homogeneous half-space

Layered half-space

Half-space with a cavity

Half-space with a trench

Concluding remarks