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

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 directapproach 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-byelement 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. 1. Problem formulation The governing equations for the elasticity problem in Laplace domain are as follows L ik ūk(x, s) + F̄i = ρ s 2 ūi (x, s), (1) ū(x, s) = ũ, x ∈ , t̄n(x, s) = t̃n, x ∈ σ , (2) where L0 ik = G + (K + G/3)grad div, u denotes Dirichlet boundary and σ – Neumann boundary, G, K are elastic moduli, F̄i is bulk body force, ρ is material density, s is the Laplace transform parameter. The boundary-value problem for full Biot’s model of linear saturated poroelastic continuum in Laplace domain concerning 4 basic functions – skeleton displacements ūi and pore pressure p̄ – takes the following form [1]: Gūi, j j + ( K + 3 ) ū j,i j − (α − β) p̄,i = s2(ρ − βρ f )ūi − F̄ (3) β sρ f p̄,i i − φ 2s R p̄ − (α − β)sūi,i = −ā, x ∈ , (4) ū′(x, s) = ũ′, x ∈ , ū′ = (ū1, ū2, ū3, p̄) , (5) t̄ ′ n(x, s) = t̃ ′ n, x ∈ σ , t̄ ′ = (t̄1, t̄2, t̄3, q̄) (6) where φ is porosity, F̄i , ā are bulk body forces, β = κρ f φ 2s φ2 + sκ(ρe + φρ f ) , α = 1− K Ks , (7) a Corresponding author: [email protected] R = φ2K f K 2 s K f (Ks − K ) + φKs(Ks − K f ) (8) – constants describe the interaction between the skeleton and filler, κ is permeability, ρ, ρe, ρ f are material density, apparent mass density and filler density respectively, Ks, K f are elastic bulk moduli of the skeleton and filler respectively. The governing equations of partially saturated poroelasticity in the Laplace domain with five unknowns – solid displacements ui , the pore wetting fluid pressure p, and the pore non-wetting fluid pressure pa – given by [2] Gūi, j j + (K + 3 )ū j,i j − (ρ − βSwρw − γ Saρa)sūi = = (α − β)Sw p̄ ,i − (α − γ )Sa p̄a ,i − F̄i , (9) −(α − β)Swsūi,i − (ζ − Saa Sw + Su)s p̄a + βSw ρws p̄ ,i i