A Time-Domain Artificial Boundary for Near-Field Wave Problem of Fluid Saturated Porous Media

Abstract

The near-field wave problem of the saturated soil involves the energy radiation effect of the truncated infinite media. A viscous spring boundary is proposed for the fluid-saturated porous media. Based on the process of wave propagation under internal point source, the stress and flow velocity boundaries are constructed by reasonable assumptions of outgoing waves and Green’s function, respectively. Without the permeability assumption, the proposed boundary avoids the low accuracy caused by the assumption of zero permeability that is widely used in the existing methods. The boundary simultaneously has a simple form, clear physical meaning, and less computational cost due to its local character. Meanwhile, a completely explicit integration algorithm considering the damping is constructed to solve the finite element equations of saturated porous media with the proposed boundary. The accuracy and high computational efficiency of the wave numerical method are verified in the examples.

1. Introduction

The wave problem of saturated two-phase porous media is an important research issue of soil dynamics and geotechnical seismic engineering. For the problem, Biot [1] established the three-dimensional quasi-static consolidation theory by considering the deformation of solid and liquid phases, as well as the fluid-solid coupling of saturated porous media. Based on this, Biot wave theory is proposed by adding the inertial terms of soil skeleton and pore fluid [2,3]. However, there are several elastic coefficients with relatively abstract meanings in the equations, and the concept of additional mass density describing the coupling of solid and fluid lacks physical meaning, which makes it relatively difficult to measure. Therefore, many scholars improved Biot theory furtherly [4,5,6,7]. The dynamic equations proposed by Zienkiewicz [7,8] is essentially equivalent to the Biot theory. In the equations, the additional mass density is ignored and assumes that dynamic permeability coefficient considers the fluid viscosity. Men et al. [9] and Chen et al. [10] considered the incompressibility and compressibility of the pore fluid, respectively, and gave the dynamic governing equations for saturated porous media under the two conditions. Wu [11] compared the dynamic equations proposed by Biot theory and Zienkiewicz, proved that the two equations are equivalent, and gave the corresponding relationships between the corresponding parameters. Zhao et al. [12] used inductive and deductive methods to review the existing wave theory of porous media. Morency and Tromp [13] presents a numerical implementation of poroelastic wave propagation using a spectral element method, based upon an averaging technique which accommodates the transition from the microscopic to the macroscopic scale. Puente and Dumbser [14] introduced a local space-time discontinuous Galerkin method with arbitrary high-order accuracy to simulate wave propagation in poroelastic material. Wang et al. [15] investigated the wave propagation in 2D periodic fluid-saturated porous media with an isotropic or a transversely isotropic matrix.

For the near-field wave problems, absorbing artificial boundary are needed to model waves propagation in infinite field, to ensure waves have no reflection effect when passing through the artificial boundary [16]. The wave transmission boundary mainly includes [17] global precise artificial boundaries [18,19,20] and local artificial boundaries [21,22,23,24,25,26,27]. The global artificial boundary mainly includes boundary element method [28], thin layer method [29], exact Kirchhoff integration method [18,19] and Dirichlet-to-Neumann method [20,30]. The boundary conditions are established based on the initial boundary value conditions of the truncated media and are gotten through rigorous analytical derivation, leading to complicated calculation process and low efficiency. Local artificial boundaries include the viscous boundaries [21,31], the viscous-spring boundaries [22,23,24,32,33,34,35], the superposition boundaries [25,36], the extrapolation boundaries [26,37], the paraxial boundaries [38,39], the multi-directional boundary [27,40], the perfectly matched layer [41,42], and the dynamic infinite element [43,44,45]. The local artificial boundary is local both in space and time domain, that is, the responses of the target node are only related to the responses of the adjacent nodes. It has less calculation amount and higher calculation efficiency.

For the fluid saturated porous media, Modaressi and Benzzenati [46,47] established paraxial boundaries for the u-p formulation and obtained the zero-order and first-order paraxial boundary conditions, where the zero-order paraxial boundary is equivalent to the viscous boundary. Akiyoshi et al. [48] established paraxial boundaries for the u-w, u-U, and u-p models of linear and isotropic saturated soils, and extended their work to the case of laterally isotropic and anisotropic saturated soils [49,50]. The above research work is based on plane waves. Liu Guanglei and Song Erxiang [51] prop osed a viscous spring artificial boundary for the u-p equation based on cylindrical waves. Du Xiuli et al. [52] and Zhao Chenggang et al. [53,54] established two viscous spring artificial boundaries based on u-U formulation. To improve the accuracy of artificial boundaries, Li Peng and Song Erxiang [55] proposed a new high-precision artificial boundary on the basis of the work of Song Erxiang [51]. Degrande and De Roeck [56,57] proposed a global artificial boundary in the frequency domain, but its computational cost is higher. Then, under the assumptions of zero and infinity permeability coefficient, Gajo et al. [58] as well as Zerfa and Loret [59] established, respectively, two viscous boundary conditions with high computational efficiency in the time domain.

The assumption of permeability coefficient leads to neglect of fluid-solid coupling, resulting in the low accuracy of dynamic responses of the fluid saturated porous media. A new viscous spring artificial boundary without the permeability coefficient assumption is proposed based on the u-p equation of the saturated porous media. Green’s function is firstly used to establish the flow boundary velocity condition due to the similar principle by describing the pore pressure field generated by the point force. A completely explicit integration algorithm considering the damping is constructed to solve the finite element equations of saturated porous media with the proposed boundary. Finally, three examples are used to show the accuracy and effectiveness of the proposed method by comparing with analysis solutions and numerical solutions of existing boundary.

2. Wave Velocity

The propagation of waves in the semi infinitely field of saturated soil under point force is shown in Figure 1, which can be described by u-p equation. Ignoring the body force, the u-p formed control equations of fluid-saturated porous media are expressed as:

$$\left(\lambda +G\right)\nabla \nabla \cdot \mathit{u}+G{\nabla }^2\mathit{u}-\alpha \nabla p=\rho \ddot{\mathit{u}}$$

(1)

$$k_f{\nabla }^{2}p-\alpha \nabla \cdot \dot{\mathit{u}}=\frac{1}{Q_b}\dot{p}$$

(2)

where λ and G are the Lamb constants of the soil. α and $Q_b$ are the compressive coefficient of the soil skeleton and pore fluid, respectively. $k_f$ is the dynamic permeability coefficient. $\rho$ represents the density of the total solid-fluid mixture. ${\nabla }^{\mathrm{T}}=\left[\begin{array}{cc}\partial /\partial x& \partial /\partial y\end{array}\right]$. u and p represent the displacement and pore pressure of saturated soil, respectively.

Equations (1) and (2) are further transformed into the frequency domain, and the following equations are obtained by introducing the parameter ${\rho }_{m1}=-\frac{1}{\overline{k}}\frac{i}{\omega }$:

$$\left(\lambda +G\right)\nabla \nabla \cdot \mathit{u}+G{\nabla }^2\mathit{u}+{\omega }^2\rho \mathit{u}-\alpha \nabla p=\mathbf{0}$$

(3)

$${\nabla }^{2}p+\frac{{\omega }^2{\rho }_m}{Q_{\mathrm{b}}}p+{\omega }^2\alpha {\rho }_m\nabla \cdot \mathit{u}=0$$

(4)

where ω represents the circular frequency, and the displacement and pore pressure are expressed by the rotational and irrotational potential functions of P1, P2, and S waves as shown in Equations (5) and (6):

$$\mathit{u}=\nabla {\phi }_{P1}+\nabla {\phi }_{P2}+\nabla \times {\psi }_s\mathit{e}$$

(5)

$$p=\left({\eta }_1-\alpha Q_{\mathrm{b}}\right){\nabla }^2{\phi }_{P1}+\left({\eta }_2-\alpha Q_{\mathrm{b}}\right){\nabla }^2{\phi }_{P2}$$

(6)

where ${\phi }_{P1}$ and ${\phi }_{P2}$ represent the potential function of P1 and P2 waves, respectively; ${\psi }_s$ represents the potential function of S wave; e is the unit vector. At the same time, ${\eta }_1$ and ${\eta }_2$ in Equation (6) are calculated as shown:

$${\eta }_{1, 2}=\frac{\alpha Q_b}{{\rho }_{m1}{V}_{P1, P2}^2-Q_{\mathrm{b}}}$$

(7)

Substituting Equations (5) and (6) into Equations (3) and (4), the wave equations expressed by the potential functions of P1, P2, and S waves are obtained as follows:

$$V_{P1, P2}^2{\nabla }^2{\phi }_{P1, P2}+{\omega }^2{\phi }_{P1, P2}=0$$

(8)

$$V_S^2{\nabla }^2{\psi }_S+{\omega }^2{\psi }_S=0$$

(9)

where, $V_{P1}$, $V_{P2}$, and $V_S$, shown in Equation (10), are the nominal wave velocity of P1, P2, and S waves [60], respectively.

$$V_{P1, P2}=\sqrt{\frac{\lambda +2G}{{\rho }_{1, 2}}}, V_S=\sqrt{\frac{G}{\rho }}$$

(10)

$$\left\{\begin{array}{l}{\rho }_{1,2}=\rho \left(d_2\mp \sqrt{d_2^2-d_1{d}_3}\right)\\ d_1=\frac{\lambda +2G}{Q_{\mathrm{b}}}\\ d_2=\frac{1}{2\rho }\left(\rho +\frac{\lambda +2G+{\alpha }^2{Q}_{\mathrm{b}}}{Q_{\mathrm{b}}}{\rho }_{m1}\right)\\ d_3=\frac{{\rho }_{m1}}{\rho }\end{array}\right.$$

(11)

As shown in Equation (10), all nominal wave velocities of three body waves are complex numbers, which are uniformly expressed in the following complex form:

$$V=V^r+V^{i}i$$

(12)

The real part $V^r$ of the wave velocity in the equation represents the actual wave velocity of the wave, and the imaginary part $V^i$ reflects the attenuation of wave. Therefore, the real part $V^r$ of the wave velocity corresponding to the P1, P2, and S waves is the actual wave velocity of the three types of waves in saturated soil.

3. Viscous-Spring Artificial Boundary

In Figure 1, the surface of the semi-infinite saturated soil space is a free drainage boundary, and the artificial boundary $\mathsf{\Gamma }$ is introduced to divide the semi-infinite space into two parts: the interior domain ${\mathsf{\Omega }}_{\mathrm{I}}$ and the exterior domain ${\mathsf{\Omega }}_{\mathrm{E}}$. The interior domain may have complex geometry and material properties, and convenient for solving using the finite element method. The infinite exterior domain is approximated as a uniform media of linear elasticity, and the effect of the exterior domain on the interior domain is generally achieved through artificial boundaries. For this problem, the displacement u and pore pressure p, the total stress σ, and the fluid phase flow velocity Φ of the interior domain and exterior domain should satisfy the continuous conditions, shown as follows:

$$\left\{\begin{array}{ll}{\mathit{u}}_{\mathrm{I}}={\mathit{u}}_{\mathrm{E}}& \mathrm{on}{\mathsf{\Gamma }}_u\\ p_{\mathrm{I}}=p_{\mathrm{E}}& \mathrm{on}{\mathsf{\Gamma }}_p\\ {\mathit{\sigma }}_{\mathrm{I}}+{\mathit{\sigma }}_{\mathrm{E}}=0& \mathrm{on}{\mathsf{\Gamma }}_{\sigma }\\ {\Phi }_{\mathrm{I}}={\Phi }_{\mathrm{E}}& \mathrm{on}{\mathsf{\Gamma }}_{\Phi }\end{array}\right.$$

(13)

where σ represents the total stress of soil, and Φ is the flow velocity of pore fluid in saturated soil. Therefore, the flow velocity condition and stress condition should be satisfied on the artificial boundary.

3.1. Flow Velocity Boundary Condition

The wave propagation problem under the action of a point source on the surface of the interior domain is similar to the significance of Green’s function of the diffusion equation, that is, the field distribution generated by a unit point source. At the same time, the Lamb problem [61] analyzes the distribution of a variable field under the action of a concentrated force or point source on the surface of infinite space, such as pore water pressure in a saturated soil field under a point force. To obtain the pore pressure value on the artificial boundary, Green’s function constructs the pore pressure field distribution of fluid-saturated porous media under the action of interior concentrated force. Therefore, Equation (2) is converted into the equation as follows in two-dimensional plane polar coordinates without considering the exterior loading and ignoring the coupling term $\alpha \nabla \cdot \dot{\mathit{u}}$:

$$\frac{\partial p}{\partial t}=k_f{Q}_b\frac{{\partial }^{2}p}{\partial r^2}$$

(14)

where r represents the outer normal direction of the artificial boundary, and t is time. It can be seen that Equation (14) is the standard form of the diffusion equation. In the infinite domain, based on Equation (14), the Green’s function $G\left(r,t\right)$ [62] for the pore pressure p is obtained, and its distribution form is shown as follows:

$$p=G\left(r,t\right)=\frac{1}{4\pi k_f{Q}_{b}t}{e}^{-\frac{r^2}{4k_{f}Q\cdot t}}$$

(15)

By utilizing the relationship between the flow velocity and the pore pressure $\Phi =-k_f\frac{\partial p}{\partial r}$, the flow velocity boundary condition expressed in terms of pore pressure is shown below:

$$\Phi =\left(\frac{l_r}{2Q_b\cdot t}\right)p$$

(16)

where p and Φ are the pore pressure and the flow velocity in the outward normal direction of the artificial boundary. Since the seepage process is a time-varying diffusion process of pore fluid in saturated porous media, only when the wave propagates to the artificial boundary is there a pressure gradient between the artificial boundary and the internal domain of soil. Therefore, Equation (16) can be further written as:

$$\Phi =\left\{\begin{array}{cc}0& \left(t<t_0\right)\\ \left(\frac{l_r}{2Q_b\cdot t}\right)p& \left(t\ge t_0\right)\end{array}\right.$$

(17)

where l_r is defined as the distance from the geometric center of the interior domain to the point on the artificial boundary; $t_0$ is the shortest time for the wave to propagate to the artificial boundary.

3.2. Stress Boundary Condition

Since Equations (8) and (9) are standard wave equations, to simplify the analysis, based on Equations (5)–(9), the outgoing wave of displacement can be assumed to be:

$$u_r\left(r,t\right)=\left(k_{1}lw+k_2\right)f\left(v_{1}t-r\right)$$

(18)

$$u_{\perp }\left(r,t\right)=\left(k_{1}lw+k_2\right)f\left(v_{2}t-r\right)$$

(19)

where n and ⊥ represent the outer normal and tangent direction of the artificial boundary, respectively; $u_r$ and the $u_{\perp }$ are solid phase displacement in the outer normal and tangent directions of the artificial boundary, respectively; v₁ and v₂ are the actual wave velocities in the normal and tangential of the artificial boundary, respectively; f represents the arbitrary waveform function; k₁ and k₂ represent wave numbers of attenuated and non-attenuated waves, respectively; l is the dimensional coordination factor; and $w=1/\sqrt{r}$ represents the geometric attenuation factor.

Using the constitutive relationship and geometric relationship of the saturated two-phase porous media, the relationship between the total stress, displacement, and the pore pressure of saturated porous media is given:

$$\left\{\begin{array}{l}{\sigma }_r=\left[\lambda \left(\frac{\partial u_{\perp }}{\partial \perp }+\frac{\partial u_r}{\partial r}\right)-\alpha p\right]+2G\frac{\partial u_r}{\partial r}\\ {\sigma }_{\perp }=\left[\lambda \left(\frac{\partial u_{\perp }}{\partial \perp }+\frac{\partial u_r}{\partial r}\right)-\alpha p\right]+2G\frac{\partial u_{\perp }}{\partial \perp }\\ \tau =G\left(\frac{\partial u_{\perp }}{\partial r}+\frac{\partial u_r}{\partial \perp }\right)\end{array}\right.$$

(20)

where ${\sigma }_r$ and ${\sigma }_{\perp }$ are the normal and tangential stresses of the solid-phase media, respectively; τ is the shear stress of the solid-phase media. In two dimensions, when the outgoing wave propagates along the outer normal direction of the artificial boundary, the stress at the artificial boundary, i.e., Equation (20), can be further simplified to:

$${\sigma }_r=\left(\lambda +2G\right)\frac{\partial u_r}{\partial r}-\alpha p$$

(21)

$$\tau =G\frac{\partial u_{\perp }}{\partial r}$$

(22)

Substitute Equation (18) into Equation (21) to obtain the normal stress represented by the waveform function f:

$${\sigma }_r=-\left(\lambda +2G\right)\left(k_{1}l{r}^{-\frac{1}{2}}+k_2\right)f^{\prime }\left(v_{1}t-r\right)-\frac{1}{2}\left(\lambda +2G\right)al{r}^{-\frac{3}{2}}f\left(v_{1}t-r\right)-\alpha p$$

(23)

By calculating the first derivative of Equation (18) with respect to time t, the relationship between the normal velocity and the waveform function f is obtained as follows:

$${\dot{u}}_r\left(r,t\right)=c_1\left(k_{1}l{r}^{-\frac{1}{2}}+k_2\right)f^{\prime }\left(v_{1}t-r\right)$$

(24)

Using Equations (18), (23), and (24) to eliminate the waveform function f, the relationship between the normal stress, displacement, velocity, and pore pressure is expressed as:

$${\sigma }_r=-\frac{\left(\lambda +2G\right)}{2r\left(1+\frac{k_2}{k_{1}l}\sqrt{r}\right)}{u}_r-\frac{\left(\lambda +2G\right)}{v_1}{\dot{u}}_r-\alpha p$$

(25)

By introducing dimensionless parameters $A=\frac{k_2}{k_{1}l}\sqrt{r}$ and $v_1=\frac{{\rho }_3^2}{B}{V}_{P1}^n$, the normal stress boundary condition is further written as:

$${\sigma }_n=-\frac{\lambda +2G}{2l_r\left(1+A\right)}{u}_r-B\rho V_{P1}^r{\dot{u}}_r-\alpha p$$

(26)

where the parameter A reflects the propagation characteristics of the outgoing wave, namely the amplitude ratio between the plane wave and the scattering wave; the parameter B represents the average wave velocity characteristics of the incident multi-sub-wave at different angles, namely, the relationship between the physical wave velocity and the apparent wave velocity. Through a large number of numerical calculations, the optimal values of A and B are obtained. A = 0.8, B = 1.1 [16].

Using a similar calculation process, the tangential stress boundary condition is shown below:

$$\tau =-\frac{G}{2l_r\left(1+A\right)}{u}_{\perp }-B\rho V_S^n{\dot{u}}_{\perp }$$

(27)

where the definitions of parameters A and B are the same as in Equation (26).

3.3. Finite Element Discretization of Artificial Boundary

The artificial boundary Equations (17), (26), and (27) are written in discrete form as follows:

$${\mathit{f}}_{uBi}^{\left(l\right)}=-{\mathit{K}}_{Bi}^{\infty \left(l\right)}{\mathit{u}}_{Bi}^{\left(l\right)}-{\mathit{C}}_{Bi}^{\infty \left(l\right)}{\dot{\mathit{u}}}_{Bi}^{\left(l\right)}-{\mathit{Q}}_{Bi}^{\infty \left(l\right)}{p}_{Bi}^{\left(l\right)}$$

(28)

$$f_{qBi}^{\left(l\right)}=J_{Bi}^{\infty \left(l\right)}{p}_{Bi}^{\left(l\right)}$$

(29)

where the superscript (l) represents the artificial boundary; ∞ represents the exterior domain; Bi represents the ith node on the boundary; ${\mathit{f}}_{uBi}^{\left(l\right)}$ and $f_{qBi}^{\left(l\right)}$ are the stress and the flow velocity of the ith node on the boundary, respectively; ${\mathit{u}}_{Bi}^{\left(l\right)}$ and $p_{Bi}^{\left(l\right)}$ are the displacement and the pore pressure of the ith node on the boundary, respectively. The variable matrices and scalar coefficients in Equations (28) and (29) are expressed in the following form:

$${\mathit{K}}_{Bi}^{\infty \left(l\right)}=L_i\left[\begin{array}{cc}{K}_{BN}^{\infty }& 0\\ 0& K_{BT}^{\infty }\end{array}\right], \left\{\begin{array}{l}{K}_{BN}^{\infty }=\frac{\lambda +2G}{2l_r\left(1+A\right)}\\ K_{BT}^{\infty }=\frac{G}{2l_r\left(1+A\right)}\end{array}\right.$$

(30)

$${\mathit{C}}_{Bi}^{\infty \left(l\right)}=L_i\left[\begin{array}{cc}{C}_{BN}^{\infty }& 0\\ 0& C_{BT}^{\infty }\end{array}\right], \left\{\begin{array}{l}{C}_{BN}^{\infty }=B\rho V_{P1}^r\\ C_{BT}^{\infty }=B\rho V_S^r\end{array}\right.$$

(31)

$${\mathit{Q}}_{Bi}^{\infty \left(l\right)}=L_i\left[\begin{array}{c}{Q}_B^{\infty }\\ 0\end{array}\right], Q_B^{\infty }=\alpha$$

(32)

$$J_{Bi}^{\infty \left(l\right)}=L_i{J}_B^{\infty }, J_B^{\infty }=\frac{l_r}{2Q_{b}t}$$

(33)

where L_i represents the boundary length corresponding to the boundary node i. Converting Equations (28) and (29) from the local coordinate system into the global coordinate system, we get:

$${\mathit{f}}_{uBi}=-{\mathit{K}}_{Bi}^{\infty }{\mathit{u}}_{Bi}-{\mathit{C}}_{Bi}^{\infty }{\dot{\mathit{u}}}_{Bi}-{\mathit{Q}}_{Bi}^{\infty }{p}_{Bi}$$

(34)

$$f_{qBi}=J_{Bi}^{\infty }{p}_{Bi}$$

(35)

In Equations (34) and (35), the physical meanings of each matrix and scalar coefficient are the same as that in Equations (28) and (29), and the coordinate transformations are carried out through the following relationships:

$$\left\{\begin{array}{l}{\mathit{K}}_{Bi}^{\infty }={\mathit{W}}^{\mathrm{T}}{\mathit{K}}_{Bi}^{\infty \left(l\right)}\mathit{W}\\ {\mathit{C}}_{Bi}^{\infty }={\mathit{W}}^{\mathrm{T}}{\mathit{C}}_{Bi}^{\infty \left(l\right)}\mathit{W}\\ {\mathit{Q}}_{Bi}^{\infty }={\mathit{W}}^{\mathrm{T}}{\mathit{Q}}_{Bi}^{\infty \left(l\right)}\end{array}\right.$$

(36)

$$J_{Bi}^{\infty }=J_{Bi}^{\infty \left(l\right)}$$

(37)

where $\mathit{W}=\left[\begin{array}{cc}{l}_{rx}& l_{ry}\\ l_{\perp x}& l_{\perp y}\end{array}\right]$ is the coordinate transformation matrix for transforming local coordinates to global coordinates. $l_{rx}=\cos (r,x)$ is the cosine of the angle between the positive direction of the local coordinate r-axis and the positive direction of the global coordinate x-axis. The rest of the parameters are similar to the definition of l_xy.

After discretization by the Galerkin finite element method, the dynamic governing equations of the saturated two-phase porous media, Equations (1) and (2), are represented by block matrices of internal domain and boundary:

$$\left[\begin{array}{cc}{\mathit{M}}_I& {\mathit{M}}_{IB}\\ {\mathit{M}}_{BI}& {\mathit{M}}_B\end{array}\right]\left\{\begin{array}{c}{\ddot{\mathit{u}}}_I\\ {\ddot{\mathit{u}}}_B\end{array}\right\}+\left[\begin{array}{cc}{\mathit{K}}_I& {\mathit{K}}_{IB}\\ {\mathit{K}}_{BI}& {\mathit{K}}_B\end{array}\right]\left\{\begin{array}{c}{\mathit{u}}_I\\ {\mathit{u}}_B\end{array}\right\}-\left[\begin{array}{cc}{\mathit{Q}}_I& {\mathit{Q}}_{IB}\\ {\mathit{Q}}_{BI}& {\mathit{Q}}_B\end{array}\right]\left\{\begin{array}{c}{\mathit{p}}_I\\ {\mathit{p}}_B\end{array}\right\}=\left\{\begin{array}{c}{\mathit{f}}_{uI}\\ {\mathit{f}}_{uB}\end{array}\right\}$$

(38)

$$\left[\begin{array}{cc}{\mathit{S}}_I& {\mathit{S}}_{IB}\\ {\mathit{S}}_{BI}& {\mathit{S}}_B\end{array}\right]\left\{\begin{array}{c}{\dot{\mathit{p}}}_I\\ {\dot{\mathit{p}}}_B\end{array}\right\}+\left[\begin{array}{cc}{\mathit{J}}_I& {\mathit{J}}_{IB}\\ {\mathit{J}}_{BI}& {\mathit{J}}_B\end{array}\right]\left\{\begin{array}{c}{\mathit{p}}_I\\ {\mathit{p}}_B\end{array}\right\}+\left[\begin{array}{cc}{\mathit{Q}}_I^T& {\mathit{Q}}_{BI}^T\\ {\mathit{Q}}_{IB}^T& {\mathit{Q}}_B^T\end{array}\right]\left\{\begin{array}{c}{\dot{\mathit{u}}}_I\\ {\dot{\mathit{u}}}_B\end{array}\right\}=\left\{\begin{array}{c}-{\mathit{f}}_{qI}\\ -{\mathit{f}}_{qB}\end{array}\right\}$$

(39)

where the subscripts B and I correspond to the nodes at the boundary and the internal domain, respectively; M is the mass matrix; K is the stiffness matrix; Q is the coupling matrix; S is the fluid compression matrix; J is the fluid permeability matrix. f_uI and f_qI represent the loading and fluid injection point source of the internal domain, respectively. f_uB and f_qB represent the effect of the exterior domain on the internal domain, namely the artificial boundary conditions (34) and (35), respectively.

At the same time, substituting Equations (34) and (35) into Equations (38) and (39), the finite element discrete equations of saturated porous media considering the boundary conditions are obtained:

$$\left[\begin{array}{cc}{\mathit{M}}_I& {\mathit{M}}_{IB}\\ {\mathit{M}}_{BI}& {\mathit{M}}_B\end{array}\right]\left\{\begin{array}{c}{\ddot{\mathit{u}}}_I\\ {\ddot{\mathit{u}}}_B\end{array}\right\}+\left[\begin{array}{cc}{\mathit{K}}_I& {\mathit{K}}_{IB}\\ {\mathit{K}}_{BI}& {\mathit{K}}_B+{\mathit{K}}_B^{\infty }\end{array}\right]\left\{\begin{array}{c}{\mathit{u}}_I\\ {\mathit{u}}_B\end{array}\right\}+\left[\begin{array}{cc}\mathbf{0}& \mathbf{0}\\ \mathbf{0}& {\mathit{C}}_B^{\infty }\end{array}\right]\left\{\begin{array}{c}{\dot{\mathit{u}}}_I\\ {\dot{\mathit{u}}}_B\end{array}\right\}-\left[\begin{array}{cc}{\mathit{Q}}_I& {\mathit{Q}}_{IB}\\ {\mathit{Q}}_{BI}& {\mathit{Q}}_B-{\mathit{Q}}_B^{\infty }\end{array}\right]\left\{\begin{array}{c}{\mathit{p}}_I\\ {\mathit{p}}_B\end{array}\right\}=\left\{\begin{array}{c}{\mathit{f}}_{uI}\\ \mathbf{0}\end{array}\right\}$$

(40)

$$\left[\begin{array}{cc}{\mathit{S}}_I& {\mathit{S}}_{IB}\\ {\mathit{S}}_{BI}& {\mathit{S}}_B\end{array}\right]\left\{\begin{array}{c}{\dot{\mathit{p}}}_I\\ {\dot{\mathit{p}}}_B\end{array}\right\}+\left[\begin{array}{cc}{\mathit{J}}_I& {\mathit{J}}_{IB}\\ {\mathit{J}}_{BI}& {\mathit{J}}_B+{\mathit{J}}_B^{\infty }\end{array}\right]\left\{\begin{array}{c}{\mathit{p}}_I\\ {\mathit{p}}_B\end{array}\right\}+\left[\begin{array}{cc}{\mathit{Q}}_I^T& {\mathit{Q}}_{BI}^T\\ {\mathit{Q}}_{IB}^T& {\mathit{Q}}_B^T\end{array}\right]\left\{\begin{array}{c}{\dot{\mathit{u}}}_I\\ {\dot{\mathit{u}}}_B\end{array}\right\}=\left\{\begin{array}{c}-{\mathit{f}}_{qI}\\ \mathbf{0}\end{array}\right\}$$

(41)

where ${\mathit{K}}_B^{\infty }$, ${\mathit{C}}_B^{\infty }$, ${\mathit{Q}}_B^{\infty }$, and ${\mathit{J}}_B^{\infty }$ are calculated by Equations (36) and (37), respectively. The finite element discrete equations of saturated porous media can be calculated by different time domain integration methods. The proposed artificial boundary actually only changes the values of the corresponding boundary nodes on the diagonal of the coefficient matrices in Equations (40) and (41). The finite element discrete equations of saturated porous media considering the boundary conditions can be solved by a completely explicit integration algorithm efficiently, which is shown in Appendix A.

The proposed artificial boundary has a simple form and is easily applied to the finite element method. Since the boundary has a local approximate form both in the space and time domains, it has low computational cost and high computational efficiency. Effective calculation accuracy can be guaranteed when the artificial boundary is set at a sufficient distance from the internal source.

4. Numerical Studies

4.1. Example 1

The computational model of one-dimensional saturated soil is shown in Figure 2. Both lateral sides of the model are fixed impermeable boundaries. The top surface is a permeable boundary and is applied a uniform constant loading of 1.0 Pa. The models (a)–(c) have different bottom boundaries, such as remote boundary, fixed boundary, and artificial boundary, respectively. For the model (a), the truncated boundary is set far enough away from the load center, and within the effective calculation time, the reflected wave generated by the boundary cannot propagate to the concerned calculation area. The numerical solutions of the model (a) can be used as reference solutions for comparison. For model (b) and model (c), the truncation boundaries have the same distance from the load center and are much closer than the remote boundary. However, the truncation boundary of model (b) is a fixed boundary, while the truncated boundary of model (c) is set to the proposed artificial boundary. The material parameters of the model, taken from Simon [63] are shown in

Table 1. Material parameters of the linear elastic saturated soil.

Parameter	Value
ES	3000 Pa		Young’s modulus
R	0.306 kg/m3		Density of two-phase media
ρf	0.2977 kg/m3		Density of pore fluid
np	0.333		Porosity
C	0.2		Poisson’s ratio
kf	0.004883 m3s/kg		Dynamic permeability coefficient
L	833.3 Pa		Lame constant
G	1250 Pa		Modulus of shear
Material Number	Value 1	Value 2
Kf	0.3999 × 105 Pa	0.6106 × 105 Pa	Pore fluid volume modulus
Ks	∞	0.5005 × 104 Pa	Solid-phase soil skeleton volume modulus
Qb	0.1201 × 106 Pa	0.1385 × 105 Pa	Pore fluid compressibility coefficient
Wave Velocity
Actual Wave Velocity	Material No.
	1	2
Cp	635.12 m/s	176.15 m/s	P wave velocity
Cs	63.92 m/s	63.92 m/s	S wave velocity

for the values of Material 1. The analytical solutions of the one-dimensional problem of saturated soil proposed by Simon [63] are used as the reference solution and are compared with the results of the remote boundary and the fixed boundary to verify the correctness of the artificial boundary proposed in this paper.

The time-history responses of nodes 5 m and 45 m away from the load center are, respectively, selected for comparison. The time-history results of the displacement and pore pressure calculated by the three models are shown in Figure 3. It can be seen from the figure that the analytical solutions and the calculation results of the far boundary are almost completely resumed. Both results are used as the reference solutions. However, the difference between the responses of the fixed boundary and the reference solution is large, indicating that the bottom fixed boundary produces an unreal reflected wave propagating to the target node, which affects the real responses of the node. The vertical displacements and pore pressure of the model with the proposed boundary are consistent with the reference solutions, which verifies the correctness of the proposed boundary. Although the pore pressure response of the node at 5 m depth from the soil surface has some oscillations at the initial stage of the calculation, the oscillation gradually decreases as time goes on.

4.2. Example 2

The calculation model of saturated soil in two-dimensional semi-infinite space is shown in Figure 4. A semi-infinite domain model including the calculation domain with an intercepted area of 20 m × 20 m, as shown in Figure 4b. The top surface of the model is the drainage boundary and is loaded with locally uniform sine loading. The viscous spring boundaries proposed by Liu et al. [51], the remote boundary, fixed boundary, and paraxial boundary proposed by Akiyoshi et al. [48], as well as the boundary proposed in this paper, are respectively applied at the intercepted boundary. By comparing with the results of the artificial boundaries mentioned above, the accuracy and advantages of the proposed boundary are illustrated. The position of the remote boundary is selected according to the following method as an example. During the time 2.0 s of the sine loading, namely, the whole calculation time, the wave propagates at the faster velocity of P wave in the soil and is reflected by the remote boundaries. However, the reflected wave is not transmitted to the node of interest within the calculation time. The locations of the remote boundaries are set at the minimum computational domain range that does not affect the dynamic responses of the interest nodes. The material parameters in the example refer to Simon [63] and are shown in Table 1 for the values of Material No. 2.

In Figure 5, taking the numerical results of the model using the remote boundary as the reference solution, the displacement and pore pressure time histories of the fixed boundary are far from the results of the reference solution. The calculation results of the proposed boundary, the paraxial boundary, and the existing viscous spring boundaries [51] are in good agreement with the reference solutions. Moreover, the proposed boundary is more accurate than the viscous spring boundary [51], which shows the accuracy of the artificial boundary proposed in this paper.

Under the same calculation conditions and calculation method in time domain, the total computational times of the models with different boundaries mentioned above are shown in Figure 6. Among them, when the time step of the paraxial boundary is set to 5 × 10⁻⁴ s, the calculation does not converge. In the case of the remote boundary, the element number of the model is 4000, and the calculation time is 5841.81 s. Comparing several boundaries, the proposed boundary takes the shortest calculation time, which is 65.01 s. It can be seen that the proposed method has significant advantages in shortening the calculation time and improving the calculation efficiency.

4.3. Example 3

The development of displacement and pore pressure of saturated soil changed under point loading is analyzed by Example 3. A concentrated impulse loading acted on the free-draining surface of the computational model, as shown in Figure 7, and an area of 20 m × 20 m is intercepted by three different boundary conditions, namely the proposed boundary, the viscous spring boundary [51] and the fixed boundary. The changes of displacement and pore pressure in the calculation area at different time points are analyzed. Moreover, the correctness of the proposed method is further verified by comparing the three boundary conditions.

Figure 8 and Figure 9 show the distribution of displacement and pore pressure under three boundary conditions at 0.2 s, 0.5 s, and 1.0 s, respectively. From the figures, it can be seen that the influence area under the impulsive loading continues to increase with the calculation time. The formed displacement and pore pressure field continue to spread out from the point loading, and the displacement and pore pressure reach the maximum value at 0.5 s, which is the peak time of the impulsive loading. By comparing the numerical results of computational models considering the three kinds of boundary conditions, the numerical results of the proposed boundary and viscous spring boundary are the same, but the displacement and pore pressure results of the fixed boundary are completely different from that of the viscous spring boundaries. The forms of boundary conditions have a greater impact on the results of the interior domain.

5. Conclusions

For the energy radiation problem caused by the finite processing of the infinite model, a viscous spring boundary is proposed for the fluid-saturated porous media. The proposed method is illustrated the computational advantages and accuracy by theoretical method and numerical computation.

• The viscous spring boundary is composed of the stress and flow velocity boundary conditions, which are constructed by the reasonable outgoing wave assumption and Green’s function. The boundary has a simple form and clear physical meaning. Since the overall system equations considering artificial boundaries only need to change the values of the corresponding boundary nodes on the diagonal of the coefficient matrices, the boundary can be easily applied to the finite element method.
• Without the assumption of permeability and using the real wave velocity, the proposed boundary has higher accuracy than the existing boundaries with the assumption of impermeability.
• After considering the viscous spring boundary, the computational system is expressed as lumped mass equations with damping. A complete explicit integration algorithm with second-order accuracy is constructed to solve the equations.

The proposed method can be used to analyze the saturated soil-structure dynamic interaction in finite domain efficiently. The artificial boundary controls the number of degrees of freedom of the site model, and the explicit method considering the damping term also improves the calculation efficiency.