Study of an artificial boundary condition based on the damping-solvent extraction method

Abstract

A new artificial boundary condition for time domain analysis of a structure-unlimited-foundation system was proposed. The boundary condition was based on the damping-solvent extraction method. The principle of the damping-solvent extraction method was described. An artificial boundary condition was then established by setting two spring-damper systems and one artificial damping limited region. A test example was developed to verify that the proposed boundary condition and model had high precision. Compared with the damping-solvent extraction method, this boundary condition is easier to be applied to finite element method (FEM)-based numerical calculations.

Keywords

damping-solvent extraction method / structure-unlimited-foundation system / spring-damper system / artificial damping limited region / finite element method

Introduction

The simulation of infinite foundations was one of the core issues for structure-foundation dynamic interaction analysis. The most commonly used technique for applying a finite element method (FEM) while taking a semi-infinite medium into account was setting an artificial boundary condition on intercepted finite fields. Artificial boundary conditions [1] included local artificial boundary conditions, such as viscous artificial boundary conditions, viscous-spring artificial boundary conditions, paraxial artificial boundary conditions, multi-directional transmitting artificial boundary conditions, superposition artificial boundary conditions, transmitting artificial boundary conditions and those used in the damping solvent extraction method (DSEM), and global artificial boundary conditions, such as those used in the boundary element method (BEM), the infinite element method (IEM), and the scaled boundary finite element method (SBFE). The viscous artificial boundary condition proposed by Lysmer and Kuhlemeyer [2,3] was based on unidirectional wave theory and was the simplest artificial boundary condition to use. However, the direction of stress wave propagation was typically not unidirectional, which led to significant calculation errors when a viscous artificial boundary condition was used in a dynamic analysis [4]. The viscous-spring artificial boundary condition proposed by Deeks [5] was based on cylindrical wave theory and could simulate geometric attenuation in diffusion processes and help solve low-frequency stability problems. Some studies [6-8] showed that the viscous-spring artificial boundary condition was superior to the viscous artificial boundary condition. However, the classical viscous-spring artificial boundary condition ignored the influence of the lateral boundary displacement in tangential direction and so missed the turbulence effects. The paraxial artificial boundary condition proposed by Clayton and Engquist [9,10] was based on acoustic wave equation theory and had high calculation precision for systems with incident waves with incident angles with measures less than 45 degrees. However, the method had low calculation precision for systems with incident waves with incident angles with measures greater than 45 degrees. The multi-directional transmitting boundary condition proposed by Higdon [11] was based on the work of Clayton and Engquist and could handle systems with incident waves having differing angles; however, the resulting model was too complicated to be readily embedded in FEM code [12].The superposition artificial boundary condition was proposed by Smith [13,14] and was based on the principle of virtual images, The superposition artificial boundary condition had a high calculation precision for all incident waves but a low calculation efficiency [15]. The DSEM proposed by Wolf and Song [16] made use of artificial damping to weaken the incident wave energy, but the method affected the calculation of dynamic system characteristics when used with finite fields. As was the case for the paraxial artificial boundary condition, the transmitting boundary condition proposed by Liao [17,18] had a high calculation precision and efficiency for incident waves with incident angles with measures less than 45 degrees but a low calculation precision for incident waves with incident angles with measures greater than 45 degrees.

In this paper, an artificial boundary condition was proposed based on the damping-solvent extraction method, but the condition was designed to make the method easy to apply in finite element method (FEM)-based numerical calculations.

Brief description of the damping solvent extraction method

We considered a structure such as a gravity dam in the time domain with linearly elastic material, as shown in Fig. 1. The motion equation of the structure was written as

(1)\left[\begin{array}{cc}{\tilde{M}}_{ss}& \mathrm{0}\\ \mathrm{0}& {\tilde{M}}_{bb}^s\end{array}\right]\left\{\begin{array}{c}{\ddot{u}}_s^t\\ {\ddot{u}}_b^t\end{array}\right\}+\left[\begin{array}{cc}{\tilde{C}}_{ss}& {\tilde{C}}_{sb}\\ {\tilde{C}}_{bs}& {\tilde{C}}_{bb}^s\end{array}\right]\left\{\begin{array}{c}{\dot{u}}_s^t\\ {\dot{u}}_b^t\end{array}\right\}+\left[\begin{array}{cc}{\tilde{K}}_{ss}& {\tilde{K}}_{sb}\\ {\tilde{K}}_{bs}& {\tilde{K}}_{bb}^s\end{array}\right]\left\{\begin{array}{c}{u}_s^t\\ u_b^t\end{array}\right\}=\left\{\begin{array}{c}\mathrm{0}\\ -R_b(t)\end{array}\right\}+\left\{\begin{array}{c}{P}_s\\ P_b\end{array}\right\},

where [\tilde{M}] , [\tilde{C}] and [\tilde{K}] were the mass, damping and stiffness matrices of the structure, respectively; \left\{{\ddot{u}}^t\right\} , \left\{{\dot{u}}^t\right\} and \{u^t\} were the acceleration, velocity and displacement of the structure, respectively; {P} was the equivalent nodal force vector for the load with the seismic load omitted; and {R_b(t)} was the interaction force between the structure and the unlimited foundation. The subscript b represented the nodes on the junction surface, while the subscript s represented the other nodes in the structure.

The motion equation for the unlimited foundation in the time domain was written as

(2)\left[\begin{array}{cc}{\overline{M}}_{nn}^{\infty }& \mathrm{0}\\ \mathrm{0}& {\overline{M}}_{bb}^{\infty f}\end{array}\right]\left\{\begin{array}{c}{\ddot{u}}_n\\ {\ddot{u}}_b\end{array}\right\}+\left[\begin{array}{cc}{\overline{C}}_{nn}^{\infty }& {\overline{C}}_{nb}^{\infty }\\ {\overline{C}}_{bn}^{\infty }& {\overline{C}}_{bb}^{\infty f}\end{array}\right]\left\{\begin{array}{c}{\dot{u}}_n\\ {\dot{u}}_b\end{array}\right\}+\left[\begin{array}{cc}{\overline{K}}_{nn}^{\infty }& {\overline{K}}_{nb}^{\infty }\\ {\overline{K}}_{bn}^{\infty }& {\overline{K}}_{bb}^{\infty f}\end{array}\right]\left\{\begin{array}{c}{u}_n\\ u_b\end{array}\right\}=\left\{\begin{array}{c}\mathrm{0}\\ R_b(t)\end{array}\right\},

where \left[{\overline{M}}^{\infty }\right] , \left[{\overline{C}}^{\infty }\right] and \left[{\overline{K}}^{\infty }\right] were the mass, damping and stiffness matrices of the unlimited foundation, respectively, and \{\ddot{u}\} , \{\dot{u}\} and \{u\} were the acceleration, velocity and displacement of the unlimited foundation, respectively. The subscript n represented nodes that were not located on the junction surface between the structure and the unlimited foundation.

The motion equation for the limited region near the structure shown in Fig. 1 was written as

(3)\left[\begin{array}{cc}{\overline{M}}_{mm}& \mathrm{0}\\ \mathrm{0}& {\overline{M}}_{bb}^f\end{array}\right]\left\{\begin{array}{c}{\ddot{u}}_m\\ {\ddot{u}}_b\end{array}\right\}+\left[\begin{array}{cc}{\overline{C}}_{mm}& {\overline{C}}_{mb}\\ {\overline{C}}_{bm}& {\overline{C}}_{bb}^f\end{array}\right]\left\{\begin{array}{c}{\dot{u}}_m\\ {\dot{u}}_b\end{array}\right\}+\left[\begin{array}{cc}{\overline{K}}_{mm}& {\overline{K}}_{mb}\\ {\overline{K}}_{bm}& {\overline{K}}_{bb}^f\end{array}\right]\left\{\begin{array}{c}{u}_m\\ u_b\end{array}\right\}=\left\{\begin{array}{c}\mathrm{0}\\ {\overline{R}}_b(t)\end{array}\right\},

where [\overline{M}] , [\overline{C}] and [\overline{K}] were the mass, damping and stiffness matrices of the limited region near the structure, respectively, and \{\ddot{u}\} , \{\dot{u}\} and \{u\} were the acceleration, velocity and displacement of the limited region near the structure, respectively. The subscript m represented the nodes not located on the junction surface.

When using an additional damping matrix \mathrm{2}\varsigma [\overline{M}] and an additional stiffness matrix {\varsigma }^{\mathrm{2}}[\overline{M}] , the motion equation for the artificial damping limited region near the structure was rewritten as

(4)\left[\begin{array}{cc}{M}_{mm}& \mathrm{0}\\ \mathrm{0}& M_{bb}^f\end{array}\right]\left\{\begin{array}{c}{\ddot{u}}_m\\ {\ddot{u}}_b\end{array}\right\}+\left[\begin{array}{cc}{C}_{mm}& C_{mb}\\ C_{bm}& C_{bb}^f\end{array}\right]\left\{\begin{array}{c}{\dot{u}}_m\\ {\dot{u}}_b\end{array}\right\}+\left[\begin{array}{cc}{K}_{mm}& K_{mb}\\ K_{bm}& K_{bb}^f\end{array}\right]\left\{\begin{array}{c}{u}_m\\ u_b\end{array}\right\}=\left\{\begin{array}{c}\mathrm{0}\\ R_{\varsigma }(t)\end{array}\right\},

in which

(5)[M]=[\overline{M}],[C]=[\overline{C}]+\mathrm{2}\varsigma [\overline{M}],[K]=[\overline{K}]+{\varsigma }^{\mathrm{2}}[\overline{M}],

In Eqs. 4 and 5, [M], [C] and [K] were the mass, damping and stiffness matrices for the artificial damping limited region near the structure, respectively, and {R_ζ(t)} was the mutual force vector between the structure and the artificial damping limited region near the structure. The dimension of the damping coefficient ζ was the same as c_s/r₀ where c_s was the shear wave velocity and r₀ was the characteristic length. It was possible that ζ could be larger than 1. The spring and damper were installed on a unit area of the external boundary. The stiffness and damping were given by ζρc and ρc, respectively, where c was the compression velocity c_p when the wave was compression and was the shear velocity c_s when the wave was shear. Eqs. (4) and (5) constituted one of the two spring-damper systems.

In the artificial damping limited region near the structure, artificially high damping was used to consume the energy of vibrational waves. This meant that the artificial damping limited region was equivalent to an unlimited foundation. Without considering the original damping of the limited region, the relation between the frequency-domain stiffness matrix [Y^∞] for an unlimited foundation and [Y_ζ] for the artificial damping limited region near the structure was written as

(6)[Y^{\infty }(\omega )]={\displaystyle \frac{G}{G^{*}}}\left\{[Y_{\varsigma }(\omega )]+{[Y_{\varsigma }(\omega )]}_{,\omega }{\displaystyle \frac{(a_{\mathrm{0}}-a_{\mathrm{0}}^{*})}{a_{\mathrm{0},\omega }^{*}}}\right\},

where G and G* were the shear modulus for an unlimited foundation and the shear modulus for the artificial damping limited region, respectively, and a₀ and a_0^{*} were dimensionless frequency parameters for the unlimited foundation and the artificial damping limited region, respectively. The parameters a₀ and a_0^{*} were given by

(7)a_{\mathrm{0}}=\omega r_{\mathrm{0}}/c_s,a_{\mathrm{0}}^{*}=\omega r_{\mathrm{0}}/c_s^{*},

where c_s and c_s^{*} were the shear velocities for the unlimited foundation and the artificial damping limited region, respectively.

The frequency-domain stiffness matrix [Y_ζ] for the artificial damping limited region near the structure was written as

(8)[Y_{\varsigma }(\omega )]=\left([\overline{K}]+{\varsigma }^{\mathrm{2}}[\overline{M}]\right)+\mathrm{2}i\omega \varsigma [\overline{M}]-{\omega }^{\mathrm{2}}[\overline{M}]=[\overline{K}]-{(\omega -i\varsigma )}^{\mathrm{2}}[\overline{M}],

(9)a_{\mathrm{0}}^{*}=(\omega -i\varsigma )r_{\mathrm{0}}/c_s,

Substituting Eqs. (8) and (9) into (6) and setting G = G*,the frequency-domain stiffness matrix [Y^∞] for an unlimited foundation was written as

(10)[Y^{\infty }(\omega )]=[Y_{\varsigma }(\omega )]+i\varsigma {[Y_{\varsigma }(\omega )]}_{,\omega },

The relation between the frequency-domain and time-domain stiffness matrices was written as

(11)[Y_{\varsigma }(\omega )]={\displaystyle {\int }_{-\infty }^{\infty }[S_{\varsigma }(t)]}{\mathrm{e}}^{-\mathrm{i}\omega t}\mathrm{d}t,

(12)[Y^{\infty }(\omega )]={\displaystyle {\int }_{-\infty }^{\infty }[S^{\infty }(t)]}{\mathrm{e}}^{-\mathrm{i}\omega t}\mathrm{d}t,

where [S^∞(t)] and [S_ζ(t)] were the time-domain stiffness matrices for the unlimited foundation and the artificial damping limited region near the structure, respectively.

After applying a Fourier transform, the relation between [S^∞(t)] and [S_ζ(t)] was written as

(13)[S^{\infty }(t)]=(\mathrm{1}+\varsigma t)[S_{\varsigma }(t)],

The relation between {R_b(t)} and [S^∞(t)] was written as

(14)\left\{\begin{array}{c}\mathrm{0}\\ R_b(t)\end{array}\right\}={\displaystyle {\int }_{\mathrm{0}}^t[S^{\infty }(t-\tau )]}\{u(\tau )\}d\tau ,

Substituting Eq. (13) into (14), the relation between {R_b(t)} and {R_ζ(t)} became

(15)\{R_b(t)\}=\{R_{\varsigma }(t)\}+\varsigma \{R_r(t)\},

in which

(16)\{R_{\varsigma }(t)\}={\displaystyle {\int }_{\mathrm{0}}^t[S_{\varsigma }(t-\tau )]}\{u(\tau )\}\mathrm{d}\tau ,

(17)\{R_r(t)\}={\displaystyle {\int }_{\mathrm{0}}^t[S_{\varsigma }(t-\tau )]}(t-\tau )\{u(\tau )\}\mathrm{d}\tau ,

Establishing an artificial boundary condition based on the damping-solvent extraction method

Through the above-described analysis, it could be observed that Eqs. (15)- (17) could describe the interaction between a structure and an unlimited-foundation in the time domain. Based on the above discussion, an artificial boundary condition was established. Derivation of the required equations was performed as shown below.

Numerical implementation for {R_ζ(t)}

Eqs. (4) and (5) showed that {R_ζ(t)}was the force determined by Eq. (4). To implement Eq. (4), [\overline{M}] was taken to be a diagonal lumped mass matrix. Because additional damping and stiffness for nodes of the artificial damping limited region were not coupled, a spring-damper system could be applied to create the system determined by Eq. (4). The normal and tangential damping and stiffness for a spring-damper system for these nodes were given as

(18)k_{ix}=k_{iy}={\varsigma }^{\mathrm{2}}{m}_i,c_{ix}=c_{iy}=\mathrm{2}\varsigma m_i,

where m_i was the concentrated mass for node i.

Numerical implementation for {R_r(t)}

The major derivation for {R_r(t)} was performed by rewriting Eqs. (16) and (17) as

(19)\left\{\begin{array}{c}\mathrm{0}\\ R_{\varsigma }(t)\end{array}\right\}={\displaystyle {\int }_{\mathrm{0}}^t[S_{\varsigma }(\theta )]}\{u(t-\theta )\}\mathrm{d}\theta =[K]\{u(t)\}+\mathrm{2}\varsigma [\overline{M}]\left\{\dot{u}(t)\right\}+[\overline{M}]\left\{\ddot{u}(t)\right\},

(20)\left\{\begin{array}{c}\mathrm{0}\\ R_r(t)\end{array}\right\}={\displaystyle {\int }_{\mathrm{0}}^t[S_{\varsigma }(\theta )]}\theta \{u(t-\theta )\}\mathrm{d}\theta ,

The first time derivative for Eq. (20) was written as

(21){\displaystyle \frac{\mathrm{d}\left\{\begin{array}{c}\mathrm{0}\\ R_r(t)\end{array}\right\}}{\mathrm{d}t}}={\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}{\displaystyle {\int }_{\mathrm{0}}^t[S_{\varsigma }(\theta )]}\theta \{u(t-\theta )\}\mathrm{d}\theta ={\displaystyle {\int }_{\mathrm{0}}^t[S_{\varsigma }(\theta )]}\theta {\displaystyle \frac{\partial }{\partial t}}\{u(t-\theta )\}\mathrm{d}\theta +[S_{\varsigma }(t)]t\{u(\mathrm{0})\},

At the initial time

(22)\{u(\mathrm{0})\}=\{\mathrm{0}\},

and {u(t)} was therefore written as

(23)\{u(t)\}\approx \{u(t-\theta )\}+{\displaystyle \frac{\partial }{\partial t}}\{u(t-\theta )\}(t-(t-\theta )),

Equation (23) was rewritten as

(24){\displaystyle \frac{\partial }{\partial t}}\{u(t-\theta )\}\theta =\{u(t)\}-\{u(t-\theta )\},

Eqs. (22) and (24) were substituted into Eq. (21) to obtain

(25){\displaystyle \frac{\mathrm{d}\left\{\begin{array}{c}\mathrm{0}\\ R_r(t)\end{array}\right\}}{\mathrm{d}t}}=\{u(t)\}{\displaystyle {\int }_{\mathrm{0}}^t[S_{\varsigma }(\theta )]}\mathrm{d}\theta -{\displaystyle {\int }_{\mathrm{0}}^t[S_{\varsigma }(\theta )]}\{u(t-\theta )\}\mathrm{d}\theta ,

When ω = 0, Eqs. (5) and (11) gave

(26){\displaystyle {\int }_{-\infty }^{\infty }[S_{\varsigma }(\theta )]}\mathrm{d}\theta =[Y_{\varsigma }(\mathrm{0})]=[\overline{K}]+{\varsigma }^{\mathrm{2}}[\overline{M}]=[K],

When time t was long, Eq. (26) gave

(27){\displaystyle {\int }_{\mathrm{0}}^t[S_{\varsigma }(\theta )]}\mathrm{d}\theta \approx {\displaystyle {\int }_{-\infty }^{\infty }[S_{\varsigma }(\theta )]}\mathrm{d}\theta =[K],

Eqs. (19) and (27) were substituted into Eq. (25) to obtain

(28){\displaystyle \frac{\mathrm{d}\left\{\begin{array}{c}\mathrm{0}\\ R_r(t)\end{array}\right\}}{\mathrm{d}t}}=[K]\{u(t)\}-\left([K]\{u(t)\}+\mathrm{2}\varsigma [\overline{M}]\left\{\dot{u}(t)\right\}+[\overline{M}]\left\{\ddot{u}(t)\right\}\right)=-\mathrm{2}\varsigma [\overline{M}]\left\{\dot{u}(t)\right\}-[\overline{M}]\left\{\ddot{u}(t)\right\},

Because [\overline{M}] and 2ζ [\overline{M}] were diagonal matrices, Eq. (28) became

(29){\displaystyle \frac{\mathrm{d}\{R_r(t)\}}{\mathrm{d}t}}=-\left(\mathrm{2}\varsigma \left[{\overline{M}}_{bb}\right]\right)\left\{{\dot{u}}_{bb}(t)\right\}-\left[{\overline{M}}_{bb}\right]\left\{{\ddot{u}}_{bb}(t)\right\},

Integrating Eq. (29) gave

(30)\{R_r(t)\}=-\left(\mathrm{2}\varsigma \left[{\overline{M}}_{bb}\right]\right)\{u_{bb}(t)\}-\left[{\overline{M}}_{bb}\right]\left\{{\dot{u}}_{bb}(t)\right\},

Eqs. (18) and (30) showed that the normal and tangential damping and stiffness for a spring-damper system for nodes on the junction surface between the structure and the artificial damping limited region were

(31)k_{bix}=k_{biy}=({\varsigma }^{\mathrm{2}}+\mathrm{2}\varsigma )m_{bi},c_{bix}=c_{biy}=(\mathrm{2}\varsigma +\mathrm{1})m_{bi},

The Eq. (31) results were the second spring-damper system and were shown in Fig. 2.

Numerical example

A plane-strain model for a rock foundation was adopted as was shown as Fig. 4. The elastic modulus, Poisson ratio and density of rock foundation were 3.90 × 10¹⁰ Pa, 0.3 and 2700 kg/m³, respectively. A step load, an impact load and a harmonic periodic load (5 Hz) were applied to the local part of the model, as shown in Fig. 3. The damping coefficients for the artificial damping limited region were taken as ζ₁ = 0.6, ζ₂ = 0.8 and ζ₃ = 1.5 for comparison. The Newmark method (α = 0.5, β= 0.25) was used in the time-domain analysis.

The vertical displacement at points A, B and C under step loads were shown as Figs. 5, 6 and 7, respectively, while the vertical displacement at points A, B and C under impact loads were shown as Figs. 8, 9 and 10, respectively. The vertical displacement at points A, B and C under harmonic periodic loads (5 Hz) were shown as Figs. 11, 12 and 13, respectively.

Discussion

An analysis of Figs. 5-13 demonstrated that under step loads, the vertical displacement was very close to the reference value, while under impact loads, the vertical displacement was slightly higher than the reference value. With the harmonic period loads, the vertical displacement was slightly higher than the reference value during the first period and was very close to the reference value at other times. These results indicate that the artificial boundary had sufficiently high precision to meet general engineering requirements.

The boundary condition used in this work was primarily designed for a homogeneous unlimited foundation. Consequently, modifications were required to use the boundary condition for a non-homogeneous case. Using the Mori-Tanaka model, the bulk modulus K and shear modulus G of an equivalent homogeneous unlimited foundation were obtained as

(32){\displaystyle \frac{K}{K_{\mathrm{0}}}}=\mathrm{1}+{\displaystyle \frac{v_{\mathrm{1}}(K_{\mathrm{1}}-K_{\mathrm{0}})}{K_{\mathrm{0}}+(\mathrm{1}-v_{\mathrm{1}})\left[{\displaystyle \frac{K_{\mathrm{1}}-K_{\mathrm{0}}}{K_{\mathrm{0}}+\mathrm{4}{G}_{\mathrm{0}}/\mathrm{3}}}\right]K_{\mathrm{0}}}},

(33){\displaystyle \frac{G}{G}}=\mathrm{1}+{\displaystyle \frac{v_{\mathrm{1}(G_{\mathrm{1}}-G_{\mathrm{0}})}}{G_{\mathrm{0}}+(\mathrm{1}-v_{\mathrm{1}})\left[{\displaystyle \frac{G_{\mathrm{1}}-G_{\mathrm{0}}}{\mathrm{1}+{\displaystyle \frac{\mathrm{9}{K}_{\mathrm{0}}+\mathrm{8}{G}_{\mathrm{0}}}{\mathrm{6}(K_{\mathrm{0}}+\mathrm{2}{G}_{\mathrm{0}})}}}}\right]}},

where K₀ and G₀ were the bulk modulus and the shear modulus of rock, respectively; K₁ and G₁ were the bulk modulus and shear modulus of the weak plane, respectively; and v₁ was the volume percentage of the weak plane in the rock foundation.

Conclusion

An artificial boundary condition was proposed based on the damping-solvent extraction method to make damping-solvent extraction method easily used in finite element method (FEM)-based numerical calculations. A test example was given to verify that the proposed model had a sufficiently high precision to meet general engineering requirements.

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