地震学报2025，Vol.47Issue(4)：556-568,13.DOI:10.11939/jass.20240044
一维饱和两相介质地震反应的显式积分格式
Explicit integration formula for seismic response of one-dimensional saturated two-phase media
摘要
Abstract
The propagation of seismic waves in the saturated two-phase media and the dynamic response of two-phase media under seismic waves are important branches of dynamic research,which is of great significance in fields such as soil dynamics,geotechnical engineering,seis-mic engineering,geophysics,etc.The finite element method is usually used to solve the dynamic response of a saturated two-phase medium,especially for large,complex,strongly nonlinear,and low-frequency seismic dynamic response problems,and the key to the accuracy of finite element dynamic response calculation is the time-domain numerical integration format.Many researchers have proposed a series of fully explicit time-domain calculation methods for different saturated two-phase media models,such as the central difference combined with the Newmark constant average acceleration method,the central difference combined with the time-domain finite element method,and the central difference combined with the precise time-integration method,etc. Throughout the above research,this paper will further investigate the integral format that considers the acceleration input for solving the dynamic equations of saturated two-phase me-dia.The specific method is as follows.Firstly,the dynamic equations of one-dimensional com-pression and shear waves are established based on the soil mechanics model of a saturated two-phase medium.Secondly,for these dynamic equations,the Galerkin weighted residual method is used for spatial discretization to get the weak form of the discrete Galerkin.After that,the time domain step-by-step integration method that combines central difference method with the Newmark linear acceleration method(γ=1/2,β=1/6)is used for time discretization.The cent-ral difference method is used to solve the solid-fluid phase displacement,of which any node i at a discrete time p+1 can be represented by the velocities and displacements at the time p.The Newmark linear acceleration method is used to calculate the velocities,of which any node i ata discrete time p+1 can be calculated from the acceleration,velocity,and displacement at the time p and the displacement at the time p+1.Then the dynamic balance equation method calcu-lates the corresponding accelerations from the displacements and velocities at the time p+1.To this extent,the time-domain recursive formulas for the dynamic response of internal nodes have been established.Thirdly,the multi-transmitting artificial boundary is introduced to limit the finite element research area.The dynamic response of artificial boundary nodes in the two-phase media is obtained by interpolating the dynamic response of internal nodes.So far the near-field wave motion problem in the saturated two-phase media can be solved by combining the time domain recurrence formulas with the multi-transmitting artificial boundary.Finally the rationality of the algorithm is verified by using two numerical examples,one is internal wave,and the other is external scattering.The former compares the numerical solution of one-dimen-sional soil column under step load and sinusoidal load with the corresponding analytical solu-tion,and the calculation results of the two are consistent.The latter uses the numerical algorithm in this paper to calculate the seismic response of a saturated half-space.Namely the elastic dynamic response of the two-phase media at the vertical incidence of the Wolong wave is calculated and analyzed by using this method.The time history curves of displacement,velo-city,and acceleration of four points at different depths(e.g.,A,B,C,and D points)are drawn.The calculated results are in accordance with the elastic wave theory.This is sufficient to prove the correctness and effectiveness of the explicit finite element method derived in this paper.The following conclusions are obtained:①The time-history curves of displacement,velocity,and acceleration in solid and liquid phases are the same when the seismic wave is incident vertically in the form of a shear wave.The waveforms of the displacement,velocity,and acceleration are the same as the corresponding waveforms of the input seismic wave,but the peak values differ;②The peak displacement of the point A at the free surface of the two-phase medium is 15.558 cm and-19.34 cm,which is close to twice that of the incident seis-mic wave such as 7.958 cm and-9.754 cm.This conforms to the theory of elastic waves in two-phase media and proves the applicability of the explicit finite element method in solving the elastic seismic response of saturated two-phase media in this paper;③The elastic dynamic response of the two-phase media solved in this paper decreases with the decrease in the intensity of input seismic wave field because the solid phase constitutive relationship is linear elastic.This provides an effective method for solving the structural dynamic and wave equations,and also lays a foundation for future research on the numerical simulation of nonlinear waves in two-phase media.关键词
饱和两相介质/土力学模型/显式有限元方法/纽马克(Newmark)线性加速度法/多次透射人工边界Key words
saturated two-phase medium/soil mechanics model/explicit finite element method/Newmark linear acceleration method/multi-transmitting artificial boundary分类
天文与地球科学引用本文复制引用
张波,陈学良,董捷,李铁飞..一维饱和两相介质地震反应的显式积分格式[J].地震学报,2025,47(4):556-568,13.基金项目
国家重点研发计划(2019YFC1509403,2023YFC3007305)、国家自然科学基金(52478566,51978633)、中国地震局地球物理研究所基本科研业务费专项(DQJB24Z01,DQJB24R31)、河北省教育厅科学研究项目(QN2025419)和河北省省属高等学校基本科研业务费研究项目(2022QNJS15)共同资助. （2019YFC1509403,2023YFC3007305）
