Evaluation On Effectiveness Of Continued Fraction Expansion For Dynamic Stiffness Of Infinite Domain

The problem of initial-boundary value in the truncated infinite domain is needed to solve to obtain the dynamic stiffness relationship between force and displacement for the analysis of soil-structure dynamic interaction.Dynamic stiffness in the truncated infinite domain and its time-domain realization forms are high accuracy artificial boundary conditions in frequency domain and time domain,respectively.Continued fraction expansion is developed to express the dynamic stiffness in recent years.Firstly,continued fraction expansion is able to be used as frequency domain artificial boundary condition.Secondly,continued fraction expansion can be converted as time domain boundary condition by introducing auxiliary variable.However,continued fraction expansion has not been used widely at present because its development time is short.So we need further study the effectiveness of continued fraction expansion.The effectiveness can be divided into three classifications:(1)whether the continued fraction can express the dynamic stiffness,namely whether the coefficient of continued fraction can be solved and whether the continued fraction has sufficient accuracy.(2)whether time domain artificial boundary condition based on the continued fraction is stable.(3)whether the coupled equation of artificial boundary condition and finite element method is stable.We study the effectiveness of continued fraction expansion for scalar wave in horizontal layered medium and in radial layered medium based on semi-discrete numerical method.Specific research work is as follows:1.Continued fraction of dynamic stiffness matrixFor horizontal layered problem,the new continued fraction proposed by the research group is extended in scalar form to matrix form.The derivations of coefficient matrix of new continued fraction,high-frequency continued fraction and doubly asymptotic continued fraction are given.For radial layered problem,the derivation of coefficient matrix of high-frequency continued fraction is given.2.Time domain artificial boundary condition and its stability,coupled equation of and artificial boundary condition and finite element method and its stabilityFor the problems of horizontal layered and radial layered,by introducing auxiliary variables and Fourier transform,the dynamic stiffness based on continued fraction is equivalently converted into first order ordinary differential equations in time,namely time domain artificial boundary condition.The coupled equation of time domain artificial boundary condition and finite element method is derived.We reference stability theory of ordinary differential equations and give the theories to evaluation on the stability of time domain artificial boundary condition and of the coupled equations.3.The study of effectiveness of continued fraction expansion in dynamic stiffness through numerical testsFor the problems of horizontal layered and radial layered,the effectiveness of continued fraction expansion in dynamic stiffness is studied through numerical tests.Numerical results show that the continued fraction expansion is effective for both horizontal and radial single-layered problems,but only a small number of lower orders of continuous fraction expansion is effective for both horizontal and radial multi-layered problems.