Continued-fraction-based High-order Radiation Boundary Conditions For Explicit Finite Element Analysis Of Wave Propagation Problem

The major civil infrastructure engineering under dynamic loading need to consider the dynamic interaction between structure and infinite foundation, leading to the wave propagation problems in infinite domain media. It can be solved using time-domain numerical method combining finite element method with artificial boundary condition. The explicit finite element method is suited to solve wave propagation problems. The explicit time integration algorithm of dynamic finite element equation should have second-order accuracy and be a single-step method. It should can consider the non-diagonal damping matrix, and have good stability. High-order time-domain artificial boundary conditions can accurately simulate the radiation damping characteristics of the truncate infinite domain. It is a hot and difficult issues in the field of engineering numerical computation. The current main difficulty is that the time convolution of high computation and storage costs arise when the artificial boundary conditions based on analytical solutions in frequency domain are transformed directly into the time domain. An existing solution is to approximate dynamic stiffness using a rational function in the frequency domain. The coefficients of the rational function are determined by using the least squares optimization algorithm. The dynamic stiffness relation described by the rational function is transformed into a temporal second-order dynamic equation system, which can be coupled with the finite element equation. The resulting equation can be solved by explicit time integration method. The deficiency of this solution is that the optimization algorithm is inefficient or even hard to converge when there are many rational functions coefficients. Recently the more effective method is using continued fraction to approximate dynamic stiffness. The coefficients of the continued fraction can be determined by solving the algebraic equations. The optimization algorithm is avoided. In this paper, the out-of-plane SH wave(two-dimensional scalar wave) problem in the half-space is considered. A new explicit time integration algorithm and a new dynamic stiffness continued fraction are proposed. The time-domain simulation method that combined explicit finite element with high-order artificial boundary condition are developed. The works are as follows.1. A new explicit time integration method to solve dynamic finite element equation is proposed. The displacement formula of this method is the second-order Taylor expansion of the former time response. The velocity formula is predictor-corrector form. The predictor velocity is the first order Taylor expansion of the response of the previous time. The predictor acceleration is the dynamic equation of predictor velocity at the present time. The correction velocity is the trapezoidal formula by using the predictor acceleration. The acceleration formula is the dynamic equation at the current time. The method is second-order explicit method when the mass matrix is diagonal, and it can consider the non-diagonal damping matrix and complex nonlinear internal force term. The method is suitable for variable time step size analysis and has the simple starting. The stability is better than existing several methods. By computing the linear and nonlinear of single and multiple degrees of freedom problems and comparing with the existing methods, the validity of the proposed method is verified. The FORTRAN finite element program of this method is written. Its correctness is verified by the comparison with the ABAQUS finite element software.2. A new dynamic-stiffness continued fraction is proposed, and the corresponding high-order time-domain artificial boundary condition is developed. The existing several high-order time-domain artificial boundary conditions are proved equivalent to the high-frequency continued fraction of dynamic stiffness in the frequency domain. The high-frequency continued fraction has inaccurate zero-frequency(static) stiffness. The doubly asymptotic continued fractions proposed by Song et al. have the unknown parameters that are determined difficultly. A continued fraction with an accurate zero-frequency stiffness is proposed. By transforming the continued fraction to rational function, the stability of the continued fraction is proved. Based on the method that transform the dynamic stiffness described by rational function to the temporally second-order dynamic equation system, the high-order time-domain artificial boundary condition is developed based on the dynamic-stiffness continued fraction. The equation coupling the boundary condition with finite element equations can be solved using the proposed explicit time integration method. By computing the numerical example of wave propagation problem and comparing with the results of finite element model with large domain, the effectiveness of the proposed high-order continued fraction artificial boundary condition is verified.