Artificial boundary condition for Klein-Gordon equation by constructing mechanics structure

An innovative local artificial boundary condition is proposed to numerically solve the Cauchy problem of the Klein-Gordon equation in an unbounded domain.Initially,the equation is considered as the axial wave prop-agation in a bar supported on a spring foundation.The numerical model is then truncated by replacing the half-infinitely long bar with an equivalent mechanical structure.The effective frequency-dependent stiffness of the half-infinitely long bar is expressed as the sum of rational terms using Pade approximation.For each term,a corresponding substructure composed of dampers and masses is constructed.Finally,the equivalent mechan-ical structure is obtained by parallelly connecting these substructures.The proposed approach can be easily implemented within a standard finite element framework by incorporating additional mass points and damper elements.Numerical examples show that with just a few extra degrees of freedom,the proposed approach effec-tively suppresses artificial reflections at the truncation boundary and exhibits first-order convergence.

3D viscous-spring artificial boundary in time domain

After a brief review of studies on artificial boundaries in dynamic soil-structure interaction, a three-dimensional viscous-spring artificial boundary （VSAB） in the time domain is developed in this paper. First, the 3D VSAB equations in the normal and tangential directions are derived based on the elastic wave motion theory. Secondly, a numerical simulation technique of wave motion equations along with the VSAB condition in the time domain is studied. Finally, numerical examples of some classical elastic wave motion problems are presented and the results are compared with the associated theoretical solutions, demonstrating that high precision and adequate stability can be achieved by using the proposed 3D VSAB. The proposed 3D VSAB can be conveniently incorporated in the general finite element program, which is commonly used to study dynamic soil-structure interaction problems.

A Method for Setting the Artificial Boundary Conditions of Groundwater Model

Numerical simulation technology is nowadays an important means for groundwater issues because of its efficiency and economical advantages. But in case of natural hydrogeological boundaries are not within the interest area, it may be a big trouble to set boundary conditions of the model artificially without enough field investigation information. This paper introduced a method for solving such problem applying field pumping test and recovery test. The method was applied to build an in-situ leaching of uranium model. Results showed that the model boundary conditions can be set satisfactorily, and also the calculated heads matched the observed data well in both two models.

New artificial boundary condition for saturated soil foundations

Anew artificial boundary model based on multi-directional transmitting and viscous-spring artificial boundary theories is proposed to absorb stress waves in a saturated soil foundation in dynamic analysis. Since shear waves （S-waves） are the same in a saturated soil foundation and a single-phase medium foundation, a tangential visco-elastic boundary condition for a single-phase medium foundation can also be used for saturated soil foundations. Thus, the purpose of the artificial boundary proposed in this paper is primarily to absorb two types of P-waves in a saturated soil foundation. The main idea is that the stress of the P-waves in the saturated soil foundation is decomposed into two types. The first type of stress, δra＇ is absorbed by the first artificial boundary. The second type of stress, δrb, is balanced by the stress generated by the second artificial boundary. Ultimately, both types of P-waves （fast-P-waves and slow-P-waves） are absorbed by the artificial boundary model proposed in this paper. In particular, note that the fast-P-waves and slow-P-waves are absorbed at the position of the first boundary. Thus, the artificial boundary model proposed herein can simultaneously absorb P-fast waves, P-slow waves and shear waves. Finally, a numerical example is given to examine the proposed artificial boundary model, and the results show that it is very accurate.

An Artificial Boundary Condition for the Vortex Movements in Two Dimensions

An approximate artificial boundary condition based on a boundary integral equa- tion is designed for the vortex movements. Point vortex and cloud in cell methods are used in numerical simulation of vortex motions. The numerical experiments show that the ap- proximate artificial boundary condition is useful and su?ciently accurate in hydrodynamics.

Application of the double absorbing boundary condition in seismic modeling

We apply the newly proposed double absorbing boundary condition（DABC）（Hagstrom et al., 2014） to solve the boundary reflection problem in seismic finite-difference（FD） modeling. In the DABC scheme, the local high-order absorbing boundary condition is used on two parallel artificial boundaries, and thus double absorption is achieved. Using the general 2D acoustic wave propagation equations as an example, we use the DABC in seismic FD modeling, and discuss the derivation and implementation steps in detail. Compared with the perfectly matched layer（PML）, the complexity decreases, and the stability and fl exibility improve. A homogeneous model and the SEG salt model are selected for numerical experiments. The results show that absorption using the DABC is considerably improved relative to the Clayton–Engquist boundary condition and nearly the same as that in the PML.

Model test to investigate reasonable reactive artificial boundary in shaking table test with a rigid container

When conducting dynamic tests of underground structure by a rigid container, reasonable boundary conditions are one of the essential factors related to the accuracy of test results, especially the artificial boundary perpendicular to the excitation direction. On the basis of numerous studies, shaking table tests with four different typical boundaries are performed in this study. The tests consider the seismic intensity and seismic wave types. Then, the simulation effects of the four boundary conditions are evaluated from four aspects as follows: the differential rate of peak acceleration, acceleration curve, similarity of Fourier frequency spectra, and uneven soil settlement in rigid containers. Results show that the simulation effects of the boundary conditions are not only affected by the nature of the boundary material but also related to the seismic intensity, types of seismic waves, and filter characteristic of the filling medium in containers. In comparison with the other three types of boundary condition, foamed polyethylene shows the best simulation effect and its effect decreases gradually with the increase in earthquake intensity. Finally, on the basis of existing studies, the evaluation criteria of boundary effect, the principle for the selection of boundary material type and the thickness of boundary material are discussed and summarized, and the corresponding design methods and suggestions are then provided.

ARTIFICIAL BOUNDARY METHOD FOR BURGERS＇ EQUATION USING NONLINEAR BOUNDARY CONDITIONS

This paper discusses the numerical solution of Burgers＇ equation on unbounded domains. Two artificial boundaries are introduced and boundary conditions are obtained on the artificial boundaries, which are in nonlinear forms. Then the original problem is reduced to an equivalent problem on a bounded domain. Finite difference method is applied to the reduced problem, and some numerical examples are given to show the effectiveness of the new approach.

APPROXIMATION, STABILITY AND FAST EVALUATION OF EXACT ARTIFICIAL BOUNDARY CONDITION FOR THE ONE-DIMENSIONAL HEAT EQUATION

In this paper we consider the numerical solution of the one-dimensional heat equation on unbounded domains. First an exact semi-discrete artificial boundary condition is derived by discretizing the time variable with the Crank-Nicolson method. The semi-discretized heat equation equipped with this boundary condition is then proved to be unconditionally stable, and its solution is shown to have second-order accuracy. In order to reduce the computational cost, we develop a new fast evaluation method for the convolution operation involved in the exact semi-discrete artificial boundary condition. A great advantage of this method is that the unconditional stability held by the semi-discretized heat equation is preserved. An error estimate is also given to show the dependence of numerical errors on the time step and the approximation accuracy of the convolution kernel. Finally, a simple numerical example is presented to validate the theoretical results.

THE APPROXIMATIONS OF THE EXACT BOUNDARY CONDITION AT AN ARTIFICIAL BOUNDARY FOR LINEARIZED INCOMPRESSIBLE VISCOUS FLOWS

We consider the linearized incompressible Navier-Stokes (Oseen) equations in a flat channel. A sequence of approximations to the exact boundary condition at an artificial boundary is derived. Then the original problem is reduced to a boundary value problem in a bounded domain, which is well-posed. A finite element approximation on the bounded domain is given, furthermore the error estimate of the finite element approximation is obtained. Numerical example shows that our artificial boundary conditions are very effective.

THE GLOBAL ARTIFICIAL BOUNDARY CONDITIONS FOR NUMERICAL SIMULATIONS OF THE 3D FLOW AROUND A SUBMERGED BODY

We consider the numerical approximations of the three-dimensional steady potential flow around a body moving in a liquid of finite constant depth at constant speed and distance below a free surface in a channel. One vertical side is introduced as the upstream artificial boundary and two vertical sides are introduced as the downstream artificial boundaries. On the artificial boundaries, a sequence of high-order global artificial boundary conditions are given. Then the original problem is reduced to a problem defined on a finite computational domain, which is equivalent to a variational problem. After solving the variational problem by the finite element method, we obtain the numerical approximation of the original problem. The numerical examples show that the artificial boundary conditions given in this paper are very effective.

Artificial Boundary Conditions for Nonlocal Heat Equations on Unbounded Domain

This paper is concerned with numerical approximations of a nonlocal heat equation define on an infinite domain.Two classes of artificial boundary conditions(ABCs)are designed,namely,nonlocal analog Dirichlet-to-Neumann-type ABCs(global in time)and high-order Pad´e approximate ABCs(local in time).These ABCs reformulate the original problem into an initial-boundary-value(IBV)problem on a bounded domain.For the global ABCs,we adopt a fast evolution to enhance computational efficiency and reduce memory storage.High order fully discrete schemes,both second-order in time and space,are given to discretize two reduced problems.Extensive numerical experiments are carried out to show the accuracy and efficiency of the proposed methods.

Split Local Artificial Boundary Conditions for the Two-Dimensional Sine-Gordon Equation on R^(2)

In this paper the numerical solution of the two-dimensional sine-Gordon equation is studied.Split local artificial boundary conditions are obtained by the operator splitting method.Then the original problem is reduced to an initial boundary value problem on a bounded computational domain,which can be solved by the finite differencemethod.Several numerical examples are provided to demonstrate the effectiveness and accuracy of the proposed method,and some interesting propagation and collision behaviors of the solitary wave solutions are observed.

Exact Artificial Boundary Condition for the Poisson Equation in the Simulation of the 2D Schrodinger-Poisson System

We study the computation of ground states and time dependent solutions of the Schr¨odinger-Poisson system(SPS)on a bounded domain in 2D(i.e.in two space dimensions).On a disc-shaped domain,we derive exact artificial boundary conditions for the Poisson potential based on truncated Fourier series expansion inθ,and propose a second order finite difference scheme to solve the r-variable ODEs of the Fourier coefficients.The Poisson potential can be solved within O(M NlogN)arithmetic operations where M,N are the number of grid points in r-direction and the Fourier bases.Combined with the Poisson solver,a backward Euler and a semi-implicit/leap-frog method are proposed to compute the ground state and dynamics respectively.Numerical results are shown to confirm the accuracy and efficiency.Also we make it clear that backward Euler sine pseudospectral(BESP)method in[33]can not be applied to 2D SPS simulation.

Mixed Finite Element Method and Higher-Order Local Artificial Boundary Conditions for Exterior 3-D Poisson Equation

The mixed finite element method is used to solve the exterior Poisson equations with higher-order local artificial boundary conditions in 3-D space. New unknowns are introduced to reduce the order of the derivatives of the unknown to two. The result is an equivalent mixed variational problem which was solved using bilinear finite elements. The primary advantage is that special finite elements are not needed on the adjacent layer of the artificial boundary for the higher-order derivatives. Error estimates are obtained for some local artificial boundary conditions with prescibed orders. A numerical example demonstrates the effectiveness of this method.

A POSTERIORI ERROR ESTIMATES OF A NON-CONFORMING FINITE ELEMENT METHOD FOR PROBLEMS WITH ARTIFICIAL BOUNDARY CONDITIONS

An a posteriori error estimator is obtained for a nonconforming finite element approximation of a linear elliptic problem, which is derived from a corresponding unbounded domain problem by applying a nonlocal approximate artificial boundary condition. Our method can be easily extended to obtain a class of a posteriori error estimators for various conforming and nonconforming finite element approximations of problems with different artificial boundary conditions. The reliability and efficiency of our a posteriori error estimator are rigorously proved and are verified by numerical examples.

Artificial Boundary Conditions for Time-Fractional Telegraph Equation

In this paper,we study the numerical solution of the time-fractional telegraph equation on the unbounded domain.We first introduce the artificial boundariesГ±to get a finite computational domain.On the artificial boundariesГ±,we use the Laplace transform to construct the exact artificial boundary conditions(ABCs)to reduce the original problem to an initial-boundary value problem on a bounded domain.In addition,we propose a finite difference scheme based on the L_(1−2)formule for the Caputo fractional derivative in time direction and the central difference scheme for the spatial directional derivative to solve the reduced problem.In order to reduce the effect of unsmoothness of the solution at the initial moment,we use a fine mesh and low-order interpolation to discretize the solution near t=0.Finally,some numerical results show the efficiency and reliability of the ABCs and validate our theoretical results.

无界区域上分数阶Klein-Gordon方程的近似人工边界条件

为了研究无界区域上时间分数阶Klein-Gordon方程,利用Laplace变换和Lagrange插值,将无界区域上时间分数阶Klein-Gordon方程近似转化为无界区域上整数阶偏微分方程。在此基础上,利用人工边界方法得到3种不同情形下无界区域上整数阶偏微分方程的人工边界条件,从而将无界区域上时间分数阶Klein-Gordon方程近似转化成有界区域上人工边界条件下整数阶偏微分方程的初边值问题,并证明了人工边界条件下整数阶偏微分方程的稳定性。最后,构造了人工边界条件下整数阶偏微分方程的有限差分格式,并通过数值例子验证该格式的有效性。

EXACT NONREFLECTING BOUNDARY CONDITIONS FOR AN ACOUSTIC PROBLEM IN THREE DIMENSIONS

In this paper, nonreflecting artificial boundary conditions are considered for an acoustic problem in three dimensions. With the technique of Fourier decomposition under the orthogonal basis of spherical harmonics, three kinds of equivalent exact artificial boundary conditions are obtained on a spherical artificial boundary. A numerical test is presented to show the performance of the method.

A survey on artificial boundary method Dedicated to Professor Shi Zhong-Ci on the Occasion of his 80th Birthday

The artificial boundary method is one of the most popular and effective numerical methods tor solving partial differential equations on unbounded domains, with more than thirty years development. The artificiM boundary method has reached maturity in recent years. It has been applied to various problems in scientific and engineering computations, and the theoretical issues such as the convergence and error estimates of the artificial boundary method have been solved gradually. This paper reviews the development and discusses different forms of the artificial boundary method.