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.

FINITE ELEMENT-ARTIFICIAL TRANSMITTING BOUNDARY METHOD FOR WAVE SCATTERING FROM IRREGULAR CYLINDER

The finite element artificial transmitting boundary method is employed here to treat the near field scattering of a cylindrical wave from an irregular cylinder. A comparison is made between this method and the analytical one. And then examples are given to demonstrate the solution of several problems of the irregular object scattering. The method can not only produce clear physical pictures, but can efficiently handle many complicated scattering problems.

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.

BOUNDARY ELEMENT ANALYSIS OF CONTACT PROBLEMS USING ARTIFICIAL BOUNDARY NODE APPROACH

An improved version of the regular boundary element method, the artificial boundary node approach, is derived. A simple contact algorithm is designed and implemented into the direct boundary element, regular boundary element and artificial boundary node approaches. The exisiting and derived approaches are tested using some case studies. The results of the artificial boundary node approach are compared with those of the existing boundary element program, the regular element approach, ANSYS and analytical solution whenever possible. The results show the effectiveness of the artificial boundary node approach for a wider range of boundary offsets.

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.

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.

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 conditions for Euler–Bernoulli beam equation

In a semi-discretized Euler-Bernoulli beam equa- tion, the non-nearest neighboring interaction and large span of temporal scales for wave propagations pose challenges to the effectiveness and stability for artificial boundary treat- ments. With the discrete equation regarded as an atomic lattice with a three-atom potential, two accurate artificial boundary conditions are first derived here. Reflection co- efficient and numerical tests illustrate the capability of the proposed methods. In particular, the time history treatment gives an exact boundary condition, yet with sensitivity to nu- merical implementations. The ALEX （almost EXact） bound- ary condition is numerically more effective.

Seismic wave input method for three-dimensional soil-structure dynamic interaction analysis based on the substructure of artificial boundaries

The method of inputting the seismic wave determines the accuracy of the simulation of soil-structure dynamic interaction. The wave method is a commonly used approach for seismic wave input, which converts the incident wave into equivalent loads on the cutoff boundaries. The wave method has high precision, but the implementation is complicated, especially for three-dimensional models. By deducing another form of equivalent input seismic loads in the fi nite element model, a new seismic wave input method is proposed. In the new method, by imposing the displacements of the free wave fi eld on the nodes of the substructure composed of elements that contain artifi cial boundaries, the equivalent input seismic loads are obtained through dynamic analysis of the substructure. Subsequently, the equivalent input seismic loads are imposed on the artifi cial boundary nodes to complete the seismic wave input and perform seismic analysis of the soil-structure dynamic interaction model. Compared with the wave method, the new method is simplifi ed by avoiding the complex processes of calculating the equivalent input seismic loads. The validity of the new method is verifi ed by the dynamic analysis numerical examples of the homogeneous and layered half space under vertical and oblique incident seismic waves.

Comparison of perfectly matched layer and multi-transmitting formula artificial boundary condition based on hybrid finite element formulation

The theory of perfectly matched layer （PML） artificial boundary condition （ABC）, which is characterized by absorption any wave motions with arbitrary frequency and arbitrarily incident angle, is introduced. The construction process of PML boundary based on elastodynamic partial differential equation （PDE） system is developed. Combining with velocity-stress hybrid finite element formulation, the applicability of PML boundary is investigated and the numerical reflection of PML boundary is estimated. The reflectivity of PML and multi-transmitting formula （MTF） boundary is then compared based on body wave and surface wave simulations. The results show that although PML boundary yields some reflection, its absorption performance is superior to MTF boundary in the numerical simulations of near-fault wave propagation, especially in comer and large angle grazing incidence situations. The PML boundary does not arise any unstable phenomenon and the stability of PML boundary is better than MTF boundary in hybrid finite element method. For a specified problem and analysis tolerance, the computational efficiency of PML boundary is only a little lower than MTF boundary.

Stability conditions of explicit integration algorithms when using 3D viscoelastic artificial boundaries

Viscoelastic artificial boundaries are widely adopted in numerical simulations of wave propagation problems.When explicit time-domain integration algorithms are used,the stability condition of the boundary domain is stricter than that of the internal region due to the influence of the damping and stiffness of an viscoelastic artificial boundary.The lack of a clear and practical stability criterion for this problem,however,affects the reasonable selection of an integral time step when using viscoelastic artificial boundaries.In this study,we investigate the stability conditions of explicit integration algorithms when using three-dimensional(3D)viscoelastic artificial boundaries through an analysis method based on a local subsystem.Several boundary subsystems that can represent localized characteristics of a complete numerical model are established,and their analytical stability conditions are derived from and further compared to one another.The stability of the complete model is controlled by the corner regions,and thus,the global stability criterion for the numerical model with viscoelastic artificial boundaries is obtained.Next,by analyzing the impact of different factors on stability conditions,we recommend a stability coefficient for practically estimating the maximum stable integral time step in the dynamic analysis when using 3D viscoelastic artificial boundaries.

Stable Boundary Layer Height Parameterization: Learning from Artificial Neural Networks

Artificial neural networks (ANN) are employed using different combinations among the surface friction velocity u*, surface buoyancy flux Bs, free-flow stability N, Coriolis parameter f, and surface roughness length z0 from large-eddy simulation data as inputs to investigate which variables are essential in determining the stable boundary layer(SBL) height h. In addition, the performances of several conventional linear SBL height parameterizations are evaluated. ANN results indicate that the surface friction velocity u* is the most predominant variable in the estimation of SBL height h. When u* is absent, the secondly important variable is the surface buoyancy flux Bs. The relevance of N, f, and z0 to h is also discussed;f affects more than N does, and z0 shows to be the most insensitive variable to h.

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.

Wave Radiation by a Floating Body in Water of Finite Depth Using an Exact DtN Boundary Condition

The present paper focuses on the wave radiation by an oscillating body with six degrees of freedom by using the DtN artifi-cial boundary condition.The artificial boundary is usually selected as a circle or spherical surface to solve various types of fields,such as sound waves or electromagnetic waves,provided that the considered domain is infinite or unbounded in all directions.However,the substantial wave motion is considered in water of finite depth,that is,the fluid domain is bounded vertically but unbounded horizon-tally.Thus,the DtN boundary condition is given on an artificial cylindrical surface,which divides the water domain into an interior and exterior region.The boundary integral equation is adopted to implement the present model.In the case of a floating cylinder,the results of hydrodynamic coefficients of a chamfer box are discussed.

斜入射地震波下单层球面网壳土-结构相互作用及其地震响应分析

为分析斜入射地震波下单层球面网壳土-结构相互作用及其地震响应,采用等效节点力实现地震波输入,通过黏弹性人工边界处理无穷远辐射条件,分析了地震波类型、土体参数、入射角度等因素对单层球面网壳结构的土-结构相互作用及其地震响应影响。结果表明:单层球面网壳结构在地震波入射一侧出现翘起,网壳总体沿入射方向发生旋转,当P波斜入射时支座位移差最大达0.514 m,为网壳跨度的1/250。当P波斜入射时,软弱土情况下网壳顶点位移比中硬土和中软土大,顶点位移随入射角增大呈现先增大后减小的趋势;当SV波入射时,中软土情况下网壳顶点位移比中硬土和软弱土大,顶点位移随入射角增大而增大。斜入射地震波下,考虑土-结构相互作用后网壳顶点位移增幅最大达5.5倍,网壳外圈位移增幅大于网壳跨中增幅。

基于动态人工波速的透射边界

人工边界处理是近场地震波动数值模拟的关键问题,其中透射边界(multi-transmitting formula,MTF)是一种应用较为广泛的人工边界方法,但它也存在计算精度与稳定性相互冲突的问题.为提高稳定性较好的低阶边界的模拟精度,发展出一种动态人工波速透射边界,能够适应复杂波动中外行波透射角随时间大幅度变化的情形,并探讨了动态人工波速识别这一关键问题.通过在当前时刻波场中沿人工边界附近划定观察窗口,并在一段时间之前某时刻波场中构建出一组与边界法线夹角在-90°~90°范围变化的筛选窗口,设计出一种对这两个时刻局部窗口中的波场进行比对的图像处理方法来识别外行波的透射角度.将识别得到的透射角度用于动态更新透射边界的人工波速,实现基于动态人工波速透射边界的波动模拟.在SH波动数值模拟中分别以单一脉冲波的简单波动和多个平面波叠加的复杂波动为例,证明所发展的动态人工波速透射边界对于不同透射角的波动成分均具有较好的适应性,模拟精度显著优于采用经验固定人工波速的传统透射边界.该方法为发展和完善高精度且计算稳定的人工边界提供了一个全新的研究思路.

SPH模拟螺杆挤出的粒子穿透边界问题研究

采用光滑粒子流体动力学(smoothed particle hydrodynamics,SPH)模拟双螺杆挤出时,常会出现粒子穿透边界的问题,这是由于螺杆挤出过程存在“挤压流”这一特殊流动现象而造成的。过去的“人工声速判据”不适用于解决“挤压流”这类问题。为了修正“人工声速判据”的缺陷,针对挤压过程进行推算,提出了一种理论计算“特征速度”的方法。为了验证该方法的有效性,对SPH的模拟结果与传统商业软件POLYFLOW的模拟结果进行了比较。结果显示,新计算方案的运用可以有效避免粒子穿透边界的问题。这种计算方案可为其他存在“挤压流”现象的复杂加工模拟解决粒子穿透边界问题提供参考。

基于改进人工势场法的通用航空路径规划

通用航空飞机在飞行过程除了要面对障碍物和天气系统的威胁,同时也受到飞行空域的限制,往往飞机只能在一块固定的空域或狭小的飞行走廊内飞行,一旦飞出限制空域,则有可能引发飞行冲突,甚至发生严重的安全事故。针对这些问题,在人工势场法斥力势场函数中添加航空器和目标点之间的相对距离函数,首先解决传统人工势场法存在的目标不可达问题和局部极小点问题,然后在算法中加入航路边界斥力影响因子,使飞机避障后在边界斥力的作用下重回航路中心。经过MATLAB仿真验证,试验结果表明,改进后的人工势场法能够实现主动避障和飞机航迹的规划。

考虑SSI效应的带地下室RC框架结构抗震性能研究

土-结构相互作用效应(Soil-Structure Interaction,SSI)是带地下室钢筋混凝土(Reinforced Concrete,RC)框架结构抗震性能分析的重要因素。基于模态和动力时程分析,通过有限元分析软件系统探究了地下室顶板嵌固、基础顶面嵌固和考虑SSI效应地下室周围土体嵌固等3种不同嵌固方式的RC框架结构抗震性能。结果表明:考虑SSI效应带地下室RC框架结构的自振周期明显延长,其中心柱剪力、各层加速度和层间位移角均有所减小;基础顶面嵌固模型未考虑SSI效应,其计算结果较为保守;SSI效应对带地下室RC框架结构会产生较大的减震影响。

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.