Mapped infinite elements for three-dimensional multi-region boundary element analysis

A formulation for an infinite boundary element (BE) is presented, which allows the modelling of infinite surfaces. The concept is that the finite surface is mapped to an infinite surface using special mapping functions. Using such mapping functions together with linear and quadratic interpolation for the displacements and the tractions, respectively, the desired decay behaviour can be modelled. The implementation of the proposed infinite elements becomes straightforward, since the Cauchy principal value, as well as the free term, are evaluated for the finite and infinite BEs with exactly the same techniques. The element developed can be used in a multi-region BE analysis of piecewise homogeneous domains, or for domains with joints and faults. The accuracy of the element is tested on some benchmark problems. Finally, a practical application in tunnelling is shown. Copyright © 2004 John Wiley & Sons, Ltd.     

A 2.5D finite/infinite element approach for modelling visco‐elastic bodies subjected to moving loads

The objective of this study is to propose a 2.5D finite/infinite element procedure for dealing with the ground vibrations induced by moving loads. Besides the two in‐plane degrees of freedom (DOFs) per node conventionally used for plane strain elements, an extra DOF is introduced to account for the out‐of‐plane wave transmission. The profile of the half‐space is divided into a near field and a semi‐infinite far field. The near field containing loads and irregular structures is simulated by the finite elements, while the far field covering the soils extending to infinity by the infinite elements with due account taken of the radiation effects for moving loads. Enhanced by the automated mesh expansion procedure proposed previously by the writers, the far field impedances for all the lower frequencies are generated repetitively from the mesh created for the highest frequency considered. Finally, the accuracy of the proposed method is verified through comparison with a number of analytical solutions. Copyright © 2001 John Wiley & Sons, Ltd.     

Simulation of transient heat conduction using one‐dimensional mapped infinite elements

Many engineering problems exist in physical domains that can be said to be infinitely large. A common problem in the simulation of these unbounded domains is that a balance must be met between a practically sized mesh and the accuracy of the solution. In transient applications, developing an appropriate mesh size becomes increasingly difficult as time marches forward. The concept of the infinite element was introduced and implemented for elliptic and for parabolic problems using exponential decay functions. This paper presents a different methodology for modeling transient heat conduction using a simplified mesh consisting of only two‐node, one‐dimensional infinite elements for diffusion into an unbounded domain and is shown to be applicable for multi‐dimensional problems. A brief review of infinite elements applied to static and transient problems is presented. A transient infinite element is presented in which the element length is time‐dependent such that it provides the optimal solution at each time step. The element is validated against the exact solution for constant surface heat flux into an infinite half‐space and then applied to the problem of heat loss in thermal reservoirs. The methodology presented accurately models these phenomena and presents an alternative methodology for modeling heat loss in thermal reservoirs. Copyright © 2010 John Wiley & Sons, Ltd.     

Infinite elements in the time domain using a prolate spheroidal multipole expansion

Many phenomena in acoustically loaded structural vibrations are better understood in the time domain, particularly transient radiation, shock, and problems involving non-linearities, cavitation, and bulk structural motion. In addition, the geometric complexity of structures of interest drives the analyst toward domain-discretized solution methods, such as finite elements or finite differences, and large numbers of degrees of freedom. In such methods, efficient numerical enforcement of the Sommerfeld radiation condition in the time domain becomes difficult. Although a great many methodologies for doing so have been demonstrated, there seems to exist no consensus on the optimal numerical implementation of this boundary condition in the time domain. Here, we present theoretical development of several new boundary operators for conventional finite element codes. Each proceeds from successful domain-discretized, projected field-type harmonic solutions, in contrast to boundary integral equation operators or those derived from analyses of outgoing waves. We exploit the separable prolate-spheroidal co-ordinate system, which is sufficiently general for a large variety of problems of naval interest, to obtain finite element-like operators (matrices) for the boundary points. Use of this co-ordinate system results in element matrices that can be analytically inverse transformed from the frequency to the time domain, without imposing continuity requirements on the solution above those imposed by the underlying partial differential equation. In addition, use of element-like boundary operators does not alter the banded structure of assembled system matrices. Results presented here include theoretical derivation of the infinite elements, resolution of the Fourier inversion issues, and element matrices for the boundary operators which introduce no new continuity requirements on the fluid field variable. The simplest infinite elements are verified in a coupled three-dimensional context against DAA2 and Helmholtz integral equation results. © 1998 John Wiley & Sons, Ltd.     

冲击波作用下水下结构载荷计算方法

针对冲击波激励下水下结构与无限声学水域的流固耦合问题,建立了冲击波及无限水域作用在结构上的载荷计算方法。该方法采用场分离技术将水下结构载荷分解为冲击波压力及散射波压力,而散射波压力又分解成反射波压力和辐射波压力。基于比例边界有限元法,建立反射波压力计算方法,而辐射波压力由有限元商业软件的声固耦合技术确定。该文利用该方法在商业有限元软件中分析了一经典算例,分析结果表明该方法能有效模拟冲击波作用下水下结构载荷及其响应,避免了冲击波在无限流体中传播的有限元分析,降低了分析规模。     

A coupled interpolating meshfree method for computing sound radiation in infinite domain

In this paper, the coupling of the improved interpolating element‐free Galerkin (IIEFG) method and the variable‐order infinite acoustic wave envelope element (WEE) method is studied. A coupled IIEFG‐WEE method for computing sound radiation is proposed to make use of their advantages while evading their disadvantages. The coupling is achieved by constructing the hybrid shape function of continuity and compatibility on the interface between the IIEFG and WEE domains. In the IIEFG domain, the improved interpolating moving least‐squares (IIMLS) method is used to form the shape functions satisfying the Kronecker delta condition while nonsingular weight functions can be used. The impacts of the size of the influence domain and the shape parameter on the performance of this coupled method are investigated. The numerical results show that the coupled IIEFG‐WEE method can take full advantage of both the IIEFG and WEE methods and that it not only can achieve higher accuracy but also has a faster convergence speed than the conventional method of the finite element coupled with the WEE. The experimental results show that the method is very flexible for acoustic radiation prediction in the infinite domain.     

三维双向及三向映射无限元

不同于以往的单向映射无限元,本文提出了一种能分别从两个和三个方向延伸到无穷远的多向映射无限元,它们不仅能很好地与三维等参数有限单元耦合,保持公共边界的连续性,而且具有物理概念明确、节点数目少、精度高、无需改动单元形状、减少计算准备工作量、便于网格自动形成等诸多优点,能广泛应用于岩土工程领域及其他力学问题的数值分析工作中.     

无限元边界条件下的浅海海底地震波场建模

针对模拟无限大的海底地震波传播问题,应用无限元理论,提出了一种基于无限元边界条件的地震波建模方法,仿真了海底应力场,与高阶交错网格有限差分法仿真结果比对,验证建模方法的正确性.进一步分析无限元边界对海底地震波的吸收效果,结果表明无限元边界对纵波、横波的吸收效果好,可应用于海底地震波场建模中.     

Hermite infinite elements and graded quadratic B-spline finite elements

Quadratic B-spline finite elements are defined for a graded mesh. Hermite infinite elements are proposed to extend the applicability of these finite elements to unbounded regions. Test problems used to compare this technique with published procedures show that the quadratic B-spline finite element solution has, as expected, lower error bounds than a linear element solution. These experiments also demonstrate that the Hermite infinite elements used to close the B-spline finite element arrays lead to error norms comparable in size with other infinite element formulations. The generation of solitary waves in a semi-infinite shallow channel by boundary forcing is modelled by the Korteweg-de Vries equation using an array of graded elements closed by a zero pole infinite element. The resulting simulation of solitary wave motion across a non-uniform mesh confirms existing work and illustrates the effectiveness of the present formulation.     

A unified approach for the formulation of interaction problems by the boundary element method

A unified formulation is presented, based on boundary element method, in a form suitable for performing the interaction analyses by substructure method for solid–solid and soil–structure problems. The proposed formulation permits the evaluation of all the elements of impedance and input motion matrices simultaneously at a single step in terms of system matrices of the boundary element method without solving any special problem, such as, unit displacement or load problem, as required in conventional methods. It eliminates further the complicated procedure and the need for using scattering analysis in the evaluation of input motion functions. To explain the formulation, it is first given for an inclusion interacting with an infinite surrounding medium under the influence of a seismic input, where both the inclusion and surrounding medium are treated as viscoelastic. It is shown that the formulation for a rigid inclusion may be obtained from that for flexible inclusion as a special case through a transformation. Then, the formulation is extended to other types of interaction problems: a multi‐inclusion problem and an interaction problem involving a foundation embedded in a viscoelastic half‐space. It is found that the proposed formulation remains essentially the same for all kinds of interaction problems and it can be used not only in regular interaction analysis, but also in the analysis involving diffraction of waves in a medium containing holes. Copyright © 2005 John Wiley & Sons, Ltd.     

一种新型的时域动力无穷单元

针对频域动力无穷元不便与时域有限元结合的缺陷,采用时域的波传函数,推导出一种新型的时域动力无穷单元,并采用UPFs工具对ANSYS进行二次开发,将其嵌入到ANSYS当中。然后采用经典算例对所提出的新型动力无穷单元进行验证,并将其结果与远置边界、粘性边界、粘弹性边界的结果进行比较。验证结果表明,所提出的新型时域动力无穷单元是成功的,并可方便地实现与时域的有限单元结合。     

薄板弯曲分析的多边形流形单元

一般的数值流形方法均采用三角形、四边形单元进行计算。对于工程中的有些实际问题, 多边形单元能更好的适应复杂计算域形状。为此, 研究了采用多边形流形单元进行数值计算的方法。采用任意几何区域的Delaunay三角网格构造出新的凸多边形网格, 并以此单元作为计算的流形单元。采用改进的Wachspress插值函数作为多边形流形单元的权函数。为说明该方法的有效性, 将该流形方法应用于薄板弯曲计算, 推导出用于薄板弯曲分析的流形格式和单元矩阵。计算结果表明:较一般有限元法, 计算精度和收敛速度有很大提高。     

Finite–infinite element for dynamic analysis of axisymmetrically saturated composite foundations

The formulations of axisymmetrically infinite elements for dynamic analysis of vertical vibration problems in unbounded saturated composite foundations are presented. The theoretical basis as well as the implementation of the elements is discussed, and the element decay functions are derived using the analytical solutions of axially symmetric configurations. Using the proposed finite–infinite element method, the surface vertical displacements of air‐saturated soil (‘dry’ soil) and of water‐saturated soil with extremely low permeability subjected to a surface point excitation (called as the Lamb's problem) are calculated and the results agree very well with the existing theoretical solutions of single‐phase elastic media. As an application, the velocity admittances of a concrete block resting on cement mixing‐pile or gravel‐pile saturated composite foundations are calculated. The influence of soil permeability and pile rigidity on the dynamic response of the composite foundations is investigated. The method proposed by this paper is a simple and reliable numerical one that could be used to study axisymmetrically dynamic problems of layered saturated media and to get the mechanism of dynamic testing on single‐pile saturated composite foundations. Copyright © 2006 John Wiley & Sons, Ltd.     

预报振动噪声的径向基点插值无网格与无限元耦合方法

为克服传统的有限元耦合无限元方法中的单元匹配问题,研究了径向基点插值法和无限元法的耦合规律,提出了一种预报无限域结构振动噪声的径向基点插值无网格与可变阶无限声波包络单元耦合方法,推导了预报声压的计算公式。为提高声场预报精度和满足声波在无限域的自由衰减,结构外部无限声场分为使用无网格表示的近场和可变阶声波包络单元离散的远场。在该耦合方法中,通过在近场与远场之间的交界面上配置虚拟网格来构造具有连续性的声压形函数,确保了声压的连续与一致性。采用数值仿真和试验对该耦合方法进行了验证,结果表明该耦合方法拥有无网格法的高精度和可变阶声波包络单元法满足声波自由衰减的特点,具有良好的精度和收敛性,可用于实际噪声预报。     

Viscoplastic plane‐strain analysis of infinite solids using elastic supports

A highly efficient novel Finite Element Boundary Element Method (FEBEM) is proposed for the elasto‐viscoplastic plane‐strain analysis of displacements and stresses in infinite solids. The proposed method takes advantage of both the Finite Element Method (FEM) and the Boundary Element Method (BEM) to achieve higher efficiency and accuracy by using the concept of elastic supports to simulate the effects of unbounded solid mass surrounding the region of interest. The BEM is used to compute the stiffnesses of elastic supports and to estimate the location of the truncation boundary for the finite element model. As compared to the conventional coupled FEBEM, the proposed method has three main computational advantages. Firstly, the symmetrical and highly banded form of the standard finite element stiffness matrix is not disturbed. Secondly, the proposed technique may be implemented simply by using standard codes for elasto‐viscoplastic finite element analysis and elastic boundary element analysis. Thirdly, the yielded zone is approximately located in advance by using the BEM and hence, an unnecessarily large extent of the domain does not have to be discretized for the finite element modelling. The efficiency and accuracy of the proposed method are demonstrated by computing elastic and elasto‐plastic displacements and stresses around ‘deep’ underground openings in rock mass subject to hydrostatic and non‐hydrostatic in situ stresses. Results obtained by the proposed method are compared with ‘exact’ solutions and with those obtained by using a BEM and a coupled FEBEM. Copyright © 1999 John Wiley & Sons, Ltd.     

结构—地基—结构相互作用地震反应分析

针对成层地基，用样条有限元与半解析无限元耦合法分析水平方向无限伸展的地基土，对结构－地基－结构进行了地震反应分析，并在分析中考虑了地震波存在的衰减和相位差，得到了不同结构－地基－结构相互作用之间的一些规律     

The scaled boundary finite element method in structural dynamics

The scaled boundary finite element method is extended to solve problems of structural dynamics. The dynamic stiffness matrix of a bounded (finite) domain is obtained as a continued fraction solution for the scaled boundary finite element equation. The inertial effect at high frequencies is modeled by high‐order terms of the continued fraction without introducing an internal mesh. By using this solution and introducing auxiliary variables, the equation of motion of the bounded domain is expressed in high‐order static stiffness and mass matrices. Standard procedures in structural dynamics can be applied to perform modal analyses and transient response analyses directly in the time domain. Numerical examples for modal and direct time‐domain analyses are presented. Rapid convergence is observed as the order of continued fraction increases. A guideline for selecting the order of continued fraction is proposed and validated. High computational efficiency is demonstrated for problems with stress singularity. Copyright © 2008 John Wiley & Sons, Ltd.     

一种新的声无限元法

提出了一种求解外声场领域新的无限元法。这种无限元方法使用文献 [1]中无限元的形函数 ,权函数为形函数的共轭 ,再乘以一个附加权因子 ,这样求解过程就避免了无穷的指数积分。采用椭球坐标。本文考虑了文献 [1]中的所忽略的一项。耦合有限元 ,这种新的无限元 ,理论上可以求解任意形状、任意频率的声源的声辐射问题。文中首次提出 ,通过检验无限元求解的精度 ,而不是耦合的有限元和无限元的总体精度来验证无限元方法的可行性。我们使用这种方法分析了一个摆动球和一个椭球例子 ,结果表明这种方法是可行的。     

Modelling of short wave diffraction problems using approximating systems of plane waves

This paper describes a finite element model for the solution of Helmholtz problems at higher frequencies that offers the possibility of computing many wavelengths in a single finite element. The approach is based on partition of unity isoparametric elements. At each finite element node the potential is expanded in a discrete series of planar waves, each propagating at a specified angle. These angles can be uniformly distributed or may be carefully chosen. They can also be the same for all nodes of the studied mesh or may vary from one node to another. The implemented approach is used to solve a few practical problems such as the diffraction of plane waves by cylinders and spheres. The wave number is increased and the mesh remains unchanged until a single finite element contains many wavelengths in each spatial direction and therefore the dimension of the whole problem is greatly reduced. Issues related to the integration and the conditioning are also discussed. Copyright © 2002 John Wiley & Sons, Ltd.     

Hierarchal triangular elements using orthogonal polynomials

Hierarchal elements are finite elements which have the useful property that elements with different polynomial orders can be used together in the same mesh without causing discontinuities. This paper introduces a new hierarchal triangular element in which the basis functions are constructed from orthogonal polynomials—Jacobi polynomials. The resulting element is shown to be better conditioned than the earlier hierarchal element of Rossow and Katz.¹ Recursive formulas allow the complete set of basis functions for an element to be efficiently evaluated at a given point. In addition, the formulas can be used to generate pre-computed (universal) matrices. Examples are given of universal matrices, up to order 4, for the generalized Helmholtz equation. An electromagnetic problem involving a length of transmission line is used to show the usefulness of the new elements.