Mapped infinite elements in soil consolidation

The problems of consolidation often involve unbounded continua. The most common solution is then achieved by limiting the studied area through the introduction, at finite distance, of a boundary on which adequate conditions on the field variables are imposed. In this paper mapped infinite elements are used to model the far-field solution. Spatial discretization is hence performed on the basis of both finite and infinite elements. In two examples, solutions involving finite and infinite elements are compared with known analytical solutions and it is shown that an excellent agreement can be achieved by the use of mapped infinite elements.     

Mapped infinite elements for exterior wave problems

Recently, a new type of infinite elements which uses r^?1 decay was proposed. They were applied to exterior wave problems and good results were obtained. In two-dimensional problems, however, it was necessary to move the origin of the r^?1 decay in order to model the outgoing wave more accurately, because it decays roughly as r^?1/2. In this paper, the mapped infinite elements with r^?1/2 decay and the necessary numerical integration procedure are presented. These elements do not require any artificial movement of the origin. Several example problems are solved. The results show that the infinite elements with r^?1/2 decay here give much more accurate values than the infinite elements with exponential decay and any damper elements.     

Infinite elements with 1/rn type decay

This paper presents a method of developing a family of 1/rⁿ type infinite elements for the analysis of problems definite in unbounded media. The proposed method is a direct extension of the conventional finite element method. The resulting improper integrals are integrated exactly over the infinite element domains. Two numerical examples in elastic half-space static problems are investigated to illustrate the applicability and accuracy of the method. The use of the proposed infinite elements yields excellent results and preserves all the advantages of the finite element method.     

Solution of three-dimensional, anisotropic, nonlinear problems of magnetostatics using two scalar potentials, finite and infinite multipolar elements and automatic mesh generation

The two-scalar potentials idea has been used with success for the computation of static magnetic fields in the presence of nonlinear isotropic magnetic materials by the finite element method. In this communication we formulate the two-scalar-potentials method for anisotropic materials and present a computer program and the solution of an example problem. The use of infinite multipolar elements is also discussed. Several advanced methods and ideas are employed by the program: scalar potentials, rather than vector potentials, giving only one unknown quantity; the finite element method, in which the solution is approximated by a continuous function; the Galerkin method to solve the differential equations; accurate infinite elements, which avoid the introduction of an artificial boundary for unbounded problems; automatic mesh generation, which means that the user can construct a large mesh and represent a complicated geometry with little effort; automatic elimination of nodes outside the iron, which restricts the iterations to the nonlinear anisotropic region with economy of computer time; use of sparse matrix technology, which represents a further economy in computer time when assembling the linear equations and solving them by either Gauss elimination or iterative techniques such as the conjugated gradient method, etc. The combination of these techniques is very convenient.     

Three-dimensional piezoelectric boundary element analysis of transversely isotropic half-space

In this paper, we present a boundary element method (BEM) solution technique for studying the three-dimensional transversely-isotropic piezoelectric half-space problems. The use of mixed alternative point force solutions for half and full-space problems presented are necessary to overcome the computation difficulties especially in the calculation of the derivatives with respect to z. Infinite boundary elements are introduced to model the surface of the half-space only when stresses at the internal points are required to be evaluated. The integration over the infinite boundary elements is bounded and some limitations of the infinite element construction are relaxed. Closed-form solutions for uniformly distributed mechanical and electrical loads acting on a circular area on the surface of half-space are derived. This theoretical work serves as a good verification tool for numerical computation. In this paper, the numerical and theoretical results show good agreement. Numerical analysis via the finite element method (FEM) is also carried out using the commercial solver ANSYS. These FEM results are used to verify against the accuracy of the BEM solution. Finally, numerical results for the case of Hertzian pressure applied to an imperfect half-space are presented. The effects of the coupled mechanical–electrical influences as well as the presence of voids are examined. This work was supported by NTU Academic Research Funds. The finite element simulation using the ANSYS code was conducted by Mr. Ji Ren. Also, the authors wish to acknowledge the journal editor and anonymous reviewers for their helpful suggestions and comments leading to improvement of the paper.     

Boundary integral equation analysis for radiation and scattering of elastic waves in three dimensions

Abstract

In this paper, the boundary integral equation (BIE) method is employed to investigate the radiation and scattering of time‐harmonic elastic waves by obstacles of arbitrary shape embedded in an infinite medium. Based on the vector BIE, entirely free of Cauchy principal value integrals, an efficient numerical scheme using quadratic isoparametric boundary elements is proposed. Furthermore, the difficulty of non‐uniquess of a solution inherent with BIE formulations for exterior elastodynamic problems is studied numerically and analytically. The counterparts of the combined Helmholtz integral formulation method for elastodynamics together with the least‐square or Lagrange‐multiplier technique are derived and applied to overcome this difficulty successfully. In addition, the elastic‐wave fields radiated or scattered by either a spherical cavity or a rigid sphere in an infinite medium are calculated and the results are compared with the analytical solutions to demonstrate the accuracy and versatility of the proposed numerical scheme.     

FREQUENCY-INDEPENDENT INFINITE ELEMENTS FOR ANALYSING SEMI-INFINITE PROBLEMS

Two drawbacks exist with the infinite elements used for simulating the unbounded domains of semi-infinite problems. The first is the lack of an adequate measure for calculating the decay parameter. The second is the frequency-dependent characteristic of the finite/infinite element mesh used for deriving the impedance matrices. Based on the properties of wave propagation, a scheme is proposed in this paper for evaluating the decay parameter. In addition, it is shown that by the method of dynamic condensation, the far-field impedance matrices for waves of lower frequencies can be obtained repetitively from the one for waves of the highest frequency, using exactly the same finite/infinite element mesh. Such an approach ensures that accuracy of the same degree can be maintained for waves of all frequencies within the range of consideration. Effectiveness of the proposed method is demonstrated in the numerical examples through comparison with previous results.     

Diffraction and refraction of surface waves using finite and infinite elements

The wave problem is introduced and a derivation of Berkhoff's surface wave theory is outlined. Appropriate boundary conditions are described, for finite and infinite boundaries. These equations are then presented in a variational form, which is used as a basis for finite and infinite elements. The elements are used to solve a wide range of unbounded surface wave problems. Comparisons are given with other methods. It is concluded that infinite elements are a competitive method for the solution of such problems.     

A dynamic infinite element for three-dimensional infinite-domain wave problems

A three-dimensional dynamic infinite element which satisfies the following requirements: (1) displacement compatibility on the interface between finite and infinite elements; (2) definition of the wave propagation and amplitude attenuation behaviours in the infinite element using wave propagation functions; (3) convergence of the generalized integrals related to mass and stiffness matrices of the infinite element: and (4) displacement continuity along the common boundary of neighbouring infinite elements in the case of simulating multiple material layers or multiple wave numbers within the foundation, is presented in this paper. Since P-waves, S-waves and R-waves in the foundation can be simulated Simultaneously in the present infinite element, the seismic response of an arch-dam-foundation system, especially a thin double-curvature arch-dam-foundation system where the boundary element loses its competitive capacity with the finite element, can be economically calculated by coupling this infinite element with conventional finite elements. The good accuracy obtained using the present infinite element and finite element coupling model to simulate foundation wave problems has been proven by comparing the current numerical results with previous analytical results.     

Computation of the stress intensity factors of sharp notched plates by the fractal‐like finite element method

The fractal‐like finite element method (FFEM) is an accurate and efficient method to compute the stress intensity factors (SIFs) of different crack configurations. In the FFEM, the cracked/notched body is divided into singular and regular regions; both regions are modelled using conventional finite elements. A self‐similar fractal mesh of an ‘infinite’ number of conventional finite elements is used to model the singular region. The corresponding large number of local variables in the singular region around the crack tip is transformed to a small set of global co‐ordinates after performing a global transformation by using global interpolation functions. In this paper, we extend this method to analyse the singularity problems of sharp notched plates. The exact stress and displacement fields of a plate with a notch of general angle are derived for plane‐stress/strain conditions. These exact analytical solutions which are eigenfunction expansion series are used to perform the global transformation and to determine the SIFs. The use of the global interpolation functions reduces the computational cost significantly and neither post‐processing technique to extract SIFs nor special singular elements to model the singular region are needed. The numerical examples demonstrate the accuracy and efficiency of the FFEM for sharp notched problems. Copyright © 2008 John Wiley & Sons, Ltd.     

Green's functions for the Laplace equation in a 3-layer medium,boundary element integrals and their application to cathodic protection

We report in this paper a set of nine Green's functions for the Laplace equation for an infinite 3-layer medium in which a layer of finite width is sandwiched between two semi-infinite domains. Typical 3D plots of these Green's functions are computed and presented. Taking an offshore platform as a prime example of a structure in a 3-layer medium (atmosphere, ocean and soil), we work out the boundary element integrals using macro elements such as the tubulars. Constant elements reduce several of these boundary integrals to analytical forms. As an application, we discuss the cathodic protection modelling of offshore structures using the ‘boundary element method’.     

Comparison between a finite element and a composite method for a three-dimensional potential problem

This paper is concerned with the solution of the three-dimensional potential problem for electromagnetic river gauging. It extends previous ideas of joining finite elements in an interior region to one infinite external element treated by the boundary integral method^1,2 to this case where there are two external infinite elements representing the river and the ground. The boundary continuity conditions on the infinite river–ground interface, as well as the internal–external interfaces, are dealt with by introducing a variational principle with relaxed continuity requirement³.     

单层波纹管的轴对称振动

本文基于小应变、小位移的假设,用有限元法分析了单层波纹管在轴对称振动时的自振频率。文中采用三节点曲边壳元,位移、转角作为独立变量在总体坐标系下进行插值。     

Coping with non-convex domains in the boundary element analysis

In this paper, it is shown that the boundary element method leads to correct results regardless the convexity of the actual domain, if the discretisations are sufficiently refined, and enough elements have been used to approximate infinite boundary geometries.The examples, drawn from acoustic wave propagations in 2-D domains, are presented in order to demonstrate this fact.     

A semi-analytical discretization method for the near-field analysis of unsymmetrically laminated multimaterial junctions

In the present contribution, the theoretical basics of a novel semi-analytical discretization procedure are described. The method relies on the discretization of the structural situation into an arbitrary number of sectorial elements in which adequate displacement formulations are postulated. Speaking in terms of a cylindrical coordinate system, the sectorial elements are supposed to be of infinite dimensions in the radial direction, hence the actual discretization takes place in the circumferential direction only. While linear shape functions are employed in each of the infinite sectorial elements in the circumferential direction, a set of unknown displacement functions is postulated in each of the resultant interfaces between the individual elements with respect to the radial coordinate. The principle of minimum elastic potential yields the governing Euler–Lagrange equations straightforwardly which allow for closed-form solutions for the unknown interface displacement functions. Since the method yields closed-form solutions for all state variables with respect to the radial coordinate and employs a discretization in the circumferential direction exclusively, we may actually speak of a semi-analytical methodology. Examples are presented for the near-field analysis of unsymmetrically laminated multimaterial notches. The presented semi-analytical method proves to be of high accuracy. Furthermore, while this novel discretization procedure clearly outperforms purely numerical analysis methods like FEM in terms of computational time and effort, it works with comparable accuracy which makes it very attractive for any practical application purpose with involved localization effects where reliable results need to be computed with low computational effort.     

Boundary infinite elements for the Helmholtz equation in exterior domains

A novel approach to the development of infinite element formulations for exterior problems of time-harmonic acoustics is presented. This approach is based on a functional which provides a general framework for domain-based computation of exterior problems. Special cases include non-reflecting boundary conditions (such as the DtN method). A prominent feature of this formulation is the lack of integration over the unbounded domain, simplifying the task of discretization. The original formulation is generalized to account for derivative discontinuities across infinite element boundaries, typical of standard infinite element approximations. Continuity between finite elements and infinite elements is enforced weakly, precluding compatibility requirements. Various infinite element approximations for two-dimensional configurations with circular interfaces are presented. Implementation requirements are relatively simple. Numerical results demonstrate the good performance of this scheme. © 1998 John Wiley & Sons, Ltd.     

Infinite boundary elements for electromagnetics

Consider a two‐dimensional plane wave transverse magnetic mode scattering from a perfectly electric conducting ground plane. Let the ground plane be of infinite extent and comprise two regions, a near field and a far field. In the far field, let the ground plane be flat and let us choose the co‐ordinates (x, y) such that it lies on the axis y=0. Over the interior region, let the profile of the ground plane change such that it can lie partially above and also partially below the axis y=0. Finally, let us assume that the source of the excitation lies above the ground plane. To model this general class of problems, a method of moments electric field integral equation formulation is proposed which uses infinite boundary elements to model the far field and boundary elements to model the near field. In the far field, the field variable is approximated by the highest order terms in the far‐field asymptotic expansion. The integrals over the infinite boundary elements are infinite in extent and contain oscillatory terms and hence require special integration rules. The formulation is tested for the specific problem of a semi‐circular cylindrical protrusion of radius a lying above an infinite flat ground plane, such that ka=1 where k is the wave number. This problem is chosen because it has an analytic solution in the form of a Bessel function expansion; hence, the accuracy of the formulation can be tested. In particular, the radar cross section results for various angles of incidence of the plane wave source are calculated and compared with analytic results. Copyright © 2007 John Wiley & Sons, Ltd.     

Transient infinite elements for contaminant transport problems

A time-dependent infinite element which can be used to simulate contaminant transport problems in infinite media is presented in this paper. Since this transient infinite element is constructed in a global co-ordinate system instead of a local one, a closed-form solution for the property matrices of the element has been derived from an advection–diffusion/dispersion problem in a homogeneous, anisotropic infinite medium. The numerical results from the present transient infinite element have excellent agreements with the corresponding analytical solutions. Compared with the previous infinite elements, the present infinite element has the following special characteristics: (1) both space and time variables were explicitly considered in the formulation; (2) its property matrices were expressed in a closed form; (3) it can be used to represent the far field of a mass/contaminant transport problem in a homogeneous, anisotropic infinite medium; (4) it was constructed in a global co-ordinate system. Therefore, it is highly recommended that the transient infinite element be used for the numerical simulation of contaminant transport problems in infinite media.     

The generating analytic element approach with application to the modified Helmholtz equation

In this paper a new method for obtaining functions with a given singular behavior that satisfy a class of partial differential equations is presented. Differential equations of this class contain operators of the form , where n is a positive integer. The method uses Wirtinger calculus which enables one to invert the Laplacian in combination with the decomposition method introduced by Adomian at the end of the twentieth century. The procedure uses a singular holomorphic function as its basis, and constructs the solution term by term as an infinite series of functions; the process consists of an infinite number of steps of integration. This method is applied to construct a number of singular solutions to the modified Helmholtz equation in the context of groundwater flow. These functions are discharge potentials, which are two-dimensional functions by definition. The gradient of the discharge potential is the vertically integrated flow over the thickness of an aquifer, or water-bearing layer. The discharge potentials of interest here are those used in the analytic element method. This method, as originally conceived, relies on the superposition of suitably chosen holomorphic functions, and is a form of a method known as the Trefftz method, not to be confused with the Trefftz method applied to finite element techniques. The main analytic elements used are singular line elements, characterized by either a jump along the element in the tangential or the normal component of the discharge vector. The analytic line elements for the case of divergence-free irrotational flow are well established and many of these are forms of singular Cauchy integrals. Application of the analytic element method to more general cases of flow, governed for example by the modified Helmholtz equation (flow in systems of aquifers separated by leaky layers) and the heat equation (transient flow) is possible using the method presented in this paper. The latter application is beyond the scope of this paper, but it is worth noting that for that case the constant that occurs in the modified Helmholtz is replaced by a general function of time and application of Laplace transforms can be avoided. A method for constructing such functions is presented; the procedure for constructing these functions is referred to as the generating analytic element approach. Application of this approach requires the existence of the holomorphic singular line element. The approach is discussed and an example for the case of a line-sink for a system of two aquifers separated by a leaky layer and bounded above by in impermeable boundary is presented.     

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。