A boundary element formulation for the substation grounding design

A Boundary Element approach for the numerical computation of substation grounding systems is presented. In this general formulation, several widespread intuitive methods (such as Average Potential Method (APM)) can be identified as the result of specific choices for the test and trial functions and suitable assumptions introduced in the Boundary Element Method (BEM) formulation to reduce computational cost. While linear and parabolic leakage current elements allow to increase accuracy, computing time is drastically reduced by means of new completely analytical integration techniques and semi-iterative methods for solving linear equations systems. This BEM formulation has been implemented in a specific Computer Aided Design system for grounding analysis developed in the last years. The feasibility of this new approach is demonstrated with its application to a real problem.     

A partition of unity enriched dual boundary element method for accurate computations in fracture mechanics

We introduce a novel enriched Boundary Element Method (BEM) and Dual Boundary Element Method (DBEM) approach for accurate evaluation of Stress Intensity Factors (SIFs) in crack problems. The formulation makes use of the Partition of Unity Method (PUM) such that functions obtained from a priori knowledge of the solution space can be incorporated in the element formulation. An enrichment strategy is described, in which boundary integral equations formed at additional collocation points are used to provide auxiliary equations in order to accommodate the extra introduced unknowns. In addition, an efficient numerical quadrature method is outlined for the evaluation of strongly singular and hypersingular enriched boundary integrals. Finally, results are shown for mixed mode crack problems; these illustrate that the introduction of PUM enrichment provides for an improvement in accuracy of approximately one order of magnitude in comparison to the conventional unenriched DBEM.     

针对声屏障的轨道交通的降噪研究

该文针对目前轨道交通中普遍使用的声屏障的降噪方案,提出了在轻轨轨道间增加双面吸声声屏障的新方案,以此来达到进一步降噪的目的。运用边界元方法,建立轨道声屏障降噪的边界元分析模型,利用边界元仿真软件SYSNOISE得到仿真结果,在此基础上预测声屏障的降噪效果。并研究分析了声屏障高度对降噪性能的影响,通过对比不同高度的降噪性能,得到最佳声屏障高度。最终通过对两种方案的降噪结果的对比,说明新的方案具有更好的降噪效果。     

Structural design sensitivity using boundary elements and polynomial response function

This main issue of this paper is a conjunction of the structural design sensitivity analysis using the Boundary Element Method with the polynomial response function determination. The procedure is so general that it enables sensitivity analysis for potential and elasticity problems within both homogeneous and heterogeneous plane and 3D problems. The essential difference with respect to the previous approaches like the Direct Differentiation Method or the Adjoint Variable Method is in discrete evaluation of the structural response using the response polynomials of some state parameters and design variable as the independent parameter. Such a determination is carried out via the several solutions of the given boundary value problem, where design parameter mean value is regularly perturbed in each of the solutions to cover the closest neighborhood of this mean value. Those few solutions make it possible to recover the polynomial response function from node-to node within the boundary elements, so that further symbolic differentiation using MAPLE returns the sensitivity gradients particular values. The entire procedure is tested here twice—first example deals with the homogeneous cantilever beam, where comparison against pure analytical differentiation is done and, separately, for two-component composite cantilever, where such a comparison is made against the central difference method linked with the same BEM solution.     

Asymptotic simplifications for hybrid BEM/GO/PO/PTD techniques based on a Generalized Scattering Matrix approach

In this paper we describe a new hybrid weak coupling of asymptotic GO/PO/PTD techniques, Boundary Element Methods (BEM) and Finite Element Method (FEM) based on the FACTOPO Domain Decomposition Methodology (DDM). Thus, the modular domain decomposition approach already assessed with exact techniques such as BEM and FEM is conserved, with the utilization of GO/PO/PTD techniques, resulting in an important reduction of CPU time during parametric studies. As the coupling scheme between asymptotic and exact methods is based on the Lorentz reciprocity theorem, the external structure of the large object is considered perfectly conducting. The accuracy and efficiency of this technique is assessed by performing the computation of the diffraction and radiation by several test-objects in a multi-domain way, cross compared with reference integral equation results.     

基于边界面法的完整实体应力分析理论与应用

提出基于边界面法（Boundary Face Method,BFM）的完整实体应力分析方法.在该分析中,避免对结构作几何上的简化,结构的所有局部细节都按实际形状尺寸作为三维实体处理.以边界积分方程为理论基础的BFM是完整实体应力分析的自然选择.在该方法中,边界积分和场变量插值都在实体边界曲面的参数空间里实现.高斯积分点的几何数据,如坐标、雅可比和外法向量都直接由曲面算得,而不是通过单元插值近似获得,从而避免几何误差.该方法的实现直接基于边界表征的CAD模型,可做到与CAD软件的无缝连接.线弹性问题的应用实例表明,该方法可以简单有效地模拟具有细小特征的复杂结构,并且计算结果的应力精度比边界元法（Boundary Element Method,BEM）和有限元法（Finite Element Method,FEM）高.     

A fast multigrid solution of boundary integral equations

A new iterative method is developed for solving the boundary integral equation of steady-state potential problems supplied with mixed boundary conditions. Based on this iteration, we define a simple multigrid technique in connection with the Boundary Integral Equation Method. As we show it through an example, this technique seems to be extremely efficient. It is possible to reduce the total computational cost even by one order of magnitude in certain cases. More precisely, it is shown that if the boundary is discretized by N points, the number of arithmetic operations required is proportional to N² while using the traditional Boundary Integral Equation Method, this number is proportional to N³.     

A parallel multipolar boundary element method for internal stokes flows

A multipolar expansion technique is applied to the Boundary Element Method (direct and indirect formulation) in order to solve the two-dimensional internal Stokes Flow with first kind boundary conditions. The algorithm is based on a multipolar expansion for the far field and numerical evaluation for the inner field. Due to the nature of the algorithm, it is necessary to resort to the use of an iterative solver for the resulting algebraic linear system of equations. In comparison with the direct BEM formulation, the indirect formulation is more stable with iterative solvers, and does not need to be preconditioned to obtain a fast rate of convergence. A parallel implementation is designed to take advantage of the natural domain decomposition of fast multipolar techniques and bring further improvement. A good result in memory saving and computing time is obtained that enable us to run huge examples which are prohibitive for traditional BEM implementations.     

Single- and multi-objective shape design of Y-noise barriers using evolutionary computation and boundary elements

The optimum shape design of Y-noise barriers is carried out using single and multi-objective evolutionary algorithms and the Boundary Element Method (BEM). Reduction of noise impact efficiency (using the insertion loss-IL-magnitude) and cost of the barrier (using its total length magnitude) are considered. A two-dimensional problem of sound propagation in the frequency domain is handled, defined by a fixed position emitting source, which pulses in a frequency range, and receptor. A noise barrier (limiting its maximum effective height) is situated between both. Its shape is modified to minimize the receptor measured sound level, which is calculated using BEM. Results of an inverse problem using the IL barrier curve as reference are successfully performed to validate the methodology. The proposed methodology is then used to obtain Y-barriers with 15% and 30% improved IL spectrum. Finally, six non-dominated solutions of the multi-objective optimum design problem are presented in detail.     

Implementation of spectral basis functions in BEM/FEM/GSM Domain Decomposition Methods devoted to scattering and radiation applications

In this paper, we discuss several improvements of a substructuring Domain Decomposition Method (DDM) devoted to Electromagnetic computations, based on the Boundary Element Method (BEM) and the Finite Element Method (FEM). This computation procedure is applied to the analysis of antenna performance on board vehicles as well as Radar Cross Section (RCS). The benefits of the subdomain Computational Electromagnetic Method are mainly the ability to deal with collaborative studies involving several companies, and the reduction of the computation costs by one or more orders of magnitude, especially in the context of parametric studies. Furthermore, this paper proposes a Spectral Basis Function (SBF) defined on fictitious surfaces surrounding equipment, to deal with both the computation of antenna far field patterns and RCS in a multi-domain mode. By masking the complexity of the equipment (wires, thin surfaces, materials, supply network, weapons) the external domain of the vehicle can be closed so that the Combined Field Integral Equation (CFIE) can be used, which is better conditioned than the Electric Field Integral Equation (EFIE). This calculation procedure leads to a faster convergence when using iterative Multi Level Fast Multiple Algorithms (MLFMA). The accuracy and efficiency of this technique is assessed by performing the computation of the diffraction and radiation of several test-objects in a multi-domain way cross compared with reference integral equation results.     

Boundary element analysis of capped and uncapped pile groups

A conventional application of the Boundary Element Method (BEM) to the elastic analysis of sizeable capped pile groups rapidly leads to large computational execution time. This paper develops a BEM formulation for solving such problems more efficiently, and with adequate precision, in which the traction along each pile in a group is represented by a polynomial function. With this approach the tractions need only a few nodes along the shaft to be represented and all the integrals involved can be analytically evaluated. The pile is supposed to be rigid but the formulation can be easily extended to the inclusion of its flexibility. Only vertical displacement compatibility between the soil, the piles and the smooth, rigid cap is enforced. The cap–soil interface is divided into triangular elements, each with three nodes, across which contact pressures vary linearly. Numerical results are presented for single piles and for pile groups, with and without ground-contacting caps. In all these examples the pile loads and group stiffnesses are close to those obtained from other formulations.     

边界元法中区域积分的降维计算方法

§1.引 言 边界元方法是在经典的积分方程法和有限元离散化技术的基础上发展起来的求解偏微分方程的数值计算方法.由于它在几何上的广泛适应性,输入数据的简单性以及在数值上的确定性,这种方法已广泛地应用于不同学科领域及各种工程技术问题的数值计算,其基本的思     

On the use of a SIMD vector extension for the fast evaluation of Boundary Element Method coefficients

The paper reports in detail a methodology to fully exploit the potential of a SIMD (Single Instruction Multiple Data) vector extension in the evaluation of certain type of integrals, which occur in the numerical solution of a Boundary Integral Equation (BIE) through the Boundary Element Method (BEM). Specifically, we present an algorithm for the fast evaluation of the integral coefficients appearing in the assembly of the BEM system matrices, which represents an extremely time-consuming task. The numerical scheme is tailored to the specific structure of the integrals associated to a wave propagation phenomenon, governed, in the time domain, by the D’Alembert equation. The reason of this choice resides in the critical importance achieved by this class of problems in many engineering applications. In particular, the application framework this work belongs to is the design of environmentally friendly commercial aircraft, for which the regulation and certification restrictions are, nowadays, a key constraint effecting even the conceptual phase of the design process. For the sake of generality, we used here only the basic features of the SIMD vector extension, common to all the specific architectures available on the market. Particular attention is payed to the accuracy-related issues arising from the use of the low-latency approximations of some of the operators involved. The resulting algorithm minimizes the number of operations involving operands belonging to the same register (“horizontal” or “intra-register” operations). Preliminary numerical results reveal a remarkable speed-up of this highly-demanding part of the solution process, close, in most of the cases, to the theoretical peak. Standard multithreading techniques are additionally introduced to further increase the performance on multiprocessors machines.     

The development of the Trefftzian methodology for engineering design analysis

This paper reviews the developments in the Trefftzian Methodology, which have been undertaken by the authors at the University of Sheffield during the past fifteen years and the application of these developments to engineering design analysis. Initially, in the late 1970s, this work concentrated on the Direct Boundary Element Method (DBEM) and the Indirect Boundary Element Method (IBEM). Unfortunately these methods, as they are normally formulated, give rise to singular integrals, which require special mathematical treatment, when the source and field points coincide on the boundary of the component being analysed. These singular integrals can however be eliminated by placing the source boundary outside the domain of the problem being analysed so that the field and source points never coincide. This technique is known as either the Regular Direct Boundary Element Method (RDBEM) or the Regular Indirect Boundary Element Method (RIBEM) In a further development of the RIBEM, based on the Trefftz Method, the continuous distribution of sources is replaced with sources distributed at discrete points on the source boundary. This modified Trefftz Method eliminates the integrations in the solution procedure, it provides a series solution in terms of the fundamental solution of the problem being analysed and is referred to as the Indirect Discrete Boundary Method (IDBM). The emergence of the IDBM provided the opportunity to develop a combined Boundary Element Finite Element technique which enables these methods to be used simultaneously in a single calculation, thereby exploiting their strengths and minimising their weaknesses. A number of case studies will be discussed in the paper to illustrate the developments in the Trefftzian Methodology and its application to engineering design analysis.     

基于边界元素法的柔软物体变形模拟

在计算机动画和虚拟现实技术中，基于物理的建模方法是高真实感地模拟物体受力变形和运动的有效途径．近年来基于边界元的物理模型方法因其简捷的计算模式而受到关注，该文针对当前边界元模型在视觉效果和计算量上的一些缺陷，分别提出了两方面的改进方法，基于LOD的动态自适应多分辨率网格边界元模型和近似的非线性边界元的物理模型，分别用于在不损失视觉效果的前提下减少计算量以及模拟物体大变形，并提出了相应的加速算法，取得了较好的效果．     

基于Matlab的径向基函数数值计算软件包

为推动基于无网格方法的计算软件的发展,介绍基于Matlab自主开发的径向基函数（Radial Basis Function,RBF）数值计算软件包,阐述软件的理论基础、设计思路以及该软件包的功能和特点,并结合边界节点法（Boundary Knot Method,BKM）的数值实例给出软件的使用过程．该软件包可以根据不同的数学物理模型选择合适的数值算法来求解多种实际物理问题,也可对不同数值算法得到的结果进行比较．最后,总结应用Matlab进行数值计算软件开发的优缺点．     

二维任意形状寄生电阻电容的边界元计算

由于集成电路技术高速发展,精确提取任意形状寄生电阻电容变得十分重要,本文以直接边界元素法为基础,利用圆弧样条插值近似任意曲线边界,使插值曲线具有整体一阶连续性,并克服了大挠度与多值的困扰。在直线边界使用线性连续元,在曲线边界使用二次连续无。对两邻国为直线的角点,我们曾提出处理角点处存在多重法的导数的一种方法。这里,它被推广到角点邻边含曲线段的情形。数值结果表明模拟器是可行的。     

Meshfree Particle Methods in the Framework of Boundary Element Methods for the Helmholtz Equation

In this paper, we study electromagnetic wave scattering from periodic structures and eigenvalue analysis of the Helmholtz equation. Boundary element method (BEM) is an effective tool to deal with Helmholtz problems on bounded as well as unbounded domains. Recently, Oh et al. (Comput. Mech. 48:27–45, 2011) developed reproducing polynomial boundary particle methods (RPBPM) that can handle effectively boundary integral equations in the framework of the collocation BEM. The reproducing polynomial particle (RPP) shape functions used in RPBPM have compact support and are not periodic. Thus it is not ideal to use these RPP shape functions as approximation functions along the boundary of a circular domain. In order to get periodic approximation functions, we consider the limit of the RPP shape function as its support is getting infinitely large. We show that the basic approximation function obtained by the limit of the RPP shape function yields accurate solutions of Helmholtz problems on circular, or annular domains as well as on the infinite domains.     

Analysis of three-dimensional anisotropic heat conduction problems on thin domains using an advanced boundary element method

In this paper, an advanced boundary element method (BEM) is developed for solving three-dimensional (3D) anisotropic heat conduction problems in thin-walled structures. The troublesome nearly singular integrals, which are crucial in the applications of the BEM to thin structures, are calculated efficiently by using a nonlinear coordinate transformation method. For the test problems studied, promising BEM results with only a small number of boundary elements have been obtained when the thickness of the structure is in the orders of micro-scales (10^?6), which is sufficient for modeling most thin-walled structures as used in, for example, smart materials and thin layered coating systems. The advantages, disadvantages as well as potential applications of the proposed method, as compared with the finite element method (FEM), are also discussed.     

Dual BEM for crack growth analysis on distributed-memory multiprocessors

The Dual Boundary Element Method (DBEM) has been presented as an effective numerical technique for the analysis of linear elastic crack problems [Portela A, Aliabadi MH. The dual boundary element method: effective implementation for crack problems. Int J Num Meth Engng 1992;33:1269–1287]. Analysis of large structural integrity problems may need the use of large computational resources, both in terms of CPU time and memory requirements. This paper reports a message-passing implementation of the DBEM formulation dealing with the analysis of crack growth in structures. We have analyzed the construction of the system and its resolution. Different data distribution techniques have been studied with several problems. Results in terms of scalability and load balance for these two stages are presented in this paper.