Quadrature for parabolic Galerkin BEM with moving surfaces

The successful implementation of the Galerkin Boundary Element Method hinges on the accurate and effective quadrature of the influence coefficients. For parabolic boundary integral operators quadrature must be performed in space and time where integrals have singularities when source- and evaluation points coincide. For problems where the surface is fixed, the time integration can be performed analytically, but for moving geometries numerical quadrature in space and time must be used. For this case a set of transformations is derived that render the singular space–time integrals into smooth integrals that can be treated with standard tensor product Gauss quadrature rules. This methodology can be applied to the heat equation and to transient Stokes flow.     

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.     

Global and local Trefftz boundary integral formulations for sound vibration

The sound-pressure field harmonically varying in time is governed by the Helmholtz equation. The Trefftz boundary integral equation method is presented to solve two-dimensional boundary value problems. Both direct and indirect BIE formulations are given. Non-singular Trefftz formulations lead to regular integrals counterpart to the conventional BIE with the singular fundamental solution. The paper presents also the local boundary integral equations with Trefftz functions as a test function. Physical fields are approximated by the moving least-square in the meshless implementation. Numerical results are given for a square patch test and a circular disc.     

Recent developments in the Trefftz method for finite element and boundary element applications

In 1926 E. Trefftz published a paper about a variational formulation which utilizes boundary integrals. Almost half a century later researchers became interested again in the ideas of Trefftz when the potential advantage of the Trefftz-method for an efficient use in numerical application on a computer was recognized. The concept of Trefftz can be used both for finite element and boundary element applications. A crucial ingredient of the Trefftz- method is a set of linearly independent trial functions which a priori satisfy the governing differential equations under consideration. In this paper an overview of some recent developments to construct trial functions for the Trefftz-method in a systematic manner is given. Using different types of approximation functions (singular or non-singular) we can obtain very accurate finite element and boundary element algorithms.     

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.     

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     

Trefftz direct method

This paper discusses a so-called Trefftz Direct Method (TDM), in which the non-singular boundary integral equation is used as the starting formulation and complete sets of solution are used as weighting functions. Several relevant characteristics and problems of this approximation are also discussed.     

A semi-analytical algorithm for the evaluation of the nearly singular integrals in three-dimensional boundary element methods

The nearly singular integrals occur in the boundary integral equations when the source point is close to an integration element (as compared to its size) but not on the element. In this paper, the concept of a relative distance from a source point to the boundary element is introduced to describe possible influence of the singularity of the integrals. Then a semi-analytical algorithm is proposed for evaluating the nearly strongly singular and hypersingular integrals in the three-dimensional BEM. By using integration by parts, the nearly singular surface integrals on the elements are transformed to a series of line integrals along the contour of the element. The singular behavior, which appears as factor, is separated from remaining regular integrals. Consequently standard numerical quadrature can provide very accurate evaluation of the resulting line integrals. The semi-analytical algorithm is applied to analyzing the three-dimensional elasticity problems, such as very thin-walled structures. Meanwhile, the displacements and stresses at the interior points very close to its bounding surface are also determined efficiently. The results of the numerical investigation demonstrate the accuracy and effectiveness of the algorithm.     

Transformation Methods for the Numerical Integration of Three-Dimensional Singular Functions

The implementations of the eXtended Finite Element Method and the Boundary Element Method need to face the challenge of integrating singular functions. Since standard quadrature techniques usually produce inaccurate results, a number of specific algorithms have been developed to address this problem. We present a general framework for the systematic formulation of the three-dimensional case. The classical cubic transformation is also considered, including an analytical optimization of its parameters for improved practical efficiency.     

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.     

Trefftz-Herrera domain decomposition

The author's algebraic theory of boundary value problems has permitted systematizing Trefftz method and expanding its scope. The concept of TH-completeness has played a key role for such developments. This paper is devoted to revise the present state of these matters. Starting from the basic concepts of the algebraic theory, Green-Herrera formulas are presented and Localized Adjoint Method (LAM) derived. Then the classical Trefftz method is shown to be a particular case of LAM. This leads to a natural generalization of Trefftz method and a special class of domain decomposition methods: Trefftz-Herrera domain decomposition.     

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．     

A natural stress boundary integral equation for calculating the near boundary stress field

The stress computational accuracy of internal points by conventional boundary element method becomes more and more deteriorate as the points approach to the boundary due to the nearly singular integrals including nearly strong singularity and hyper-singularity. For calculating the boundary stress, a natural boundary integral equation in which the boundary variables are the displacements, tractions and natural boundary variables was established in the authors’ previous work. Herein, a natural stress boundary integral equation (NSBIE) is further proposed by introducing the natural variables to analyze the stress field of interior points. There are only nearly strong singular integrals in the NSBIE, i.e., the singularity is reduced by one order. The regularization algorithm proposed by the authors is taken over to deal with these singular integrals. Consequently, the NSBIE can analyze the stress field closer to the boundary. Numerical examples demonstrated that two orders of magnitude improvement in reducing the approaching degree can be achieved by NSBIE compared to the conventional one when the near boundary stress field is evaluated. Furthermore, this new way is extended to the multi-domain elasticity problem to calculate the stress field near the boundary and interface.     

Use of Trefftz functions in non-linear BEM/FEM

Simulation of many practical problems requires to use non-linear formulations with large displacements, large strains and large rotations. It is well known that the use of Trefftz (T-) functions (i. e. the functions satisfying the governing equations inside the domain) as weighting, or interpolation functions leads to more efficient formulations than those obtained by classical methods. In this paper we will show the use of T-functions and especially T-polynomials, Kelvin, or Kupradze and Boussinesq functions (Green functions with singularity points defined outside of the domain) and their combination in connection with the total Lagrangian formulation for multi-domain BEM (reciprocity based FEM) analysis of displacements and for the post-processing phase in the analysis (evaluation of both gradient of displacements and stress fields). The formulation results in non-singular boundary integrals which has numerical advantages over other formulations using singular boundary integral equations.     

Direct Expansion Method of Boundary Condition for solving 3D elliptic equations with small parameters in the irregular domain

In this article, a new methodology, Direct Expansion Method of Boundary Condition (DEMBC), is developed to solve 3D elliptic equations in the irregular domain. First, the previous Rational Differential Quadrature Method (Rational Spectral Collocation Method in (Berrut et al. 2005) [8]), developed by Berrut et al. (2005) [8], has been generalized to solve 3D elliptic equations. Second, it is showed that Direct Expansion Method of Boundary Condition is capable of handling boundary problems with higher efficiency. Finally, with the help of conformal mapping (Tee and Trefethen, 2006) [9] and domain decomposition method, DEMBC and 3D-RDQM are able solve three kinds of 3D elliptic equations with small parameters in the irregular domain. Numerous test results justify the accuracy and efficiency of our approach.     

Analytical integral algorithm applied to boundary layer effect and thin body effect in BEM for anisotropic potential problems

A new completely analytical integral algorithm is proposed and applied to the evaluation of nearly singular integrals in boundary element method (BEM) for two-dimensional anisotropic potential problems. The boundary layer effect and thin body effect are dealt with. The completely analytical integral formulas are suitable for the linear and non-isoparametric quadratic elements. The present algorithm applies the analytical formulas to treat nearly singular integrals. The potentials and fluxes at the interior points very close to boundary are evaluated. The unknown potentials and fluxes at boundary nodes for thin body problems with the thickness-to-length ratios from 1E−1 to 1E−8 are accurately calculated by the present algorithm. Numerical examples on heat conduction demonstrate that the present algorithm can effectively handle nearly singular integrals occurring in boundary layer effect and thin body effect in BEM. Furthermore, the present linear BEM is especially accurate and efficient for the numerical analysis of thin body problems.     

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.     

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.     

Evaluation of singular integrals in the symmetric Galerkin boundary element method

The implementation of the symmetric Galerkin boundary element method (SGBEM) involves extensive work on the evaluation of various integrals, ranging from regular integrals to hypersingular integrals. In this paper, the treatments of weak singular integrals in the time domain are reviewed, and analytical evaluations for the spatial double integrals which contain weak singular terms are derived. A special scheme on the allocation of Gaussian integration points for regular double integrals in the SGBEM is developed to improve the efficiency of the Gauss–Legendre rule. The proposed approach is implemented for the two-dimensional elastodynamic problems, and two numerical examples are presented to verify the accuracy of the numerical implementation.