Fast parallel solvers for symmetric boundary element domain decomposition equations

Summary. The boundary element method (BEM) is of advantage in many applications including far-field computations in magnetostatics and solid mechanics as well as accurate computations of singularities. Since the numerical approximation is essentially reduced to the boundary of the domain under consideration, the mesh generation and handling is simpler than, for example, in a finite element discretization of the domain. In this paper, we discuss fast solution techniques for the linear systems of equations obtained by the BEM (BE-equations) utilizing the non-overlapping domain decomposition (DD). We study parallel algorithms for solving large scale Galerkin BE–equations approximating linear potential problems in plane, bounded domains with piecewise homogeneous material properties. We give an elementary spectral equivalence analysis of the BEM Schur complement that provides the tool for constructing and analysing appropriate preconditioners. Finally, we present numerical results obtained on a massively parallel machine using up to 128 processors, and we sketch further applications to elasticity problems and to the coupling of the finite element method (FEM) with the boundary element method. As shown theoretically and confirmed by the numerical experiments, the methods are of algebraic complexity and of high parallel efficiency, where denotes the usual discretization parameter. Received August 28, 1996 / Revised version received March 10, 1997     

Time Domain BEM Based on Generalised Convolution Quadrature

We present the application of the generalised convolution quadrature (gCQ) technique to the time domain boundary element method (BEM) which solves the retarded potential boundary integral equation (RPBIE). Our result allows to employ the multi-stage Runge-Kutta method as the time stepping scheme for the generalised convolution quadrature which is used in the time domain BEM for acoustic problems in either a bounded three-dimensional domain or in its unbounded exterior. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

一种有限元-边界元耦合分域算法

提出了一种有限元-边界元耦合分域算法．该算法将所分析问题的区域分解成有限元和边界元子域,在满足两子域界面上位移和面力协调连续的条件下,通过迭代求解得到问题的解．在迭代求解过程中,引入动态松弛系数,使收敛得以加速．该方法在两子域界面上有限单元结点和边界单元结点的位置相互独立,无需协调一致,对诸如裂纹扩展过程的模拟具有独特的优势．用所提出的耦合算法分析算例,得到的结果与有限元法、边界元法和另一种耦合算法的数值计算结果一致,验证了这种算法的正确性和可行性．     

利用边界元法求解一类重调和方程

边界元法(BEM)和多重互易法(MRM)相结合求解一类重调和方程.通过重调和基本解序列给出的MRM-方法和BEM, 推导出该类问题的MRM-边界变分方程, 用边界元法求解该变分方程, 从而得到重调和方程的近似解, 并给出了解的存在唯一性证明.通过数值算例说明了MRM-方法具有收敛速度快、计算精度高, 易编程等优点, 为使用边界元法数值求解重调和方程提供了方法和理论依据.适合于工程中的实际运算.     

三维弹塑性结构下限分析的边界元方法

基于极限分析的下限定理,建立了用常规边界元方法进行三维理想弹塑性结构极限分析的求解算法.下限分析所需的弹性应力场可直接由边界元方法求得.所需的自平衡应力场由一组带有待定系数的自平衡应力场基矢量的线性组合进行模拟,这些自平衡应力场基矢量由边界元弹塑性迭代计算得到.下限分析问题最终被归结为一系列未知变量较少的非线性数学规划子问题并通过复合形法进行求解.给出的计算结果表明该算法有较高的精度和计算效率.     

A double-layer interpolation method for implementation of BEM analysis of problems in potential theory

A double-layer interpolation method (DLIM) is proposed to improve the performance of the boundary element method (BEM). In the DLIM, the nodes of an element are sorted into two groups: (i) nodes inside the element, called source nodes, and (ii) nodes on the vertices and edges of the element, called virtual nodes. With only source nodes, the element becomes a conventional discontinuous element. Taking into account both source and virtual nodes, the element becomes a standard continuous element. The physical variables are interpolated by continuous elements (first-layer interpolation), while the boundary integral equations are collocated at the source nodes only. We further established additional constraint equations between source and virtual nodes using a moving least-squares (MLS) approximation (second-layer interpolation). Using these constraints, a square coefficient matrix of the overall system of linear equations was finally achieved. The DLIM keeps the main advantages of MLS, such as significantly alleviating the meshing task, while providing much better accuracy than the traditional BEM. The method has been used successfully for solving potential problems in two dimensions. Several numerical examples in comparison with other methods have demonstrated the accuracy and efficiency of our method.     

A boundary element method in elastodynamics for geometrically axisymmetric problems

A direct boundary element method (BEM) in elastodynamics is developed for geometrically axisymmetric problems subjected to arbitrary external loads. Traction and displacement components are expressed in terms of Fourier series. Unlike classical BEMs, the kernel functions in the resulting integral governing equations in the present method are expanded as implicit functions of the difference of the polar angles at two field points. The new BEM is therefore capable of solving a wider variety of elastodynamic problems. Two numerical examples in foundation engineering show that the present BEM is also accurate and computationally efficient.     

An insight into the rotational stall delay

Blade element momentum (BEM) theory which is based on the two-dimensional (2D) aerodynamic properties of airfoil blade element is the most common computational engineering method for the prediction of loads and power curves of wind turbines. Although most BEM models yield acceptable results for high tip-speed ratios where the local angles of attack are small, no generally accepted model exists up to date that consistently predicts the loads and power in stall regime for stall-controlled turbines. Understanding of the stall delay phenomenon on wind turbines remains, to this day, incomplete. The lack of a conceptual model for the complex three-dimensional (3D) flow field on the rotor blade, where stall is begins, how it progresses and where stall is practically terminated, has hindered the finding of a unanimously accepted solution. The paper aims at giving a better understanding of the delayed stall events and a reasonably simple correction model that complements the 2D airfoil characteristics used to a BEM method. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Adaptive coupling of FEM and BEM: Simple error estimators and convergence

A posteriori error estimators and adaptive mesh-refinement have themselves proven to be important tools for scientific computing. For error control in finite element methods (FEM), there is a broad variety of a posteriori error estimators available, and convergence as well as optimality of adaptive FEM is well-studied in the literature. This is, however, in sharp contrast to the boundary element method (BEM) and to the coupling of FEM and BEM. In our contribution, we present an easy-to-implement error estimator for some FEM-BEM coupling which, to the best of our knowledge, has not been proposed in the literature before. The derived mesh-refining algorithm provides the first adaptive coupling procedure which is mathematically proven to converge. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Solution of Poisson’s equation by analytical boundary element integration

The solution of Poisson’s equation is essential for many branches of science and engineering such as fluid-mechanics, geosciences, and electrostatics. Solution of two-dimensional Poisson’s equations is carried out by BEM based on Galerkin Vector Method in which the integrals that appear in the boundary element method are expressed by analytical integration. In this paper, the Galerkin vector method is developed for more general case than presented in the previous works. The integrals are computed for constant and linear elements in BEM. By employing analytical integration in BEM computation, the numerical schemes and coordinate transformations can be avoided. The presented method can also be used for the multiple domain case. The results of the analytical integration are employed in BEM code and the obtained analytical expression will be applied to several examples where the exact solution exists. The produced results are in good agreement with the exact solution.     

On the coupling of regularization techniques and the boundary element method for a hemivariational inequality modelling a delamination problem

In this paper, we couple regularization techniques of nondifferentiable optimization with the h‐version of the boundary element method (h‐BEM) to solve nonsmooth variational problems arising in contact mechanics. As a model example, we consider the delamination problem. The variational formulation of this problem leads to a hemivariational inequality with a nonsmooth functional defined on the contact boundary. This problem is first regularized and then discretized by an h‐BEM. We prove convergence of the h‐BEM Galerkin solution of the regularized problem in the energy norm, provide an a priori error estimate and give a numerical examples. Copyright © 2016 John Wiley & Sons, Ltd.     

关于薄板的无网格局部边界积分方程方法中的友解

无网格局部边界积分方程方法是最近发展起来的一种新的数值方法,这种方法综合了伽辽金有限元、边界元和无单元伽辽金法的优点,是一种具有广阔应用前景的、真正的无网格方法．把无网格局部边界积分方程方法应用于求解薄板问题,给出了薄板无网格局部边界积分方程方法所需要的友解及其全部公式．     

BEM‐FEM coupling for the 1D Klein–Gordon equation

A transmission (bidomain) problem for the one‐dimensional Klein–Gordon equation on an unbounded interval is numerically solved by a boundary element method‐finite element method (BEM‐FEM) coupling procedure. We prove stability and convergence of the proposed method by means of energy arguments. Several numerical results are presented, confirming theoretical results. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 2042–2082, 2014     

On the adaptive coupling of FEM and BEM in 2–d–elasticity

Summary. This paper concerns the combination of the finite element method (FEM) and the boundary element method (BEM) using the symmetric coupling. As a model problem in two dimensions we consider the Hencky material (a certain nonlinear elastic material) in a bounded domain with Navier–Lamé differential equation in the unbounded complementary domain. Using some boundary integral operators the problem is rewritten such that the Galerkin procedure leads to a FEM/BEM coupling and quasi–optimally convergent discrete solutions. Beside this a priori information we derive an a posteriori error estimate which allows (up to a constant factor) the error control in the energy norm. Since information about the singularities of the solution is not available a priori in many situation and having in mind the goal of an automatic mesh–refinement we state adaptive algorithms for the –version of the FEM/BEM–coupling. Illustrating numerical results are included. Received April 15, 1994 / Revised version received January 8, 1996     

A meshless scaling iterative algorithm based on compactly supported radial basis functions for the numerical solution of Lane‐Emden‐Fowler equation

In this article, we combine the compactly supported radial basis function (RBF) collocation method and the scaling iterative algorithm to compute and visualize the multiple solutions of the Lane‐Emden‐Fowler equation on a bounded domain Ω ? R² with a homogeneous Dirichlet boundary condition. This novel method has the advantage over traditional methods, which approximate the spatial derivatives using either the finite difference method (FDM), the finite element method (FEM), or the boundary element method (BEM), because it does not require a mesh over the domain. As a result, it needs less computational time than the globally supported RBF collocation method. When compared with the reference solutions in (Chen, Zhou, and Ni, Int J Bifurcation Chaos 10 (2000), 565–1612), our numerical results demonstrate the accuracy and ease of implementation of this method. It is therefore much more suitable for dealing with the complex domains than the FEM, the FDM, and the BEM. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 554‐572, 2012     

Numerical evaluation of various levels of singular integrals, arising in BEM and its application in hydrofoil analysis

The sound implementation of the boundary element method (BEM) is highly dependent on an accurate numerical integration of singular integrals. In this paper, a set of various types of singular domain integrals with three-dimensional boundary element discretization is evaluated based on a transformation integration technique. In the BEM, the integration domain (body surface) needs to be discretized into small elements. For each element, the integral I(x^p, x) is calculated on the domain dS. Several types of integrals IB_α and IC_α are numerically and analytically computed and compared with the relative error. The method is extended to evaluate singular integrals which arise in the solution of the three-dimensional Laplace’s equation. An example of the elliptic hydrofoil is performed to study the physical accuracy. The results obtained using both numerical and analytical methods are shown in good agreement with the experimental data.     

正交各向异性半平面内作用集中载荷的弹性解及边界元法常单元基本公式

本文采用镜相法,推导出了正交各向异性半平面作用集中载荷的理论解,给出了常单元系数矩阵表达式,为采用边界元法求解半平面问题提供了必要的公式.特解表达形式简洁,对边界元间接法常单元和高次单元各积分均可求出其原函数,可避免计算程序中的定积分数值计算过程.     

Interface problem in holonomic elastoplasticity

The three-dimensional interface problem with the homogeneous Lamé system in an unbounded exterior domain and holonomic material behaviour in a bounded interior Lipschitz domain is considered. Existence and uniqueness of solutions of the interface problem are obtained rewriting the exterior problem in terms of boundary integral operators following the symmetric coupling procedure. The numerical approximation of the solutions consists in coupling of the boundary element method (BEM) and the finite element method (FEM). A Céa-like error estimate is presented for the discrete solutions of the numerical procedure proving its convergence.     

Elastic-plastic stress analysis of cracked structures using the boundary element method

The application of the boundary element method (BEM) for the 3D-stress analysis of cracked structures considering elastic-plastic material behavior is presented. For separating the coincident crack surfaces the DUAL-BEM is utilized. The relevant boundary integral equations (BIE) – the strongly singular displacement BIE and the hypersingular traction BIE – are evaluated in the framework of a collocation procedure. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An Improved Implementation of An Iterative Method in Boundary Identification Problems

In this paper, an inverse problem of determining geometric shape of a part of the boundary of a bounded domain is considered. Based on a conjugate gradient method, employing the adjoint equation to obtain the descent direction, an identification scheme is developed. The implementation of the method based on the boundary element method (BEM) is also discussed. This method solves the inverse boundary problem accurately without a priori information about the unknown shape to be estimated.