A new a posteriori error estimator in adaptive direct boundary element methods: the Dirichlet problem

In this paper we propose a new a posteriori error estimator for a boundary element solution related to a Dirichlet problem with a second order elliptic partial differential operator. The method is based on an approximate solution of a boundary integral equation of the second kind by a Neumann series to estimate the error of a previously computed boundary element solution. For this one may use an arbitrary boundary element method, for example, a Galerkin, collocation or qualocation scheme, to solve an appropriate boundary integral equation. Due to the approximate solution of the error equation the proposed estimator provides high accuracy. A numerical example supports the theoretical results. Received: June 1999 / Accepted: September 1999     

A symmetric variational finite element-boundary integral equation coupling method

This paper is concerned with the development of a mixed variational principle for coupling finite element and boundary integral methods in interface problems, using the generalized Poisson's equation as a prototype situation. One of its primary objectives is to compare the performance of fully variational procedures with methods that use collocation for the treatment of boundary integral equations. A distinctive feature of the new variational principle is that the discretized algebraic equations for the coupled problem are automatically symmetric since they are all derived from a single functional. In addition, the condition that the flux remain continuous across interfaces is satisfied naturally. In discretizing the problem within inhomogeneous or loaded regions, domain finite elements are used to approximate the field variable. On the other hand, only boundary elements are used for regions where the medium is homogeneous and free of external agents. The corresponding integral equations are discretized both by fully variational and by collocation techniques. Results of numerical experiments indicate that the accuracy of the fully variational procedure is significantly greater than that of collocation for the complete interface problem, especially for complex disturbances, at little additional computational cost. This suggests that fully variational procedures may be preferable to collocation, not only in dealing with interface problems, but even for solving integral equations by themselves.     

The exterior problem for the Helmholtz equation with mixed boundary conditions in three dimensions

In this article, a new integral equation is derived to solve the exterior problem for the Helmholtz equation with mixed boundary conditions in three dimensions, and existence and uniqueness is proven for all wave numbers. We apply the boundary element collocation method to solve the system of Fredholm integral equations of the second kind, where we use constant interpolation. We observe superconvergence at the collocation nodes and illustrate it with numerical results for several smooth surfaces.     

Richardson extrapolation applied to boundary element method results in a Dirichlet problem for the Laplace equation

Richardson extrapolation is used to improve the accuracy of the numerical solutions for the normal boundary flux and for the interior potential resulting from the boundary element method. The boundary integral equations arise from a direct boundary integral formulation for solving a Dirichlet problem for the Laplace equation. The Richardson extrapolation is used in two different applications: (i) to improve the accuracy of the collocation solution for the normal boundary flux and, separately, (ii) to improve the solution for the potential in the domain interior. The main innovative aspects of this work are that the orders of dominant error terms are estimated numerically, and that these estimates are then used to develop an a posteriori technique that predicts if the Richardson extrapolation results for applications (i) and (ii) are reliable. Numerical results from test problems are presented to demonstrate the technique.     

Green element computation of the Sturm-Liouville equations

A mathematical derivation of a new numerical procedure called the Green element method (GEM) is presented and applied to the solution of Sturm-Liouville problems. The GEM is a numerical technique which expands the scope of application of the boundary element method (BEM) by implementing the singular boundary integral theory in an element-by-element fashion; and like the finite element method (FEM) gives rise to a banded coefficient matrix which is easy to handle numerically. For this application, the location of both the field and the source nodes within the same element makes it possible for integrations to be carried out accurately, thereby enhancing the accuracy of discrete equations. The method is therefore easy to apply and, because of its domain based implementation, it maintains the flexibility of the FEM. We apply the GEM to the solution of boundary value differential equations which represent the form of Sturm-Liouville problems, and its capability is demonstrated by comparing the results with those of the finite element methods available in the literature. Satisfactory results and a second-order accuracy were found to be exhibited.     

A coupled formulation of finite and boundary element methods in two-dimensional elasticity

A symmetric stiffness formulation based on a boundary element method is studied for the structural analysis of a shear wall, with or without cutouts. To satisfy compatibility requirements with finite beam elements and to avoid problems due to the eventual discontinuities of the traction vector, different interpolation schemes are adopted to approximate the boundary displacements and tractions. A set of boundary integral equations is obtained with the collocation points on the boundary, which are selected by the error minimization technique proposed in this paper, and the stiffness matrix is formulated with those equations and symmetric coupling techniques of finite and boundary element methods. The newly developed plane stress element can have the openings in its interior domain and can be easily linked with the finite beam/column elements.     

等几何分析与有限元直接耦合法

等几何分析(IsoGeometric Analysis,IGA)具有几何模型精确,分析精度高和收敛速度快等优点,但积分效率不高、边界条件处理复杂.因此,提出一种IGA与有限元直接耦合的方法,将有限元与IGA交界处节点自由度用等几何控制节点直接表示出来,从而形成IGA与有限元的耦合单元.算例分析表明,该方法与常规有限元法相比具有精度高的特点,与IGA相比具有施加边界条件方便的优点.     

A modified boundary integral evolution formulation for the wave equation

We apply a modified boundary integral formulation otherwise known as the Green element method (GEM) to the solution of the two-dimensional scalar wave equation.GEM essentially combines three techniques namely: (a) finite difference approximation of the time term (b) finite element discretization of the problem domain and (c) boundary integral replication of the governing equation. These unique and advantageous characteristics of GEM facilitates a direct numerical approximation of the governing equation and obviate the need for converting the governing partial differential equation to a Helmholtz-type Laplace operator equation for an easier boundary element manipulation. C₁ continuity of the computed solutions is established by using Overhauser elements. Numerical tests show a reasonably close agreement with analytical results. Though in the case of the Overhauser GEM solutions, the level of accuracy obtained does not in all cases justify the extra numerical rigor.     

Bi-expansion approximation of the Bessel functions applied to analytical integration based BEM for Reissner plates

The paper deals with the boundary element analysis of plates modeled by the Reissner theory. The integral equations are obtained in standard fashion, by boundary collocation. The fundamental solution is expressed in terms of modified Bessel functions, using different series expansions for small and large arguments. The transition from one to the other form is regulated by an error-based criterion for selecting the number of terms which have to be used in the series expansion. Attention is paid to the computational aspects influencing the accuracy of the numerical model. The boundary variables are described by using a quadratic B-spline interpolation and the evaluation of the boundary integrals is developed almost entirely in analytical form. Some numerical experiments test and compare the performances of the numerical model. In particular the solution within the boundary layer is studied.     

The coupling of boundary integral and finite element methods for the exterior steady Oseen problem

In this article, we present a new numerical method for solving the steady Oseen equations in an unbounded plane domain. The technique consists in coupling the boundary integral and the finite element methods. An artificial smooth boundary is introduced separating an interior inhomogeneous region from an exterior homogeneous one. The solution in exterior region is represented by an integral equation over the artificial boundary. This integral equation is incorporated into a velocity-pressure formulation for the interior region, and a finite element method is used to approximate the resulting variational problem. Finally, the optimal error estimates of the numerical solution are derived.Computer results will be discussed in a forthcoming paper.     

SMB色谱模型数值计算的时空守恒元解元方法

为了探寻出一种求解SMB色谱模型的快速数值求解方法,并试图通过比较得出时空守恒元/解元(CE/SE)方法确实是快速数值求解方法,因而采用该方法对SMB色谱模型进行数值求解,并在数值方法的计算效率和精确度两个方面与有限差分法和正交配置有限元法进行了比较,最终得出了CE/SE方法是具有高计算效率和高精确度特性的快速数值求解方法.通过两个实例的模拟仿真,结果表明了该方法在高计算效率和高精确度方面的优越性.     

Combination of the boundary integral equation method and the extrapolation method

Many numerical methods including the boundary integral equation method start with division of the domain of calculation into intervals. The accuracy of their results can be improved considerably by extrapolation. To be able to apply the extrapolation method it is necessary to know the asymptotic expansion of the error.In this paper the principle of the extrapolation method and subjects important for its application are described. Above all it is shown how to determine the asymptotic expansions numerically by trial and error. In the first sections the matter is explained in a general manner to encourage users of various numerical methods—among them users of the finite element method—to try to extrapolate their results. Then the investigations are exemplified in detail by the boundary integral equation method. The accuracy of approximate solutions of integral equations for plane elastostatic problems with prescribed boundary tractions and displacements is improved by extrapolation. Particular attention is paid to boundary tractions and displacements with discontinuous derivatives.To induce also practically orientated readers without specialized mathematical knowledge to think about applying the extrapolation method the basic topics are represented in an extensive manner and illustrated by simple examples. (For a survey of this paper see end of Section 1.)     

An explicit-implicit water-level model for tidal computations of rivers

The combined use of explicit and implicit time integration methods leads to a numerical model, based on the finite element method, that overcomes the often claimed advantages of finite difference methods. The implicit part of the model shows the whole range of flexibility the finite element method is famous for, whereas the explicit part reduces to simple difference quotients for regular and irregular discretizations. The stability of the numerical solution is ensured and high numerical damping is avoided by means of a generalized collocation method for the integrations in space.     

求解应力强度因子的样条虚边界元交替法

为求解平面裂纹问题的应力强度因子,提出基于Muskhelishvili基本解和样条虚边界元法的样条虚边界元交替法.该方法将平面内带裂纹有限域问题分解成带裂纹无限域问题与不带裂纹有限域问题的叠加.带裂纹无限域问题利用Muskhelishvili基本解法直接得出,不带裂纹有限域问题采用样条虚边界元法求解.利用该方法对复合型中心裂纹方板和I型偏心裂纹矩形板进行分析.数值结果表明该方法精度高且适用性强.     

Boundary element method for membrane and torsion crack problems

Here we present as an application of [12] an improved Galerkin method for the boundary integral equations governing a plane interface problem. Membrane and torsion crack problems can be treated by slight modifications.A tedious analysis incorporating the Mellin transform shows that the coupled system of integral equations—with some Fredholm equations of the second kind and some of the first kind—on the boundary curve Γ is strongly elliptic, i.e., there holds a Gårding inequality. This property implies convergence of almost optimal order of the Galerkin procedure. The use of singularity functions together with regular finite elements on Γ provides convergence results in a scale of Sobolev spaces and even quasi-optimal asymptotic error estimates for the stress intensity factors. These factors are computed directly by our Galerkin scheme, i.e., no additional computations are needed.The Galerkin method is implemented by an appropriate numerical integration leading to a Galerkin collocation. The latter is a modified collocation method which can be easily implemented on computing machines.     

油藏数值模拟有限元并行计算方法研究

针对目前油藏数值模拟普遍采用的有限差分法计算精度低的问题,提出了兼顾计算精度、计算速度问题的有限元油藏数值模拟方法,即在建立了油藏数值模拟数学模型的基础上通过有限元数值分析方法建立有限元数值模型,但有限元在油藏数值模拟时存在单机计算困难、计算时间长的问题,为此提出了利用区域分解技术的油藏数值模拟并行计算方法,最后将该方法通过实例进行检验,取得了良好的加速比和并行效率。     

Convergence and accuracy of the path integral approach for elastostatics

This paper addresses convergence rate and accuracy of a numerical technique for linear elastostatics based on a path integral formulation [Int. J. Numer. Math. Eng. 47 (2000) 1463]. The computational implementation combines a simple polynomial approximation of the displacement field with an approximate statement of the exact evolution equations, which is designated as functional integral method. A convergence analysis is performed for some simple nodal arrays. This is followed by two different estimations of the optimum parameter ζ: one is based on statistical arguments and the other on inspection of third order residuals. When the eight closest neighbors to a node are used for polynomial approximation the optimum parameter is found to depend on Poisson's ratio and lie in the range 0.5<ζ<1.5. Two well established numerical methods are then recovered as specific instances of the FIM. The strong formulation––point collocation––corresponds to the limit ζ=0 while bilinear finite elements corresponds exactly to the choice ζ=0.5. The use of the optimum parameter provides better precision than the other two methods with similar computational cost. Other nodal arrays are also studied both in two and three dimensions and the performance of the FIM compared with the corresponding finite element and collocation schemes. Finally, the implementation of FIM on unstructured meshes is discussed, and a numerical example solving Laplace equation is analyzed. It is shown that FIM compares favorably with FEM and offers a number of advantages.     

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.     

Numerical investigation on convergence of boundary knot method in the analysis of homogeneous Helmholtz, modified Helmholtz, and convection-diffusion problems

This paper concerns a numerical study of convergence properties of the boundary knot method (BKM) applied to the solution of 2D and 3D homogeneous Helmholtz, modified Helmholtz, and convection-diffusion problems. The BKM is a new boundary-type, meshfree radial function basis collocation technique. The method differentiates from the method of fundamental solutions (MFS) in that it does not need the controversial artificial boundary outside physical domain due to the use of non-singular general solutions instead of the singular fundamental solutions. The BKM is also generally applicable to a variety of inhomogeneous problems in conjunction with the dual reciprocity method (DRM). Therefore, when applied to inhomogeneous problems, the error of the DRM confounds the BKM accuracy in approximation of homogeneous solution, while the latter essentially distinguishes the BKM, MFS, and boundary element method. In order to avoid the interference of the DRM, this study focuses on the investigation of the convergence property of the BKM for homogeneous problems. The given numerical experiments reveal rapid convergence, high accuracy and efficiency, mathematical simplicity of the BKM.     

Numerical results of Emden–Fowler boundary value problems with derivative dependence using the Bernstein collocation method

In this paper, we propose an efficient numerical technique based on the Bernstein polynomials for the numerical solution of the equivalent integral form of the derivative dependent Emden–Fowler boundary value problems which arises in various fields of applied mathematics, physical and chemical sciences. The Bernstein collocation method is used to convert the integral equation into a system of nonlinear equations. This system is then solved efficiently by suitable iterative method. The error analysis of the present method is discussed. The accuracy of the proposed method is examined by calculating the maximum absolute error and the \(L_{2}\) error of four examples. The obtained numerical results are compared with the results obtained by the other known techniques.