On the symmetric boundary element method and the symmetric coupling of boundary elements and finite elements

The finite element method and the boundary element method areamong the most frequently applied tools in the numerical treatmentof partial differential equations. However, their propertiesappear to be complementary: while the boundary element methodis appropriate for the most important linear partial differentialequations with constant coefficients in bounded or unboundeddomains, the finite element method seems to be more appropriatefor inhomogeneous or even nonlinear problems. but is somehowrestricted to bounded domains. The symmetric coupling of thetwo methods inherits the advantages of both methods. This paper treats the symmetric coupling of finite elementsand boundary elements for a model transmission problem in twoand three dimensions where we have two domains: a bounded domainwith nonlinear, even plastic material behaviour, is surroundedby an unbounded, exterior, domain with isotropic homogeneouslinear elastic material. Practically. the coupling is performedsuch that the boundary element method contributes a macro-element,like a large finite element, within a standard finite elementanalysis program. Emphasis is on two-dimensional problems wherethe approach using the Poincaré-Steklov operator seemsto be impossible at first glance. E-mail: cc{at}numerik.uni-kiel.de     

Application of orthogonal collocation on finite elements for solving non-linear boundary value problems

A simple, convenient and easy approach to solve non-linear boundary value problems (BVP) using orthogonal collocation on finite elements (OCFE) is presented. The algorithm is the conjunction of finite element method (FEM) and orthogonal collocation method (OCM). The stability of the numerical results is checked by a novel algorithm which not only justifies the stability of the results but also checks the convergence of the method. The method is applied to the non-symmetric boundary value problems having Dirichlet’s and mixed Robbin’s boundary conditions.     

A mixed finite element method for fourth order eigenvalue problems

A mixed finite element method for approximating eigenpairs of IV order elliptic eigenvalue problems with Dirichlet boundary conditions has been given. The method can be applied to the vibration analysis of anisotropic/orthotropic/isotropic/biharmonic plates. Computer implementation procedures for this mixed method are given along with the results of numerical experiments.     

Interaction of multiple cracks in a rectangular plate

This paper is concerned with interaction of multiple cracks in a finite plate by using the hybrid displacement discontinuity method (a boundary element method). Detail solutions of the stress intensity factors (SIFs) of the multiple-crack problems in a rectangular plate are given, which can reveal the effect of geometric parameters of the cracked body on the SIFs. The numerical results reported here illustrate that the boundary element method is simple, yet accurate for calculating the SIFs of multiple crack problems in a finite plate.     

Three-dimensional simulation of flat rolling through a combined finite element–boundary element approach

This paper presents an innovative approach for analysing three-dimensional flat rolling. The proposed approach is based on a solution resulting from the combination of the finite element method with the boundary element method. The finite element method is used to perform the rigid–plastic numerical modelling of the workpiece allowing the estimation of the roll separating force, rolling torque and contact pressure along the surface of the rolls. The boundary element method is applied for computing the elastic deformation of the rolls. The combination of the two numerical methods is made using the finite element solution of the contact pressure along the surface of the rolls to define the boundary conditions to be applied on the elastic analysis of the rolls. The validity of the proposed approach is discussed by comparing the theoretical predictions with experimental data found in the literature.     

Canonical Boundary Element Method for Plane Elasticity Problems

In this paper, we apply the canonical boundary reduction, suggested by Feng Kang, to the plane elasticity problems, find the expressions of canonical integral equations and Poisson integral formulas in some typical domains. We also give the numerical method for solving these equations together with their convergence and error estimates. Coupling with classical finite element method, this method can be applied to other domains.     

Enriched finite element spaces for transient conduction heat transfer

This paper presents an alternative approach via finite elements to treat numerically the thermal shocks in heat transfer finite element analysis. The method consists in using the standard enriched finite element approaches with time-interpolation. It will be applied here to the transient conduction heat equation where the classical Galerkin method is shown to be unstable. The proposed method consists in adding and eliminating bubbles to the finite element space and then to interpolate the solution to the real time step. This modification is equivalent to the addition of a stabilizing term tuned by a local time-dependent stability parameter, which ensures an oscillating-free solution. To validate this approach, the numerical results obtained in classical 2D and 3D benchmark problems are compared with the Galerkin and the analytical solutions.     

The Dual Singular Function Method for 2D Boundary Integral Equations

The accuracy of standard boundary element methods for elliptic boundary value problems deteriorates if the boundary of the domain contains corners or if the boundary conditions change along the boundary. Here we first investigate the convergence behaviour of standard spline Galerkin approximation on quasi-uniform meshes for boundary integral equations on polygonal domains. It turns out, that the order of convergence depends on some constant describing the singular behaviour of solutions near corner points of the boundary. In order to recover the full order of convergence for the Galerkin approximation we propose the dual singular function method which is often used for improving the accuracy of finite element methods. The theoretical convergence results are confirmed and illustrated by a numerical example.     

Stability of a finite element method for 3D exterior stationary Navier-Stokes flows

We consider numerical approximations of stationary incompressible Navier-Stokes flows in 3D exterior domains, with nonzero velocity at infinity. It is shown that a P1-P1 stabilized finite element method proposed by C. Rebollo: A term by term stabilization algorithm for finite element solution of incompressible flow problems, Numer. Math. 79 (1998), 283–319, is stable when applied to a Navier-Stokes flow in a truncated exterior domain with a pointwise boundary condition on the artificial boundary.     

Hybrid-conventional finite element for gradient-dependent plasticity

The hybrid-conventional finite element method is applied to the analysis of quasi-static, gradient-dependent elastoplastic problems in solid mechanics. The stresses within the element domain and the displacements on the boundary are simultaneously and independently approximated using Trefftz constraints, which lead to boundary integrals. The plastic multipliers are conventionally approximated with regard to C₀ continuity of the multiplier field of the gradient-dependent plasticity. The finite element formulation is derived using a Galerkin-weighted residual approach. The plastic boundary conditions are examined and plastic radiations are set to zero on the plastic boundaries. The effectiveness of the present method is demonstrated with three numerical applications.     

复合材料层合板的弯曲与振动

本文应用应力杂交有限元方法分析了复合材料层合板的弯曲与振动.在本文中,首先根据修正的余能变分原理,构造了一个适合于复合材料层合板特点的矩形应力杂交板弯曲单元.在单元内,分层假设应力参数,在单元的边界上,根据YNS理论的假设确定边界位移场.这样使得构造出来的单元不仅能够考虑横向剪切变形的影响和局部扭曲效应,而且具有较少的自由度数.其次,用此单元求解了层合板的弯曲与振动问题,并将计算结果与精确解进行了比较,比较表明二者非常接近.这说明了在计算方面本文单元具有较高的精确度.     

On the solution a of second order singularly-perturbed boundary value problem by the Sinc-Galerkin method

Sinc methods are now recognized as an efficient numerical method for problems whose solutions may have singularities, or infinite domains, or boundary layers. This work deals with the Sinc-Galerkin method for solving second order singularly perturbed boundary value problems. The method is then tested on linear and nonlinear examples and a comparison with spline method and finite element scheme is made. It is shown that the Sinc-Galerkin method yields better results.     

On the use of physical spline finite element method for acoustic scattering

This paper presents the comparison of physical spline finite element method (PSFEM), in which differential equations are incorporated into interpolations of basic elements, with least-squares finite element method (LSFEM) and mixed Galerkin finite element method (MGFEM) on the numerical solution of one dimensional Helmholtz equation applied to an acoustic scattering problem. Firstly, all three methods are explained in detail and then it is shown that PSFEM reaches higher precision in a shorter time with fewer nodes than the other methods. It is also observed that this method is well suited for high frequency acoustic problems. Consequently, the results of PSFEM point out better efficiency in terms of number of unknowns and accuracy level.     

无界区域上基于自然边界归化的一种区域分解算法

无界区域上基于自然边界归化的一种区域分解算法余德浩（中国科学院计算中心）ＡＤＯＭＡＩＮＤＥＣＯＭＰＯＳＩＴＩＯＮＭＥＴＨＯＤＢＡＳＥＤＯＮＴＨＥＮＡＴＵＲＡＬＢＯＵＮＤＡＲＹＲＥＤＵＣＴＩＯＮＯＶＥＲＵＮＢＯＵＮＤＥＤＤＯＭＡＩＮ￥ＹｕＤｅ－ｈａｏ（...     

一种新的声椭球无限单元

提出了一种新的声椭球无限单元．这种声无限单元基于一种新的声压表达式,这种声压表达式能够更准确地代表着椭球声场的声传播模式．这种新方法的形函数类似于Burnett方法,而权函数定义为形函数和一个附加因子的乘积．因为仅需要一维的数值积分,这种新方法的代码生成十分容易,就像处理一维单元一样．耦合标准的有限元程序,这种声无限单元理论上能够高效地求解任何形状的声源的声辐射和声散射现象．简要地推导了这种新方法,并给出了这种方法详尽的推导结果．为更有效地检验该无限元方法的可行性,文中例子仅考虑无限元求解的精度,而不包括相应的有限元．使用这种新方法,精确地推导出了摆动球的理论计算公式．而长旋转椭球的例子则表明了这种方法优于边界元方法和其他声椭球无限元方法．这些例子表明了这种新方法是切实可行的．     

Analysis of traffic flow by the finite element method

A finite elernent methodology is developed for the numerical solution of traffic flow problems encountered in arterial streets. The simple continuum traffic flow model consisting of the equation of continuity and an equilibrium flow-density relationship is adopted. A Galerkin type finite element method is used to formulate the problem in discrete form and the solution is obtained by a step-by-step time integration in conjunction with the Newton-Raphson method. The proposed finite element methodology, which is of the shock capturing type, is applied to flow traffic problems. Two numerical examples illustrate the method and demonstrate its advantages over other analytical or numerical techniques.     

A full multigrid method for semilinear elliptic equation

A full multigrid finite element method is proposed for semilinear elliptic equations. The main idea is to transform the solution of the semilinear problem into a series of solutions of the corresponding linear boundary value problems on the sequence of finite element spaces and semilinear problems on a very low dimensional space. The linearized boundary value problems are solved by some multigrid iterations. Besides the multigrid iteration, all other efficient numerical methods can also serve as the linear solver for solving boundary value problems. The optimality of the computational work is also proved. Compared with the existing multigrid methods which need the bounded second order derivatives of the nonlinear term, the proposed method only needs the Lipschitz continuation in some sense of the nonlinear term.     

Six combinations of the Ritz-Galerkin and finite element methods for elliptic boundary value problems

Overall comparisons are made for six efficient combinations of the Ritz-Galerkin and finite element methods for solving elliptic boundary value problems with singularities or interfaces. The comparisons are done by using theoretical analysis and numerical experiments. Significant relations among the six combinations are also found. A survey of the six combinations and their coupling strategies are given. These combinations are important not only for matching the Ritz-Galerkin method and the finite element method but also for matching other numerical methods such as the Ritz-Galerkin method and the finite difference method.     

NUMERICAL SOLUTIONS OF PARABOLIC PROBLEMS ON UNBOUNDED 3-D SPATIAL DOMAIN

In this paper,the numerical solutions of heat equation on 3-D unbounded spatial do-main are considered. n artificial boundary Γ is introduced to finite the computationaldomain.On the artificial boundary Γ,the exact boundary condition and a series of approx-imating boundary conditions are derived,which are called artificial boundary conditions.By the exact or approximating boundary condition on the artificial boundary,the originalproblem is reduced to an initial-boundary value problem on the bounded computationaldomain,which is equivalent or approximating to the original problem.The finite differencemethod and finite element method are used to solve the reduced problems on the finitecomputational domain.The numerical results demonstrate that the method given in thispaper is effective and feasible.     

A Lyapunov function for a two-chemical species version of the chemotaxis model

Since the accuracy of finite element solutions of partial differential equations is generally mesh dependent, especially when solutions have singularities and discontinuities, a proper mesh generation is often important and sometimes crucial for an accurate numerical approximation of such problems. In this paper, the mesh transformation method is applied to the boundary value problems of elliptic partial differential equations, and it is proved that the method leads to the optimal finite element solutions. AMS subject classification (2000) 73C50, 65K10, 65N12, 65N30