A new boundary element technique for solving plane problems of linear elasticity: improved theory and an application to fracture mechanics

An improved boundary element method for solving plane problems of linear elasticity theory is described. The method is based on the Muskhelishvili complex variable representation for the displacement and stress fields. The paper shows how to take account of symmetry about the x and/or y axes.The potential accuracy of the method is illustrated by its application to the calculation of stress intensity factors associated with cracks in both a square and a circular plate. The crack problem is solved using a Gauss-Chebyshev representation of a singular integral equation by a set of linear algebraic equations. The integral equation involves an analytic function which takes account of the presence of the external boundary. This function is determined directly using the boundary element method.Numerical results are believed to be more accurate than the existing published values which are quoted to four significant figures.     

An application of a boundary element method to natural convection

The direct boundary element method is applied to the numerical modelling of thermal fluid flow in a transient state. The Navier-Stokes equations are considered under the Boussinesq approximation and the viscous thermal flow equations are expressed in terms of stream function, vorticity, and temperature in two dimensions. Boundary integral equations are derived using logarithmic potential and time-dependent heat potential as fundamental solutions. Boundary unknowns are discretized by linear boundary elements and flow domains are divided into a series of triangular cells. Charged points are translated upstream in the numerical evaluation of convective terms. Unknown stream function, vorticity, and temperature are staggered in the computational scheme.

Simple iteration is found to converge to the quasi steady-state flow. Boundary solutions for two-dimensional examples at a Reynolds number 100 and Grashoff number 10⁷ are obtained.     

A BOUNDARY ELEMENT METHOD FOR A NONLINEAR BOUNDARY VALUE PROBLEM IN STEADY-STATE HEAT TRANSFER IN DIMENSION THREE

In this paper, a new method of boundary reduction is proposed, which reduces thesteady-state heat transfer equation with radiation. Moreover, a boundary element method is pre-sented for its solution and the error estimates of the numerical approximations are given.     

Superelements for the finite element solution of two-dimensional elliptic problems with boundary singularities

A novel singular superelement (SSE) formulation has been developed to overcome the loss of accuracy encountered when applying the standard finite element schemes to two-dimensional elliptic problems possessing a singularity on the boundary arising from an abrupt change of boundary conditions or a reentrant corner. The SSE consists of an inner region over which the known analytic form of the solution in the vicinity of the singular point is utilized, and a transition region in which blending functions are used to provide a smooth transition to the usual linear or quadratic isoparametric elements used over the remainder of the domain. Solution of the finite element equations yield directly the coefficients of the asymptotic series, known as the flux/stress intensity factors in linear heat transfer or elasticity theories, respectively. Numerical examples using the SSE for the Laplace equation and for computing the stress intensity factors in the linear theory of elasticity are given, demonstrating that accurate results can be attained for a moderate computational effort.     

Numerical solution of boundary integral equations by means of attenuation factors

We consider first-kind boundary integral equations with logarithmickernel such as those arising from solving Dirichlet problemsfor the Laplace equation by means of single-layer potentials.The first-kind equations are transformed into equivalent equationsof the second kind which contain the conjugation operator andwhich are then solved with a degenerate-kernel method basedon Fourier analysis and attenuation factors. The approximationswe consider, among them spline interpolants, are linear andtranslation invariant. In view of the particularly small kernel,the linear systems resulting from the discretization can besolved directly by fixed-point iteration.     

The fast multipole method for the symmetric boundary integral formulation

^** Email: of{at}mathematik.uni-stuttgart.de^*** Email: o.steinbach{at}tugraz.at^**** Email: wendland{at}mathematik.uni-stuttgart.de A symmetric Galerkin boundary-element method is used for thesolution of boundary-value problems with mixed boundary conditionsof Dirichlet and Neumann type. As a model problem we considerthe Laplace equation. When an iterative scheme is employed forsolving the resulting linear system, the discrete boundary integraloperators are realized by the fast multipole method. While thesingle-layer potential can be implemented straightforwardlyas in the original algorithm for particle simulation, the double-layerpotential and its adjoint operator are approximated by the applicationof normal derivatives to the multipole series for the kernelof the single-layer potential. The Galerkin discretization ofthe hypersingular integral operator is reduced to the single-layerpotential via integration by parts. We finally present a correspondingstability and error analysis for these approximations by thefast multipole method of the boundary integral operators. Itis shown that the use of the fast multipole method does notharm the optimal asymptotic convergence. The resulting linearsystem is solved by a GMRES scheme which is preconditioned bythe use of hierarchical strategies as already employed in thefast multipole method. Our numerical examples are in agreementwith the theoretical results.     

Quadrature formulas for periodic functions and their application to the boundary element method

Two-dimensional and axisymmetric boundary value problems for the Laplace equation in a domain bounded by a closed smooth contour are considered. The problems are reduced to integral equations with a periodic singular kernel, where the period is equal to the length of the contour. Taking into account the periodicity property, high-order accurate quadrature formulas are applied to the integral operator. As a result, the integral equations are reduced to a system of linear algebraic equations. This substantially simplifies the numerical schemes for solving boundary value problems and considerably improves the accuracy of approximation of the integral operator. The boundaries are specified by analytic functions, and the remainder of the quadrature formulas decreases faster than any power of the integration step size. The examples include the two-dimensional potential inviscid circulation flow past a single blade or a grid of blades; the axisymmetric flow past a torus; and free-surface flow problems, such as wave breakdown, standing waves, and the development of Rayleigh-Taylor instability.     

The coupling of boundary integral and finite element methods for the nonstationary exterior flow

In this article, we represent a new numerical method for solving the nonstationary Navier–Stokes equations in an unbounded domain. The technique consists of coupling the boundary integral and the finite element method. The variational formulation and the well-posedness of the coupling method are obtained. The convergence and optimal error estimates for the approximate solution are provided. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 549–565, 1998     

A boundary element analysis for stress intensity factors of multiple circular arc cracks in a plane elasticity plate

In this paper, a numerical approach for analyzing interacting multiple cracks in infinite linear elastic media is presented. By extending Bueckner’s principle suited for a crack to a general system containing multiple interacting cracks, the original problem is divided into a homogeneous problem (the one without cracks) subjected to remote loads and a multiple crack problem in an unloaded body with applied tractions on the crack surfaces. Thus, the results in terms of the stress intensity factors (SIFs) can be obtained by considering the latter problem, which is analyzed easily by means of the displacement discontinuity method with crack-tip elements proposed recently by the author. Test examples are given to illustrate that the numerical approach is very accurate for analyzing interacting multiple cracks in an infinite linear elastic media under remote uniform stresses. In addition, the displacement discontinuity method with crack-tip elements is used to analyze a multiple crack problem in a finite plate. It is found that the boundary element method is also very accurate for investigating interacting multiple cracks in a finite plate. Specially, a generalization of Bueckner’s principle and the displacement discontinuity method with crack-tip elements are used to analyze multiple circular arc crack problems in infinite plate in tension (including: Two Collinear Circular Arc Cracks, Three Collinear Circular Arc Cracks, Two Parallel Circular Arc Cracks, Three Parallel Circular Arc Cracks and Two Circular Arc Cracks) in a plane elasticity plate. Many results are given.     

三维Helmholtz方程边值问题的新型边界积分-微分方程及其边界元解法

用双层位势表示的二维Neumann边值问题的边界归化方法,将原始问题归化为新型边界积分-微分方程,由此导出一种新的既能保持原始问题的自伴性,又具有可积弱奇性积分核的边界变分方程.本文将此法推广到三维Helmholtz方程Neumann边值问题,并给出最优能量模误差估计和内部最大模超收敛估计.     

Development of boundary element method to dynamic problems for porous media

Biot's consolidation theory is extended to a general class of viscoelastic bodies defined by Riemann-Stieltjes integral convolutions. From a new reciprocity theorem, proved for the governing equations including the inertia terms, the basic integral representations of the displacement fields and pore pressure are obtained. It is shown that, in the absence of internal inputs, a formulation of the dynamic problem in terms of the boundary unknown fields only is possible.     

求解二维多区域声波传播问题的一种边界元方法

针对多区域中声波的传播问题,其中每个散射区域的介质是相同的,将散射区域内的声波用一种单双层混合位势的形式来表示,再应用Green定理表示出外部介质区域中的声波,并形成相应的边界积分方程.如果区域个数为M时,传统的边界元方法最终将形成2M个边界积分方程并对应2M个未知函数,而本文的边界元方法最终只形成M个边界积分方程以及对应M个未知函数,从而使得求解的方程和未知数的个数都减少了一倍.最后,通过对数值算例的求解,验证了该方法的可行性及精确性.     

A new method for solving singular fourth-order boundary value problems with mixed boundary conditions

In [1], [2], [3], [4], [5], [6], [7] and [8], it is very difficult to get reproducing kernel space of problem (1). This paper is concerned with a new algorithm for giving the analytical and approximate solutions of a class of fourth-order in the new reproducing kernel space. The numerical results are compared with both the exact solution and its n-order derived functions in the example. It is demonstrated that the new method is quite accurate and efficient for fourth-order problems.     

Solution of some plate bending problems using the boundary element method

Some fundamental aspects of the boundary element method of the Kirchhoff theory of thin plate flexure are given. The direct boundary integral equation method with higher conforming properties (using first-order Hermitian interpolation for plate displacement ω, and zero-order Hermitian interpolation for angle of rotation θ, the moment M andthe equivalent shear V) are used for several computational examples. They are: square plate with simply-supported or clamped edges, the same square plate with square central opening and the cantilevered triangular plates. The results of computation as compared with some known experimental and theoritical results showed that the numerical schemes seemed to be satisfactory for the practical applications.     

A hybrid method using wavelets for the numerical solution of boundary value problems on the interval

In this work, various aspects of wavelet-based methods for second order boundary value problems under Galerkin framework are investigated. Based on the B-spline multiresolution analysis (MRA) on the line we propose a hybrid method on the interval which combines different treatments for interior and boundary splines. By using this procedure, the MRA structure was conserved and hierarchical representations of the solution at different scales were obtained without much computational effort. Numerical examples are given to verify the effectiveness of the proposed method and the comparison with other techniques is presented.     

A new boundary element technique for solving plane problems of linear elasticity: 1. Theory

A new boundary elements technique for solving plane problems of linear elasticity theory is described. The method is based upon the Muskhelishvili complex variable representation of the displacement and stress fields involving two independent complex functions. These functions are represented by complex Cauchy integrals where the path of integration is taken around the external boundary of the solid. Two complex density functions appearing in the integrands of the Cauchy integrals are represented by spline functions and these are determined by the application of appropriate boundary conditions. The theory presented is suitable only for bounded simply-connected regions.     

A system of boundary integral equations for polygonal plates with free edges

We consider the problem of a polygonal plate with free edges. It is a boundary value problem for the biharmonic operator on a polygon with Neumann boundary conditions. Its resolution is studied via boundary integral equations. A variational formulation of the boundary problem obtained by a double-layer potential is given. Finally, we implement the method and give numerical results. © 1998 B. G. Teubner Stuttgart–John Wiley & Sons Ltd.