NEW NUMERICAL INTEGRATION SCHEMES FOR APPLICATIONS OF GALERKIN BEM TO 2-D PROBLEMS

We consider hypersingular integral formulation of some elasticity and potential boundary value problems on 2-D domains. In particular, we consider all integrals whose evaluation is required when the equations are solved by a Galerkin BEM based on piecewise polynomial approximants of arbitrary local degrees. In order to compute these integrals, we use very efficient formulas recently proposed, which require the user to define a mesh, not necessarily uniform, on the boundary and specify the local degrees of the approximant. These rules are quite suitable for the construction of h–p version of the BEM. Implementation of h−, p− and h–p methods are applied to some classical problems and several numerical results are presented. © 1997 by John Wiley & Sons, Ltd.     

Numerical integration in 3D Galerkin BEM solution of HBIEs

 We consider hypersingular boundary integral equations associated with 3D problems defined on polygonal domains, whose solutions are approximated with a Galerkin boundary element method, related to a given triangulation of the boundary. At first, for linear shape functions, the most frequently used basis functions, we give explicit results of the analytical inner integrations. Then, after an analysis of the singularities arising in the whole integration process, we propose suitable quadrature schemes to evaluate integrals required to form the Galerkin matrix elements. Several numerical results are presented. Received 6 November 2000     

On the validity of conforming BEM algorithms for hypersingular boundary integral equations

The widely held notion that the use of standard conforming isoparametric boundary elements may not be used in the solution of hypersingular integral equations is investigated. It is demonstrated that for points on the boundary where the underlying field is C ^1,α continuous, a class of rigorous nonsingular conforming BEM algorithms may be applied. The justification for this class of algorithms is interpreted in terms of some recent criticism. It is shown that the numerical integration in these conforming BEM algorithms using relaxed regularization represents a finite approximation to the standard two-sided Hadamard finite part interpretation of hypersingular integrals. It is also shown that the integration schemes in this class of algorithms are not based upon one-sided finite part interpretations. As a result, the attendant ambiguities associated with the incorrect use of the one-sided interpretations in boundary integral equations pose no problem for this class of algorithms. Additionally, the distinction is made between the analytic discontinuities in the field which place limitations on the applicability of the conforming BEM and the discontinuities resulting from the use of piece-wise C ^1,α interpolations.     

Two trigonometric quadrature formulae for evaluating hypersingular integrals

Two trigonometric quadrature formulae, one of non‐interpolatory type and one of interpolatory type for computing the hypersingular integral are developed on the basis of trigonometric quadrature formulae for Cauchy principal value integrals. The formulae use the cosine change of variables and trigonometric polynomial interpolation at the practical abscissae. Fast three‐term recurrence relations for evaluating the quadrature weights are derived. Numerical tests are carried out using the current formula. As applications, two simple crack problems are considered. One is a semi‐infinite plane containing an internal crack perpendicular to its boundary and the other is a centre cracked panel subjected to both normal and shear tractions. It is found that the present method generally gives superior results. Copyright © 2002 John Wiley & Sons, Ltd.     

Evaluations of certain hypersingular integrals on interval

In this paper, we investigate a hypersingular integral on an interval. The definition of Hadamard's finite‐part integrals and some of its properties are given. Some numerical methods on approximate computation of the finite‐part integrals are constructed. The new method is very simple, easy to implement, reliable, and above all, not affected by the location of singular point. Some numerical experiments are carried out using the current formulae, and numerical results show that the current methods are feasible and effective. Copyright © 2001 John Wiley & Sons, Ltd.     

Evaluations of hypersingular integrals using Gaussian quadrature

A Gaussian quadrature formula for hypersingular integrals with second‐order singularities is developed based on previous Gaussian quadrature formulae for Cauchy principal value integrals. The formula uses classical orthonormal polynomials, and the formula is then specialized to the case of Legendre and Chebyshev polynomials. Numerical experiments are carried out using the current formula and a previous formula developed by Kutt. It is found that the two methods generally give similar results, and in some cases the current method works better. It has also been shown that the current method allows the choice of an appropriate weight which can increase the convergence rate and the accuracy of the results. Copyright © 1999 John Wiley & Sons, Ltd.     

Numerical solutions of hypersingular integral equation for curved cracks in circular regions

In this paper, numerical solutions of a hypersingular integral equation for curved cracks in circular regions are presented. The boundary of the circular regions is assumed to be traction free or fixed. The suggested complex potential is composed of two parts, the principle part and the complementary part. The principle part can model the property of a curved crack in an infinite plate. For the case of the traction free boundary, the complementary part can compensate the traction on the circular boundary caused by the principle part. Physically, the proposed idea is similar to the image method in electrostatics. By using the crack opening displacement (COD) as the unknown function and traction as right hand term in the equation, a hypersingular integral equation for the curved crack problems in the circular regions is obtained. The equation is solved by using the curve length coordinate method. In order to prove that the suggested method can be used to solve more complicated cases of the curved cracks, several numerical examples are given.     

Complex hypersingular integral equation for the piece-wise homogeneous half-plane with cracks

New complex hypersingular integral equation (CHSIE) is derived for the half-plane containing the inclusions (which can have the different elastic properties), holes, notches and cracks of the arbitrary shape. This equation is obtained by superposition of the equations for each homogeneous region in a half-plane. The last equations follow from the use of complex analogs of Somigliana's displacement and stress identities (SDI and SSI) and Melan's fundamental solution (FS) written in a complex form. The universal numerical algorithm suggested before for the analogous problem for a piece-wise homogeneous plane is extended on case of a half plane. The unknown functions are approximated by complex Lagrange polynomials of the arbitrary degree. The asymptotics for the displacement discontinuities (DD) at the crack tips are taken into account. Only two types of the boundary elements (straight segments and circular arcs) are used to approximate the boundaries. All the integrals involved in CHSIE are evaluated in a closed form. A wide range of elasticity problems for a half-plane with cracks, openings and inclusions are solved numerically.     

Numerical solution of the variation boundary integral equation for inverse problems

In this paper a procedure to solve the identification inverse problems for two‐dimensional potential fields is presented. The procedure relies on a boundary integral equation (BIE) for the variations of the potential, flux, and geometry. This equation is a linearization of the regular BIE for small changes in the geometry. The aim in the identification inverse problems is to find an unknown part of the boundary of the domain, usually an internal flaw, using experimental measurements as additional information. In this paper this problem is solved without resorting to a minimization of a functional, but by an iterative algorithm which alternately solves the regular BIE and the variation BIE. The variation of the geometry of the flaw is modelled by a virtual strainfield, which allows for greater flexibility in the shape of the assumed flaw. Several numerical examples demonstrate the effectiveness and reliability of the proposed approach. Copyright © 2000 John Wiley & Sons, Ltd.     

Volume integration in the hypersingular boundary integral equation

The evaluation of volume integrals that arise in conjunction with a hypersingular boundary integral formulation is considered. In a recent work for the standard (singular) boundary integral equation, the volume term was decomposed into an easily computed boundary integral, plus a remainder volume integral with a modified source function. The key feature of this modified function is that it is everywhere zero on the boundary. In this work it is shown that the same basic approach is successful for the hypersingular equation, despite the stronger singularity in the domain integral. Specifically, the volume term can be directly evaluated without a body-fitted volume mesh, by means of a regular grid of cells that cover the domain. Cells that intersect the boundary are treated by continuously extending the integrand to be zero outside the domain. The method and error results for test problems are presented in terms of the three-dimensional Poisson problem, but the techniques are expected to be generally applicable.     

Numerical solution of the integral equations for optimal sequential filtering

Inhomogeneous Fredholm integral equations occur frequently in communication theory where it is desired to determine optimal receivers and filters for signal detection and estimation. In this paper an initial value method is utilized to determine the Fredholm resolvent and the solution of the integral equation. Numerical results are given for a simple example. The method is of particular interest where sequential solutions are desired.     

A Galerkin boundary integral method for multiple circular elastic inclusions

The problem of an infinite, isotropic elastic plane containing an arbitrary number of circular elastic inclusions is considered. The analysis procedure is based on the use of a complex singular integral equation. The unknown tractions at each circular boundary are approximated by a truncated complex Fourier series. A system of linear algebraic equations is obtained by using the classical Galerkin method and the Gauss–Seidel algorithm is used to solve the system. Several numerical examples are considered to demonstrate the effectiveness of the approach. Copyright © 2001 John Wiley & Sons, Ltd.     

Numerical integration schemes for evaluation of (hyper)singular integrals in 2D BEM

We consider hypersingular integral equations associated with 2D boundary value problems and defined on domains given by piecewise smooth parametric representations. In particular, given any (polynomial) local basis, we consider all integrals whose evaluation is required when the equations are solved by Galerkin BEM. In order to compute these integrals we use very efficient numerical schemes, recently proposed, which require the user to define a mesh, not necessarily uniform, on the boundary and to specify the local degrees of the approximant. Therefore these rules are quite suitable for the construction of p- and h-p versions of Galerkin BEM.     

Numerical analysis of dissolution processes in cementitious materials using discontinuous and continuous Galerkin time integration schemes

The present paper is concerned with the numerical integration of non‐linear reaction–diffusion problems by means of discontinuous and continuous Galerkin methods. The first‐order semidiscrete initial value problem of calcium leaching of cementitious materials, based on a phenomenological dissolution model, an electrolyte diffusion model and the spatial p‐finite element discretization, is used as a highly non‐linear model problem. A p‐finite element method is used for the spatial discretization. In the context of discontinuous Galerkin methods the semidiscrete mass balance and the continuity of the primary variables are weakly formulated within time steps and between time steps, respectively. Continuous Galerkin methods are obtained by the strong enforcement of the continuity condition as special cases. The introduction of a natural time co‐ordinate allows for the application of standard higher order temporal shape functions of the p‐Lagrange type and the well‐known Gauss–Legendre quadrature of associated time integrals. It is shown, that arbitrary order accurate integration schemes can be developed within the framework of the proposed temporal p‐Galerkin methods. Selected benchmark analyses of calcium dissolution demonstrate the robustness of these methods with respect to pronounced changes of the reaction term and non‐smooth changes of Dirichlet boundary conditions. Copyright © 2006 John Wiley & Sons, Ltd.     

Nearly singular approximations of CPV integrals with end- and corner-singularities for the numerical solution of hypersingular boundary integral equations

A local numerical approach to cope with the singular and hypersingular boundary integral equations (BIEs) in non-regularized forms is presented in the paper for 2D elastostatics. The approach is based on the fact that the singular boundary integrals can be represented approximately by the mean values of two nearly singular boundary integrals and on the techniques of distance transformations developed primarily in previous work of the authors. The nearly singular approximations in the present work, including the normal and the tangential distance transformations, are designed for the numerical evaluation of boundary integrals with end-singularities at junctures between two elements, especially at corner points where sufficient continuity requirements are met. The approach is very general, which makes it possible to solve the hypersingular BIE numerically in a non-regularized form by using conforming C⁰ quadratic boundary elements and standard Gaussian quadratures without paying special attention to the corner treatment.With the proposed approach, an infinite tension plate with an elliptical hole and a pressurized thick cylinder were analyzed by using both the formulations of conventional displacement and traction boundary element methods, showing encouragingly the efficiency and the reliability of the proposed approach. The behaviors of boundary integrals with end- and corner-singular kernels were observed and compared by the additional numerical tests. It is considered that the weakly singularities remain but should have been cancelled with each other if used in pairs by the corresponding terms in the neighboring elements, where the corner information is included naturally in the approximations.     

On the removal of the non‐uniqueness in the solution of elastostatic problems by symmetric Galerkin BEM

A study of the removal of the non‐uniqueness in the solution of elastostatic problems by means of the symmetric Galerkin boundary element method is presented. The paper focuses on elastic problems defined on domains with cavities, where cavity boundaries are subjected to traction boundary conditions. A simple method consisting in a direct application of support conditions and several methods based on the Fredholm theory of linear operators are introduced, implemented and analysed. Numerical examples demonstrate the performance of the proposed methods and accuracy of their results, a comparative evaluation of the methods developed being finally presented. Copyright © 2005 John Wiley & Sons, Ltd.     

Smoothness–relaxation strategies for singular and hypersingular integral equations

Three stages are involved in the formulation of a typical direct boundary element method: derivation of an integral representation; taking a Limit To the Boundary (LTB) so as to obtain an integral equation; and discretization. We examine the second and third stages, focussing on strategies that are intended to permit the relaxation of standard smoothness assumptions. Two such strategies are indicated. The first is the introduction of various apparent or ‘pseudo-LTBs’. The second is ‘relaxed regularization’, in which a regularized integral equation, derived rigorously under certain smoothness assumptions, is used when less smoothness is available. Both strategies are shown to be based on inconsistent reasoning. Nevertheless, reasons are offered for having some confidence in numerical results obtained with certain strategies. Our work is done in two physical contexts, namely two-dimensional potential theory (using functions of a complex variable) and three-dimensional elastostatics. © 1998 John Wiley & Sons, Ltd.     

Numerical stability in regular BEM schemes for elastostatic problems

The regular boundary element method is employed for the static analysis of boundary value problems of elasticity. This method allows one to reduce a given boundary value problem to a system of regular integral equations of the first kind with respect to source functions not located on the boundary. This paper is concerned with the numerical stability analysis of regular boundary element methods. In particular, the existence and stability of approximate solutions for integral equations of the first kind with continuous kernels are discussed. The special regularization technique for treating such a class of integral equations is developed. Numerical examples illustrate proposed algorithms and demonstrate their advantages.     

Numerical algorithm for the solution of linear two-dimensional integral equations of the first kind

A numerical algorithm is proposed for the solution of two-dimensional integral equations of the first kind, to which some inverse problems of heat conduction are reduced.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 33, No. 6, pp. 1103–1108, December, 1977.     

Numerical solution of Fredholm integral equations with a logarithmic singularity

Generalized quadrature is used for the numerical solution of two Fredholm integral equations which occur in electrostatics and aerodynamics.