A simple, accurate scheme for the numerical evaluation of integrals with complex singularity poles

The procedure proposed herein is to the authors' best knowledge the only mathematically consistent technique for dealing with general quasi-singularities that occur in the boundary integral equation formulations. Its implementation results in a robust code and implies in no additional computational effort.     

Elementary analytical integrals required in subtraction of singularity method for evaluation of weakly singular boundary integrals

This paper presents closed form analytical formulae for the elementary integrals required in the numerical evaluation of weakly singular boundary integrals involving logarithmic singularity using the subtraction of singularity method. Results are limited to curved quadratic elements in view of their practical usefulness in modeling complex boundaries. Continuous as well as discontinuous elements have been considered. Sample numerical results obtained using these formulae together with a hybrid subtraction of singularity–nonlinear transformation method have been included to demonstrate the usefulness of these formulae.     

An efficient iterative method for numerical evaluation of integrals over a semi-infinite range

A procedure for numerical quadrature over a semi-infinite range is described which does not require storing weights or nodes. It is applicable to a wide variety of integrands including oscillatory functions and forms with integrable singularites. A computer program and numerical examples are presented.     

Symmetric BE method in two-dimensional elasticity: evaluation of double integrals for curved elements

A procedure for the evaluation of the double integrals involved in the symmetric BE method is proposed, which consists in a regularization via integration by parts and in a successive numerical evaluation of weakly singular integrals. The procedure is applicable to curved elements and higher order shape functions and lends itself to a fairly straightforward implementation. Continuity is required for displacement modelling, while tractions may be modelled as piece-wise continuous. Some example problems characterized by the presence of curved boundaries or of cracks are analyzed using the proposed technique to demonstrate its effectiveness. This study is part of a research project supported by CNR (Italian National Research Council)     

A self-adaptive co-ordinate transformation for efficient numerical evaluation of general boundary element integrals

Almost all general purpose boundary element computer packages include a curved geometry modelling capability. Thus, numerical quadrature schemes play an important role in the efficiency of programming the technique. The present work discusses this problem in detail and introduces efficient means of computing singular or nearly singular integrals currently found in two-dimensional, axisymmetric and three-dimensional applications. Emphasis is given to a new third degree polynomial transformation which was found greatly to improve the accuracy of Gaussian quadrature scheme's within the near-singularity range. The procedure can easily be implemented into existing BE codes and presents the important feature of being self-adaptive, i.e. it produces a variable lumping of the Gauss stations toward the singularity, depending on the minimum distance from the source point to the element. The self-adaptiveness of the scheme also makes it inactive when not useful (large source distances) which makes it very safe for general usage.     

A general and effective way for evaluating the integrals with various orders of singularity in the direct boundary element method

A general and efficient technique is developed for the evaluation of the integrals with various orders of singularity, such as occur in the three-dimensional boundary element method (BEM). Generalized (extended) triangle, tetrahedron polar co-ordinate mappings together with two conditions are used to remove the singularity of the integrals, and to evaluate the corresponding non-singular ones in a new numerical space. Triangle and tetrahedron polar co-ordinates in Reference 1 are proved to be a special case of the generalized ones in this paper. With the developed idea, boundary element results converge rapidly towards the analytical solutions for the strongly singular integrals evaluated directly, and the analytical solutions can be gained in principle, even when employing higher order, triangular boundary elements and tetrahedral cells. The generality and practicability of the method are demonstrated in the case of higher order elements, discontinuous elements and large engineering problems.     

Green element method and singularity programming for numerical well test analysis

One technique available to petroleum reservoir engineers to determine the properties (such as permeability and reservoir size) of oil and gas reservoirs is well test analysis. In a well test a well undergoes a step-change in its flowrate and the resulting variation in the well pressure is carefully measured. Traditionally these pressure responses are interpreted by comparing them to analytical solutions. However these solutions are limited to homogeneous reservoirs of regular shapes. An alternative is to compare the measured data to numerical simulations of the well test. This allows for more complex reservoir geometry and heterogeneity in the reservoir permeability to be included. Traditionally reservoir engineers use finite difference methods for these fluid flow calculations. These are prone to some numerical artifacts that make well test responses difficult to compute accurately.This work explores the advantages of a hybrid boundary element method (BEM) known as the Green element method (GEM) for modeling well tests. BEMs are a natural choice for the problem because they are based on Green's functions, which are an established part of well test analysis [Soc Petrol Engr J (1973) 285]. The classical BEM is limited to single phase flow in homogeneous media. This work presents formulations, which give computationally efficient means to handle heterogeneity. The accuracy of the scheme is further enhanced by incorporating singularity programming.Comparisons of the proposed GEM approach to conventional finite difference simulation, using the same gridding and timestepping, show that finite difference simulations of well test responses do not accurately reproduce the corresponding analytical solutions. GEM can accurately reproduce analytical solutions for the pressure and its derivative even using coarse gridding. It can also efficiently handle heterogeneity.     

A numerical method for the solution of two-dimensional inverse heat conduction problems

A numerical method for the solution of inverse heat conduction problems in two-dimensional rectangular domains is established and its performance is demonstrated by computational results. The present method extends Beck's⁸ method to two spatial dimensions and also utilizes future times in order to stabilize the ill-posedness of the underlying problems. The approach relies on a line approximation of the elliptic part of the parabolic differential equation leading to a system of one-dimensional problems which can be decoupled.     

The radial integration method for evaluation of domain integrals with boundary-only discretization

In this paper, a simple and robust method, called the radial integration method, is presented for transforming domain integrals into equivalent boundary integrals. Any two- or three-dimensional domain integral can be evaluated in a unified way without the need to discretize the domain into internal cells. Domain integrals consisting of known functions can be directly and accurately transformed to the boundary, while for domain integrals including unknown variables, the transformation is accomplished by approximating these variables using radial basis functions. In the proposed method, weak singularities involved in the domain integrals are also explicitly transformed to the boundary integrals, so no singularities exist at internal points. Some analytical and numerical examples are presented to verify the validity of this method.     

Direct evaluation of singular boundary integrals in two-dimensional biharmonic analysis

Development of techniques to provide rapid and accurate evaluation of the integrations required in boundary element method (BEM) formulations are receiving more attention in the literature. In this work, a series of direct expressions for surface integrals, required for a boundary element solution of the non-homogeneous biharmonic over a general two-dimensional curvilinear surface, are presented. The concept of an isoparametric representation, usually applied to the variation of the field variables and the geometry, is extended to the parametric mapping of the curvilinear geometry. The result renders the typically complicated Jacobian function into a series of polynomial expressions based on the shape function set and several discrete Jacobian values. An application of the isoparametric approximation of the Jacobian for a quadratic element representation is developed. Implementation of this approximation significantly improves the accuracy of the boundary integral solution by eliminating error associated with numerical quadrature. Overall computational efficiency is improved by reducing the time necessary to calculate individual surface integrals and evaluate field variables at internal points. A numerical solution of the boundary integral equations of phenomena governed by the biharmonic equation is presented and compared with an exact analysis.     

A bi-cubic transformation for the numerical evaluation of the Cauchy principal value integrals in boundary methods

The numerical strategies employed in the evaluation of singular integrals existing in the Cauchy principal value (CPV) sense are, undoubtedly, one of the key aspects which remarkably affect the performance and accuracy of the boundary element method (BEM). Thus, a new procedure, based upon a bi-cubic co-ordinate transformation and oriented towards the numerical evaluation of both the CPV integrals and some others which contain different types of singularity is developed. Both the ideas and some details involved in the proposed formulae are presented, obtaining rather simple and-attractive expressions for the numerical quadrature which are also easily embodied into existing BEM codes. Some illustrative examples which assess the stability and accuracy of the new formulae are included.     

Distance transformation for the numerical evaluation of nearly singular integrals on triangular elements

The accurate numerical evaluation of nearly singular boundary integrals is a major concerned issue in the implementation of the boundary element method (BEM). In this paper, the previous distance transformation method is extended into triangular elements both in polar and Cartesian coordinate systems. A new simple and efficient method using an approximate nearly singular point is proposed to deal with the case when the nearly singular point is located outside the element. In general, the results obtained using the polar coordinate system are superior to that in the Cartesian coordinate system when the nearly singular point is located inside the element. Besides, the accuracy of the results is influenced by the locations of the nearly singular point due to the special topology of triangular elements. However, when the nearly singular point is located outside the element, both the polar and Cartesian coordinate systems can get acceptable results.     

Deconvolution method for two-dimensional spatial-response mapping of lithographic infrared antennas

The spatial impulse response of antenna-coupled infrared detectors with dimensions comparable with the wavelength is obtained from a two-dimensional scan of a tightly focused CO(2)-laser beam. The method uses an experimental setup with submicrometer resolution and an iterative deconvolution algorithm. The measured spatial response is compared with numerically computed near-field distributions of a dipole antenna, with good agreement.     

Three-dimensional harmonic functions near termination or intersection of gradient singularity lines: A general numerical method

The harmonic function u near point 0 from which a single singularity ray emanates is assumed to be dominated by the term r^λρ^ρU where r = distance from point 0, p = known constant and ρ = chosen function of angular spherical coordinates θ, ?, for which a partial differential equation with boundary conditions, especially those at the singularity rays, and a variational principle, are derived. Because grad U is nonsingular, a 1numerical solution is possible, using, e.g. the finite difference or finite element methods. This reduces the problem to finding λ of the smallest real part satisfying the equation Det (A_ij) = 0 where A_ij is a large matrix whose coefficients depend linearly on μ = λ(λ + 1). In general λ and A_ij are complex. Solutions can be obtained either by reduction to a standard matrix eigenvalue problem for μ, or by successive conversions to nonhomogeneous linear equation systems. Computer studies have confirmed the feasibility of the method and have shown that highly accurate results can be obtained. Solutions for cracks and notches ending at a plane or conical surface, and for cracks ending obliquely at a halfspace surface, are presented. In these cases, λ is real and the singularity is always weaker (λ > p) than on the singularity line and may even disappear (λ > 1). Furthermore, elastic stresses under a wedge-shaped rigid sliding stamp or at a corner of a crack edge, and also harmonic functions at three-sided pyramidal notches, have been analyzed. Here λ < p was found to occur. A simple analytical solution for one class of special cases has also been found and used to check some of the numerical results.     

A general algorithm for the numerical evaluation of nearly singular boundary integrals of various orders for two- and three-dimensional elasticity

 A general algorithm of the distance transformation type is presented in this paper for the accurate numerical evaluation of nearly singular boundary integrals encountered in elasticity, which, next to the singular ones, has long been an issue of major concern in computational mechanics with boundary element methods. The distance transformation is realized by making use of the distance functions, defined in the local intrinsic coordinate systems, which plays the role of damping-out the near singularity of integrands resulting from the very small distance between the source and the integration points. By taking advantage of the divergence-free property of the integrals with the nearly hypersingular kernels in the 3D case, a technique of geometric conversion over the auxiliary cone surfaces of the boundary element is designed, which is suitable also for the numerical evaluation of the hypersingular boundary integrals. The effects of the distance transformations are studied and compared numerically for different orders in the 2D case and in the different local systems in the 3D case using quadratic boundary elements. It is shown that the proposed algorithm works very well, by using standard Gaussian quadrature formulae, for both the 2D and 3D elastic problems. Received: 20 November 2001 / Accepted: 4 June 2002 The work was supported by the Science Foundation of Shanghai Municipal Commission of Education.