Quasi-singular Integrals Research Articles

The consistent boundary element method for potential and elasticity: Part II — machine-precision numerical evaluations for 2D problems

The collocation boundary element method is outlined in a first companion paper for the general three-dimensional, static case of elasticity. This is done in a consistent way that sheds light on some essential conceptual and implementation-oriented aspects that have been up to now in part overseen and in part mistaken in the technical literature. We show in the present developments that actually only Gauss–Legendre quadratures are needed in a two-dimensional code implementation of potential and elasticity problems, with arbitrarily high – actually only machine-precision limited – computational accuracy of all results of interest independently of a problem’s geometry or topology. In fact, the higher the effect of a quasi-singularity the more accurate a result is achievable with a – for all practical means – restricted number of quadrature points by just having the mathematical aspects properly dealt with beforehand and general preliminary evaluations stored for use as black boxes. The developments are presented for generally curved boundary elements of any high order, with the mathematics related to improper, singular, real or complex quasi-singular integrals adequately outlined, which includes the accurate evaluation of results at internal points that may be arbitrarily close to the boundary. Numerical results are shown in a second companion paper and convergence features are assessed for problems with very challenging topology issues and even for subnanometer source-field distances.

Remarks on “Application of Mixed Quadrature Rule on Electromagnetic Field Problems”

Numerical integration of one class of quasi-singular integrals appearing in the convenient formulas of electromagnetic field components and potential functions of linear source distributions is considered. Some comments to a recent paper by S.R. Jena, D. Nayak, and M.M. Acharya [Comput. Math. Modeling, 28, 267–276 (2017)] on this subject are given, as well as some illustrative numerical examples.

On the evaluation of input impedance and transient impedance for grounding electrodes using antenna theory

Purpose – The purpose of this paper is to propose a fast, accurate and efficient algorithm for assessment of input impedance and consequently the evaluation of transient impedance of the grounding electrode. Design/methodology/approach – The mathematical model is based on the thin wire antenna theory and related Pocklington integro-differential equation in the frequency domain, which is numerically treated via Galerkin-Bubnov variant of the indirect boundary element method (GB-IBEM). Two different approaches, scattered voltage method (ScVM) and induced electromotive force – boundary element method (IEMF-BEM), for input and transient impedance are discussed in detail. Extensive numerical experiments have been undertaken to analyze numerical sensitivity of the methods. Findings – Although it was widely used so far, the ScVM, was shown to be unsuitable for the grounding impedance assessment because results are dependent on the number of elements used in the numerical solution. On the other hand, the other method, IEMF-BEM is rather stable, with the respect to the number of elements used and with excellent convergence rate. In addition, IEMF-BEM is much simpler to implement as it requires only multiplication of matrices already assembled within the procedure of current distribution calculation, as opposed to the ScVM which requires numerical integration of quasi-singular integrals which, by it self, can be very demanding. Originality/value – The IEMF-BEM is originally developed by the authors and used for the first time for grounding impedance assessment. It is simple and very efficient and can easily be extended to arbitrary grounding configurations.

Application of the boundary element method for analysis of the antiplane shear of anisotropic bodies containing thin-walled structures

We have ascertained the limits of reasonable application of the classical boundary element method for the solution of the antiplane problem of the theory of elasticity in the study of bodies with thin-walled elements of structure and geometry. We have proposed an approach for the regularization of singular and quasisingular integrals, which appear inevitably in analyzing thin structures. We give also numerical examples demonstrating the reliability and efficiency of the proposed approach.

Analysis of thin-walled structural elements via 3D standard BEM with generic substructuring

This paper is concerned with the application of standard 3D Boundary Element Methods to solve thin-walled structural elements (needle-like/shell-like solids). A subregion-by-subregion data structure, incorporating iterative solvers and discontinuous boundary elements, is presented. To efficiently and accurately evaluate the quasi-singular integrals, special quadrature methods are applied. In addition, structured matrix-vector products, designed to avoid the excessive number of conditional tests during solver iterations, are proposed. Numerical results for complex thin-walled BE models are validated by comparison with FEM calculations and previously published BEM analyses.

Analysis of stiffened plates by the boundary element method

In this paper, a formulation for representation of stiffeners in plane stress by the boundary elements method (BEM) in linear analysis is presented. The strategy is to adopt approximations for the displacements in the central line of the stiffener. With this simplification the spurious oscillations in the stress along stiffeners with small thickness is prevented. Worked examples are analyzed to show the efficiency of these techniques, especially in the insertion of very narrow sub-regions, in which quasi-singular integrals are calculated, with stiffeners that are much stiffer than the main domain. The results obtained with this formulation are very close to those obtained with other formulations.

A modified Gauss quadrature formula with special integration points for evaluation of Quasi-singular integrals

It is well known in the boundary element method that integration rules fail when the integrand presents a nearby singularity. This drawback arises when the field point is near the source point, i.e. in the case of a domain with very narrow boundaries or when the field point where we try to calculate stresses or any other field variables, is near the boundaries. In the present paper a quadrature formulas for ℓ isolated singularities near the integration interval, based on ordinary or special Langrange interpolatory polynomials, is obtained. This interpolatory formulas present similarities with known formulas for the numerical evaluation of singular integrals. Quadrature formulas for regular and singular integrals with conjugate poles are also derived. Numerical examples are given and the proposed quadrature rules present the expected polynomial accuracy.

Generic domain decomposition and iterative solvers for 3D BEM problems

SUMMARY In the past two decades, considerable improvements concerning integration algorithms and solvers involved in boundary-element formulations have been obtained. First, a great deal of efficient techniques for evaluating singular and quasi-singular boundary-element integrals have been, definitely, established, and second, iterative Krylov solvers have proven to be advantageous when compared to direct ones also including non-Hermitian matrices. The former fact has implied in CPU-time reduction during the assembling of the system of equations and the latter fact in its faster solution. In this paper, a triangle-polar-co-ordinate transformation and the Telles co-ordinate transformation, applied in previous works independently for evaluating singular and quasi-singular integrals, are combined to increase the efficiency of the integration algorithms, and so, to improve the performance of the matrixassembly routines. In addition, the Jacobi-preconditioned biconjugate gradient (J-BiCG) solver is used to develop a generic substructuring boundary-element algorithm. In this way, it is not only the system solution accelerated but also the computer memory optimized. Discontinuous boundary elements are implemented to simplify the coupling algorithm for a generic number of subregions. Several numerical experiments are carried out to show the performance of the computer code with regard to matrix assembly and the system solving. In the discussion of results, expressed in terms of accuracy and CPU time, advantages and potential applications of the BE code developed are highlighted. Copyright 2006 John Wiley & Sons, Ltd.

Boundary element formulation for 2D solids with stiff and soft thin inclusions

In this work, a linear boundary element formulation is presented for analysing solids containing stiff and soft thin inclusions. A particular sub-region technique, in which the equilibrium is preserved along interfaces without traction approximation, is adapted to model thin inclusions as fibres and beams immersed in a solid. An alternative formulation, in which tractions along interfaces are preserved as unknowns, eliminating therefore displacements, is also derived and applied to the analysis of inclusions in 2D solids in general. For the case of thin inclusions, fibres and beams, the displacement field is properly approximate over the cross-sections. The quasi-singular integrals appearing in all presented formulations are computed by using closed expressions or employing a numerical scheme with sub-elements.

Two-dimensional solids reinforced by thin bars using the boundary element method

In this work, a linear boundary element formulation is presented for analyzing stiffened domains. A particular sub-region technique, in which the equilibrium is preserved along interfaces without traction approximation, is adapted to model fiber immersed in a body. The integral representation is written for a whole body, requiring only the displacement along interfaces. The sub-region is then assumed to be very thin to simulate fibers only when normal forces are taken into account. The thin sub-regions then degenerate so that they can be represented by its skeleton line. The displacement field over the fiber cross-sections is assumed to be constant. In the case of 2D problems, the degrees of freedom are reduced to two components only at each fiber node. Thus, displacement integral representations for collocations defined along the fiber skeleton are needed. The quasi-singular integrals are computed by using closed expressions or employing a numerical scheme with sub-elements. An example is solved to show that the formulation is very accurate for modeling cases of domains stiffened by fibers.

A hybrid boundary element method for shallow water acoustic propagation over an irregular bottom

A hybrid numerical model combining a boundary element method (BEM) and eigenfunction expansions is developed to solve acoustic wave propagation in shallow water. Waves are assumed harmonic and, therefore, the governing equation reduces to a Helmholtz equation. Accurate numerical integration techniques are implemented in the BEM, for calculating singular and quasi-singular integrals. For the latter, an adaptive integration technique is developed and tested for computational domains with very small aspect ratios, representative of shallow water environments. The model is validated by comparing the numerical solution to analytic solutions for problems with simple boundary geometries (e.g. rectangular, step, and sloped domains). Results indicate good agreement between the two solutions. Effects of node resolution, adaptive integrations, and number of modes in radiated fields, on the accuracy of the solution, are assessed. Finally, the model is used to study acoustic transmission over a rectangular bump in the bottom, as a function of frequency and bump geometry. Expected results are obtained below the first cut-off frequency over the bump, i.e. the transmission of energy beyond the bump through evanescent modes. A similar effect is commonly noted in layered media and is known as ‘tunneling’.

A non-linear transformation algorithm for the integration of the singular kernels in 3D BEM for elastostatics

A technique for the evaluation of the singular kernels in the boundary element method (BEM) for 3D elastostatics, based on the previous application of a non-linear co-ordinate transformation is presented in this work. The algorithm is a 3D extension of a former scheme developed for the 2D case and is able to integrate accurately all the different types of kernels appearing in the BEM (weak, CPV and quasisingular integrals).

Quasi-singular integrals in the modeling of nonlinear water waves in shallow water

The model by Grilli et al., 5,8 based on fully nonlinear potential flow equations, is used to study propagation of water waves over arbitrary bottom topography. The model combines a higher-order boundary element method for the solution of Laplace's equation at a given time, and Lagrangian Taylor expansions for the time updating of the free surface position and potential. In this paper, both the accuracy and the efficiency of computations are improved, for wave shoaling and breaking over gentle slopes, in domains with very sharp geometry and large aspect ratio, by using quasi-singular integration techniques based on modified Telles 17 and Lutz 11 methods. Applications are presented that demonstrate the accuracy and the efficiency of the new approaches.

On the efficient numerical evaluation of integrals with complex singularity poles

A simple quadrature rule is proposed for the evaluation of one-dimensional quasi-singular integrals with a complex pole. The technique is based on the concept of synthetic division of two polynomials, as a means of regularizing the quasi-singularity, followed by a quadrature using the roots of shifted Legendre polynomials as fixed abscissas. This procedure is a generalization of the technique proposed in a previous paper for singularities due to a real pole. With this technique, it is even possible to conceive of procedures for the analytical or semi-analytical evaluation of most kinds of integrals that appear in a general boundary element formulation with curved elements.