Evaluation of 2‐D Green's boundary formula and its normal derivative using Legendre polynomials,with an application to acoustic scattering problems

This paper presents the non‐singular forms, in a global sense, of two‐dimensional Green's boundary formula and its normal derivative. The main advantage of the modified formulations is that they are amenable to solution by directly applying standard quadrature formulas over the entire integration domain; that is, the proposed element‐free method requires only nodal data. The approach includes expressing the unknown function as a truncated Fourier–Legendre series, together with transforming the integration interval [a, b] to [‐1,1] ; the series coefficients are thus to be determined. The hypersingular integral, interpreted in the Hadamard finite‐part sense, and some weakly singular integrals can be evaluated analytically; the remaining integrals are regular with the limiting values of the integrands defined explicitly when a source point coincides with a field point. The effectiveness of the modified formulations is examined by an elliptic cylinder subject to prescribed boundary conditions. The regularization is further applied to acoustic scattering problems. The well‐known Burton–Miller method, using a linear combination of the surface Helmholtz integral equation and its normal derivative, is adopted to overcome the non‐uniqueness problem. A general non‐singular form of the composite equation is derived. Comparisons with analytical solutions for acoustically soft and hard circular cylinders are made. Copyright © 2001 John Wiley & Sons, Ltd.     

A comparison of techniques for overcoming non‐uniqueness of boundary integral equations for the collocation partition of unity method in two‐dimensional acoustic scattering

The Partition of Unity Method has become an attractive approach for extending the allowable frequency range for wave simulations beyond that available using piecewise polynomial elements. The non‐uniqueness of solution obtained from the conventional boundary integral equation (CBIE) is well known. The CBIE derived through Green's identities suffers from a problem of non‐uniqueness at certain characteristic frequencies. Two of the standard methods of overcoming this problem are the so‐called Combined Helmholtz Integral Equation Formulation (CHIEF) method and that of Burton and Miller. The latter method introduces a hypersingular integral, which may be treated in various ways. In this paper, we present the collocation partition of unity boundary element method (PUBEM) for the Helmholtz problem and compare the performance of CHIEF against a Burton–Miller formulation regularised using the approach of Li and Huang. Copyright © 2013 John Wiley & Sons, Ltd.     

Finite element formulation of exact non‐reflecting boundary conditions for the time‐dependent wave equation

A modified version of an exact Non‐reflecting Boundary Condition (NRBC) first derived by Grote and Keller is implemented in a finite element formulation for the scalar wave equation. The NRBC annihilate the first N wave harmonics on a spherical truncation boundary, and may be viewed as an extension of the second‐order local boundary condition derived by Bayliss and Turkel. Two alternative finite element formulations are given. In the first, the boundary operator is implemented directly as a ‘natural’ boundary condition in the weak form of the initial–boundary value problem. In the second, the operator is implemented indirectly by introducing auxiliary variables on the truncation boundary. Several versions of implicit and explicit time‐integration schemes are presented for solution of the finite element semidiscrete equations concurrently with the first‐order differential equations associated with the NRBC and an auxiliary variable. Numerical studies are performed to assess the accuracy and convergence properties of the NRBC when implemented in the finite element method. The results demonstrate that the finite element formulation of the (modified) NRBC is remarkably robust, and highly accurate. Copyright © 1999 John Wiley & Sons, Ltd.     

A comparison between a symmetric and a non-symmetric Galerkin finite element—boundary integral equation coupling for the two-dimensional exterior Stokes problem

Symmetric and non-symmetric Galerkin formulations are presented for the coupling of a finite element modelled interior region to a boundary integral supported exterior region for the two-dimensional steady state exterior Stokes problem. Both single and double-layer hydrodynamic potentials are used allowing a well conditioned symmetric matrix structure for the entire interior–exterior, velocity–pressure system when the exterior velocity boundary integral equation (VBIE) is augmented by a traction boundary integral equation (TBIE) with the pressure determined everywhere purely through the imposition of the divergence-free velocity condition. Corresponding non-symmetric formulations are obtained by additionally discretizing an associated pressure boundary integral equation (PBIE), where the associated kernel functions have singularities an order higher than in the VBIE, with a simple regularization of the new hyper-singular pressure kernel. Comparable solution convergence with mesh refinement for the symmetric and non-symmetric schemes is shown for stabilized and mixed velocity–pressure conforming finite element pairs using Lagrangian shape functions.     

Three‐dimensional BEM for scattering of elastic waves in general anisotropic media

Based on the full‐space Green's functions, a three‐dimensional time‐harmonic boundary element method is presented for the scattering of elastic waves in a triclinic full space. The boundary integral equations for incident, scattered and total wave fields are given. An efficient numerical method is proposed to calculate the free terms for any geometry. The discretization of the boundary integral equation is achieved by using a linear triangular element. Applications are discussed for scattering of elastic waves by a spherical cavity in a 3D triclinic medium. The method has been tested by comparing the numerical results with the existing analytical solutions for an isotropic problem. The results show that, in addition to the frequency of the incident waves, the scattered waves strongly depend on the anisotropy of the media. Copyright © 2003 John Wiley & Sons, Ltd.     

The development of the pFFT accelerated BEM for 3-D acoustic scattering problems based on the Burton and Miller's integral formulation

The precorrected-FFT acceleration technique is successfully applied in the boundary element method for the simulation of 3-D acoustic scattering problems. The composite Helmholtz integral equation presented by Burton and Miller is employed to overcome the nonuniqueness problem occurring in the simulation of exterior acoustic problems by the boundary element method. Since the triangular constant element is employed, the hypersingular boundary integral equation is reduced into a weakly singular boundary integral equation with the application of a modified Burton and Miller's formulation. The computational cost, the consumed memory and the convergence of the current method are demonstrated and analyzed through the simulation of a plane acoustic wave scattering from a rigid sphere and from an axisymmetrical rigid structure.     

High‐order Higdon‐like boundary conditions for exterior transient wave problems

Recently developed non‐reflecting boundary conditions are applied for exterior time‐dependent wave problems in unbounded domains. The linear time‐dependent wave equation, with or without a dispersive term, is considered in an infinite domain. The infinite domain is truncated via an artificial boundary ??, and a high‐order non‐reflecting boundary condition (NRBC) is imposed on ??. Then the problem is solved numerically in the finite domain bounded by ??. The new boundary scheme is based on a reformulation of the sequence of NRBCs proposed by Higdon. We consider here two reformulations: one that involves high‐order derivatives with a special discretization scheme, and another that does not involve any high derivatives beyond second order. The latter formulation is made possible by introducing special auxiliary variables on ??. In both formulations the new NRBCs can easily be used up to any desired order. They can be incorporated in a finite element or a finite difference scheme; in the present paper the latter is used. In contrast to previous papers using similar formulations, here the method is applied to a fully exterior two‐dimensional problem, with a rectangular boundary. Numerical examples in infinite domains are used to demonstrate the performance and advantages of the new method. In the auxiliary‐variable formulation long‐time corner instability is observed, that requires special treatment of the corners (not addressed in this paper). No such difficulties arise in the high‐derivative formulation. Published in 2005 by John Wiley & Sons, Ltd.     

CONTINUITY AND COLLOCATION EFFECTS IN THE BOUNDARY ELEMENT METHOD

The presence of singularities in the integral operators of the boundary element methods requires that the density functions must satisfy certain continuity requirements if the displacements and stresses are to be bounded. Quite often the continuity conditions, particularly on the derivatives of the density functions, are relaxed at the element ends for the sake of simplicity in approximating the unknown density functions. In this paper, a numerical study on the effects of satisfying or violating the continuity requirements and the effect of the boundary condition collocating point on three different BEM formulations is presented. Two are indirect formulations using force singularities and displacement discontinuity singularities, and the third is Rizzo's direct formulation. The two integral operators in the direct BEM appear individually in the two different formulations of the indirect BEM. This makes it possible to study the numerical error and other problems in each integral operator and the interaction of the two integral operators in the direct BEM. The impact of the study on numerical modelling for the three BEM formulations is presented in the paper. © 1997 by John Wiley & Sons, Ltd.     

A wideband fast multipole boundary element method for three dimensional acoustic shape sensitivity analysis based on direct differentiation method

This paper presents a wideband fast multipole boundary element approach for three dimensional acoustic shape sensitivity analysis. The Burton-Miller method is adopted to tackle the fictitious eigenfrequency problem associated with the conventional boundary integral equation method in solving exterior acoustic wave problems. The sensitivity boundary integral equations are obtained by the direct differentiation method, and the concept of material derivative is used in the derivation. The iterative solver generalized minimal residual method (GMRES) and the wideband fast multipole method are employed to improve the overall computational efficiency. Several numerical examples are given to demonstrate the accuracy and efficiency of the present method.     

Direct second kind boundary integral formulation for Stokes flow problems

A direct boundary element method is formulated for the Stokes flow problem based on an integral equation representation for the components of traction. For problems in which the components of velocity are prescribed on the boundary of the domain, this new formulation results in a hypersingular Fredholm integral equation of the second kind. A method of regularization to evaluate the hypersingular integral is discussed. For certain problems involving flows about particles, the integral equation representation for the tractions is not unique because of the existence of rigid body eigenmodes. A method to constrain out these rigid body modes is also discussed. Several example problems are considered in which this new formulation is compared to more traditional boundary element formulations.     

A spline wavelet finite‐element method in structural mechanics

The wavelet‐based methods are powerful to analyse the field problems with changes in gradients and singularities due to the excellent multi‐resolution properties of wavelet functions. Wavelet‐based finite elements are often constructed in the wavelet space where field displacements are expressed as a product of wavelet functions and wavelet coefficients. When a complex structural problem is analysed, the interface between different elements and boundary conditions cannot be easily treated as in the case of conventional finite‐element methods (FEMs). A new wavelet‐based FEM in structural mechanics is proposed in the paper by using the spline wavelets, in which the formulation is developed in a similar way of conventional displacement‐based FEM. The spline wavelet functions are used as the element displacement interpolation functions and the shape functions are expressed by wavelets. The detailed formulations of typical spline wavelet elements such as plane beam element, in‐plane triangular element, in‐plane rectangular element, tetrahedral solid element, and hexahedral solid element are derived. The numerical examples have illustrated that the proposed spline wavelet finite‐element formulation achieves a high numerical accuracy and fast convergence rate. Copyright © 2005 John Wiley & Sons, Ltd.     

The boundary element-linear complementarity method for the Signorini problem

The boundary element-linear complementarity method for solving the Laplacian Signorini problem is presented in this paper. Both Green's formula and the fundamental solution of the Laplace equation have been used to solve the boundary integral equation. By imposing the Signorini constraints of the potential and its normal derivative on the boundary, the discrete integral equation can be written into a standard linear complementarity problem (LCP). In the LCP, the unique variable to be affected by the Signorini boundary constraints is the boundary potential variable. A projected successive over-relaxation (PSOR) iterative method is employed to solve the LCP, and some numerical results are presented to illustrate the efficiency of this method.     

Flux and traction boundary elements without hypersingular or strongly singular integrals

The present paper deals with a boundary element formulation based on the traction elasticity boundary integral equation (potential derivative for Laplace's problem). The hypersingular and strongly singular integrals appearing in the formulation are analytically transformed to yield line and surface integrals which are at most weakly singular. Regularization and analytical transformation of the boundary integrals is done prior to any boundary discretization. The integration process does not require any change of co‐ordinates and the resulting integrals can be numerically evaluated in a simple and efficient way. The formulation presented is completely general and valid for arbitrary shaped open or closed boundaries. Analytical expressions for all the required hypersingular or strongly singular integrals are given in the paper. To fulfil the continuity requirement over the primary density a simple BE discretization strategy is adopted. Continuous elements are used whereas the collocation points are shifted towards the interior of the elements. This paper pretends to contribute to the transformation of hypersingular boundary element formulations as something clear, general and easy to handle similar to in the classical formulation. Copyright © 2000 John Wiley & Sons, Ltd.     

A new boundary integral equation method for analysis of cracked linear elastic bodies

Abstract

A novel integral equation method is developed in this paper for the analysis of two‐dimensional general anisotropic elastic bodies with cracks. In contrast to the conventional boundary integral methods based on reciprocal work theorem, the present method is derived from Stroh's formalism for anisotropic elasticity in conjunction with Cauchy's integral formula. The proposed boundary integral equations contain boundary displacement gradients and tractions on the non‐crack boundary and the dislocations on the crack lines. In cases where only the crack faces are subjected to tractions, the integrals on the non‐crack boundary are non‐singular. The boundary integral equations can be solved using Gaussian‐type integration formulas directly without dividing the boundary into discrete elements. Numerical examples of stress intensity factors are given to illustrate the effectiveness and accuracy of the present method.     

Self‐regular boundary integral equation formulations for Laplace's equation in 2‐D

The purpose of this work is to demonstrate the application of the self‐regular formulation strategy using Green's identity (potential‐BIE) and its gradient form (flux‐BIE) for Laplace's equation. Self‐regular formulations lead to highly effective BEM algorithms that utilize standard conforming boundary elements and low‐order Gaussian integrations. Both formulations are discussed and implemented for two‐dimensional potential problems, and numerical results are presented. Potential results show that the use of quartic interpolations is required for the flux‐BIE to show comparable accuracy to the potential‐BIE using quadratic interpolations. On the other hand, flux error results in the potential‐BIE implementation can be dominated by the numerical integration of the logarithmic kernel of the remaining weakly singular integral. Accuracy of these flux results does not improve beyond a certain level when using standard quadrature together with a special transformation, but when an alternative logarithmic quadrature scheme is used these errors are shown to reduce abruptly, and the flux results converge monotonically to the exact answer. In the flux‐BIE implementation, where all integrals are regularized, flux results accuracy improves systematically, even with some oscillations, when refining the mesh or increasing the order of the interpolating function. The flux‐BIE approach presents a great numerical sensitivity to the mesh generation scheme and refinement. Accurate results for the potential and the flux were obtained for coarse‐graded meshes in which the rate of change of the tangential derivative of the potential was better approximated. This numerical sensitivity and the need for graded meshes were not found in the elasticity problem for which self‐regular formulations have also been developed using a similar approach. Logarithmic quadrature to evaluate the weakly singular integral is implemented in the self‐regular potential‐BIE, showing that the magnitude of the error is dependent only on the standard Gauss integration of the regularized integral, but not on this logarithmic quadrature of the weakly singular integral. The self‐regular potential‐BIE is compared with the standard (CPV) formulation, showing the equivalence between these formulations. The self‐regular BIE formulations and computational algorithms are established as robust alternatives to singular BIE formulations for potential problems. Copyright © 2001 John Wiley & Sons, Ltd.     

Variable thermal singularity boundary elements in the study of neighbouring singularities

Two new boundary elements are presented for the simulation of variable order thermal singularities in two dimensions. The first can model the variable order temperature derivative and heat flux singularities at one end of the element. The second element can simulate the temperature derivative and heat flux singularities at both the ends of the element simultaneously. These elements are useful for studying the interaction of variable order thermal neighbouring singularities. They are employed here for the computation of stress intensity factors in the crack–crack interaction problems under thermal load. To improve the accuracy of such computations a modified crack closure integral based method is adapted. Examples of mode I and mode II centre crack, two collinear neighbouring cracks, kinked crack in a plate, and tee joint with a kinked edge crack under thermal or thermal and mechanical loading are studied to illustrate the usefulness of these elements in the study of neighbouring thermal singularities. Copyright © 2001 John Wiley & Sons, Ltd.     

An advanced boundary element method for solving 2D and 3D static problems in Mindlin's strain‐gradient theory of elasticity

An advanced boundary element method (BEM) for solving two‐ (2D) and three‐dimensional (3D) problems in materials with microstructural effects is presented. The analysis is performed in the context of Mindlin's Form‐II gradient elastic theory. The fundamental solution of the equilibrium partial differential equation is explicitly derived. The integral representation of the problem, consisting of two boundary integral equations, one for displacements and the other for its normal derivative, is developed. The global boundary of the analyzed domain is discretized into quadratic line and quadrilateral elements for 2D and 3D problems, respectively. Representative 2D and 3D numerical examples are presented to illustrate the method, demonstrate its accuracy and efficiency and assess the gradient effect on the response. The importance of satisfying the correct boundary conditions in gradient elastic problems is illustrated with the solution of simple 2D problems. Copyright © 2010 John Wiley & Sons, Ltd.     

Meshless thermo‐elastoplastic analysis by triple‐reciprocity boundary element method

In general, internal cells are required to solve thermo‐elastoplasticity problems by a conventional boundary element method (BEM). However, in this case, the merit of BEM, which is the easy preparation of data, is lost. A conventional multiple‐reciprocity boundary element method (MRBEM) cannot be used to solve elastoplasticity problems, because the distribution of initial strain or stress cannot be determined analytically. In this study, it is shown that without the use of internal cells, two‐dimensional thermo‐elastoplasticity problems can be solved by a triple‐reciprocity BEM using a thin plate spline. Initial strain and stress formulations are adopted and the initial strain or stress distribution is interpolated using boundary integral equations. A new computer program was developed and applied to solve several problems. Copyright © 2009 John Wiley & Sons, Ltd.     

A BEM sensitivity and shape identification analysis for acoustic scattering in fluid–solid problems

In this paper a boundary element formulation for the sensitivity analysis of structures immersed in an inviscide fluid and illuminated by harmonic incident plane waves is presented. Also presented is the sensitivity analysis coupled with an optimization procedure for analyses of flaw identification problems. The formulation developed utilizes the boundary integral equation of the Helmholtz equation for the external problem and the Cauchy–Navier equation for the internal elastic problem. The sensitivities are obtained by the implicit differentiation technique. Examples are presented to demonstrate the accuracy of the proposed formulations. © 1998 John Wiley & Sons, Ltd.     

A new particular solution strategy for hyperbolic boundary value problems using hybrid‐Trefftz displacement elements

A new indirect approach to the problem of approximating the particular solution of non‐homogeneous hyperbolic boundary value problems is presented. Unlike the dual reciprocity method, which constructs approximate particular solutions using radial basis functions, polynomials or trigonometric functions, the method reported here uses the homogeneous solutions of the problem obtained by discarding all time‐derivative terms from the governing equation. Nevertheless, what typifies the present approach from a conceptual standpoint is the option of not using these trial functions exclusively for the approximation of the particular solution but to fully integrate them with the (Trefftz‐compliant) homogeneous solution basis. The particular solution trial basis is capable of significantly improving the Trefftz solution even when the original equation is genuinely homogeneous, an advantage that is lost if the basis is used exclusively for the recovery of the source terms. Similarly, a sufficiently refined Trefftz‐compliant basis is able to compensate for possible weaknesses of the particular solution approximation. The method is implemented using the displacement model of the hybrid‐Trefftz finite element method. The functions used in the particular solution basis reduce most terms of the matrix of coefficients to boundary integral expressions and preserve the Hermitian, sparse and localized structure of the solving system that typifies hybrid‐Trefftz formulations. Even when domain integrals are present, they are generally easy to handle, because the integrand presents no singularity. Copyright © 2015 John Wiley & Sons, Ltd.