Conventional Boundary Integral Equation Research Articles

An equivalent boundary integral formulation for bending problems of thin plates

For the bending problems of simply supported thin plates, the governing bi-harmonic equation can be decomposed into two independent Poisson equations based upon Kirchhoff theory. The conventional boundary integral formulations based on this decomposition technique have been proved to yield non-equivalent solutions compared with the boundary value problem. In this paper, a boundary integral formulation called an equivalent boundary integral equation has been deduced from Poisson equations, and it is also numerically shown that the problem of solution non-equivalence of the conventional boundary integral equation does not exist in the equivalent boundary integral equation.

The boundary contour method based on the equivalent boundary integral equation for 2-D linear elasticity

The conventional boundary integral equation in two dimensions is non-equivalent to its corresponding boundary value problem when the scale in the fundamental solution reaches its degenerate scale values. An equivalent boundary integral equation was recently derived. This equation has the same solution as the boundary value problem of differential equations. This paper presents the boundary contour method based on the equivalent boundary integral equation for two-dimensional linear elasticity. The method requires only numerical evaluation of potential functions and gives correct equivalent results to the boundary value problem of differential equations in two dimensions. Numerical results are presented for some examples. The present approach is shown to give excellent results in illustrative examples. Meanwhile, the traction results from the BCM based on the conventional displacement boundary integral equation are incorrect.

New identities for fundamental solutions and their applications to non-singular boundary element formulations

Based on a general, operational approach, two new integral identities for the fundamental solutions of the potential and elastostatic problems are established in this paper. Non-singular forms of the conventional boundary integral equations (BIEs) are derived by employing these two identities for the fundamental solutions and the two-term subtraction technique. Both the strongly- (Cauchy type) and weakly-singular integrals existing in the conventional BIEs are removed from the BIE formulations. The existence of the non-singular forms of the conventional BIEs raises new and interesting questions about the smoothness requirement in the boundary element method (BEM), since the two-term subtraction requires, theoretically, C 1 continuity of the density function, rather than the C 0 continuity as required by the original singular or weakly-singular forms of the conventional BIEs. Implication of the non-singular BIEs on the smoothness requirement will be discussed in this paper.

Boundary integral equations of unique solutions in elasticity

The properties of the fundamental solution are derived in linear elastostatics. These properties are used to show that the conventional displacement and traction boundary integral equations yield non-unique displacement solutions in a traction boundary value, problem. The condition for the existence of unique displacement solutions is proposed for the traction boundary value problem. The degrees of freedom of the displacement solution are removed by the condition to obtain the boundary integral equations of unique solutions for the traction boundary value problems. Numerical example is presented to demonstrate the accuracy and efficiency of the present equations.

A unified boundary element method for the analysis of sound and shell-like structure interactions. I. Formulation and verification

A unified boundary element method (BEM) is developed in this paper to model both the exterior acoustic field and the elastic shell-like structure in a coupled analysis. The conventional boundary integral equation (BIE) for three-dimensional (3D) elastodynamics is applied to thin shell-like structures which can have arbitrary shapes and small thicknesses. The nearly singular integrals existing in the BIE when applied to thin bodies are transformed to nonsingular line integrals and are evaluated accurately and efficiently. For the exterior 3D acoustic domain, the Burton and Miller composite BIE formulation is employed to overcome the fictitious eigenfrequency difficulty (FED) and the thin-shape breakdown (TSB). Conforming C0 quadratic elements are employed in the discretization of the two sets of BIEs. The developed BIE formulations are valid for both radiation and scattering problems and for all wave numbers. Numerical examples using spherical and cylindrical shells, including nonuniform thickness and nondimensional wave numbers up to 12, clearly demonstrate the effectiveness and accuracy of the developed BEM approach.

Seismic soil-structure interaction analysis by direct boundary element methods

Upon establishing the mathematical framework necessary for a proper understanding of the analytical theory, a regularized form of the conventional direct boundary integral equation formulation for three-dimensional elastodynamics is presented for a general anisotropic medium. Founded on the basis of a full decomposition of the Greens functions into regular and singular parts, the alternative boundary integral equation format is compact and without demand ingmathematical and numerical complexities such as Cauchy principal values. Extended to deal with general seismic soil-structure interaction problems in semi-infinite media, the formulation is implemented computationally together with a rigorous treatment of singular dynamic multi-layered viscoelastic half-space Greens functions and interfacial boundary tractions arising in typical soil-structure-foundation configurations. A set of new benchmark numerical results are included.

Stress intensity sensitivities via Hypersingular boundary integral equations

In sensitivity analysis for problems involving thin domains or cracks, conventional boundary integral equations must be supplemented and/or replaced by hypersingular ones. Hypersingular boundary integral equations for sensitivity analysis are developed here in two forms, using both a global and a local regularization. These regularizations are easily implemented, since the singularity order of the sensitivity BIE formulae are no stronger than those of the ordinary BIEs. A justification for this work is that obtaining stress intensity factor versus crack size curves is made easier using sensitivity analysis, since the computation of a function value and a derivative is less expensive than that for two function values. Examples of a circular bar with an embedded penny-shaped crack under tension, bending, and torsion loads are shown to be accurate and verify the formulation and the computer codes developed in this paper.

Analysis of shell-like structures by the Boundary Element Method based on 3-D elasticity: formulation and verification

In this paper, the Boundary Element Method (BEM) for 3-D elastostatic problems is studied for the analysis of shell or shell-like structures. It is shown that the conventional boundary integral equation (CBIE) for 3-D elasticity does not degenerate when applied to shell-like structures, contrary to the case when it is applied to crack-like problems where it does degenerate due to the closeness of the two crack surfaces. The treatment of the nearly singular integrals, which is a crucial step in the applications of BIEs to thin shapes, is presented in detail. To verify the theory, numerical examples of spherical and ellipsoidal vessels are presented using the BEM approach developed in this paper. It is found that the system of equations using the CBIE is well conditioned for all the thickness studied for the vessels. The advantages, disadvantages and potential applications of the proposed BEM approach to shell-like structures, as compared with the FEM regarding modelling and accuracy, are discussed in the last section. Applications of this BEM approach to shell-like structures with non-uniform thickness, stiffeners and layers will be reported in a subsequent paper. © 1998 John Wiley & Sons, Ltd.

Scattering of elastic waves from thin shapes in three dimensions using the composite boundary integral equation formulation

In this paper, the composite boundary integral equation (BIE) formulation is applied to scattering of elastic waves from thin shapes with small but finite thickness (open cracks or thin voids, thin inclusions, thin-layer interfaces, etc.), which are modeled with two surfaces. This composite BIE formulation, which is an extension of the Burton and Miller’s formulation for acoustic waves, uses a linear combination of the conventional BIE and the hypersingular BIE. For thin shapes, the conventional BIE, as well as the hypersingular BIE, will degenerate (or nearly degenerate) if they are applied individually on the two surfaces. The composite BIE formulation, however, will not degenerate for such problems, as demonstrated in this paper. Nearly singular and hypersingular integrals, which arise in problems involving thin shapes modeled with two surfaces, are transformed into sums of weakly singular integrals and nonsingular line integrals. Thus, no finer mesh is needed to compute these nearly singular integrals. Numerical examples of elastic waves scattered from penny-shaped cracks with varying openings are presented to demonstrate the effectiveness of the composite BIE formulation.

Non-equivalence of the conventional boundary integral formulation and its elimination for two-dimensional mixed potential problems

For a given mixed type potential problem, the corresponding conventional boundary integral equation is shown to yield non-equivalent solutions. Numerical results show that the conventional boundary integral formulation yields incorrect potential and flux results when the distance scale in the fundamental solution approaches its degenerate value. Such a kind of non-equivalence of the conventional boundary integral equation can be eliminated by the use of the necessary and sufficient boundary integral formulation which always ensures the equivalence of solutions.

A necessary and sufficient boundary integral formulation for plane elasticity problems

With respect to a given boundary value problem, the corresponding conventional boundary integral equation is shown to yield non-equivalent solutions, which are dependent upon Poisson's ratio and geometry. In the paper a systematic method for establishing a necessary and sufficient boundary integral formulation has been proposed for two-dimensional elastostatic problems. Numerical analyses show that the conventional boundary integral equation yields incorrect results when the scale in the fundamental solution approaches a degenerate scale value. However, the results of the necessary and sufficient boundary integral equation are in good agreement with analytical solutions of the boundary value problem.

A necessary and sufficient boundary integral formulation for plane elasticity problems

With respect to a given boundary value problem, the corresponding conventional boundary integral equation is shown to yield non-equivalent solutions, which are dependent upon Poisson's ratio and geometry. In the paper a systematic method for establishing a necessary and sufficient boundary integral formulation has been proposed for two-dimensional elastostatic problems. Numerical analyses show that the conventional boundary integral equation yields incorrect results when the scale in the fundamental solution approaches a degenerate scale value. However, the results of the necessary and sufficient boundary integral equation are in good agreement with analytical solutions of the boundary value problem.

Computer-aided design of dielectric optical waveguide bends by the boundary-element method based on guided-mode extracted integral equations

We show examples of accurate computer-aided design of two-dimensional dielectric waveguide bends of finite length by using the boundary-element method (BEM) based on guided-mode extracted integral equations (GMEIE’s) that have been proposed previously. In contrast to integral equations used in previous papers, the unknown function in the GMEIE’s that we discuss is the total field, which is the same as that in the conventional boundary integral equation method. Therefore waveguide discontinuity problems in dielectric open waveguides can be perfectly regarded as problems of scattering by isolated dielectric objects. Various properties in numerical calculations of GMEIE’s are examined. The numerical results obtained satisfy the energy conservation law within an accuracy of 1.0%. It is shown that it is possible to design an optimum bend shape of a dielectric optical waveguide bend that gives small radiation losses by using the BEM based on GMEIE’s.

Degenerate scales and boundary element analysis of two dimensional potential and elasticity problems

Some widely used conventional boundary integral equations yield non-equivalent solutions compared to their corresponding boundary value problems when the scales in the fundamental solutions reach their degenerate scale values. The effects of such degenerate scales on a two-dimensional boundary element solution of potential and plane elasticity problems are investigated for different geometries. Moreover, a simple convenient method is suggested to evaluate these degenerate scales. On the other hand, using the necessary and sufficient boundary integral formulation can eliminate the non-equivalence of these conventional boundary integral formulations.

Non-equivalence of the conventional boundary integral formulation and its elimination for plane elasticity problems

Compared with a given boundary value problem of plane elasticity, the corresponding conventional boundary integral equation is shown to yield non-equivalent solutions which are dependent upon Poisson's ratio and geometry. Such a non-equivalence of solutions of boundary integral equations can be eliminated by using a necessary and sufficient boundary integral formulation proposed by He [Necessary and sufficient BIE-BEM: its theory and practice. Ph.D. Dissertation, Zhejiang University, Hangzhou, China (1993)]. Numerical analysis shows that the conventional boundary integral equation yields incorrect non-equivalent results when the scale in the fundamental solution is near its degenerate scale value. Also, this non-equivalence can be remedied by using the necessary and sufficient boundary integral equation.

A NON-SINGULAR BOUNDARY INTEGRAL EQUATION FOR ACOUSTIC PROBLEMS

A weakly singular boundar integral equation and a non-singular one are derived for acoustic problems. By using the one-dimensional wave propagation mode, the degree of the singularities appearing in the conventional boundary integral equation can be reduced. Since the strong singularity is removed in the weakly singular equation, the jump term of the field pressure expression in the very vicinity of the boundary disappears. It is shown that the weak singularity can be also removed for a smooth boundary in the non-singular representation, and thus special treatments for singular integral are no longer needed. In order to test the proposed method, several simple problems have been solved by employing the triangular element with standard Gaussian quadrature integration points. The results obtained show the accuracy of the method in the near field as well as on the boundary.

ON THE CHOICE OF A DERIVATIVE BOUNDARY ELEMENT FORMULATION USING HERMITE INTERPOLATION

This paper reports on some problems that can arise with the use of regularized derivative boundary integral equations. It concentrates on developing a formulation for the simple Laplace equation using a cubic Hermite interpolation and shows how certain combinations of derivative and conventional boundary integral equations can result in a solution scheme severely lacking in stability. With some simple two- and three-dimensional geometries, the derivative equations on their own do not provide enough information to solve a Dirichlet problem. Even combinations of the conventional and derivative equations fail for some simple geometries. We conclude that the only consistently successful combination is that of the conventional equation with the tangential derivative equation, which showed cubic convergence of results with mesh refinement. Numerical results are presented for this scheme in both two and three dimensions.

The final stage of the collapse of a cavitation bubble near a rigid wall

During the collapse of an initially spherical cavitation bubble near a rigid wall, a reentrant jet forms from the side of the bubble farthest from the wall. This re-entrant jet impacts and penetrates the bubble surface closest to the wall during the final stage of the collapse. In the present paper, this phenomenon is modelled with potential flow theory, and a numerical approach based on conventional and hypersingular boundary integral equations is presented. The method allows for the continuous simulation of the bubble motion from growth to collapse and the impact and penetration of the reentrant jet. The numerical investigations show that during penetration the bubble surface is transformed to a ring bubble that is smoothly attached to a vortex sheet. The velocity of the tip of the re-entrant jet is always directed toward the wall during penetration with a speed less than its speed before impact. A high-pressure region is created around the penetration interface. Theoretical analysis and numerical results show that the liquid-liquid impact causes a loss in the kinetic energy of the flow field. Variations in the initial distance from the bubble centre to the wall are found to cause large changes in the details of the flow field. No existing experimental data are available to make a direct comparison with the numerical predictions. However, the results obtained in this study agree qualitatively with experimental observations.

NUMERICAL COMPARISON OF TWO BOUNDARY ELEMENT METHODS FOR PLANE HARMONIC FUNCTIONS

A direct boundary element method (BEM) has been studied inthe paper based on a set of sufficient and necessary boundary integral equations(BIE) for the plane harmonic functions. The new sufficient and necessary BEM leadsto accurate results while the conventional insufficient BEM will lead to inaccurateresults when the conventional BIE has multiple solutions. Theoretical and numericalanalyses show that it is beneficial to use the sufficient and necessary BEM, to avoidhidden dangers due to non-unique solution of the conventional BIE.

Numerical comparison of two boundary element methods for plane harmonic functions

A direct boundary element method (BEM) has been studied in the paper based on a set of sufficient and necessary boundary integral equations (BIE) for the plane harmonic functions. The new sufficient and necessary BEM leads to accurate results while the conventional insufficient BEM will lead to inaccurate results when the conventional BIE has multiple solutions. Theoretical and numerical analyses show that it is beneficial to use the sufficient and necessary BEM, to avoid hidden dangers due to non-unique solution of the conventional BIE.