Conventional Boundary Integral Equation Research Articles

Boundary integral equation method for the diffraction of elastic waves using simplified green's functions

AbstractAn alternate formulation of the ‘substructure deletion method’ suggested by Dasgupta in 19791 has been successfully implemented. The idea is to utilize simple Green's functions developed for a surface problem to replace the more complicated Green's functions required for embedded problems while still being able to generate an accurate solution. Since the exterior medium is usually represented by a continuum model, the interior medium in the present approach will also be represented by a continuum model rather than a finite element model as suggested originally, thereby eliminating the incompatibility between the solutions of the interior and exterior media. Detailed studies of the method's accuracy and limitations were performed using two‐dimensional examples in wave scattering of canyons and alluvial valleys, problems which are more suitable for this method than the embedded foundation problem. The results obtained indicate that the alternate formulation gives accurate results only when the vertical dimension of the scattering object is not too large; if the aspect ratio (vertical over lateral) exceeds a certain limit, the results will not approach the known results given by boundary integral equation solutions or indirect boundary integral equations no matter what the refinement of the model may be. The greatest advantage of the present method is that the task of calculating Green's functions is reduced significantly; computational time using this new formulation is approximately five times less than for conventional boundary integral equation methods.

Hypersingular boundary integral equations for radiation and scattering of elastic waves in three dimensions

A weakly singular form of the hypersingular boundary integral equation (BIE) (traction equation) for 3-D elastic wave problems is developed in this paper. All integrals involved are at most weakly singular and except for a stronger smoothness requirement on boundary elements, regular quadrature and collocation procedures used for conventional BIEs are sufficient for the discretization of the original hypersingular BIE. This weakly singular form of the hypersingular BIE is applied to the composite BIE formulation which uses a linear combination of the conventional BIE and the hypersingular BIE to remove the fictitious eigenfrequencies existing in the conventional BIE formulation for elastic wave problems. Numerical examples employing different types of boundary elements clearly demonstrate the effectiveness and efficiency of the developed formulation.

Diffusion problems using transient discrete source superposition

AbstractDiscrete source superposition techniques seek sets of discrete, singular source strengths which satisfy the boundary conditions of a particular problem, and, necessarily, the governing differential equation. The sum of the fields produced by such sources thus provides an approximation to the required solution. In many respects these techniques are a special case of conventional boundary integral equation methods, where density distributions of smeared sources are sought. In this paper an attempt is made to extend the discrete source superposition technique to transient problems. The example addressed is the transient diffusion equation. The method developed does seem to have sòme promise. It is found that even coarse spatial discretization can yield quite accurate results, and that surprisingly large timesteps can be employed and accuracy maintained. Further, although no direct comparisons are made, the absence of spatial integrations indicates that the computational effort per timestep is likely to be less than for conventional BIE.

A weakly singular form of the hypersingular boundary integral equation applied to 3-D acoustic wave problems

The composite boundary integral equation (BIE) formulation, using a linear combination of the conventional BIE and the hypersingular BIE, emerges as the most effective and efficient formula for acoustic wave problems in an exterior medium which is free of the well-known fictitious eigen-frequency difficulty. The crucial part in implementing this formulation is dealing with the hypersingular integrals. Various regularization procedures in the literature give rise, in general, to integrals which are still difficult and/or extremely time consuming to evaluate or are limited to the use of special, usually flat, boundary elements. In this paper, a general form of the hypersingular BIE is developed for 3-D acoustic wave problems, which contains at most weakly singular integrals. This weakly singular form can be derived by employing certain integral identities involving the static Green's function. It is shown that the discretization of this weakly singular form of the hypersingular BIE is straightforward and the same collocation procedures and regular quadrature as that used for conventional BIEs are sufficient to compute all the integrals involved. Computing times are only slightly longer than with conventional BIEs. The C 1 smoothness requirement imposed on the density function for existence of the hypersingular BIEs and the possibility of relaxing this requirement are discussed. Three kinds of boundary elements, having different smoothness features, are employed. Numerical results are given for scattering from a rigid sphere at the fictitious frequencies, for values of wavenumber from π to 5π. In essence, with the methodology in this paper the fictitious eigenfrequency difficulty, long associated with the BIE for exterior problems, should no longer be a troublesome issue.

Some identities for fundamental solutions and their applications to weakly-singular boundary element formulations

Some integral identities for the fundamental solutions of potential and elastostatic problems are established in this paper. With these identities it is shown that the conventional boundary integral equation (BIE), which is generally expressed in terms of singular integrals in the sense of the Cauchy principal value (CPV), and the derivative BIE, which is similarly expressed in terms of hypersingular integrals in the sense of the Hadamard finite-part (HFP), can both be written as weakly-singular integral equations in a systematic approach. Discretization of the weakly-singular BIE leads to the weakly-singular boundary element formulation equivalent to the method of using the rigid body displacement to determine the diagonal submatrices, which involve the CPV terms and the geometric matrix C, in the conventional BEM. The discretization of the weakly-singular derivative BIE possesses a similar feature, i.e. no CPV and HFP are involved. All these suggest that the practice of calculating CPV or HFP (for boundary integrals) and the geometric matrix C, either analytically or numerically, is unnecessary in the BEM. The approach developed in this paper is applicable to other problems such as plate bending, acoustics and elastodynamics.

Boundary integral equations in three-dimensional elastostatics using the Boussinesq-Cerruti fundamental solution

A so-called traction-free fundamental solution rather than the conventional Kelvin fundamental solution is used to formulate the boundary integral equations for problems in three-dimensional elastostatics. For an isotropic material, the traction-free fundamental solution is a combination of the Boussinesq and Cerruti solutions. This formulation has two advantages over the conventional one. The first is that the boundary integral equations are less singular than the conventional formulation and converge in the normal sense rather than in the Cauchy principal value sense. The second is that a formal differentiation of the boundary integral representation for displacement leads to a valid integral representation for the in-plane stress components on the boundary in terms of the boundary displacements and tractions only. The analogous representation obtained from the conventional boundary integral equations does not converge.

Three‐dimensional crack analysis using singular boundary elements

AbstractA multi‐domain method of solving three‐dimensional elastic crack problems in an infinite elastic body using the boundary element method is proposed. The displacement and traction behaviours near a crack front are incorporated in special crack elements. The elimination of singularities arising from the term combined with Kelvin's kernel for displacement in the integrals is discussed in detail. Stress intensity factors of modes I, II and III are obtained directly from crack‐front nodal values, without any extrapolation as in some other methods. No differentiation of conventional boundary integral equations (with Kelvin's tensor kernels) is necessary in the current approach. This method is applicable to cracks of arbitrary shape. Infinite bodies are modelled precisely as such, not approximated as large finite bodies. Numerical solutions of stress intensity factors are given for several problems involving a penny‐shaped crack.

Suppression of irregular frequency effects in fluid–structure interaction problems using a combined boundary integral equation method

AbstractIt is well known that at certain discrete frequencies the conventional boundary integral equation formulation of free surface fluid–structure interaction analyses breaks down. At such ‘irregular’ frequencies the BIE method fails to provide either an acceptable or a unique solution. Having established the existence of irregular frequencies, a review of the different approaches adopted to remedy this problem is presented.A very simple modification of the BIE method is also presented to eliminate the irregular frequency problem. The proposed procedure, designated the combined boundary integral equation method (CBIEM), can be categorized as a modified integral domain method. A description of the CBIEM formulation is presented and its ability to provide a unique solution at all frequencies is demonstrated. Predictions of 3D hydrodynamic reactive coefficients of added mass and fluid damping for a Series 60 hull form and an ellipsoid based on the CBIEM procedure are presented. These predictions are compared with results generated using conventional integral equation methods. The numerical studies demonstrate that the CBIEM is both a practical and effective method of suppressing irregular frequencies. In particular, the procedure is easy to implement in existing BIE computer codes with minimal additional computational effort.

Direct BIE formulation for skin and proximity effect calculation with and without the use of the surface impedance approximation

The authors illustrate a new, direct BIE (boundary integral equation) formulation for skin and proximity effect calculations. The main advantage of the present formulation is that it is a logical extension of the well-known TM (transverse magnetic) induction formulation. The authors introduce the impedance boundary condition (IBC) approximation of the skin and proximity effect problems in terms of the conventional Fredholm boundary integral equations. Unlike previous formulations, the IBC approximation can be used with the external electric field equation, so the skin and proximity effect problem is formulated as an exterior problem. The method has been applied to test problems; reliable results were obtained both with and without the use of the IBC approximation. >

Application of integral equations with superstrong singularity to steady state heat conduction

In this paper, we present the theory of dual integral equations for steady state heat conduction. There are four kernel functions with different orders of singularity in the two equations. Using the first equation with weaker singularity, the conventional direct boundary integral equation method (BIEM) was proposed long ago. An important characteristics of the first equation is that its kernels are of the Riemann and Cauchy types. The purpose of this paper is to present a method based on the second equation with stronger singularity kernels to solve the steady state heat conduction problems. Whereas the kernels of the second equation are of the Cauchy and Hadamard types. It is further shown that combination of the two equations can be used to solve problems with degenerate boundary which have long suffered from lack of a general formulation of the BIEM. For concreteness, an illustrative example is performed numerically to see the validity of the theory.