Nonsingular Boundary Integral Equations Research Articles

A new global and direct integral formulation for 2D potential problems

A new global and direct integral formulation (GDIF) is presented for 2D potential problems. The 'global' and 'direct' mean that Gaussian quadrature can be applied directly to the entire body surface if its geometry description is mathematically available. This concept is simple and time-honored. The method has been long pursued by several researchers thanks to its accuracy and efficiency. However, the GDIF is based on the boundary integral equations (BIEs). The most crucial but difficult part in this method is to eliminate the singularities in BIEs, especially the source singularity. In this study, new non-singular boundary integral equations (NSBIEs) with indirect unknowns are developed in association with the average source technique without using the equi-potential method for source singularity. The integrands of all integrals in the NSBIEs are finite at any point on the body surface, which allows them to be considered as a normal function for computation. Based on this, with collocation points chosen in the NSBIEs being exactly the same as Gaussian points, an arbitrary order Gaussian quadrature can be directly applied to evaluate the integrals over the global elements. Three benchmark examples are tested to verify the efficiency and convergence of the proposed scheme.

A fast multipole boundary element method for three‐dimensional acoustic problems in a subsonic uniform flow

AbstractA fast multipole boundary element method (FMBEM) in a subsonic uniform flow is presented. It is based on the boundary integral equation (BIE) in a subsonic uniform flow. The convected Green's function complicates its multipole expansion as well as the implementation of the computer code. Although the Lorentz transformation allows the Helmholtz equation in the uniform flow to be reduced to the standard Helmholtz equation, the deformation of the domain complicates the boundary conditions and may cause the elements' distortion. In this work, the analytical evaluations of singular integrals are achieved. Then a nonsingular BIE in a subsonic uniform flow is obtained and is incorporated in building FMBEM with the plane wave multipole expansion of Green's function directly. Details on the implementation of the algorithm are described. Numerical examples including a pulsating sphere radiation problem, a multibody scattering problem and an aircraft model are performed to validate the accuracy and efficiency of the proposed method. Results show that FMBEM solutions are in good agreement with analytical solutions. The difference between the analytical moments and numerical moments is also investigated carefully in the implementation of the fast multipole method. Dramatical improvements on solution efficiency are observed by comparing the developed algorithm with the CBEM.

On uniqueness of numerical solution of boundary integral equations with 3-times monotone radial kernels

The uniqueness of solution of boundary integral equations (BIEs) is studied here when geometry of boundary and unknown functions are assumed piecewise constant. In fact we will show BIEs with 3-times monotone radial kernels have unique piecewise constant solution. In this paper nonnegative radial function Fδ3 is introduced which has important contribution in proving the uniqueness. It can be found from the paper if δ3 is sufficiently small then eigenvalues of the boundary integral operator are bigger than Fδ3/2. Note that there is a smart relation between δ3 and boundary discretization which is reported in the paper, clearly. In this article an appropriate constant c0 is found which ensures uniqueness of solution of BIE with logarithmic kernel ln(c0r) as fundamental solution of Laplace equation. As a result, an upper bound for condition number of constant Galerkin BEMs system matrix is obtained when the size of boundary cells decreases. The upper bound found depends on three important issues: geometry of boundary, size of boundary cells and the kernel function. Also non-singular BIEs are proposed which can be used in boundary elements method (BEM) instead of singular ones to solve partial differential equations (PDEs). Then singular boundary integrals are vanished from BEM when the non-singular BIEs are used. Finally some numerical examples are presented which confirm the analytical results.

Evaluation of Singular and Nearly Singular Integrals in the BEM with Exact Geometrical Representation

The geometries of many problems of practical interest are created from circular or elliptic arcs. Arc boundary elements can represent these boundaries exactly, and consequently, errors caused by representing such geometries using polynomial shape functions can be removed. To fully utilize the geometry of circular boundary, the non-singular boundary integral equations (BIEs) and a general nonlinear transformation technique available for arc elements are introduced to remove or damp out the singular or nearly singular properties of the integral kernels. Several benchmark 2D elastostatic problems demonstrate that the present algorithm can effectively handle singular and nearly singular integrals occurring in the boundary element method (BEM) for boundary layer effect and thin-walled structural problems. Owing to the employment of exact geometrical representation, only a small number of elements need to be divided along the boundary and high accuracy can be achieved without increasing other more computational efforts.

Nonsingular boundary integral equation for two‐dimensional electromagnetic scattering problems

AbstractThis paper is concerned with the application of the nonsingular boundary integral equation (NSBIE) for 2D electromagnetic scattering problems in the frequency domain. In the proposed NSBIE, the conventional treatment of the singular integral for the boundary integral equation is circumvented by employing the Gauss flux theorem and the concept of equipotential. Then the NSBIE for the Helmholtz equation is used to predict the scattering of plane wave by a perfectly electric conducting circular cylinder of infinite length at normal incidence. Based on the good results, as compared with solutions of the analytical and boundary‐element methods, it is verified that the NSBIE is a highly simple and efficient numerical scheme for 2D electromagnetic scattering simulations. © 2006 Wiley Periodicals, Inc. Microwave Opt Technol Lett 48: 760–765, 2006; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.21467

Multi-Variable Non-Singular BEM for 2-D Potential Problems

A multi-variable non-singular boundary element method (MNBEM) is presented for 2-D potential problems. This method is based on the coincident collocation of non-singular boundary integral equations (BIEs) of the potential and its derivatives, where the nodal potential derivatives are considered independent of the nodal potential and flux. The system equation is solved to determine the unknown boundary potentials and fluxes, with high accuracy boundary nodal potential derivatives obtained from the solution at the same time. A modified Gaussian elimination algorithm was developed to improve the solution efficiency of the final system equation. Numerical examples verify the validity of the proposed algorithm.

A multi-variable non-singular BEM in 2D elasticity

A new BEM algorithm is proposed in the paper. This algorithm is based on the coincident collocation of the non-singular boundary integral equations (BIEs) of displacement and its derivatives, and the basic variables of the new BEM system equation are the boundary nodal displacements, tractions and displacement derivatives. Due to the non-singular character of the BIEs, all integrals in the algorithm can be evaluated by the Gaussian quadrature. Furthermore, the conforming element can be used for this algorithm as the displacement derivative can be considered independent of the displacement. Once the new BEM system equation is solved, the boundary stress can be evaluated immediately from the obtained displacement derivatives. Numerical examples show the validity of the algorithm.

Nonsingular Boundary Integral Equations for Two-Dimensional Anisotropic Elasticity

Nonsingular boundary integral equations for two-dimensional anisotropic elasticity problems are developed. The integral equations can be solved numerically by Gaussian quadratures. A numerical example is given to illustrate the effectiveness of the integral equations. [S0021-8936(00)00303-2]

Sensor proximity error in the conformal near-field acoustic holography and its solution by using the nonsingular boundary integral equation

In the conformal near-field acoustic holography (NAH) using the boundary element method (BEM), the transfer matrix relating the vibro-acoustic properties of source and field depends solely on the geometrical condition of the problem. This kind of NAH is known to be very powerful in dealing with the sources having irregular boundaries. When the vibro-acoustic source field is reconstructed by using this conformal NAH, one tends to position the sensors as close as possible to the source surface in order to get rich information on the nonpropagating wave components. The conventional acoustic BEM based on the Kirchhoff–Helmholtz integral equation has the singularity problem in the very-near field of the source surface: computation error increases very rapidly as approaching from the far field. This problem originated from the singular kernel of the fundamental solution of the BIE and can influence the reconstruction accuracy. In this paper, the holographic BIE is reformulated to remove the singularity by introducing an additional propagating plane wave. Reconstructed results by the conventional and the nonsingular BIE are compared for a simple radiator model. It is observed that the nonsingular formulation can yield accurate vibro-acoustic transfer matrix and thus improve the resolution of the reconstructed source field.

Numerical modelling of spillway flow with free drop and initially unknown discharge

The difficulties in solving the spillway flow arise not only from the determination of the critical point, which has been used to divide the subcritical and supercritical flows, but also from the fact that either the boundary or the discharge is not known a priori. It is much more difficult to deal with such flow pattern with free drop. In this paper, by using the analytic functional boundary-value theory and the substitution of variables, the nonsingular boundary integral equations in the physical plane are derived. A synchronous iterative method for the discharge and the flow pattern is presented according to the consistency between the discharge and the uniform velocity at far upstream. Flows over different spillway profiles are treated. The discharge, the profiles of the free drop and the pressure distributions on the walls are calculated. The numerical results are in good agreement with the measured ones.

Elastodynamic Thin Plate Bending Analysis by Boundary Element Method with Laplace Transform

The boundary element method(BEM) is developed for the dynamic analysis of thin elastic plate bending problems with arbitrary boundary conditions. The formulation employs Laplace-transform technique, where the boundary integral equations are obtained on the Laplace transformed domain with the fundamental solutions derived from Kelvin's functions. The accuracy of the numerical results mainly depends on those of numerical estimation of the singular integral derived from the static term of the fundamental solutions. In the present paper, an BEM formulation based on a single boundary integral equation of the deflection, which employes a source point on both the boundary and outer region, is discussed in detail. A non-singular boundary integral equation is introduced on the transformed domain, which is obtained by superposition of the analyzed field and the referenced field with a uniform gradient of deflection. Numerical results obtained by the proposed method are compared with the analytical solutions and the other numerical solutions by means of several numerical examples. These examples also serve to illustrate the use of the proposed method.

Trefftz direct method

This paper discusses a so-called Trefftz Direct Method (TDM), in which the non-singular boundary integral equation is used as the starting formulation and complete sets of solution are used as weighting functions. Several relevant characteristics and problems of this approximation are also discussed.

Multiple reciprocity method for harmonic vibration of thin elastic plates

In this work, the boundary element method formulation is developed for solution of problems of harmonic vibrations of thin elastic plates (including viscous damping) by using a static fundamental solution of the biharmonic operator. The multiple reciprocity method is applied to the transformation of the domain integrals of unknown deflection and/or its gradient to boundary integrals. A short discussion of the numerical implementation of the derived nonsingular boundary integral equations is included.

Computation of Boundary Stresses by Solving Nonsingular Boundary Integral Equations

This paper concerns the derivation of the nonsingular ordinary boundary integral equations for the tangential derivatives of boundary displacements. Having solved these equations after computation of tractions on the boundary, one can evaluate the stress tensor components at any boundary point. The present formulation can be implemented numerically using standard C0-continuous boundary elements for all the boundary densities. The number of integral equations which are to be solved is decreased as compared with the other formulations. Moreover, the formulatlon is presented in the most general form applicable to two- and three-dimensional problems.

Nonsingular BEM formulations for thin-walled structures and elastostatic crack problems

This paper presents a unique approach, named “mathematical regularization”, to remove the nearly-singular and singular integrals occurring in the boundary integral formulations for the solution of the boundary value problems with a “pathologica” integration boundary due to the very near or coinciding parts of the total boundary. Nonsingular boundary integral equations are derived for thin-walled structure problems. In the case of crack-like problems, we present two kinds of the nonsingular integral representations of the secondary fields and the derivative boundary integral equations.