Norio Kamiya

Distance transformation for the numerical evaluation of near singular boundary integrals with various kernels in boundary element method

The accurate numerical solution of near singular boundary integrals was an issue of major concern in most of the boundary element analysis next to the singular boundary integrals. The problem was solved in this paper by a kind of non-linear transformation, namely, the distance transformation for the accurate evaluation of near singular boundary integrals with various kernels for both the two- and three-dimensional problems incorporated with the distance functions defined in the local intrinsic coordinate systems. It is considered that two effects play the role in the transformation. They are the damping out of the near singularity and the rational redistribution of integration points. The actual numerical computation can be performed by standard Gaussian quadrature formulae and can be easily included in the existing computer code, along with its insensitivity to the kind of the boundary elements. Numerical results of potential problem were presented, showing the effectiveness and the generality of the algorithm, which makes it possible, for the first time, to observe the behaviors of various boundary integral values with numerical means, when the source point is moving across the boundary with fine steps.

Error estimation and adaptive mesh refinement in boundary element method, an overview

Abstract Further to the previous review article (Adv Engng Software 19(1) (1994) 21–32), this paper reviews more recent studies on the same subject by citing more than one hundred papers. The adaptive mesh refinement process is composed of three processes; the error estimation, the adaptive tactics and the mesh refinement processes. Therefore, in this paper, the existing studies are classified and discussed according to the processes. The error estimation schemes are classified into the residual-type, the interpolation-type, the integral equation-type, the node sensitivity-type and the solution difference type. The mesh refinement schemes are classified into h-, p-, r-schemes and the others. The adaptive tactics are closely related to the mesh refinement schemes. Therefore, they are discussed individually. The discussion presented herein is an extension of the previous article and focuses our principal attention on the following points. Some interesting studies for the error estimation scheme are added; e.g. new schemes named as ‘nodal design sensitivity’, ‘hyper-singular residual type’ and ‘solution difference type’. Some studies for the adaptive tactics are added; e.g. the tactics based on the convergence property of the error, the extension of the extended error indicator to r- and hr-adaptive schemes and so on.

A new complex-valued formulation and eigenvalue analysis of the Helmholtz equation by boundary element method

By considering the close relationship between the multiple reciprocity boundary element formulation and that of the fundamental solution of the Helmholtz differential operator, we present a new complex-valued integral equation formulation for the eigenvalue analysis of the scalar-valued Helmholtz equation. Eigenvalues are determined as local minima of the determinant of the coefficient matrix of the discretized equation iteratively by the Newton scheme. The necessary recurrence formula is derived and computed with high efficiency, due to polynomial representation of the matrix components. Some example computations demonstrate the utility of the proposed formulation and eigenvalue determination scheme, and construction of adaptive boundary elements for the eigenvalue determination is attempted.

Parallel implementation of boundary element method with domain decomposition

Algorithms for the parallel boundary element computation of the domain-decomposed problem are considered in this paper. In this method, boundary conditions on virtual internal boundaries are assumed, and then the boundary element analysis for each subdomain is performed in parallel. The assumed conditions are modified according to the numerical results and the process returns to parallel computation. The Uzawa and the Schwarz methods without domain overlap are applied as alternative schemes on the internal boundary. The methods are implemented on the cluster computing system in order to examine the schemes from the view-point of the convergency of the solution.

A general algorithm for accurate computation of field variables and its derivatives near the boundary in BEM

Abstract A general algorithm was proposed in the paper for the accurate computation of the field variables and its derivatives at domain points near the boundary in attempt to solve the so-called boundary layer effect in the boundary element method. The algorithm is based on the parameter, including modified Gauss–Tschebyscheff quadrature formula with the aid of the approximate distance function introduced, where the parameter is defined as the ratio of the minimum distance of the domain point to the boundary and the length of the boundary element. The algorithm is not only numerically stable because the singular part of the integrand serves as the weight function in the modified Gauss–Tschebyscheff quadrature formula but also independent of the kind of boundary elements. The method can be extended to the three-dimensional case with little modifications. Numerical examples of the potential problem and the elastic problem of plane strain were given by using the cubic and the quadratic boundary elements, respectively, showing the feasibility and the effectiveness of the proposed algorithm.

Application of a direct Trefftz method with domain decomposition to 2D potential problems

Abstract This article presents an application of a direct Trefftz method with domain decomposition method to the two-dimensional potential problem. In the direct Trefftz methods, regular T-complete functions satisfying the governing equations are taken as the weighting functions and then, the boundary integral equations are derived from the weighted residual expressions of the governing equations. Since the T-complete functions are regular, the final equations are also regular and therefore, much simpler than the ordinary boundary element methods employing the singular fundamental solutions. Their computational accuracy, however, is dependent on the condition number of the coefficient matrices of the algebraic system of equations. So, for improving the accuracy, we introduce the domain decomposition method to the direct Trefftz methods. The present method is applied to the two-dimensional potential problem in order to confirm the validity.

Eigenvalue analysis by the boundary element method: new developments☆

Abstract Recent developments in boundary element eigenvalue analysis are reviewed, focusing on the Helmholtz equation in terms of a scalar-valued function. The problem is of fundamental importance in the framework of the sophisticated boundary element scheme, in general devised for the non-homogeneous differential equation. The most popular approach using domain cells for domain integration and some transformation methods, such as the dual reciprocity method (DRM) and the multiple reciprocity method (MRM), are discussed. Two key issues are the solution without domain integration and the standard routine eigenvalue search in contrast to the conventional domain cell discretization and the direct eigenvalue search using distribution of the magnitude of the determinant, which are the marked item of the boundary element method for numerical efficiency and are preffered over other methods.

An adaptive BEM by sample point error analysis

Abstract A new adaptive boundary element method (BEM) is developed in this paper based on the concept ‘sample point error analysis’. The method relies on an estimation of the discretization error on each boundary element appearing in the boundary integral equation, through the magnitude of inconsistency of the intermediate solution, except at the boundary nodes for the prescribed discretization. This paper deals with the h -version of the adaptive boundary mesh refinement scheme with some simple numerical examples of the two-dimensional Laplace equation.

Parallel computing for the combination method of BEM and FEM

A new algorithm for the parallel computing of the boundary-element and finite-element combination method is presented in this paper. By introducing the domain decomposition of an entire domain under consideration into the boundary-element and finite-element subdomains, each analysis is performed independently and in parallel. A renewal iterative scheme for parallel computing is the Schwarz method which was previously adopted to the domain decomposition parallel scheme in boundary-element analysis. A cluster parallel computing system by workstations connected by the LAN is constructed and employed aiming at efficient analysis. Convergence and accuracy of solutions on the internal virtual boundaries are shown through some sample examples.

Indirect Trefftz method for boundary value problem of Poisson equation

Trefftz method is the boundary-type solution procedure using regular T-complete functions satisfying the governing equation. Until now, it has been mainly applied to numerical analyses of the problems governed with the homogeneous differential equations such as the two- and three-dimensional Laplace problems and the two-dimensional elastic problem without body forces. On the other hand, this paper describes the application of the indirect Trefftz method to the solution of the boundary value problems of the two-dimensional Poisson equation. Since the Poisson equation has an inhomogeneous term, it is generally difficult to determine the T-complete function satisfying the governing equation. In this paper, the inhomogeneous term containing an unknown function is approximated by a polynomial in the Cartesian coordinates to determine the particular solutions related to the inhomogeneous term. Then, the boundary value problem of the Poisson equation is transformed to that of the Laplace equation by using the particular solution. Once the boundary value problem of the Poisson equation is solved according to the ordinary Trefftz formulation, the solution of the boundary value problem of the Poisson equation is estimated from the solution of the Laplace equation and the particular solution. The unknown parameters included in the particular solution are determined by the iterative process. The present scheme is applied to some examples in order to examine the numerical properties.