Eisuke Kita

Trefftz method: an overview

Abstract The aim of this paper is to review the existing formulations of ‘Trefftz method’. The Trefftz formulations are classified into the direct and the indirect formulations and then, compared with other boundary-type solution procedures, such as boundary element, singularity, charge simulation and surface charge methods, in order to establish the identity of the method.

Recent studies on adaptive boundary element methods

Abstract This is a review of the recent studies on adaptive mesh refinement schemes for boundary element methods. We suppose that the adaptive mesh refinement schemes are constructed of error estimation, adaptive tactics and mesh refinement processes. The error estimation process estimates the error of numerical solution. The existing studies on error estimation are classified into residual type, interpolation error type, boundary integral equation error type, and others. The adaptive tactics process selects boundary elements to be refined and the related mesh refinement scheme. The mesh refinement process generates the data of the refined mesh. The existing studies on mesh refinement are classified into h-, p- and r-refinement, and their combination schemes. Various schemes proposed will be discussed from these three points of view.

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.

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.

Stock price forecast using Bayesian network

Bayesian network is a probabilistic graphical model that represents a set of random variables and their conditional dependencies via a directed acyclic graph. This paper describes the price earnings ratio (P/E ratio) forecast by using Bayesian network. Firstly, the use of clustering algorithm transforms the continuous P/E ratio to the set of digitized values. The Bayesian network for the P/E ratio forecast is determined from the set of the digitized values. NIKKEI stock average (NIKKEI225) and Toyota motor corporation stock price are considered as numerical examples. The results show that the forecast accuracy of the present algorithm is better than that of the traditional time-series forecast algorithms in comparison of their correlation coefficient and the root mean square error.

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.

Shape optimization of continuum structures by genetic algorithm and boundary element method

This paper presents a new scheme for the shape optimization of continuum structures by using genetic algorithm (GA) and boundary element methods (BEM). The profiles of the objects under consideration are represented by the free-form deformation (FFD) technique. The chromosomes for the profiles are defined by considering as the genes the position vectors of the FFD control points. The population, which is constructed of many chromosomes, is modified by applying genetic operations such as the selection, the crossover and the mutation for determining a better profile for the design objectives. Besides, the objective functions are estimated by the BEM. Three schemes are presented in order to improve the computational efficiency of the present scheme. Finally, a cantilever beam under uniformly distributed load is considered as a numerical example.

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.

Search performance improvement of Particle Swarm Optimization by second best particle information

In the original Particle Swarm Optimization (PSO), the particle position vectors denote the potential solutions of the optimization problem. Then, the position vectors are updated from the information of the global best and the personal best particles, which denote the best particle which has been ever found by all particles and the best particle which has been ever found by each particle, respectively.The aim of this study is to discuss that, in addition to the information of the global and personal best particles, the use of the information of the second global best and second personal best particles improves the search performance of the original PSO. Firstly, two algorithms are explained. One updates the particle positions by the positions of the global best, the personal best and second global best particles. Another uses second personal best particles instead of second global best particle. The present algorithms are compared with 6 PSO algorithms in 11 test functions. The results show that the present algorithms have the faster convergence speed and find better optimal solution than other algorithms. Therefore, it is concluded that the use of the second best particles can improve the search performance of the original PSO algorithm.