Error Analysis of Band Matrix Method

Numerical error in the solution of the band matrix method based on the elimination method in single precision is investigated theoretically and experimentally, and the behaviour of the truncation error and the roundoff error is clarified. Some important suggestions for the useful application of the band solver are proposed by using the results of above error analysis

Fundamental Study of the Fill-in Minimization Problem

In this paper the fill-in minimization problem which arises at the application of the sparse matrix method for a large sparse set of linear equations is discussed from the graph-theoretic viewpoint and also through the numerical experiments. Therefore, this investigation consists of two parts, and in the former part the author shows, at first, that the elimination process of a sparse matrix is equivalently replaced to the vertex eliminations for a graph obtained from the matrix, and by use of some concepts in the theory of graph he proves that the vertex elimination process for the minimum fill-in is equivalent to the vertex eliminations for vertices in each subgraph which is obtained by the appropriate dissection of whole graph, and that there are only two types of vertex eliminations through the process. This results in the proposal of a new model of the vertex elimination process. The latter part of this investigation is used for the verification of the results from the theoretic investigation. Through the numerical experiments he concludes that the new model of the vertex elimination process is valid, at least, for a graph like a regular finite element mesh. Furthermore, he shows that this model coincides with Nested Dissection Method which can give the minimum value of fill-in, at present

Two-Dimensional Automatic Mesh Generator for Finite Element Analysis

In this study a new automatic mesh generator for 2- dimensional finite element analysis is proposed, and its effectivity is surveyed through a number of test examples. Proposed one is for a micro-computer, and the program is written in BASIC. The user needs no preparation for making finite element model in advance. All of the neccessary informations are displayed on CRT display and its user may answer for questions. It is expected that the cost neccessary for preparing the input-data for finite element analysis is largely decreased

Recognition of Surface of 3-dimensional Body and Its Input to Computer - Use of Delaunay Triangulation -

CAD system is useful to define and input any artificial 3-dimensional configuration into computer, but the input of arbitrary natural 3-dimensional configuration is still serious for computer users. This paper includes a method to input arbitrary 3-dimensional configuration into computer. The method is based on 3-dimensioanl Delaunay triangulation which is a geometrical subdivision of arbitrary 3-dimensional convex domain into tetrahedra, and the auther shows a method to find triangles which cover whole surface of 3-dimensional body using the result of Delaunay triangulation

A GRAPH-THEORETIC STUDY OF THE MINIMUM FILL-IN PROBLEM FOR SPARSE MATRIX METHOD

In this paper the minimum fill-in problem which arises at the application of the sparse matrix method for linear sparse systems is discussed from the graphtheoretic viewpoint and the author gives some results which can be directly introduced in the design of, so called, the optimal elimination ordering algorithm which gives the minimum fill-in(the number of zeros in coefficient matrix which become non-zero during the elimination process). Through this investigation only graphs are treated instead of the coefficient matrices for linear systems, and the elimination process for a matrix is equivalated to the vertx eliminations for the graph. Then, the results by the theoretical investigation are summarized as following: 1. Optimal elimination for each subgraph which is subdivided appropriately from whole graph leads to the global optimum. 2. In each subgraph there are only two kind of eliminations. Furthermore, some numerical experiments show the characteristics of the subset of vertices, which subdivide a subgraph from the residual

Bandwidth Minimization Algorithm for Finite Element Mesh

Renumbering algorithms commonly in use for the band solver are generally applicable for any kind of linear equations, and, therefore, we may say that they cann't effectively utilize the characteristics of the finite element mesh. In this paper we investigate the characteristics of the finite element mesh systems, and introduce them into Taniguchi-Shiraishi Algorithm which already introduced some properties of FEM mesh systems. And through several numerical experiments it is proved that this improved algorithm is one of the fastest one

Convergence Condition of Explicit Finite Element Method for Heat Transfer Problem

The convergence condition of the explicit difference method for the heat transfer problem is aiready obtained. On the other hand, if the problem is formulated by using the weighted residual method for spatial axis, we have no tool to estimate the critical timestep width. In this paper, the estimation method is theoretically presented, and its propriety is examined through a number of numerical experiments

On the use of Delaunay triangulation as 3D finite element modeler

Delaunay triangulation, a geometric subdivision of any convex domain, is often used as a finite element modeling method, but there are still several problems, which originally come from the characteristics of Delaunay triangulation. One problem appears when we remove some nodes which are already introduced for the triangulation. In this case we aim to obtain the triangulation without nodes by partial modification of the Delaunay triangulation with the node. Another problem occurs when tetrahedra with zero volume are generated by Delaunay triangulation. In this case they must be removed for the numerical analysis in order to guarantee the numerical stability and good numerical solutions. In this paper these two problems occuring at the use of Delaunay triangulation are theoretically discussed

On Robust Incomplete Choleski-Conjugate Gradient Method And Its Modification

This paper includes a solver for a large sparse set of linear algebraic equations which are obtained by the application of the finite element method to static structural problems. Proposed method is a modification of Robust Incomplete Choleski-Conjugate Gradient Method, which belongs to Preconditioned Conjugate Gradient Method suitable for supercomputers. Through a number of numerical experiments the authors show that Robust Incomplete Choleski-Conjugate Gradient Method sometimes fails in to obtain the solutions, secondly they clarify the reason of the failures from the theoretical viewpoint, and finally they propose a modification of the robust method by the introduction of the theoretical result. Proposed method is as effective as the original, and it can overcome the demerit of Robust Method which is clarified through numerical experiments

Estimation Method of the Optimum Relaxation Factor for the Successive Overrelaxation Method

New estimation method of the optimum relaxation factor for the successive overrelaxation method (SOR) is proposed, and the efficiency of the new method is surveyed through a number of numerical experiments. This method can a priori determine the value of the factor by using only the topological properties of the problem, and it is valid for a sparse set of linear equations obtained by using the five-point difference scheme for any rectangular area with arbitrary boundary conditions. The experiments clarify that the method can estimate good approximate value of the factor