Introduction

Many engineering's tasks changed in connection with technical progress: they became more difficult, and their decision requires introduction of new concepts. Approach changed to the practical engineering's tasks. If an engineer could before, coming from the examined physical phenomenon or technical problem, to set the problem and give it a solution mathematician-calculator, that now business is differently. In many engineering's tasks the construction of calculation model so closely interlaces with the process of calculations, that to divide these processes at times is not possible.

In this connection lately a mass appeal was purchased by the method of eventual elements. Presently the method of eventual elements is used at the decision of the most various tasks of mathematical physics, although the first works on the method of eventual elements were executed specialists on structural mechanics. This circumstance was reflected on terminology of method and his primary interpretation.

This interpretation consists of the following: a continuous environment is replaced some equivalent joint system, and technique of calculation statically indefinite joint systems well known.

Popularity of method of eventual elements is explained simplicity of his physical interpretation and mathematical form. And the use of computer allows to get the close decisions of many technical tasks. The method of eventual elements already is now used as an ordinary engineering method in many organizations of designers.

Appearance of the parallel computer systems gave development of method new direction. Use of parallel computers for a calculation can promote design exactness the method of eventual elements and accelerate the design of difficult physical processes and phenomena. At the same time parallel realization of method of eventual elements is a thorny scientific problem, requiring comprehensive research.

Essence of finite element method (FEM)

Engineering's constructions can be examined as some aggregate of structural elements, united in the eventual number of key points. If correlations are known between forces and moving for every element,, using the known receptions of structural mechanics, it is possible to describe properties and explore the conduct of construction on the whole.

In a continuous environment the number of points is endless, and exactly it presents basic difficulty of receipt of numeral results in the theory of resiliency. The concept of eventual element allows to tide over this difficulty. A continuous body is broken up on separate elements, interactive between itself only in key points. If such approaching it admits, a continuous task can be taken to the ordinary task of structural mechanics and can be decided numeral.

Possibilities of concurrency FEM

We will consider a simple framework construction, resulted on a picture 1 as an example.

Picture 1 is the Simple framework construction

Every eventual element of I can be described the system of linear equalizations of A[I].x* == B[I]. Where A[I] is a matrix of inflexibility of I of eventual element. Calculation component of matrices of inflexibility for each of elements independently and it can be executed parallel.

Got set of matrices of inflexibility of ansambling in the single matrix of inflexibility of A. The got system of linear equalizations has a band nxn matrix of coefficients and can be decided on the parallel computer system.

Purpose of work

The purpose of this work is realization of existent parallel algorithms of decision of the band systems of linear equalizations and analysis of their efficiency as it applies to the method of eventual elements.

The great number of direct and iterational methods, successfully deciding the similar systems is developed for today, however much the questions of estimation of complication of one or another methods, their efficiency and acceleration, mapping to the computer systems of different structure, and also their applicability, to FEM remain opened.

Intermediate job performances

This work is unfinished, the planned date of completion of work is January, 2007. However now got results of research of parallel algorithms of multiplying of matrices are.

An operation of multiplying of matrices is one of basic operations, which is executed enormous amount of one times in other, more difficult algorithms. For this reason efficiency of the decided tasks depends on the choice of effective parallel algorithm of multiplying of matrices.

The analysis of algorithms of coarse-grained of Kenon's is conducted, Sectional multiplying of matrices, fine-grained systole algorithm and the estimations of acceleration, efficiency, calculable and of communication complication of the mentioned algorithms are got. The transferred algorithms were realized with the use of library of MPI. The experimentally got descriptions of fast-acting were confirmed by theoretical estimations which can be used in further work above FEM.

Conclusions

A receipt of theoretical and experimental estimations of application of parallel algorithms of decision of the band systems of linear algebraic equalizations (SLAE) on the method of eventual elements is of certain interest. The separate area of researches can be mixing of direct methods of decision of SLAE with iterative, which, possibly, will allow to multiply the fast-acting of method with saving of acceptable exactness of calculations.

This job performances can be used for conducting of engineering experiments, decision of differential equalizations with regional terms, research of the physical phenomena in 2d space, testing and ground of choice of eventual elements of difficult form.

The explored algorithms, possessing the studied parameters of efficiency, can be used for the estimation of efficiency of work of geterogennykh clusters.

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