İki boyutlu iki gruplu nötron difüzyon denkleminin lineer sınır elemanları ile çözümü

Abstract

ÖZET Bu çalışmada amaç, matematiksel fizik ve mühendislik alanlarında kullanımı giderek yaygınlaşan sınır elemanları yönteminin nötron difüzyonu sahasındaki etkinliğinin irdelenmesidîr. Nötron difüzyonu problemlerinin çözümlerinde sınır elemanları uygulamaları 1985'li yıllarda başladı. İlk olarak tek gruplu tek bölgeli difüzyon problemlerinin çözümünde sabit sınır elemanları kullanıldı. Daha sonraki çalışmalarda gruplu ve çok gruplu difüzyon teorisi uygulamaları sabit sınır elemanları kullanılarak gerçekleştirildi. Bu çalışmada, iki gruplu nötron difüzyon denkleminin çözümlerinde lineer sınır elemanları yöntemi kullanılmıştır. Çalışmanın giriş bölümünde sınır elemanları yönteminin tarihsel gelişimi gözden geçirilip alternatif sayısal tekniklerle kısa bir karşılaştırması yapılmıştır, ikinci bölümde ise iki gruplu iki boyutlu nötron difüzyon denkleminin sınır elemanları yöntemiyle ayrıklaştırılması yapılmıştır. Yöntem hem sabit kaynak hem de kritiklik problemlerine uygulanmıştır. 3. bölümde, 2. bölümdeki formülasyon BENDL2G adlı FORTRAN programına uyarlanmıştır. Yine bu bölümde analitik sonuçlan bulunabilen değişik problemler için BENDL2G programının doğrulanması yoluna gidilmiştir. Son bölümde ise sonuçların irdelenmesi ve önerilerden oluşmaktadır. ıx

SUMMARY The problems of mathematical physics, such as electrostatics, quantum mechanics, elasticity theory, hydrodynamics etc... usually lead to partial differential equations, more rarely to ordinary differential equations. These equations have to be integrated under the initial or boundary conditions of a specific problem. The necessity of solving these problems as accurately as possible led in the past two decades, to the development of very powerful numerical solution techniques. Engineers and physical scientists have become very conversant with numerical techniques of analysis. When an engineer or scientist constructs a mathematical model of almost any kind of system, he usually starts by establishing the behaviour of an infinitesimally differential element of it based on assumed relationships between the major variables involved. This leads to a description of the system in the form of a set of differential equations. Once the basic model has been constructed and the properties of the particular differential equation understood, subsequent efforts are then directed towards obtaining a solution of the equations within a particular region which is often of a very complicated shape and composed of zones of different materials each with complex properties. Various conditions will have been specified on the boundaries of the region and these may be either constant or variable with time, etc. In such cases we must use numerical solution techniques. The numerical methods most widely used at present tackle the differential equations directly in the form in which they were derived, without any further mathematical manipulation, in one of two ways: either by approximating the differential operators in the equations by simpler localized algebraic ones validat a series of nodes within the region or by representing the region itself by noninfinitesimal elements of material which are assembled to provide an approximation to the real system. The most known methods are Finite Difference Method (FDM), Finite Element Methods (FEM) and Boundary Element Method (BEM). The first method, FDM is the progenitor of the former approach and was very popular until about fifteen years ago, when it began to be superseded by the other methods. FDM is based on approximation of differential operators with local algebraic operators albeit with some predetermined error. The most important point is that this method can be applied to every differential equation as long as the geometry is regular. The second method, FEM has attracted the attention of the analysts largely due to its property of dividing the continuum into a series of elements, which can be associated with physical parts. The method can sometimes be based on variational principles, more generally on weighted residual expressions or reformulation of the problem as a minimization of a functional and subdivision of the problem domain. The subdivision of the problem into finite physical parts may result in a number of equations. The elements, represent the whole problem. This method may be called an element library. The method can be applied to complex or irregular geometries and this shows the robustness method. One of the greatest advantages of the method is the easiness of the application of the boundary conditions and it is widely used. The other weakest aspect is that discrezitation of distant boundaries in three dimesional problems and this cause excessive computer time in problem solving. The great interest that the Finite Element Method attracted at the beginning of the I960' s had two important consequences: First, it orginated an xiimpressive amount of work in computational techniques and efficient engineering software. Second, substantial research into basic physical pirinciples, such as variational techniques and weighted residuals, was stimulated. The first of the above points comes as a natural consequence of the emergence of new and powerful computers. The third method is the BEM. The mathematical background of BEM has been known for nearly hundred years. Indeed, some of the boundary integral formulations for elastic, elastodynamic, wave and potential flow equations have existed in the literature for at least fifty years. With the emergence of digital computers the method had begun to gain popularity as `the boundary integral equation method`, `panel method`, `integral equation method`, etc. during the 1960's. BEM is based on the change of the boundary value problems into classical integral equation formulation. This formulation is regarded as theoretical but on the other hand the progress in the computer technology led to usage of BEM in problem solving. In another words, BEM is based on the change of partial differential equation into boundary integral equations with a transformation. The boundary element method is used on the boundary integral equation and the principle of weighted residuals, where the fundamental solution is chosen as the weighting function. The value of the function and its normal derivative along the boundary are assumed to be unknowns. By using discretization, similar to that used in the finite element method, the boundary integral equation is transformed into a set of algebraic equations for the nodes at the boundary. Then the value of the function and its normal derivative are xuobtained simultaneously by solving the matrix equation. The method contains the following steps. 1) The boundary is discretized into a number of elements over which the unknown function and its normal derivative are assumed by the interpolation functions. 2) According to the error minimization principle of weighted residuals, the fundamental solution as the weighting function is chosen to form the matrix equation. 3) After the integrals over each element are evaluated analytically or numerically, the coefficients of the matrix equation are evaluated. 4) Setting the proper boundary conditions to the given nodes, a set of linear algebraic equations are then obtained. The solutions of these equations result in the boundary value of the field variable and its normal derivatives. Hence the field strength of interest on the boundary is computed directly from the matrix equation. 5) The value of the function at any interior point can be calculated once all the function values and their normal derivatives on the boundary are known. The Historical Development of Boundary Element Methods A lots of methods can be traced to precomputer times and involve different ways of solving the governing equations of a problem, i.e..., Galerkin collocation, least squares, line techniques, matrix progression or transfer, the combination of different techniques, etc. In spite of the crucial xiiistudies on the differential equations in 19th century, with the begining of the computer development, the engineers have figured out the area of numerical solutions of `Fredholm Type Integral Equations`. The researches in `Fredholm Type Integral` was issued in 1905. The formal understanding of integral equations has been established more recently by Mikhlin. Integral equation techniques were, until recently, considered to be a different type of analytical method, somewhat unrelated to approximate methods. They became known in Europe through the work of a series of Russian authors, such as Muskhelishvili, Mikhlin, Kupradze and Smirnov but were not very popular with engineers. A predecessor of some of this work was Kellogg who applied integral equations for the solution of Laplace-type problems. Integral equation techniques were mainly used in fluid mechanics and general potential problems. Work on this method continued throughout the 1960's and 1970's in the pioneering work of Jaswon, Symm, Massonet, Hess and many others. It is difficult to point out precisely who was the first to propose this method. It is found in a different form in Kupradze' s book. It seems fair, however, from the engineering point of view, to accept that the method orginated in the work of Cruse and Rizzo in elastostacis. The direct method is the one which will be used mainly in this work as it is the most appealing to engineers and physical scientists. Since the early I960' s a small research group at Southampton University worked on the applications of integral equations to solve stress analysis problems. Unfortunately, the presentation of the problem, the difficulty of defining the appropriate Green's functions, and the parallel emergence of the finite element method all contributed to minimize importance of this work. At the begining of the 1970's recent developments in finite elements started to find their way into the formulation of boundary integral equations and contributed to the development of general curved elements. Still, the xivquestion of how effectively one relates the boundary integral equations to other approximate techniques was unresolved. This was done by Brebbia who in the 1970's worked on the relationship of different approximate methods. BEM was used for the first time in two papers by Brebbia and Dominguez in 1977. These works culminated with the first book in 1978 for which the title `Boundary Elements` was used. Same year the first conference was held. Three important international conferences were held at Southampton University in 1978, 1980 and 1982 and another one in California in 1981. During this time many papers, theses, books, textbooks and advanced level monographs have appeared which give a very comprehensive and thorough account of the existing literature on the method. In recent years BEM developed rapidly. It has been expanded so as to include time-dependent and non-linear problems. At first this method was almost exclusively in the domain of the mathematicians and physicists with very little work being done to apply them to realistic engineering problems. The development of BEM has now reached a stage when attempts can be made to analyse realistic complex engineering problems. Although all BEM have a common origin they divide naturally into three different but closely related categories. 1) The Direct Formulation of BEM In this formulation the unknown functions appearing in the integral equations are the actual physical variables of the problem. This method was used by Cruse, Lachat, Rizzo, Shaw, Watson and others. xv2) Semi-Direct Formulation of BEM Alternatively, the integral equations can be formulated in terms of unknown functions. This method was used by Henry, Jaswon, Ponter, Rim and Symm. 3) Indirect Formulation of BEM The integral equations are expressed entirely in terms of a unit singular solution of the original differential equations distributed at a specific density over the boundaries of the region of interest. This method was used by Banerjee, Butterfield, Hess, Jaswon, Massonnet, Oliviera, Symm, Tomlin, Watson and others. With the development of the computer field the usage of BEM has been increased. BEM can be adapted in all kinds of problem, such as elliptic, parabolic, hyperbolic, partial differential equations and eigenvalue problems. BEM can be used in some engineering fields such as, an advanced stress analysis system, elastic and inelastic analysis of solids, elastic and inelastic dynamic analysis of solids, gas turbine engine structures, poroelastic and thermoelastic analysis, automotive industry, fracture mechanics, elastodynamic wave propagation problems, elastostatic, plasticity, acoustics, heat conduction, fluid mechanic, shape optimization, potential flow. Comparison of BEM and FEM Many problems in engineering can be solved more easily with BEM than FEM. The number of equation in BEM is less than those in FEM. And that's why, time that is spent in matrix solution in BEM is less than in FEM, xvihowever as a result of having a symmetrical feature in matrix formulation FEM is more advantageous than BEM. After the determination of the boundaries, the values of variables may be found out at the inside points. It's much more proper to use FEM with small homogenous areas and BEM with large ones. The percent of error is seen to decrease faster in FEM. The most important reasons of using BEM are; high precision in results, reduction of dimension; such as transformation of three-dimensional problem into two dimensional boundary integral equation, low cost, easy usage in many problems. The most important disadvantages of BEM are mentioned as follows the method cannot be used directly for non-linear problems, the fundamental solution of the governing equation is difficult in some problems, a great number of integrations are required and the singularities of the integral must be considered. Hence the calculation of the coefficient matrix requires more time than that of FEM. The aim of this work is to apply BEM to the two dimensional and two group neutron diffusion equation by using linear boundary elements. This work contains four parts. In the first part of this work, the basic concepts and history of BEM have been presented briefly with comparison against other numerical methods. In the second part, the application of BEM to two dimensional and two group neutron diffusion equation has been presented. In the third part of the program BENDL2G has been developed according to the formulations which have been obtained in the second part. The numerical results that have been obtained by using this program have been compared with the analytical results. In xviithe final part, BENDL2G program has been developed which solves two dimensional and two group neutron diffusion equation. This program also can solve one group neutron diffusion equation. The developed program has been written in FORTRAN and all tests are made under the LINUX operating system on personal computers with PENTIUM processor. xvm