İki Boyutlu İki Gruplu Nötron Difüzyon Denkleminin Çözümünde Sınır Elemanları Yönteminin Kullanılması

Abstract

Sınır Elemanları Yöntemi'ni (Boundary Element Method-BEM), iki boyutlu iki gruplu nötron diftizyon denklemine uygulamak amacıyla gerçekleştirilen bu çalışmanın ilk bölümünde, BEM'in temel kavramları gözden geçirilmiş ve diğer sayısal yöntemlerle kısa bir karşılaşürması yapılmıştır. Ardından gelen ikinci bölüm, BEM'in, iki boyutlu iki gruplu nötron diftizyon denklemine uygulanmasıdır. Nötron diftjzyon denkleminin sınır integral denklemine dönüştürülüp, ardından BEM'in uygulanması sırasında ikinci grup denkleminde değişken olarak farklı bir yaklaşım kullanılmıştır. İkinci bölümdeki bu inceleme, serbest kaynak ve her iki grup için de fısyon kaynağı olması durumları için ayrı ayrı yapılmıştır. Gerçekleştirilen formülasyonlara uygun olarak, iki boyutlu tek gruplu nötron diftizyon denklemini çözen BENffS yanlımı temel alınarak BEMG2 yazılımı geliştirilmiştir. Yazılımın doğrulanması, ele alınan problemlerin yazılımın çalıştırılmasından elde edilen sonuçlarının, analitik çözümleriyle ve uygun başka yazılımların sonuçlarıyla karşılaştırılmasıyla yapılmıştır. BEMG2 yanlımı sadece iki guplu problemlerin çözümünde değil, tek gruplu problemlerin çözümünde de kullanılabilmektedir. Bu, üçüncü bölümde test edilen uygulamalarda da görülebilmektedir. Son bölümde ise genel bir değerlendirme yapılmaktadır. BEMG2, Fortran77'de yazılmış ve tüm testler Inte1486 işlemcili kişisel bilgisayarlarda MSDOS6.2 altında yapılmıştır. Yazılım listesi ve yanlımda kullanılan değişkenlerin tablosu EK.C'de verilmiştir.

in most of the problems which can be simulated by differential equations, analytic analysis becomes impossible since the conceraed geometries are not regular. Therefore, the numerical solution techniques are to be used. Numerical methods have been developed in two different ways. The first is to approximate the differential operators with local algebraic operators albeit with some predetermined error. The other depends on the reformulation of the problem as a minimization of a fimctional. Two of most known methods are Finite Difference Method-FDM and Finite Element Method-FEM. FDM is based on the first way and can be applied to every differential equation, but it is not efficient when the geometry is irregular. FEM which is based on the subdivision of the problem domain can be easily applied to irregular geometries. Continuity among subdivided parts, i. e. the elenıents, is provided by means of selected points called nodes. Because any changes in geometry require reidentification of the elements, this method is slow in some applications. Although Fredholm announced some solutions of equations which were mentioned as 'Fredholm Integral Equations' in 1903, boundary integral equation techniques were thought for a long time as not having any relations with approximate solution methods and remained as an analytical method in the domain of mathematicians and physicists. Although there were several numerical applications of the boundary integral equations before, the BEM like solutions began to be developed as a result of the developments in computer techniques at the beginning of 1960. Unlike FEM which uses suitable approximate fünctions to boundary conditions, BEM uses fimdamental solutions which are unsuitable to boundary conditions but suitable to goveraed equations. Based on boundary integral equations, unlike FEM that requires volume discretization, BEM requires surface discretization which makes it an effective method. These discretizations, made only on the surface of the concemed domain, reduce the number of unknowns and consequently the dimension of the coefficient matrbt is diminished. However, the coefficient matrbc which assumes a banded form in FEM and FDM, transforms to a full matrix in BEM. This fullness in matrix is the majör disadvantage of BEM. FDM is only used in discretization of regular meshes, but FEM and BEM are used in discretization of irregular domains by using isoparametric elements. Boundary discretization in BEM makes the method easily applicable to irregular geometries; also makes it suitable to geometrical changes and usable in infinite geometries and reduces problem dimensions. x[ The BEM, although formulated for homogeneous domains in this work, could also be applied to problems with a heteregenous structure.The internal variables could be calculated in BEM, after the evaluation of boundary values, by using another integral equation. in spite of BEM's advantages, since it does not use approximate numerical techniques, it becomes very öpen to error. Due to the difficulty of getting required integrals in closed form, the error comes into existence on ör near the boundary. If accurate numerical integration methods is used, the error can be reduced as desired. Most of BEM's advantages come from its complex mathematical background. This background makes the writting of BEM codes difficult. But BEM codes can produce very accurate results and be easily modified. BEM which is applicable to elliptic, parabolic, hyperbolic partial differential equations and eigenvalue problems is used in many engineering fields; i. e heat conduction, fluid mechanics, thermoelasticity, shape optimization ete. [1],[2],[3],[4] There are two majör difficulties in the application of BEM to the neutron diffusion equation. The fîrst öne, is the generation of an internal mesh for the calculation of the source term integrals. The second öne, arises in eigenvalue problems which have to be solved for finding criticality. in iteration with the classical fission source, it is necessary to calculate the internal fluxes every time. There are several researches concerned with preventing this drawback. xi [5],[6],[7],[8] The aim of this work is to apply BEM, which has been used in engineering since the beginning of 1980, to the two-dimensional and two-group neutron diffusion equation. In the first chapter, the basic concepts of BEM have been presented briefly and compared with other numerical methods. The subject of the second chapter is the application of BEM to two- dimensional and two-group neutron diffusion equation. In this chapter the neutron diffusion equation is transformed to integral equations for both energy groups and a novel approach is developed for the solution of the second group equation. The developed formulation is capable of handling both fixed source and criticality problems for one and two energy groups. BEMG2 is developed from its predecessor program BEMFS [7]. The results obtained with BEMG2 have been compared with known analytical solutions and the results of other computer programms. Thus, BEMG2 is validated. BEMG2 is utilized in the solution of both one and two group problems. The relevant applications are presented in the third chapter. In the final chapter xn there is a general evaluation. BEMG2 has been written in Fortran77 and all tests are made under the MSDOS6.2 on personal computer with Intel486 processor. The program listing and the definitions of the variables used are given in Appendix C.