Sonlu elemanlar metodunun spreadsheed programı yardımıyla nötron difüzyon denklemine uygulanması

Abstract

ÖZET Sayısal analiz tekniklerinden biri olan sonlu elemanlar metodu, bilim ve mühendislik alanındaki problemlerin birçok çeşidine yaklaşık çözümler elde etmek için kullanılan çok güçlü bir tekniktir. Ancak basit problemlerin çözümü için Fortran, Basic, Pascal gibi geleneksel yüksek seviye programlama dillerinden birine gerek duyulması metodun öğretilmesini ve öğrenilmesini zorlaştırır. Hem metodun öğretilmesi ve öğrenilmesi hem de yüksek seviye programlama dili kullanılması durumunda programın karmaşıklığı içinde dalınarak metodtan tamamen uzaklaşılır hale gelinir. Oysa Spreadsheet programlan çok az programlama tecrübesi gerektiren bir ortam sağlar. Böylece spreadsheet programlan, hem metodun öğretilmesini ve öğrenilmesini çok kolay hale getirir hem de yüksek seviye programlama dillerine göre problemleri daha çabuk ve daha hızlı çözer. Bu tezde, sonlu elemanlar metodunun çok yaygın olarak kullanılan metodu olan Galericin metodu nötron difuzyon denklemine uygulanmıştır. Lineer üçgen eleman tipi kullanılarak elde edilen sonlu elemanlar formülasyonu, Excel 5'te kullanılan spreadsheet programına uygun notasyonda yazılarak nötron difuzyon denkleminin çözümü yapılmıştır. Karşılaştırma olarak, sonlu farklar metodunda kullanılan eşit boyutlu ızgara, dikdörtgensel bölge için ideal olan spreadsheete rahatça uygulanabilmesine rağmen, sonlu elemanlar metodunda farklı boyutlu ızgaranın yapılabilmesiyle çözümün daha hızlı değiştiği gösterilmiştir.

IMPLEMENTATION OF FINITE ELEMENT METHOD ON A SPREADSHEET TO NEUTRON DIFFUSION EQUATION SUMMARY The Finite Element is a numerical analysis technique for obtaining approximate solutions to a wide variety of engineering and science problems. Analytical solutions to the engineering and science problems are possible only if the geometry and boundary conditions of the problem are simple. Otherwise it is necessary to use an approximate solution such as Finite Element Method (FEM). At the past, the most commonly used method is the Finite Difference Method (FDM). A finite difference model gives a pointwise approximation to the governing equations. The disadvantages of the finite difference are; boundary conditions are difficult to satisfy and accuracy of the result is usually poor. The difficulty of FEM is that in order to solve even very simple problems a computational approach is needed. Students frequently find that they get bogged down in the complexities of a specific programming language and lose sight of the finite element method itself. The use of a spreadsheet, however, has made the teaching and learning of the method considerably easier. The interpretative nature of the environment, together with the straightforward method of placing formulae in cells, allows students to implement the method for problems of moderate size in a manner which is much quicker than was the case with a conventional programming language. When learning advanced numerical techniques such as the finite element method it is essential that students have the opportunity of working through each step in order to get a feel for what is going on. Very simple problems amenable to hand calculation are useful but to a limited extent only. The major deficiency is that the problems are so simple as to bear almost no relation whatsoever to more realistic problems which require a computational solution. The spreadsheet provides an excellent compromise. It is still necessary to get right down to the detail of the algorithm; the insertion of the algorithm into appropriate cells is very much easy to writing down the expression by hand. The finite element method provides a model of the solution region with many small, interconnected subregions or elements. A finite element model of a problem gives a piecewise approximation to the governing equations. With this method, a solution region can be analytically modelled or approximated by replacing it with an assemblage of discrete elements. These elements can be used to represent exceedingly complex shapes. A continuum problem is one with an infinite number of unknowns. The finite element discretization procedures reduce the problem to one of a finite number of unknowns. By means of this method, solution region is divided into elements.The unknown field XIvariable is expressed in terms of the values of the field variables at specified points called nodes. Nodes usually lie on the element boundaries. But some elements may have a few interior nodes. In this way, the problem is stated in terms of these nodal values as new unknowns. Once these unknowns are found, the interpolation functions define the field variable thorough at the assemblage of elements. The nature of the solution and the degree of approximation depend on the size and number of the elements and interpolation functions. Interpolation functions must satisfy certain compatibility conditions. Often functions are chosen so that the field variable and its derivatives are continuous across adjoining element boundaries. In FEM, we can formulate the solution for individual elements. Then we put them together to obtain solution for entire domain. There are basically four different approaches to formulate the properties of individual elements which are direct approach, variational approach, weighted residuals approach and energy balance approach. Regardless of the approach used to find the element properties, the solution of a continuum problem by the FEM always follows are orderly step-by-step process: 1. Domain discretization by triangular or quadrilateral elements. 2. Interpolation of the unknown over each element. 3. Evaluation of element matrices. 4. Assembly of element matrices. 5. Introduction of the boundary conditions. 6. Solution of the algebreaic equations. 7. Evaluation of any subsidiary element quantities. Galerkin method is widely used solving method of FEM and usually presented as one of the weighted residual methods. In the Galerkin method, the weighting functions coj are chosen from the basis functions [NJ used for constructing approximate solution Oe. This approach is the basis of the finite element method for problems involving first derivative terms. This method yields the same result as the variational method when applied to differential equations that are self-adjoint. Galerkin' s method is used to develop the finite element equations for the field problems. In this thesis, Galerkin's method is applied to the two-dimensional and one group `neutron diffusion equation`. The neutron diffusion equation can be written as: -V.D(x,y)V.O(x,y) + G(x,y)0(x,y) = S(x,y) where D(x,y) diffusion constant, o(x,y) absorption cross section and S(x,y) the neutron source. The flux distribution <D(x,y) is to be calculated in two dimensional rectangular geometry of the nuclear system. XllThe classical restrictions on 0(x,y) are: analyticity of 0(x,y) in each homogeneous region and continuity of the flux (3>(x,y)), and the normal component of the current (-Dd$>/dri) across material interfaces. Either the zero flux boundary condition (Dirichlet) ®(x,y) = 0 or the symmetry boundary condition (Neumann) 8Q>/8n = 0 are assumed at system boundaries. Suppose that A is subdivided into m elements, Ae <e,NT)AC=0 e=-DV20 + o<D~S J[N]T(-DV2<D + aO - S)dA = 0 where [N] is shape function. Each of O is interpolated from its nodal value {O6}, there being n nodes in total. This gives us a piecewise polynomial approximation to O of the form, ®e = [N]{#e The finite element method leads to a set of linear algebraic equations for the nodal values. And a system of algebraic equation having the following form: KO + MO = F where K and M are called the stiffness matrix and mass matrix. The global matrix K and M are obtained from element matrices k by accumulating into the i, j position those terms from the elements which contain both nodes i and j. The global right hand side vector F is developed in a similar manner. The terms in the element matrices have the following forms: K = J( A CK 8k 8k 8y 8y )dxdy m]; = JuNjNidxdy A' q = JcpNjdxdy Ae The finite element method really comes into its own when dealing with problems which have complicated geometry. As far as a spreadsheet implementation is concerned it is not possible to set-up a template which will deal with an arbitrary region. However, regions that are essentially rectangular are ideal for the spreadsheet. Although finite differences are alsowell-suited to a rectangular mesh, with consequent appropriate spreadsheet implementations, the finite element method is more versatile Xlllin the handling of graded meshes which are required in regions where the solution may be expected to change rapidly. We use a linear triangle for the solution of the neutron diffusion equation with the function S being constant. We define, for each element, the parameters, ai = x2y3 - x3y2 bi = y2 - y3 ci = x3 - x2 with a2, a3, b2, b3, c2, c3 obtained cyclic permutation of the subscripts. Then the element matrices are given by A ki^(bJb.+V«) m«=H' 12 A where A is the area of an triangular element. The parent triangular element is shown in below figure (x3, y3) (xi, yi) (x2, yi) -* x Our element matrices are as follows, element stiffness matrix; ke 2hk k2+h2 -k2 -h2 -k2 k2 0 -h2 0 h2 1 1 - + r -- -r r r -I I o r r -r 0 r where r = h/k is the aspect ratio of the element. xivElement mass matrix, m - u. The element source vector, A fe=(p- 3 Consider now the equation from set KO + M<D = F corresponding to O. The mesh in the neighbourhood of node /' is shown in following figure. Using the notation of above figure, the finite element equation corresponding to node i is xv_(0 + 0) + ^(AA+AA)^)p+^(-rA-rB) + ^(AA+AB)_pN 4 +Y^(aa+ad) $ w -f ^tS+ + - + rc + +-^(2AA +2AA +2AB +2AC +2AC +2AD) 12 a>; + if- L 1 _2V I`b W + ^(AB+AC) G>E + (-rc-rD) + ^(Ac+AD)j^ 4[(o+o)+^(ac + ac)]d1 f h?, h* h^ hfl q> 27L + 71 + 271 + 71hT(2AA+AB+2Ac+AD) We note that for a uniform rectangular mesh, with rA = rB = rc = rD = 1 and AA = AB = Ac = AD = h2/2, this reduces to <d, = (1-^X*k+*w+*b+*,)~(*p+*q) Since the equations are strictly diagonally dominant an iterative solution will converge and we write the equations in a form suitable for iteration as follows: <fc = -{^(Aa + AA)0P -{i(-rA -rB) +^(aa + AB)}d>: (-rc-rD) + ^(Ac+AD)}«i,-{^(Ac+Ac)}*. V rB rcj + ^(AB+Ac)k +f(2AA+AB+2Ac+AD) ^K7+rA + lrB + rB + - + rc + +j^(2AA +2AA +2Ab +2Ac +2Ac +2Ad) XVIAbove equation can be easily implemented to a spreadsheet. As a result, the spreadsheet offers a more straightforward approach to the development of numerical methods and is more suitable for learning a programming language in order to study numerical methods. We have seen that the finite element method is easily implemented in a spreadsheet environment. My experience is that students are able to get finite element problems up and running much more quickly than they can in a traditional environment using a conventional programming language. The consequence is that they are able to develop solutions to problems in a time much shorter than is usually the case. xvn