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Elektrostatik alanların sonlu farklar yöntemiyle incelenmesi

Electrostatic field analysis by finite difference methods

  1. Tez No: 22069
  2. Yazar: SUPHİ YARICI
  3. Danışmanlar: PROF. DR. MUZAFFER ÖZKAYA
  4. Tez Türü: Yüksek Lisans
  5. Konular: Elektrik ve Elektronik Mühendisliği, Electrical and Electronics Engineering
  6. Anahtar Kelimeler: Belirtilmemiş.
  7. Yıl: 1992
  8. Dil: Türkçe
  9. Üniversite: İstanbul Teknik Üniversitesi
  10. Enstitü: Fen Bilimleri Enstitüsü
  11. Ana Bilim Dalı: Belirtilmemiş.
  12. Bilim Dalı: Belirtilmemiş.
  13. Sayfa Sayısı: 86

Özet

ÖZET Bu tez çalışmasında, teknik uygulamalarda yaygın bir şekilde kullanılan sayısal yöntemlerden biri olan Sonlu Farklar Yöntemi ve yöntemin statik elektrik alan incelemelerinde kullanım olanakları ayrıntılı bir şekilde incelenmiştir. Sonlu Farklar Yönteminin ilkesi, incelenecek olan bölgenin kare, dikdörtgen,... şeklinde gözlerden oluşan bir ızgara şeklinde küçük alt bölgelere ayrılması ve ızgara üzerindeki düğümlere ilişkin potansiyel bağıntılarının yazılmasına dayanır. Çalışmada, öncelikle sonlu fark işleçleri tanıtılarak, çeşitli koordinat sistemlerinde Laplace denkleminin sonlu fark denklemleri şeklinde yazılması ele alınmıştır. Daha sonra, oluşturulan sonlu fark denklemlerinin çözümü için çeşitli çözüm yöntemleri ayrıntılı bir şekilde incelenmiştir. Son bölümde ise Sonlu Farklar Yöntemi (SFY) kullanılarak yüksek gerilim tekniğinde karşılaşılan temel elektrot sistemlerinden bazıları için alan incelemeleri yapılmıştır. Burada, incelenecek olan elektrot sistemine uygun algoritma çıkartılmış ve potansiyel değerleri, ha zırlanan bilgisayar programları yardımıyla hesaplanmış tır. Bulunan düğüm potansiyellerine lineer interpolasyon işlemi uygulanarak eşpotansiyel noktalar, dolayısıyla eş- potansiyel çizgiler elde edilmiştir. Bu şekilde düzlem- düzlem ve sivri uç-düzlem elektrot sistemleri ile içinde gaz boşluğu bulunan bir katı yalıtkan için elektrik alan incelemeleri yapılmış ve yöntemin alan incelemelerindeki kullanım zorlukları ve olanakları görülmüştür.

Özet (Çeviri)

SUMMARY ELECTROSTATIC FIELD ANALYSIS BY FINITE DIFFERENCE METHOD In recent years, several numerical methods for solving partial differential equations and thus also Laplace and Poisson equations have become available. There are inherent difficulties in solving partial dif ferential equations and thus in Laplace or Poisson equa tions for general two or three dimensional fields with sophisticated boundary conditions, or for insulating materials with different permittivities and/or conductiv ities. Each of the different numerical methods, however, has inherent advantages or disadvantages, depending upon the actual problem to be solved and thus the methods are to some extent complementary. Some of these numerical methods are Monte Carlo Method (MGM), Moment Method (MM), Charge Simulation Method (CSM), Boundary Element Method (BEM), Finite Element Method (FEM) and Finite Difference Method (FDM). FEM and FDM are based on the solution of Laplace equation in a differential form. CSM and MM are integral methods. The numerical electric field calculation by computer is still a subject of research and development. Before digital computers have been available, the elec tric field strength had been obtained either by approxi mating a given electrode by an analytically computable model or by field mapping, which estimates it from the equipotential distributions. The Finite Difference Method (FDM) has been used for field computations since before the computer age. However, without a computer, it is very painstaking work to treat regular calculations such as solving a simulta neous equation by iterative procedure even in a small model. Hence it is quite reasonable that the FDM was initially examined in a computer associated electric field simulation. Many important scientific and engineering prob lems fall into the field of partial differential equa tions. The mathematical formulation of most problems in science involving rates of change with respect to two or more independent variables, usually representing time, length or angle, leads either to a partial differential 9 viequation or to a set of such equations. Special cases of the two dimensional second order equation dx2 dxdy dy2 dx oy (1) where a, b, c, d, e, f and g may be functions of the independent variables x and y and of the dependent vari able V, occur more frequently than any other because they are often the mathematical form of one of the conserva tion principles of physics. Equation (1) is said to be elliptic when b2-4ac0. Finite Difference Method is found to be discrete technique wherein the domain of interest is represented by a set of points or nodes and information between these points is commonly obtained using Taylor series expan sions. To understand finite difference method it is first necessary to consider the nomenclature and funda mental concepts encountered in this form of approximation theory. The basic concepts are quite simple. The domain of solution of the given Laplace equation is first subdi vided by a net with a finite number of mesh points. An example of a square grid with its sides parallel to the x or y axis is shown in Figure 1. The derivative at each point is then replaced by a finite difference approxima tion. Also, boundary conditions must have known to solve the partial differential equations exactly. There are three types of boundary conditions. These are Dirichlet (first type) boundary condition, Neumann (second type) boundary condition and Robbins or hybrid (third type) boundary condition. yj+2 yj+i yj-i yj-2 X Ax- Ay i-2 Ai-1 Xi Xi+1 Xi+2 Figure 1. An example of a regular grid for finite differ ence method, indicating the node numbers. VllLaplace equation for two dimensional field in Cartesian coordinates is a^ + üZ-o. (2) dx2 dy2 Replacing the derivatives by difference quotients which approximate the derivatives at the point {x.ityt) by tak ing Ax=Ay, Equation (2) yields (3) v2v^ = -^[v1 + v-2 + v3 + v4-4V0] =0 Equation (3) is the finite difference form of Laplace equation for a two dimensional field in Cartesian coordi nates. In this thesis, using of electrical engineering of Finite Difference Method which is commonly used in scientific and engineering problems was investigated. Electric field distributions with this method were exam ined for some of the basic electrode systems which are important in high voltage technology. In Chapter 2, finite difference operators, prin ciple of Finite Difference Method and the representation of Laplace equation by finite difference equations for regular and irregular regions with two and three dimen sional in some coordinates were investigated. Laplace equations in the different coordinate systems were given in Appendix A. The basis of Finite Difference Method is the replacement of a continuous domain representing the entire space surrounding the high voltage electrodes with a rectangular or polar grid of discrete“ nodes ”at which the value of unknown potential is to be computed. Thus, the derivatives are replaced describing Laplace equation with“divided difference”approximations ob tained as functions of the nodal values. In Chapter 3, the solution methods belonging to the obtained sets of finite difference equations were investigated. Usually, a convenient classification is direct and indirect (or iterative) methods. The term“ direct methods ”usually refers to techniques that in volve a fixed number of arithmetic operations to reach an answer. On the other hand,“indirect methods”involve the repetition of certain processes for an unknown number of times until a required accuracy in the answer is achieved. Some of these solutions methods are Gaussian Elimination, LU Decomposition, Gauss-Seidel Method and viiiSuccessive Over Relaxation Method (SOR). In Chapter 4, studies made with Finite Difference Method were explained. During the studies, some comput ers programs were improved and these programs were given in Appendix B. Successive Over Relaxation Method (SOR) was used as a solution method. In developed programs, acceleration parameter or factor

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