Statik elektrik alanlarının sınır elemanları yöntemiyle hesabı
Başlık çevirisi mevcut değil.
- Tez No: 75561
- Danışmanlar: YRD. DOÇ. DR. ÖZCAN KALENDERLİ
- Tez Türü: Yüksek Lisans
- Konular: Elektrik ve Elektronik Mühendisliği, Electrical and Electronics Engineering
- Anahtar Kelimeler: Belirtilmemiş.
- Yıl: 1998
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Elektrik Mühendisliği Ana Bilim Dalı
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: 102
Özet
ÖZET Bu çalışmada, son yıllarda çeşitli mühendislik problemlerinin çözümünde gittikçe artan bir hızla yaygınlaşan sayısal yöntemlerin, ortaya çıkış bakımından olmasa da kullanıcılar tarafından ilgi görmeye başlaması açısından en genç yöntemi olan, Sınır Elemanları Yöntemi (SINEY) incelenmiştir. Yöntemin temelleri ve denklemleri geniş bir şekilde ele alınarak, iki boyutlu statik elektrik alan problemlerinin çözümleri araştırılmıştır. 1. Bölümde, elektromagnetik alan kavramına ve sayısal yöntemlere giriş niteliğinde değinilmektedir. 2. Bölümde, temel olarak yöntemin matematiği incelenmiştir. İlk olarak, iki boyutlu potansiyel problemlerinin sınır integral denklemlerine dönüştürülmesi gösterilmektedir. Ağırlıklı integral yöntemi baz alınarak belirli bir işleç için çözüm araştırılmaktadır. İki boyutlu Laplace işleci kullanılarak, iç nokta integral ve sınır integral yaklaşımları tanıtılmaktadır. Bundan sonra, çeşitli sınır elemanlarına göre, sınırın elemanlara bölünmesi gösterilmektedir. Özellikle sabit ve doğrusal elemanlı yaklaşımlar. için sınır ve iç nokta integral denklemlerinin ayrıklaştırılması gerçekleştirilmektedir. Her iki yaklaşım için de sınır matrislerinin toplanması ayrı ayrı incelenmektedir. Son olarak, sabit elemanlı yaklaşım için {h} ve {g} sınır vektör elemanlarının hesabı verilmektedir. 3. Bölümde, sabit elemanlı yaklaşım ile SINEY 'in statik elektrik alan problemlerine uygulanması bir örnek üzerinde ele alınmaktadır. 4. bölümde, yöntemin genel bir değerlendirilmesi yapılmaktadır. Çalışmanın genelinde, açık bir matematiksel yapının kurulmasına ve programlamaya uygun bir notasyon oluşturulmasına özen gösterilmiştir.
Özet (Çeviri)
SUMMARY NUMERICAL COMPUTATION OF ELECTROSTATIC FIELDS BY THE BOUNDARY ELEMENT METHOD Electrostatic field calculations are one of the most important topic in electrical engineering to achieve economy, reliability and well balanced design of high voltage apparatus. The designer of high voltage equipment is usually confronted with multi- dielectric and sophisticated field problems. Many of the insulation materials used in isolating conductors from ground potential are very sensitively react to changes in the magnitude, direction and distribution of the field stress on a surface. A typical example for such material is pressurized sulfurhexafluorid (SF6). This gas is widely used today because of its high insulation strength [11,12]. Computational methods have made significant contributions in all fields of engineering. Before using numerical methods, the electric field strength had been obtained by either analytical or analog methods. Analytical solutions are available only for very simple electrode configurations. Conformal mapping can also be applied two-dimensional electrodes in some special geometry. Analog methods have been used extensively, employing electrolytic tanks, conductive paper or resistance networks. But the both methods are inaccurate, inconvenient and expensive. They are also limited in their applications [5,10]. Numerical methods have become more and more attractive with the increasing availability of modern high-speed digital computers and associated equipment. Some of these methods are the Finite Difference Method (FDM), the Monte Carlo Method (MCM), the Charge Simulation Method (CSM), the Finite Element Method (FEM) and the Boundary Element Method (BEM). The most popular commercial one of these methods is the Finite Element Method. But the finite elements have been proved to be inadequate or inefficient in many engineering applications and what is perhaps more important is in many cases cumbersome to use and hence difficult to implement in Computer Aided Engineering (CAE) systems. Finite element analysis is still a comparatively slow process due to the need to define or redefine meshes in the piece or domain under study [5-7]. The Boundary Element Method have emerged as a powerful alternative to the finite element method particularly in cases where better accuracy is required due to problems such as stress concentration or where the domain extends to infinity [1,5,7]. The most important features of boundary elements however is that it only requires discretization of the surface rather than the volume. Hence boundary element codes are easier to use with existing solid modellers and mesh generators. This advantage is particularly important for designing as the process usually involves a series of vuimodifications which are more difficult to carry out using finite elements. Meshes can easily be generated and design changes do not require a complete remeshing [6,1 1]. It is difficult to say who was the pioneer of the boundary element method [2,5]. The mathematical background of the boundary element method has been known for nearly one hundred years. Indeed, some of the boundary integral formulations for elastic, elastodynamic wave propagation and potential flow equations have existed in the literature for at least fifty years. With the emergence of digital computers the method had began to gain popularity as 'the panel method', 'the boundary integral equation method' and 'the integral equation method' during sixties. The name was changed to 'the boundary element method (BEM)' by Banerjee and Butterfield in 1975, so as to make it more popular in the engineering analysis community [2]. The first book entitled Boundary Elements was published in 1978. After that BEM developed rapidly. It has been expanded so as to include time-dependent and non-linear problems. During this time many papers, theses and books have been published. The method is now regarded as important as FEM. An international conference to discuss BEM is held every year and the edited proceedings are valuable references [1,2,5]. There exist two basic types of BEM, the indirect and the direct methods [2,5]. In the indirect boundary element methods, the solution is sought by superposition of the sources being distributed continuously over its boundary. The intensity of the distribution, which generally varies from point to point of the boundary, is usually known as the density functions; the partial differential equation is automatically satisfied at every interior point of the domain, and all that is required is to satisfy the boundary conditions by suitable choice of that function. It is found that for the boundary conditions to be satisfied, the density function must be the solution of an equation over the boundary, the form of which depends upon the type of boundary condition. The boundary integral equation can not in general be solved exactly; instead an approximate solution is obtained by proposing some variation of the density function over the boundary in terms of a finite number of parameters. Once the integral equation has been solved, physically meaningful results at boundary and interior points of the domain are computed by integration over the boundary. In the direct boundary element methods, an integral equation is obtained from the divergence theorem, which states that the integral over a domain of the divergence of a vector field equals the integral over the boundary of the domain of the outward normal component of that field. By making appropriate substitutions into the divergence theorem, formulae such as Green's symmetric identity and Maxwell-Betti reciprocal theorem are obtained. By taking one of the two arbitrary functions in such a formula to be the function to be computed, and the other to be a fundamental solution with the source point located on the boundary, an integral equation over the boundary is obtained. This is known as the boundary integral equation of the direct method because the functions appearing in the equation are physically meaningful functions (such as displacement and traction for elasticity) rather than fictitious density functions. Solution of the integral equation therefore immediately yields the desired results on the boundary. If results at interior points of the domain are required, they can than be computed from formulae obtained in the same way as the integral equation, but with the source point of the fundamental solution located at interior points. IXThe method basically contains the following steps [1]; 1- The boundary S is discretized into a number of elements over which the unknown function and its normal derivative are assumed by 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 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 potential and its normal derivatives. Hence the field strength of the most interest on the boundary is computed 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. In this theses, the direct boundary element method is illustrated. As mentioned above, the BEM is based 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 of along the boundary are assumed to be the unknowns. The method of weighted residuals may be applied for arbitrary operators. It requires knowledge of the governing equation and the corresponding boundary conditions. For a boundary value problem [1-4] ; LV=b inB V\s=V0on^ (1) âV ân q0 on S2 where L; linear differential operator, V; unknown function, b; arbitrary known function, B, domain, S = Sı + & the periphery of B. Assume that the solution of the governing equation is approximated by a function as shown in equation (2), i.e. N V = Za*fa (2) 4=1where a^ are the unknown parameters and fa are linearly independent functions such as i(x), ^ 2(x),, ^ n(x). These functions are usually chosen in such a way as to satisfy certain given conditions, called admissible conditions. Substituting Eq.(2) into Eq.(l), residuals or called errors are unavoidable, i.e, R(V)=LV-b inB Rj(V)= V-V0 on Sj (3). âV R2(V) = j^-q0onS2 In order to make the errors approach zero, the average error distribution principle is used, i.e. JR(V) wdB + JR^V) wxds + j R2(V) w2 ds = 0 (4) B Si S2 aw where w, Wj, w2 are weighting functions. By assuming W] = - - (otherwise the equation will not have the correct dimension) and Wj = - w, Eq.(4) is simplified to \R(V)wdB = - /^(V) - -ds + \R2(V)wds (5) B Si dYl S2 Considering the governing equation is Poisson's equation, then Eq.(5) is changed to J(fW - bw)dB = - jq0wds - \-wds + j>0 -ds + \V- ds (6) b s2 s, °n $ °n S2“Tl where V is the approximate solution. This is a weak formulation of Eq.(6), as it reduces the order of the derivative of the unknown function. Hence the requirement of continuity of the approximate function of F is reduced. Choosing the fundamental solution F, which satisfies Eq.(7), as the weighting function V2F = -Si(r-r') (7) xiwhere S, is a Dirac's delta function has the property that J W2F dB = - jvS,(r - r')dB = -Vt (8) while point ”/'“ is in the domain B, then Eq.(6).âV ”, r“ âF, c”dF Vt + \bFdB = \q0Fds+ \-Fds- [vo-ds- \V-ds İ ; {ân % ân i ân D ı>2 uı Oj ı>2 (9) The compact form of Eq.(9) is; i* İT âV _âF CM = J I F TT- ^^T \ds-jbFdB ân B (10) where = \ dF «j ân ds.“-J- gJ = J F& (14) (15) Eq.(12) becomes 1 N N Z j=l H rdV^ ân) (16) In the above equations, F is the fundamental solution of the governing equation for two-dimensional Laplace's equation; F-±^ ' In \r - r'\ (17) As a simplification, it is defined that hAi = h”i*J 1 hn+- i = j 2 J (18) Then Eq.(16) is written as; N N j=l " J=l 2>,Jvi = Z*,,«J (19) X1UThe matrix form of Eq. (19) is [H] {V} = [G] {q} (20) This is the normal form of the boundary element equation. Here [H], [G] are matrices of the order of NxN, these are full matrices and in general they are asymmetrical. {V}, {q} are two unknown column matrices of the order N. They are potentials and its normal derivative at each nodes. Substitute the known boundary conditions of the first and the second kind into Eq. (20) and rearrange the unknowns on both sides of the equations. Gauss 's elimination or Cholesky's decomposition methods are used for solving the matrix equation (algebraic equation). A.X = B (21) The solution set gives the unknowns of the equation. In conclusion; 1- It is obvious that the main characteristic of the BEM is reduction of the basic process dimension by one; i.e. for two-dimensional problems the method generates a one-dimensional boundary integral equation and for three-dimensional problems, only the surface of the domain needs to be discretized hence it produces a much smaller number of algebraic equations. 2- It is especially attractive that the data preparation is simple because the tedious domain discretization is avoid. 3- The post-processing of data is also simpler than in the domain methods. Only the required values are calculated. 4- The method is well suited to solve problems with open boundaries. 5- Finally, the solution of the derivatives of the unknown function are as accurate as the function itself. Disadvantages of the BEM are; 1- A great number of integrations are required and singularities of the integral must be considered. Hence the calculation of the coefficient matrix requires more time than for FEM. 2- The fundamental solution of the governing equation is difficult in some problems. 3- The method can not be used directly for non-linear problems. XIV
Benzer Tezler
- Design and finite element analysis of 100 ton double girder overhead crane
100 ton kapasiteli çift köprülü köprü tipi krenin tasarımı ve sonlu elemanlar yöntemiyle yapısal analizi
GÜRKAN TAŞDEMİR
Yüksek Lisans
İngilizce
2020
Makine Mühendisliğiİstanbul Teknik ÜniversitesiMakine Mühendisliği Ana Bilim Dalı
DOÇ. DR. SERPİL KURT HABİBOĞLU
- Girdap akımlarının ve deri etkisinin modellenmesi
Modelling of eddy currents and skin effect
SERKAN ÖZÇETİN
Yüksek Lisans
Türkçe
2001
Elektrik ve Elektronik Mühendisliğiİstanbul Teknik ÜniversitesiPROF.DR. NURDAN GÜZELBEYOĞLU
- İran ekonomisinde ithal ikamesi ve yapısal değişmenin ekonometrik analizi
Başlık çevirisi yok
HEDAYAT MONTAKHAB
- Deprem etkisindeki yapıların aktif kontrolü
Active control of structures under seismic excitation
BEKİR BORA GÖZÜKIZIL
- Finite element solution for 2-D and 3-D acoustic scattering problem
Akustik saçılma probleminin sonlu elemanlar yöntemi ile 2 ve 3 boyutta çözümü
DUYGU KAN
Yüksek Lisans
İngilizce
2017
Elektrik ve Elektronik Mühendisliğiİstanbul Teknik ÜniversitesiHesaplamalı Bilimler ve Mühendislik Ana Bilim Dalı
PROF. DR. İBRAHİM AKDUMAN