3-boyutlu maxwell denklemlerinin çözünümü
Başlık çevirisi mevcut değil.
- Tez No: 66755
- Danışmanlar: PROF. DR. ÜLGEN GÜLÇAT
- Tez Türü: Yüksek Lisans
- Konular: Astronomi ve Uzay Bilimleri, Astronomy and Space Sciences
- Anahtar Kelimeler: Belirtilmemiş.
- Yıl: 1997
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Uzay Bilimleri ve Teknolojisi Ana Bilim Dalı
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: 70
Özet
ÖZET Uçan cisimlerin aerodinamik performanslarının yanısıra radar kesitlerinin de önemli bir dizayn kriteri olarak kabullenilmesi, radar kesitinin hesaplanması ihtiyacım gündeme getirmiş, bunun için de deneysel yöntemlerle karşılaştınlınca daha az zahmet ve masraf isteyen sayısal yöntemler daha çok tercih edilir hale gelmiştir. Bu çalışmada, radar kesiti hesaplanmak istenen 3-boyutlu geometriler, sekiz köşeli elemanlardan meydana gelen bir ağ yapısıyla modellenmiştir. Ardından bu ağ üzerindeki tekil noktalar ve elde edilecek sonuçlan olumsuz yönde etkileyebilecek bazı düzensizlikler, eliptik diferansiyel denklemlerin SIP (Strongly Implicit Procedure) ile çözümünü içeren bir bilgisayar programı ile düzeltilmiştir. Daha sonra, bu ağ ile temsil edilen geometrilerin saçılma karakteristiklerini belirlemek üzere, sonlu farklar yaklaşımı ile Maxwell denklemlerinin çözümünü verecek bir bilgisayar programı geliştirilip kullanılmıştır. Geliştirilen ağ üretici program kullanılarak elde edilen ağ örnekleri ve basit matematik denklemler yardımıyla üretilen bir küre etrafında Maxwell Denklemleri 'nin çözülmesiyle elde edilen elektrik alana ait grafikler tezde sunulmuştur. vıu
Özet (Çeviri)
SUMMARY Since the early days of aviation, shaping for good. aerodynamic characteristic has been an integral part of aircraft design. However, with the introduction of aircraft, such as the Lockheed -F-117A and the Northrop B-2,.it is clear that low radar signature is also rapidly becoming an important criterion in the design of new aircraft. Thus, today designer is not only faced with shaping for low radar cross section (RCS). Because of that, accurate analyses of aerodynamics and res characteristics must be determined simultaneously. A smooth grid is required to obtain radar cross section of a 3-dimensional. object. In this thesis elleptic partial diferantial equation is used to generate 3- dimensional grid. The grid in the physical domain is mapped into computational space by transforming x-y-z coordinates into £- 77- £ coordinates. The general transformation is governed by a set of Poisson Equation in the following form : The inhomogenous terms P, Q and R control the distribution of nodes inside the domain of interest. Transformation equation. above to the curvilinear space produces system of poisson equation which is the most common PDE used for grid generation as follows: (hxX^a^x^^x^+a^x^a^^+^x^-I^Px^ +Qxn+Rx() Writing this equation in finite difference form using second order centered difference approximations of the exact partial derrivatives yield the following FDE : IXXi+lJJc IXjJJc +XlAJJt ? Xj-HJ-tiJc Xi-lJ+\Jc XMJ-Ut +Xi-iJ-lk XMJMl ~Xi-l,jJc+l ~Xi+yjc-\ +Xt-UJc-l Xi.j\ijk ~^Xij\k +XtJ-Xk a^J* AAÇAT ^*u* Arj2 XiJ+lk-il ~Xi,j-U+l ~XiJ+U-l +XiJ-U-\ Xi.jMl ~~XiJJc +XijM _ Tz \XM,jM~Xi-lJJi p, XiJ+lk ~XiJ-Xk ^ XtJJ:-a ~XİJJc-l n (i) 'u* v 2zkf 'J* 2Arj >}* 2A;, In the equation above specific functional forms must be chosen for P, Q and R to achieve the desired interior grid point distribution. The method developed by Steger and Sorenson ( 1979 ) and Hilgenstock ( 1988 ) consists of determining the values P, Q and R on the boundaries by imposing the follwing two constraints : 1. The spacing in the physical space along the transverse coordinate lines between the boundary coordinate line and first interior coordinate line a specified priori 2. The angle of intersection of the transverse coordinate lines with the boundary coordinate lines is specified a priori. After the value of P, Q and R determined on the boundaries, they must be extrapolate into the interior of the domain. Exponantially extrapolate the boundary values P, Q and R into the interior is one of the method. Since the terms P, Q and R required to produce the desired grid are unknown initially they have to be determined iteratively as follows : P*+' = P“+AP R”+I = R“ + AR Here n denotes the number of iteration level. The initial values of P, Q and R can be set zero for the first time. The determination of AP, AQ and AR are boundary dependent. Angle control at a surface £ = const, is given as follows : a”= cos -1 Where T, and T are the tangents described in fig. 3.2 with a required angle a - in this case 90° - the AQ correction is defined as:A Q = tan a“-a a The same procedure is used to determine AP. The distance between the boundary point at the Ç = const plane and adjecent grid point in ğ direction is : As = \xi-ij,k ~ xu,k ) + \yt-ij,k ~ yij,k ) + \zi-u,k - ztj,k ) -Iİ/2 with a required distance As* one obtains the following relation for the correction of the corresponding source term P : A P = tan i (As”-As As* Solution of large linear equation systems uses most computing time in computational fluid dynamics (CFD ) codes. Due to the non-linearity and coupling of partial diferantial equations, outer iteration must be used to update the coefficients and source terms in the linear equation systems. These are also solved iteratively and iteration within the solver is called inner iteration. One of the most popular iterative solvers for ( CFD ) is the Stone's strongly imlicit ( SIP ) solver. In this thesis ( SIP ) is used to solve poisson equation. Finite difference formulation for poisson equation can be written in matrix form as follows : The iterative solution of the equation above requires an iteration matrix [M] and the proces of solution proceeds as follows : [M]{r^\ = M-[A-M]{r-} or Where Sm is the change in variable r from iteration m to m+1 and \0m\ is the resudual after m'th iteration, {öm; = {^}-[^]{r m). For an efficient iterative solution method, the matrix [M] should be as closed to [A] as possible. The SIP method uses a product of two triangular matrices [/,] and [if] ( for lower and upper ) as the iteration matrix : siFor the rapid converge, [N]. must be very small. After calculation of the element of the matrix [L] and [u] at ijk th node, auxilary vector matrix {V } can be calculated as follows : {Fm} = [zr']{0ra} And the increment vector matrix {S } : {sm}=[u-1]{vm} These two equations are solved easily by forward and bacward substitution The RCS analysis is based on the numerical solution of the time domain Maxwell's curl equation. The Maxwell's equations for free space are given by Ampere's and Faraday's laws as follows : -* SF VAH-e- = 0 ât -* âlî ^ ât In three dimensions, the Maxwell equations involve three components of the electric field (Ex, Ey, Ez ) and the three components of the magnetic field (Hx, Hy, H2 ). The three dimensional Maxwell equation in cartesian coordinates can be written in conservation form as follows : 0 = F3 = E* 0 Hy -Hx 0 As a result of the transformation from the cartesian coordinates to the generalized curvilinear coordinate system the equations above become Ql + Fl4 + F2,, + F3c = 0 (2) XII(2) The relationship between the Q and F vectors as follows Fl = AQ F2=BQ F3 = CQ Fl,F2,F3 and Q vectors are called flux vectors. /=/,...,5 ve J=l,...,6, v âFHl), v â¥2(l) f x âVHl) 4^^)=-^ 5(/^)=-^t( c(i,j)= k ) dQÜ) ÖQ{J) dQU) So, A, B and C jakobyen matrix are given below: A = XIII5 = C = The equation (2) is a system of hyperbolic equations. This kind of equations can be solved numerically by using any of numerous CFD based finite difference schemes. In this thesis, Lax-Wendroff scheme is used. The Lax- Wendroff scheme has a second order accuracy in both time and space and is an explicit scheme. The final form of Lax-Wendroff scheme for Maxwell's equations as follows: & = QIj* ~ "(?! + K + F? ) + ^-{4?* + f1* + f$ + ^{fİ+fÎ +Fİ)+ BİFl + F^ + FÜ + Bfâ+Ft+F*) + C(Fİ + Fi + F^) + C,(f' + F2 + F/ )} For numerical solution second order finite difference formulation is used in the equation above. And finally, two computer codes are devolped in ftn77 two modify any grid and to solve Maxwell's equations. The first computer code is used to modify any grid generated by any method for RCS analyses of an object and RCS analyses obtained by this modified grid has more accuracy. In this thesis, RCS analyses is obtained around a sphere. The sphere is generated by gebric method. And the results are presented in chapter 6. Some grids which is modified by computer code are also presented in chapter 6. XIV
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