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Nötron difüzyon hesapları için bir genel geometri

Başlık çevirisi mevcut değil.

  1. Tez No: 35655
  2. Yazar: YASEMİN KABADAYI
  3. Danışmanlar: YRD. DOÇ. DR. BİLGE ÖZGENER
  4. Tez Türü: Yüksek Lisans
  5. Konular: Fizik ve Fizik Mühendisliği, Nükleer Mühendislik, Physics and Physics Engineering, Nuclear Engineering
  6. Anahtar Kelimeler: Belirtilmemiş.
  7. Yıl: 1994
  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: 110

Özet

ÖZET Bu çalışmada nötron difüzyon hesaplara için bir geometri sonlu elemanlar yazılımı geliştirilmiştir. Çok- gruplu difüzyon denklemi kendine- ek olduğu için varyasyonel prensipten hareket edilmiştir. Lineer ve kuadratik üçgen elemanlar ile bilineer ve bikuadratik serendipity dörtgen elemanların kullanımına izin verilmiştir. Ayraca ikinci derece izoparametrik elemanlar kullanılarak, eğrisel kenarların modellenmesi de mümkün olmuştur. Son yıllardaki bilgisayar bellek ve hızındaki gelişmeler dikkate alınarak; genel geometri özelliği de istendiğinden iteratif çözüme gidilmemiş, grup-içî denklem lerinin çözümünde dolaysız yarı-bant genişlikli simetrik depo lama lı Cholesky ayrışımı yöntemi kullanalmaştar. Saçılma ve fisyon matrislerinin oluşumu ise uyumlu kaynak yaklaşımı ile gerçekleştirilmiştir. Geliştirilen FEND yazılımı, bazı problemlerin analitik çözümleri ile program sonuçlara karşılaştırılarak doğrulanmıştır. Ayrıca gerçekçi bir reaktör sisteminin çözümlenmesi için kullanılmış ve elde edilen sonuçlar, başka program sonuçları ile de karşılaştırılmıştar. ıx

Özet (Çeviri)

SUMMARY The numerical solution of the wi thin-group diffusion equation involves the spatial discretization of the problem domain. During the last quarter century, a variety of discretization methods have been proposed, applied and assessed. Among those, finite differences and finite elements stand out with the virtue of having a solid mathematical foundation. In contrast to finite differences, the method of finite elements is based on a variational principle instead of the neutron-diffusion equation itself. The variational principle states that the function which gives the minimum point of a functional is ideniicaJ to the solution of the neutron diffusion equation. Thus, the minimization of the functional is equivalent to the solution of the diffusion equation. The domain of the functional is the space of functions with square-integrable first derivatives, a Hilbert space. In the finite element approximation, the problem domain is divided into a number of spatial regions with a regular geometrical shape which ^re termed as“finite elements”. The approximate minimum of the functional is sought in a“ Sobolev space ”, whose members are continuous functions which are low-order Lagrange type interpolatory polynomials in each of the elements. When the approximatesolution is sought in the afore mentioned Sobolev space, the probJem is transformed into the solution of a lineer system, whose solution gives the approximate fluxes at the nodes, which Are specially defined points on the interelement boundaries. The choice of the Sobolev space determines the accuracy and convergence characteristics of the finite element solution. By increasing the degree of the afore- mentioned Lagrange type polynomial, the accuracy of the solution can be enhanced. The finite elements usually assume the shape of quadrilaterals and triangels in two-dimensional discretizations. But the sides of the chosen elements are not restricted to be line segments They could be chosen as curves with the introduction of appropriate coordinate transformations. The elements with curved sides ar& termed isoparametric elements. Their usage renders the discretization of systems with curved interfaces possible. This constitutes one of the major advantages of the finite element method over other methods of discretization. The first application of the finite element method to the neutron diffusion dates back to 1970 *s. From that time on, quite a number of software have been developed for the finite element solution of the neutron diffusion and neutron transport equations. Most of the developed programs are limited to a certain geometry f i.e. Cartesian, cylindrical etc.) and to a certain element type XIt' i. e serendipity type quadrilaterals, linear triangles etc.) and used analytical integration for the calculation of the coefficient matrix. The use of isoparametric type elements was quite limited. The recent developments in the computer hardware and software have resulted in radical changes in computer implementation of numerical methods for the solution of equations of mathematical physics. In many areas (i.e. heat transf er, solid mechanics), the newly developed software renders the solution in any arbitrary geometry possible The use of isoparametric elements and arbitrary element degree have become common. In this work, we developed, applied and assessed a program, which solves the neutron diffusion equation in arbitrary two-dimensional geometry using a variety of element types, including the isoparametric ones. The developed program is named FEND t“ finite element neutron diffusion). FEND has the capability to solve criticality eigenvalue problems and fixed sources problems in multigroup diffusion theory. The wi thin-group equations are solved by Cholesky decomposition. In criticality eigenvalue problems, the classical fission source iteration is employed. The program is written in FORTRAN 77. The program uses the technique of variable dimensioning so that all data is stored in a single array. All group- dependent matrices are stored on random access devices İ disk). This data is transf orred to the memory only when needed. All the test cases are run on personal xxxcomputers with INTEL 466 processor under the MS-DOS öper a t i ng s y s t em. At the validation stage of the code, bare and reflected reactor geometries with one and two-group diffusion are used. The effective multiplication constant, power densities and pointwise fluxes calculated by FEND are compared to analytical solutions. The convergence to the analytical solution is observed in all cases as the mesh is progressively refined. For the test of isoparametric capability, a square unit cell with a cylindrical fuel rod is used. The obtained multiplication constant and average fluxes are compared to the results of a finite element neutron transport code. The results are identical When the PI approximation is used in the latter. The capabilities of FEND are tested by running a benchmark problem. A foui - group model of a fast reactor I”namely SNR-300 ) is used for this purpose. Both linear and quadratic elements are employed in the discretization. The obtained results Are compared with the results of a number of programs, namely CITATION, HEXAGA II and TRIFT. XlllThe effective multiplication factor obtained by FEND is consistent with the values obtained by these codes. In conclusion we could state that FEND is a“'state -» of- the- art ”program which could solve realistic reactor problems with speed and accuracy. Nevertheless, FEND has one deficiency. At the moment, the classical fission source iteration is used in criticality eigenvalue calculation. This unaccelerated iteration is known to be somewhat slow, especially in problems with a complex geometry. The use of Chebyshev acceleration or coarse mesh rebalance is expected to enhance the speed of the solution in problems with complex domains. It is recommended that further research is pursued in that direction. XIV

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