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|Başlık:||Zamandan bağımsız ve zamana bağlı nötron transport denkleminin sayısal çözümleri için süreksiz sonlu elemanlar yöntemleri|
|Diğer Başlıklar:||Discontinuous finite element methods for the numerical solution of time-independent and time-dependent neutron transport equation|
|Anahtar kelimeler:||Neutron transport equation|
Nötron transport denklemi
Finite elements method
Sonlu elemanlar yöntemleri
|Yayıncı:||İstanbul Teknik Üniversitesi, Enerji Enstitüsü|
|Atıf:||Mercimek, M. (2014) Zamandan bağımsız ve zamana bağlı nötron transport denkleminin sayısal çözümleri için süreksiz sonlu elemanlar yöntemleri. (Yayımlanmamış doktora tezi). İstanbul : İstanbul Teknik Üniversitesi, Enerji Enstitüsü|
|Özet:||Bu çalışmada, zamandan bağımsız ve zamana bağlı nötron transport denklemini küresel bir nükleer reaktörde sayısal olarak çözmek için uzaysal farklamada kullanılmak üzere doğrusal ve kuadratik süreksiz Galerkin sonlu elemanlar yöntemi geliştirilmiştir. Bunların yanında uzaysal farklamada elmas farklaması ve başka bir doğrusal süreksiz sonlu elemanlar yöntemi kullanılmıştır. Bu yöntemlerin türetimi yapılmış ve dört farklı uzaysal ayrıklaştırma yöntemi ile iki farklı zaman ayrıklaştırması yöntemi kullanılarak bilgisayar programları geliştirilmiştir. Zaman farklaması olarak kapalı ve elmas farklaması, yönsel ayrıklaştırma yöntemi olarak ayrık ordinatlar yöntemi kullanılmıştır. Yazılan programlar analitik çözümü bilinen zamandan bağımsız ve zamana bağlı farklı tip test problemleri ile doğrulanmış ve yöntemler karşılaştırılmıştır. Zamandan bağımsız problemlerde kuadratik süreksiz sonlu elemanlar yöntemi aynı nokta sayısı ile karşılaştırma yapıldığında etkin çoğaltma katsayısını diğer yöntemlere oranla daha doğru hesaplayabilmiş, ayrıca hesaplama yükü olarak bakıldığında daha kısa sürede daha hatasız sonuçlar üreterek iyi bir performans göstermiştir. Zamana bağlı reaktör problemlerinde sekiz ayrı yöntem arasından uzaysal farklamada kuadratik süreksiz sonlu elemanlar yöntemi, zaman farklamasında ise elmas farklaması kullanıldığında daha doğru sonuçlar elde edilmiştir. Bu yöntemlerin hesaplamada kullandığı iç ve dış iterasyonların hızlandırılmasında kaba ızgara yeniden dengeleme hızlandırma yöntemi kullanılmış ve performansı test edilmiştir. Özellikle zamana bağlı problemler için bu hızlandırma yöntemi geliştirilmiş ve özgün olarak geliştirilen iki süreksiz sonlu elemanlar yönteminin hesaplama performansı arttırılmıştır. Sonuç olarak bu yöntemin uygun reaktör problemlerinde ve kaba ızgara başına ince ızgara sayısının uygun seçildiği durumlarda etkili bir hızlandırma yaptığı gösterilmiştir.|
The Discontinuous Finite Element Method is among the most flexible numerical methods in discrete ordinates formulations of neutral particle transport. The method was first introduced for the solution of the hyperbolic neutron transport equation in the early 1970s. It uses piecewise polynomial spaces similar to the (Continuous) Finite Element Method but with relaxed continuity conditions at interelement boundaries. Trial functions can be chosen so that the field variable, its derivative or both are discontinuous across interelement boundaries. The method includes as its subsets both the Finite Element and Finite Difference methods. Hence, it has many good features of both methods. An important distinction between the Discontinuous and Continuous Finite Element Methods is that in the Discontinuous Methods the resulting equations are local to the generating element. The solution within each element is not reconstructed by looking to neighbouring elements. Its compact formulation can be applied near boundaries without special treatment, which greatly increases the robustness and accuracy of any boundary condition implementation. Because of the local formulation, a discontinuous finite element algorithm will not result in an assembled global matrix and thus in-core memory demand is not as strong. Also, the local formulation makes it very easy to parallelize the algorithm, taking advantage of either shared memory parallel computing or distributed parallel computing. In this Thesis, Linear and Quadratic Discontinuous Finite Element methods have been developed for the solution of time-independent and timedependent neutron transport equation in spherical geometry. Several computer programs have been developed for neutron transport using the Linear Discontinuous Finite Element Method. ONETRAN and TIMEX are examples of those codes for the solution of time-independent and time-dependent transport, respectively. In this Thesis, a FORTRAN computer program named SPDOT has been written. It includes several discretization methods for the solution of space, angle, energy and time dependent neutron transport equation in spherical geometry. All developed formulations used in the SPDOT code are based on the prompt neutron transport equation. Hence delayed neutron effects are not treated in the code. Discrete Ordinates Method is used as an angular discretization method in this code. First, this code was developed using Diamond Differencing Method in space for the solution of time-independent transport equation. In order to validate first version of SPDOT, some benchmark problems of time-independent transport equation and PARTISN code which employs Diamond Differencing in space too were used. Second, Linear Discontinuous Finite Element Method similar to the method used in TIMEX code were developed and implemented in another version of SPDOT. The method employed in TIMEX code has been reformulated by using newly developed methodology. As the first objective of this study, a new and unique Linear xxii Discontinuous Finite Element Method has been developed. After comparison of two linear discontinuous methods, it has been shown that newly developed linear discontinuous method gives more accurate results than the older method. Hence, we used the same methodology to develop Quadratic Discontinuous Finite Element Method. Higher-order Discontinuous Finite Element Method has been extensively investigated in other disciplines. But limited research has been carried out using elements of higher degree in neutron transport. The second objective of this Thesis is to develop a higher-order Discontinuous Finite Element formulation for the solution of neutron transport problems in spherical geometry both for the time-independent and time dependent variety. As a higher-order Discontinuous Finite Element Method, we employ the Quadratic Discontinuous Galerkin Finite Element Method. In this Thesis, we present the derivations of linear systems of equations for TIMEX type, newly developed Linear Discontinuous Finite Element Method and Quadratic Discontinuous Finite Element Method for the solution of time-independent and timedependent prompt neutron transport equation in spherical geometry. We use the classical Galerkin method in which the weight functions are taken to be the same as the shape functions which are Lagrange type polynomials with compact support in the newly developed discontinuous methods. Weighted Residual Method is used to obtain matrix equations. Depending on the sign of the angular variable, discontinuity is introduced differently. Hence different matrix equations are obtained for rightward and leftward sweeps. Also for the starting direction equations, there are two options in the SPDOT, starting with equation similar to the equations used in the plane geometry and step-start method. Derivations for the special direction given in this study are based on the former. This is used in all numerical applications of this study. Second option for starting direction equations is step-starting similar to the method described in TIMEX code. Difference in computational accuracy is very small between two methods. Linear Discontinuous Finite Element Method solves a 2x2 linear system, while Quadratic Discontinuous Finite Element Method must solve 3x3 linear system in every iteration. In order to solve 3x3 linear system in Quadratic Discontinuous Finite Element Method, there are two options in the SPDOT as Gauss Elimination with Partial Pivoting and Cramer’s Rule. The Cramer’s rule runs about two-times faster than the Gauss Elimination with Partial Pivoting. An analysis with time-independent benchmark problems has shown that CPU times for discontinuous methods are longer than Diamond Difference Method for the same number of elements. But for the fixed number of points, Quadratic Discontinuous Finite Element Method results in the smallest CPU times. If the number of points in all methods is adjusted to get the same accuracy in effective multiplication factor, Quadratic Discontinuous Finite Element Method with Cramer’s rule is the most computationally efficient method. Also numerical analysis in this study reveals that Quadratic Discontinuous Finite Element Method developed in this study is well suited for large-scale time-dependent computations in which high accuracy is required. SPDOT can carry out four different spatial discretization methods as Diamond Difference, two Linear Discontinuous Finite Element Methods and Quadratic Discontinuous Finite Element Methods. In addition to inner and outer iterations, reflective boundary condition iteration is also available in the code. Vacuum boundary condition is used in time-independent and finite medium time-dependent xxiii transport benchmark problems, while reflective boundary condition is employed in infinite medium time-dependent problems. In addition, two possible options are described as central boundary condition in this study. Also negative flux fix up has been implemented to the code. Benchmark problems used in this study include bare and reflected spheres, isotropic and anisotropic scattering regions, multigroup and multiregional problems, critical, multiplying regions and fixed source problems. Numerical applications running SPDOT for different benchmark problems reveals that Quadratic Discontinuous Finite Element Method is the best method among others when accuracy and computational performance of the methods are considered. Also, newly developed Linear Discontinuous Finite Element Method is the second most accurate method in time independent problems. Another advantage of newly developed discontinuous methods is that these methods show no flux dip near centre of the sphere while the other methods have flux dips. Implicit and Diamond Difference Methods are used for the temporal differencing in time-dependent problems. SPDOT has been run for several time-dependent finite and infinite media benchmark problems to assess the performance of various spatial discretization methods in the solutions of neutron transport problems by the discrete ordinates method. This Thesis demonstrates that Quadratic Discontinuous Galerkin Finite Element Method is the best method for time-dependent problems. In addition, it is possible to increase quadrature order up to S256 in SPDOT, hence refinement in SN order is possible in the code. The refinement in the temporal-spatial-angular mesh results in convergence in this method for time-dependent benchmark problems. It is found in numerical calculations that Diamond Difference Method is more accurate method than the Implicit Method especially infinite media problems. As a result, the combination of quadratic discontinuous in space-diamond difference in time has the highest performance if accuracy and running times are considered. Every time step in a time-dependent neutron transport problem requires many numbers of iterations. In addition, it is evident that convergence is very slow in optically thick and scattering regions. This causes large computer CPU time. Therefore, acceleration methods are required in this kind of problems. The timemarching scheme using the implicit method requires the solution of the equivalent steady-state subcritical, external source problem in each time step. The solution of this problem is realized by fission source iteration (outer iteration). Scattering source iteration (inner iteration) is also necessary for each energy group. Both of these iterative solution methods require the onset of acceleration techniques in many problems. Rebalance methods normalize the discrete transport solution so that converged solution must satisfy the angularly-integrated neutron balance equation over individual cells (fine mesh rebalance) or groups of fine-mesh cells (coarse mesh rebalance). Coarse Mesh Rebalance can accelerate effectively if the size of the coarse meshes is properly chosen. Thus, the code user must select an optimal coarse mesh to maximize efficiency. The Coarse Mesh Rebalance equation is formulated in terms of rebalance factors. These factors are then used to update the solution in each finemesh cell. As the solution converges, the rebalance factors approaches unity. First, derivations of the equations for this method are given in this Thesis. Then, this acceleration method has been implemented to the code. Both acceleration of outer iterations which include eigenvalue problems and subcritical reactor or timedependent transport problems and inner iterations are possible in SPDOT. Performance of acceleration method has been assessed via numerical calculations for some test problems. The results show that the suggested acceleration procedure is xxiv effective in reducing the number of iterations and computing time requirements if it is applied appropriate problems and number of fine mesh cells in each coarse mesh cell is properly chosen. Especially, acceleration is valuable tool for Discontinuous Finite Element Methods since they produce stable and accurate results in optically thick or thin regions. On the other hand, Course Mesh Rebalance Acceleration Method is a conditionally stable method and it can be unstable in such regions. Hence, it has been shown in this study that it is an efficient method to accelerate iterations in Discontinuous Finite Element Methods. Finaly, descriptions of the subroutines and main program of SPDOT are given as an Appendix. There are 19 subroutines in the code. Input and output files for benchmark problems are also included. Another computer program for the calculations of Linear Discontinuous Finite Element Method previously developed (based on formulations similar to the TIMEX code) is given in a different folder.
|Koleksiyonlarda Görünür:||Tez 2010-2019 yılları|
DSpace'deki bütün öğeler, aksi belirtilmedikçe, tüm hakları saklı tutulmak şartıyla telif hakkı ile korunmaktadır.