I= AN ADAPTIVE FINITE ELEMENT FORMULATION OF THE BOLTZMANN-TYPE NEUTRON TRANSPORT EQUATION
Pakistan Research Repository Home
 

Title of Thesis
AN ADAPTIVE FINITE ELEMENT FORMULATION OF THE BOLTZMANN-TYPE NEUTRON TRANSPORT EQUATION

Author(s)
Shaukat Iqbal
Institute/University/Department Details
Faculty of Computer Sciences and Engineering/ Ghulam Ishaq Khan Institute of Engineering Sciences and Technology, Topi
Session
2007
Subject
Computer Sciences and Engineering
Number of Pages
154
Keywords (Extracted from title, table of contents and abstract of thesis)
boltzmann-type neutron transport equation, even parity boltzmann transport equation, finite elements, neutron angular flux, legendre expansion, scattering kernel, least squares approach, odd parity flux, even parity flux

Abstract
Adaptive grid refinement strategies have been formulated to solve the even parity Boltzmann transport equation. The application of continuous and discontinuous finite elements for approximating the spatial dependence of neutron angular flux along with the use of spherical harmonics for directional representation was investigated for some novel neutron transport problems.

The study of the conventional variational approaches employing finite elements for solving neutron transport equation has shown that such schemes only ensure a global particle balance over the whole system. Adaptive finite elements have been found to be superior not only in enforcing local particle conservation but also in being more suitable for modelling abrupt changes in angular flux.

A residual based a posteriori error estimation scheme has been utilized for checking the approximate solutions for various finite element grids. The local particle balance has been considered as an error assessment criterion. To implement the adaptive approach, a computer program has been developed to solve the second order even parity Boltzmann transport equation 'using variational principles for different geometries. The program has a core module, which employs Lagrange polynomials as spatial basis functions for the finite element formulation and Legendre polynomials for the directional dependence of the solution. The core module is called in by the adaptive grid generator to determine local gradients and residuals to explore the possibility of grid refinements in appropriate regions of the problem. The a posteriori error estimation scheme has been implemented in the outer grid refining iteration module. Numerical experiments indicate that local errors are large in regions where the flux gradients are large. A comparison of the spatially adaptive grid refinement approach with that of uniform meshing approach for various benchmark cases, confirms its superiority in greatly enhancing the accuracy of the solution without increasing the number of unknown coefficients. This scheme is then combined with a discontinuous finite element based composite scheme. This has given the added advantage of automatically generating the orders of angular approximations to be used in different elements/regions in the method of composite solutions. A reduction in the local errors of the order of 102 has been achieved using the new approach in some cases
Download Full Thesis
2015.21 KB
S. No. Chapter Title of the Chapters Page Size (KB)
1 0 Contents
187.82 KB
2 1 Introduction 18
145.78 KB
  1.1 The Importance Of Numerical Methods 18
  1.2 Finite Element Method In Neutron Transport 21
  1.3 Mesh Adaptivity 24
  1.4 Motivation Of The Work 27
  1.5 Contribution To The Work 28
  1.6 Layout Of Thesis 29
3 2 Transport Theory 31
231.39 KB
  2.1 Introduction 31
  2.2 Introductory Definitions 32
  2.3 Derivation Of Particle Transport Equation 37
  2.4 Neutron Transport Equation 41
  2.5 Boundary Conditions 42
  2.6 Legendre Expansion Of The Scattering Kernel 44
  2.7 Second Order From Of The Transport Equation 45
4 3 Variational Treatment Of The Boltzmann Transport Equation
186.01 KB
  3.1 Need For A Generalized Least Squares Approach 49
  3.2 Generalized Least Squares Approach 50
  3.3 A Generalized Least Squares Identity For The First Order Boltzmann Equation 50
  3.4 Measures Of Errors Of Approximate Solutions 52
  3.5 Least-Squares Functionals 55
  3.6 Variational Principles Admitting Continuous Trial Functions 58
  3.7 Variational Principles Admitting Discontinuous Trial Functions 61
  3.8 The Role Of The Penalty Parameter Λ 63
5 4 Finite Elements Formulation 65
468.6 KB
  4.1 Introduction 65
  4.2 Trial Functions For Directional Dependence Of The Angular Flux 66
  4.3 Finite Element Trial Functions For Spatial Domain 69
  4.4 Even And Odd Parity Trial Functions 75
  4.5 The Discretized Functionals 76
  4.6 Derivation Of Odd Parity Flux From Even Parity Flux 89
  4.7 Derivation Of Even Parity Flux From Odd Parity Flux 89
  4.8 Determination Of Integral Quantities Of Interest 90
6 5 Adaptive Finite Element Approach
323.52 KB
  5.1 Error Estimation 92
  5.2 Adaptive Strategies 94
  5.3 A Posteriori Error Estimates For Adaptive Grid Refinement Approaches 97
  5.4 Computational Aspects And Implementation 99
  5.5 Numerical Results And Discussion 101
7 6 Development Of A Combined Approaches For Spatial Mesh And Angular Approximation Adaptivity
451.95 KB
  6.1 Introduction 111
  6.2 Composite Solution Case Studies 113
  6.3 Adaptive Construction Of Composite Solutions 128
  6.4 Numerical Case Studies For Adaptive Composite Approach 130
8 7 Conclusions And Future Recommendations
45.86 KB
  7.1 Conclusions 136
  7.2 Future Recommendations 138
9 8 References 140
159.86 KB
  8.1 Appendix 151