Transverse-integrated nodal di usion methods currently represent the standard in full core neutronic simulation. The primary shortcoming of this approach is the utilization of the quadratic transverse leakage approximation. This approach, although proven to work well for typical LWR problems, is not consistent with the formulation of nodal methods and can cause accuracy and convergence problems. In this work, an improved, consistent quadratic leakage approximation is formulated, which derives from the class of higher-order nodal methods developed some years ago. In this thesis a number of iteration schemes are developed around this consistent quadratic leakage approximation which yields accurate node average results in much improved calculational times. The most promising of these iteration schemes results from utilizing the consistent leakage approximation as a correction method to the standard quadratic leakage approximation. Numerical results are demonstrated on a set of benchmark problems and further applied to realistic reactor problems for particularly the SAFARI-1 reactor operating at Necsa, South Africa. The nal optimal solution strategy is packaged into a standalone module which may be simply coupled to existing nodal di usion codes, illustrated via coupling of the module to the OSCAR-4 code system developed at Necsa and utilized for the calculational support of a number of operating research reactors around the world. / Thesis(PhD (Reactor Science))--North-West University, Potchefstroom Campus, 2013

Transverse-integrated nodal di usion methods currently represent the standard in full core neutronic simulation. The primary shortcoming of this approach is the utilization of the quadratic transverse leakage approximation. This approach, although proven to work well for typical LWR problems, is not consistent with the formulation of nodal methods and can cause accuracy and convergence problems. In this work, an improved, consistent quadratic leakage approximation is formulated, which derives from the class of higher-order nodal methods developed some years ago. In this thesis a number of iteration schemes are developed around this consistent quadratic leakage approximation which yields accurate node average results in much improved calculational times. The most promising of these iteration schemes results from utilizing the consistent leakage approximation as a correction method to the standard quadratic leakage approximation. Numerical results are demonstrated on a set of benchmark problems and further applied to realistic reactor problems for particularly the SAFARI-1 reactor operating at Necsa, South Africa. The nal optimal solution strategy is packaged into a standalone module which may be simply coupled to existing nodal di usion codes, illustrated via coupling of the module to the OSCAR-4 code system developed at Necsa and utilized for the calculational support of a number of operating research reactors around the world. / Thesis(PhD (Reactor Science))--North-West University, Potchefstroom Campus, 2013

Let us consider the singularly perturbed model problem Lu := -epsilon laplace u-bu_x+cu = f with homogeneous Dirichlet boundary conditions on the unit-square (0,1)^2. Assuming that b > 0 is of order one, the small perturbation parameter 0 < epsilon << 1 causes boundary layers in the solution. In order to solve above problem numerically, it is beneficial to resolve these layers. On properly layer-adapted meshes we can apply finite element methods and observe convergence. We will consider standard Galerkin and stabilised FEM applied to above problem. Therein the polynomial order p will be usually greater then two, i.e. we will consider higher-order methods. Most of the analysis presented here is done in the standard energy norm. Nevertheless, the question arises: Is this the right norm for this kind of problem, especially if characteristic layers occur? We will address this question by looking into a balanced norm. Finally, a-posteriori error analysis is an important tool to construct adapted meshes iteratively by solving discrete problems, estimating the error and adjusting the mesh accordingly. We will present estimates on the Green’s function associated with L, that can be used to derive pointwise error estimators.

Let us consider the singularly perturbed model problem Lu := -epsilon laplace u-bu_x+cu = f with homogeneous Dirichlet boundary conditions on the unit-square (0,1)^2. Assuming that b > 0 is of order one, the small perturbation parameter 0 < epsilon << 1 causes boundary layers in the solution. In order to solve above problem numerically, it is beneficial to resolve these layers. On properly layer-adapted meshes we can apply finite element methods and observe convergence. We will consider standard Galerkin and stabilised FEM applied to above problem. Therein the polynomial order p will be usually greater then two, i.e. we will consider higher-order methods. Most of the analysis presented here is done in the standard energy norm. Nevertheless, the question arises: Is this the right norm for this kind of problem, especially if characteristic layers occur? We will address this question by looking into a balanced norm. Finally, a-posteriori error analysis is an important tool to construct adapted meshes iteratively by solving discrete problems, estimating the error and adjusting the mesh accordingly. We will present estimates on the Green’s function associated with L, that can be used to derive pointwise error estimators.