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Abstract: Computer modeling of radiation deep penetration problems is historically based on the discrete ordinates (S N) formulation. For efficiency reasons, besides accuracy, coarse-mesh spatial discretization is desirable. The spectral Green''s function (SGF) methods form a class of accurate coarse-mesh numerical methods as they use polynomial approximations only for the node-edge transverse leakage terms; the scattering source terms are treated analytically in the numerical algorithm. Therefore, algebraic work and the computational algorithms of the spectral nodal methods are rather complicated. To alleviate this negative feature, we offer in this paper a composite spatial grid SGF nodal method for the numerical solution of one-speed deep penetration S N problems with isotropic scattering in X,Y geometry. This method uses a rectangular coarse spatial grid, that is coincident with the material region distribution within the shielding structure. We first transverse integrate the S N equations separately in the x- and y-coordinate directions inside each material region, and then we introduce flat approximations for the transverse leakage terms. Furthermore, we use a fine spatial grid to discretize each set of “one-dimensional”S N nodal equations. As the spatial directions are coupled by the transverse leakage terms, we use an explicit alternate direction technique to converge the numerical solution. In order to verify the offered method''s accuracy, we present numerical results for typical model problems. Moreover, we compare the computing performance of this method with the conventional SGF-constant nodal method. [Copyright &y& Elsevier]

In this paper, we present a new approach of-spline curve fitting to a set of ordered points, which is motivated by an insight that properly selected points called dominant points can play an important role in producing better curve approximation. The proposed approach takes four main steps: parameterization, dominant point selection, knot placement, and least-squares minimization. The approach is substantially different from the conventional approaches in knot placement and dominant point selection. In the knot placement, the knots are determined by averaging the parameter values of the dominant points, which basically transforms-spline curve fitting into the problem of dominant point selection. We describe the properties of the knot placement including the property of local modification useful for adaptive curve refinement. We also present an algorithm for dominant point selection based on the adaptive refinement paradigm. The approach adaptively refines a-spline curve by selecting fewer dominant points at flat regions but more at complex regions. For the same number of control points, the proposed approach can generate a-spline curve with less deviation than the conventional approaches. When adopted in error-bounded curve approximation, it can generate a-spline curve with far fewer control points while satisfying the desired shape fidelity. Some experimental results demonstrate its usefulness and quality. [Copyright &y& Elsevier]