Nodal expansion model

GPT is a method of evaluating the effects of cross-section perturbations on quantities that can be formulated as integral responses, such as reactivity and power density. An initial requirement is an exact solution of a reactor physics model for a reference core configuration. In FORMOSA-P the reference neutronics model is a two-dimensional Cartesian [x-y] geometry implementation of the nodal expansion method (NEM) to solve the two-group, steady-state neutron diffusion equation ... 

The described direct derivation of shape functions by the formulation and solution of algebraic equations in terms of nodal coordinates and nodal degrees of freedom is tedious and becomes impractical for higher-order elements. Furthermore, the existence of a solution for these equations (i.e. existence of an inverse for the coefficients matrix in them) is only guaranteed if the elemental interpolations are based on complete polynomials. Important families of useful finite elements do not provide interpolation models that correspond to complete polynomial expansions. Therefore, in practice, indirect methods are employed to derive the shape functions associated with the elements that belong to these families. 

A consistent numerical scheme produces a system of algebraic equations which can be shown to be equivalent to the original model equations as the grid spacing tends to zero. The truncation error represents the difference between the discretized equation and the exact one. For low order finite difference methods the error is usually estimated by replacing all the nodal values in the discrete approximation by a Taylor series expansion about a single point. As a result one recovers the original differential equation plus a remainder, which represents the truncation error. 

In Ref. [123], we propose an entirely different numerical model of fluid film dynamics from those, which can be derived from the NS approach or its asymptotic expansions. The model is based on the DPD particle model and can be used for simulating thin-film dynamics in the mesos-cale. Instead of changes of film thickness in nodal points in time according to the evolution equation discretized in both space and time, the temporal evolution of DPD particle system is governed by Newtonian laws of motion Equation (26.1)-Equation (26.4). 

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