Nodal expansion

Since the pioneer work of Mayer, many methods have become available for obtaining the equilibrium properties of plasmas and electrolytes from the general formulation of statistical mechanics. Let us cite, apart from the well-known cluster expansion 22 the collective coordinates approach, the dielectric constant method (for an excellent summary of these two methods see Ref. 4), and the nodal expansion method.23... 

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 ... 

E. Meeron, Nodal expansions III, Exact integral equations for particle correlation functions, /. Math. Phys. 1, 192-201 (1960). 

Modem nuclear design methods for commercial LWRs have been based on nodal methods the nodal expansion method (NEM), analytic nodal method (ANM), and analytic function expansion nodal method (AFEN). The nodal method treats a fuel assembly as a node, and an intra-node neutron flux is expressed as a synthesis of a polynomial expansion (NEM) or an analytic solution for each direction (ANM), or its combined expansion (AFEN), which provides very fast solutions for core design. 

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. 

The direct correlation function c is the sum of all graphs in h with no nodal points. The cluster expansions for the correlation functions were first obtained and analyzed in detail by Madden and Glandt [15,16]. However, the exact equations for the correlation functions, which have been called the replica Ornstein-Zernike (ROZ) equations, have been derived by Given and Stell [17-19]. These equations, for a one-component fluid in a one-component matrix, have the following form... 

First, it is possible to simplify the secular equation (2) by means of symmetry. It can be shown by group theory (140) that, in general, the integrals Hi and Si are nonzero only if the orbitals < , and j have the same transformation properties under all the symmetry elements of the molecule. As a simple example, the interaction between an s and a pn orbital which have different properties with respect to the nodal plane of the pn orbital is clearly zero. Interaction above the symmetry plane is cancelled exactly by interaction below the plane (Fig. 13). It is thus possible to split the secular determinant into a set of diagonal blocks with all integrals outside these blocks identically zero. Expansion of the determinant is then simply the product of those lower-order determinants, and so the magnitude of the... 

Applying a Taylor expansion to give the velocity at the first nodal point away from the wall in terms of the velocity at the wall, i.e., zero, gives on neglecting the higher order terms ... 

To have an idea about the magnitude of the local discretization error, consider the Taylor serie.s expansion of the temperature at a specified nodal point m about time... 

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. 

The results for the repulsive 2pcru interaction are summarized in Figs. 21 and 22. As the guiding principles predict, all the density redistributions A/, A/, and A/ show expansions whose magnitudes increase monotoni-cally with the decrease of R. In the antibonding state, the presence of the nodal plane pz = 0 (see Fig. 16) also works to enhance the expansion. The resultant energy curves again show the predominance of the atom and parallel parts when partitioned. 

ITT-type basis functions, however, transform like a set of unit vectors because the intrinsic nodal plane in each one defines a particular direction. Stone recognized that appropriate LCAO combinations may be formed using vector surface harmonics, which have both magnitude and direction at any given point in space. Hence, instead of LCAO expansion coefficients, which are Just the values of spherical harmonics evaluated at the atom sites, the coefficients for jr-type basis functions define not only an amplitude but also the direction in which the 7T-functions point. ... 

Inserting the expansion, Eq. (14), into the Schrodinger equation (6) at the nodal points together with the normalization condition (16) leads to a system of 2N coupled uni-dimensional differential equations for the radial functions and t i(0-... 

Like all other meshless methods, the first step in GFD is to scatter nodal points in the computational domain and along the boimdary. To each node (point), a collection of neighboring nodes are associated which is called star. The number and the position of nodes in each star are decisive factors affecting the finite difference approximation. Particular node patterns can lead to ill-conditioned situations and ultimately degenerated solutions. Using the Taylor s series expansion, the value of any sufficiently differentiable smooth function u at the central node of star, uq, can be expressed in terms of the value of the same function at the rest of nodes, with i = 1,. .N where N is the total number of neighboring nodes in the star and is one less than the total number of nodes in it. In two dimensions, a second-order accurate Taylor series expansion can be written as... 

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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