Grid-point models

Buzzi-Ferraris et al., (1983, 1984) proposed the use of a more powerful statistic for the discrimination among rival models however, computationally it is more intensive as it requires the calculation of the sensitivity coefficients at each grid point of the operability region. Thus, the simple divergence criterion of Hunter and Reimer (1965) appears to be the most attractive. 

Step 2. For each grid point of the operability region, compute the Weighted Divergence, D, given by Equation 12.12. In the computations we consider only the models which are adequate at the present time (not all the rival models). 

If instead of precise parameter estimation, we are designing experiments for model discrimination, the best grid point of the operability region is chosen by maximizing the overall divergence, defined for dynamic systems as... 

In the IBM, the presence of the solid boundary (fixed or moving) in the fluid can be represented by a virtual body force field -rp( ) applied on the computational grid at the vicinity of solid-flow interface. Considering the stability and efficiency in a 3-D simulation, the direct forcing scheme is adopted in this model. Details of this scheme are introduced in Section II.B. In this study, a new velocity interpolation method is developed based on the particle level-set function (p), which is shown in Fig. 20. At each time step of the simulation, the fluid-particle boundary condition (no-slip or free-slip) is imposed on the computational cells located in a small band across the particle surface. The thickness of this band can be chosen to be equal to 3A, where A is the mesh size (assuming a uniform mesh is used). If a grid point (like p and q in Fig. 20), where the velocity components of the control volume are defined, falls into this band, that is... 

In order to simplify the formulations, we model the field as a cross-connected grid as in Fig. 1 The field model consists of the grid points, the starting point and the destination point. 

Using the field model described in section 1, detection probabilities are to be computed for each grid point to find the breach probability. The optimal decision rule that maximizes the detection probability subject to a maximum allowable false alarm rate a is given by the Neyman-Pearson formulation [20]. Two hypotheses that represent the presence and absence of a target are set up. The Neyman-Pearson (NP) detector computes the likelihood ratio of the respective probability density functions, and compares it against a threshold which is designed such that a specified false alarm constraint is satisfied. 

In order to solve the weakest breach path problem, where we construct a graph to model the field, Dijkstra s shortest path algorithm can be employed [40]. The detection probabilities associated with the grid points cannot be directly used as weights of the grid points, and consequently they must be transformed to a new measure dv. Specifically, let... 

Let us return for the moment to Eq. (2.2). In atmospheric problems it is impossible to solve the equations of motion analytically. Under these conditions information about the instantaneous velocity field u is available only from direct measurements or from numerical simulations of the fluid flow. In either case we are confronted with the problem of reconstructing the complete, continuous velocity field from observations at discrete points in space, namely the measuring sites or the grid points of the numerical model. The sampling theorem tells us that from a set of discrete values, only those features of the field with scales larger than the discretization interval can be reproduced in their entirety (Papoulis, 1%5). Therefore, we decompose the wind velocity in the form... 

---