Cubic grid

Whenever a cubic grid is mandatory—either due to coding limitations from the part of academic groups or due to the inherent properties of, e.g., LB techniques—and a staircase approach is to be avoided (e.g., for a revolving impeller axis) one can take refuge to some immersed boundary method (see, e.g., Mittal and Iaccarino, 2005). One may distinguish between... 

Fields as property values in 3-D space can either be evaluated and encoded on a regular (often cubic) grid [71], or approximated by certain distribution functions. Most often, Gaussians [83] have been employed here, as they have some desirable mathematical properties and can usually approximate the original field reasonably well with not too many parameters. 

In the case of a cubic grid (h1 p-dimensional cube we might have... 

Figure 21. The (10,3)-a net in its most symmetrical form, i.e. cubic, displayed on a cubic grid.

The net conveniently referred to as the a-Po net (41263 topology) consists essentially, in its geometrically most regular form, of a simple cubic grid in which all nodes are connected to six octahedrally disposed neighbors. Examples of 2-fold and 3-fold interpenetrating a-Po-like nets are known. 

It is obvious that a uniform cubic grid of points is not a good choice since in this case many points are required for the adequate description of the MEP map. For uniform cubic grid there will be too many points at large distances where the MEP maps are poor in information. As a consequence, an immense number of points would be needed to obtain stable EP charges [69]. 

Fig. 5. A contour plot representing a two-dimensional slice of the molecular electrostatic potential for vinyl sulfone in the plane of the molecule with a sample uniform cubic grid superimposed grid points would be placed at the intersections of the perpendicular lines. (Reproduced from [71] copyright-John Wiley Sons)...

The results summarized above were obtained through the analysis of spatial free energy density profiles using the following methods. Spatial distribution functions for ion occupancies around the macroion are computed on a cubic grid. These distribution functions can be converted to spatial free energy density profiles using the relation ... 

In a practical implementation, the domain on which the phase volume functions are specified is typically a cubic grid of Nx x Ny x Nz voxels, which corresponds to real dimensions of Lx — hNx, Ly - hNy, and Lz — hNz, where h is the voxel size. We will further call this region of real space the computational unit cell. The relationship between the unit cell and the multiphase medium of interest depends on the absolute dimensions of the medium and on the spatial resolution at which the medium is represented (feature dimensions). The unit cell can either contain the entire medium and some void space surrounding it, as in the case of virtual granules described in Section IV.D below, or be a sample of a much larger (theoretically infinite) medium, as in the case of transport properties calculation, described in Section II.E below. 

Figure 1 Two-dimensional cross-section of a Vycor glass realization generated by the Gaussitm random field method. The cubic grid size is 10 A.

The PB equation may be solved numerically for macromolecules (for reviews, see References 36-38. The finite difference, finite element, and multigrid methods are used most commonly to solve the PB equation. Usually, this technique is performed by mapping the molecules onto a three-dimensional cubic grid. To solve the PB equation, a suitable interior relative dielectric constant and definition of the dielectric boundary should be assigned (39, 40). 

Consider a given molecular contour surface G(a). If the size s of the cubes is chosen small enough, then any finite polycube P can fit within G(a). As in the two-dimensional case, we do not consider orientation constraints and we assume that the contour surface G(a) and polycube P may be translated and rotated with respect to one another the relative orientation of G(a) and the cubic grid is not fixed. In this model, the identity of a polycube is independent of its orientation. Two polycubes P and F are regarded identical if and only if they can be superimposed on one another by translation and rotation in 3D space. Note, however, that the polycube method of shape analysis and determination of resolution based similarity measures can be augmented with orientation constraints, suitable for the study of molecular recognition and shape problems in external fields or within enzyme cavities [240,243]. 

Many properties are calculated by introducing a theoretical probe molecule or group of atoms at each point in the lattice, and determining the energy of the interaction between the probe and the molecules under study. For example, an ion might be used to determine the electrostatic interaction, or a neutral atom used to determine the intermolecular forces. Such calculations rely on molecular mechanics methods and can be performed extremely rapidly, so that the interaction of every molecule with a probe at each grid point can be ascertained. Suppose a cubic grid ofnx XB points were used and the interaction with two probes at every point calculated. There would now be 2n3 variables to fit to the Molecule in a typical 3D-QSAR... 

Figure 20 Schematic view of the k spectrum sampled on a three-dimensional cubic grid (bottom) and on a skewed grid (top). The Fourier transform of a function / is contained in the sphere pQ. Sampling the function / on a discrete grid produces copies of f(K), each containing a sphere with radius Kmax. These spheres should be distinct for optimal sampling.

Figure 21 Comparison of sampling efficiency of a cubic to a skewed grid as a function of dimension. The stars represent the cubic grid, the open circles the skewed grid.

The standard error obtained from the procedure described above can be diminished, for the same n, by using stratified sampling. The domain is first divided up into Ns subdomains, or strata, according to some prescription for example, the division could be into a cubic grid. Each subdomain is sampled uniformly with nk samples (which will generally not be equal to one another). The integral is then given by a sum of stratum estimates [104,108] ... 

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