Finite lattice method

The methods of series generation and analysis is particularly powerful in determining lattice dependent properties. For example, the growth constants of S.AW on various lattices is most accurately detennined by series methods. Universal properties, such as critical exponents seem to be equally well determined by high-class Monte Carlo work or high class series work. In two dimensions, where finite-lattice methods (FLM) can be applied [50.64], series methods are extremely powerful, and likely to be better than all but the most exhaustive Monte Carlo calculations. Where interactions become very complicated, or df lon range, so that FLM type methods are difficult or impossible... 

These apparent restrictions in size and length of simulation time of the fully quantum-mechanical methods or molecular-dynamics methods with continuous degrees of freedom in real space are the basic reason why the direct simulation of lattice models of the Ising type or of solid-on-solid type is still the most popular technique to simulate crystal growth processes. Consequently, a substantial part of this article will deal with scientific problems on those time and length scales which are simultaneously accessible by the experimental STM methods on one hand and by Monte Carlo lattice simulations on the other hand. Even these methods, however, are too microscopic to incorporate the boundary conditions from the laboratory set-up into the models in a reahstic way. Therefore one uses phenomenological models of the phase-field or sharp-interface type, and finally even finite-element methods, to treat the diffusion transport and hydrodynamic convections which control a reahstic crystal growth process from the melt on an industrial scale. 

The numerical results reviewed above were obtained for infinite lattices. How do the various quantities of interest behave near the percolation threshold in a large but finite lattice This problem has been studied by renormalization methods, which are essentially equivalent to finite-size scaling. For finite lattices the percolation transition is smeared out over a range of p, and one must expect a similar trend in other functions, including the conductivity. Computer simulations by the Monte Carlo method have been carried out for bond percolation on a three-dimensional simple cubic lattice by Kirkpatrick (1979). Five such experimental curves are shown in Fig. 40, each of which corresponds to a cube of size b, containing bonds. In Fig. 40 the vertical axis gives the fraction p of such samples that percolate (i.e., have opposite faces con-... 

It follows from the results obtained above that the regularities in the statistical properties of a percolation cluster can be studied even at length intervals of the order of the lattice constant. Hence, we shall analyze in greater detail the finite scales method [78] which was implemented on the smallest models. Thus, we separate the set of all possible initial cells in two-dimensional space into classes according to the following criterion A model belongs to class C n) if the difference lx — /, for the model equals n, where n C Z. 

To solve the PB equation for arbitrary geometries requires some type of discretization, to convert the partial differential equation into a set of difference equations. Finite difference methods divide space into a cubic lattice, with the potential, charge density, and ion accessibility defined at the lattice points (or grid points ) and the permittivity defined on the branches (or grid lines ). Equation [1] becomes a system of simultaneous equations referred to as the finite difference Poisson-Boltzmann (FDPB) equation ... 

Sankaranarayanan, K., I.G. Kevrekidis, S. Sundaresan, J. Lu and G. Tryggvason. A Comparative Study of Lattice Boltzmann and Front-Tracking Finite-Difference Methods for Bubble Simulations. Int. J. Multiphase Flow 29 109-116 (2003). 

Complementarily, several numerical approaches have also been proposed either at the molecular scale, using molecular dynamics [17], or at larger mesoscopic scales using finite element methods, lattice-Boltzmann simulations, or phase-field... 

In the following sections we briefly describe a few technical details, the finite lattice algorithm for improving the DMRG accuracy for a particular system size, and extensions of the DMRG method. 

Several different DNS methods have been used to simulate blood flow. These include the finite element method (FEM), boundary element method (BEM), and the lattice Boltzmann method (LBM). For example, Kaazempur-Mofrad et al. " used the FEM to simulate blood flow in an artery. For rough calculation purposes, this may be acceptable. As noted above, however, one has to assume that the blood behaves as a Newtonian fluid to perform such simulations. Moreover, the simulations do not provide information about the spatial distribution of the hematocrit. 

Equation [21] represents the essence of the finite difference method.It shows that the electrostatic potential at each point is linearly related to the potentials at the neighboring points. In the finite difference method, the macro-molecule] s) and a region of the surrounding solvent are mapped onto a cubic lattice each lattice point represents a small region of either the molecule(s) or the solvent. At each point, values for the charge, the dielectric constant, and the ionic strength parameters are assigned for the Poisson-Boltzmann equation, and self-consistent potential values must be found by an iterative method. 

Mori-Tanaka model Kalpin-Tsai model Lattice-spring model Finite element method Equivalent continuum approach Seif-similar approach... 

Consequently, numerical solution of the equations of change has been an important research topic for many decades, both in solid mechanics and in fluid mechanics. Solid mechanics is significantly simpler than fluid mechanics because of the absence of the nonlinear convection term, and the finite element method has become the standard method. In fluid mechanics, however, the finite element method is primarily used for laminar flows, and other methods, such as the finite difference and finite volume methods, are used for both laminar and turbulent flows. The recently developed lattice-Boltzmann method is also being used, primarily in academic circles. All of these methods involve the approximation of the field equations defined over a continuous domain by discrete eqnalions associated with a finite set of discrete points within the domain and specified by the user, directly or through an antomated algorithm. Regardless of the method, the numerical solution of the conservation equations for fluid flow is known as computational fluid dynamics (CFD). 

The last term of Eq. (4.12), q, is the thermal force that satisfied the fiuctuation-dissipation theorem. Equations (4.12) and (4.13) may be solved numerically on a square lattice using a finite difference method vith central difference for spatial steps and forward difference temporal steps. A temperature gradient field can be created between the top and bottom walls. To express the external temperature gradient field, the dimensionless model parameter k may be rescaled with ... 

Rock properties are computed from the micro, to plug and whole core scale. The absolute permeability, for example, can be computed using Lattice-Boltzmann simulations, while the calculation of the formation resistivity factor is based on a solution of the Laplace equation with charge conservation. The elastic properties are calculated with the finite element method. 

However, integrabihty imposes a criterion for obtaining DBs analytically. DBs are obtained analytically for integrable systems, while for non-integrable systems it is obtained by various numerical methods viz. spectral collocation method, finite-difference method, finite element method, Floquet analysis, etc. As evident from many numerical experiments, DBs mobility is achieved by an appropriate perturbation [42]. From the practical application perspective, dissipative DBs are more relevant than their Hamiltonian counterparts. The latter with the character of an attractor for different initial conditions in the corresponding basin of attraction may appear whenever power balance, instead of energy conservation, governs the nonlinear lattice dynamics. The attractor character for dissipative DBs allows for the existence of quasi-periodic and even chaotic DBs [54, 55]. 

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