Volume-based numerical method

At the same time it is worth to notice that in modern numerical methods of a solution of boundary value problems, based on replacement of differential equations by finite difference, these steps are performed simultaneously. In accordance with the theorem of uniqueness, the field inside the volume V is defined by a distribution of masses inside this volume and boundary conditions, and correspondingly it is natural to derive an equation establishing this link. With this purpose in mind we will again proceed from Gauss s theorem,... 

The computational code used in solving the hydrodynamic equation is developed based on the CFDLIB, a finite-volume hydro-code using a common data structure and a common numerical method (Kashiwa et al., 1994). An explicit time-marching, cell-centered Implicit Continuous-fluid Eulerian (ICE) numerical technique is employed to solve the governing equations (Amsden and Harlow, 1968). The computation cycle is split to two distinct phases a Lagrangian phase and a remapping phase, in which the Arbitrary Lagrangian Eulerian (ALE) technique is applied to support the arbitrary mesh motion with fluid flow. 

For most numerically solved models, a control-volume approach is used. This approach is based on dividing the modeling domain into a mesh. Between mesh points, there are finite elements or boxes. Using Taylor series expansions, the governing equations are cast in finite-difference form. Next, the equations for the two half-boxes on either side of a mesh point are set equal to each other hence, mass is rigorously conserved. This approach requires that all vectors be defined at half-mesh points, all scalars at full-mesh points, and all reaction rates at quarter-mesh points. The exact details of the numerical methods can be found elsewhere (for example, see ref 273) and are not the purview of this review article. The above approach is essentially the same as that used in CFD packages (e.g.. Fluent) or discussed in Appendix C of ref 139 and is related to other numerical methods applied to fuel-cell modeling. ... 

Transistors based on a-Si H do not follow the Borkan-Weimer equation (Borkan and Weimer, 1963) since the characteristics of an a-Si H transistor depend on the density of states in the gap. For an exponential density of tail state distribution, appropriate theories were formulated as early as 1975 (Neudeck and Malhotra, 1975, 1976) that have subsequently been developed further (Kishida et al., 1983). If the density of states varies in a nonexponential manner, numerical methods must be used to derive the transistor characteristics. The inverse problem, a derivation of the density of states from field-effect measurements, is discussed in Chapter 2 by Cohen of Volume 21C. 

To numerically solve equations of the above mathematical models, the general computational gas dynamics is adopted in the present work. The general differential equations (2.7) and (2.31) are then discretized by the control volume-based finite difference method, and the resulting set of algebraic equations is iteratively solved. The numerical solver for the general differential equations can be repeatedly appUed for each scale variable over a controlled volume mesh. This process must be conducted extremely carefully to avoid the influence of slight changes in the accuracy of discretization. 

When the governing model is given by the convection-diffusion equation (no electrical migration effects are considered), well-established numerical methods can be used directly in electrochemical cell design. When using commercial software, it should be remembered that the code has probably been benchmarked for applications different from those found in metallization, where spatial distributions of flux at high Schmidt numbers may be of more interest than the spatial average flux. Freitas has recently provided a comparison of several commercial CFD codes. Many of these codes are based on a finite-volume method (FVM) or a finite-element method. West jj yg discussed the application of... 

The kinetic equation (11.1) is a nonlinear integro-differential equation, general theory of which does not exist. Its known exact solutions are based on the use of operational methods with reference to a case of linear dependence K(V,w) on each of drop volumes [2]. To solve the equation (11.1) with more general kernel, the approximate methods are used - parametric methods and method of moments, and also numerical methods. Parametric methods and method of moments are based on transforming the kinetic equation into a system of equations for the moments of drop distribution over volumes. However, the resulting system of equations is, as a rule, incomplete, since, apart from the integer moments,... 

Implicit solvation models developed for condensed phases represent the solvent by a continuous electric field, and are based on the Poisson equation, which is valid when a surrounding dielectric medium responds linearly to the charge distribution of the solute. The Poisson equation is actually a special case of the Poisson-Boltzmann (PB) equation PB electrostatics applies when electrolytes are present in solution, while the Poisson equation applies when no ions are present. Solving the Poisson equation for an arbitrary equation requires numerical methods, and many researchers have developed an alternative way to approximate the Poisson equation that can be solved analytically, known as the Generalized Born (GB) approach. The most common implicit models used for small molecules are the Conductor-like Screening Model (COSMO) [96,97], the Dielectric Polarized Continuum Model (DPCM) [98], the Conductor-like modification to the Polarized Continuum Model (CPCM) [99], the Integral Equation Formalism implementation of PCM (lEF-PCM) [100] PB models and the GB SMx models of Cramer and Truhlar [52,57,101,102]. The newest Miimesota solvation models are the SMD (universal Solvation Model based on solute electron Density [57]) and the SMLVE method, which combines the surface and volume polarization for electrostatic interactions model (SVPE) [103-105] with semiempirical terms that account for local electrostatics [106]. Further details on these methods can be found in Chapter 11 of reference 52. 

Up to now, a vast number of studies on micro-and nanofluidics have been performed based on the theoretical, numerical, and experimental approaches. For the numerical analysis, however, finite-volume or finite-difference schemes have been mosfly used. In the future, spectral methods will contribute more to the studies on micro- and nanofluidics that require numerical methods with high spatial accuracy, for example, electrokinetic and capillary effects. 

Kietzmann C, Van der Walt JP, Morsi YS (1998) A free-front tracking algorithm for a control-volume-based Hele-Shaw method. Int J Numer Methods Eng 41 253-269 Kim SH, Kim CH, Oh H, Choi CH, Kim BY, Youn JR (2007) Residual stresses and viscoelastic deformation of an injection molded automotive part. Korea-Australia Rheol J 19 183-190 Klein DH, Leal LG, Garcfa-Cervera CJ, Ceniceros HD (2008) Three-dimensional shear-driven dynamics of polydomain textures and disclination loops in liquid crystalline polymers. J Rheol 52 837-863... 

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