Finite Cell Current, Numerical Solution

In this section, it is assumed that 1. As discussed in the section Large s Jq  

The steady-state distribution of overpotential is given by Equation 4.121. With 

An analytical solution to Equation 5.111 is hopelessly cumbersome. A numerical solution to the complex Equation 5.111 can be obtained if one substitutes ) = re + In equation and separates equations for the real and imaginary 

It is advisable to calculate the static differential charge transfer resistivity Ra of the CCL. Differentiating Equations 4.130 and 4.133 overyb yields  

in both cases, the charge transfer resistivity is inversely proportional to the cell current density. However, the numerator in Equation 5.112 changes from 1 to 2 as the current increases. 

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Fixed Coordinate Approaches. In the fixed coordinate approach to airshed modeling, the airshed is divided into a three-dimensional grid for the numerical solution of some form of (7), the specific form depending upon the simplifying assumptions made. We classify the general methods for solution of the continuity equations by conventional finite difference methods, particle in cell methods, and variational methods. Finite difference methods and particle in cell methods are discussed here. Variational methods involve assuming the form of the concentration distribution, usually in terms of an expansion of known functions, and evaluating coeflBcients in the expansion. There is currently active interest in the application of these techniques (23) however, they are not yet suflBciently well developed that they may be applied to the solution of three-dimensional time-dependent partial differential equations, such as (7). For this reason we will not discuss these methods here. 

The modeling of the current distribution in a general-geometry cell nearly always requires a numerical solution. The following discussion focuses on the thin boundary layer approximation, with the overpotential components lumped within a thin boundary layer which may be of a varying thickness. The Laplace equation for the potential with nonlinear boundary conditions must be solved. Similar considerations typically apply to the more comprehensive solution of the Nernst-Planck equation (10) however, the need to account for the convective fluid flow in the latter case makes the application of the boundary methods more complex. We focus our brief discussion on the most common methods the finite-difference method, the finite-element method, and the boundary-element method, schematically depicted in Fig. 4. Since the finite-difference method is the simplest to implement and the best known technique, it is discussed in somewhat more detail. 

For numerical investigations of stress localizations in laminates, the discretizational effort can be reduced significantly if only the boundary needs to be discretized, as it is for e -ample the case in the classical boundary element method (BEM). But in this method a fundamental solution is needed which is in many cases difficult to achieve or even unknown. The Boundary Finite Element Method (BFEM) to be presented here does not require such a fundamental solution, because the element formulation is based on the finite element method (FEM), Thus the BFEM can be characterized to be a finite element based boundary discretization method. This method was originally developed from Wolf and Song [10] under the name Consistent Finite Element Cell Method for time-dependent problems in soil-mechanics. The basic assumption of this method is that a stiffness matrix describing the force-displacement relation at discrete degrees of freedom at the boundary of the continuum is scalable with respect to one point in three-dimensional space, the so-called similarity center, if similar contours within the continuum are considered. In contrast to this, the current work deals with the case of equivalent cross-sectional properties, i.e., that cross-sections parallel to the boundary can be described by the same stiffness matrix, which is the appropriate formulation for the case of the free-edge effect and the matrix crack problem. The boundary stiffness matrix results from a Matrix-Riccati equation. The field quantities inside of the continuum can be calculated from an ordinary differential equation. 

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