Finite difference method complex geometries

With the development of high-speed personal computers, it is very convenient to use numerical techniques to solve heat transfer problems. The finite-difference method and the finite-element method are two popular and useful methods. The finite-element method is not as direct, conceptually, as the finite-difference method. It has some advantages over the finite-difference method in solving heat transfer problems, especially for problems with complex geometries. 

One major advantage of finite element methods over finite difference methods is the way they naturally incorporate non-uniform meshes. They can therefore be applied to problems with a complex geometry (Stevens et al, 1997), for example elevated and recessed electrodes (Ferrigno et ah, 1997), and, in principle, simulation of rough electrodes. On the downside, finite element methods are more complex to program, especially when simulating chemical steps, and result in a linear system of equations which is not neatly banded. 

The complexity of the modelling method may influence the choice of geometry. This is summarized in Table 9, together with suggested finite difference methods which should provide efficient simulations. 

K. Asami, Dielectric dispersion in biological cells of complex geometry simulated by the three-dimensional finite difference method. J. Phys. D Appl. Phys., 39, 492-499 (2006). 

Because of the complex geometries and the nonlinear boundary conditions involved in the current distribution problems, there are few analytical solutions. The primary concurrent distribution profiles for various geometries have been calculated and tabulated in an excellent series of papers by Kojima [8,9] and Klingert et al. [10]. Prentice and Tobias [11] reviewed current distribution problems solved by numerical methods in the literature. The finite difference method and the finite element method are widely used for determining current distribution profiles. 

Model 3-num (Numerical Approach). Finding analytical solutions for PDEs, such as eq 2, becomes difficult if complex geometries or special boundary conditions are involved. An alternative is to solve such equations nmnerically. We used the finite-difference method (FDM)... 

Fulian et al. [75] introduced the boundary element method (BEM) for the niunerical solution of SECM diffusion problems and showed that it is more suitable for problems with regions of complex or rapidly changing geometries than finite difference methods employed in earlier SECM publications. The BEM was used to simulate current responses for a nondisk tip approaching a flat substrate a disk-shaped tip over a hemispherical or a sphere-cap substrate, or a tilted substrate a lateral scan of a disk-shaped tip over an insulating/conductive boundary. 

The advantage of the finite difference method is the simple computer implementation of the procedures and thus, it is easy to write ovra codes and to implement or consider new features. The drawback can be the consideration of the boundary conditions for complex shaped geometries and the consideration of the symmetry of the stiffiiess matrix might be difficult. Thus, many applications of the finite difference method are restricted to simple geometries. To overcome these problems, the so called finite difference energy method was developed (Bushnell et al. 1971) where the displacement derivatives in the total potential energy of a system are approximated by finite differences and the minimum condition of the potential energy is used to calculate the unknown displacements. 

This chapter focuses upon real-space methods, in which a computational grid is overlaid upon the domain. The BVP is then converted into a set of ODEs for a time-dependent problem or a set of algebraic equations for a steady problem. This technique can be used even whenno analytical solution exists, and can be extended to BVPs with multiple equations or complex domain geometries. Here, the focus is upon the methods of finite differences, finite volumes, and finite elements. These methods have many characteristics in common therefore, particular attention is paid to the finite difference method, as it is the easiest to code. The finite volume and finite element methods also are discussed however, as the reader is most likely to use these in ftie context of prewritten software, the emphasis is upon conceptual understanding as opposed to implementation. 

Above, our focus has been on the finite difference method, which is easy to implement in domains of rather simple geometry. In complex domains, it is difficult to place a grid and keep track of neighbors when the grid points are required to he along the coordinate axes. Here, we discuss another method that is not subject to this condition. We again consider the 2-D Poisson equation but now instead of the microscopic equation... 

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. 

Commercially available CFD codes use one of the three basic spatial discretization methods finite differences (FD), finite volumes (FV), or finite elements (FE). Earlier CFD codes used FD or FV methods and have been used in stress and flow problems. The major disadvantage of the FD method is that it is limited to structured grids, which are hard to apply to complex geometries and... 

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