1. DELIMITACIÓN DEL PROBLEMA
4.3 MARCO TEÓRICO
In gas discharge problems, the Poisson equation needs to be solved to calculate the electric potential as well as the electric eld. To date, many dierent techniques have been proposed for solving this equation and can be classied into two dierent categories: integral-based and dierential-based.
Boundary Element Method (BEM) and Charge Simulation Method (CSM) are two examples of integral-based techniques; Finite Dierence Method (FDM), Fi- nite Element Method (FEM), and Finite Volume Method (FVM) are examples of dierential-based techniques used for solving this equation. A brief comparison of these two categories is shown in Table 3.1.
The rst proposed technique for calculating the electric eld was FDM [39]. In this technique, the derivatives in the partial dierential equation are replaced with nite dierences. The whole region is discretized and a mesh is formed. The nite dierence approximation is then applied to every node of a mesh. As a result, the dierential equations are transformed to linear algebraic equations. Simplicity of this technique is one of the advantages of FDM; however, since this technique needs a rectangular mesh, it is dicult to apply FDM to problems for irregular domains. For problems with complicated geometry, FDM must use a large mesh which will be time
consuming. Another disadvantage of this technique is that it can not handle the sharp geometry of the discharge electrode very well [39]; therefore, using this technique for modelling corona discharge is not recommended.
FEM was the next proposed technique and eventually it became a dominant technique for solving Laplace and Poisson equations [40]-[43]. FEM is based on mini- mizing the energy of the system instead on direct solution of the equations. Therefore, it can determine the energy related parameters with a much better accuracy. To use FEM, the whole domain should be discretized into a set of triangular, quadrilateral or other type of elements depending on the problem conguration. A simple ma- trix equation for each element is then obtained by minimizing a properly formulated functional, which can be obtained from the variational principle. By assembling these matrix equations, a global set of algebraic equations can be formulated. After intro- ducing the boundary conditions, this global set of equations is solved to obtain the values of the unknown functions at each node. A detailed description of this technique will be presented later in this Chapter.
The advantage of FEM over FDM is its ability in solving problems with com- plicated congurations. It can use unstructured grids as compared with structured grids required for FDM which results in a smaller mesh and makes the calculations less time consuming. By increasing the number of elements or the order of interpolating polynomials in each element, accuracy of the technique can be easily improved.
Singer and Steinbigler [44] introduced CSM, the rst integral-based technique for the calculation of high voltage elds and applied it to two and 3D elds with rotational symmetry. Since then, many renements to the original method have been proposed by Horenstein [45], Castle [46], Malik [47] and Elmoursi [48], and the CSM
has been successfully used to solve a variety of eld problems.
This technique is based on the fact that the electric eld in a considered domain can be uniquely determined by the problem boundary conditions. Therefore, the original problem can be replaced by introducing a set of ctional charges that match the boundary conditions. In this technique, some number of point charges are used instead of continuous surface charge density on electrodes [47]. Values of the discrete charges are determined in order to satisfy the boundary conditions of the original problem. Knowing the values and positions of these substitute charges, the potential and eld distribution can be easily computed anywhere in the region. If the charges are set properly by the user, CSM can be fast and accurate. However, since there is no specic rule for setting these charges, nding their best number, value and location is not easy. This technique is not suitable for problems with complicated geometries or in the presence of several dierent materials in the domain.
BEM was the next integral-based technique proposed for solving the Poisson equation [49]-[52]. Similarly to CSM, BEM introduces a set of eld sources but they are located on the boundaries. These sources are selected so that the continuity and boundary conditions are satised. The advantage of BEM over CSM is that in BEM the charges are on the surface of the electrode while in CSM the user should locate the charges somewhere inside the electrode and the accuracy of the technique is dependent on these locations. The main disadvantage of BEM is that it is time consuming for problems with space charge and it is impossible to use it for non-linear problems.
FVM is a more recent technique, proposed for corona discharge problems [52]. This technique also converts partial dierential equations into algebraic equations.
Similarly to the FDM, values of unknown solution are calculated at discrete nodes of a mesh. Finite volume refers to the small volume surrounding each node. In the FVM, the volume integrals in a partial dierential equation containing diver- gence terms are converted to surface integrals using the divergence theorem. These terms are then evaluated as uxes at the surfaces of each nite volume. Because the ux entering a given volume is identical to that leaving the adjacent volume, these methods are conservative. Another advantage of the FVM is that it can be easily formulated for both structured and unstructured meshes. This method is used in many computational uid dynamics packages.
Since each of these techniques has its own advantages and disadvantages, it is useful to combine dierent methods in one algorithm. The integral-based and dier- ential based techniques have some complementary characteristics; therefore, hybrid techniques which take the advantage of the individual techniques are proposed by many researchers. As an example, combination of BEM-FEM is used by Adamiak and Atten [53]. In their approach, FEM was used to calculate the Poissonian and BEM the Laplacian components of the electric eld.