technique which caused divergence in the calculation process. All attempts to solve the equation set resulted in an uncontrollable increase in time of the charge density amplitudes. Although the algorithm was tested for single-species modelling (Chapter 5) with the results being reasonable and consistent with the physical expectations, the three-species simulation was not convergent. A thorough search for nding the cause of the problem was conducted.
After doing a detailed investigation of dierent routines of the code and con- rming the solution of the Poisson equation by comparing the code results with a commercial FEM solver (COMSOL Multiphysics), it was found that FEM-FCT al- gorithm used for solving the charge continuity equations was the cause. Two steps of this algorithm were the most critical: the high order and low order solutions. The process of nding the reasons for the lack of convergence and modifying the technique is described below:
1. The high-order solution:
The charge continuity equation for electrons (4.3) can be written in a simpler form as: ∂Ne ∂t +∇ · ⃗ Je= (vi−va)Ne− NeNpβ e0 (4.9)
in which, i e and a e
After discretizing (4.9) the following equation is obtained:
∑ j tij∂Nej ∂t = ∑ j ( cijNej)+∑ j (( viav−vaav−β Npav e0 ) tijNej ) (4.10) in which cij = kij + ( ke ε0 (
Npav−Neav−Nnav
))
tij −Desij, and Npav, Neav, Nnav, viav and vaav refer to the average values of Np, Ne, Nn, vi and va over an element.
To solve this equation, the rst step is to nd the high-order solution related to the transport term by solving:
[T] ( ∂Nei ∂t ) = [K] (Nei) (4.11)
In the previous Chapter, it was explained that in order to avoid using the Gaus- sian elimination technique, the distributed mass matrix[T]can be approximated
by a simpler lumped mass matrix [M] in which mi =∑jtij.
If the same approximation is used for the three species modelling, the numer- ical technique would diverge. The space charge densities will have oscillatory behaviour and will increase uncontrollably up to very large values. The large densities of space charge increase the amplitude of the corona current to very large values as well and the program eventually diverges. The origin of this
To x this, two solutions were proposed:
• Solve the equation: [M] (∂N ei ∂t ) = [K] (Nei) + ([M]−[T]) (∂N ei ∂t ) which can be represented as:
Nein+1−Nein = m1 i
(
∆t[[K] (Nei) + ([M]−[T])(Nein+1−nein)]) In the literature, it is mentioned that the above equation usually leads to a convergent result after three iterations. However, this was not conrmed in the discussed case and a larger number of iterations was required for this purpose.
• Use matrix [T] (instead of [M]) and the Gaussian elimination inversion
technique to directly solve (4.11).
These two approaches were tested and both of them were able to x the con- vergence problems although the second approach was less time-consuming, as the number of iterations needed for the rst approach to converge was rather large (10-12 iterations) and it was much more time-consuming in comparison with the exact solving of the equation set.
2. The low-order solution: Since the high-order solution is oscillatory, this step adds some articial diusion which is needed to remove these oscillations from the charge distributions.
losi=hosi+∆t
mi
∑
j
eliminating all negative o-diagonal elements from the coecient matrix re- moves the oscillations and smoothes the charge distributions. The necessary approach in a case when the coecient matrix ([C]) is negative dominant has
never been discussed in detail.
For negative ions and electrons, the coecient matrix is mostly negative and removing all negative o-diagonal elements will cause convergence problems in these cases. The low-order solution is not supposed to create negative values for charge density distributions; however, during the simulation this step was generating such values. It was suspected that removing the positive o-diagonal elements from the coecient matrix of electrons and negative ions equations would alleviate the problem.
Therefore, the next important step was to verify if changing the coecient ma- trix ([C]), so that all o-diagonal entries are negative for electron and negative
ions would improve the results. This hypothesis has been conrmed and the results were convergent and consistent with the expectations.
3. Limiting procedure: Since in the second step the added diusion is not fully needed, in the at (nonoscillatory) parts of the distributions the extra diusion should be removed. Diusion is only needed in the oscillatory parts and an exact amount of diusion should be left just to remove the oscillations.
mei =losi+ ∆t mi ∑ j ( −αdij(Nej−Nei)) (4.13)
term are calculated, mei. However, to nd the nal distributions, there is still a term related to reactions, ionization, attachment and recombination which needs to be added to the charge densities. Therefore, the nal result for the density of electron at node iwould be:
Nei =mei+ ∆t
mi
∑
j
teijmej (4.14)
in which mei is the calculated electron density resulting from the transport term. Nei is the total electron density obtained from the superposition of the transport term and the reaction terms.