Donor cell approach to charge-coupled electrostatics problems

This dissertation discusses work in the donor cell method of solving charge-coupled electrostatic problems. Briefly stated, these are problems in which space charge influences the electric field through Poisson's equation, while the electric field simultaneously serves to drive a current density in the region. An example of such a system is the electrostatic particle precipitator. The objective is to find steady state field and charge density structures that are self-consistent with each other; that is Poisson's equation and current continuity are simultaneously satisfied.; Conservation of charge is imposed by employing the donor cell method, originally developed in the domain of fluid mechanics. The donor cell method converges in many situations where the more typical central differencing algorithms do not. It uses upstream values of the quantity in flux, taken here to be charge carriers, to describe the flux through boundaries of control volumes. The charge conservation equation in integral form is thus easily imposed for each control volume.; The second part of the problem is to solve Poisson's equation for a given charge density. This is accomplished by standard finite element techniques. The triangular mesh is generated by the Delaunay algorithm, which produces as a geometric dual the Voronoi polygons, used as the control volumes mentioned above. Thus the designer has little more to do than specify the problem geometry and operating characteristics.; It may be easily seen that an iterative relaxation loop is formed. Given an initial guess of charge density and appropriate electric boundary conditions, the field structure is evaluated. This field structure then determines the next guess of charge through numerical approximation of the conservation equation. This loop may be traced until a desired convergence criterion is met. An alternative formulation involves quasilinearizing the entire system using Newton's method and solving for all the variables at once. This too is an iterative procedure with attractive convergence properties albeit at higher computer costs.; This formulation is aimed at including physical mechanisms beyond unipolar ion drift. Specifically, bipolar charge migration is considered, and it is shown that including a diffusive term in the definition of current density introduces only a minor complication to the conservation equation. This extension to previous work seems to be the next logical step towards understanding of the interactions between transport phenomena in the context of charge-coupled electrostatics.