Current-density approximation for efficient computation of the electrostatic field in wire-plate precipitators

The problem is defined by a reduced set of the Maxwell equations. The efficiency of numerical iterative computations is significantly improved by deriving an initial field as close as possible to the final solution from an approximation of the current density field J. Different techniques to approximate J are proposed. A first analytical approximation is derived, which verifies by construction the boundary conditions of the problem, and, in particular, gives the correct average value at the plate. A second analytical approximation is also considered, which contains a free parameter that can be computed by an optimization procedure based on the known value of the potential at the wire. Finally, proper orthogonal decomposition (POD) is used and the current density field is expressed. The coefficients can be determined again by an optimization algorithm. Starting from these approximated J fields, a procedure is proposed to obtain an estimate of the complete electrostatic field. It is shown that this estimate, which is obtained at negligible computational cost, is in all cases much closer to the exact solution than guesses typically employed in the literature. Hence, it can be used as initialization for standard numerical solvers, leading to a significant gain in the efficiency of the numerical algorithm, especially when the POD decomposition of J is considered.