Application of conjugate gradient method for static problems involving conductors of arbitrary shape

In this paper, two implementations of the Conjugate Gradient Method (CGM) for the solution of problems in electrostatics involving conductors of arbitrary shapes have been discussed. The first method uses a least squares approximation for the computation of the pertinent integral operator and is referred to as LSD. A second implementation, referred to as Point Matching Discretisation (PMD) effects considerable saving in the computer time since it uses the midpoint rule for the integral arising in LSD. Both these techniques require O(N) storage, where N is the number of patches used to model the conductor. Further, a matrix interpretation of the present formulation has been derived. This has facilitated the comparison of the techniques described in this paper with the well known Method of Moments (MoM) formulation and has led to better understanding of the convergence of the results. Using illustrative examples of canonical (square and circular discs) and arbitrary shape (a pyramid mounted on a cube), the convergence of and the computer time for both the implementations have been investigated. It has been shown that both the techniques yielded monotonically convergent results for all the examples considered and that the LSD offers better estimate of the capacitance than PMD with lower number of patches