Abstract
When a straight cylindrical conductor of finite length is electrostatically charged, its electrostatic potential ϕ depends on the electrostatic charge qe, as expressed by the equation L(qe)=ϕ, where L is an integral operator. Method of moments (MoM) is an excellent candidate for solving L(qe)=ϕ numerically. In fact, considering qe as a piece-wise constant over the length of the conductor, it can be expressed as a finite series of weighted basis functions, qe=∑n=1Nαnfn (with weights αn and N, number of the subsections of the conductor) defined in the L domain so that ϕ becomes a finite sum of integrals from which, considering testing functions suitably combined with the basis functions, one obtains an algebraic system Lmnαn=gm with dense matrix, equivalent to L(qe)=ϕ. Once solved, the linear algebraic system gets αn and therefore qe is obtainable so that the electrostatic capacitance C=qe/V, where V is the external electrical tension applied, can give the corresponding electrostatic capacitance. In this paper, a comparison was made among some Krylov subspace method-based procedures to solve Lmnαn=gm. These methods have, as a basic idea, the projection of a problem related to a matrix A∈Rn×n, having a number of non-null elements of the order of n, in a subspace of lower order. This reduces the computational complexity of the algorithms for solving linear algebraic systems in which the matrix is dense. Five cases were identified to determine Lmn according to the type of basis-testing functions pair used. In particular: (1) pulse function as the basis function and delta function as the testing function; (2) pulse function as the basis function as well as testing function; (3) triangular function as the basis function and delta function as the testing function; (4) triangular function as the basis function and pulse function as the testing function; (5) triangular function as the basis function with the Galerkin Procedure. Therefore, five Lmn and five pair qe and C were computed. For each case, for the resolution of Lmnαn=gm obtained, GMRES, CGS, and BicGStab algorithms (based on Krylov subspaces approach) were implemented in the MatLab® Toolbox to evaluate qe and C as N increases, highlighting asymptotical behaviors of the procedures. Then, a particular value for N is obtained, exploiting both the conditioning number of Lmn and considerations on C, to avoid instability phenomena. The performances of the exploited procedures have been evaluated in terms of convergence speed and CPU-times as the length/diameter and N increase. The results show the superiority of BcGStab, compared to the other procedures used, since even if the number of iterations increases significantly, the CPU-time decreases (more than 50%) when the asymptotic behavior of all the procedures is in place. This superiority is much more evident when the CPU-time of BicGStab is compared with that achieved by exploiting Gauss elimination and Gauss–Seidel approaches.
Highlights
As it is well known, electrostatic problems are classified according to the equations that describe them
We present the study carried out that exploits the pulse function as the basis function, and as the testing function, the delta function will be considered as required in the point-matching approach; In the second case, following the pseudo-Galerkin procedure, the pulse function will be exploited as the basis function as well as the testing function; In the third case, the triangular function will be exploited as the basis function, while the delta function will be exploited as the testing function; In the fourth case, the triangular function will be considered the basis function while the pulse function will be exploited as the testing function; 5
The third approach leads to the bi-conjugate gradient (Bi-Conjugate Gradient (CG)) and quasi-minimal residual (QMR) methods while the fourth is not immediate, but for A = A T, it leads to the symmetric LQ (SYMMLQ) method
Summary
As it is well known, electrostatic problems are classified according to the equations that describe them. These equations could be differential (ordinary or partial derivative), integral, or both. For solving the problem (1), it is usually transformed into an equivalent algebraic problem Lα = g, (2). In which L is an algebraic matrix [2,3,4] As it is well known, modern computers have created the conditions for developing numerical resolution methods for a wide class of problems lacking an analytical solution. Among them is the MoM which, initially introduced by Harrington [1,2], solves (1), transforming it into an equivalent algebraic formulation
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