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Minimizing Computations Required to Fill a Matrix Describing Relation Ships Between Cells on the Faces of a Rectangular Bar

IP.com Disclosure Number: IPCOM000043785D
Original Publication Date: 1984-Sep-01
Included in the Prior Art Database: 2005-Feb-05
Document File: 3 page(s) / 37K

Publishing Venue

IBM

Related People

Weeks, WT: AUTHOR [+2]

Abstract

This article discloses an alternative method for calculating the potential coefficients required to fill a potential coefficient matrix, as employed in the calculation of the capacitance of a rectangular bar. The disclosed solution to this problem reduces the computation time required to fill the coefficient matrix by one third to one half over the usual approaches employed in the literature. Typically, this involves dividing the faces of the bar into grids of rectangular cells, letting Qi be the charge on the i-th cell, and solving the system of equations shown in the equation below for the Qi, given the values of (Image Omitted) Vi . Before this can be done, however, the potential coefficients, Pij, must be calculated.

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Minimizing Computations Required to Fill a Matrix Describing Relation Ships Between Cells on the Faces of a Rectangular Bar

This article discloses an alternative method for calculating the potential coefficients required to fill a potential coefficient matrix, as employed in the calculation of the capacitance of a rectangular bar. The disclosed solution to this problem reduces the computation time required to fill the coefficient matrix by one third to one half over the usual approaches employed in the literature. Typically, this involves dividing the faces of the bar into grids of rectangular cells, letting Qi be the charge on the i-th cell, and solving the system of equations shown in the equation below for the Qi, given the values of

(Image Omitted)

Vi . Before this can be done, however, the potential coefficients, Pij, must be calculated. This is done by averaging the values of a Greens' function (often a truncated infinite series) over the cells i and j, divided up among the faces of the rectangular bar shown in Fig. 1. The

(Image Omitted)

matrix in the equation develops from the introduction of a system of rectangular coordinates illustrated in the figure so that each face of the rectangular bar is parallel to one of the coordinate planes. The face names R, L, B, T, F and P obtain as shown thereon, and the potential coefficients Pij form a symmetric matrix, i.e., Pij = Pji for all i and j, thereby making it necessary only to evaluate the upper triangle in the matrix in the equation and eliminating redundant evaluations which contribute to increased computation time. The upper triangle of the Pij matrix can be ordered so that 21 face-to-face submatrices can be identified as shown in Fig. 2. The 21 submatrices are denoted by P(F1,F2), etc., where F1 and F2 can be any of the face names indicated in Fig. 1, e.g., R, L, B, T, F and P. For example, submatrix 15 in Fig. 2 will be called P(P,L). It will be assumed that a complete submatrix, P(F1,F2), will be calculated in a buffer called BUF(Fig. 2) and then copied into the proper...