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# Use of Symmetry to Reduce Computations Required to Fill a Matrix Describing Interactions Between Cells on two Rectangular Parallel Planes

IP.com Disclosure Number: IPCOM000045639D
Original Publication Date: 1983-Apr-01
Included in the Prior Art Database: 2005-Feb-07
Document File: 5 page(s) / 71K

IBM

## Related People

Weeks, WT: AUTHOR [+2]

## Abstract

The preceding article deals with the calculation of interactions between cells in the same plane, as part of the computation of LSI package electrical parameters. The present article is an extension of the former and addresses the calculation of interactions between cells on two rectangular parallel planes. Again, symmetry is used to avoid computation of an element having the same value as a previously calculated element.

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Use of Symmetry to Reduce Computations Required to Fill a Matrix Describing Interactions Between Cells on two Rectangular Parallel Planes

The preceding article deals with the calculation of interactions between cells in the same plane, as part of the computation of LSI package electrical parameters. The present article is an extension of the former and addresses the calculation of interactions between cells on two rectangular parallel planes. Again, symmetry is used to avoid computation of an element having the same value as a previously calculated element.

INTRODUCTION.

Consider two parallel rectangular faces oriented so that their corresponding edges are parallel, as shown in Fig. 1a. The two faces need not lie in the same plane. Divide each face into meshes of rectangular cells. Number the cells in the order shown in Fig. 1b.

In Fig. 1b, identify columns of cells, such as 1, 2, 3 or 4, 5, 6 in which consecutive cell numbers differ by one. Also, identify rows of cells, such as 1,4 or 2,5 or 3,6, in which consecutive cell numbers differ by the number of cells in a column. Consider any matrix, P, whose elements describe a physical interaction between cells on one face and cells on the other face. That is, P(ij) represents the interaction between cell i on the first face and cell j on the second face. The number of rows in P is equal to the number of cells on the first face, and the number of columns is equal to the number of cells on the second face. It is important to distinguish between rows and columns of cells on a face and rows and columns of elements in the P matrix.

Because of the geometric symmetry of the two faces and meshes of cell divisions, not all of the matrix elements of P will be different. Many are merely copies of others. This article uses symmetry to avoid the calculation of matrix elements that are equal to previously calculated elements.

In the following it will be assumed that P(ij) is a function only of: (A-1) center to center distance between cells i and j. (A-2) lengths and widths of cells i and j. (A-3) relative orientation of cells i and i.

Many physical interactions satisfy these three conditions.

BASIC SYMMETRY OPERATIONS Consider four basic symmetry operations: 1. I - The Identity Operations. Maps each cell onto itself. 2. C - The Column Reversal Operation Reverses the order of the columns on the faces.

Example : 3. R - The Row Reversal Operation. Reverses the order of the rows on the faces. 4. T - The Transposition Operation. Interchanges corresponding cell numbers on the two faces.

(Unlike I, C, and R, T is applicable

only if both faces have the same number of cells.)

1

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Example :

Operations C and R are similar to the operations V and H, respectively, of the aforementioned preceding article except that C and R operate on pairs of faces while V and H operate on a single face. The operation T is a new addition.

From the four basic operations I, C, R and T it is possible to form compound ope...