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Table-Lookup Algorithm for Solving Problems Involving Nonuniform Grids

IP.com Disclosure Number: IPCOM000099882D
Original Publication Date: 1990-Feb-01
Included in the Prior Art Database: 2005-Mar-15
Document File: 2 page(s) / 87K

Publishing Venue

IBM

Related People

Rubin, BJ: AUTHOR

Abstract

Disclosed is an algorithm for reducing the time required to calculate the matrix elements in physical problems associated with nonuniform grids. Consider problems that require solution of the equation Ax=b, where A is a square matrix of order N and employ lookup tables to eliminate redundant calculation of matrix elements. Each line of such a table includes the value of a unique matrix element and sufficient physical parameters to identify the element for subsequent fill-in of the matrix A. A lookup table associated with a sufficiently fine grid is first generated and then, through appropriate matrix operations, used to fill-in the matrix associated with the nonuniform grid.

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Table-Lookup Algorithm for Solving Problems Involving Nonuniform Grids

       Disclosed is an algorithm for reducing the time required
to calculate the matrix elements in physical problems associated with
nonuniform grids.  Consider problems that require solution of the
equation Ax=b, where A is a square matrix of order N and employ
lookup tables to eliminate redundant calculation of matrix elements.
Each line of such a table includes the value of a unique matrix
element and sufficient physical parameters to identify the element
for subsequent fill-in of the matrix A.  A lookup table associated
with a sufficiently fine grid is first generated and then, through
appropriate matrix operations, used to fill-in the matrix associated
with the nonuniform grid. Because a uniform grid has fewer degrees of
freedom than a nonuniform one, this procedure generally reduces the
size of the lookup table and offers a substantial savings in computer
computations.  An example follows.

      To find the capacitance of a rectangular plate, the plate may
be divided into rectangular patches of constant charge density and a
point matching procedure employed to generate a matrix equation
having charges as the unknowns (*).  Summing these charges gives the
desired capacitance. Consider the plate in the figure, which is
sectioned into a large center rectangle (shown shaded) and 16 smaller
ones along the perimeter.  Each of the 17 rectangles has a constant
charge associated with it.  Such a nonuniform grid formed by these
rectangles leads to a solution with fewer unknown charges than one
where the grid is uniform (rectangles all the same size).  A 17-by-17
matrix equation must be generated.  From physical principles, the
matrix elements depend only on the x and y distances between the
rectangles, and on the sides of the rectangles; only 36 of the 289
matrix elements are unique. Now consider a...