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Rapid Evaluation of Multiple Integrals Over All Cells in a Mesh

IP.com Disclosure Number: IPCOM000043408D
Original Publication Date: 1984-Aug-01
Included in the Prior Art Database: 2005-Feb-04
Document File: 3 page(s) / 22K

Publishing Venue

IBM

Related People

Weeks, WT: AUTHOR [+2]

Abstract

A significant reduction is achieved in the time required for the computer evaluation of multiple integrals over all circuit cells in a mesh of cells by the use of an algorithm for performing multiple integrals at the grid (mesh) vertices of all the cells being considered. Let x1, x2, ..., xn be a system of n orthogonal curvilinear coordinates, and consider the mesh of cells generated by the constant coordinate surfaces. xk = ak (ik) = constant wherek = 1, 2, ..., n and ik = 0, 1, ..., Nk . Suppose it is required to evaluate the integral (Image Omitted) for every cell in the mesh, as is often the case in the numerical solution of integral equations or in the application of the finite element method. If a function f(x1,x2,...,xn) can be found such that which is a well-known result.

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Rapid Evaluation of Multiple Integrals Over All Cells in a Mesh

A significant reduction is achieved in the time required for the computer evaluation of multiple integrals over all circuit cells in a mesh of cells by the use of an algorithm for performing multiple integrals at the grid (mesh) vertices of all the cells being considered. Let x1, x2, ..., xn be a system of n orthogonal curvilinear coordinates, and consider the mesh of cells generated by the constant coordinate surfaces. xk = ak (ik) = constant wherek = 1, 2, ..., n and ik = 0, 1, ..., Nk . Suppose it is required to evaluate the integral

(Image Omitted)

for every cell in the mesh, as is often the case in the numerical solution of integral equations or in the application of the finite element method. If a function f(x1,x2,...,xn) can be found such that which is a well-known result. The traditional method of evaluation is to apply equation (3) cell by cell computing each value of f as needed. Let T be the mean time required to evaluate the function, f. The evaluation of equation (3) for one cell requires 2n evaluations of f.

The number of cells for which equation (3) must be evaluated is

(Image Omitted)

where Ni is the number of cell divisions along the i-th coordinate direction. Therefore, the time required to evaluate equation (3) for all cells in the mesh is

(Image Omitted)

This method of evaluation is inefficient.

If a1(j1), a2(j2), an(jn) is an interior vertex of the mesh, it is shared by 2n adjacent cells. Therefore, f(a1(j1), a2(j2), ..., an(jn)) will be evaluated, and re- evaluated, a total of 2n times, once for each of the adjacent cells which share...