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Obtaining Finite Difference Formulas for Neumann Boundary Conditions along Irregularly Shaped Boundaries

IP.com Disclosure Number: IPCOM000076405D
Original Publication Date: 1972-Feb-01
Included in the Prior Art Database: 2005-Feb-24
Document File: 4 page(s) / 36K

Publishing Venue

IBM

Related People

Alsop, LE: AUTHOR [+2]

Abstract

A method to obtain finite difference formulas for Neumann boundary conditions along irregularly shaped boundaries is described. Some description of Friedrichs and Keller's tent function (Numerical Solutions of Partial Differential Equations, Academic Press, New York) and the finite method, as presented in Zienkiewiz (The Finite Element Method in Structural and Continuum Mechanics, McGraw-Hill, London, 1967, Chapters 1-3), is necessary to provide the context in which to explain the method.

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Obtaining Finite Difference Formulas for Neumann Boundary Conditions along Irregularly Shaped Boundaries

A method to obtain finite difference formulas for Neumann boundary conditions along irregularly shaped boundaries is described.

Some description of Friedrichs and Keller's tent function (Numerical Solutions of Partial Differential Equations, Academic Press, New York) and the finite method, as presented in Zienkiewiz (The Finite Element Method in Structural and Continuum Mechanics, McGraw-Hill, London, 1967, Chapters 1-3), is necessary to provide the context in which to explain the method.

Let a rectangular grid in the x,z plane be given with the diagonals from lower left to upper right drawn in to yield a triangularization of the plane. The quantities to be calculated are the nodal displacements in a region R of the plane with boundary beta on which Neumann conditions are given. Assume that within each triangle the displacements are a linear function of the displacements at the vertices of the triangle. If one imagines the displacements to be vertical displacements, then the plane of the triangle is displaced to a plane passing through the displaced vertices.

Next, consider the planes formed by partial displacements such that the displacement at any two nodes is set equal to zero, while the displacement at the remaining node assumes its previous value. There are three such planes. The sum of the partial displacements is the original displacement, since the sum must be linear and has the same values at three points, the vertices.

An alternate formulation is obtained using Friedrichs and Keller's tent function, shown in Fig. 1. The function is identically zero everywhere, except in the six triangles numbered with Roman numerals. The central mesh point is numbered zero, while the six outside mesh points are numbered in a counterclockwise sense starting at the positive x axis. The ten function is a continuous function made up of linear functions defined over the six triangles, such that the value of every function is unity at the central mesh point and zero at the outside mesh points. Assume similar functions centered about the other mesh points. Then the three tent functions centered about the points 0, 1, and 2 and no others will have nonzero values over triangle I. Thus the displacement at a point (x,z) of the triangular element I may be written as

(Image Omitted)

where delta i is the displacement at the apex i and T(i)(x z) is the value of the tent function about the apex i at the point (x,z).

Friedrichs and Keller used these tent functions centered on the grid points as trial functions for solving elliptic partial differential equations by the Ritz method. These functions are used herein in the usual engineering procedure for deriving stiffness matrices, since the method for handling irregular boundaries is more easily seen in this way.

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Following Zienkiewicz, the B matrix is first obtained, which when multiplied ...