# NUMERICAL SOLUTION OF A MODEL PROBLEM FROM COLLAPSE LOAD ANALYSIS

Original Publication Date: 1984-Dec-31

Included in the Prior Art Database: 2005-Sep-15

## Publishing Venue

Software Patent Institute

## Related People

Michael L. Overton: AUTHOR [+2]

## Abstract

In collapse load analysis the following problem arises (see Strang min f ovua u Q subject to the Dirichlet boundary conditions [Equation ommitted] on and the constraint [Equation ommitted] Here 1lvull denotes the Euclidean norm of the gradient of [Equation ommitted] We shall restrict n to be a square with side two. The functions f xandy care usually smooth functions, defined on an and n respectively. Sometimes the con-straint (l.lc) does not appear, but this is required to obtain a nontrivial. solution if, for example, [Equation ommitted] . In order for a minimum to be achieved, the appropriate space of functions to consider is BV, the functions of bounded varia-tion. Strang shows, using a result of Fleming [9], that when [Equation ommitted] the solution u(x,y) is a characteristic function with a jump disconti-nuity along a curve r in n. Specifically, u(x,y) has constant positive value inside r and zero value outside r in the following figure, where r is the solid curve and the broken lines represent an: The curve r consists of four circular arcs of radius [Equation ommitted] , plus parts of the sides of the square. Once it is recognized that the solution is a characteristic function, (1.1) reduces to a classical isoperimetric problem, since the quantity to be minimized reduces to measuring the length of r while the constraint fixes the area inside r. The solution therefore defines the region in the square with minimal perimeter given fixed area (maximal area given fixed perimeter). In this paper we are concerned with obtaining the numerical solution of a discrete approximation to (1.1). Let us triangulate n as follows: [Equation ommitted]

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__Page 1 of 15__THIS DOCUMENT IS AN APPROXIMATE REPRESENTATION OF THE ORIGINAL.

**NUMERICAL SOLUTION OF A MODEL PROBLEM FROM COLLAPSE
LOAD ANALYSIS* **

By

Michael L. Overton

February, 1984

Report 4100 *This paper will appear in the Proceedings of the Sixth International Symposium on `Computing Methods in Engineering and Applied Sciences", sponsored by the Institut National de Recherche en Informatique et en Automatique (INRTA), Versailles, France, December 12- 16, 1983, to be published by North-Holland (Amsterdam, 1984). NUMERICAL SOLUTION,OF A MODEL PROBLEM FROM COLLAPSE LOAD ANALYSIS Michael L. Overton

Courant Institute of Mathematical Sciences New York University New York, N. Y. 10012 U.S.A.

We consider a model problem from collapse load analysis discussed recently by Strang. The analytic solution is a characteristic function with a jump discontinuity. We develop a method for solving a discretized version of the model problem, which requires the minimization of a convex piecewise differentiable function. Numerical results are presented.

**INTRODUCTION **

In collapse load analysis the following problem arises (see Strang min f ovua u Q subject to the Dirichlet boundary conditions

(Equation Omitted)

on and the constraint

(Equation Omitted)

Here 1lvull denotes the Euclidean norm of the gradient of

(Equation Omitted)

We shall restrict n to be a square with side two. The functions f xandy care usually smooth functions, defined on an and n respectively. Sometimes the con-straint (l.lc) does not appear, but this is required to obtain a nontrivial. solution if, for example,

(Equation Omitted)

. In order for a minimum to be achieved, the appropriate space of functions to consider is BV, the functions of bounded varia-tion. Strang shows, using a result of Fleming [9], that when

(Equation Omitted)

New York University Page 1 Dec 31, 1984

__Page 2 of 15__NUMERICAL SOLUTION OF A MODEL PROBLEM FROM COLLAPSE LOAD ANALYSIS

the solution u(x,y) is a characteristic function with a jump disconti-nuity along a curve r in n. Specifically, u(x,y) has constant positive value inside r and zero value outside r in the following figure, where r is the solid curve and the broken lines represent an: The curve r consists of four circular arcs of radius

(Equation Omitted)

, plus parts of the sides of the square. Once it is recognized that the solution is a characteristic function, (1.1) reduces to a classical isoperimetric problem, since the quantity to be minimized reduces to measuring the length of r while the constraint fixes the area inside r. The solution therefore defines the region in the square with minimal perimeter given fixed area (maximal area given fixed perimeter).

In this paper we are concerned with obtaining the numerical solution of a discrete approximation to (1.1). Let us triangulate n as follows:

(Equation Omitted)

Let

(Equation Omitted)

be the mesh size, where there are N mesh points in each direc-tion. We replace a in (1.1) by a piecewise linear finite element appro...