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Tracking Structural Optima as a Function of Available Resources by a Homotopy Method

IP.com Disclosure Number: IPCOM000128717D
Original Publication Date: 1988-Dec-31
Included in the Prior Art Database: 2005-Sep-16
Document File: 10 page(s) / 35K

Publishing Venue

Software Patent Institute

Related People

Yung S. Shin: AUTHOR [+6]

Abstract

Optimization problems are typically solved by starting with an initial estimate and proceed-ing iteratively to improve it until the optimum is found.. The design points along the path from the initial estimate to the optimum are usually of no value. However, this need not be the case. In many applications, it is of interest to find the family of optima obtained by varying an input parameter such as the amount of available resources. If one member of the family is known, it may be possible to use it as a starting point and to follow an optimization path that goes through the other members of the family. A first step in tracing a family of optima is the application of sensitivity information to extrapolate from one member of the family to another. The present paper proposes the use of the sensitivity information to formulate the path of optima as the trajectory of a differential equation, a procedure known as a homotopy technique. The basic theory of globally convergent (convergent from an arbitrary starting point) hornotopy methods was developed in 1976 [l, 2',]. Since then, the method has been used in a wide range of scientific and engineering problems. It has been successfully applied to nonlinear complementarity problems (3], nonlinear two-point boundary value problems [4], fluid dynamics problems [5, 6], and nonlinear elastica problems (7, 8]. References [9,10] show * This work was supported by NASA grant NAG-1-168 and AFOSR grant 85-0250. 0045-7825/88/$3.50 C~ 1988, Elsevier Science Publishers BY. (North-Holland) <<...>> the application to optimum structural design problems discretized by plane stress finite elements. Reference [9] shows that an appropriate homotopy method is globally convergent i for an optimum design problem. For nonconvex problems the global convergence may be to = . v only a local optimum design. In this paper, the original globally convergent homotopy method is adapted to the design of an elastic foundation for maximizing the buckling load of a column. This problem has been solved before [11] for a limited range of resource (i.e., total foundation stiffness). The present paper shows how the solution process can start from the minimum amount of resources which is required for a feasible solution to the highest value that may be of interest.

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THIS DOCUMENT IS AN APPROXIMATE REPRESENTATION OF THE ORIGINAL.

Tracking Structural Optima as a Function of Available Resources by a Homotopy Method

Yung S. Shin, Raphael T. Haftka, Layne T. Watson and Raymond H. Plaut

TR 88 COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING 70 (1988) 151-164 NORTH-HOLLAND

TRACING STRUCTURAL OPTIMA AS A FUNCTION OF AVAILABLE RESOURCES BY A HOMOTOPY METHOD*

Yung S. SHIN, Raphael T. HAhT'KA, Layne T. WATSON and Raymond H. PLAUT Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, U.S.A.

Received 27 June 1987 Revised manuscript received 8 February 1988

Optimization problems are typically solved by starting with an initial estimate and proceeding iteratively to improve it until the optimum is found. The design points along the path from the initial estimate to the optimum are usually of no value. The present work proposes a strategy for tracing a path of optimum solutions parameterized by the amount of available resources. The paper specifically treats the optimum design of a structure to maximize its buckling load. Equations for the optimum path are obtained using Lagrange multipliers, and solved by a homotopy method. The solution path has several branches due to changes in the active constraint set and transitions from unimodal to bimodal solutions. The Lagrange multipliers and second-order optimality conditions are used to detect branching points and to switch to the optimum solution path. The procedure is applied to the design of a foundation which supports a column for maximum buckling load. Using the total available foundation stiffness as a homotopy parameter, a set of optimum foundation designs is obtained.

1. Introduction

Optimization problems are typically solved by starting with an initial estimate and proceed-ing iteratively to improve it until the optimum is found.. The design points along the path from the initial estimate to the optimum are usually of no value. However, this need not be the case. In many applications, it is of interest to find the family of optima obtained by varying an input parameter such as the amount of available resources. If one member of the family is known, it may be possible to use it as a starting point and to follow an optimization path that goes through the other members of the family. A first step in tracing a family of optima is the application of sensitivity information to extrapolate from one member of the family to another. The present paper proposes the use of the sensitivity information to formulate the path of optima as the trajectory of a differential equation, a procedure known as a homotopy technique. The basic theory of globally convergent (convergent from an arbitrary starting point) hornotopy methods was developed in 1976 [l, 2',]. Since then, the method has been used in a wide range of scientific and engineering problems. It has been successfully applied to nonlinear complementarity problems (3], nonlinear two-point b...