The implementation of continuously updated sharing in the simple Genetic Algorithm code, and its application to the optimal placement of elastic supports on a simply supported plate

Abstract

A sharing method is incorporated into the simple Genetic Algorithm code using tournament selection. Several examples of the performance of this algorithm are given. Finally, the code is applied to the maximization of the first system natural frequency of a simply-supported square plate by optimally locating a number of elastic supports, where the discrete-distributed system is modeled analytically using a dynamic flexibility formulation. Jntroduction Genetic algorithms (GAS) have been applied to a wide variety of problems in the last several decades, including some in the area of structural ~ ~ t i m i z a t i o n l ~ ' ~ ~ ' ~ ~ . Recent work on the optimal placement of elastic supports on a thin, simply supported square plate in order to maximize the first system natural frequency used a Steepest Descent method to find the optimal l o ~ a t i o n s ~ ~ . The purpose of this study is to (a) review the dynamics of the plate-support system, (b) modify the C source code of the simple Genetic Algorithm (sGA)'~ to implement a tournament selection routine with continuously updated sharinglg, (c) show, by giving several simple examples, that the sharing algorithm does indeed work, and finally (6) to apply this algorithm to the problem of optimal elastic support placement on a simply-supported square plate to maximize the first system eigenvalue. Optimization of support location in a continuous structure is an important and current area of research. From a structural analysis standpoint, it may be desirable to locate supports of specified stiffness to maximize one or more natural frequencies of the system or to minimize the response to a particular input. From a control synthesis viewpoint, it is more likely that particular eigenvalues andlor eigenfunctions of the system will be specified, and related actuator gains and locations must be determined. The active and passive problems have much in common, and the approach to each has been somewhat similar, insofar as the modeling of the continuum goes. In both cases, the usual approach has been to discretize the continuum using a finite element formulation. In the control problem, the actuator locations are defined a priori, which "freezes" the mesh. Standard algorithms are applied to determine the required gains1'. In general, one eigenvalue or eigenfunction can be stipulated per actuator. While controls designed in this manner will satisfy the objective, * Research Assistant, Aeronautical and Astronautical Engineering Department they may not do so optimally; i.e., at minimum control cost. In fact, to find the optimal control will require the parameterization of the actuator locations over the continuum, and this is generally not cost effective in the context of finite element analyses. In the structural analysis problem, the objective is relatively simple. However, the finite element model forces the optimization to be posed as a discrete problem, with candidate support locations constrained to be at the element nodes. Thus, for distributed systems that may exhibit relatively complex dynamics, the possibility of missing the optimum altogether or at least misinterpreting results becomes a factor. This is the case with structures as simple as thin rectangular plates. In a recent papeg, the problem is addressed by using classical optimization theory on a two-dimensional nonlinear response surface that is the least-squares fit of natural frequency (as determined by a finite element analysis) versus support location for the case of four symmetrically-positioned elastic supports, where the support locations were chosen from a small number of possible positions. The existence and uniqueness of such optimal solutions were investigated in another recent paper7. Structural Svstem Dvnarnic~ The undamped, free transverse vibration of a thin, rectangular, simply supported plate, having Young's Modulus E , Poisson's ratio v , density p , and thickness h , with R elastic supports at locations ci, i = 1, . . ., R , as shown in Fig. 1, is described by the initial-boundary value problem1,2 W . ~ , ~ , (0, x2, t ) = w ,X IX~ (a, x2, t) = 0 Figure 1: Simply supported plate with elastic supports. Copyright O 1994 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved where w = a 2 w / a t 2 , the stiffness operator, L,, is given by and the forces of constraint due to the elastic supports are given by In order to facilitate the comparison of results to follow and to allow the interaction of system parameters to be better understood, the system equations of motion are nondimensionalized using the following relations Substituting relations (4) into the equation of motion and boundary conditions ( I ) , the dimensionless form becomes