Stacking Sequence Optimization of Laminated Plates by Genetic Algorithm

The use of mathematical programming techniques for engineering design optimization is becoming increasingly more effective with today's powerful computing resources. Design optimization is concerned with selecting parameters for a particular design in order to satisfy certain predetermined criteria such as yield strength, weight, and material considerations. These parameters are varied until an acceptable tolerance between the actual and the optimal solution is met. Several different methods of design optimization can be utilized: linear programming, penalty functions, and gradient projection, to name a few. In the past twenty years, a new technique involving genetic algorithms (GA's) has arisen and made it possible to investigate far more potential designs than conventional algorithms allow.

Genetic algorithms seek to mimic the biological processes of reproduction and natural selection. Natural selection determines which members of a population survive to reproduce, and reproduction ensures that the species will continue. To employ the genetic algorithm for engineering design optimization, the parameters of the design are usually encoded into a string of binary digits. The genetic algorithm is then applied to a population of randomly generated binary strings. The "fitness" of each string is determined according to required design specifications. High-quality strings "mate" by swapping portions of their strings and produce two offspring, thus beginning a new "generation." And just as biological organisms have less chance of reproducing if they have low fitness, so do low-quality strings. However, they are replaced by the offspring of high-quality strings to keep the total population size constant. This process occurs for a specified number of generations, or until no progress in fitness appears to be possible.

In structural design, one of the primary objectives is to minimize weight. Laminated composite materials are a natural choice as replacements for metallic structural applications, where high stiffness-to-weight and strength-to-weight ratios are required. These laminated plates, or panels, are typically composed of layers, or plies, of lightweight composite materials, such as graphite- epoxy, stacked one on top of the other. When designing these panels, one generally wants to optimize the ply orientations (the angle of the fibers in each ply) as to increase the buckling load and maximize strength.

A new technique for design population visualization (DPV) was developed as a tool for monitoring the performance of the genetic algorithm. DPV is a way of viewing an entire population of designs and their objective function values simultaneously. This data and data from other runs of the GA not presented here support the claim that for the stacking sequence problem, the GA tends to fix the outermost stacks and then adjusts the innermost (those nearest the plane of symmetry) to find the optimal and near-optimal designs. Thus, DPV is an excellent way of monitoring the performance of the GA. Firstly, it reveals at a glance the diversity of the design population (i.e., how large is the OFV range over which the designs are spread?). Secondly, it illustrates nicely how the GA converges on an optimal solution, and lastly, it allows a designer to view what features the designs in a given population have in common and perhaps identify key design variables.

Joseph Lynn Henderson (

Department of Aerospace and Ocean Engineering

Virginia Tech

Blacksburg, Virginia

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