Shape optimization of a fractional horse-power Dc-motor by stochastic methods

Automated Optimal Design (AOD) of electromagnetic devices turns out to be a task of increasing significance in the field of electrical engineering. Recent developments offer the opportunity to attack realistic problems of technical importance. The distinctive feature of this type of problem is its complexity which results from a high number of design parameters, a complicated dependence on the quality of design parameters and various constraints. Often the direct relation of desired quality of the technical product on the objective variables is unknown. Stochastic optimization methods in combination with general numerical field computation techniques like the finite element method (FEM) offer the most universal approach in AOD. The paper discusses methodology, characteristic features and behaviour of the methods used. The application to a nonlinear magneto-static problem of technical significance is demonstrated by minimizing the overall material costs of a small de-motor by optimizing the rotor and stator shape. INTRODUCTION Numerical optimization methods have to be examined with regard to the following criteria: • reliability • robustness • insensibility to stochastic disturbances • application range Transactions on the Built Environment vol 2, © 1993 WIT Press, www.witpress.com, ISSN 1743-3509 16 Optimization of Structural Systems • accuracy • performance Stochastic methods are known to fulfil the above mentioned requirements. A main advantage is their insensibility to stochastic disturbances of the objective function caused by numerical evaluation [2]. This insensibility is a consequence of non deterministic search and disuse of derivatives. A second important property is the easy treatment of constraints. Hence a complicated transformation into an unconstrained problem formulation is not necessary. To solve the nonlinear field problem a suitable method of wide application range has to be chosen. Here field calculation is accomplished by the FEM. Error estimation, adaptive mesh generation and refinement are used. This method guarantees the greatest possible facility in modelling and allows optimization without severe geometrical restrictions. OPTIMIZATION METHOD Optimizing requires the concentration of all design aims into a single function Z(x). This function depends on all design parameters and represents the quality of the specific design. The function is called objective function. Additional constraints G;(x) limit the admissible parameter variation. The general problem class to be considered is the nonlinear constrained optimization and may be expressed in mathematical terms as: Z(x) = Z(zi,Z2,...,zJ -> Min. (1) with independent variables x = {x, : i = l(l)n} in the space x e JR^ and j = 1(1)77% constraints, t \ OXx) = G;(zi,Z2,...,zJ = 0 (2) Random based search methods only evaluate the objective function itself without use of derivatives. Using numerical methods the dependence of the quality function on the objective parameters is not available in explicit form and may be very complicated. The numerical calculation of derivatives may become troublesome as well due to discretization and cancellation errors. A further uncertainty occurs from inexact objective function evaluation. So, the simplest and most reliable procedure is to use stochastic search methods for optimization. Transactions on the Built Environment vol 2, © 1993 WIT Press, www.witpress.com, ISSN 1743-3509 Optimization of Structural Systems 17 Random variations of the objective variables x, is done by the rules of evolution strategy. The introduction of the control parameter temperature in the search process aims to avoid the system getting stuck in a local minimum. Barriers of height kgT, where ks is the BOLTZMANN constant and T the temperature, can be surmounted on the way to a better solution. The evolution strategy, a Monte-Carlo method, copies the natural principles mutation and selection (survival of the fittest) of biological evolution into the technical optimization problem. The basic concept of the evolution strategy is found in the substitution of DARWIN'S notion of fitness to the quality of a technical problem. The driving force in the optimization process is the repetition of mutation and selection in successive steps. RECHENBERG [5] transformed the scheme of biological evolution into a simple algorithm. This elementary form is termed the (l + l)-strategy. One parent (valid solution vector) generates a descendant, which differs by mutation of the objective variables from the parent. After evaluation of the objective function the parameter vector with the better quality is chosen to form the parent of the following generation etc.. Based on the (l-fl)-strategy more general and powerful strategies were introduced by SCHWEFEL [6]. To illustrate the simplicity and its universal applicability fig. 1 shows a scheme of an extended algorithm. The mutation of the objective variables of an initial generation of valid parameter vectors (parents) leads to a number of children. The variables of one child may depend on multiple parent variable vectors as indicated in fig. 1. The best children are selected to form the next parent generation. If is the number of parents, A the number of children and p a hereditary factor the strategies can be distinguished in the following manner. \r~ , • -y ^ — — o»/ The population of the next generation is selected from parents and A children. (/.t, A)-comma-strategy: The population of the next generation is selected from the /.i best children. Parents only live for one single generation. (—h A)-strategy: p p parents contribute to the creation of a child (p-sexuality). ^ of the hereditary factor of one parent is transferred to a child. • (-, A)-strategy: p comma-variant of the last mentioned strategy. Transactions on the Built Environment vol 2, © 1993 WIT Press, www.witpress.com, ISSN 1743-3509 18 Optimization of Structural Systems