Improved Simulated Annealing Search for Structural Optimization

Abstract

An optimization algorithm based on simulated annealing is proposed, in which the domain of the search is successively reduced based on a probability concept until the stopping criteria are satise ed. By introducing the ideas of probability cumulative distribution function and stable energy, the selection of initial temperature and equilibrium criterion in the process of simulated annealing becomes easy and effective. Numerical studies using a set of standard test functions and an example of a 10-bar truss show that the approach is effective and robust in solving both functional and structural optimization problems. HE achievement of optimum design is a goal naturally attractive to the designer. The e eld of structural design is generally characterized by a large number of variables, which are usually discrete in dimensions or properties, and these variables should satisfy all of the constraints to be a feasible structural design. Some structuraloptimizationproblemsarefairlyamenableto amathematical programming approach. In such instances, the design space is continuousand convex. The searchmay be deterministic,and methods are employed using, for example, gradient concepts. However, there is a large class of structural optimization problems with nonconvexities in the design space and with a mix of continuous and discrete variables. Under such circumstances, standard mathematical programming techniques are usually inefe cientbecause they are computationally expensive and are almost assured of locating the relative optimum close to the starting design. To overcome these dife culties, the stochastic search in structural optimization is considered. Many methods have become possible with the powerful computing facilities available in recent years. Among the stochastic algorithms, pure random search 1 is the simplest strategy for optimal design. Some modie ed versions have been suggested, such as single start, multistart, and random directions. 2,3 Methods in this class generally are quite simple to implement, but the appropriate stopping rules are very dife cult to derive. Recently, two classes of powerful search methods, which have their philosophical basis in processes found in nature, have been widely used in structural optimization. The e rst class of methods, including genetic algorithms 4 and simulated evolution, 5 is based on the spirit of Darwinian theory of evolution. The second class of methods is generally referred to as simulated annealing techniques 6 because they are qualitatively derived from a simulation of the behaviorofparticlesinthermalequilibriumatagiventemperature.Because of their global capabilities, research on the utilization of these search methods in design optimization has been undertaken. 7i 11 Somehybridtechniqueshavealsobeendevelopedbycombiningfeaturesofthesetwoalgorithms,suchasusingageneticalgorithmtodetermine better annealing schedules 12 and introducing a Boltzmanntype mutation or selection process into simulated evolution. 13,14 In this paper, we propose a method based on simulated annealing that searches from a population as in the method of simulated evolution instead of from a single point. The algorithm is called the region-reduction simulated annealing (RRSA) method because it locates the optimum by successively eliminating the regions with low probability of containing the optimum. A brief review of basic simulated annealing is given in Sec. II, which is helpful for development and explanation of the proposed