Adaptyvieji stochastiniai algoritmai mechaninių sistemų elementams optimizuoti

Abstract

Šiame darbe nagrinėjamas stochastinių algoritmų vystymas ir taikymas mechaninių sistemų elementų geometrijai optimizuoti. Inžinerinėje praktikoje geometrijos optimizacijos uždaviniai dažniausiai yra netiesiniai, turintys triukšmo komponenčių, diskretiniai – išreikšti skaičių eilutėmis, o tikslo funkcija ir apribojimai gali būti netolydūs. Jiems spręsti geriausiai tinka stochastiniai metodai. Tai įrodo daugybės sėkmingų stochastinių algoritmų (pvz., evoliucinių algoritmų) taikymas tokiems optimizacijos uždaviniams išspręsti, kuriems deterministiniai metodai (pvz., gradiento metodai) beveik netinka. Inžinerinėje praktikoje mechaninių sistemų elementų geometrija optimizuojama ją parametrizuojant – apibrėžiant optimizacijos kintamuosius ir taikant optimizacijos algoritmą – randant optimalias šių kintamųjų reikšmes. Geometrijos parametrizacijos būdo pasirinkimas lemia optimizacijos uždavinio sprendimo kokybę ir daro įtaką optimizacijos algortimo pasirinkimui. Todėl efektyvių parametrizacijos ir optimizacijos metodų plėtotė yra du lygiavertės svarbos uždaviniai, sprendžiami mechanikos inžinerijoje. Šiame darbe nagrinėjama tik optimizacijos metodų plėtra, kai optimizacijos kintamieji yra realieji – tolydūs skaičiai.

The thesis addresses the development of stochastic algorithms to mechanical-structure optimisation problems. Since structure optimisation problems encountered in mechanical engineering are usually highly nonlinear, nosy, and discrete, stochastic algorithms represent reasonable optimisation methods for them. This evidence is justified by many successful applications of stochastic algorithms (e.g. evolutionary algorithms) on those mechanical engineering problems, where deterministic methods are hardly applicable. This work embodies four new original approaches concerning about (1) the theoretical measures of the algorithmic efficiency, (2) the further development of existing and (3) the design of new stochastic algorithms. Thus, the first approach aims at predicting the run-time efficiency of evolutionary algorithms through the calculation of higher-order statistical moments, namely the skewness and the kurtosis, and the use of a new proposed statistic–the best fitness frequency. The performed statistical analysis is based on two hypothesises: (1) a population, considered as a distribution of fitness values, varies over run-time by changing its shape, and (2) such a variance of the fitness-distribution shape reflects fitness landscape regions. The optimisation results performed on some theoretical test functions support the stated hypotheses. The presented statistical efficiency analysis can be used only in population-based stochastic algorithms. The following two approaches focus on the further development of evolutionary algorithms, as a class of stochastic algorithms. Thus, a new mutation adaptation approach for evolutionary algorithms is presented. This approach includes the mutation rate adaptation and mutation step adaptation issues. The development of this adaptation approach was motivated by the fitness landscape analysis concept, presented by the fist approach, which appears to be appropriate for the mutation adaptation, as the theoretical analysis indicates. The next approach is a new mutation operator aimed at constrained optimisation problems. This operator is based on the statistical sensitivity analysis applied to find those optimisation variables which has a significant influence to response variables. Here response variables are either the objective function or imposed constraints. Then the mutation operator tend to mutate only those found correlated input variables and navigates the search towards, or from, the feasibility borders. The performed optimisations of theoretical highly nonlinear and higher-dimension test functions show the reasonability of thismutation operator. The fourth approach manifests a new stochastic algorithm–spherical stochastic algorithm for the constrained optimisation. Here the search is performed in a sphere, or hypersphere in the n-dimensional space. The centre point of the hypersphere always represents the best point found so far while the radius is an Euclidean distance between the centre point and farthest selected infeasible point. Therefore the spherical algorithm always includes feasible and infeasible solution spaces thus investigating the feasibility border. The search is performed by randomly generating points within the hypersphere and the convergence is guaranteed by shrinking of the hypersphere radius. The hypersphere shrinks when the exploration in the best point neighbourhood is faster than the relocation of the best point itself. This stochastic algorithm is most efficient if a theoretical optimum is located right at the intersection of objective and constraint functions, although its efficiency decreases with increasing number of imposed constraints, as theoretical results indicate. Finally all these approaches are applied for solving two real-world optimisation problems. For the optimisation four stochastic algorithms based on the presented approaches were employed; one of them is a plain evolutionary algorithm used for comparison purposes. The first optimisation problem is the sizing of Ducati motorbike steel-frame, where the objective is the minimum mass subject to stickness and stress constraints. The motorbike frame is parameterised through inner radius and thickness of the frame tubes resulting in 30 real-valued optimisation variables. The optimisation resulted in 21.7% of the mass reduction by the evolutionary algorithm with the statistical sensitivity analysis based mutation operator. The second real-world application is the end-plate of the fuel-stack topology optimisation problem, where the objective is the minimum mass subject to the stress constraint. The end-plate is parameterised through cross-section topology dimensions (e.g. position of ribs) resulting in 30 real-valued optimisation variables. For this problem the best optimisation method was the evolutionary algorithm with the adaptive mutation approaches giving 42.5% in the mass reduction. For both applications the spherical stochastic algorithm was behind the used evolutionary algorithms, that indicates a need for its further development.