Study On Chaos Optimization Algorithms For Structural Global Optimal Design

At present, the world is facing with the pressure of increasingly scarce resources and energy, and the design of industrial equipments and engineering structures with lightweight and high performance is urgently needed. The global optimization design for structure is an eternal pursuit of industry and academic community, but there is also a great challenge in mathematics and computational methods to realize it. The global optimal solution of the optimization problem cannot be characterized by any mathematical condition so far, while the local optimal solutions are different, which can be represented by local property such as gradient and Hessian matrix of functions.Moreover, considering the construction process and cost of engineering structure, engineer needs to choose suitable cross-sections with standard size, which is the classic optimization problem with discrete variables. Optimization algorithms (gradient algorithm for instance) for problem with continuous design variables are not applicable to the optimization problem with discrete variables, because of non-continuity of design variables, non-differentiability of objective function and constraint function as well as the inadaptability of K-T conditions. There are two categories of method to cope with this dilemma:one is to transform discrete optimization problem into continuous problem, but the local optimum or infeasible solution will be achieved; the other is the direct method, among which the heuristic algorithm is the most distinguished method for it can cope with the discrete optimization problem regardless of non-continuity of design variables and non-differentiability of objective function etc. However, it is still necessary to propose the more efficient and accurate algorithm for discrete and global optimization.In recent years, an emerging and potential method for global optimization, namely, chaos optimization algorithm (COA), which is based on nonlinear dynamical characteristics such as pseudo-randomness, ergodicity, the sensitivity on initial value and self-similar fractals of chaos etc., has been paid close attention, investigated and widely applied by the scholars of various disciplinary areas in science and technology. Classical COA does not explicitly require the gradient information and function expression of optimization problem. It is easy to program and jump out of local optimum, but it needs a large amount of calculations and converges slowly when dealing with the complex global optimization design problems. Therefore, this thesis develops the COA for global optimization design of engineering structure with improved global optimization efficiency, computational accuracy and universality, then the global optimization performance is deeply studied and the inherent nature and the important influence factors of COA is revealed. This study has important scientific significance and application perspective. The main works are presented as follows:(1) The important influence factors on the global optimization performance and mechanism of efficiency of chaos-BFGS algorithm based on chaotic search are revealed. The probability distribution and search speed of chaotic sequence are closely related to the ergodicity and pseudo-randomness of chaos etc., and they affect the global optimization capacity of chaos-BFGS algorithm (as part of the first kind of hybrid COA, which is the combination of chaotic search and gradient algorithm). Considering that the probability distribution and search speed of chaotic sequence can be respectively characterized by its probability density function (PDF) and Lyapunov exponent, the results of optimizing nonlinear multimodal functions by chaos-BFGS algorithm are compared and analyzed from two aspects:different chaotic maps with close LE and distinct PDF, and Kent map with different LE and the same PDF. The optimization results indicate that the probability distribution property, search speed of chaotic sequences from different chaotic maps and the location of the optimal solution significantly affect the global searching capability and optimization efficiency of chaos-BFGS algorithm, and COA is better than Monte Carlo-BFGS algorithm. To achieve the high efficiency of COA, it is recommended to adopt the appropriate chaotic map generating the desired chaotic sequences with uniform or nearly uniform probability distribution and large Lyapunov exponent.(2) A new optimization algorithm about the accelerated particle swarm optimization combined with different chaotic search (APSOC) is established, and an index of dispersion degree which can measure the aggregation or dispersal degree of chaotic sequence is proposed. It is hard for chaos-BFGS algorithm to deal with complicated nonlinear engineering optimization problem, in which the objective function is usually non-continuous, non-differentiable and there exist lots of local extremum. The APSOC is developed through combination of chaotic search and heuristic algorithm which has less requirement on optimization problem. Firstly, a new effective numerical method to compute Lyapunov exponent of one-dimensional piecewise chaotic map is presented, by introducing the probability measure into the definition of LE. Secondly, it is revealed through classified discussion that, the probability distribution, search speed of chaotic sequences and dispersion degree affect the performance of APSOC and other three CPSO based on chaotic search remarkably. Moreover, the influence regularity of dispersion degree and probabilistic distribution of chaotic sequences on the efficiency of CPSOs is consistent. If the PDF obeys uniform or approximately uniform distribution, the LE and the dispersion degree are large, then the optimizing success ratios of CPSOs are very high. Finally, the optimal designs of trusses with discrete variables are performed by APSOC, and the optimization scheme demonstrates the high efficiency and effectiveness of APSOC.(3) The intrinsic cause affecting the global optimization efficiency of the second hybrid COA, namely the chaos-enhanced accelerated PSO (CAPSO), which utilizes one-dimensional chaotic maps to replace the key parameter of APSO, is illustrated. CAPSO algorithm adopts the six continuous and six piecewise one-dimensional chaotic maps. And the influence of separation degree of the adjacent points and probability distribution of chaotic sequence on the performace of hybrid COA is examined. Moreover, the computational performance of CAPSO is compared with two cases based on the results of global optimization of nonlinear benchmark functions. The optimized results illustrate that the efficiency of CAPSO is obviously affected by the PDF and LE of chaotic sequence. Lastly, the ICMIC-CAPSO algorithm with the highest efficiency is adopted for optimum design of engineering systems with discrete and mixed variables.(4) Chaos optimization algorithm is extended to solve frequency optimization of the composite frame structure. Considering the requirement on load-bearing capacity and dynamic characteristic of the composite frame structure, the fiber winding angle is generally chosen as the design variable. Meanwhile, the fundamental frequency, sum of first three frequencys and first five frequencys are maximized by APSOC and CAPSO algorithm to improving the dynamic stiffness of the composite frame structure. The efficiency of the two hybride COAs is demonstrated by optimization design of discrete and continuous fiber winding angle, and APSOC achieves the optimal design scheme with faster convergence speed.(5) Chaos-fractal optimization algorithm embedding in the fractal theory is established from a new perspective. Based on the fractal property of Julia set, this paper utilizes accurately the step length of descent direction in BFGS method and conjugate gradient method by Newton-Raphson method, proposes fractal-BFGS algorithm and fractal-conjugate gradient algorithm, and combines these two algorithms with chaos optimization algorithm to improve the global optimization capacity. It is indicated that the computational efficiency of the new algorithms is higher than that of the combined algorithm using Wolf method for one dimensional inexact search to calculate the step length of descent direction, and the optimization ability of chaos-fractal with BFGS algorithm is superior to that of chaos-fractal with conjugate gradient algorithm. It is also illustrated that the local search ability of BFGS in chaos-fractal optimization algorithm is better than that of conjugate gradient algorithm.