改进的微粒群算法及其在结构拓扑优化中的应用
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摘要
群智能优化算法越来越受到人们重视。微粒群算法是一种典型的群智能优化算法,其结构简单、易于实现、寻优能力强,与传统的优化方法相比,具有明显的优越性。微粒群算法仅有两个简单的迭代公式,需要调整的参数较少。然而在优化高维、多极值的复杂优化问题时,标准微粒群算法的寻优能力较差,容易陷入局部最优。微粒群算法虽然已经得到了广泛的应用,但进一步拓展其应用领域仍是很有意义的。本文提出了一种改进的微粒群算法并应用于结构拓扑优化。
提出了一种基于遗忘特性和群体平均思想的改进微粒群算法。注意到标准微粒群算法的速度迭代公式中,个体最优位置和群体最优位置为引导每个个体向最优位置移动起重要作用。基于群智能算法的个体具有简单行为的特点,改进的微粒群算法中的个体仅具有有限的记忆力,即遗忘掉前期所经历过的位置。由于群体最优位置是从所有个体最优位置中挑选出来的,从而群体表现出遗忘性。标准微粒群算法中的群体最优位置对所有个体的吸引力过强,容易使算法陷入局部最优。群居生物中的个体具有从众心理,即个体既要跟随群体中最优者的步伐,也要兼顾到跟随群体中大多数的步伐。这样,改进的微粒算法在迭代过程中,随机地用群体平均位置替换群体最优位置,充分利用了两者的优点。同时注意到,改进措施没有明显地增加算法的复杂性。复杂函数优化的算例验证了改进算法良好的优化性能。
微粒群算法的理论研究是当前的一个难点,还没有形成一套完备的理论体系。本文考虑到微粒群算法与遗传算法有许多相似之处,而遗传算法有较完备的马氏过程理论体系,故将马氏过程理论应用于研究微粒群算法。将群体在有限步内的速度和位置构成的向量作为一个随机过程的变量,证明了该随机过程是一个齐次马尔科夫过程。
拓展微粒群算法的应用领域是当前的一个研究热点,而结构拓扑优化是工程优化领域中的一个难点。将微粒群算法应用于桁架结构拓扑优化和连续体结构拓扑优化,拓展了微粒群算法的应用领域,为结构拓扑优化开辟了新思路。在算法的程序实现上,研究了MATLAB编程语言与ANSYS软件的调用问题。
多目标优化问题是生活和工程中经常会遇到的问题。在多目标优化问题中,需要优化的目标函数不止一个,这给个体优劣性的评定带来困难。寻找问题的Pareto最优解集,是多目标优化的目的。注意到一个目标函数值特别小,而其它目标函数值又特别大的所谓最优解是不切实际的,由各函数值的差别信息定义了非劣解集实用性的标准。为得到具有良好分散性和实用性的非劣解集,借鉴群体最优位置的选取来自“群体精英解集”的思路,引入了“个体精英解集”,以从中挑选个体最优位置,并将各目标函数值差别的信息用于计算个体的适应度。多目标函数优化的算例表明,改进算法更易得到分散性和实用性好的非劣解集。
最后,对进一步研究工作的方向进行了简要的展望。
Now more and more importance has been attached to swarm intelligence optimization algorithm. Particle swarm optimization algorithm (PSO) is a typical swarm intelligence optimization algorithm. PSO has a simple structure, a strong ability to find best solution and can been realized easily. It has more evident advantage when comparing with traditional optimization methods. There are only two simple evolution formulas in PSO, and parameters needed to be adjusted are less. However, in the complicated optimization problems with multi-dimensions and many extrema, the standard PSO behaves with pool ability and is easy to find a local best solution. PSO has been applied broad, but it is significant to develop more application of PSO. The improvement and application of PSO are researched in this thesis.
An improved PSO based on the forgetting character and the average information of swarm is advanced. It can be observed that the individual best position and the global best position have important effect on piloting every particle moving to optimal position. The individual in swarm intelligence optimization algorithm possess simple behavior. So we let the memory of particle finite in the improved PSO, and the best position in former phase is forgotten. The global best position is chosen from the all individual best positions, and then the swarm has forgetting character. The global best position has too strong attraction to every particle, so the standard PSO is easy to find local best solution. The individual has mind to follow the center of swarm (here it is the center of all individual best position). Namely, the individual will follow the excellent particle and also want to follow the center of all individual best positions. So, in the evolution process the global best position is displaced randomly by the center of all individual best positions, and both virtues are taken full advantage of. At the same time, we notice that the complexity of improved PSO is not added evidently. The good performance is validated by complicated optimization functions.
The theory research for PSO is difficult currently. A mature theory system doesn't come into being. We think that PSO is similar to genetic algorithm in many ways. The genetic algorithm has mature Markov process theory system, so in this thesis PSO is researched by Markov process theory. The stochastic process variable is constituted by the vectors of all velocity and positions, and the proof that it is a homogeneous Markov process is given.
A current research hotspot on PSO is to develop its broad application. At the same time, topology optimization is difficult in engineering. The application of PSO is developed in this thesis by applying PSO to truss structural topology optimization and continuum structural topology optimization. So a new measure is pioneered for structural topology optimization. The algorithm is realized and programmed by using MATLAB language and data transfer from ANSYS to MATLAB.
The multi-objective optimization problems are often encountered in life and engineering. In the multi-objective optimization, more than one objective is needed to be optimized. So it is difficult to evaluate individual good or bad. The multi-objective optimization aims to find the Pareto solution set. We notice that the solution is impracticable which makes one objective very small but others very big. So the practicability criterion is defined by calculating the difference of objectives. Inspired by the idea that the global best position is chosen from the swarm elitism set, the individual elitism set is advanced, in order to get a solution set with good spacing and good practicability. Then the individual best position is chosen from the individual elitism set. The individual fitness is evaluated by calculating the difference of objectives. A good non-dominated solution set with good spacing and practicability can be gotten by the improved PSO seen from the examples.
Finally, the outlook about further research directions is given briefly.
提出了一种基于遗忘特性和群体平均思想的改进微粒群算法。注意到标准微粒群算法的速度迭代公式中,个体最优位置和群体最优位置为引导每个个体向最优位置移动起重要作用。基于群智能算法的个体具有简单行为的特点,改进的微粒群算法中的个体仅具有有限的记忆力,即遗忘掉前期所经历过的位置。由于群体最优位置是从所有个体最优位置中挑选出来的,从而群体表现出遗忘性。标准微粒群算法中的群体最优位置对所有个体的吸引力过强,容易使算法陷入局部最优。群居生物中的个体具有从众心理,即个体既要跟随群体中最优者的步伐,也要兼顾到跟随群体中大多数的步伐。这样,改进的微粒算法在迭代过程中,随机地用群体平均位置替换群体最优位置,充分利用了两者的优点。同时注意到,改进措施没有明显地增加算法的复杂性。复杂函数优化的算例验证了改进算法良好的优化性能。
微粒群算法的理论研究是当前的一个难点,还没有形成一套完备的理论体系。本文考虑到微粒群算法与遗传算法有许多相似之处,而遗传算法有较完备的马氏过程理论体系,故将马氏过程理论应用于研究微粒群算法。将群体在有限步内的速度和位置构成的向量作为一个随机过程的变量,证明了该随机过程是一个齐次马尔科夫过程。
拓展微粒群算法的应用领域是当前的一个研究热点,而结构拓扑优化是工程优化领域中的一个难点。将微粒群算法应用于桁架结构拓扑优化和连续体结构拓扑优化,拓展了微粒群算法的应用领域,为结构拓扑优化开辟了新思路。在算法的程序实现上,研究了MATLAB编程语言与ANSYS软件的调用问题。
多目标优化问题是生活和工程中经常会遇到的问题。在多目标优化问题中,需要优化的目标函数不止一个,这给个体优劣性的评定带来困难。寻找问题的Pareto最优解集,是多目标优化的目的。注意到一个目标函数值特别小,而其它目标函数值又特别大的所谓最优解是不切实际的,由各函数值的差别信息定义了非劣解集实用性的标准。为得到具有良好分散性和实用性的非劣解集,借鉴群体最优位置的选取来自“群体精英解集”的思路,引入了“个体精英解集”,以从中挑选个体最优位置,并将各目标函数值差别的信息用于计算个体的适应度。多目标函数优化的算例表明,改进算法更易得到分散性和实用性好的非劣解集。
最后,对进一步研究工作的方向进行了简要的展望。
Now more and more importance has been attached to swarm intelligence optimization algorithm. Particle swarm optimization algorithm (PSO) is a typical swarm intelligence optimization algorithm. PSO has a simple structure, a strong ability to find best solution and can been realized easily. It has more evident advantage when comparing with traditional optimization methods. There are only two simple evolution formulas in PSO, and parameters needed to be adjusted are less. However, in the complicated optimization problems with multi-dimensions and many extrema, the standard PSO behaves with pool ability and is easy to find a local best solution. PSO has been applied broad, but it is significant to develop more application of PSO. The improvement and application of PSO are researched in this thesis.
An improved PSO based on the forgetting character and the average information of swarm is advanced. It can be observed that the individual best position and the global best position have important effect on piloting every particle moving to optimal position. The individual in swarm intelligence optimization algorithm possess simple behavior. So we let the memory of particle finite in the improved PSO, and the best position in former phase is forgotten. The global best position is chosen from the all individual best positions, and then the swarm has forgetting character. The global best position has too strong attraction to every particle, so the standard PSO is easy to find local best solution. The individual has mind to follow the center of swarm (here it is the center of all individual best position). Namely, the individual will follow the excellent particle and also want to follow the center of all individual best positions. So, in the evolution process the global best position is displaced randomly by the center of all individual best positions, and both virtues are taken full advantage of. At the same time, we notice that the complexity of improved PSO is not added evidently. The good performance is validated by complicated optimization functions.
The theory research for PSO is difficult currently. A mature theory system doesn't come into being. We think that PSO is similar to genetic algorithm in many ways. The genetic algorithm has mature Markov process theory system, so in this thesis PSO is researched by Markov process theory. The stochastic process variable is constituted by the vectors of all velocity and positions, and the proof that it is a homogeneous Markov process is given.
A current research hotspot on PSO is to develop its broad application. At the same time, topology optimization is difficult in engineering. The application of PSO is developed in this thesis by applying PSO to truss structural topology optimization and continuum structural topology optimization. So a new measure is pioneered for structural topology optimization. The algorithm is realized and programmed by using MATLAB language and data transfer from ANSYS to MATLAB.
The multi-objective optimization problems are often encountered in life and engineering. In the multi-objective optimization, more than one objective is needed to be optimized. So it is difficult to evaluate individual good or bad. The multi-objective optimization aims to find the Pareto solution set. We notice that the solution is impracticable which makes one objective very small but others very big. So the practicability criterion is defined by calculating the difference of objectives. Inspired by the idea that the global best position is chosen from the swarm elitism set, the individual elitism set is advanced, in order to get a solution set with good spacing and good practicability. Then the individual best position is chosen from the individual elitism set. The individual fitness is evaluated by calculating the difference of objectives. A good non-dominated solution set with good spacing and practicability can be gotten by the improved PSO seen from the examples.
Finally, the outlook about further research directions is given briefly.
引文
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