结构优化中的模拟退火算法研究和应用
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摘要
结构优化设计不仅可以降低结构重量和材料成本，而且能够改进结构的强度、刚度、振动特性、屈曲稳定性等性能，是计算力学以及现代设计制造领域的重要研究方向。结构优化方法，大致有优化准则法、数学规划法、混合法、随机搜索法等。其中，遗传算法、模拟退火算法等新的随机搜索方法，在处理全局优化、离散变量、多连通可行区等困难问题中，具有传统结构优化算法不可比拟的优势。虽然它们的计算效率很低，但是在计算机计算速度不断提高的条件下，具有不可低估的发展潜力和重要的研究价值。
本文的研究工作由两部分组成：1．在研究分析基本的模拟退火算法原理和综合近期关于模拟退火算法研究进展的基础上，针对结构优化问题的特殊性质和要求，对模拟退火算法提出了若干改进措施。2．在若干类型的结构优化问题中应用本文改进的模拟退火算法，通过大量算例的数值试验和算法比较表明，模拟退火算法在结构优化的某些困难问题中具有其特点，本文的改进措施是可行和有效的。
各章节的内容安排如下：
第一章首先论述了模拟退火算法的研究进展。内容包括首先概述本文研究课题的背景及其意义；接着模拟退火算法的发展历史和特点、理论的研究、国内外研究概况及发展趋势等；最后概述了本文的研究工作。
第二章介绍了模拟退火算法的基本过程及其基本原理。首先介绍模拟退火算法的物理基础。然后介绍了模拟退火算法的基本过程及其数学基础。最后介绍了模拟退火算法的有限时间实现及其参数选择原则。
第三章提出了对模拟退火算法的改进，是本文理论算法研究的重要部分。在研究分析模拟退火算法原理和综合近期关于模拟退火算法研究进展的基础上，本文针对结构优化问题的特殊性质和要求，对模拟退火算法提出了若干改进措施。主要包括：
(1)通过一种自适应的函数变换方法，使初始温度的确定与具体问题无关，使算法具有更强的通用性和的适应性：
(2)在算法中设置了一个记忆器，使之成为有记忆的模拟退火算法；
(3)新解的确定方法结合了成功一失败法，使偏移量的产生更加有效；在连续函数中并且为了加强局部的趋化性搜索能力，在精细搜索阶段采用变尺度方法缩小搜索区间；
(4)提出了一种以相对精度为基础的结束准则，同时考虑了降温次数和可行解的个数，以提高算法的健壮性和鲁棒性；
(5)采用多目标规划的思想，将离散变量优化和连续变量优化结合起来，较好解决了离散变量优化设计时的难点。
给出了改进的模拟退火算法在一些传统优化方法难以解决的数学函数优化问题中的应用算例。
第四章研究了模拟退火算法在桁架结构连续变量尺寸优化、离散变量尺寸优化和形状优化中的应用。对连续变量的桁架尺寸优化问题，通过一组经典算例，比较了模拟退
大连理工大学硕士学位论文
火算法与遗传算法和传统方法的优化结果，表明本文算法能够得到比较满意的优化结
果，其改进方法可行、有效，计算效率和计算精度均优于遗传算法。对离散变量的析架
尺寸优化问题，本文提出了采用多目标规划的思想、将离散变量优化和连续变量优化结
合起来的求解方法。对于混合变量的析架形状优化问题，本文进一步改进了解的产生方
式。大量的算例表明，模拟退火算法能够得到问题的全局最优解，本文的改进模拟退火
算法在这些结构优化问题中比较有效，具有优于传统优化算法和遗传算法的某些特点。
第五章总结全文的工作，并展望了进一步的研究工作。
本文的研究工作是国家重点基础研究发展规划项目“大规模科学计算研究”课题“大
规模计算工程软件系统的基础理论和实施，，(编号G1999032805)和国家自然科学基金重
点项目“藕合系统的多学科优化设计理论与数值方法”(编号10032030)的部分内容。
By means of structural optimization, not only the weight of structures can be reduced, but also the strength, stiffness, vibration behavior, buckling stability, and other performances of structures can be improved efficiently. Structural optimization is an important research direction in the computational mechanics and modern design field.
The research work in the dissertation consists of two major parts. The first part proposes several improved measures on SA based on the analysis of the SA's theory and recent research of SA according to special property and requirements of structural optimization problems. The second part describes application of improved SA in some types of structural optimization problems. The numerical test and algorithmic comparison show that improved measures on S A are feasible and efficient.
The research work will be introduced as follows:
In chapter 1, the research developments of SA are surveyed, which include the questions for discussion of background and signification summarized in the first section, and then the history of development and characteristic for SA, research of the theory, a survey of internal and external research and the trend of development, the main contents of the research in this dissertation are presented in the last section.
In chapter 2, the process and fundament of simple SA are introduced. The basic of physics are presented firstly. Then the process and the basic mathematics of SA are given. Finally, implementation of SA in a limit time and the principle of choosing parameter in SA are described.
In chapter 3, improvements on SA are investigated, which is important substantial about academic algorithm of the dissertation. An improved SA is proposed by several methods being put forward. By introducing a method of adaptive conversion function, the determination of initial temperature, usually a difficult problem in SA has been solved and becomes independent to the practical problems solved. A memory is set up in order to becoming a memorial SA. Combined with the success-failure method and the variable metric method, the conception of effective shift-increment is proposed to improve the method generating new solutions. On the basis of the newly defined relative precision, a termination criterion is proposed to make better balance between the computational efficiency and the solution accuracy, and then, enhance the efficiency and robustness of the SA algorithm. Joined up optimization of continuous variable and discrete variable, the difficult was solved by using the idea of multi-objections.
The contents of the following chapter show that the improvements in this dissertation are feasible and effective by some kinds of typical examples.
In chapter 4, SA is applied to sizing optimization of truss with continuous, discrete variables and configuration optimization of truss with mixed variables. In sizing optimization of truss with continuous variables, solutions among SA, GA and the traditional methods are compared, which reveals that satisfying solutions are achieved, and that improved SA are feasible and effective, better computational efficiency and solution accuracy achieved. In sizing optimization of truss with discrete variables, joined up optimization of continuous variable and discrete variable, the idea of multi-objections is used. In configuration optimization of truss with mixed variables, the method generating new solutions is improved.
The numerical examples show that the improved SA in the dissertation are feasible and effective.
In chapter 5, the main contributions of the dissertation are summarized and the further work is suggested.
The research of this dissertation is supported by the Special Funds for National Key Basic Research of China (No. G1999032805) and the Special Funds for National Key Research of China in natural science (No. G10032030).
本文的研究工作由两部分组成：1．在研究分析基本的模拟退火算法原理和综合近期关于模拟退火算法研究进展的基础上，针对结构优化问题的特殊性质和要求，对模拟退火算法提出了若干改进措施。2．在若干类型的结构优化问题中应用本文改进的模拟退火算法，通过大量算例的数值试验和算法比较表明，模拟退火算法在结构优化的某些困难问题中具有其特点，本文的改进措施是可行和有效的。
各章节的内容安排如下：
第一章首先论述了模拟退火算法的研究进展。内容包括首先概述本文研究课题的背景及其意义；接着模拟退火算法的发展历史和特点、理论的研究、国内外研究概况及发展趋势等；最后概述了本文的研究工作。
第二章介绍了模拟退火算法的基本过程及其基本原理。首先介绍模拟退火算法的物理基础。然后介绍了模拟退火算法的基本过程及其数学基础。最后介绍了模拟退火算法的有限时间实现及其参数选择原则。
第三章提出了对模拟退火算法的改进，是本文理论算法研究的重要部分。在研究分析模拟退火算法原理和综合近期关于模拟退火算法研究进展的基础上，本文针对结构优化问题的特殊性质和要求，对模拟退火算法提出了若干改进措施。主要包括：
(1)通过一种自适应的函数变换方法，使初始温度的确定与具体问题无关，使算法具有更强的通用性和的适应性：
(2)在算法中设置了一个记忆器，使之成为有记忆的模拟退火算法；
(3)新解的确定方法结合了成功一失败法，使偏移量的产生更加有效；在连续函数中并且为了加强局部的趋化性搜索能力，在精细搜索阶段采用变尺度方法缩小搜索区间；
(4)提出了一种以相对精度为基础的结束准则，同时考虑了降温次数和可行解的个数，以提高算法的健壮性和鲁棒性；
(5)采用多目标规划的思想，将离散变量优化和连续变量优化结合起来，较好解决了离散变量优化设计时的难点。
给出了改进的模拟退火算法在一些传统优化方法难以解决的数学函数优化问题中的应用算例。
第四章研究了模拟退火算法在桁架结构连续变量尺寸优化、离散变量尺寸优化和形状优化中的应用。对连续变量的桁架尺寸优化问题，通过一组经典算例，比较了模拟退
大连理工大学硕士学位论文
火算法与遗传算法和传统方法的优化结果，表明本文算法能够得到比较满意的优化结
果，其改进方法可行、有效，计算效率和计算精度均优于遗传算法。对离散变量的析架
尺寸优化问题，本文提出了采用多目标规划的思想、将离散变量优化和连续变量优化结
合起来的求解方法。对于混合变量的析架形状优化问题，本文进一步改进了解的产生方
式。大量的算例表明，模拟退火算法能够得到问题的全局最优解，本文的改进模拟退火
算法在这些结构优化问题中比较有效，具有优于传统优化算法和遗传算法的某些特点。
第五章总结全文的工作，并展望了进一步的研究工作。
本文的研究工作是国家重点基础研究发展规划项目“大规模科学计算研究”课题“大
规模计算工程软件系统的基础理论和实施，，(编号G1999032805)和国家自然科学基金重
点项目“藕合系统的多学科优化设计理论与数值方法”(编号10032030)的部分内容。
By means of structural optimization, not only the weight of structures can be reduced, but also the strength, stiffness, vibration behavior, buckling stability, and other performances of structures can be improved efficiently. Structural optimization is an important research direction in the computational mechanics and modern design field.
The research work in the dissertation consists of two major parts. The first part proposes several improved measures on SA based on the analysis of the SA's theory and recent research of SA according to special property and requirements of structural optimization problems. The second part describes application of improved SA in some types of structural optimization problems. The numerical test and algorithmic comparison show that improved measures on S A are feasible and efficient.
The research work will be introduced as follows:
In chapter 1, the research developments of SA are surveyed, which include the questions for discussion of background and signification summarized in the first section, and then the history of development and characteristic for SA, research of the theory, a survey of internal and external research and the trend of development, the main contents of the research in this dissertation are presented in the last section.
In chapter 2, the process and fundament of simple SA are introduced. The basic of physics are presented firstly. Then the process and the basic mathematics of SA are given. Finally, implementation of SA in a limit time and the principle of choosing parameter in SA are described.
In chapter 3, improvements on SA are investigated, which is important substantial about academic algorithm of the dissertation. An improved SA is proposed by several methods being put forward. By introducing a method of adaptive conversion function, the determination of initial temperature, usually a difficult problem in SA has been solved and becomes independent to the practical problems solved. A memory is set up in order to becoming a memorial SA. Combined with the success-failure method and the variable metric method, the conception of effective shift-increment is proposed to improve the method generating new solutions. On the basis of the newly defined relative precision, a termination criterion is proposed to make better balance between the computational efficiency and the solution accuracy, and then, enhance the efficiency and robustness of the SA algorithm. Joined up optimization of continuous variable and discrete variable, the difficult was solved by using the idea of multi-objections.
The contents of the following chapter show that the improvements in this dissertation are feasible and effective by some kinds of typical examples.
In chapter 4, SA is applied to sizing optimization of truss with continuous, discrete variables and configuration optimization of truss with mixed variables. In sizing optimization of truss with continuous variables, solutions among SA, GA and the traditional methods are compared, which reveals that satisfying solutions are achieved, and that improved SA are feasible and effective, better computational efficiency and solution accuracy achieved. In sizing optimization of truss with discrete variables, joined up optimization of continuous variable and discrete variable, the idea of multi-objections is used. In configuration optimization of truss with mixed variables, the method generating new solutions is improved.
The numerical examples show that the improved SA in the dissertation are feasible and effective.
In chapter 5, the main contributions of the dissertation are summarized and the further work is suggested.
The research of this dissertation is supported by the Special Funds for National Key Basic Research of China (No. G1999032805) and the Special Funds for National Key Research of China in natural science (No. G10032030).
引文
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