连续体结构拓扑优化模型讨论及其应用
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摘要
由于在设计结构的尺寸和形状的同时可设计结构的构型,拓扑优化与传统的尺寸优化和形状优化相比具有更高的优化效益,从而成为航天航空等高技术领域结构设计的重要手段。针对不同的设计要求,建立合适的拓扑优化问题的提法和数学模型,成为重要的研究课题。
本论文在总结评述拓扑优化的理论、方法和应用方面的研究进展的基础上,开展了拓扑优化模型的研究,针对最大刚度设计和最优散热结构设计问题,提出了基于几何平均位移和几何平均温度为目标的拓扑优化设计问题的提法和求解方法；研究了具有拉压不同模量的结构拓扑优化问题,以及结构拓扑与构件间连接方式的协同设计问题。具体研究内容和成果如下。
1)连续体结构拓扑优化模型的讨论。传统结构静力拓扑优化往往集中于以柔顺性为目标函数的研究,然而实际工程设计中通常采用最大位移作为衡量结构刚度性能的指标。本论文以悬臂梁结构截面设计为例,评估了柔顺性最小设计与最大位移最小设计之间的差别；将几何平均位移作为最大位移的近似,并以此为目标函数建立新的优化模型；研究其与传统基于柔顺性的优化模型的差别与各自适用的设计范围。相似地,将几何平均温度作为最高温度的近似引入到结构散热拓扑优化设计中,并作为目标函数建立散热结构拓扑优化模型来实现最高温度最小化设计；通过比较最高温度、最大温度梯度以及温度方差(衡量温度场均匀程度的指标)等特征指标来研究基于几何平均温度的优化模型与基于热量传递势容耗散、温度方差这两个优化模型在散热结构拓扑优化设计中的差异及各自适用的设计范围。
2)拉压不同模量结构拓扑优化设计。具有拉压不同模量特性的材料如混凝土、纤维增强复合材料等已被大量应用于土木建筑以及航天航空结构中,传统基于线弹性本构的连续体拓扑优化技术不能满足此类问题的设计要求。本论文研究了拉压不同模量结构拓扑优化设计的理论和方法。针对拉压不同模量问题本构不连续的特性,采用改进的Heaviside函数光滑连续化此非线性材料本构关系,并提出基于牛顿—拉斐逊方法的有限元迭代求解格式。以此为基础,建立了拉压不同模量拓扑优化模型以及具有多相材料的结构构型设计策略,深入地讨论了各类型设计参数对结构最优构型的影响。
3)结构构型与结构间连接方式协同优化设计。通常的结构拓扑优化是在给定外部作用下的构型设计。然而对于复杂结构,各部件由连接构件相连并通过它传递外部作用。因此,连接方式影响部件承受载荷的分布,即连接构件的布局对部件的最优构型有重要影响。本论文研究结构部件构型与部件之间连接方式的协同优化问题,在连接区域引入一种特殊的材料模型,以此材料的分布来描述连接方式,并将承力结构与连接构件区域的材料密度同时作为设计变量,提出了一种基于拓扑优化思想的连接方式与结构拓扑协同设计的优化模型和相应的求解方法。
本文工作得到国家重点基础研究(973项目)计划(2006CB601205)、国家自然科学基金(90605002,10721062,90816025)和高等学校博士学科点专项科研基金(20090041110023)的资助,在此表示感谢。
Topology optimization can provide much more benefits than traditional size and shape optimization, since it can achieve the structural configuration, size and shape design at the same time. This makes it become the most important structural design method in aerospace and other high-tech fields. How to establish reasonable optimization formulations and mathematical models for different design requirements has already become an important subject in the structural optimization research field.
Based on the critical review on theory, method and application of structural topology optimization, this dissertation focused on topology optimization models. The optimization formulation and corresponding solution methods in which the geometric average displacement\temperature are considered as objective functions respectively are put forward to solve the problems in the maximum stiffness design and optimum heat dissipation design. Structural topology design with different tensile and compressive properties is proposed in this dissertation, and concurrent optimum design method of layouts of component and connection between components in the structure is also well presented for grid structure design. The main achievement and works of the dissertation are as following:
(1) The discussion on topology optimization models. Designs of structures with maximum stiffness based on the compliance have attracted much attention and many achievements have been made. However, in practical engineering the maximum displacement is the most commonly used index in measuring structural stiffness. The difference between minimal compliance design and minimal maximum displacement is pointed out through a cantilever beam section design example in this dissertation. As a good approximation of the maximum displacement, the geometric average displacement in the design region is considered as the objective function in the new static topology optimization model in order to achieve minimal maximum displacement design. The proposed model is compared with the conventional compliance design model. The differences between these two models and their scope of application are well discussed through theory and numerical analysis. Similarly, the geometric average temperature, which is used as a good approximation of the maximum temperature, is considered as the objective function in the new heat conduction topology optimization model in order to achieve minimal maximum temperature design. The comparison on the proposed model, the conventional dissipation of heat transport potential capacity design model and the nodal temperature variance based optimization model is also carried out. By comparing the maximum temperature, the maximum temperature gradient, and the nodal temperature in the design results, the differences among these three models as well as their scope of application are discussed through theory and numerical analysis.
(2) Topology optimization of continuum structures with different tensile and compressive properties. The materials of dual extension/compression modulus such as concrete and fiber reinforced composite are often encountered in the engineering problems. These materials are widely used in the construction industry, machinofacture, aircraft manufacturing, and other industries. However, conventional topology optimization with linear material can not fulfill these design requirements any longer. Consequently, the theory and method of topology optimization of continuum structures with different tensile and compressive properties have been studied thoroughly in this dissertation. In order to solve elasticity problems with dual extension/compression modulus, a technique that employ modified Heaviside function is presented to describe the nonlinear relationship of stress and material modulus smoothing the constitutive discontinuity. By utilizing Newton-Raphson algorithm, the iterative finite element numerical analysis method is proposed. Meanwhile a topology optimization model of continuum structures with different tensile and compressive properties is constructed to achieve structural layout design. Furthermore, based on this method a multimaterial model is proposed to formulate the topology optimization problem for structural topology design with multiphase materials. Two types of materials are distributed within the design domain to accommodate design need. Then, this dissertation discusses the influence of some relative design parameters in optimum design in depth.
(3) Concurrent optimum design of layouts of component and connection between components in the structure. The last part of this dissertation focuses on concurrent design of layouts of components and connection between components of structure. The conventional structural topology optimization is the layouts design of structure with fixed external loads. It is obvious for the complex construction that components joined together by connection transfer the external force, thus linkage model of components of structure affect load distribution greatly. In other words, connection distribution has great effect on optimal topology of components. The connection conditions are included in the topology optimization by introducing a new set of material model that represents connective domain through material distribution. Considering the relative material density of loading-carrying domain and connective domain as design variable, a topology optimization based approach and the corresponding solving technique has been developed to simultaneously designing structure and connection distribution.
This work is supported by National Basic Research Program (973 Program) of China (No.2006CB601205), National Natural Science Foundation of China (No.90605002, 10721062,90816025), and the Research Fund for the Doctoral Program of Higher Education of China (No.20090041110023). The financial supports are gratefully acknowledged.
本论文在总结评述拓扑优化的理论、方法和应用方面的研究进展的基础上,开展了拓扑优化模型的研究,针对最大刚度设计和最优散热结构设计问题,提出了基于几何平均位移和几何平均温度为目标的拓扑优化设计问题的提法和求解方法；研究了具有拉压不同模量的结构拓扑优化问题,以及结构拓扑与构件间连接方式的协同设计问题。具体研究内容和成果如下。
1)连续体结构拓扑优化模型的讨论。传统结构静力拓扑优化往往集中于以柔顺性为目标函数的研究,然而实际工程设计中通常采用最大位移作为衡量结构刚度性能的指标。本论文以悬臂梁结构截面设计为例,评估了柔顺性最小设计与最大位移最小设计之间的差别；将几何平均位移作为最大位移的近似,并以此为目标函数建立新的优化模型；研究其与传统基于柔顺性的优化模型的差别与各自适用的设计范围。相似地,将几何平均温度作为最高温度的近似引入到结构散热拓扑优化设计中,并作为目标函数建立散热结构拓扑优化模型来实现最高温度最小化设计；通过比较最高温度、最大温度梯度以及温度方差(衡量温度场均匀程度的指标)等特征指标来研究基于几何平均温度的优化模型与基于热量传递势容耗散、温度方差这两个优化模型在散热结构拓扑优化设计中的差异及各自适用的设计范围。
2)拉压不同模量结构拓扑优化设计。具有拉压不同模量特性的材料如混凝土、纤维增强复合材料等已被大量应用于土木建筑以及航天航空结构中,传统基于线弹性本构的连续体拓扑优化技术不能满足此类问题的设计要求。本论文研究了拉压不同模量结构拓扑优化设计的理论和方法。针对拉压不同模量问题本构不连续的特性,采用改进的Heaviside函数光滑连续化此非线性材料本构关系,并提出基于牛顿—拉斐逊方法的有限元迭代求解格式。以此为基础,建立了拉压不同模量拓扑优化模型以及具有多相材料的结构构型设计策略,深入地讨论了各类型设计参数对结构最优构型的影响。
3)结构构型与结构间连接方式协同优化设计。通常的结构拓扑优化是在给定外部作用下的构型设计。然而对于复杂结构,各部件由连接构件相连并通过它传递外部作用。因此,连接方式影响部件承受载荷的分布,即连接构件的布局对部件的最优构型有重要影响。本论文研究结构部件构型与部件之间连接方式的协同优化问题,在连接区域引入一种特殊的材料模型,以此材料的分布来描述连接方式,并将承力结构与连接构件区域的材料密度同时作为设计变量,提出了一种基于拓扑优化思想的连接方式与结构拓扑协同设计的优化模型和相应的求解方法。
本文工作得到国家重点基础研究(973项目)计划(2006CB601205)、国家自然科学基金(90605002,10721062,90816025)和高等学校博士学科点专项科研基金(20090041110023)的资助,在此表示感谢。
Topology optimization can provide much more benefits than traditional size and shape optimization, since it can achieve the structural configuration, size and shape design at the same time. This makes it become the most important structural design method in aerospace and other high-tech fields. How to establish reasonable optimization formulations and mathematical models for different design requirements has already become an important subject in the structural optimization research field.
Based on the critical review on theory, method and application of structural topology optimization, this dissertation focused on topology optimization models. The optimization formulation and corresponding solution methods in which the geometric average displacement\temperature are considered as objective functions respectively are put forward to solve the problems in the maximum stiffness design and optimum heat dissipation design. Structural topology design with different tensile and compressive properties is proposed in this dissertation, and concurrent optimum design method of layouts of component and connection between components in the structure is also well presented for grid structure design. The main achievement and works of the dissertation are as following:
(1) The discussion on topology optimization models. Designs of structures with maximum stiffness based on the compliance have attracted much attention and many achievements have been made. However, in practical engineering the maximum displacement is the most commonly used index in measuring structural stiffness. The difference between minimal compliance design and minimal maximum displacement is pointed out through a cantilever beam section design example in this dissertation. As a good approximation of the maximum displacement, the geometric average displacement in the design region is considered as the objective function in the new static topology optimization model in order to achieve minimal maximum displacement design. The proposed model is compared with the conventional compliance design model. The differences between these two models and their scope of application are well discussed through theory and numerical analysis. Similarly, the geometric average temperature, which is used as a good approximation of the maximum temperature, is considered as the objective function in the new heat conduction topology optimization model in order to achieve minimal maximum temperature design. The comparison on the proposed model, the conventional dissipation of heat transport potential capacity design model and the nodal temperature variance based optimization model is also carried out. By comparing the maximum temperature, the maximum temperature gradient, and the nodal temperature in the design results, the differences among these three models as well as their scope of application are discussed through theory and numerical analysis.
(2) Topology optimization of continuum structures with different tensile and compressive properties. The materials of dual extension/compression modulus such as concrete and fiber reinforced composite are often encountered in the engineering problems. These materials are widely used in the construction industry, machinofacture, aircraft manufacturing, and other industries. However, conventional topology optimization with linear material can not fulfill these design requirements any longer. Consequently, the theory and method of topology optimization of continuum structures with different tensile and compressive properties have been studied thoroughly in this dissertation. In order to solve elasticity problems with dual extension/compression modulus, a technique that employ modified Heaviside function is presented to describe the nonlinear relationship of stress and material modulus smoothing the constitutive discontinuity. By utilizing Newton-Raphson algorithm, the iterative finite element numerical analysis method is proposed. Meanwhile a topology optimization model of continuum structures with different tensile and compressive properties is constructed to achieve structural layout design. Furthermore, based on this method a multimaterial model is proposed to formulate the topology optimization problem for structural topology design with multiphase materials. Two types of materials are distributed within the design domain to accommodate design need. Then, this dissertation discusses the influence of some relative design parameters in optimum design in depth.
(3) Concurrent optimum design of layouts of component and connection between components in the structure. The last part of this dissertation focuses on concurrent design of layouts of components and connection between components of structure. The conventional structural topology optimization is the layouts design of structure with fixed external loads. It is obvious for the complex construction that components joined together by connection transfer the external force, thus linkage model of components of structure affect load distribution greatly. In other words, connection distribution has great effect on optimal topology of components. The connection conditions are included in the topology optimization by introducing a new set of material model that represents connective domain through material distribution. Considering the relative material density of loading-carrying domain and connective domain as design variable, a topology optimization based approach and the corresponding solving technique has been developed to simultaneously designing structure and connection distribution.
This work is supported by National Basic Research Program (973 Program) of China (No.2006CB601205), National Natural Science Foundation of China (No.90605002, 10721062,90816025), and the Research Fund for the Doctoral Program of Higher Education of China (No.20090041110023). The financial supports are gratefully acknowledged.
引文
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