材料和结构的拓扑优化关键理论与方法研究
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摘要
拓扑优化比尺寸或形状优化具有更显著的节省材料和改进结构性能的技术优势，经过近二十年的快速发展，拓扑优化的研究应用已扩展到许多领域。拓扑优化本质上具有同时优化材料和结构的能力，这为对结构重量、性能敏感的航空航天、汽车等工业领域提供了大幅提升结构性能、挖掘材料潜力的技术基础。结构宏观材料布局和材料细观微结构设计分别处在宏观、细观两个空间尺度，但具有相同的优化实质。目前，材料微结构设计与宏观结构拓扑优化处于分离状况，没有充分发挥材料性能和拓扑优化的技术潜力。因此，以拓扑优化技术为依托，开展材料／结构一体化设计新理念与相关技术的研究，设计具有多功能的材料／结构系统，消除材料、结构界限，已成为当前结构优化的发展方向。本文以此为背景开展关键基础理论与方法的研究，主要成果如下：
回顾了现有解决拓扑优化中数值计算不稳定问题的几种方法，提出适用于非规则网格的广义周长控制方法。该方法以单元间几何距离为计算参数，建立连续二次型广义周长计算表达式，方便近似处理和优化计算，消除了优化结果的棋盘格和中间密度值。根据三相材料拓扑优化中周长的多样性和特殊性，拓展周长定义，提出四种周长控制新方法，其中针对不同类别设计变量分别进行独立周长约束的方法具有最佳控制效能，能获得清晰的材料分布优化结果。大量由两相、三相材料组成的2D和3D结构的拓扑优化算例验证了方法的有效性，为材料和结构的拓扑优化设计奠定了基础。
以均匀化方法为基础，结合有限元数值计算开发实现了材料微结构的等效性能求解和灵敏度计算软件。以此为基础，综合考虑微结构等效性能，建立了多相材料微结构多目标优化模型；利用周长控制方法，开展了两相、三相材料微结构多目标优化设计，获得清晰的微结构构型。从优化模型和实现算法上分析比较了分别以极端性能和给定性能为目标的微结构设计问题，并结合单一体积约束下具有特定性能微结构的优化模型开展了具有特定泊松比的微结构反设计，获得了满足要求的微结构构型，为开展材料／结构的一体化设计铺平了道路。
以自然界多孔材料结构的组织结构特征为指导思想，建立了具有分层梯度变化的材料／结构模型，提出了材料／结构一体化设计方法。采用分步计算策略完成了材料／结构的一体化设计。首先进行结构宏观优化确定材料分布，其次进行细观微结构的精细设计获得了呈梯度分布的多孔材料结构，实现了对结构性能的显著改进。结合材料／结构一体化设计思想，比较了不同长宽比微结构体胞对优化结果的影响，验证了基于均匀化理论的优化设计方法不具备揭示尺度效应的能力。另外，对三相材料构成的材料／结构的一体化优化进一步验证了所建立的模型和方法的有效性。
提出尺度关联的材料／结构一体化设计方法。结合分层梯度的材料／结构设计模型，利用超单元方法和变量关联技术，采用分步计算方法对包含不同表征体胞数的材料／结构进行优化设计，优化结果展现了明显的尺度效应，并显示出设计域包含表征体胞的数目、表征体胞的长宽比例对优化结果的影响。发现随着设计域包含表征体胞数的增加，优化结构性能逐渐趋同于基于均匀化方法的优化结果。对圆环对称结构、圆柱夹芯结构，以及三相材料所构成的材料／结构的优化计算，进一步展现了尺度对优化结果的影响。
研究讨论了弱耦合热弹性结构的拓扑优化设计。基于弱耦合热弹性结构有限元分析计算过程，将弱耦合热弹性结构拓扑优化分解为三类优化子问题，并对子问题分别进行了研究。基于结构散热性能的优化问题比较了不同热载荷形式、不同材料模型时的优化结果，阐述了均布热流载荷下的优化模型对优化结果的影响。非耦合的多目标优化包括宏观结构拓扑优化和周期性材料／结构优化两个方面，通过变化加权因子和边界条件得到不同的优化结果，结果显示了明显的尺度效应。对已知温度场的热力耦合结构的拓扑优化，以桁架结构为例，分析了热、机械载荷的相对大小、材料惩罚模型的不同惩罚因子对优化结果的影响，提出采用不同惩罚模型应对不同类别载荷的处理思想，获得了收敛于0-1的优化解，实现了热力耦合热弹性结构的拓扑优化设计。
Topology optimization has been rapidly developing in the last two decades and is now extended in new application areas at an increasing rate, because it achieves by far greater savings and design improvement than sizing or shape optimization. Its intrinsic capacity of simultaneous optimal design of materials and structures promises to attain higher material and structural efficiency in aerospace and automobile industries, where some indexes such as weight are crucial to the performance of structures. In general, structure optimization and material design are dealt with based on the same topology optimal techniques in different scales. Currently, the potential ability of topology optimization is not fully explored, as material layout and design of microstructure are separated in the design procedure. It is necessary to investigate the integrated design methodology of materials and structures for designing the multifunctional material and structural system and removing the frontier etween materials and structures. The main research work in the dissertation will be introduced as follows:
A generalized perimeter scheme with weak rotational mesh-dependence is proposed to prevent numerical instabilities associated with checkerboards and intermediate densities in topology optimization. The scheme of quadratic form means that it is easy to establish a suitable explicit approximation and favors its implementation in convex programming. Based on the characteristics of density design variables associated with multiphase materials, the general concept of perimeter is extended and four kinds of perimeter control methods are proposed for topology optimization multiphase materials. Several 2D and 3D optimal examples made of two or three different materials phases are carried out for the compliance minimization. It shows that checkerboards and intermediate values of density variables are able to be eliminated efficiently.
Using the homogenization method and the finite element method, the effective properties and the sensitivities with respect to element densities of material microstructures are calculted. A multi-objective model is presented to optimize the topology of the microstructure with two or three-phase materials, subject to constraints on volume fraction and perimeter control. A comparative study is carried out for design of microstructures with the extreme properties and the prescribed elastic properties including the modeling and the algorithm. Several designs of microstructures with the prescribed properties are tested. These works are the basis for the integrated design of materials and structures.
Inspired by hierarchical structure of natural or biological materials, an computational model for the integrated design of materials and structures consisting of hierarchical cellular material levels or layers is described and an integrated design methodology is proposed for the global stiffness maximization of the overall structure and local design of material microstructures based on the homogenization method of multi-scale computing. The optimal design procedure includes two steps. Firstly, the material layout is figured out. Secondly, topologies of microstructures of intermediate density materials are designed. Numerical results show that the proposed method is well adapted to the design of lightweight structures. In addition to this, another issue about influences of microstructure aspect ratio on the optimal design is discussed and numerical experiences show that the integrated design methodology based on homogenization method is unable to reflect scale effect. The three-phase material integrated design is investigated and computational results illustrate the procedure.
An integrated design methodology characterizing Representative Volume Element scale is proposed. Based on the proposed concept of design element (DE), designs of materials and structures are dealt with in a unified way. By changing the periodic arrangement, the scale and aspect ratio of the DE, scale-related effects are well revealed and distinguished in the final optimal results. Furthermore, numerical design problems for 2D layered structures with cellular core, three-phase material structures, circular structures and cylindrical grid structures are investigated to illustrate the proposed approach.
Topology optimization of weakly coupled thermo-elastic problems in steady state is discussed. The optimal design is decomposed into three sub-problems. Numerical results are obtained in different thermal loading cases. Different interpolation schemes are compared for maximizing the thermal conduction efficiency. It is shown that the optimal model affects optimal design subject to distributed heating over the structure. To study the scale-effect, the multi-objective design associated with rigidity and thermal conduction is performed by considering the whole design domain and the periodic partitionof the design domain. Finally, truss structures with uniform temperature distribution are analyzed and optimized. Different penalty approaches are tested to prevent intermediate densities and influences of penalty parameter on optimal results.
回顾了现有解决拓扑优化中数值计算不稳定问题的几种方法，提出适用于非规则网格的广义周长控制方法。该方法以单元间几何距离为计算参数，建立连续二次型广义周长计算表达式，方便近似处理和优化计算，消除了优化结果的棋盘格和中间密度值。根据三相材料拓扑优化中周长的多样性和特殊性，拓展周长定义，提出四种周长控制新方法，其中针对不同类别设计变量分别进行独立周长约束的方法具有最佳控制效能，能获得清晰的材料分布优化结果。大量由两相、三相材料组成的2D和3D结构的拓扑优化算例验证了方法的有效性，为材料和结构的拓扑优化设计奠定了基础。
以均匀化方法为基础，结合有限元数值计算开发实现了材料微结构的等效性能求解和灵敏度计算软件。以此为基础，综合考虑微结构等效性能，建立了多相材料微结构多目标优化模型；利用周长控制方法，开展了两相、三相材料微结构多目标优化设计，获得清晰的微结构构型。从优化模型和实现算法上分析比较了分别以极端性能和给定性能为目标的微结构设计问题，并结合单一体积约束下具有特定性能微结构的优化模型开展了具有特定泊松比的微结构反设计，获得了满足要求的微结构构型，为开展材料／结构的一体化设计铺平了道路。
以自然界多孔材料结构的组织结构特征为指导思想，建立了具有分层梯度变化的材料／结构模型，提出了材料／结构一体化设计方法。采用分步计算策略完成了材料／结构的一体化设计。首先进行结构宏观优化确定材料分布，其次进行细观微结构的精细设计获得了呈梯度分布的多孔材料结构，实现了对结构性能的显著改进。结合材料／结构一体化设计思想，比较了不同长宽比微结构体胞对优化结果的影响，验证了基于均匀化理论的优化设计方法不具备揭示尺度效应的能力。另外，对三相材料构成的材料／结构的一体化优化进一步验证了所建立的模型和方法的有效性。
提出尺度关联的材料／结构一体化设计方法。结合分层梯度的材料／结构设计模型，利用超单元方法和变量关联技术，采用分步计算方法对包含不同表征体胞数的材料／结构进行优化设计，优化结果展现了明显的尺度效应，并显示出设计域包含表征体胞的数目、表征体胞的长宽比例对优化结果的影响。发现随着设计域包含表征体胞数的增加，优化结构性能逐渐趋同于基于均匀化方法的优化结果。对圆环对称结构、圆柱夹芯结构，以及三相材料所构成的材料／结构的优化计算，进一步展现了尺度对优化结果的影响。
研究讨论了弱耦合热弹性结构的拓扑优化设计。基于弱耦合热弹性结构有限元分析计算过程，将弱耦合热弹性结构拓扑优化分解为三类优化子问题，并对子问题分别进行了研究。基于结构散热性能的优化问题比较了不同热载荷形式、不同材料模型时的优化结果，阐述了均布热流载荷下的优化模型对优化结果的影响。非耦合的多目标优化包括宏观结构拓扑优化和周期性材料／结构优化两个方面，通过变化加权因子和边界条件得到不同的优化结果，结果显示了明显的尺度效应。对已知温度场的热力耦合结构的拓扑优化，以桁架结构为例，分析了热、机械载荷的相对大小、材料惩罚模型的不同惩罚因子对优化结果的影响，提出采用不同惩罚模型应对不同类别载荷的处理思想，获得了收敛于0-1的优化解，实现了热力耦合热弹性结构的拓扑优化设计。
Topology optimization has been rapidly developing in the last two decades and is now extended in new application areas at an increasing rate, because it achieves by far greater savings and design improvement than sizing or shape optimization. Its intrinsic capacity of simultaneous optimal design of materials and structures promises to attain higher material and structural efficiency in aerospace and automobile industries, where some indexes such as weight are crucial to the performance of structures. In general, structure optimization and material design are dealt with based on the same topology optimal techniques in different scales. Currently, the potential ability of topology optimization is not fully explored, as material layout and design of microstructure are separated in the design procedure. It is necessary to investigate the integrated design methodology of materials and structures for designing the multifunctional material and structural system and removing the frontier etween materials and structures. The main research work in the dissertation will be introduced as follows:
A generalized perimeter scheme with weak rotational mesh-dependence is proposed to prevent numerical instabilities associated with checkerboards and intermediate densities in topology optimization. The scheme of quadratic form means that it is easy to establish a suitable explicit approximation and favors its implementation in convex programming. Based on the characteristics of density design variables associated with multiphase materials, the general concept of perimeter is extended and four kinds of perimeter control methods are proposed for topology optimization multiphase materials. Several 2D and 3D optimal examples made of two or three different materials phases are carried out for the compliance minimization. It shows that checkerboards and intermediate values of density variables are able to be eliminated efficiently.
Using the homogenization method and the finite element method, the effective properties and the sensitivities with respect to element densities of material microstructures are calculted. A multi-objective model is presented to optimize the topology of the microstructure with two or three-phase materials, subject to constraints on volume fraction and perimeter control. A comparative study is carried out for design of microstructures with the extreme properties and the prescribed elastic properties including the modeling and the algorithm. Several designs of microstructures with the prescribed properties are tested. These works are the basis for the integrated design of materials and structures.
Inspired by hierarchical structure of natural or biological materials, an computational model for the integrated design of materials and structures consisting of hierarchical cellular material levels or layers is described and an integrated design methodology is proposed for the global stiffness maximization of the overall structure and local design of material microstructures based on the homogenization method of multi-scale computing. The optimal design procedure includes two steps. Firstly, the material layout is figured out. Secondly, topologies of microstructures of intermediate density materials are designed. Numerical results show that the proposed method is well adapted to the design of lightweight structures. In addition to this, another issue about influences of microstructure aspect ratio on the optimal design is discussed and numerical experiences show that the integrated design methodology based on homogenization method is unable to reflect scale effect. The three-phase material integrated design is investigated and computational results illustrate the procedure.
An integrated design methodology characterizing Representative Volume Element scale is proposed. Based on the proposed concept of design element (DE), designs of materials and structures are dealt with in a unified way. By changing the periodic arrangement, the scale and aspect ratio of the DE, scale-related effects are well revealed and distinguished in the final optimal results. Furthermore, numerical design problems for 2D layered structures with cellular core, three-phase material structures, circular structures and cylindrical grid structures are investigated to illustrate the proposed approach.
Topology optimization of weakly coupled thermo-elastic problems in steady state is discussed. The optimal design is decomposed into three sub-problems. Numerical results are obtained in different thermal loading cases. Different interpolation schemes are compared for maximizing the thermal conduction efficiency. It is shown that the optimal model affects optimal design subject to distributed heating over the structure. To study the scale-effect, the multi-objective design associated with rigidity and thermal conduction is performed by considering the whole design domain and the periodic partitionof the design domain. Finally, truss structures with uniform temperature distribution are analyzed and optimized. Different penalty approaches are tested to prevent intermediate densities and influences of penalty parameter on optimal results.
引文
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