基于能量流分析的结构动力学拓扑优化理论与方法研究
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摘要
为了提高航天、海洋等复杂装备结构的动力学特性，减轻其重量，并扩大结构的设计空间，本文系统研究了结构动力学拓扑优化的相关理论和方法。搭建了结构拓扑优化与结构能量流分析的理论融合桥梁，基于Rayleigh阻尼对比分析了频率响应、动刚度和特征频率动力学拓扑优化的建模、求解过程和优化结果，提出了基于功率流响应、模态功率流和能量有限元的结构动力学拓扑优化方法，开发了结构动力学拓扑优化的软件集成原型系统。
本文紧紧围绕结构动力学拓扑优化方法和应用面临的关键理论和技术问题展开研究，研究的主要内容包括：结构拓扑优化与结构能量流分析的理论融合；基于Rayleigh阻尼的结构频率响应、动刚度和特征频率动力学拓扑优化；基于功率流响应的结构动力学拓扑优化；基于模态功率流理论的结构动力学拓扑优化；基于能量有限元的结构能量流拓扑优化；基于拓扑优化的结构设计集成系统。论文研究取得的主要创新成果包括：
1、通过研究基于材料分布的结构拓扑优化和结构能量流分析的基本理论，搭建了两种理论的融合桥梁，为结构动力学拓扑优化开辟了一条新的研究途径。
2、基于Rayleigh阻尼全面对比分析频率响应、动刚度和特征频率动力学拓扑优化的建模、求解过程和优化结果，揭示了三种拓扑优化模型的物理内涵、优缺点及其相互联系，为基于能量流分析的结构动力学拓扑优化提供了模型参考标准和结果参考依据。
3、提出了基于功率流响应的结构动力学拓扑优化方法，推导了功率流响应的复模型表达，构建了基于功率流响应的结构动力学拓扑优化数学模型，采用伴随灵敏度分析和移动渐进线的方法对模型进行了求解，讨论了优化结果的减重特性，揭示了加载频率、材料和阻尼因子等设计参数对优化结果的影响规律。
4、提出了基于模态功率流理论的结构动力学拓扑优化方法，构建了基于模态功率流理论的最小输入模态功率流和最大单阶功率流模态的的结构动力学拓扑优化数学模型，基于移动渐进线法分别对模型进行了求解，优化结果与传统功率流分析方法进行了对比分析，讨论了模态功率流理论在结构动力学拓扑优化中的应用可行性和优势。
5、研究了基于能量有限元法的结构能量流拓扑优化设计，实现对结构能量流传播过程的控制和优化。从稳定状态的能量流控制方程出发，基于能量有限元法和材料接合面处的功率流守恒关系推导了双材料结构能量流拓扑优化数学模型。建立了材料密度、弹性模量和阻尼系数等设计参数的双材料SIMP插值函数。利用移动渐近线和拓扑灵敏度分析的方法对模型进行了求解。分析了不同加载频率下的最优拓扑结果，并与动刚度拓扑优化结果进行了定量对比。讨论了结构能量流拓扑优化与结构模态特性的关系，进一步验证了该方法的有效性和可行性。
6、以结构拓扑优化的理论研究成果为指导，构建了基于拓扑优化的结构设计集成系统的工作流程和体系结构，设计了系统的功能和数据模型，并在此基础上开发了原型系统。
总之，本文主要针对结构动力学拓扑优化方法和应用需要解决的关键理论与技术问题展开系统深入研究：进一步完善了结构动力学拓扑优化的理论方法，构建了结构拓扑优化和结构能量流分析的理论融合桥梁，提出了基于功率流响应、模态功率流和能量有限元的结构动力学拓扑优化方法，这些研究成果为面向动力学特性的结构轻量化设计应用奠定了重要的理论基础和技术支撑。
For promoting the dynamic properties, lightening the weight of aerospace/marinestructures, and extending the structural design space, both theoretical and technicalproblems of structural dynamic topology optimization were investigated thoroughly inthis dissertation. Based on comparison and analysis of material distribution basedtopology optimization methods and the structural energy flow theories, a theoreticalbridge between structural topology optimization and structural energy flow analysis wasconstructed. Methods of power flow response based structural dynamic topologyoptimization, power flow mode based structural dynamic topology optimization, andenergy finite element based structural energy flow topology optimization were proposed,and topology optimization based integrating system of structural design was developed.
The dissertation is aimed at solving some key theoretical and technical problems ofstructural dynamic topology optimization. The following five aspects have beeninvestigated: The fusing theory of structural topology optimization and structural energyflow analysis; Rayleigh damping based topological design model of “frequencyresponse”,“dynamic compliance” and “eigen-frequency”; Power flow response basedstructural dynamic topology optimization; Power flow mode theory based structuraldynamic topology optimization; Energy finite element based structural energy flowtopology optimization; Topology optimization based structural design integrated system.The main findings of this dissertation include:
1. Based on the studies of structural topology optimization and structural energyflow theories, a theoretical bridge between material distribution based topologyoptimization and energy flow based structural dynamic analysis was constructed,providing a novel approach for structural dynamic topology optimization.
2. Based on Rayleigh damping, a detail comparison of “frequency response”,“dynamic compliance” and “eigen-frequency” dynamic topology optimization modelswas carried out, and the intrinsic physical meanings, advantages and disadvantages, andcoupling relationships of these three models have been explaned.
3. The method of power flow response based structural dynamic topologyoptimization was proposed. The complex expression of power flow was deduced, basedon which the mathematic optimization model of power flow response based structuraltopological design was established. Adjoint sensitivity analysis and the method ofmoving asymptotes (MMA) were apployed for solving the model, and the effectmechanics of the loading frequency, damping factors and material parameters on theoptimum topology was discovered.
4. The method of power flow mode theory based structural dynamic topologyoptimization was proposed. The basic theory of power flow mode was reviewed, based on which the mathematic optimization models of “minimizing the input mode powerflow” and “separated power flow mode” were established. Sensitivity analysis and themethod of moving asymptotes were apployed for solving the models, and the effectmechanics of the loading frequency and damping coefficients on the optimum topologywas analyzed. The availability and advantage of power flow mode theory for thedynamic topology design were discussed.
5. Energy finite element method (EFEM) was employed for bi-material structuraltopology optimization of energy flow problems, such that the propagation process of theenergy flow can be controlled and optimized. Based on the energy flow governingequation in steady state, the discrete optimization model was formulated, where therelation of power flow conservation across the material junctions was applyed. Thebi-material interpolation model of material density, Young’s modulus, and dampingfactor are constructed using SIMP principle. The optimization model was solved usingMMA and a direct differentiation sensitivity analysis method. The method wasvalidated by comparison with dynamic compliance and eigenfrequency topologicaldesign results.
6. Based on the theoretical foundings, topology optimization based structuraldesign integrating system was developed. The system process, structure, function anddata model were designed, based on which the prototype system was developed.
In summary, this dissertation was concerned with some key theoretical andtechnical problems involved in the researches and applications of structural dynamictopology optimization. The main contributions of this dissertation include:1) atheoretical bridge between structural topology optimization and energy flow analysiswas constructed;2) power flow response based structural dynamic topologyoptimization was proposed;3) power flow mode theory based structural dynamictopology optimization was proposed;4) EFEM based energy flow topologyoptimization was proposed. The results of this dissertation provide significanttheoretical foundations and technical guidelines to support the practical.
本文紧紧围绕结构动力学拓扑优化方法和应用面临的关键理论和技术问题展开研究，研究的主要内容包括：结构拓扑优化与结构能量流分析的理论融合；基于Rayleigh阻尼的结构频率响应、动刚度和特征频率动力学拓扑优化；基于功率流响应的结构动力学拓扑优化；基于模态功率流理论的结构动力学拓扑优化；基于能量有限元的结构能量流拓扑优化；基于拓扑优化的结构设计集成系统。论文研究取得的主要创新成果包括：
1、通过研究基于材料分布的结构拓扑优化和结构能量流分析的基本理论，搭建了两种理论的融合桥梁，为结构动力学拓扑优化开辟了一条新的研究途径。
2、基于Rayleigh阻尼全面对比分析频率响应、动刚度和特征频率动力学拓扑优化的建模、求解过程和优化结果，揭示了三种拓扑优化模型的物理内涵、优缺点及其相互联系，为基于能量流分析的结构动力学拓扑优化提供了模型参考标准和结果参考依据。
3、提出了基于功率流响应的结构动力学拓扑优化方法，推导了功率流响应的复模型表达，构建了基于功率流响应的结构动力学拓扑优化数学模型，采用伴随灵敏度分析和移动渐进线的方法对模型进行了求解，讨论了优化结果的减重特性，揭示了加载频率、材料和阻尼因子等设计参数对优化结果的影响规律。
4、提出了基于模态功率流理论的结构动力学拓扑优化方法，构建了基于模态功率流理论的最小输入模态功率流和最大单阶功率流模态的的结构动力学拓扑优化数学模型，基于移动渐进线法分别对模型进行了求解，优化结果与传统功率流分析方法进行了对比分析，讨论了模态功率流理论在结构动力学拓扑优化中的应用可行性和优势。
5、研究了基于能量有限元法的结构能量流拓扑优化设计，实现对结构能量流传播过程的控制和优化。从稳定状态的能量流控制方程出发，基于能量有限元法和材料接合面处的功率流守恒关系推导了双材料结构能量流拓扑优化数学模型。建立了材料密度、弹性模量和阻尼系数等设计参数的双材料SIMP插值函数。利用移动渐近线和拓扑灵敏度分析的方法对模型进行了求解。分析了不同加载频率下的最优拓扑结果，并与动刚度拓扑优化结果进行了定量对比。讨论了结构能量流拓扑优化与结构模态特性的关系，进一步验证了该方法的有效性和可行性。
6、以结构拓扑优化的理论研究成果为指导，构建了基于拓扑优化的结构设计集成系统的工作流程和体系结构，设计了系统的功能和数据模型，并在此基础上开发了原型系统。
总之，本文主要针对结构动力学拓扑优化方法和应用需要解决的关键理论与技术问题展开系统深入研究：进一步完善了结构动力学拓扑优化的理论方法，构建了结构拓扑优化和结构能量流分析的理论融合桥梁，提出了基于功率流响应、模态功率流和能量有限元的结构动力学拓扑优化方法，这些研究成果为面向动力学特性的结构轻量化设计应用奠定了重要的理论基础和技术支撑。
For promoting the dynamic properties, lightening the weight of aerospace/marinestructures, and extending the structural design space, both theoretical and technicalproblems of structural dynamic topology optimization were investigated thoroughly inthis dissertation. Based on comparison and analysis of material distribution basedtopology optimization methods and the structural energy flow theories, a theoreticalbridge between structural topology optimization and structural energy flow analysis wasconstructed. Methods of power flow response based structural dynamic topologyoptimization, power flow mode based structural dynamic topology optimization, andenergy finite element based structural energy flow topology optimization were proposed,and topology optimization based integrating system of structural design was developed.
The dissertation is aimed at solving some key theoretical and technical problems ofstructural dynamic topology optimization. The following five aspects have beeninvestigated: The fusing theory of structural topology optimization and structural energyflow analysis; Rayleigh damping based topological design model of “frequencyresponse”,“dynamic compliance” and “eigen-frequency”; Power flow response basedstructural dynamic topology optimization; Power flow mode theory based structuraldynamic topology optimization; Energy finite element based structural energy flowtopology optimization; Topology optimization based structural design integrated system.The main findings of this dissertation include:
1. Based on the studies of structural topology optimization and structural energyflow theories, a theoretical bridge between material distribution based topologyoptimization and energy flow based structural dynamic analysis was constructed,providing a novel approach for structural dynamic topology optimization.
2. Based on Rayleigh damping, a detail comparison of “frequency response”,“dynamic compliance” and “eigen-frequency” dynamic topology optimization modelswas carried out, and the intrinsic physical meanings, advantages and disadvantages, andcoupling relationships of these three models have been explaned.
3. The method of power flow response based structural dynamic topologyoptimization was proposed. The complex expression of power flow was deduced, basedon which the mathematic optimization model of power flow response based structuraltopological design was established. Adjoint sensitivity analysis and the method ofmoving asymptotes (MMA) were apployed for solving the model, and the effectmechanics of the loading frequency, damping factors and material parameters on theoptimum topology was discovered.
4. The method of power flow mode theory based structural dynamic topologyoptimization was proposed. The basic theory of power flow mode was reviewed, based on which the mathematic optimization models of “minimizing the input mode powerflow” and “separated power flow mode” were established. Sensitivity analysis and themethod of moving asymptotes were apployed for solving the models, and the effectmechanics of the loading frequency and damping coefficients on the optimum topologywas analyzed. The availability and advantage of power flow mode theory for thedynamic topology design were discussed.
5. Energy finite element method (EFEM) was employed for bi-material structuraltopology optimization of energy flow problems, such that the propagation process of theenergy flow can be controlled and optimized. Based on the energy flow governingequation in steady state, the discrete optimization model was formulated, where therelation of power flow conservation across the material junctions was applyed. Thebi-material interpolation model of material density, Young’s modulus, and dampingfactor are constructed using SIMP principle. The optimization model was solved usingMMA and a direct differentiation sensitivity analysis method. The method wasvalidated by comparison with dynamic compliance and eigenfrequency topologicaldesign results.
6. Based on the theoretical foundings, topology optimization based structuraldesign integrating system was developed. The system process, structure, function anddata model were designed, based on which the prototype system was developed.
In summary, this dissertation was concerned with some key theoretical andtechnical problems involved in the researches and applications of structural dynamictopology optimization. The main contributions of this dissertation include:1) atheoretical bridge between structural topology optimization and energy flow analysiswas constructed;2) power flow response based structural dynamic topologyoptimization was proposed;3) power flow mode theory based structural dynamictopology optimization was proposed;4) EFEM based energy flow topologyoptimization was proposed. The results of this dissertation provide significanttheoretical foundations and technical guidelines to support the practical.
引文
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