基于结构和材料一体化的多尺度动力优化设计
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摘要
本文采用结构优化技术研究结构和材料动力性能优化的一体化设计方法。为实现这一目标,针对有广泛应用前景的有序多孔材料结构和纤维增强复合材料结构,提出多尺度动力优化方法,得到考虑动力性能指标的材料微观拓扑和结构宏观拓扑的最优形式。
本文的主要研究内容如下：
1.提出、改进和实现了由周期性微结构的材料构成的结构的等效分析方法。包括：对于正交各向异性三角形和Kagome单胞蜂窝材料,推导了其等效刚度和强度的解析表达式,给出了初始屈服函数和近似弹性屈曲强度,讨论了等效刚度与各向异性率和相对密度的关系；采用数值均匀化方法对三维类桁架材料组成的结构进行等效弹塑性分析,并和离散建模的空间桁架计算进行对比,说明等效分析可以大大节省计算时间,缩小计算规模；对蜂窝材料等效为微极连续介质的多种不同方法进行总结,注意到对同一种单胞不同文献得到不同本构关系参数,提出了一种基于边界位移连续和内部平衡约束的推导微极等效本构参数的新方法。以正方形和六角形单胞制成的结构为例,在多种条件下比较了离散完全计算、经典连续介质等效和不同微极连续体等效本构的计算结果,建议了较好的微极本构参数值。
2.通过回顾经典的给定材料体积约束下最小化结构柔顺性(最大刚度)拓扑优化问题,对比不同的优化求解方法得到的拓扑优化结果最优KKT (Karush-Kuhn-Tucker)条件的满足情况。进而,基于广义最优准则和动力优化连续化方法,考虑激励频率在一个频率带内变化的简谐激励作用下,振动的双材料板动态柔顺性和材料使用量协同最小化设计问题。
3.基于结构和材料一体化设计概念,研究复合材料结构的振动与声学优化设计问题。考虑具有指定频率和振幅的简谐激励作用下复合材料层合板的振动,设计目标是最小化从层合板表面辐射到周围声学介质中的声功率。考虑到复合材料层合板具有较平的表面构型,采用近似的Rayleigh积分计算从板表面辐射的声功率。采用离散材料设计方法实现纤维角度、铺层顺序及多材料选择的优化。
4.提出一种以基频最大化为目标的宏观结构和微观单胞并发优化的方法。在此方法中,具有两类独立的设计变量,分别为宏观和微观单元密度,通过均匀化方法两个尺度的优化统一到同一框架下,基体材料自动实现在宏微观两个层次的分布。并且假设宏观结构由均一的微观单胞周期性排列形成,方便制造。论文还将此两尺度并发设计方法推广到多孔板的面外振动问题,说明本方法的广泛适用性。
本论文工作得到国家自然科学基金(编号：90816025,10332010)和国家重点基础研究发展计划(编号2006CB601205)的资助,在此表示感谢。
This dissertation aims at extending structural optimization techniques to the integrated design of structures and materials used in dynamic environments. In order to achieve this, methods are proposed to realize the multi-scale dynamic design optimization of structures made of promising cellular material and composite material. The ultralight microstructure in the microscale and structural configuration in the macroscale with optimum dynamic performances are obtained simultaneously by the multi-scale optimization. The main works of this dissertation are as follows:
1. For materials with periodic microstructures, equivalent analysis methods are proposed and improved in order to obtain effective properties of the material and realize multiscale analysis of the macrostructure made of the periodic material. First, effective elastic stiffness and initial yield strength and elastic buckling limits are derived in analytical forms for the periodic honeycomb structures with orthotropic isosceles triangle and Kagome cells. The dependence of the effective stiffness on the shape anisotropic ratio and relative density is discussed. Second, based on the numerical homogenization method, the elasto-plastic behavior is equivalently analyzed for the three dimensional (3D) heterogeneous structure made of truss-like material with periodic microstructures. The comparison with results of discrete modeling in time consumption and precision verifies the advantages of the effective multiscale analysis. Finally, methods in literatures to realize the micropolar equivalent continuum modeling of cellular materials are summarized. Different effective micropolar constants are observed in different papers for the same microstructure. A new approach is proposed to formulate the equivalent micropolar constitutive relation of 2-D periodic cellular materials. The new approach takes both displacement compatibility on the boundary and equilibrium inside the cell into account in the micromechanical analysis of a cell structure. The solutions from the classical Cauchy-type continuum and three kinds of micropolar continuum modelings using different effective micropolar constants by different authors are compared with the exact discrete simulations of the same structure under various conditions. It is found that the micropolar constants developed in this dissertation give satisfying results of equivalent analysis for square, triangular and hexagonal cells.
2. For the well-known minimum compliance problem with the constraint of given material amount, the satisfactions of the Karush-Kuhn-Tucker necessary conditions for optimality are checked for different optimum solutions obtained from different optimization algorithms. Furthermore, using the extended optimality and continuation technique, the problem of minimizing the product of the dynamic compliance and the total cost of material is considered for vibrating bi-material plate structures subjected to time-harmonic external mechanical loading with a given band of external excitation frequencies. This objective appears to facilitate reduction of vibration by driving the resonance frequencies of the structure as far away as possible from the prescribed range of excitation frequencies while simultaneously minimizing the amount of material to be used for the structure.
3. Based on the integrated design of structure and material, the problem of vibro-acoustic optimization of laminated composite plates is considered. The vibration of the laminated plate is excited by time-harmonic external mechanical loading with prescribed frequency and amplitude, and the design objective is to minimize the total sound power radiated from the surface of the laminated plate to the surrounding acoustic medium. Instead of solving the Helmholtz equation for evaluation of the sound power, advantage is taken of the fact that the surface of the laminated plate is flat, which implies that Rayleigh's integral approximation can be used to evaluate the sound power radiated from the surface of the plate. The Discrete Material Optimization (DMO) formulation has been applied to achieve the design optimization of fiber angles, stacking sequence and selection of material for laminated composite plates.
4. A two-scale optimization method is developed to realize the integrated design of macro-structure and micro-structure of cellular material for maximizing structural fundamental eigenfrequency. In this method macro and micro densities are introduced as independent design variables for the macrostructure and microstructure respectively: Optimizations at two scales are integrated into one system through homogenization theory and base material is distributed between the two scales automatically with the optimization model. Microstructure of materials is assumed to be homogeneous at the macro scale to meet today's manufacture practice and reduce manufacturing cost. Plane structure with homogeneous cellular material and perforated plate are studied. Numerical experiments validate the proposed method and computational model.
The research of this dissertation was partially supported by the National Natural Science Foundation of China (Grant No.90816025,10332010), and National Basic Research Program of China (Grant No.2006CB601205). This support is gratefully acknowledged by the author.
本文的主要研究内容如下：
1.提出、改进和实现了由周期性微结构的材料构成的结构的等效分析方法。包括：对于正交各向异性三角形和Kagome单胞蜂窝材料,推导了其等效刚度和强度的解析表达式,给出了初始屈服函数和近似弹性屈曲强度,讨论了等效刚度与各向异性率和相对密度的关系；采用数值均匀化方法对三维类桁架材料组成的结构进行等效弹塑性分析,并和离散建模的空间桁架计算进行对比,说明等效分析可以大大节省计算时间,缩小计算规模；对蜂窝材料等效为微极连续介质的多种不同方法进行总结,注意到对同一种单胞不同文献得到不同本构关系参数,提出了一种基于边界位移连续和内部平衡约束的推导微极等效本构参数的新方法。以正方形和六角形单胞制成的结构为例,在多种条件下比较了离散完全计算、经典连续介质等效和不同微极连续体等效本构的计算结果,建议了较好的微极本构参数值。
2.通过回顾经典的给定材料体积约束下最小化结构柔顺性(最大刚度)拓扑优化问题,对比不同的优化求解方法得到的拓扑优化结果最优KKT (Karush-Kuhn-Tucker)条件的满足情况。进而,基于广义最优准则和动力优化连续化方法,考虑激励频率在一个频率带内变化的简谐激励作用下,振动的双材料板动态柔顺性和材料使用量协同最小化设计问题。
3.基于结构和材料一体化设计概念,研究复合材料结构的振动与声学优化设计问题。考虑具有指定频率和振幅的简谐激励作用下复合材料层合板的振动,设计目标是最小化从层合板表面辐射到周围声学介质中的声功率。考虑到复合材料层合板具有较平的表面构型,采用近似的Rayleigh积分计算从板表面辐射的声功率。采用离散材料设计方法实现纤维角度、铺层顺序及多材料选择的优化。
4.提出一种以基频最大化为目标的宏观结构和微观单胞并发优化的方法。在此方法中,具有两类独立的设计变量,分别为宏观和微观单元密度,通过均匀化方法两个尺度的优化统一到同一框架下,基体材料自动实现在宏微观两个层次的分布。并且假设宏观结构由均一的微观单胞周期性排列形成,方便制造。论文还将此两尺度并发设计方法推广到多孔板的面外振动问题,说明本方法的广泛适用性。
本论文工作得到国家自然科学基金(编号：90816025,10332010)和国家重点基础研究发展计划(编号2006CB601205)的资助,在此表示感谢。
This dissertation aims at extending structural optimization techniques to the integrated design of structures and materials used in dynamic environments. In order to achieve this, methods are proposed to realize the multi-scale dynamic design optimization of structures made of promising cellular material and composite material. The ultralight microstructure in the microscale and structural configuration in the macroscale with optimum dynamic performances are obtained simultaneously by the multi-scale optimization. The main works of this dissertation are as follows:
1. For materials with periodic microstructures, equivalent analysis methods are proposed and improved in order to obtain effective properties of the material and realize multiscale analysis of the macrostructure made of the periodic material. First, effective elastic stiffness and initial yield strength and elastic buckling limits are derived in analytical forms for the periodic honeycomb structures with orthotropic isosceles triangle and Kagome cells. The dependence of the effective stiffness on the shape anisotropic ratio and relative density is discussed. Second, based on the numerical homogenization method, the elasto-plastic behavior is equivalently analyzed for the three dimensional (3D) heterogeneous structure made of truss-like material with periodic microstructures. The comparison with results of discrete modeling in time consumption and precision verifies the advantages of the effective multiscale analysis. Finally, methods in literatures to realize the micropolar equivalent continuum modeling of cellular materials are summarized. Different effective micropolar constants are observed in different papers for the same microstructure. A new approach is proposed to formulate the equivalent micropolar constitutive relation of 2-D periodic cellular materials. The new approach takes both displacement compatibility on the boundary and equilibrium inside the cell into account in the micromechanical analysis of a cell structure. The solutions from the classical Cauchy-type continuum and three kinds of micropolar continuum modelings using different effective micropolar constants by different authors are compared with the exact discrete simulations of the same structure under various conditions. It is found that the micropolar constants developed in this dissertation give satisfying results of equivalent analysis for square, triangular and hexagonal cells.
2. For the well-known minimum compliance problem with the constraint of given material amount, the satisfactions of the Karush-Kuhn-Tucker necessary conditions for optimality are checked for different optimum solutions obtained from different optimization algorithms. Furthermore, using the extended optimality and continuation technique, the problem of minimizing the product of the dynamic compliance and the total cost of material is considered for vibrating bi-material plate structures subjected to time-harmonic external mechanical loading with a given band of external excitation frequencies. This objective appears to facilitate reduction of vibration by driving the resonance frequencies of the structure as far away as possible from the prescribed range of excitation frequencies while simultaneously minimizing the amount of material to be used for the structure.
3. Based on the integrated design of structure and material, the problem of vibro-acoustic optimization of laminated composite plates is considered. The vibration of the laminated plate is excited by time-harmonic external mechanical loading with prescribed frequency and amplitude, and the design objective is to minimize the total sound power radiated from the surface of the laminated plate to the surrounding acoustic medium. Instead of solving the Helmholtz equation for evaluation of the sound power, advantage is taken of the fact that the surface of the laminated plate is flat, which implies that Rayleigh's integral approximation can be used to evaluate the sound power radiated from the surface of the plate. The Discrete Material Optimization (DMO) formulation has been applied to achieve the design optimization of fiber angles, stacking sequence and selection of material for laminated composite plates.
4. A two-scale optimization method is developed to realize the integrated design of macro-structure and micro-structure of cellular material for maximizing structural fundamental eigenfrequency. In this method macro and micro densities are introduced as independent design variables for the macrostructure and microstructure respectively: Optimizations at two scales are integrated into one system through homogenization theory and base material is distributed between the two scales automatically with the optimization model. Microstructure of materials is assumed to be homogeneous at the macro scale to meet today's manufacture practice and reduce manufacturing cost. Plane structure with homogeneous cellular material and perforated plate are studied. Numerical experiments validate the proposed method and computational model.
The research of this dissertation was partially supported by the National Natural Science Foundation of China (Grant No.90816025,10332010), and National Basic Research Program of China (Grant No.2006CB601205). This support is gratefully acknowledged by the author.
引文
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