超轻金属结构与材料性能多尺度分析与协同优化设计
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摘要
结构物的轻量化设计对于降低产品生产和使用成本、减少长期服役能耗、提高产品性能都具有重要的意义，随着当代能源与资源的短缺及竞争的加剧，轻量化设计受到各方的关注。而随着制备工艺的成熟，超轻金属多孔材料(点阵类桁架材料、线性金属蜂窝材料、泡沫金属)越来越多的应用于工程实践，其卓越的比刚度、比强度及多孔连通性使其成为新一代轻质多功能的结构功能材料。
本论文围绕微结构具有周期性排布特点的超轻金属多孔材料，针对结构与材料性能分析方法与协同优化设计两方面展开了一系列的研究工作。具体内容如下：
1．描述并实现了适用于类桁架点阵材料等效性能预测的均匀化方法及列式：研究了基于Dirichlet型、Neumann型及周期性边界条件下的代表体元法预测类桁架点阵材料等效弹性模量方法；对代表体元法所预测的弹性性能随参与计算的单胞个数n变化而变化的尺寸效应进行了研究，指出Neumman边界条件下的单胞边界变形协调性或者Dilichlet边界条件下的边界节点力的平衡性，是产生上述尺寸效应的本质原因，也可以作为判断是否产生尺寸效应的简单判据。开展了基于均匀化理论的2D桁架材料极值剪切性能的形状优化研究，并对其中出现的奇异现象进行了分析。(第二章)
2．利用数值模拟，定量地对比了将LCAs(一种重要的类桁架点阵材料)材料等效为经典的柯西介质与微极连续体等效介质的计算精度，发现由柯西介质模型计算得到的位移和应力都存在较大的误差，具有非局部本构的微极连续体等效模型是较为合理的选择。基于能量法等效分析的结果提出了一种映射计算单胞构件微观应力的快速算法。
将具有正方形单胞的LCAs材料等效为微极连续介质，运用拓扑优化思想，以反映材料宏观特性的材料相对密度ρ和微观特性的微单胞孔径L为设计变量，进行结构应力优化。并对经典的小孔应力集中算例，分别以最小化孔边应力、结构最大应力最小、最小化孔边应力与材料屈服强度比值为目标，给出了结构与材料一体化协同设计结果，同时探讨了材料铺角对优化结果的影响；最后根据连续体等效介质模型优化的结果，建立了细致的刚架模型，通过离散建模计算验证了本文方法的有效性。(第三章)
3．针对可以通过基本设计模块周期性拼装而成的结构，研究了此类结构和模块协同优化设计的方法和模型，同时考察了基本设计模块的绝对尺寸对优化结果的影响。通过在结构和设计模块两个层次上分别引入独立的密度变量，实现了基于最优设计模块拼装的宏微观协同优化设计，采用拓扑优化技术和子结构分析方法，探讨了此种情况下最优的设计模块构形以及这种模块在结构尺度上的最优分布。(第四章)
4．基于可制造性考虑，研究了由宏观上均匀的多孔材料制成的结构与材料协同优化设计问题。待设计的结构受到给定的外力和温度载荷作用，优化设计旨在给定允许使用的材料体积约束下，设计宏观结构的拓扑及多孔材料的微结构，使得结构柔度最小。建立了一种宏观结构与微观单胞构型协同优化设计的模型和方法，在此方法中，我们引入宏观密度和微观密度两类设计变量，在微观层次上采用带惩罚的实心各向同性材料法(SIMP：Solid Isotropic Material with Penalty)，在宏观层次上采用带惩罚的多孔各向异性材料法(PAMP：Porous Anisotropic Material with penalty)，借助均匀化方法建立两个层次间的联系，通过优化方法自动确定实体材料在结构与材料两个层次上的分配，得到优化设计。讨论了温度变化、材料体积及计算参数对优化结果的影响。研究结果表明只有机械载荷作用时，基于柔顺性指标的最优微单胞构形往往是各向同性的实体材料；而同时考虑热和机械载荷时，采用多孔材料可以降低结构柔顺性。(第五章)
5．针对工程中常见的旋转对称结构，将它划分为有限个基本设计模块，而在设计模块内应用基于均匀化方法的结构与材料协同设计优化设计策略，对同时作用有集中力与离心力的旋转对称结构，给出了最优的模块构型以及构成这种模块的材料的最优微结构形式。研究了给定材料用量、不同载荷组合以及非可设计域对协同优化结果的影响，发现当同时作用有离心力与集中力时，多孔材料可以有效的提高系统刚度。(第六章)
Light weight design of structures can reduce products' mamufacturing cost and energy consumption, which greatly improve the quality of products. And it has been the focus and forever topics of researchers and engineers with shortage of energy and resources and intense competitions. With the rapid developments in mamufacturing techniques, more and more ultra-light metal porous materials (truss-like materials, linear celluar alloys, metal foams) are utilized in practice engineerings. They have received increasing attentions for their high stiffness-weight and strength-weight ratios together with potential of multifunctional applications.
A serial of researches on multiscale analysis and concurrent optimization for ultra-light metal structures and materials have been carried out. And the main works are as follows.
1. This paper compares the representative volume element (RVE) method based on Diriehlet and Neumann boundary conditions with the homogenization method for predicting the effective elastic property of truss-like materials with periodic microstructure. The formula of homogenization method are developed and implemented for truss-like materials. Numerical experiments show that, with increase of the number of unit cell, n, the results of RVE method under the Dirichlet and Neumann boundary conditions converge towards those obtained with homogenization method from the above and below sides respectively. For some specific types of the unit cell, RVE method gives the same results as those obtained with homogenization method even if only one unit cell is included. For RVE method, a simple criterion for judging existence of scale effects is whether the equilibrium of the boundary nodal forces is guaranteed under the Dirichlet boundary conditions or whether the deformation compatibility at the unit cell boundaries is satisfied under the Neumann boundary conditions. Finally, shape optimization technique is applied to find the optimal geometric shape of unit cell for truss-Iike materials with the maximum and minimum shear stiffness and the numerical singularity involved is discussed as well. (Chapter Two)
2. For LCAs (Linear Celluar Alloys), the precision of classic Cauchy effective continumm and micropolar effective contimumm are compared with the results from exact discrete models in wich each cell wall is modeled as a beam element. Numerical simulations show that micropolar effective continumm is preferred considering either from displacement or stress results. And a fast mapping algorithm for micro-stress distribution in truss-like materials is developmented based on results with micropolar continuum representation obtained by energy method.
Baed on the above analysis, optimum stress distribution for hollow plates composed of LCAs is investigated. To reduce the computational cost we model the material as micropolar continua representation. Two classes of design variables, relative density and cell size distribution of truss-like materials are to be determined by optimization under given total material volume constraint. And the concurrent designs of material and structure are obtained for three different optimization formulations. For the first formulation, we aim at minimization of the maximum stress which appears at the initial uniform design; for the second formulation, we minimize the highest stress within the specified point set. Since the yield strength of truss-like material is dependent on the relative material density, we minimize the ratio of stress over the corresponding yield strength along the hole boundary in our third formulation, which maximizes the strength reserve and seems more rational. And the influence of ply angle on the optimum result is discussed. The dependence of optimum design on finite element meshes is also investigated. An approximate discrete model is established to verify the method proposed in this paper and the stress concentration near a hole is reduced significantly. (Chapter Three)
3. The optimization model and technique for modular structures and materials are discussed. The effects of acutral dimensions of basic design modules on optimization results are investigated for those structures that are composed by periodic distribution of basic design modules. The concurrent optimizations of macro structures and design modules based on modules assembling are implemented by introducing independent densities in macro and micro scale as design variables. The optimum configuration and distribution of basic design modules are discussed via to topology optimization technique and sub-structure method. (Chapter Four)
4. An optimization technique for structures composed of uniform porous materials in macro scale is developed based on manufacturing requirements. The optimization aims at to obtain optimal configurations of macro scale structures and microstructure of materials under certain mechanical and thermal loads with specific base material volume. A concurrent topology optimization method is proposed for structures and materials to minimize compliance of thermoelastic structures. In this method macro and micro densities are introduced as the design variables for structure and material microstructure independently. Penalization approaches are adopted at both scales to ensure clear topologies, i.e. SIMP (Solid Isotropic Material Penalization) in micro-scale and PAMP (Porous Anisotropic Material Penalization) in macro-scale. Optimizations in two scales are integrated into one system with homogenization theory and distribution of base material between two scales can be decided automatically by the optimization model. Microstructure of materials is assumed to be uniform at macro scale to reduce manufacturing cost. The proposed method and computational model are validated by the numerical experiments. The effects of temperature differential, volume of base material, numerical parameters on the optimum results are also discussed. At last, for cases in which only mechanical load apply, the optimum configuration of micro structure is isotropic solid materials; for eases in which both mechanical and thermal loads apply, the configuration of porous material can help to reduce the system compliance. (Chapter Five)
5. Cyclic-symmetry structures with both mechanical and centrifugal loads are divided into finite design modules, which have the identical configurations. Concurrent optimization techniques based on homogenization method and penalty technique are applied for cyclic-symmetry structures assembled by basic design modules considering the real dimension of design modules. The effects of solid material volume, different loads combination and nondesign fields on optimum results are discussed. Numerical simulations show that for cases in which both mechanical and centrifugal loads apply, the configuration of porous material can help to reduce the system compliance.
本论文围绕微结构具有周期性排布特点的超轻金属多孔材料，针对结构与材料性能分析方法与协同优化设计两方面展开了一系列的研究工作。具体内容如下：
1．描述并实现了适用于类桁架点阵材料等效性能预测的均匀化方法及列式：研究了基于Dirichlet型、Neumann型及周期性边界条件下的代表体元法预测类桁架点阵材料等效弹性模量方法；对代表体元法所预测的弹性性能随参与计算的单胞个数n变化而变化的尺寸效应进行了研究，指出Neumman边界条件下的单胞边界变形协调性或者Dilichlet边界条件下的边界节点力的平衡性，是产生上述尺寸效应的本质原因，也可以作为判断是否产生尺寸效应的简单判据。开展了基于均匀化理论的2D桁架材料极值剪切性能的形状优化研究，并对其中出现的奇异现象进行了分析。(第二章)
2．利用数值模拟，定量地对比了将LCAs(一种重要的类桁架点阵材料)材料等效为经典的柯西介质与微极连续体等效介质的计算精度，发现由柯西介质模型计算得到的位移和应力都存在较大的误差，具有非局部本构的微极连续体等效模型是较为合理的选择。基于能量法等效分析的结果提出了一种映射计算单胞构件微观应力的快速算法。
将具有正方形单胞的LCAs材料等效为微极连续介质，运用拓扑优化思想，以反映材料宏观特性的材料相对密度ρ和微观特性的微单胞孔径L为设计变量，进行结构应力优化。并对经典的小孔应力集中算例，分别以最小化孔边应力、结构最大应力最小、最小化孔边应力与材料屈服强度比值为目标，给出了结构与材料一体化协同设计结果，同时探讨了材料铺角对优化结果的影响；最后根据连续体等效介质模型优化的结果，建立了细致的刚架模型，通过离散建模计算验证了本文方法的有效性。(第三章)
3．针对可以通过基本设计模块周期性拼装而成的结构，研究了此类结构和模块协同优化设计的方法和模型，同时考察了基本设计模块的绝对尺寸对优化结果的影响。通过在结构和设计模块两个层次上分别引入独立的密度变量，实现了基于最优设计模块拼装的宏微观协同优化设计，采用拓扑优化技术和子结构分析方法，探讨了此种情况下最优的设计模块构形以及这种模块在结构尺度上的最优分布。(第四章)
4．基于可制造性考虑，研究了由宏观上均匀的多孔材料制成的结构与材料协同优化设计问题。待设计的结构受到给定的外力和温度载荷作用，优化设计旨在给定允许使用的材料体积约束下，设计宏观结构的拓扑及多孔材料的微结构，使得结构柔度最小。建立了一种宏观结构与微观单胞构型协同优化设计的模型和方法，在此方法中，我们引入宏观密度和微观密度两类设计变量，在微观层次上采用带惩罚的实心各向同性材料法(SIMP：Solid Isotropic Material with Penalty)，在宏观层次上采用带惩罚的多孔各向异性材料法(PAMP：Porous Anisotropic Material with penalty)，借助均匀化方法建立两个层次间的联系，通过优化方法自动确定实体材料在结构与材料两个层次上的分配，得到优化设计。讨论了温度变化、材料体积及计算参数对优化结果的影响。研究结果表明只有机械载荷作用时，基于柔顺性指标的最优微单胞构形往往是各向同性的实体材料；而同时考虑热和机械载荷时，采用多孔材料可以降低结构柔顺性。(第五章)
5．针对工程中常见的旋转对称结构，将它划分为有限个基本设计模块，而在设计模块内应用基于均匀化方法的结构与材料协同设计优化设计策略，对同时作用有集中力与离心力的旋转对称结构，给出了最优的模块构型以及构成这种模块的材料的最优微结构形式。研究了给定材料用量、不同载荷组合以及非可设计域对协同优化结果的影响，发现当同时作用有离心力与集中力时，多孔材料可以有效的提高系统刚度。(第六章)
Light weight design of structures can reduce products' mamufacturing cost and energy consumption, which greatly improve the quality of products. And it has been the focus and forever topics of researchers and engineers with shortage of energy and resources and intense competitions. With the rapid developments in mamufacturing techniques, more and more ultra-light metal porous materials (truss-like materials, linear celluar alloys, metal foams) are utilized in practice engineerings. They have received increasing attentions for their high stiffness-weight and strength-weight ratios together with potential of multifunctional applications.
A serial of researches on multiscale analysis and concurrent optimization for ultra-light metal structures and materials have been carried out. And the main works are as follows.
1. This paper compares the representative volume element (RVE) method based on Diriehlet and Neumann boundary conditions with the homogenization method for predicting the effective elastic property of truss-like materials with periodic microstructure. The formula of homogenization method are developed and implemented for truss-like materials. Numerical experiments show that, with increase of the number of unit cell, n, the results of RVE method under the Dirichlet and Neumann boundary conditions converge towards those obtained with homogenization method from the above and below sides respectively. For some specific types of the unit cell, RVE method gives the same results as those obtained with homogenization method even if only one unit cell is included. For RVE method, a simple criterion for judging existence of scale effects is whether the equilibrium of the boundary nodal forces is guaranteed under the Dirichlet boundary conditions or whether the deformation compatibility at the unit cell boundaries is satisfied under the Neumann boundary conditions. Finally, shape optimization technique is applied to find the optimal geometric shape of unit cell for truss-Iike materials with the maximum and minimum shear stiffness and the numerical singularity involved is discussed as well. (Chapter Two)
2. For LCAs (Linear Celluar Alloys), the precision of classic Cauchy effective continumm and micropolar effective contimumm are compared with the results from exact discrete models in wich each cell wall is modeled as a beam element. Numerical simulations show that micropolar effective continumm is preferred considering either from displacement or stress results. And a fast mapping algorithm for micro-stress distribution in truss-like materials is developmented based on results with micropolar continuum representation obtained by energy method.
Baed on the above analysis, optimum stress distribution for hollow plates composed of LCAs is investigated. To reduce the computational cost we model the material as micropolar continua representation. Two classes of design variables, relative density and cell size distribution of truss-like materials are to be determined by optimization under given total material volume constraint. And the concurrent designs of material and structure are obtained for three different optimization formulations. For the first formulation, we aim at minimization of the maximum stress which appears at the initial uniform design; for the second formulation, we minimize the highest stress within the specified point set. Since the yield strength of truss-like material is dependent on the relative material density, we minimize the ratio of stress over the corresponding yield strength along the hole boundary in our third formulation, which maximizes the strength reserve and seems more rational. And the influence of ply angle on the optimum result is discussed. The dependence of optimum design on finite element meshes is also investigated. An approximate discrete model is established to verify the method proposed in this paper and the stress concentration near a hole is reduced significantly. (Chapter Three)
3. The optimization model and technique for modular structures and materials are discussed. The effects of acutral dimensions of basic design modules on optimization results are investigated for those structures that are composed by periodic distribution of basic design modules. The concurrent optimizations of macro structures and design modules based on modules assembling are implemented by introducing independent densities in macro and micro scale as design variables. The optimum configuration and distribution of basic design modules are discussed via to topology optimization technique and sub-structure method. (Chapter Four)
4. An optimization technique for structures composed of uniform porous materials in macro scale is developed based on manufacturing requirements. The optimization aims at to obtain optimal configurations of macro scale structures and microstructure of materials under certain mechanical and thermal loads with specific base material volume. A concurrent topology optimization method is proposed for structures and materials to minimize compliance of thermoelastic structures. In this method macro and micro densities are introduced as the design variables for structure and material microstructure independently. Penalization approaches are adopted at both scales to ensure clear topologies, i.e. SIMP (Solid Isotropic Material Penalization) in micro-scale and PAMP (Porous Anisotropic Material Penalization) in macro-scale. Optimizations in two scales are integrated into one system with homogenization theory and distribution of base material between two scales can be decided automatically by the optimization model. Microstructure of materials is assumed to be uniform at macro scale to reduce manufacturing cost. The proposed method and computational model are validated by the numerical experiments. The effects of temperature differential, volume of base material, numerical parameters on the optimum results are also discussed. At last, for cases in which only mechanical load apply, the optimum configuration of micro structure is isotropic solid materials; for eases in which both mechanical and thermal loads apply, the configuration of porous material can help to reduce the system compliance. (Chapter Five)
5. Cyclic-symmetry structures with both mechanical and centrifugal loads are divided into finite design modules, which have the identical configurations. Concurrent optimization techniques based on homogenization method and penalty technique are applied for cyclic-symmetry structures assembled by basic design modules considering the real dimension of design modules. The effects of solid material volume, different loads combination and nondesign fields on optimum results are discussed. Numerical simulations show that for cases in which both mechanical and centrifugal loads apply, the configuration of porous material can help to reduce the system compliance.
引文
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