基于骨重建理论的连续体结构拓扑优化方法
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摘要
结构优化的目的是使得结构达到给定性能条件下尽可能地降低耗费。在过去的几十年里,结构优化广泛地应用于如建筑、机械、化工、航空航天等工程设计领域。目前,连续体结构拓扑优化是结构优化的热点之一。在产品/结构的早期设计阶段,即概念性设计阶段,拓扑优化已成为产品/结构设计的主要手段。经过近二十年的研究,现已产生多种拓扑优化方法。对于结构复杂、设计变量数目庞大的连续体拓扑优化问题,是一项费时费力的工作。为了降低计算量、缩短产品设计周期,需选取合理的优化算法。本文基于骨重建理论和构造张量理论给出了连续体结构拓扑优化仿生方法,所提出的方法的主要思想是,将待优化的结构看作是遵从Wolff法则重建/生长的骨骼。当骨骼达到重建平衡状态时,结构中材料分布达到了优化点。其中,Wolff法则是指骨骼内部微结构的分布随其外部荷载变化而变化,局部材料的刚度主方向与主应力方向趋于一致。
本文的各章节内容安排如下:
第一章,综述连续体结构拓扑优化的研究概况及骨重建理论。主要包括:结构优化的研究历程、当前连续体结构拓扑优化的主要方法、设计变量更新的手段、仿生优化方法和股骨重建模型等。
第二章,介绍描述非均质材料的构造张量理论。主要内容包括构造张量的测定(平均截取长度法MIL)、三维非均质材料的微结构扫描及其有限元模型的构建和材料宏观弹性本构与构造张量间的关系模型。
第三章,给出连续体结构拓扑优化的固定参考应变区间法。1)在该方法中采用构造张量中的元素作为设计变量,利用构造张量描述材料的微结构及宏观弹性张量;2)在多孔材料的构造张量-弹性张量关系给定条件下,为了得到其相对密度,提出等刚度凝缩法。利用等刚度凝缩法将材料点的相对密度用构造张量的不变量表示;3)建立了优化模型:优化过程中设计变量的更新也称为生长规律,它是指对于设计域内任意一点处构造张量特征主方向(即材料的弹性对称面法向或材料主方向)由当前的应力主方向确定;构造张量特征主值的增量,即材料的生长速度,由对应方向上主应变的绝对值与给定的参考应变区间上、下确界的比较确定。当生长过程中构造张量特征对均需要更新时,该过程称为各向异性生长;当构造张量与单位张量成比例时,该过程称为各向同性生长;4)通过数值算例分析了算法中各参数,如参考区间、生长速度和初始材料分布等对算法收敛性的影响。
第四章,根据固定区间法分析多种不同类型结构优化问题。内容包括:1)二维连续体结构的各向同性与各向异性生长结果比较,给出各向异性生长后结构中构造张量的分布;2)三维连续体结构的各向同性生长;3)板结构的各向同性生长;4)股骨头密度分布预测及其结果与Stanford模型结果的比较。
第五章,提出浮动参考区间法用于解决具有非应变约束的优化问题。1)这些约束包括结构的体积约束、位移约束和/或应力约束;2)浮动区间法与固定区间法的主要区别在于,在生长过程中,结构的参考应变区间不断变化,即当任意一个应变主值的绝对值超过当前参考应变(区间)时,构造张量需更新;3)参考区间的更新由优化问题的主动约束控制;4)针对二维和三维结构中材料点的相对密度差异,分别采用两种方式搜索参考应变(区间);5)通过数值算例表明所提出的算法具备分析单约束或多约束优化问题的能力。
第六章给出双参考应变区间法分析具有不同拉伸与压缩性能材料的连续体结构拓扑优化问题。材料的拉伸与压缩性能不同是指两个方面,一是指结构中材料仅能抗拉(如索膜结构)或抗压(如砖墙)。另一是指材料的拉伸与压缩弹性性能不同。材料拉伸与压缩性能差异对局部材料分布及结构拓扑均有影响。若采用传统的分析方法,无论在结构分析阶段还是在设计变量更新阶段,计算量都很大。为了降低计算量并分析这种差异对结果的影响,在本文提出的方法中用具有各向同性固的相多孔材料替代了原拉压不同性能的材料、给出两个浮动参考应变区间(拉伸应变区间和压缩应变区间)用于控制多孔材料的更新。其中拉伸参考区间用于确定处于拉伸状态下材料用量在相应方向上的变化;压缩参考区间控制处于压缩状态下材料用量在相应方向上的变化。从而将材料的拉伸与压缩性能差异转化为拉伸与压缩参考区间的差异。该方法的优点在于解除了拉压性能不同材料的非线性行为,降低了结构分析的计算量。通过理论和数值算例考察了材料拉压性能差异和替代材料的拉压参考区间差异对局部材料的分布影响的等效性。指出对于同一个初始设计域,相同的优化问题,结构中材料的拉伸与压缩性能不同时,结构的最优拓扑可能会产生较大差异。结构优化时考虑这种差异可以得到更实用的最优结构。
第七章给出浮动参考应变能密度法,即采用应变能密度区间控制设计变量的更新。通过几个算例考察了算法的可靠性。利用该方法分析了受均布荷载作用的板结构的设计依赖性。分析结果对工程实践具有指导意义。
结论部分对全文内容进行总结,并展望了下一步工作的内容。
Due to the limitation of the available resources, the efficiency of using the resources has to be improved for the sustainable development. Structural optimization is an effective method to design the best structural performances. With the growth of computer power, the structural optimization methods have been developed greatly. In the last several decades, structural optimization is used widely in engineering fields, such as architectural engineering, mechanical engineering, chemical engineering, aerospace and spaceflight engineering. In structural optimization field, topology optimization is a hot topic and several approaches has been developed, which has become an effective approach in the initial or concept design phase to shorten the design time for a product. For topology optimization of a complicated structure with large sum of design variables, it is difficult to solve the problem efficiently by any method. With the consideration of such aspects, a new bionics approach is proposed in this research based on bone remodeling theory and the fabric tensor theory, whose major idea is to consider a structure as a piece of 'bone'. The optimization for structure is equivalent to the 'bone' remodeling following Wolff's law which states bone microstructure and local stiffness tend to align with the stress principal directions to adapt to the mechanical environments. As the 'bone' reaches the equilibrium state of remodeling, the final optimal structure is obtained. The thesis is orgnized as follows:
In chapter 1, a review on continuum structural topology optimization and the bone remodeling theories is provided, which is composed of the history and advance of structural optimization, some major approaches for continuum structural topology optimization, and the methods of updating the design variables. The bionics optimization approaches and some bone remodeling theories are introduced particularly.
In chapter 2, the fabric tensor theory for describing the microstructure of heterogeneous material is introduced, which consist of the measurement of the fabric tensor of a heterogeneous material (i.e. the Mean Intercept Length (MIL) approach), the micro-finite element model of three-dimensional porous media, and the theories on the relation between the fabric tensor and the elastic constitutive model of porous media.
In chapter 3, the fixed reference strain interval approach to solve continuum structural topology optimization is developed, in which fabric tensor is introduced as the design variables of porous media in design domain. To obtain the relative density of a porous material point having an imposed fabric tensor-elastic tensor relation, Stiffness Equivalent Condensation (SEC) method is proposed, by which the relative density is expressed as the function of the invariables of its fabric tensor. The update rule of the design variables, called as growth law, is established as: 1) during the iteration process of the optimization of a structure, the eigenvectors of the stress tensor at any material point in the present step are those of the fabric tensor used in the next step based on Wolff's law. 2) the increments of the eigenvalues of the fabric tensor are dependable with the principal strains and the interval of reference strain corresponding to the dead zone in bone mechanics. When the eigenpairs of the fabric tensors need to update, the process is called as anisotropic growth. And when all the fabric tensors are proportional to the second order identity tensor in the simulation, the process is called as isotropic growth. Numerical examples demonstrate the significance of the algorithm parameters, such as the interval of reference strain, increments of the eigenvalues of the fabric tensors and initial material distribution parameter.
In chapter 4, several numerical examples are provided to illustrate the applicability of the proposed bionics approach, like the isotropic and anisotropic growthes of two-dimensional structures, the isotropic growth of the three-dimensional structures and plates. As an application in biomechanics, the density distribution of proximal femur is predicted. Compared with Stanford models in the isotropic and anisotropic cases, the proposed method is validated.
In chapter 5, the floating reference strain interval approach extends the fixed reference strain interval approach to solve structural topology optimization problems with the constraints, like structural volume constraint, displacement constraint(s) and/or stress constraint. In the floating reference strain interval approach, the fabric tensor is changed, when anyone of the absolute values of the principal strains at a material point is out of the current reference interval. In optimization, the interval of reference strain is changed and the update rule of the interval depends on the active constaint of an optimization problem. Numerical results demonstrate the validity of the method developed.
In chapter 6, the double floating reference strain intervals approach is proposed to optimize structures with Different Tension and Compression Properties (DTCPs), in which there are two kinds of differences: 1) the difference on the tensile and compressive properties of material, such as the material can only resist tension (e.g. cable, membrane) or only resist compression (e.g. brick wall), or the compressive and tensile elasticity of the material are not identical. 2) two different floating reference strain intervals in tensile and compressive states to control the growth of material point are introduced in the present method. To solve the topology optimization of the strucutures with DTCPs with traditional approach, the efficiency of computation is too low. To overcome the difficulty, the structure with DTCPs is replaced with the structure with isotropic porous media and the growth of the porous media is controlled by two different intervals of reference strain in terms of the local stress strate. The equivalence of these two differences is verified with the theoretical analysis and in numerical simulations, which reveals that the different TCPs lead to different topologies even if the same objective and the character constraint functions are used in the same initial configuration. The more practical results can be obtained if the differences as mentioned above are considered in structural optimization.
In chapter 7, the floating interval of reference strain energy density (SED) approach is proposed, which provide a floating interval of SED to control the update of design variables and whose validity is verified with some numerical caculations. The mesh sensitivity of the plate subjected to uniform pressure is studied, which is useful to guide the practical engineering with this method.
Finally, the main contributions of the dissertation are concluded and some further aspects of the work are previewed.
本文的各章节内容安排如下:
第一章,综述连续体结构拓扑优化的研究概况及骨重建理论。主要包括:结构优化的研究历程、当前连续体结构拓扑优化的主要方法、设计变量更新的手段、仿生优化方法和股骨重建模型等。
第二章,介绍描述非均质材料的构造张量理论。主要内容包括构造张量的测定(平均截取长度法MIL)、三维非均质材料的微结构扫描及其有限元模型的构建和材料宏观弹性本构与构造张量间的关系模型。
第三章,给出连续体结构拓扑优化的固定参考应变区间法。1)在该方法中采用构造张量中的元素作为设计变量,利用构造张量描述材料的微结构及宏观弹性张量;2)在多孔材料的构造张量-弹性张量关系给定条件下,为了得到其相对密度,提出等刚度凝缩法。利用等刚度凝缩法将材料点的相对密度用构造张量的不变量表示;3)建立了优化模型:优化过程中设计变量的更新也称为生长规律,它是指对于设计域内任意一点处构造张量特征主方向(即材料的弹性对称面法向或材料主方向)由当前的应力主方向确定;构造张量特征主值的增量,即材料的生长速度,由对应方向上主应变的绝对值与给定的参考应变区间上、下确界的比较确定。当生长过程中构造张量特征对均需要更新时,该过程称为各向异性生长;当构造张量与单位张量成比例时,该过程称为各向同性生长;4)通过数值算例分析了算法中各参数,如参考区间、生长速度和初始材料分布等对算法收敛性的影响。
第四章,根据固定区间法分析多种不同类型结构优化问题。内容包括:1)二维连续体结构的各向同性与各向异性生长结果比较,给出各向异性生长后结构中构造张量的分布;2)三维连续体结构的各向同性生长;3)板结构的各向同性生长;4)股骨头密度分布预测及其结果与Stanford模型结果的比较。
第五章,提出浮动参考区间法用于解决具有非应变约束的优化问题。1)这些约束包括结构的体积约束、位移约束和/或应力约束;2)浮动区间法与固定区间法的主要区别在于,在生长过程中,结构的参考应变区间不断变化,即当任意一个应变主值的绝对值超过当前参考应变(区间)时,构造张量需更新;3)参考区间的更新由优化问题的主动约束控制;4)针对二维和三维结构中材料点的相对密度差异,分别采用两种方式搜索参考应变(区间);5)通过数值算例表明所提出的算法具备分析单约束或多约束优化问题的能力。
第六章给出双参考应变区间法分析具有不同拉伸与压缩性能材料的连续体结构拓扑优化问题。材料的拉伸与压缩性能不同是指两个方面,一是指结构中材料仅能抗拉(如索膜结构)或抗压(如砖墙)。另一是指材料的拉伸与压缩弹性性能不同。材料拉伸与压缩性能差异对局部材料分布及结构拓扑均有影响。若采用传统的分析方法,无论在结构分析阶段还是在设计变量更新阶段,计算量都很大。为了降低计算量并分析这种差异对结果的影响,在本文提出的方法中用具有各向同性固的相多孔材料替代了原拉压不同性能的材料、给出两个浮动参考应变区间(拉伸应变区间和压缩应变区间)用于控制多孔材料的更新。其中拉伸参考区间用于确定处于拉伸状态下材料用量在相应方向上的变化;压缩参考区间控制处于压缩状态下材料用量在相应方向上的变化。从而将材料的拉伸与压缩性能差异转化为拉伸与压缩参考区间的差异。该方法的优点在于解除了拉压性能不同材料的非线性行为,降低了结构分析的计算量。通过理论和数值算例考察了材料拉压性能差异和替代材料的拉压参考区间差异对局部材料的分布影响的等效性。指出对于同一个初始设计域,相同的优化问题,结构中材料的拉伸与压缩性能不同时,结构的最优拓扑可能会产生较大差异。结构优化时考虑这种差异可以得到更实用的最优结构。
第七章给出浮动参考应变能密度法,即采用应变能密度区间控制设计变量的更新。通过几个算例考察了算法的可靠性。利用该方法分析了受均布荷载作用的板结构的设计依赖性。分析结果对工程实践具有指导意义。
结论部分对全文内容进行总结,并展望了下一步工作的内容。
Due to the limitation of the available resources, the efficiency of using the resources has to be improved for the sustainable development. Structural optimization is an effective method to design the best structural performances. With the growth of computer power, the structural optimization methods have been developed greatly. In the last several decades, structural optimization is used widely in engineering fields, such as architectural engineering, mechanical engineering, chemical engineering, aerospace and spaceflight engineering. In structural optimization field, topology optimization is a hot topic and several approaches has been developed, which has become an effective approach in the initial or concept design phase to shorten the design time for a product. For topology optimization of a complicated structure with large sum of design variables, it is difficult to solve the problem efficiently by any method. With the consideration of such aspects, a new bionics approach is proposed in this research based on bone remodeling theory and the fabric tensor theory, whose major idea is to consider a structure as a piece of 'bone'. The optimization for structure is equivalent to the 'bone' remodeling following Wolff's law which states bone microstructure and local stiffness tend to align with the stress principal directions to adapt to the mechanical environments. As the 'bone' reaches the equilibrium state of remodeling, the final optimal structure is obtained. The thesis is orgnized as follows:
In chapter 1, a review on continuum structural topology optimization and the bone remodeling theories is provided, which is composed of the history and advance of structural optimization, some major approaches for continuum structural topology optimization, and the methods of updating the design variables. The bionics optimization approaches and some bone remodeling theories are introduced particularly.
In chapter 2, the fabric tensor theory for describing the microstructure of heterogeneous material is introduced, which consist of the measurement of the fabric tensor of a heterogeneous material (i.e. the Mean Intercept Length (MIL) approach), the micro-finite element model of three-dimensional porous media, and the theories on the relation between the fabric tensor and the elastic constitutive model of porous media.
In chapter 3, the fixed reference strain interval approach to solve continuum structural topology optimization is developed, in which fabric tensor is introduced as the design variables of porous media in design domain. To obtain the relative density of a porous material point having an imposed fabric tensor-elastic tensor relation, Stiffness Equivalent Condensation (SEC) method is proposed, by which the relative density is expressed as the function of the invariables of its fabric tensor. The update rule of the design variables, called as growth law, is established as: 1) during the iteration process of the optimization of a structure, the eigenvectors of the stress tensor at any material point in the present step are those of the fabric tensor used in the next step based on Wolff's law. 2) the increments of the eigenvalues of the fabric tensor are dependable with the principal strains and the interval of reference strain corresponding to the dead zone in bone mechanics. When the eigenpairs of the fabric tensors need to update, the process is called as anisotropic growth. And when all the fabric tensors are proportional to the second order identity tensor in the simulation, the process is called as isotropic growth. Numerical examples demonstrate the significance of the algorithm parameters, such as the interval of reference strain, increments of the eigenvalues of the fabric tensors and initial material distribution parameter.
In chapter 4, several numerical examples are provided to illustrate the applicability of the proposed bionics approach, like the isotropic and anisotropic growthes of two-dimensional structures, the isotropic growth of the three-dimensional structures and plates. As an application in biomechanics, the density distribution of proximal femur is predicted. Compared with Stanford models in the isotropic and anisotropic cases, the proposed method is validated.
In chapter 5, the floating reference strain interval approach extends the fixed reference strain interval approach to solve structural topology optimization problems with the constraints, like structural volume constraint, displacement constraint(s) and/or stress constraint. In the floating reference strain interval approach, the fabric tensor is changed, when anyone of the absolute values of the principal strains at a material point is out of the current reference interval. In optimization, the interval of reference strain is changed and the update rule of the interval depends on the active constaint of an optimization problem. Numerical results demonstrate the validity of the method developed.
In chapter 6, the double floating reference strain intervals approach is proposed to optimize structures with Different Tension and Compression Properties (DTCPs), in which there are two kinds of differences: 1) the difference on the tensile and compressive properties of material, such as the material can only resist tension (e.g. cable, membrane) or only resist compression (e.g. brick wall), or the compressive and tensile elasticity of the material are not identical. 2) two different floating reference strain intervals in tensile and compressive states to control the growth of material point are introduced in the present method. To solve the topology optimization of the strucutures with DTCPs with traditional approach, the efficiency of computation is too low. To overcome the difficulty, the structure with DTCPs is replaced with the structure with isotropic porous media and the growth of the porous media is controlled by two different intervals of reference strain in terms of the local stress strate. The equivalence of these two differences is verified with the theoretical analysis and in numerical simulations, which reveals that the different TCPs lead to different topologies even if the same objective and the character constraint functions are used in the same initial configuration. The more practical results can be obtained if the differences as mentioned above are considered in structural optimization.
In chapter 7, the floating interval of reference strain energy density (SED) approach is proposed, which provide a floating interval of SED to control the update of design variables and whose validity is verified with some numerical caculations. The mesh sensitivity of the plate subjected to uniform pressure is studied, which is useful to guide the practical engineering with this method.
Finally, the main contributions of the dissertation are concluded and some further aspects of the work are previewed.
引文
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