基于拓扑优化的结构刚度和渗流多功能材料设计
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摘要
设计具有优异性能的结构一直是工程师追求的目标。结构优化方法是寻找工程设计问题最优解的强大工具。相比于尺寸优化和形状优化,拓扑优化能够获得创新结构构型,得到了越来越广泛的应用。
多孔介质中的流体输运具有广泛的应用背景,比如过滤器、由多孔材料制成的具有发汗冷却功能的火箭发动机推力室壁。为了设计更高效的结构,需要研究能够同时考虑结构的流体渗流性能和刚度性能的优化设计方法,通过设计材料的微结构实现宏观结构的渗流和刚度多功能设计。围绕这一材料／结构多功能多尺度优化设计问题,我们针对结构拓扑优化和材料微结构优化设计中现有算法的困难,改进和提出了多个有效算法,从多个角度提高了结构拓扑优化和材料微结构算法的效率和收敛性。
SIMP (Solid Isotropic Material with Penalization)模型具有实现简单,适用范围广易于和现有有限元程序衔接的优点,是目前使用最为广泛的拓扑优化模型。通常采用正则化方法避免拓扑优化中的棋盘格式和网格相关性。线性密度过滤方法是一种有效的正则化方法,得到了广泛应用,其不足之处在于优化结果的材料边界处有灰色密度区域存在,增加了拓扑优化结果提取的难度。Heaviside非线性密度过滤函数能够消除材料边界的灰色区域,但是不能保证密度非线性变换前后材料体积守恒,优化迭代过程不稳定。本文提出了一种体积守恒非线性密度过滤函数,能够保证密度非线性变换前后材料体积守恒,优化迭代过程稳定,优化效率也得到了改进。
对于三维结构拓扑优化,以单元(结点)密度为设计变量时,优化问题规模很大,难以求解。自适应有限元方法通过合理地分布有限元网格密度,显著减少有限元模型位移未知数的数量,既提高有限元计算精度,又兼顾了计算效率。本文将网格自适应技术应用到结构拓扑优化,优化初期采用稀疏网格得到材料大致分布趋势,根据材料分布结果对材料边界处的网格进行加密,达到用尽量少的设计变量描述结构拓扑的效果,提高了结构分析和优化的效率。
材料设计是拓扑优化的一个重要应用领域。采用逆均匀化方法进行材料设计时,初始密度分布和优化控制参数对优化结果影响很大。初始密度的选取非常困难,往往依赖于设计者的经验。本文模拟人工培养晶体时为了加速晶体的生长引入籽晶的办法,提出了材料设计的晶核法。研究了晶核位置、密度幂指数、密度过滤影响域以及目标函数形式对晶核法的影响。采用晶核法设计了指定材料性能和极值材料性能相应的最优材料微结构形式。
考虑结构刚度和渗流多功能设计时,作为探索性工作,我们考虑的宏观结构是由宏观均匀的、具有周期性微结构的材料组成。通过设计材料的微结构进行宏观结构性能优化。由具有周期性微结构的材料制成的结构的刚度和渗流性质,一般的说,两者之间没有简单的关系,它们分别取决于材料的宏观等效弹性性质和渗透系数,这两者都和材料的微结构形式有关。微观材料的微结构形式和宏观结构的刚度和渗流性能通过材料的等效弹性张量和渗透系数张量联系起来。同时考虑结构刚度和渗流性能,寻求组成结构的材料的最优微结构是一个多功能多尺度优化设计问题,在理论和应用两方面都富有挑战性。在本文中,所研究的宏观结构为由具有周期性微结构的宏观均匀材料制成的线弹性结构,其内部流体宏观流动服从Darcy方程。在宏观结构刚度和渗流性能驱动下,采用逆均匀化方法设计材料的微结构。采用伴随法推导了出口流量对等效渗透系数和等效渗透系数对单元密度的灵敏度。基于Darcy-Stokes模型,将晶核法和基于自适应网格的结构拓扑优化方法应用于材料单胞最优流体流道设计,用较小的计算规模,得到了精度较高的最大各向同性等效渗透系数和相应的最优材料微结构。构造了结构刚度和渗流性能的多目标优化模型和指定出口流量下的结构刚度优化模型,针对多组优化设计参数,得到了相应的最优三维材料微结构形式。指定出口流量下的结构刚度优化算例表明体积守恒非线性密度过滤函数对于满足指定出口流量要求,稳定优化迭代过程具有重要作用。
Engineers are always seeking for structural design with better performance. Structural op-timization provides them a powerful approach to find the optimal design of engineering prob-lem. In comparison with sizing and shape optimization, topology optimization is being used extensively to obtain a creative structural configuration at the conceptual design stage.
Fluid transfer in porous media is a very universal phenomenon and has a strong background of applications such as filter and transpiration-cooled thrust chamber which are made of porous media. It is needed to develop an optimization method to design a high effective structural con-figuration considering the performances of seepage and stiffness simultaneously. Focusing on the material/structure multifunctional and multiscale optimization problem, we have improved and proposed some effective algorithms from multiple angles aiming at improving the efficiency and convergence of algorithms in structural topology optimization and material microstructure design.
SIMP (Solid Isotropic Material with Penalization) model is currently the most widely used in many kinds of topology optimization problems because it is very easy to be implemented and fits for many kinds of optimization problems. Another very important advantage is that it can be linked with the general finite element packages very easily. Density filter is a very effective regularization scheme to deal with the numerical difficulty such as checkerboard and mesh dependency in topology optimization. However, when density filter is used, some grey el-ements often emerge along the material boundary which cause difficulty in extracting structural boundary from the optimal topology result. Heaviside nonlinear density filter can eliminate grey domain along material boundary, but it can not maintain the material volume which causes oscillation during optimization iteration. A new volume preserving nonlinear density filter has been proposed which can guaranty material volume preserving before and after nonlinear trans-formation of density field. With the volume preserving nonlinear density filter, the iterative process of optimization is very stable and the optimization efficiency is improved.
For 3D (three dimensional) structural topology optimization with element or node density as the design variable, the optimization problem is very large in size and difficult to solve. AMR (Adaptive Mesh Refinement) finite element method reduces the number of displacement variables in finite element model remarkably and increases the accuracy and efficiency of finite element analysis. Following the idea of AMR finite element method, a AMR based topology optimization method was proposed to optimize the field of design variable for decreasing the amount of design variables in topology optimization. At the initial stage of optimization, the tendency of material distribution can be obtained with a very coarse mesh efficiently. Then, the mesh is remeshed with AMR rule based on the information of density field of topology optimization. The amount of mesh around material boundary is increased and that of mesh far away from material boundary is decreased. In comparison with the traditional topology optimization, AMR based topology optimization can obtain the optimal material distribution which having smooth material boundary with fewer design variables and higher efficiency.
Material design is an important application field for topology optimization. The optimal result is influenced by both the type of initial density distribution and the value of optimiza-tion control parameters when designing material microstructure with inverse homogenization method. The type of initial density distribution which often depends on the experience of de-signer is very difficult to be determined. Crystallon is usually used to accelerate crystal's growth when making cultured crystal. Inspired from this physical process, a new method called Crys-tal Nucleus Method (CNM) was proposed to determine the initial density distribution. Some issues about this method were studied which including the position of crystal nucleus, the value of power exponent in SIMP model, the density filter domain and the type of object function. Optimal material microstructures with described and extreme performances were successfully obtained with this method.
The microstructure of material is designed to optimize the multifunctional performances of stiffness and seepage of macrostructure which is made of porous media with periodic mi-crostructure. Generally speaking, there is no simple relationship between stiffness and seepage performances of structure made of material with periodic microstructure. The performances of stiffness and seepage are determined by the effective elastic property and permeability coeffi-cient which are both related to the microstructure of material. The microstructure of material and the stiffness and seepage performances of macro structure are connected through the ef-fective elastic tensor and permeability tensor of material. To design optimal microstructure for the stiffness and seepage performances of macrostructure is a multifunctional and multi-scale optimization problem. It is a challenging research topic in theory and application. For the present study, the macrostructure made of macro homogeneous material with periodic mi-crostructure is considered as linear elastic structure and the macro flow of fluid is governed by Darcy equation. Driven by the stiffness and seepage performances of macrostructure, the opti-mal mi crostructure was obtained with the method of inverse homogenization. Adjoint method was used to calculate the sensitivity of flow rate and effective permeability coefficient with re-spect to the element density. Based on the Darcy-Stokes model, optimal material microstructure for maximization of effective permeability under the given amount of solid material is obtained with reasonable precision by designing fluid channel in the domain of material cell by CNM and AMR based topology optimization. For multifunctional material design considering structural stiffness and seepage performance, two optimization models were proposed which were multi-objective design of structural stiffness and seepage performances and structural stiffness design under specified seepage flow rate. Optimal microstructures of 3D material were obtained with different design parameters. For the optimization of maximizing structural stiffness under the prescribed flow rate, the volume preserving nonlinear density filter can make the optimization process more stable.
多孔介质中的流体输运具有广泛的应用背景,比如过滤器、由多孔材料制成的具有发汗冷却功能的火箭发动机推力室壁。为了设计更高效的结构,需要研究能够同时考虑结构的流体渗流性能和刚度性能的优化设计方法,通过设计材料的微结构实现宏观结构的渗流和刚度多功能设计。围绕这一材料／结构多功能多尺度优化设计问题,我们针对结构拓扑优化和材料微结构优化设计中现有算法的困难,改进和提出了多个有效算法,从多个角度提高了结构拓扑优化和材料微结构算法的效率和收敛性。
SIMP (Solid Isotropic Material with Penalization)模型具有实现简单,适用范围广易于和现有有限元程序衔接的优点,是目前使用最为广泛的拓扑优化模型。通常采用正则化方法避免拓扑优化中的棋盘格式和网格相关性。线性密度过滤方法是一种有效的正则化方法,得到了广泛应用,其不足之处在于优化结果的材料边界处有灰色密度区域存在,增加了拓扑优化结果提取的难度。Heaviside非线性密度过滤函数能够消除材料边界的灰色区域,但是不能保证密度非线性变换前后材料体积守恒,优化迭代过程不稳定。本文提出了一种体积守恒非线性密度过滤函数,能够保证密度非线性变换前后材料体积守恒,优化迭代过程稳定,优化效率也得到了改进。
对于三维结构拓扑优化,以单元(结点)密度为设计变量时,优化问题规模很大,难以求解。自适应有限元方法通过合理地分布有限元网格密度,显著减少有限元模型位移未知数的数量,既提高有限元计算精度,又兼顾了计算效率。本文将网格自适应技术应用到结构拓扑优化,优化初期采用稀疏网格得到材料大致分布趋势,根据材料分布结果对材料边界处的网格进行加密,达到用尽量少的设计变量描述结构拓扑的效果,提高了结构分析和优化的效率。
材料设计是拓扑优化的一个重要应用领域。采用逆均匀化方法进行材料设计时,初始密度分布和优化控制参数对优化结果影响很大。初始密度的选取非常困难,往往依赖于设计者的经验。本文模拟人工培养晶体时为了加速晶体的生长引入籽晶的办法,提出了材料设计的晶核法。研究了晶核位置、密度幂指数、密度过滤影响域以及目标函数形式对晶核法的影响。采用晶核法设计了指定材料性能和极值材料性能相应的最优材料微结构形式。
考虑结构刚度和渗流多功能设计时,作为探索性工作,我们考虑的宏观结构是由宏观均匀的、具有周期性微结构的材料组成。通过设计材料的微结构进行宏观结构性能优化。由具有周期性微结构的材料制成的结构的刚度和渗流性质,一般的说,两者之间没有简单的关系,它们分别取决于材料的宏观等效弹性性质和渗透系数,这两者都和材料的微结构形式有关。微观材料的微结构形式和宏观结构的刚度和渗流性能通过材料的等效弹性张量和渗透系数张量联系起来。同时考虑结构刚度和渗流性能,寻求组成结构的材料的最优微结构是一个多功能多尺度优化设计问题,在理论和应用两方面都富有挑战性。在本文中,所研究的宏观结构为由具有周期性微结构的宏观均匀材料制成的线弹性结构,其内部流体宏观流动服从Darcy方程。在宏观结构刚度和渗流性能驱动下,采用逆均匀化方法设计材料的微结构。采用伴随法推导了出口流量对等效渗透系数和等效渗透系数对单元密度的灵敏度。基于Darcy-Stokes模型,将晶核法和基于自适应网格的结构拓扑优化方法应用于材料单胞最优流体流道设计,用较小的计算规模,得到了精度较高的最大各向同性等效渗透系数和相应的最优材料微结构。构造了结构刚度和渗流性能的多目标优化模型和指定出口流量下的结构刚度优化模型,针对多组优化设计参数,得到了相应的最优三维材料微结构形式。指定出口流量下的结构刚度优化算例表明体积守恒非线性密度过滤函数对于满足指定出口流量要求,稳定优化迭代过程具有重要作用。
Engineers are always seeking for structural design with better performance. Structural op-timization provides them a powerful approach to find the optimal design of engineering prob-lem. In comparison with sizing and shape optimization, topology optimization is being used extensively to obtain a creative structural configuration at the conceptual design stage.
Fluid transfer in porous media is a very universal phenomenon and has a strong background of applications such as filter and transpiration-cooled thrust chamber which are made of porous media. It is needed to develop an optimization method to design a high effective structural con-figuration considering the performances of seepage and stiffness simultaneously. Focusing on the material/structure multifunctional and multiscale optimization problem, we have improved and proposed some effective algorithms from multiple angles aiming at improving the efficiency and convergence of algorithms in structural topology optimization and material microstructure design.
SIMP (Solid Isotropic Material with Penalization) model is currently the most widely used in many kinds of topology optimization problems because it is very easy to be implemented and fits for many kinds of optimization problems. Another very important advantage is that it can be linked with the general finite element packages very easily. Density filter is a very effective regularization scheme to deal with the numerical difficulty such as checkerboard and mesh dependency in topology optimization. However, when density filter is used, some grey el-ements often emerge along the material boundary which cause difficulty in extracting structural boundary from the optimal topology result. Heaviside nonlinear density filter can eliminate grey domain along material boundary, but it can not maintain the material volume which causes oscillation during optimization iteration. A new volume preserving nonlinear density filter has been proposed which can guaranty material volume preserving before and after nonlinear trans-formation of density field. With the volume preserving nonlinear density filter, the iterative process of optimization is very stable and the optimization efficiency is improved.
For 3D (three dimensional) structural topology optimization with element or node density as the design variable, the optimization problem is very large in size and difficult to solve. AMR (Adaptive Mesh Refinement) finite element method reduces the number of displacement variables in finite element model remarkably and increases the accuracy and efficiency of finite element analysis. Following the idea of AMR finite element method, a AMR based topology optimization method was proposed to optimize the field of design variable for decreasing the amount of design variables in topology optimization. At the initial stage of optimization, the tendency of material distribution can be obtained with a very coarse mesh efficiently. Then, the mesh is remeshed with AMR rule based on the information of density field of topology optimization. The amount of mesh around material boundary is increased and that of mesh far away from material boundary is decreased. In comparison with the traditional topology optimization, AMR based topology optimization can obtain the optimal material distribution which having smooth material boundary with fewer design variables and higher efficiency.
Material design is an important application field for topology optimization. The optimal result is influenced by both the type of initial density distribution and the value of optimiza-tion control parameters when designing material microstructure with inverse homogenization method. The type of initial density distribution which often depends on the experience of de-signer is very difficult to be determined. Crystallon is usually used to accelerate crystal's growth when making cultured crystal. Inspired from this physical process, a new method called Crys-tal Nucleus Method (CNM) was proposed to determine the initial density distribution. Some issues about this method were studied which including the position of crystal nucleus, the value of power exponent in SIMP model, the density filter domain and the type of object function. Optimal material microstructures with described and extreme performances were successfully obtained with this method.
The microstructure of material is designed to optimize the multifunctional performances of stiffness and seepage of macrostructure which is made of porous media with periodic mi-crostructure. Generally speaking, there is no simple relationship between stiffness and seepage performances of structure made of material with periodic microstructure. The performances of stiffness and seepage are determined by the effective elastic property and permeability coeffi-cient which are both related to the microstructure of material. The microstructure of material and the stiffness and seepage performances of macro structure are connected through the ef-fective elastic tensor and permeability tensor of material. To design optimal microstructure for the stiffness and seepage performances of macrostructure is a multifunctional and multi-scale optimization problem. It is a challenging research topic in theory and application. For the present study, the macrostructure made of macro homogeneous material with periodic mi-crostructure is considered as linear elastic structure and the macro flow of fluid is governed by Darcy equation. Driven by the stiffness and seepage performances of macrostructure, the opti-mal mi crostructure was obtained with the method of inverse homogenization. Adjoint method was used to calculate the sensitivity of flow rate and effective permeability coefficient with re-spect to the element density. Based on the Darcy-Stokes model, optimal material microstructure for maximization of effective permeability under the given amount of solid material is obtained with reasonable precision by designing fluid channel in the domain of material cell by CNM and AMR based topology optimization. For multifunctional material design considering structural stiffness and seepage performance, two optimization models were proposed which were multi-objective design of structural stiffness and seepage performances and structural stiffness design under specified seepage flow rate. Optimal microstructures of 3D material were obtained with different design parameters. For the optimization of maximizing structural stiffness under the prescribed flow rate, the volume preserving nonlinear density filter can make the optimization process more stable.
引文
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