渐进结构优化方法的改进策略及应用
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摘要
科学技术的发展对结构的性能提出了越来越高的要求,建立合理的方法去设计结构的构型,尤其是依赖计算机与数值技术设计出超越人类工程经验的更新更好的结构构型是结构设计领域倍受关注的问题。结构拓扑构形选择的恰当与否,决定了产品设计的主要性能,因而,在复杂结构的选型和轻量化设计中,拓扑优化比后续的形状和尺寸优化更有价值。
本文主要针对目前拓扑优化的重要方法之一的渐进结构优化方法,通过对其进行改进以提高其合理性、通用性以及工程实际应用的能力。将改进的渐进结构优化方法应用于最小柔顺性优化和热传导结构拓扑优化,在热传导结构优化中不仅考虑了散热问题,还考虑了研究较少的隔热问题。
1.以传热结构的性能为目标,针对工程结构设计中对于传热性能越来越高的要求,基于渐进结构优化方法,建立了以热传递势容耗散为目标函数的优化模型,分别针对结构的最佳散热性能与最佳隔热性能,给出了问题的提法和求解方法。给出了多种荷载与边界条件工况下二维方形板的拓扑设计,并对比优化前后结构的温度分布,证明经过ESO优化后结构的传热性能可以获得极大的提升。
2.传统BESO方法中,随着在高效单元附近添加含有材料的新单元,需要进行网格重新剖分。这给BESO方法的程序实现和实际应用带来了困难;而且传统的BESO方法需要依赖经验给出一个初始设计,而不同的初始设计可能会得到不同的优化结果。本文提出固定网格的BESO方法,并应用于热传导结构拓扑优化设计。在设计初始时,只需给定设计域而不需要依赖经验给出初始设计。在初始的设计域上,所有的单元都是不含材料的孔洞单元,基于仿生的理念,结构逐步生长,不断提高性能,最终得到优化设计。在整个优化过程中,不需要进行网格重剖分,易于程序实现以及工程应用。
3.常规的进化方法通过逐渐删除低效材料(通常称为ESO方法)或逐渐在高效区域添加材料(称为AESO方法)两种策略实现结构构型的进化。AESO的基本思想是在结构的高效单元上逐渐添加材料,从而获得优化设计。本文(第四章)分析了基于敏度的AESO方法的特点,通过算例发现了该方法寻优能力的不足,说明了引起这项不足的主要原因是敏度所描述的是设计变量变化非常小时目标函数的增量,而在进化算法当中,当单元密度从O(或一非常小的数)直接变化到1时,由于敏度并非恒定不变,因此其对目标函数增量的描述是不准确的。尤其在采用添加材料的AESO算法时,当结构中低密度单元较多,荷载传输路径不够清晰的时候,这种情况更为明显。针对AESO算法的缺陷,我们提出了渐进密度AESO方法并应用于连续体热传导优化问题,可以有效地缩小敏度计算误差,从而使得AESO方法和BESO方法中的AESO过程更加稳定可靠。
4.本文分析了ESO方法应用于一些优化问题求解失效的原因,提出有效的改进策略。指出,在以目标函数对设计变量的敏度作为进化准则的ESO优化中,敏度分析过程存在较大的误差从而可能导致ESO方法不能获得合理的优化结果。尤其在网格划分并不十分细密的情况下,敏度计算误差的影响更为明显。文中引入了奇异单元的概念,将各单元按照一定准则去分成奇异单元和常规单元,对常规单元采用传统的梯度方法求敏度,对于奇异单元采用差分法计算目标函数对奇异单元的精确敏度,以提高结构中各单元敏度的计算精度,从而保证了优化的合理性。给出了能够在网格较为稀疏的情况下仍可保证优化设计的合理性、可行性及高效性的改进的ESO方法的实现算法。通过算例说明该改进算法在保持ESO方法的原有优点的同时,拥有更高的稳定性和可靠性,使得ESO方法在工程实际中的应用更加便利。
5.在传统渐进结构优化算法(ESO)及含惩罚的变密度法(SIMP)的基础上,建立了将二者相结合的基于SIMP插值模型的渐进结构优化算法。该方法通过缩小传统ESO算法中的进化步长,从而缩小了由于进化步长过大而导致的敏度评估误差,对于密度处于0—1之间的单元,采用SIMP模型构建本构关系。同时,通过合理设置惩罚函数及进化率,可以有效的控制灰色单元的数量,从而保证了拓扑结果清晰,拥有良好的可制造性。并针对荷载与拓扑相关的优化问题,给出了问题的提法与分析过程,对于外部荷载与自重类的拓扑相关荷载耦合作用下的结构优化能够取得比传统ESO方法更加良好的结果,同时保证了较高的计算效率和良好的通用性。
The quest for performance of structure has been intensified in recent years due to the rapid development of science and technology.Especially the methods to obtain new design beyond engineering experiences are explored by researchers in the structural optimization fields.The topology of structures determines the main performance of structures,so in the configuration design and lightweight design of complex structures,topology optimization is more valuable than shape and size optimization.
This paper focused focuses on the improved approaches and application of evolutionary structural optimization.The frustrated reason of traditional ESO method in some stiffness optimal topology design problems is analyzed and an improved ESO method is proposed.The improved ESO method is also applied in compliance optimization and thermal optimization. Not only the thermal dissipation problem but also the thermal insulation problem is considered.
1.The thermal topology optimization method is proposed based on ESO by choosing the heat resistance as the design object,and element relative densities as design variables.To obtain the best performance of thermal dissipation and thermal insulation,the topology optimization results of a 2D square plate under different work conditions are presented.By this method,the heat performance of structures can be improved greatly.
2.A Bi-Directional Evolutionary Structural Optimization(BESO) method for topology optimization of heat conduction structures is presented.In BESO method the elements are allowed to be added as well as removed.To improve the heat performance of structure,the additive criterion and rejection criterion are proposed respectively.With the limit volume of the high conductive material,the optimal layout of structure with high efficiency of heat dissipation and uniform temperature distribution can be obtained by BESO procedure.During the procedure of optimization,the re-mesh is avoided,which makes the BESO method more efficient,
3.A new version of the additive evolutionary structural optimization(AESO) procedure based on sensitivity analysis for topology optimization of continuum structures is proposed. Illustrative examples have proved that the one-step AESO algorithm based on sensitivity analysis can't obtain good optimal results in the optimization of some continuum structures. The reason is pointed out as:the relationship of objective function and design variables can not be described accurately when the design variables are changed significantly(For example, from 0 to 1).And a new version of so called progressive AESO algorithm based on sensitivity analysis is proposed and is demonstrated by illustrative examples of topology optimization of heat conduction problems.The strategy of adding material in the bi-directional ESO algorithm based on sensitivity analysis is provided.
4.The reason of failure of traditional ESO method in some topology design of structure stiffness problems is analyzed and an improved ESO method is proposed.The reason of failure of traditional ESO method in some cases is that sensitivity analysis is inaccurate.In traditional ESO method,the sensitivity of objective function(the difference between the objective function values at material densities of element being 1 and 0) is approximated as the derivative of the objective function with respect to the material density at the density being 1.This approximation may lead to large errors in sensitivity calculation in the case that the derivative of the objective function varies significantly with the change of material density of element from 1 to 0.Due to the large computational cost to improve the accuracy of the sensitivity analysis by the global difference method,a concept of singular element is introduced and a new improved ESO method is proposed.In this method,all the elements are divided into singular and ordinary elements based on the sensitivity of objective function.The sensitivities of objective function with respect to the densities of singular elements are calculated by the difference method.Numerical examples demonstrate the efficiency and validity of the method.
5.The improved ESO method based on SIMP is presented,which is combination of ESO and SIMP.The step length of evolution is reduced so that the accuracy of sensitivity analysis can be improved.The SIMP method is employed to describe the relationship between the relative density and the stiffness of grey elements.With logical parameter the numbers of grey dements can be controlled,so the topology can be clearer and easier to manufacture. Topology optimization of continuum structures considering self-weight is carried out by this method,and more optimal results are obtained than those by traditional ESO method.
本文主要针对目前拓扑优化的重要方法之一的渐进结构优化方法,通过对其进行改进以提高其合理性、通用性以及工程实际应用的能力。将改进的渐进结构优化方法应用于最小柔顺性优化和热传导结构拓扑优化,在热传导结构优化中不仅考虑了散热问题,还考虑了研究较少的隔热问题。
1.以传热结构的性能为目标,针对工程结构设计中对于传热性能越来越高的要求,基于渐进结构优化方法,建立了以热传递势容耗散为目标函数的优化模型,分别针对结构的最佳散热性能与最佳隔热性能,给出了问题的提法和求解方法。给出了多种荷载与边界条件工况下二维方形板的拓扑设计,并对比优化前后结构的温度分布,证明经过ESO优化后结构的传热性能可以获得极大的提升。
2.传统BESO方法中,随着在高效单元附近添加含有材料的新单元,需要进行网格重新剖分。这给BESO方法的程序实现和实际应用带来了困难;而且传统的BESO方法需要依赖经验给出一个初始设计,而不同的初始设计可能会得到不同的优化结果。本文提出固定网格的BESO方法,并应用于热传导结构拓扑优化设计。在设计初始时,只需给定设计域而不需要依赖经验给出初始设计。在初始的设计域上,所有的单元都是不含材料的孔洞单元,基于仿生的理念,结构逐步生长,不断提高性能,最终得到优化设计。在整个优化过程中,不需要进行网格重剖分,易于程序实现以及工程应用。
3.常规的进化方法通过逐渐删除低效材料(通常称为ESO方法)或逐渐在高效区域添加材料(称为AESO方法)两种策略实现结构构型的进化。AESO的基本思想是在结构的高效单元上逐渐添加材料,从而获得优化设计。本文(第四章)分析了基于敏度的AESO方法的特点,通过算例发现了该方法寻优能力的不足,说明了引起这项不足的主要原因是敏度所描述的是设计变量变化非常小时目标函数的增量,而在进化算法当中,当单元密度从O(或一非常小的数)直接变化到1时,由于敏度并非恒定不变,因此其对目标函数增量的描述是不准确的。尤其在采用添加材料的AESO算法时,当结构中低密度单元较多,荷载传输路径不够清晰的时候,这种情况更为明显。针对AESO算法的缺陷,我们提出了渐进密度AESO方法并应用于连续体热传导优化问题,可以有效地缩小敏度计算误差,从而使得AESO方法和BESO方法中的AESO过程更加稳定可靠。
4.本文分析了ESO方法应用于一些优化问题求解失效的原因,提出有效的改进策略。指出,在以目标函数对设计变量的敏度作为进化准则的ESO优化中,敏度分析过程存在较大的误差从而可能导致ESO方法不能获得合理的优化结果。尤其在网格划分并不十分细密的情况下,敏度计算误差的影响更为明显。文中引入了奇异单元的概念,将各单元按照一定准则去分成奇异单元和常规单元,对常规单元采用传统的梯度方法求敏度,对于奇异单元采用差分法计算目标函数对奇异单元的精确敏度,以提高结构中各单元敏度的计算精度,从而保证了优化的合理性。给出了能够在网格较为稀疏的情况下仍可保证优化设计的合理性、可行性及高效性的改进的ESO方法的实现算法。通过算例说明该改进算法在保持ESO方法的原有优点的同时,拥有更高的稳定性和可靠性,使得ESO方法在工程实际中的应用更加便利。
5.在传统渐进结构优化算法(ESO)及含惩罚的变密度法(SIMP)的基础上,建立了将二者相结合的基于SIMP插值模型的渐进结构优化算法。该方法通过缩小传统ESO算法中的进化步长,从而缩小了由于进化步长过大而导致的敏度评估误差,对于密度处于0—1之间的单元,采用SIMP模型构建本构关系。同时,通过合理设置惩罚函数及进化率,可以有效的控制灰色单元的数量,从而保证了拓扑结果清晰,拥有良好的可制造性。并针对荷载与拓扑相关的优化问题,给出了问题的提法与分析过程,对于外部荷载与自重类的拓扑相关荷载耦合作用下的结构优化能够取得比传统ESO方法更加良好的结果,同时保证了较高的计算效率和良好的通用性。
The quest for performance of structure has been intensified in recent years due to the rapid development of science and technology.Especially the methods to obtain new design beyond engineering experiences are explored by researchers in the structural optimization fields.The topology of structures determines the main performance of structures,so in the configuration design and lightweight design of complex structures,topology optimization is more valuable than shape and size optimization.
This paper focused focuses on the improved approaches and application of evolutionary structural optimization.The frustrated reason of traditional ESO method in some stiffness optimal topology design problems is analyzed and an improved ESO method is proposed.The improved ESO method is also applied in compliance optimization and thermal optimization. Not only the thermal dissipation problem but also the thermal insulation problem is considered.
1.The thermal topology optimization method is proposed based on ESO by choosing the heat resistance as the design object,and element relative densities as design variables.To obtain the best performance of thermal dissipation and thermal insulation,the topology optimization results of a 2D square plate under different work conditions are presented.By this method,the heat performance of structures can be improved greatly.
2.A Bi-Directional Evolutionary Structural Optimization(BESO) method for topology optimization of heat conduction structures is presented.In BESO method the elements are allowed to be added as well as removed.To improve the heat performance of structure,the additive criterion and rejection criterion are proposed respectively.With the limit volume of the high conductive material,the optimal layout of structure with high efficiency of heat dissipation and uniform temperature distribution can be obtained by BESO procedure.During the procedure of optimization,the re-mesh is avoided,which makes the BESO method more efficient,
3.A new version of the additive evolutionary structural optimization(AESO) procedure based on sensitivity analysis for topology optimization of continuum structures is proposed. Illustrative examples have proved that the one-step AESO algorithm based on sensitivity analysis can't obtain good optimal results in the optimization of some continuum structures. The reason is pointed out as:the relationship of objective function and design variables can not be described accurately when the design variables are changed significantly(For example, from 0 to 1).And a new version of so called progressive AESO algorithm based on sensitivity analysis is proposed and is demonstrated by illustrative examples of topology optimization of heat conduction problems.The strategy of adding material in the bi-directional ESO algorithm based on sensitivity analysis is provided.
4.The reason of failure of traditional ESO method in some topology design of structure stiffness problems is analyzed and an improved ESO method is proposed.The reason of failure of traditional ESO method in some cases is that sensitivity analysis is inaccurate.In traditional ESO method,the sensitivity of objective function(the difference between the objective function values at material densities of element being 1 and 0) is approximated as the derivative of the objective function with respect to the material density at the density being 1.This approximation may lead to large errors in sensitivity calculation in the case that the derivative of the objective function varies significantly with the change of material density of element from 1 to 0.Due to the large computational cost to improve the accuracy of the sensitivity analysis by the global difference method,a concept of singular element is introduced and a new improved ESO method is proposed.In this method,all the elements are divided into singular and ordinary elements based on the sensitivity of objective function.The sensitivities of objective function with respect to the densities of singular elements are calculated by the difference method.Numerical examples demonstrate the efficiency and validity of the method.
5.The improved ESO method based on SIMP is presented,which is combination of ESO and SIMP.The step length of evolution is reduced so that the accuracy of sensitivity analysis can be improved.The SIMP method is employed to describe the relationship between the relative density and the stiffness of grey elements.With logical parameter the numbers of grey dements can be controlled,so the topology can be clearer and easier to manufacture. Topology optimization of continuum structures considering self-weight is carried out by this method,and more optimal results are obtained than those by traditional ESO method.
引文
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