基于水平集方法的分布式柔顺机构拓扑优化设计研究
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摘要
柔顺机构以其易实现高精度、微型化等独特的优良性能在微纳操作、精密测量等许多尖端领域得到了广泛应用，已成为近年来国际上的研究热点之一，给机构学带来了革命性的冲击，是现代机构和精密机械装备领域发展的新方向。但是，要真正使得柔顺机构得到广泛应用和长足发展，必须以夯实其理论基础、完善其设计方法为前提。本文正是在这一背景下应用水平集方法对柔顺机构拓扑优化设计的基本理论和方法进行研究。
论文首先研究了在应用水平集方法求解拓扑优化问题时的若干难点，具体内容包括：根据符号距离函数的基本性质，构造了一种距离保持泛函，并将其嵌入到柔顺机构拓扑优化设计模型的目标泛函中，这不仅完全避免了耗时的重新初始化，且大大简化了水平集函数的初始化工作；在整合了水平集方法及渐进结构优化算法的基础上，探索了在优化过程中的拓扑调整策略以拓展寻优空间；针对基于水平集法的拓扑优化方法计算复杂性大、效率低等缺点，提出了速度场预测——矫正构造方法，并对所构造的速度方向为目标函数的可行下降方向进行了数学证明；根据水平集算法具体实施过程中速度场在设计域内均为已知量及其分布特点，提出了形状敏度过滤法以增大单个迭代步中零水平集的演化幅度；分别应用上述技术对经典的结构和柔顺机构设计算例进行了优化和分析，验证了上述方法在避免重新初始化、拓展寻优空间以及提高计算效率等方面的可行性和有效性。其次，针对传统柔顺机构拓扑优化中的类铰链(de factohinge)问题，揭示了产生类铰链的内在机理，提出了一种最小化性能差建模方法。在此基础上，应用线性加权组合方式及比例组合方式构造了用于分布式柔顺机构拓扑优化的两类数学模型，并针对加权组合模型中加权因子难以选择的问题，提出了加权因子的自适应调整策略；综合应用不同的设计案例验证了两类数学模型在抑制类铰链产生从而获得分布式柔顺机构方面的可行性和有效性，并应用不同的拓扑优化设计方法对两类模型进行求解验证了其通用性。再次，针对某些特殊工况下柔顺机构拓扑优化设计时的病态问题，在优化模型中引入了输入端和输出端柔度，结合加权因子的自适应调整策略，构造了一种加权和式目标函数，从而在将设计问题由病态转成适定的同时，有效抑制了类铰链的产生，并应用此方法的基本原理针对单输入多输出柔顺机构的设计进行扩展。最后，总结全文并对未来的工作进行了展望。
本文的完成对分布式柔顺机构拓扑优化的理论建模以及基于水平集方法的求解算法实现提供了切实可行的思路和方法，对发展柔顺机构拓扑优化领域的理论水平有重要的意义和价值。
Compliant mechanisms have been paid more attention and become one of the internationalresearch hotspots due to their inherent advantages and potential of being used in the micro-nano-manipulation or precision measurement area. To broaden compliant mechanisms’ applications,consolidating their theoretical basis and design methods is the key. That is the background ofthis dissertation in which the focus is on designing distributed compliant mechanisms usinglevel set methods.
Firstly, several methods are developed to overcome the inherent shortcomings of the con-ventional level set method, include:(I) presenting a new distance-remaining functional for-mulation that forces the level set function to be close to a signed distance function during theoptimization process, and therefore completely eliminates the need of the costly re-initializationprocedure;(II) presenting a topology-adjusting scheme which allows new holes being automat-ically generated in the design domain and obtains optimized configurations even being startedfrom a minimum possible initial guess;(III) presenting a predictor-corrector scheme for con-structing velocity field and a shape sensitivity filtering scheme which both can converge withless iterations and thus improve the overall computational efciency.
Secondly, the underlying reason of de facto hinges of using spring model to design com-pliant mechanisms is analyzed. A performance diference minimization method for the topol-ogy optimization of compliant mechanisms aimed at eliminating de facto hinges is developed.Based on the method, several alternative formulation are developed. The validity and versatilityof the method are examined using several topology optimization approaches, such as the levelset methods and SIMP method.
Thirdly, for designing compliant mechanisms without using the output spring, a method isdeveloped by inserting the input and output mean compliance into the optimization model. Anew objective function is developed using the weighting sum method. The weighting factorsare set based on the information that is obtained from the previous iteration and automaticallyupdated with each optimization iteration step. The validity and versatility of the method are ex-amined using several topology optimization approaches to design compliant mechanisms withboth single input-output behaviour and multiple outputs behaviour.
Finally, conclusions and future works are put forward
This work will surely provide the theoretical support both in developing more powerfuland efcient level set-based topology optimization method and in consolidating the theoreticalbasis of designing compliant mechanisms.
论文首先研究了在应用水平集方法求解拓扑优化问题时的若干难点，具体内容包括：根据符号距离函数的基本性质，构造了一种距离保持泛函，并将其嵌入到柔顺机构拓扑优化设计模型的目标泛函中，这不仅完全避免了耗时的重新初始化，且大大简化了水平集函数的初始化工作；在整合了水平集方法及渐进结构优化算法的基础上，探索了在优化过程中的拓扑调整策略以拓展寻优空间；针对基于水平集法的拓扑优化方法计算复杂性大、效率低等缺点，提出了速度场预测——矫正构造方法，并对所构造的速度方向为目标函数的可行下降方向进行了数学证明；根据水平集算法具体实施过程中速度场在设计域内均为已知量及其分布特点，提出了形状敏度过滤法以增大单个迭代步中零水平集的演化幅度；分别应用上述技术对经典的结构和柔顺机构设计算例进行了优化和分析，验证了上述方法在避免重新初始化、拓展寻优空间以及提高计算效率等方面的可行性和有效性。其次，针对传统柔顺机构拓扑优化中的类铰链(de factohinge)问题，揭示了产生类铰链的内在机理，提出了一种最小化性能差建模方法。在此基础上，应用线性加权组合方式及比例组合方式构造了用于分布式柔顺机构拓扑优化的两类数学模型，并针对加权组合模型中加权因子难以选择的问题，提出了加权因子的自适应调整策略；综合应用不同的设计案例验证了两类数学模型在抑制类铰链产生从而获得分布式柔顺机构方面的可行性和有效性，并应用不同的拓扑优化设计方法对两类模型进行求解验证了其通用性。再次，针对某些特殊工况下柔顺机构拓扑优化设计时的病态问题，在优化模型中引入了输入端和输出端柔度，结合加权因子的自适应调整策略，构造了一种加权和式目标函数，从而在将设计问题由病态转成适定的同时，有效抑制了类铰链的产生，并应用此方法的基本原理针对单输入多输出柔顺机构的设计进行扩展。最后，总结全文并对未来的工作进行了展望。
本文的完成对分布式柔顺机构拓扑优化的理论建模以及基于水平集方法的求解算法实现提供了切实可行的思路和方法，对发展柔顺机构拓扑优化领域的理论水平有重要的意义和价值。
Compliant mechanisms have been paid more attention and become one of the internationalresearch hotspots due to their inherent advantages and potential of being used in the micro-nano-manipulation or precision measurement area. To broaden compliant mechanisms’ applications,consolidating their theoretical basis and design methods is the key. That is the background ofthis dissertation in which the focus is on designing distributed compliant mechanisms usinglevel set methods.
Firstly, several methods are developed to overcome the inherent shortcomings of the con-ventional level set method, include:(I) presenting a new distance-remaining functional for-mulation that forces the level set function to be close to a signed distance function during theoptimization process, and therefore completely eliminates the need of the costly re-initializationprocedure;(II) presenting a topology-adjusting scheme which allows new holes being automat-ically generated in the design domain and obtains optimized configurations even being startedfrom a minimum possible initial guess;(III) presenting a predictor-corrector scheme for con-structing velocity field and a shape sensitivity filtering scheme which both can converge withless iterations and thus improve the overall computational efciency.
Secondly, the underlying reason of de facto hinges of using spring model to design com-pliant mechanisms is analyzed. A performance diference minimization method for the topol-ogy optimization of compliant mechanisms aimed at eliminating de facto hinges is developed.Based on the method, several alternative formulation are developed. The validity and versatilityof the method are examined using several topology optimization approaches, such as the levelset methods and SIMP method.
Thirdly, for designing compliant mechanisms without using the output spring, a method isdeveloped by inserting the input and output mean compliance into the optimization model. Anew objective function is developed using the weighting sum method. The weighting factorsare set based on the information that is obtained from the previous iteration and automaticallyupdated with each optimization iteration step. The validity and versatility of the method are ex-amined using several topology optimization approaches to design compliant mechanisms withboth single input-output behaviour and multiple outputs behaviour.
Finally, conclusions and future works are put forward
This work will surely provide the theoretical support both in developing more powerfuland efcient level set-based topology optimization method and in consolidating the theoreticalbasis of designing compliant mechanisms.
引文
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