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基于变密度法的结构动响应拓扑优化研究
【作者】 李翔；
【导师】 王皓；
【作者基本信息】 复旦大学 ， 工程力学， 2011， 硕士
【摘要】 本文以连续体结构拓扑优化的变密度法及其在动响应拓扑优化中的应用为研究对象。在变密度法方面,从经典变密度法的灰度单元问题以及由此带来的收敛速度过慢问题出发,提出了带灰度过滤函数的变密度法；在应用方面,研究了连续体结构在基础简谐激励下的响应拓扑优化问题,以及连续体在频带激励下的整体响应优化问题。首先,本文介绍了变密度法连续体拓扑优化的基本理论。详细介绍了拓扑优化的SIMP插值模型,从Kuhn-Tucker条件出发严格推导了最优准则法,讨论了变密度法优化的棋盘格现象及Sigmund过滤方法。其中,定义了一个结构平均密度指标,以此评价优化结果的灰度收敛性。从惩罚因子对优化结果的影响、棋盘格现象、计算效率及灰度收敛性等方面展开讨论。其次,针对经典变密度法的优化结果中灰度单元过多,以及由灰度收敛性太慢引起的优化效率过低问题,提出了带灰度过滤函数的变密度拓扑优化方法。探讨了经典变密度法产生灰度单元的原因,归纳了灰度过滤函数满足的性质并给出两个可行的过滤函数,由此提出了一种带灰度过滤函数的拓扑优化方法。针对经典的静刚度问题展开讨论,并选择文献中常见的其它拓扑优化算例验证方法的通用性和实用性。第三,研究了连续体基础简谐激励下的动响应拓扑优化问题。给出了基础简谐激励响应优化的数学模型,并推导了目标和约束的灵敏度；该模型为有阻尼模型,灵敏度的推导采用伴随法实现。阐明了SIMP插值模型不能合理描述低密度区域材料属性的,从而出现附属效应现象,并探讨了避免该现象的策略。此外,讨论了基础激励频率对优化结果的影响。第四,对结构在频带激励下的整体响应优化问题进行研究。采用目标频带内各模态峰值响应的q-范数作为整体响应水平的衡量指标,将等效目标函数的灵敏度表示为固有频率和振型灵敏度的函数,进而采用基于梯度的迭代方法求解,实现结构在宽频激励下的整体动响应优化,该方法适合于多频带的响应控制。此外,在频带响应优化模型的基础上,通过增加频率约束,使之能应用于密集频率结构,避免迭代过程中的密频和重频现象。数值算例上先以较简单的桁架尺寸和形状优化为背景考察方法的有效性,最后以连续体拓扑优化的为例展开讨论。
【Abstract】 This thesis focuses on structural topology optimization using variable density method and its applications in dynamics response optimization. Form theoretical aspect, a new variable density method with gray-scale filter function is adopted compared with the classic variable density method. As applications, dynamic response topology optimization under harmonic foundation excitation and structural bandwidth frequency response optimization are investigated.Firstly, the fundamental theory of continuum topological optimization by variable density approach is discussed, where the SIMP density-stiffness interpolation scheme as well as the optimally criteria method based on Kuhn-Tucker optimal condition are mainly concerned. Numerical instabilities, such as checkboards, mesh dependencies and intermediate densities in optimization result are discussed. A structural mean density index is proposed to check the gray-scale element in final optimized structure and its convergence ability during iteration.Secondly, a new gray-scale filter function was added on traditional variable density approach for topology optimization, in order to solve the presence of gray-scale material in optimization result and low convergence efficiency during optimization procedure. Based on the properties of filter function, two proper gray-scale filter functions were adopted. Illustrated examples are presented both for verification and application for the filter method. The newly proposed gray-scale filter function method takes effect on enhancing the 0-1 convergence of the optimization result and decreasing iteration number caused by low convergence efficiency of gray elements.Thirdly, structural dynamic response optimization under foundation harmonic excitation is investigated. First, the optimization model is established with an aim for minimization concerned point’s dynamic response under global volume constraint. Based on the optimization model, sensitivity of aim function is derived then optimized by optimality criteria method. In order to improve the low convergence ability of gray density variables, a gray variable filter method is adapted in this paper. Illustrated examples are presented both for verification and application for the dynamic response optimization and the effect of new gray variable filter method.Finally, a structural dynamics response optimization under bandwidth frequency excitation is considered. A differentiable q-norm form of response function’s peak value is introduced according to properties of frequency response function. This equivalent aim function can be used in multiple-bandwidth situation. Sensitivities of equivalent aim function then can be derived from structure’s eigen-sensitivity. Besides, frequency constraints were introduced for prohibiting clustered or repeated frequencies during optimization. Illustrated examples of truss sizing and shape optimization and continuum topology optimization are presented for verification and application for proposed method.