复杂细长结构动力模型降价及优化研究
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摘要
现代工程计算中常遇到类似火箭、火车车厢、船舶和高层建筑等这类超大型复杂细长结构的动力学性能分析和优化设计。随着CAD和有限元技术的进步使得我们能够很方便的建立模拟这些结构的规模很大、网格很密、详尽地描写了结构细节的有限元模型用于结构静力分析,但是由于大型复杂结构动力学分析的计算工作量非常大,动力优化工作量更大,几乎不可能基于这类有限元模型进行动力分析和优化。因此在对此类大型复杂结构建立非常精细的有限元模型进行分析和设计的同时,用很小的工作量获得一个能够近似预测结构动力性能,满足工程初步设计、优化、模拟等任务的物理降阶计算模型,在很多设计部门仍然非常有价值。本论文的研究正是基于这些需求。在对动力学模型降阶方法综述的基础上,主要包括了以下研究工作：
1.改进了局部刚体化动力模型降阶方法。在模型降阶位移转换阵中考虑了原结构的转角自由度,提高了复杂空间结构降阶模型的精度,由简到难的两个算例表明了本文改进后方法的有效性。将降阶模型和变量分离方法结合,建立了基于降阶模型的高效动力学优化方法,分别研究了基于给定壳体结构材料用量,以壳体厚度为设计变量,最大化结构基频和前两阶频率差值的优化问题,并与基于原结构的优化及基于响应面方法的优化,结果表明了基于局部刚体化动力降阶模型的优化不仅结果精度高,而且计算效率也高。
2.基于改进的局部刚体化模型降阶方法建立了空间复杂大型桁架的物理降阶模型,在此基础上设计了对桁架结构振动主动控制的LQG最优控制律。通过一个典型例题的数值仿真与控制理论中经常使用的内平衡降阶方法进行了比较,结果表明：本文提出的物理降阶模型不仅计算效率更高,而且在相同的控制力作用下,基于物理降阶模型具有更高的控制效率
3.针对纵向尺度显著大于横向的火箭等复杂梁式结构,提出了基于梁平截面假设的模型降阶方法。基于梁理论中的平截面假设,将结构每个截面上的有限元节点通过位移转换矩阵凝聚到该截面的形心,从而快速建立了用于模型降阶的减缩基向量。对于具有大开口的结构,用数值方法得到表示截面翘曲变形的翘曲基向量,弥补了平截面假定的不足。利用这些减缩基向量,可以快速得到自由-自由结构的降阶模型。自由-自由降阶模型可以用于实现多种边界条件下结构的频率分析和考虑频率约束的轻量化设计,具体算例证明了本方法的有效性。
4.引入有限元方法中梁单元的位移插值函数并建立新的模型降阶转换矩阵,改进了基于平截面假设的动力模型降阶方法。这一改进解决了基于平截面假定的动力模型降阶法要求有限元模型具有规整网格的苛刻条件、对于网格过密的模型所建立的转换矩阵过大这些问题,使该方法适用于任意网格及几何形状较为复杂的模型。利用降阶模型计算得到的模态向量是原结构特征向量的高精度近似,基于瑞利-里茨法进行一次逆迭代即可得到高精度的特征值。通过与Krylov子空间法相比较,进一步说明了本文模型降阶方法的计算量更低。在降阶模型基础上进行复杂结构时程响应计算,结果表明降阶模型不仅计算精度可满足实际工程计算的需要,并且大大的减少了计算量。
5.研究了将原梁式结构模型划分为不同的梁段,利用基于梁平截面及位移插值函数的动力模型降阶方法把每一梁段降阶为一个梁超单元,并按照一定次序组装成超梁模型。通过引入横向剪切变形修正了梁超单元的刚度阵,给出了快速计算或者优化获得剪切修正系数的过程,具体算例表明了引入剪切修正后的超梁模型具有较高的计算精度和效率。对于蒙皮加筋、蒙皮格栅圆柱壳模型的具体算例,将该方法与等效厚度梁模型、静力等效梁模型和等效刚度梁模型进行了对比,考虑了具有不同长度,结构不均匀的计算模型,频率分析结果表明超梁模型不仅计算精度和效率高,而且模型建立灵活,可以处理不同情况下的结构模型,适用范围广。
本人在论文工作初期参与动车组铝合金车体结构型材断面优化设计研究,优化结果实现了大型复杂车体结构减重,同时提升了车体整体刚度,降低了关键部位的应力水平,并完成会议论文一篇。这部分研究工作是本人第一次接触到实际大型复杂细长结构,也是本论文选题的应用背景之一。因此将这部分研究工作列入本论文附录中,希望能够反映本人学习相关内容的一个过程。
In modern engineering calculations, we often have to deal with dynamic analysis, or even optimization of dynamic performance for complicated slender structures, such as rockets, railroad cars, ships and high-rise buildings. With the development of CAD technology and finite element preprocessor, FEM model with huge number of degrees of freedom is often easily constructed and used for static analysis of such complicated structures. However, due to time-consuming computations and huge storage requirement of dynamic analysis and optimization, it is almost impossible to optimize structural dynamic performances based on the model which has huge number of degrees of freedom. Therefore, based on the fine FEM model of such complicated structures, constructing an approximate physical model with a small amount of work, which can be used to predict the structural dynamic performance and meet the requirement of design, optimization, simulation and other tasks in the preliminary design stage, is still very valuable in many design departments. To meet the need, the present work starts from the summary of dynamic model reduction methods and completes the following study:
1. The reduction method based on the assumption of block-wise rigid body motion is further modified and studied. By considering the rotational freedom of the original structure, the displacement transfer matrix is modified and the method is used for model reduction of complex spatial structure, two examples show the effectiveness of the modified method in this paper. By combining the reduced model with the variable decomposition approach, an efficient dynamic optimization method based on reduced model was developed. Then two optimization model are established, in which the constraint to keep the structural mass constant is imposed, the shell thickness is design variables, and the optimization objective is to maximize the fundamental frequency of the original structure and the frequency gap between the first two consecutive vibration frequencies respectively. By comparing with optimization based on response surface model and original model, the results show that the optimization based on the reduced model has high computational accuracy and efficiency.
2. By using the modified reduction method which is based on the assumption of block-wise rigid body motion, a physical reduced model of a large complex space truss is established. LQG optimal control law is designed to achieve the purpose of a truss structure vibration active control based on the physical reduced model. By comparing with the internal balanced model reduction method, the numerical simulation results show that the physical reduced model not only has higher computational efficiency, but also has higher control efficiency.
3. A new model reduction method is presented for the structures whose longitudinal dimension is significantly larger than transverse one. Based on the plane section assumption of the beam theory, the displacement of FEM nodes in each cross section is approximated by the motion of centroid of the cross section through the displacement transformation matrix, resulting in the localized base vectors. When the structure has large openings, structures'cross-sectional warping deformation mode is obtained by using numerical methods, which is considered as additional reduced base vectors. Then a free-free reduced model with high accuracy is constructed. Based on the reduced model, optimization model to minimize the structural weight considering frequency constrain in a variety of boundary conditions is studied. Numerical examples demonstrate the effectiveness of the proposed method.
4. The model reduction method based on plane cross-section assumption is further improved by including displacement interpolation function of beam and constructing a new physical reduced base vectors. This improvement solves the deficiency of the model reduction method based on the plane cross-section assumption, such as restrictions on regular finite element grid and large size of the reduced model, and makes the method applicable for more complicated FEM models of arbitrary meshes and geometry. By using the vibration modal vectors of the reduced model and constructing an initial guess of vibration modes of the original structure, one Rayleigh-Ritz iteration results in the vibration frequency of high accuracy. The method is compared with the Krylov subspace method, the examples show the validity and efficiency of this method. Transient dynamic analysis of a complicated example structure is carried out, the results show that the method meets the accuracy of practical engineering and greatly reduces the computational cost.
5. Based on the presented model reduction method, slender structure is divided into several beam part and each part is reduced to a super beam element, resulting in a free-free reduced super beam model. The shear deformation is included to modify the super beam for improving computational accuracy. A shear correction factor is quickly calculated or obtained through an optimization process. The numerical examples demonstrate the feasibility and efficiency of the proposed super beam model for fast approximate frequency analysis of complicated slender structures. Examples of specific grid stiffened cylindrical shell model are given to compare the reduced super beam model with the beam model based on equivalent thickness, the static response and equivalent stiffness. Different length, nonuniform structure models are tested. The frequency analysis results show that the reduced super beam not only has high computational accuracy and efficiency, and can handle different situations for a wide range.
At the early stage of this dissertation, author attended to the optimization study for aluminum alloy structure of CRH train and published one conference paper. For its difference to the main work of this dissertation, it will be presented in the appendix.
1.改进了局部刚体化动力模型降阶方法。在模型降阶位移转换阵中考虑了原结构的转角自由度,提高了复杂空间结构降阶模型的精度,由简到难的两个算例表明了本文改进后方法的有效性。将降阶模型和变量分离方法结合,建立了基于降阶模型的高效动力学优化方法,分别研究了基于给定壳体结构材料用量,以壳体厚度为设计变量,最大化结构基频和前两阶频率差值的优化问题,并与基于原结构的优化及基于响应面方法的优化,结果表明了基于局部刚体化动力降阶模型的优化不仅结果精度高,而且计算效率也高。
2.基于改进的局部刚体化模型降阶方法建立了空间复杂大型桁架的物理降阶模型,在此基础上设计了对桁架结构振动主动控制的LQG最优控制律。通过一个典型例题的数值仿真与控制理论中经常使用的内平衡降阶方法进行了比较,结果表明：本文提出的物理降阶模型不仅计算效率更高,而且在相同的控制力作用下,基于物理降阶模型具有更高的控制效率
3.针对纵向尺度显著大于横向的火箭等复杂梁式结构,提出了基于梁平截面假设的模型降阶方法。基于梁理论中的平截面假设,将结构每个截面上的有限元节点通过位移转换矩阵凝聚到该截面的形心,从而快速建立了用于模型降阶的减缩基向量。对于具有大开口的结构,用数值方法得到表示截面翘曲变形的翘曲基向量,弥补了平截面假定的不足。利用这些减缩基向量,可以快速得到自由-自由结构的降阶模型。自由-自由降阶模型可以用于实现多种边界条件下结构的频率分析和考虑频率约束的轻量化设计,具体算例证明了本方法的有效性。
4.引入有限元方法中梁单元的位移插值函数并建立新的模型降阶转换矩阵,改进了基于平截面假设的动力模型降阶方法。这一改进解决了基于平截面假定的动力模型降阶法要求有限元模型具有规整网格的苛刻条件、对于网格过密的模型所建立的转换矩阵过大这些问题,使该方法适用于任意网格及几何形状较为复杂的模型。利用降阶模型计算得到的模态向量是原结构特征向量的高精度近似,基于瑞利-里茨法进行一次逆迭代即可得到高精度的特征值。通过与Krylov子空间法相比较,进一步说明了本文模型降阶方法的计算量更低。在降阶模型基础上进行复杂结构时程响应计算,结果表明降阶模型不仅计算精度可满足实际工程计算的需要,并且大大的减少了计算量。
5.研究了将原梁式结构模型划分为不同的梁段,利用基于梁平截面及位移插值函数的动力模型降阶方法把每一梁段降阶为一个梁超单元,并按照一定次序组装成超梁模型。通过引入横向剪切变形修正了梁超单元的刚度阵,给出了快速计算或者优化获得剪切修正系数的过程,具体算例表明了引入剪切修正后的超梁模型具有较高的计算精度和效率。对于蒙皮加筋、蒙皮格栅圆柱壳模型的具体算例,将该方法与等效厚度梁模型、静力等效梁模型和等效刚度梁模型进行了对比,考虑了具有不同长度,结构不均匀的计算模型,频率分析结果表明超梁模型不仅计算精度和效率高,而且模型建立灵活,可以处理不同情况下的结构模型,适用范围广。
本人在论文工作初期参与动车组铝合金车体结构型材断面优化设计研究,优化结果实现了大型复杂车体结构减重,同时提升了车体整体刚度,降低了关键部位的应力水平,并完成会议论文一篇。这部分研究工作是本人第一次接触到实际大型复杂细长结构,也是本论文选题的应用背景之一。因此将这部分研究工作列入本论文附录中,希望能够反映本人学习相关内容的一个过程。
In modern engineering calculations, we often have to deal with dynamic analysis, or even optimization of dynamic performance for complicated slender structures, such as rockets, railroad cars, ships and high-rise buildings. With the development of CAD technology and finite element preprocessor, FEM model with huge number of degrees of freedom is often easily constructed and used for static analysis of such complicated structures. However, due to time-consuming computations and huge storage requirement of dynamic analysis and optimization, it is almost impossible to optimize structural dynamic performances based on the model which has huge number of degrees of freedom. Therefore, based on the fine FEM model of such complicated structures, constructing an approximate physical model with a small amount of work, which can be used to predict the structural dynamic performance and meet the requirement of design, optimization, simulation and other tasks in the preliminary design stage, is still very valuable in many design departments. To meet the need, the present work starts from the summary of dynamic model reduction methods and completes the following study:
1. The reduction method based on the assumption of block-wise rigid body motion is further modified and studied. By considering the rotational freedom of the original structure, the displacement transfer matrix is modified and the method is used for model reduction of complex spatial structure, two examples show the effectiveness of the modified method in this paper. By combining the reduced model with the variable decomposition approach, an efficient dynamic optimization method based on reduced model was developed. Then two optimization model are established, in which the constraint to keep the structural mass constant is imposed, the shell thickness is design variables, and the optimization objective is to maximize the fundamental frequency of the original structure and the frequency gap between the first two consecutive vibration frequencies respectively. By comparing with optimization based on response surface model and original model, the results show that the optimization based on the reduced model has high computational accuracy and efficiency.
2. By using the modified reduction method which is based on the assumption of block-wise rigid body motion, a physical reduced model of a large complex space truss is established. LQG optimal control law is designed to achieve the purpose of a truss structure vibration active control based on the physical reduced model. By comparing with the internal balanced model reduction method, the numerical simulation results show that the physical reduced model not only has higher computational efficiency, but also has higher control efficiency.
3. A new model reduction method is presented for the structures whose longitudinal dimension is significantly larger than transverse one. Based on the plane section assumption of the beam theory, the displacement of FEM nodes in each cross section is approximated by the motion of centroid of the cross section through the displacement transformation matrix, resulting in the localized base vectors. When the structure has large openings, structures'cross-sectional warping deformation mode is obtained by using numerical methods, which is considered as additional reduced base vectors. Then a free-free reduced model with high accuracy is constructed. Based on the reduced model, optimization model to minimize the structural weight considering frequency constrain in a variety of boundary conditions is studied. Numerical examples demonstrate the effectiveness of the proposed method.
4. The model reduction method based on plane cross-section assumption is further improved by including displacement interpolation function of beam and constructing a new physical reduced base vectors. This improvement solves the deficiency of the model reduction method based on the plane cross-section assumption, such as restrictions on regular finite element grid and large size of the reduced model, and makes the method applicable for more complicated FEM models of arbitrary meshes and geometry. By using the vibration modal vectors of the reduced model and constructing an initial guess of vibration modes of the original structure, one Rayleigh-Ritz iteration results in the vibration frequency of high accuracy. The method is compared with the Krylov subspace method, the examples show the validity and efficiency of this method. Transient dynamic analysis of a complicated example structure is carried out, the results show that the method meets the accuracy of practical engineering and greatly reduces the computational cost.
5. Based on the presented model reduction method, slender structure is divided into several beam part and each part is reduced to a super beam element, resulting in a free-free reduced super beam model. The shear deformation is included to modify the super beam for improving computational accuracy. A shear correction factor is quickly calculated or obtained through an optimization process. The numerical examples demonstrate the feasibility and efficiency of the proposed super beam model for fast approximate frequency analysis of complicated slender structures. Examples of specific grid stiffened cylindrical shell model are given to compare the reduced super beam model with the beam model based on equivalent thickness, the static response and equivalent stiffness. Different length, nonuniform structure models are tested. The frequency analysis results show that the reduced super beam not only has high computational accuracy and efficiency, and can handle different situations for a wide range.
At the early stage of this dissertation, author attended to the optimization study for aluminum alloy structure of CRH train and published one conference paper. For its difference to the main work of this dissertation, it will be presented in the appendix.
引文
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