大规模有限元系统频率响应及其灵敏度分析的高效算法研究

The frequency response analysis is used to determine the steady-state periodic responses of structures under the harmonically varying loads. This kind of forced vibrations will result in precision reduction, life decline, instability, or even damage and other serious consequences of engineering structures. The frequency response and its sensitivity analysis to the design parameters also plays a vital role in structural finite element model updating, damage identification, dynamic optimization, vibration control and etc. However, for a frequency band of interest, the frequency response and its sensitivity analysis of a complex structure require solving a series of sets of (complex) linear equations depending on frequency of excitation load, the computational process is very time-consuming. So the research on calculation methods for frequency response and its sensitivity analysis of a complex structure has important engineering significance and broad application prospects. The project is focused on the computational problems of frequency response and its sensitivity analysis for a large scale finite element system. We will establish the corresponding efficient algorithms, analyze their convergence, and develop the software module using Intel's MKL large sparse matrix calculation platform.. Research results of this project will enable computational cost of frequency response and its sensitivity analysis for large-scale finite element system to be remarkbly reduced, and can also promote development of structural finite element model modification, damage recognition, dynamic optimization and vibration control and other related fields.

频率响应分析用于确定结构在简谐载荷作用下产生的稳态周期响应。这类强迫振动会直接造成工程结构精度降低、寿命下降、失稳、甚至损坏等严重后果。频率响应及其对设计参数的灵敏度在结构有限元模型修正、损伤识别、动力优化、振动控制等领域也起着至关重要的作用。然而，在一个感兴趣的频带计算复杂结构的频率响应及其灵敏度需求解依赖于激励荷载频率的多个（复）线性方程组，非常耗时。因此研究复杂结构的频率响应及其灵敏度分析的数值方法具有重要的工程意义和广阔的应用前景。本项目将针对大规模有限元系统频率响应及其灵敏度分析问题展开研究，建立相应的高效算法并分析其收敛性，然后基于Intel 的MKL大型稀疏矩阵计算平台开发相应的软件模块。. 本项目研究成果可显著缩减大规模有限元系统频率响应及灵敏度分析的计算成本，亦能促进结构有限元模型修正、损伤识别、动力优化与振动控制等相关领域的发展。

频率响应分析用于确定结构在简谐载荷作用下产生的稳态周期响应。这类强迫振动会直接造成工程结构精度降低、寿命下降、失稳、甚至损坏等严重后果。频率响应及其对设计参数的灵敏度在结构有限元模型修正、损伤识别、动力优化、振动控制等领域也起着至关重要的作用。然而，在一个感兴趣的频带计算复杂结构的频率响应及其灵敏度需求解依赖于激励荷载频率的多个（复）线性方程组，非常耗时。因此研究复杂结构的频率响应及其灵敏度分析的数值方法具有重要的l理论意义和广阔的工程应用前景。本项目针对大规模有限元系统频率响应及其灵敏度分析问题开展了相关研究，建立了相应的高效算法并证明了所提出算法的收敛性，然后基于Intel 的MKL大型稀疏矩阵计算平台编写了相应的软件模块。此外， 我们还将这些高效算法成功用于结构受一个频段激励以及随机激励的结构拓扑优化问题。. 本项目建立的方法可显著缩减大规模有限元系统频率响应及灵敏度分析的计算成本，研究成果能促进结构有限元模型修正、损伤识别、动力优化与振动控制等相关领域的发展。

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