离散奇异卷积法-基本原理及应用
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摘要
离散奇异卷积法是一种相对新型的数值计算方法，该法同时具有全域方法的精确性和局部方法的适应性，在许多工程问题分析中被证明是有效的。离散奇异卷积法能够处理复杂的几何，并被成功地用于求解一些有挑战性的问题，例如，具有不规则内部支持的板的自由振动问题和高模态的振动问题。
离散奇异卷积法成功应用的一个关键是如何利用边界条件消除区域外的虚节点。但是，如果有自由边界的情况，则在应用离散奇异卷积法时目前仍然存在一些困难，所以，文献中报道的成功应用绝大部分只涉及简支或者固支边界。论文的主要目的就是通过研究提出一种能够有效处理自由边界条件的方法，并采用发展了的离散奇异卷积法来求解含各种边界包括自由边界的线性和非线性问题，拓展离散奇异卷积法在工程分析中的应用范围。
本文的内容包括：一开始先对离散奇异卷积法及其应用的进展情况作一综述，根据其在应用方面存在的问题确定本文的研究内容。然后，为了论文的完整性简要介绍离散奇异卷积法的基本原理，提出一种施加自由边界条件的新方法--基于泰勒级数展开法，同时将离散奇异卷积法与微分求积法进行了比较。接着是各种各样的应用，包括：静力和屈曲分析、自由振动和波的传播分析、非线性分析，重点放在含应力边界或自由边界、几何和载荷不连续变化的结构元件的力学分析。论文给出了详细的公式推导和求解过程，并将采用离散奇异卷积法获得的结果与解析解、文献中有报道的数值解或者采用有限元法和微分求积法得到的结果进行了比较。最后，对全文的工作进行了总结，给出了一些结论和论文的创新点，同时，还给出了一些需要进一步研究的内容。
The discrete singular convolution (DSC) is a relatively new numerical method. The methodpossesses both the accuracy of the global method and the flexibility of the local method, and has beenproved efficient for analyzing many engineering problems. The DSC can handle complex geometry,and has been successfully used in solving some challenge problems, such as the free vibration ofplates with irregular internal supports, and vibrating at higher-order modes.
One of the key issues for successfully applying the DSC algorithm is the elimination of thefictitious points outside the real structural domains by using the boundary conditions. It is observed,however, that there still exist some difficulties in using DSC if the free boundary is encountered.Therefore, most reported research work in using DSC only involved the simply supported or/andclamped edges. The main objectives of this dissertation are to investigate an efficient way forapplying the free boundary conditions, use the developed DSC algorithm for solutions of linear andnonlinear problems with various boundary conditions, including the free boundary conditions, andextend the application range of the DSC algorithm in engineering analysis.
The contents of the dissertation are as follows. Firstly, the research development on the DSCalgorithm and its applications is summarized. Based on the existing shortage in applications of theDSC algorithm, the research contents are presented. Then the basic principle of the DSC algorithm isbriefly presented for completeness considerations. A new way to applying the free boundaryconditions, called Taylor series expansion based method, is proposed. Comparisons are made with thedifferential quadrature method (DQM). After that, various applications in using the DSC algorithmare presented, including static and buckling analysis, free vibration and wave propagation analysis,and nonlinear analysis. Emphasize has been paid to the structural components with stress boundaryconditions or free boundary conditions, with geometric discontinuity, as well as load discontinuity.Detained formulations and solution procedures are presented. The DSC results are compared to eithertheoretical solutions, or existing numerical results or results obtained by using finite element methodor by the DQM. Finally, the research work is briefly summarized. A few conclusions are drawn andthe innovations are pointed out. Some research topics are given for future investigations.
离散奇异卷积法成功应用的一个关键是如何利用边界条件消除区域外的虚节点。但是，如果有自由边界的情况，则在应用离散奇异卷积法时目前仍然存在一些困难，所以，文献中报道的成功应用绝大部分只涉及简支或者固支边界。论文的主要目的就是通过研究提出一种能够有效处理自由边界条件的方法，并采用发展了的离散奇异卷积法来求解含各种边界包括自由边界的线性和非线性问题，拓展离散奇异卷积法在工程分析中的应用范围。
本文的内容包括：一开始先对离散奇异卷积法及其应用的进展情况作一综述，根据其在应用方面存在的问题确定本文的研究内容。然后，为了论文的完整性简要介绍离散奇异卷积法的基本原理，提出一种施加自由边界条件的新方法--基于泰勒级数展开法，同时将离散奇异卷积法与微分求积法进行了比较。接着是各种各样的应用，包括：静力和屈曲分析、自由振动和波的传播分析、非线性分析，重点放在含应力边界或自由边界、几何和载荷不连续变化的结构元件的力学分析。论文给出了详细的公式推导和求解过程，并将采用离散奇异卷积法获得的结果与解析解、文献中有报道的数值解或者采用有限元法和微分求积法得到的结果进行了比较。最后，对全文的工作进行了总结，给出了一些结论和论文的创新点，同时，还给出了一些需要进一步研究的内容。
The discrete singular convolution (DSC) is a relatively new numerical method. The methodpossesses both the accuracy of the global method and the flexibility of the local method, and has beenproved efficient for analyzing many engineering problems. The DSC can handle complex geometry,and has been successfully used in solving some challenge problems, such as the free vibration ofplates with irregular internal supports, and vibrating at higher-order modes.
One of the key issues for successfully applying the DSC algorithm is the elimination of thefictitious points outside the real structural domains by using the boundary conditions. It is observed,however, that there still exist some difficulties in using DSC if the free boundary is encountered.Therefore, most reported research work in using DSC only involved the simply supported or/andclamped edges. The main objectives of this dissertation are to investigate an efficient way forapplying the free boundary conditions, use the developed DSC algorithm for solutions of linear andnonlinear problems with various boundary conditions, including the free boundary conditions, andextend the application range of the DSC algorithm in engineering analysis.
The contents of the dissertation are as follows. Firstly, the research development on the DSCalgorithm and its applications is summarized. Based on the existing shortage in applications of theDSC algorithm, the research contents are presented. Then the basic principle of the DSC algorithm isbriefly presented for completeness considerations. A new way to applying the free boundaryconditions, called Taylor series expansion based method, is proposed. Comparisons are made with thedifferential quadrature method (DQM). After that, various applications in using the DSC algorithmare presented, including static and buckling analysis, free vibration and wave propagation analysis,and nonlinear analysis. Emphasize has been paid to the structural components with stress boundaryconditions or free boundary conditions, with geometric discontinuity, as well as load discontinuity.Detained formulations and solution procedures are presented. The DSC results are compared to eithertheoretical solutions, or existing numerical results or results obtained by using finite element methodor by the DQM. Finally, the research work is briefly summarized. A few conclusions are drawn andthe innovations are pointed out. Some research topics are given for future investigations.
引文
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