【作者】 周浩洋；

【导师】 李锡夔；

【作者基本信息】 大连理工大学 ， 固体力学， 2006， 博士

【摘要】 弹性波传播问题的研究在许多科学和技术领域都有着广泛的应用。例如:通过研究弹性波传播中的衍射现象来解释和研究结构中的动应力集中问题:通过研究真实或人工地震产生的波动以了解地球的内部结构;通过研究地下间断的反射波可以大概地知道可能含油的地层;对材料和结构进行无损探伤;在土木工程领域对地基和地下建筑进行强度和结构分析;在医学上对人体物理信息探测所使用的最普遍的B超和CT等等都与弹性波传播理论有着密切的关系。 有限单元法是求解波传播问题的主要数值方法之一。虽然它有很多优点,并成功地模拟了很多波传播问题,但同样存在许多不足之处。事实上,Zienkiewicz把短波问题的数值模拟视为有限元法尚未解决的两个主要问题之一。例如,为使结果达到可接受的程度,一般说来低阶的有限单元每个波长需要至少布置10个节点。由此导致计算时需要的内存较大,耗时较多,计算效率低下。并且,低阶的有限单元有比较严重的频散特性,高阶的有限单元则可能产生虚假的波动。单位分解有限元法(PUFEM)是近十年来发展起来的数值方法,它使得有限元插值空间中可以包含所求问题解的已知解析信息。因此它可以胜任许多传统有限元方法不能处理得很好或者需要非常大计算量的问题。 本文首先回顾了PUFEM的理论基础和现有工作,然后针对传统有限元法模拟短波传播问题的严重局限性,利用PUFEM插值空间中可包含所求问题解的已知信息的特点,主要进行了以下工作:1.首次发展了一种用于瞬态弹性波传播数值模拟的单位分解有限元模型,有限元空间由形成单位分解的标准有限单元形函数乘以定义为局部子空间基函数的简谐振荡形函数构成。2.针对PUFEM单元矩阵中的被积函数具有强烈的振荡特性,应用直角坐标下的标准有限元形函数和单元内的波动方向知识提出了一种新的单元矩阵解析积分方案。3.将PUFEM应用于反平面剪切波的传播和散射问题中。对已知波传播方向时单位分解有限元如何选择局部子空间中的波数k给出了建议。4.在Zienkiewicz和Shiomi的用于高速动力过程分析的饱和多孔介质广义Biot理论u-U公式的基础上,推导和建立了基于PUFEM的饱和多孔介质动力问题的离散方程。5.在大型通用有限元分析程序LAGAMINE的框架下,编制了用于二维波传播数值模拟的PUFEM程序。数值例题显示在相同精度下,PUFEM的计算效率明显高于传统的有限元法。解析积分在计算效率上也比高斯—勒让德数值积分有大幅度的提高。各章的内容安排如下: 第一章首先对有限差分法、伪谱法、有限元法、无限元法、边界元法、谱单元法和格子法等各种用于波传播的数值方法做了简单的回顾,接下来着重介绍了以单位分更多还原

【Abstract】 The theory of elastic wave propagation can be applied to a variety of fields of science and technology, such as dynamic stress concentration in the structure, earthquake in seismology, oil exploration in geophysics, nondestructive evaluation of materials, strength and structure analysis in civil engineering, CT in medicine etc.The finite element method is one of the most effective numerical methods to simulate wave propagation problems. But there still have some unsolved problems. Indeed Zienkiewicz regarded the numerical simulation of the short wave problem as one of the two main unsolved problems of FEM. It is known that the choice of around 10 nodal points per wavelength is usually recommended to obtain an acceptable level of accuracy in order to solve the elastic wave equation. Consequently, this issue presents a severe limitation on the application of the FE procedure to the numerical simulation of the short wave problems in practice since the computational time needed is prohibitively expensive. In addition, it was reported that low-order finite elements exhibit poor dispersion properties, while higher-order finite elements raise some troublesome problems like the occurrence of spurious waves.Partition of unity finite element method (PUFEM), which was developed and studied in last decade, has the ability to include a priori knowledge about the local behavior of the solution in the finite element space. It enables us to construct finite element spaces that perform very well in cases where the classical finite element methods fail or are prohibitively expensive.In the dissertation, the mathematical foundation and recent work of PUFEM are reviewed firstly. Then, PUFEM was used to study the numerical simulation of elastic wave propagation as follows. (1) A PUFEM model is developed for the numerical simulation of transient elastic wave propagation. The finite element spaces are constructed by multiplying the standard finite element shape functions, which form a partition of unity, with the harmonic shape functions defined as the bases of the subspaces, which consist of a set of plane waves traveling in prescribed directions. (2) A special integration scheme, which is analytic for the elements with straight edges and semi-analytic for the elements with curved edges, for computing element matrices with the oscillatory nature of the integrands is developed. (3) The PUFEM model was used to simulate the wave propagation and scattering problem of anti-plane wave. The proper choice of the wave number k in the subspace of discretization approximation is discussed when the directions of wave propagation areknown. (4) A PUFEM model in the frame of generalized Biot u-U formulations proposed by Zienkiewicz and Shiomi is developed to simulate the problem of wave propagation in the saturated porous media when the fluid is compressible. (5) A program of PUFEM was developed to simulate the wave propagation in two dimensions in the framework of LAGAMINE, which is a general FEM code. The results of numerical examples exhibit that PUFEM is more efficient than FEM and the derived analytic integration scheme used for PUFEM saves a great deal of computational time as compared with standard Gauss-Legendre integration scheme. The present dissertation is outlined with the following chapters.In chapter 1, the numerical methods for simulating wave propagation are surveyed, such as finite difference method, pseudo-spectral method, finite element method, infinite element method, boundary element method, spectral element method and grid method etc. The PUFEM and its recent developments are discussed. Some existing work used to simulate wave propagation in the saturated porous media is also briefly reviewed. At the last of the chapter, the contents of the dissertation are outlined.In chapter 2, the theory and application of PUFEM are surveyed briefly. The concept of PUFEM is introduced after the discussion on the property of FEM approximation. The mathematical foundation of PUFEM and the relation of PUFEM to other numerical methods are presented. The applications of PUFEM to both Crack and the wave propagation problems are introduced in detail.In chapter 3, the PUFEM model for simulating transient elastic wave propagation is investigated. It is remarked that the existing work on PUFEM for simulations of wave propagations are only limited to steady-state scalar problems. The Shape function spaces of PUFEM for transient analysis are constructed, that makes the model able to directly handle the essential boundary conditions as it does in FEM. In each local shape function subspace, the wave direction and the wave number can be specified individually for a nodal point. The ill-conditioning problem of the effective stiffness matrix is also discussed in this chapter.In chapter 4, an analytic integration scheme for the element matrix in PUFEM is studied. The integrand of the matrix has highly oscillatory property; few existing integration schemes are successful in view of both accuracy and efficiency in computation. Here, a special integration scheme for computing the oscillatory function is developed. The integration scheme is analytic for the elements with straight edges and semi-analytic for the elements with curved edges. Therefore, the computational efficiency of the PUFEM is further enhanced as compared with the PUFEM using standard Gauss-Legendre integration scheme.In chapter 5, PUFEM is used to simulate the propagation and scattering problem of anti-plane wave, and the choice of the wave numbers k in the harmonic subspaces is suggested when the directions of wave propagation are known. When the directions of the wave propagation are known, the harmonic shape function subspaces of PUFEM can be enriched by taking different values of the wave number in the given few directions to obtain the numerical results superior to those obtained by the standard FEM in accuracy and efficiency, while the ill-conditioning of the effective stiffness matrix can be alleviated and even avoided. It is noted that the PUFEM can not only well simulate the propagation of a single harmonic wave but also simulate that of the complex wave composed of a number of harmonic waves.In chapter 6, the PUFEM is used to simulate the wave propagation problem in the saturated porous media. A PUFEM model in the frame of generalized Biot u-U formulations proposed by Zienkiewicz and Shiomi is developed to simulate the problem of wave propagation in the saturated porous media when the fluid is compressible. The PUFEM model is used to simulate the elastic wave propagation problems in drained and undrained saturated porous media.In chapter 7, the PUFEM program and data structures of the PUFEM code for simulating the wave propagation in two dimensions are described. The program was developed in the framework of the general FEM code LAGAMINE. Particularly, the functions of main subroutines, the flow charts of the PUFEM analysis, and the format of the data flow are explained and presented.The main contributions of the dissertation are summarized and the further work is suggested in the chapter 8.The work carried out in the dissertation is supported by National Key Basic Research and Development Program (973 Program, No. 2002CB412709) and the National Natural Science Foundation of China (No. 19832010).更多还原