光滑粒子法与有限元的耦合算法及其在冲击动力学中的应用
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摘要
本文较为系统和全面的介绍了Lagrange有限元法、光滑粒子法和一种光滑粒子与有限元的耦合算法的基本原理及离散思想。详细讨论了这三种方法中的一些基本问题，如有限元法中的离散格式、滑移面问题，光滑粒子法中的离散格式、核函数、光滑长度等问题，以及耦合算法中有限元单元向光滑粒子的转换问题、光滑粒子与有限元交界面上的滑移计算和交界面附近光滑粒子的计算等问题。最后将耦合算法应用于一维应变波传播和高速碰撞计算，通过与有限元及纯光滑粒子法的对比计算说明耦合算法的特点和长处。
首先介绍了采用Lagrange有限元离散连续介质力学守恒方程的基本思想和计算步骤。讨论了有限元法在高速侵彻模拟计算中出现的滑移面问题和单元大畸变问题，给出了相应的处理方法。
然后利用插值理论，给出光滑粒子算法中函数及函数各阶导数的核估计，分析了光滑粒子法离散连续介质力学守恒方程组的基本方法及相应的离散形式。此外还讨论了光滑粒子法中的一些基本问题，比如核函数的性质、光滑长度的选取、人工粘性、守恒光滑法、本构关系以及该方法计算的主要步骤和时间步长选取等。
接着介绍了一种光滑粒子与有限元的耦合算法，这种算法在初始时刻用有限元方法建模，随着变形的增大，大变形区域的有限元单元将自动转换成光滑粒子进行模拟计算，从而保证计算的正常进行。讨论和分析了耦合算法中的三个主要难点，有限元单元向光滑粒子的转换问题、光滑粒子与有限元交界面上的滑移计算问题和交界面附近光滑粒子的计算问题，提出了解决的方法并给出了全部算式。
粒子搜索算法在耦合算法和光滑粒子法中都起着非常重要的作用，直接决定了计算效率，以此为目标，利用核函数影响域的局域性，提出了一种搜索简便、计算效率获得很大提高的分区搜索算法。边界的确定也是耦合算法与光滑粒子法的一个难点。本文通过引入粒子边界圆的概念，提出了依据边界圆的覆盖方式区分边界点和内点，有效地解决了边界难以确定的问题。
将耦合算法应用于一维应变波计算是冲击力学数值方法研究中的重要内容之一。一维应变波算例的结果显示，耦合算法完全可以用于弹塑性波传播和演化规律的数值模拟，其精度与有限元法和纯光滑粒子法相当，但其计算效率要比光滑粒子法高。这为材料和结构冲击响应的数值方法研究积累了经验，拓宽了耦合算法的应用范围。
最后本文研究了二维轴对称条件下光滑粒子法的离散思想、方法和全部算式，并将耦合算法成功应用于高速碰撞和侵彻贯穿模拟计算。通过Taylor碰撞的考核计算，验证了耦合算法计算高速碰撞和大变形问题的能力和精度。在高速碰撞和侵彻贯穿计算中，比较了有限元法、纯光滑粒子法和耦合算法的计算结果，讨论了三种
In this thesis the basis theory and discretization method of Lagrange Finite Element Method(FEM), Smoothed Particle Hydrodynamics(SPH) and a coupling method (linking of Smoothed Particle Hydrodynamics methods to standard finite element methods) are described. Some related problems are discussed, such as discretization forms and sliding surface in FEM; discretization forms, kernel functions and smoothing length in SPH; conversion of elements to particles, sliding algorithms on the interface between elements and particles and the computation of the particles near the interface in coupling method. Finally, the coupling method is applied to the study of one-dimensional strain wave propagation and hyperveiocity impact phenomenon, the results show that the method has good performance in the numerical study of impact dynamics.
First, the basic idea of the discretization method of conservation equations in continuum mechanics with Lagrange Finite Element Method and the flowchart of computations are described. The problems of sliding surface and severely distorted element in hyperveiocity impact simulations are discussed, and the corresponding solving ways are proposed.
Kernel estimate of a function and its derivatives are the key point in SPH method and their various forms are obtained by kernel interpolation theory. Based on the analysis of SPH method, the discretization formations of conservation equations in continuum mechanics are established. In addition, several topics, such as kernel function properties, the choice of smoothing length, artificial viscosity, constitution relations, conservative smoothing method and flowchart of computations, are discussed in details.
Then a coupling method (linking of Smoothed Particle Hydrodynamics methods to standard finite element methods) is proposed, which creates the model with finite element at the beginning and automatically convert distorted elements into smoothed particles during dynamic deformation. Three difficulties in the coupling method are discussed, such as the conversion of elements to particles, the contact and sliding algorithms between elements and particles and the computation of the particles near the interface, and some ways are proposed to solve these problems.
Particle searching algorithm plays a very important role in coupling method and SPH computations since it takes most computational time and hence it determines the computation efficiency. Considering kernel function as a strong compact support function, a subdomain searching algorithm is suggested, which is a simple method and increases computation efficiency greatly. One problem in coupling method and SPH is hard to define the boundary. In order to solve the problem, we use the concept of particle boundary circle to determine it and the boundary particle is identified by whether the particle circle is
首先介绍了采用Lagrange有限元离散连续介质力学守恒方程的基本思想和计算步骤。讨论了有限元法在高速侵彻模拟计算中出现的滑移面问题和单元大畸变问题，给出了相应的处理方法。
然后利用插值理论，给出光滑粒子算法中函数及函数各阶导数的核估计，分析了光滑粒子法离散连续介质力学守恒方程组的基本方法及相应的离散形式。此外还讨论了光滑粒子法中的一些基本问题，比如核函数的性质、光滑长度的选取、人工粘性、守恒光滑法、本构关系以及该方法计算的主要步骤和时间步长选取等。
接着介绍了一种光滑粒子与有限元的耦合算法，这种算法在初始时刻用有限元方法建模，随着变形的增大，大变形区域的有限元单元将自动转换成光滑粒子进行模拟计算，从而保证计算的正常进行。讨论和分析了耦合算法中的三个主要难点，有限元单元向光滑粒子的转换问题、光滑粒子与有限元交界面上的滑移计算问题和交界面附近光滑粒子的计算问题，提出了解决的方法并给出了全部算式。
粒子搜索算法在耦合算法和光滑粒子法中都起着非常重要的作用，直接决定了计算效率，以此为目标，利用核函数影响域的局域性，提出了一种搜索简便、计算效率获得很大提高的分区搜索算法。边界的确定也是耦合算法与光滑粒子法的一个难点。本文通过引入粒子边界圆的概念，提出了依据边界圆的覆盖方式区分边界点和内点，有效地解决了边界难以确定的问题。
将耦合算法应用于一维应变波计算是冲击力学数值方法研究中的重要内容之一。一维应变波算例的结果显示，耦合算法完全可以用于弹塑性波传播和演化规律的数值模拟，其精度与有限元法和纯光滑粒子法相当，但其计算效率要比光滑粒子法高。这为材料和结构冲击响应的数值方法研究积累了经验，拓宽了耦合算法的应用范围。
最后本文研究了二维轴对称条件下光滑粒子法的离散思想、方法和全部算式，并将耦合算法成功应用于高速碰撞和侵彻贯穿模拟计算。通过Taylor碰撞的考核计算，验证了耦合算法计算高速碰撞和大变形问题的能力和精度。在高速碰撞和侵彻贯穿计算中，比较了有限元法、纯光滑粒子法和耦合算法的计算结果，讨论了三种
In this thesis the basis theory and discretization method of Lagrange Finite Element Method(FEM), Smoothed Particle Hydrodynamics(SPH) and a coupling method (linking of Smoothed Particle Hydrodynamics methods to standard finite element methods) are described. Some related problems are discussed, such as discretization forms and sliding surface in FEM; discretization forms, kernel functions and smoothing length in SPH; conversion of elements to particles, sliding algorithms on the interface between elements and particles and the computation of the particles near the interface in coupling method. Finally, the coupling method is applied to the study of one-dimensional strain wave propagation and hyperveiocity impact phenomenon, the results show that the method has good performance in the numerical study of impact dynamics.
First, the basic idea of the discretization method of conservation equations in continuum mechanics with Lagrange Finite Element Method and the flowchart of computations are described. The problems of sliding surface and severely distorted element in hyperveiocity impact simulations are discussed, and the corresponding solving ways are proposed.
Kernel estimate of a function and its derivatives are the key point in SPH method and their various forms are obtained by kernel interpolation theory. Based on the analysis of SPH method, the discretization formations of conservation equations in continuum mechanics are established. In addition, several topics, such as kernel function properties, the choice of smoothing length, artificial viscosity, constitution relations, conservative smoothing method and flowchart of computations, are discussed in details.
Then a coupling method (linking of Smoothed Particle Hydrodynamics methods to standard finite element methods) is proposed, which creates the model with finite element at the beginning and automatically convert distorted elements into smoothed particles during dynamic deformation. Three difficulties in the coupling method are discussed, such as the conversion of elements to particles, the contact and sliding algorithms between elements and particles and the computation of the particles near the interface, and some ways are proposed to solve these problems.
Particle searching algorithm plays a very important role in coupling method and SPH computations since it takes most computational time and hence it determines the computation efficiency. Considering kernel function as a strong compact support function, a subdomain searching algorithm is suggested, which is a simple method and increases computation efficiency greatly. One problem in coupling method and SPH is hard to define the boundary. In order to solve the problem, we use the concept of particle boundary circle to determine it and the boundary particle is identified by whether the particle circle is
引文
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