二维弹性力学中的元胞自动机—有限元耦合算法

发布时间：2018-06-10 00:27

本文选题：耦合算法 + 有限元法　；参考：《西南科技大学》2017年硕士论文

【摘要】：元胞自动机是一种时间、空间都离散的动力系统,作为一种数值计算工具已经被广泛应用许多领域。本研究基于元胞自动机思想提出一种用于求解二维弹性力学问题的无网格方法。并将该无网格方法与有限元法进行耦合,形成一种新的耦合算法,用于计算二维固体模型的位移与应力。该耦合算法将二维弹性域分成两个子区域,靠近边界的为有限元子区域,其余的为元胞自动机子区域。有限元子区域用有限元法进行网格划分,元胞自动机子区域用一系列随机的元胞结点进行离散。在元胞自动机子区域内,每个元胞结点都有一个正六边形的影响域,影响域内包含若干个结点,从中选择6个结点作为中心结点的邻居。元胞的正六边形影响域可以分成6个正三角形,用有限元三角形插值可以建立邻居结点与影响域顶点、中心结点间的位移关系。通过三角形的刚度系数矩阵可以建立影响域顶点和中心结点间的关系,利用系数矩阵的转置消去影响域顶点的位移,则得到元胞自动机区域内任一结点与其邻居结点间位移的关系。在靠近边界的有限元子区域,任一结点与其相关结点间的关系直接通过有限元法中的总体刚度矩阵建立。两个子区域都在元胞自动机的框架下求解,将结点位移求解式定义为元胞自动机的局部演化规则,利用元胞自动机的动态演化进行求解。用这种方法可以将有限元子区域和元胞自动机子区域无缝连接,实现算法的自然耦合。对二维弹性力学模型,通过元胞自动机的动态演化求得结点的位移后,结点的应力可以利用传统有限元方法得到。元胞自动机的演化求解可以从任意位置以任意顺序进行,这使得该耦合算法在并行计算方面有很大优势。本文中也提出了一种简单的并行计算方法。算例证明了该耦合算法的正确性。
[Abstract]:Cellular automata (CA) is a dynamic system with discrete time and space. As a numerical tool, Cellular Automata (CA) has been widely used in many fields. This paper presents a meshless method for solving two-dimensional elastic problems based on cellular automata. The meshless method is coupled with the finite element method to form a new coupling algorithm, which can be used to calculate the displacement and stress of the two-dimensional solid model. The coupling algorithm divides the two-dimensional elastic domain into two sub-regions, the finite element subregion near the boundary, and the other sub-area for the cellular automatic machine. Finite element method is used to divide the finite element subregion and a series of random cellular nodes are used to discretize the automatic cell subregion. In the sub-region of cellular automatic machine, each cell node has a hexagonal influence domain, which contains several nodes, from which 6 nodes are selected as the neighbors of the central node. The regular hexagonal influence domain of a cell can be divided into six regular triangles, and the displacement relationship between neighbor nodes, vertices and center nodes of the domain can be established by means of finite element triangle interpolation. By means of the stiffness coefficient matrix of the triangle, the relationship between the vertex of the influence domain and the central node can be established, and the displacement of the vertex of the influence domain can be eliminated by the transposition of the coefficient matrix. Then the relationship between the displacement of any node in the cellular automata region and its neighbor node is obtained. In the finite element subregion near the boundary, the relationship between any node and its related nodes is directly established by the total stiffness matrix in the finite element method. Both subregions are solved under the framework of cellular automata. The solution of node displacement is defined as the local evolution rule of cellular automata, and the dynamic evolution of cellular automata is used to solve the problem. By using this method, the finite element subregion and the cellular automatic sub-region can be seamlessly connected to realize the natural coupling of the algorithm. For the two-dimensional elastic model, the stress of the node can be obtained by the traditional finite element method after the dynamic evolution of the cellular automata is used to obtain the displacement of the node. The evolutionary solution of cellular automata can be carried out in any order from any position, which makes the coupled algorithm have a great advantage in parallel computing. A simple parallel computing method is also proposed in this paper. An example is given to prove the correctness of the coupled algorithm.
【学位授予单位】：西南科技大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O343.1

【相似文献】