【作者】 于宁；

【导师】 孙吉贵；

【作者基本信息】 吉林大学 ， 软件工程， 2006， 硕士

【摘要】 有限元方法是一种解决工程与数学物理问题的数值方法。其基本求解思想是把计算域划分为有限个互不重叠的单元,在每个单元内,选择一些合适的节点作为求解函数的插值点,将微分方程中的变量改写成由各变量或其导数的节点值与所选用的插值函数组成的线性表达式,借助于变分原理或加权余量法,将微分方程离散求解。在工程领域应用有限元方法来分析物理系统就是我们所常说的有限元分析。今天,计算机技术日益完善,有限元方法也在各个工程领域中不断得到深入应用。本文的主要工作就是从有限元的理论入手,介绍如何对一类特定的微分方程:Poisson方程应用有限元方法,并用MATLAB来实现这样一个有限元求解器。最后模拟一个简单的工程问题,用这个有限元求解器来解决。具体如下:提出适用与本文中的有限元方法求解器的Poisson方程,并就这些类型的方程阐述有限元算法的实现。其中包括有weak form的推导,有限元的划分及类型的选择,针对不同类型的基函数的选择,局部刚性矩阵和局部负荷向量的生成,高斯求积公式的应用等。描述有限元求解器的结构,用MATLAB对这个求解器实现以及每部分实现的功能的描述,其中包括一些重要细节的处理方法。用已知精确解的Poisson方程来验证求解器的可用性。对比用求解器求出的数值解和精确解,计算误差并分析。然后用它来解决一个简单的工程问题,给出数值解。更多还原

【Abstract】 Finite Element Method (FEM) is a numerical method for solving problems in eigineeringand mathematical physics. In numerical analysis, FEM is used for solving partialdifferential equations (PDE) approximately. It’s main idea is to devied the computionaldomain into finite number of elements that don’t overlap with each other, and in eachelement, choose some proper nodes as the interpolation points for solving the unknownfunction, with the help of variational principle and weighted residual method, discretethe PDE to get the solution. Implementing FEM in engineering fieldto do analyses on physical systems is what we called Finite Element Analyses(FEA). FEMis first used in Structrual Mechanics, and later with the development of computerscience it is used in Fluid Dynamics gradually. Nowadays, as the computer technologybecoming more advanced day by day, the implemention of FEM becomes more and moreimportant in every engineering field. Implementing FEM in the design of industrialproducts’ structure makes the design of industrial products’ structure optimizeremarkablely, and uses the theoretical design instead of the experiential analogydesign.The main task of this thesis is to begin with the theory of FEM and introduce howto implement FEM in a certain type of differential equations: Poisson Equations, thenuse MATLAB to implement such a FEM solver, in the end use the FEM solver to slove asimple engineering problem. Details are as follows:1. Derive FEM form the solution of a physical model problem, and introduce how theFEM works in general problems. 2. Give the Poisson Equations that can be solved with the FEM solver, and expatiatehow to use FEM to solve these equations, which includes the derivation of weak form,how to generate the finite elements and how to choose the type of the element. Andaccording to different types of elements how to choose different basis functions, howto generate local stiff matrix and local load vector and how to use the Gauss quadrature. 3. Describ the structure of the FEM solver, use MATLAB to implement the FEM solverand describ the function each part holes. And methods of dealing with some importantdetails are included.4. Use a Poisson Equation whose analytical solution is already known to test theusablity of the FEM solver. Contrast the analytical solution with the numerical solutiongot form the FEM solver, compute for the error and do analyses.This thesis only does primary discussion and attempt to implement FEM. At presenta large number of software for FEM are produced, but lacking of currency, so still thereis a lot of research work to do in this field. Till now FEM is still developing, andis becoming more and more mature in theory and application continuously. Thus thedevelopment of large current FEM programs which are based on the powerful computationalcapability of modem computers has a bright future.更多还原