The Implementation Of A FEM Solver To Poisson Equations Within Two Dimensions Using MATLAB

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.