Analisis De Elementos Finitos

CSCI, MATH 6860 FINITE ELEMENT ANALYSIS Lecture Notes: Spring 2000
Joseph E. Flaherty Amos Eaton Professor Department of Computer Science Department of Mathematical Sciences Rensselaer Polytechnic Institute Troy, New York 12180

c 2000, Joseph E. Flaherty, all rights reserved. These notes are intended for classroom use by Rensselaer students taking courses CSCI, MATH 6860. Copying ordownloading by others for personal use is acceptable with noti cation of the author.

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CSCI, MATH 6860: Finite Element Analysis
Spring 2000

Outline

1. Introduction 1.1. Historical Perspective 1.2. Weighted Residual Methods 1.3. A Simple Finite Element Problem 2. One-Dimensional Finite Element Methods 2.1. 2.2. 2.3. 2.4. 2.5. 2.6. Introduction Galerkin's Method and Extremal PrinciplesEssential and Natural Boundary Conditions Piecewise Lagrange Approximation Hierarchical Bases Interpolation Errors

3. Multi-Dimensional Variational Principles 3.1. Galerkin's Method and Extremal Principles 3.2. Function Spaces and Approximation 3.3. Overview of the Finite Element Method 4. Finite Element Approximation 4.1. 4.2. 4.3. 4.4. 4.5. 4.6. Introduction Lagrange Bases on Triangles LagrangeBases on Rectangles Hierarchical Bases Three-dimensional Bases Interpolation Errors

5. Mesh Generation and Assembly 5.1. Introduction iii

5.2. 5.3. 5.4. 5.5. 5.6.

Mesh Generation Data Structures Coordinate Transformations Generation of Element Matrices and Their Assembly Assembly of Vector Systems

6. Numerical Integration 6.1. Introduction 6.2. One-Dimensional Gaussian Quadrature 6.3.Multi-Dimensional Gaussian Quadrature 7. Discretization Errors 7.1. Introduction 7.2. Convergence and Optimality 7.3. Perturbations 8. Adaptivity 8.1. Introduction 8.2. h-Re nement 8.3. p- and hp-Re nement 9. Parabolic Problems 9.1. 9.2. 9.3. 9.4. 9.5. Introduction Semi-Discrete Galerkin Methods: The Method of Lines Finite Element Methods in Time Convergence and Stability Convection-Di usion Systems10. Hyperbolic Problems 10.1. Introduction 10.2. Flow Problems and Upwind Weighting 10.3. Arti cial Di usion iv

10.4. Streamline Weighting 11. Linear Systems Solution 11.1. 11.2. 11.3. 11.4. 11.5. Introduction Banded Gaussian Elimination and Pro le Techniques Nested Dissection and Domain Decomposition Conjugate Gradient Methods Nonlinear Problems and Newton's Method

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viBibliography
1] A.K. Aziz, editor. The Mathematical Foundations of the Finite Element Method with Applications to Partial Di erential Equations, New York, 1972. Academic Press. 2] I. Babuska, J. Chandra, and J.E. Flaherty, editors. Adaptive Computational Methods for Partial Di erential Equations, Philadelphia, 1983. SIAM. 3] I. Babuska, O.C. Zienkiewicz, J. Gago, and E.R. de A. Oliveira, editors.Accuracy Estimates and Adaptive Re nements in Finite Element Computations. John Wiley and Sons, Chichester, 1986. 4] K.-J. Bathe. Finite Element Procedures. Prentice Hall, Englewood Cli s, 1995. 5] E.B. Becker, G.F. Carey, and J.T. Oden. Finite Elements: An Introduction, volume I. Prentice Hall, Englewood Cli s, 1981. 6] M.W. Bern, J.E. Flaherty, and M. Luskin, editors. Grid Generation and AdaptiveAlgorithms, volume 113 of The IMA Volumes in Mathematics and its Applications, New York, 1999. Springer. 7] C.A. Brebia. The Boundary Element Method for Engineers. Pentech Press, London, second edition, 1978. 8] S.C. Brenner and L.R. Scott. The Mathematical Theory of Finite Element Methods. Springer-Verlag, New York, 1994. 9] G.F. Carey. Computational Grids: Generation, Adaptation, and SolutionStrategies. Series in Computational and Physical Processes in Mechanics and Thermal science. Taylor and Francis, New York, 1997. 10] G.F. Carey and J.T. Oden. Finite Elements: A Second Course, volume II. Prentice Hall, Englewood Cli s, 1983. 11] G.F. Carey and J.T. Oden. Finite Elements: Computational Aspects, volume III. Prentice Hall, Englewood Cli s, 1984. vii

12] P.G. Ciarlet. The Finite...