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Finite Elements and Approximation

ISBN-10: 0486453014
ISBN-13: 9780486453019
Edition: 2006
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Description: A powerful tool for the approximate solution of differential equations, the finite element is extensively used in industry and research. This book offers students of engineering and physics a comprehensive view of the principles involved, with  More...

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Book details

List price: $19.95
Copyright year: 2006
Publisher: Dover Publications, Incorporated
Publication date: 10/6/2006
Binding: Paperback
Pages: 350
Size: 5.25" wide x 8.25" long x 0.75" tall
Weight: 0.792
Language: English

A powerful tool for the approximate solution of differential equations, the finite element is extensively used in industry and research. This book offers students of engineering and physics a comprehensive view of the principles involved, with numerous illustrative examples and exercises. 1983 edition.

Continuum Boundary Value Problems and the Need for Numerical Discretization. Finite Difference Methods
Introduction
Some Examples of Continuum Problems
Finite Differences in One Dimension
Derivative Boundary Conditions
Nonlinear Problems
Finite Differences in More Than One Dimension
Problems Involving Irregularly Shaped Regions
Nonlinear Problems in More Than One Dimension
Approximation and Convergence
Concluding Remarks
References
Suggested Further Reading
Weighted Residual Methods: Use of Continuous Trial Functions
Introduction-Approximation by Trial Functions
Weighted Residual Approximations
Approximation to the Solutions of Differential Equations and the Use of Trial Function-Weighted Residual Forms. Boundary Conditions Satisfied by Choice of Trial Functions
Simultaneous Approximation to the Solutions of Differential Equations and to the Boundary Conditions
Natural Boundary Conditions
Boundary Solution Methods
Systems of Differential Equations
Nonlinear Problems
Concluding Remarks
References
Suggested Further Reading
Piecewise Defined Trial Functions and the Finite Element Method
Introduction-The Finite Element Concept
Some Typical Locally Defined Narrow-Base Shape Functions
Approximation to Solutions of Differential Equations and Continuity Requirements
Weak Formulation and the Galerkin Method
Some One-Dimensional Problems
Standard Discrete System. A Physical Analogue of the Equation Assembly Process
Generalization of the Finite Element Concepts for Two- and Three-Dimensional Problems
The Finite Element Method for Two-Dimensional Heat Conduction Problems
Two-Dimensional Elastic Stress Analysis Using Triangular Elements
Are Finite Differences a Special Case of the Finite Element Method?
Concluding Remarks
References
Suggested Further Reading
Higher Order Finite Element Approximation
Introduction
Degree of Polynomial in Trial Functions and Convergence Rates
The Patch Test
Standard Higher Order Shape Functions for One-Dimensional Elements with C[superscript 0] Continuity
Hierarchical Forms of Higher Order One-Dimensional Elements with C[superscript 0] Continuity
Two-Dimensional Rectangular Finite Element Shape Functions of Higher Order
Two-Dimensional Shape Functions for Triangles
Three-Dimensional Shape Functions
Concluding Remarks
References
Suggested Further Reading
Mapping and Numerical Integration
The Concept of Mapping
Numerical Integration
More on Mapping
Mesh Generation and Concluding Remarks
References
Suggested Further Reading
Variational Methods
Introduction
Variational Principles
The Establishment of Natural Variational Principles
Approximate Solution of Differential Equations by the Rayleigh-Ritz Method
The Use of Lagrange Multipliers
General Variational Principles
Penalty Functions
Least-Squares Method
Concluding Remarks
References
Suggested Further Reading
Partial Discretization and Time-Dependent Problems
Introduction
Partial Discretization Applied to Boundary Value Problems
Time-Dependent Problems Via Partial Discretization
Analytical Solution Procedures
Finite Element Solution Procedures in the Time Domain
References
Suggested Further Reading
Generalized Finite Elements, Error Estimates, and Concluding Remarks
The Generalized Finite Element Method
The Discretization Error in a Numerical Solution
A Measure of Discretization Error
Estimate of Discretization Error
The State of the Art
References
Suggested Further Reading
Index

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