Jun 28, 2017

EBOOK - A First Course in Finite Elements (Jacob Fish & Ted Belytschko)

This book is written to be an undergraduate and introductory graduate level textbook, depending on whether the more advanced topics appearing at the end of each chapter are covered. Without the advanced topics, the book is of a level readily comprehensible by junior and senior undergraduate students in science and engineering. With the advanced topics included, the book can serve as the textbook for the firstcourse in finite elements at the graduate level. The text material evolved from over 50 years of combined teaching experience by the authors of graduate and undergraduate finite element courses.

The book focuses on the formulation and application of the finite element method. It differs from other elementary finite element textbooks in the following three aspects:
1. Itis introductory and self-contained.Only a modestbackground in mathematics and physics is needed, allof which is covered in engineering and science curricula in the firsttwo years. urthermore, many of the specific topics in mathematics, such as matrix algebra, some topics in differential equations, and mechanics and physics, such as conservation laws and constitutive equations, are reviewed prior to their application.
2. It is generic. While most introductory finite element textbooks are application specific, e.g. focusing on linear elasticity, the finite element method in this book is formulated as a general purpose numerical procedure for solving engineering problems governed by partial differential equations. The methodology for obtaining weak forms for the governing equations, a crucial step in the development and understanding of finite elements, is carefully developed. Consequently, students from various engineering and science disciplines will benefit equally from the exposition of the subject.
3. Itis a hands-on experience.The book integrates finite elementtheory, finite elementcode development and the application of commercial software package. Finite element code development is introduced through MATLAB exercises and a MATLAB program, whereas ABAQUS is used for demonstrating the use of commercial finite element software.

1 Introduction 1
1.1 Background 1
1.2 Applications of Finite elements 7
References 9
2 Direct Approach for Discrete Systems 11
2.1 Describing the Behavior of a Single Bar Element 11
2.2 Equations for a System 15
2.2.1 Equations for Assembly 18
2.2.2 Boundary Conditions and System Solution 20
2.3 Applications to Other Linear Systems 24
2.4 Two-Dimensional Truss Systems 27
2.5 Transformation Law 30
2.6 Three-Dimensional Truss Systems 35
References 36
Problems 37
3 Strong and Weak Forms for One-Dimensional Problems 41
3.1 The Strong Form in One-Dimensional Problems 42
3.1.1 The Strong Form for an Axially Loaded Elastic Bar 42
3.1.2 The Strong Form for Heat Conduction in One Dimension 44
3.1.3 Diffusion in One Dimension 46
3.2 The Weak Form in One Dimension 47
3.3 Continuity 50
3.4 The Equivalence Between the Weak and Strong Forms 51
3.5 One-Dimensional Stress Analysis with Arbitrary Boundary Conditions 58
3.5.1 Strong Form for One-Dimensional Stress Analysis 58
3.5.2 Weak Form for One-Dimensional Stress Analysis 59
3.6 One-Dimensional Heat Conduction with Arbitrary Boundary Conditions 60
3.6.1 Strong Form for Heat Conduction in One Dimension with Arbitrary Boundary Conditions 60
3.6.2 Weak Form for Heat Conduction in One Dimension with Arbitrary Boundary Conditions 61
3.7 Two-Point Boundary Value Problem with Generalized Boundary Conditions 62
3.7.1 Strong Form for Two-Point Boundary Value Problems with Generalized Boundary Conditions
3.7.2 Weak Form for Two-Point Boundary Value Problems
with Generalized Boundary Conditions 63
3.8 Advection–Diffusion 64
3.8.1 Strong Form of Advection–Diffusion Equation 65
3.8.2 Weak Form of Advection–Diffusion Equation 66
3.9 Minimum Potential Energy 67
3.10 Integrability 71
References 72
Problems 72
4 Approximation of Trial Solutions, Weight Functions
and Gauss Quadrature for One-Dimensional Problems 77
4.1 Two-Node Linear Element 79
4.2 Quadratic One-Dimensional Element 81
4.3 Direct Construction of Shape Functions in One Dimension 82
4.4 Approximation of the Weight Functions 84
4.5 Global Approximation and Continuity 84
4.6 Gauss Quadrature 85
Reference 90
Problems 90
5 Finite Element Formulation for One-Dimensional Problems 93
5.1 Development of Discrete Equation: Simple Case 93
5.2 Element Matrices for Two-Node Element 97
5.3 Application to Heat Conduction and Diffusion Problems 99
5.4 Development of Discrete Equations for Arbitrary Boundary Conditions 105
5.5 Two-Point Boundary Value Problem with Generalized Boundary Conditions 111
5.6 Convergence of the FEM 113
5.6.1 Convergence by Numerical Experiments 115
5.6.2 Convergence by Analysis 118
5.7 FEM for Advection–Diffusion Equation 120
References 122
Problems 123



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