Fish J., Belytschko T. A First Course in Finite Elements

John Wiley & Sons Ltd, 2007. 336 p. ISBN:0470035803Developed from the authors, combined total of 50 years undergraduate and graduate teaching experience, this book presents the finite element method formulated as a general-purpose numerical procedure for solving engineering problems governed by partial differential equations.
Focusing on the formulation and application of the finite element method through the integration of finite element theory, code development, and software application, the book is both introductory and self-contained, as well as being a hands-on experience for any student.Background
Applications of Finite elementsDirect Approach for Discrete Systems
Describing the Behavior of a Single Bar Element
Equations for a System
Applications to Other Linear Systems
Two-Dimensional Truss Systems
Transformation Law
Three-Dimensional Truss SystemsStrong andWeak Forms for One-Dimensional Problems
The Strong Form in One-Dimensional Problems
TheWeak Form in One Dimension
Continuity
The Equivalence Between theWeak and Strong Forms
One-Dimensional Stress Analysis with Arbitrary Boundary Conditions
One-Dimensional Heat Conduction with Arbitrary Boundary Conditions
Two-Point Boundary Value Problem with Generalized Boundary Conditions
Advection–Diffusion
Minimum Potential Energy
IntegrabilityApproximation of Trial Solutions,Weight Functions and Gauss Quadrature for One-Dimensional Problems
Two-Node Linear Element
Quadratic One-Dimensional Element
Direct Construction of Shape Functions in One Dimension
Approximation of theWeight Functions
Global Approximation and Continuity
Gauss QuadratureFinite Element Formulation for One-Dimensional Problems
Development of Discrete Equation: Simple Case
Element Matrices for Two-Node Element
Application to Heat Conduction and Diffusion Problems
Development of Discrete Equations for Arbitrary Boundary Conditions
Two-Point Boundary Value Problem with Generalized Boundary Conditions
Convergence of the FEM
FEM for Advection–Diffusion EquationStrong andWeak Forms for Multidimensional Scalar Field Problems
Divergence Theorem and Green’s Formula
Strong Form
Weak Form
The Equivalence BetweenWeak and Strong Forms
Generalization to Three-Dimensional Problems
Strong andWeak Forms of Scalar Steady-State Advection–Diffusion in Two DimensionsApproximations of Trial Solutions,Weight Functions and Gauss Quadrature for Multidimensional Problems
Completeness and Continuity
Three-Node Triangular Element
Four-Node Rectangular Elements
Four-Node Quadrilateral Element
Higher Order Quadrilateral Elements
Triangular Coordinates
Completeness of Isoparametric Elements
Gauss Quadrature in Two Dimensions
Integration Over Quadrilateral Elements
Integration Over Triangular Elements
Three-Dimensional ElementsFinite Element Formulation for Multidimensional Scalar Field Problems
Finite Element Formulation for Two-Dimensional Heat Conduction Problems
Verification and Validation
Advection–Diffusion EquationFinite Element Formulation for Vector Field Problems – Linear Elasticity
Linear Elasticity
Strong andWeak Forms
Finite Element Discretization
Three-Node Triangular Element
Generalization of Boundary Conditions
Discussion
Linear Elasticity Equations in Three DimensionsFinite Element Formulation for Beams
Governing Equations of the Beam
Strong Form toWeak Form
Finite Element Discretization
Theorem of Minimum Potential Energy
Remarks on Shell ElementsCommercial Finite Element Program ABAQUS TutorialsPreliminaries
Creating a Part
Creating a Material Definition
Defining and Assigning Section Properties
Assembling the Model
Configuring the Analysis
Applying a Boundary Condition and a Load to the Model
Meshing the Model
Creating and Submitting an Analysis Job
Viewing the Analysis Results
Solving the Problem Using Quadrilaterals
Refining the Mesh
Copying the Model
Modifying the Material Definition
Configuring the Analysis
Applying a Boundary Condition and a Load to the Model
Meshing the Model
Creating and Submitting an Analysis Job
Viewing the Analysis Results
Creating a New Model
Creating a Part
Creating a Material Definition
Defining and Assigning Section Properties
Assembling the Model
Configuring the Analysis
Applying a Boundary Condition and a Load to the Model
Meshing the Model
Creating and Submitting an Analysis Job
Viewing the Analysis Results
Refining the MeshAppendix
Rotation of Coordinate System in Three Dimensions
Scalar Product Theorem
Taylor’s Formula with Remainder and the Mean Value Theorem
Green’s Theorem
Point Force (Source)
Static Condensation
Solution Methods
Direct Solvers
Iterative Solvers
Conditioning