Finite Element Analysis. With Numeric and Symbolic Matlab

Contents
Preface
About the Author
List of Examples
List of Matlab Scripts
List of Useful Tables
1. Overview
	1.1 Introduction
	1.2 Finite Element Data
	1.3 Matrix Notation∗ (an optional section)
	1.4 Matrix Partitions
	1.6 Finite Element Integrals
	1.7 Linear Spring Networks (Work–Energy)
	1.8 Layout of this Book
	1.9 Summary and Notation
	1.10 Review
2. Calculus Review
	2.1 Introduction
	2.2 Jacobian Matrix
		2.2.1 Linear triangle geometry
		2.2.2 Linear tetrahedron
	2.3 Inverse Jacobian
	2.4 Integration by Parts
	2.5 Evaluation of Integrals with Constant Jacobian
	2.6 Summary
	2.7 Review
3. Terminology from Differential Equations
	3.1 Introduction
	3.2 Differential Equation Jargon∗
	3.3 Boundary Conditions
	3.4 Adjoint Operator∗
	3.5 Model Second-Order Elliptic PDE
	3.6 Point Singularities∗
	3.7 Eigenproblems
	3.8 Summary
	3.9 Review
4. Parametric Interpolation
	4.1 Types of Interpolation
	4.2 Lagrange One-Dimensional Interpolation
	4.3 Natural Coordinates∗
	4.4 Hermite One-Dimensional Interpolation∗
	4.5 Lagrangian Quadrilateral Elements
	4.6 Lagrangian Triangular Elements
	4.7 Tetrahedral Elements in Unit Coordinates∗
	4.8 Serendipity Quadrilaterals∗
	4.9 Why Parametric Coordinates∗
	4.10 Most Common One-Dimensional Element Integrals
	4.11 Summary
5. Numerical Integration
	5.1 One-Dimensional Quadratures
	5.2 Two- and Three-Dimensional Quadratures
	5.3 Summary
	5.4 Review
6. Equivalent Integral Forms
	6.1 Introduction
	6.2 Galerkin’s Method
	6.3 Methods of Weighted Residuals
	6.4 Common Weighting Methods
		6.4.1 The least square method
		6.4.2 The method of moments
		6.4.3 The collocation methods
		6.4.4 Galerkin weights
	6.5 Euler’s Theorem*
	6.6 Summary
7. Matrix Procedures for Finite Elements
	7.1 Introduction
	7.2 Equation Numbers for Gather and Scatter
	7.3 Vector Subscripts
	7.4 Partitioning the System Equations
	7.5 Numerically Equivalent Process
	7.6 EBC by a Penalty Method*
	7.7 Multiple Point Constraints* (MPC)
	7.8 Equation Factorization*
	7.9 Skyline Sparse Storage*
	7.10 Locate Full Locations in a Sparse Matrix*
	7.11 Summary
	7.12 Review
8. Applications of One-Dimensional Lagrange Elements
	8.1 Introduction
	8.2 Constant Coefficient Matrices
	8.3 Variable Source Terms
	8.4 Mixed Boundary Conditions
	8.5 Automating Lagrange Element Solutions
	8.6 Numerically Integrated Elements
		8.6.1 Bars
		8.6.2 Shafts
		8.6.3 Hydrodynamic lubrication
		8.6.4 Variable coefficients
	8.7 Symbolic Solutions*
	8.8 Symmetry and Anti-Symmetry
	8.9 Patch Test∗
	8.10 Summary
	8.11 Review
9. Truss Analysis
	9.1 Planar Truss
	9.2 Space Truss
	9.3 Summary
	9.4 Review
10. Applications of One-Dimensional Hermite Elements
	10.1 Introduction
	10.2 General Case Fourth-Order Beam Equation
	10.3 Integral Form
	10.4 Element Arrays
	10.5 C1 Element Models
	10.6 Classic Beams
	10.7 Structural Symmetry
	10.8 Multiple Span Beams*
	10.9 Beam on Elastic Foundation*
	10.10 Summary
	10.11 Review
11. Frame Analysis
	11.1 Planar Frames
	11.2 Frame Member Reactions
	11.3 Space Frames*
	11.4 Numerically Integrated Frame Members*
	11.5 Summary
12. Scalar Fields and Thermal Analysis
	12.1 Introduction
	12.2 General Field Problem
	12.3 Common Flux Components
	12.4 Galerkin Integral Form
	12.5 Corresponding Element and Boundary Matrices
	12.6 Governing Matrix Form
	12.7 The Classic Linear Triangle Field Element
	12.8 Jacobian Matrix for Equal Space Dimensions
	12.9 Field Flux Vector at a Point
	12.10 Evaluation of Integrals with Constant Jacobian
	12.11 Symmetry and Anti-Symmetry
	12.12 Torsion of Non-Circular Shafts
	12.13 Planar Heat Transfer
	12.14 Axisymmetric Fields
	12.15 Three-Dimensional Fields
	12.16 Summary
	12.17 Review
13. Elasticity
	13.1 Introduction
	13.2 Linear Springs
	13.3 Mechanical Work
	13.4 Displacements and Mechanical Strains
	13.5 Strain Energy
	13.6 Material Properties
	13.7 Interpolating Displacement Vectors:
	13.8 Mechanical Work in Matrix Form
	13.9 The Strain–Displacement Matrix
	13.10 Stiffness Matrix
	13.11 Work Done by Initial Strains∗
	13.12 Matrix Equilibrium Equations
	13.13 Kinetic Energy and Dynamics*
	13.14 Plane–stress and Plane–strain
	13.15 Plane–Stress Q10 Application
	13.16 Axisymmetric Solid
	13.17 Discussion
	13.18 Symmetry and Cyclic-symmetry*
	13.19 Summary
	13.20 Review
14. Eigenanalysis
	14.1 Introduction
	14.2 Finite Element Eigenproblems
	14.3 Spring-mass Systems
	14.4 Vibrating String
	14.5 Torsional Vibrations
	14.6 Beam Vibrations
	14.7 Membrane Vibration
	14.8 Structural Buckling*
	14.9 Beam Frequency with an Axial Load*
	14.10 Plane–Frame Modes and Frequencies*
	14.11 Modes and Frequencies of Two-Dimensional Continua
	14.12 Principal Stresses*
	14.13 Mohr’s Circle for Principal Stresses*
	14.14 Summary
	14.15 Review
15. Transient and Dynamic Solutions
	15.1 Introduction to Transient Systems
	15.2 Generalized Trapezoidal Algorithms
	15.3 One-Dimensional Transient Responses
	15.4 Two-Dimensional Transient Responses
	15.5 Accuracy and Control
	15.6 Introduction to Dynamic Solutions
	15.7 Wilson Method*
	15.8 Summaries
		15.8.1 Transient summary
Index