Koutromanos I. Fundamentals of Finite Element Analysis. Linear Finite Element Analysis

Hoboken: John Wiley & Sons, 2018. – 710 p.Physical Processes and Mathematical Models
Approximation, Error, and Convergence
Approximate Solution of Differential Equations and the Finite Element Method
Brief History of the Finite Element Method
Finite Element Software
Significance of Finite Element Analysis for Engineering
Typical Process for Obtaining a Finite Element Solution for a Physical Problem
A Note on Linearity and the Principle of SuperpositionStrong and Weak Form for One-Dimensional Problems
Strong Form for One-Dimensional Elasticity Problems
General Expressions for Essential and Natural B.C. in One-Dimensional Elasticity Problems
Weak Form for One-Dimensional Elasticity Problems
Equivalence of Weak Form and Strong Form
Strong Form for One-Dimensional Heat Conduction
Weak Form for One-Dimensional Heat ConductionFinite Element Formulation for One-Dimensional Problems
Introduction—Piecewise Approximation
Shape (Interpolation) Functions and Finite Elements
Discrete Equations for Piecewise Finite Element Approximation
Finite Element Equations for Heat Conduction
Accounting for Nodes with Prescribed Solution Value (“Fixed” Nodes)
Examples on One-Dimensional Finite Element Analysis
Numerical Integration—Gauss Quadrature
Convergence of One-Dimensional Finite Element Method
Effect of Concentrated Forces in One-Dimensional Finite Element AnalysisMultidimensional Problems: Mathematical Preliminaries
Basic Definitions
Green’s Theorem—Divergence Theorem and Green’s Formula
Procedure for Multidimensional ProblemsTwo-Dimensional Heat Conduction and Other Scalar Field Problems
Strong Form for Two-Dimensional Heat Conduction
Weak Form for Two-Dimensional Heat Conduction
Equivalence of Strong Form and Weak Form
Other Scalar Field Problems
Two-Dimensional Potential Fluid Flow
Fluid Flow in Porous Media
Chemical (Molecular) Diffusion-ReactionFinite Element Formulation for Two-Dimensional Scalar Field Problems
Finite Element Discretization and Piecewise Approximation
Three-Node Triangular Finite Element
Four-Node Rectangular Element
Isoparametric Finite Elements and the Four-Node Quadrilateral (4Q) Element
Numerical Integration for Isoparametric Quadrilateral Elements
Higher-Order Isoparametric Quadrilateral Elements
Isoparametric Triangular Elements
Continuity and Completeness of Isoparametric Elements
Concluding Remarks: Finite Element Analysis for Other Scalar Field ProblemsMultidimensional Elasticity
Definition of Strain Tensor
Definition of Stress Tensor
Representing Stress and Strain as Column Vectors—The Voigt Notation
Constitutive Law (Stress-Strain Relation) for Multidimensional Linear Elasticity
Coordinate Transformation Rules for Stress, Strain, and Material Stiffness Matrix
Stress, Strain, and Constitutive Models for Two-Dimensional (Planar) Elasticity
Strong Form for Two-Dimensional Elasticity
Weak Form for Two-Dimensional Elasticity
Equivalence between the Strong Form and the Weak Form
Strong Form for Three-Dimensional Elasticity
Using Polar (Cylindrical) CoordinatesFinite Element Formulation for Two-Dimensional Elasticity
Piecewise Finite Element Approximation—Assembly Equations
Accounting for Restrained (Fixed) Displacements
Postprocessing
Continuity—Completeness Requirements
Finite Elements for Two-Dimensional Elasticity
Three-Node Triangular Element (Constant Strain Triangle)
Quadrilateral Isoparametric Element
Example: Calculation of Stiffness Matrix and Equivalent Nodal Forces for Four-Node Quadrilateral Isoparametric ElementFinite Element Formulation for Three-Dimensional Elasticity
Weak Form for Three-Dimensional Elasticity
Piecewise Finite Element Approximation—Assembly Equations
Isoparametric Finite Elements for Three-Dimensional Elasticity
Eight-Node Hexahedral Element
Numerical (Gaussian) Quadrature for Hexahedral Isoparametric Elements
Calculation of Boundary Integral Contributions to Nodal Forces
Higher-Order Hexahedral Isoparametric Elements
Tetrahedral Isoparametric Elements
Three-Dimensional Elements from Collapsed (Degenerated) Hexahedral Elements
Concluding Remark: Continuity and Completeness Ensured by Three-Dimensional Isoparametric Elements and Use for Other ProblemsTopics in Applied Finite Element Analysis
Concentrated Loads in Multidimensional Analysis
Effect of Autogenous (Self-Induced) Strains—The Special Case of Thermal Strains
The Patch Test for Verification of Finite Element Analysis Software
Subparametric and Superparametric Elements
Field-Dependent Natural Boundary Conditions: Emission Conditions and Compliant Supports
Treatment of Nodal Constraints
Treatment of Compliant (Spring) Connections Between Nodal Points
Symmetry in Analysis
Axisymmetric Problems and Finite Element Analysis
A Brief Discussion on Efficient Mesh RefinementConvergence of Multidimensional Finite Element Analysis, Locking Phenomena in Multidimensional Solids and Reduced Integration
Convergence of Multidimensional Finite Elements
Effect of Element Shape in Multidimensional Analysis
Incompatible Modes for Quadrilateral Finite Elements
Volumetric Locking in Continuum Elements
Uniform Reduced Integration and Spurious Zero-Energy (Hourglass) Modes
Resolving the Problem of Hourglass Modes: Hourglass Stiffness
Selective-Reduced Integration
The B-bar Method for Resolving LockingMultifield (Mixed) Finite Elements
Multifield Weak Forms for Elasticity
Mixed (Multifield) Finite Element Formulations
Two-Field (Stress-Displacement) Formulations and the Pian-Sumihara Quadrilateral Element
Displacement-Pressure (u-p) Formulations and Finite Element Approximations
Stability of Mixed u-p Formulations—the inf-sup Condition
Assumed (Enhanced)-Strain Methods and the B-bar Method as a Special Case
A Concluding Remark for Multifield ElementsFinite Element Analysis of Beams
Basic Definitions for Beams
Differential Equations and Boundary Conditions for 2D Beams
Euler-Bernoulli Beam Theory
Strong Form for Two-Dimensional Euler-Bernoulli Beams
Weak Form for Two-Dimensional Euler-Bernoulli Beams
Finite Element Formulation: Two-Node Euler-Bernoulli Beam Element
Coordinate Transformation Rules for Two-Dimensional Beam Elements
Timoshenko Beam Theory
Strong Form for Two-Dimensional Timoshenko Beam Theory
Weak Form for Two-Dimensional Timoshenko Beam Theory
Two-Node Timoshenko Beam Finite Element
Continuum-Based Beam Elements
Extension of Continuum-Based Beam Elements to General Curved Beams
Shear Locking and Selective-Reduced Integration for Thin Timoshenko Beam ElementsFinite Element Analysis of Shells
Stress Resultants for Shells
Differential Equations of Equilibrium and Boundary Conditions for Flat Shells
Constitutive Law for Linear Elasticity in Terms of Stress Resultants and Generalized Strains
Weak Form of Shell Equations
Finite Element Formulation for Shell Structures
Four-Node Planar (Flat) Shell Finite Element
Coordinate Transformations for Shell Elements
“Clever” Way to Approximately Satisfy C1 Continuity Requirements for Thin Shells—The Discrete Kirchhoff Formulation
Continuum-Based Formulation for Nonplanar (Curved) ShellsFinite Elements for Elastodynamics, Structural Dynamics, and Time-Dependent Scalar Field Problems
Strong Form for One-Dimensional Elastodynamics
Strong Form in the Presence of Material Damping
Weak Form for One-Dimensional Elastodynamics
Finite Element Approximation and Semi-Discrete Equations of Motion
Three-Dimensional Elastodynamics
Semi-Discrete Equations of Motion for Three-Dimensional Elastodynamics
Structural Dynamics Problems
Dynamic Beam Problems
Dynamic Shell Problems
Diagonal (Lumped) Mass Matrices and Mass Lumping Techniques
Mass Lumping for Continuum (Solid) Elements
Mass Lumping for Structural Elements (Beams and Shells)
Strong and Weak Form for Time-Dependent Scalar Field (Parabolic) Problems
Time-Dependent Heat Conduction
Time-Dependent Fluid Flow in Porous Media
Time-Dependent Chemical Diffusion
Semi-Discrete Finite Element Equations for Scalar Field (Parabolic) Problems
Solid and Structural Dynamics as a “Parabolic” Problem: The State-Space FormulationAnalysis of Time-Dependent Scalar Field (Parabolic) Problems
Single-Step Algorithms
Linear Multistep Algorithms
Adams-Bashforth Methods
Adams-Moulton Methods
Predictor-Corrector Algorithms—Runge-Kutta Methods
Convergence of a Time-Stepping Algorithm
Stability of Time-Stepping Algorithms
Error, Order of Accuracy, Consistency, and Convergence
Modal Analysis and Its Use for Determining the Stability for Systems with Many Degrees of FreedomSolution Procedures for Elastodynamics and Structural DynamicsModal Analysis: What Will NOT Be Presented in Detail
Proportional Damping Matrices—Rayleigh Damping Matrix
Step-by-Step Algorithms for Direct Integration of Equations of Motion
Explicit Central Difference Method
Newmark Method
Hilber-Hughes Taylor (HHT or Alpha) Method
Stability and Accuracy of Transient Solution Algorithms
Application of Step-by-Step Algorithms for Discrete Systems with More than One Degrees of FreedomVerification and Validation for the Finite Element MethodCode Verification
Order of Accuracy Testing
Systematic Mesh Refinement
Exact Solutions
Solution Verification
Iterative Error
Discretization Error
Numerical Uncertainty
Sources and Types of Uncertainty
Validation Experiments
Validation Metrics
Extrapolation of Model Prediction Uncertainty
Predictive CapabilityNumerical Solution of Linear Systems of EquationsDirect Methods
Gaussian Elimination
The LU Decomposition
Iterative Methods
The Jacobi Method
The Conjugate Gradient Method
Parallel Computing and the Finite Element Method
Efficiency of Parallel Algorithms
Parallel Architectures
Parallel Conjugate Gradient MethodAppendix
Concise Review of Vector and Matrix Algebra
Review of Matrix Analysis for Discrete Systems
Minimum Potential Energy for Elasticity—Variational Principles
Calculation of Displacement and Force Transformations for Rigid-Body Connections