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 |  موضوع: كتاب An Introduction to Nonlinear Finite Element Analysis السبت 19 مايو 2018, 11:08 am | |
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An Introduction to Nonlinear Finite Element Analysis
J. N. Reddy
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Contents
Preface xvn
1 Introduction
1.1 Mathematical Models
1.2 Numerical Simulations
1.3 The Finite Element Method
1.4 Nonlinear Analysis
1.4.1 Introduction
1.4.2 Classification of Nonlinearities
1.5 The Big Picture
References
2 The Finite Element Method: A Review 13
2.1 Introduction
2.2 One-Dimensional Problems
2.2.1 Governing Differential Equation
2.2.2 Finite Element Approximation
2.2.3 Derivation of the Weak Form
2.2.4 Interpolation Functions
2.2.5 Finite Element Model
2.3 Two-Dimensional Problems
2.3.1 Governing Differential Equation
2.3.2 Finite Element Approximation
2.3.3 Weak Formulation
2.3.4 Finite Element Model
2.3.5 Interpolation Functions
2.3.6 Assembly of Elements
2.4 Library of Two-Dimensional Finite Elements
2.4.1 Introduction
2.4.2 Triangular Elements
2.4.3 Rectangular Elements
2.5 Numerical Integration
2.5.1 Preliminary Comments
2.5.2 Coordinate Transformations
2.5.3 Integration Over a Master Rectangular Element
2.5.4 Integration Over a Master Triangular Element
2.6 Computer Implementation
2.6.1 General Comments
2.6.2 One-Dimensional Problems
2.6.3 Two-Dimensional Problems
2.7 Closure
Problems
References
3 Heat Transfer and Other Field Problems in One Dimension
3.1 Model Differential Equation
3.2 Weak Formulation
3.3 Finite Element Model
3.4 Solution Procedures
3.4.1 General Comments
3.4.2 Direct Iteration Procedure
3.4.3 Newton’s Iteration Procedure
3.5 Computer Implementation
3.5.1 Introduction
3.5.2 Preprocessor Unit
3.5.3 Processor Unit
3.6 Closing Remarks
Problems
References
4 Nonlinear Bending of Straight Beams
4.1 Introduction
4.2 Euler-Bernoulli Beams
4.2.1 Basic Assumptions
4.2.2 Displacement Field and Strains
4.2.3 Weak Forms
4.2.4 Finite Element Model
4.2.5 Iterative Solutions of Nonlinear Equations
4.2.6 Load Increments
4.2.7 Membrane Locking
4.2.8 Computer Implementation
4.3 Timoshenko Beams
4.3.1 Displacement Field and Strains
4.3.2 Weak Forms
4.3.3 General Finite Element Model
4.3.4 Shear and Membrane Locking
4.3.5 Tangent Stiffness Matrix
Problems 124
References 126
5 Heat Transfer and Other Fields Problems in Two Dimensions 127
5.1 Model Equation
5.2 Weak Form
5.3 Finite Element Model
5.4 Solution Procedures
5.4.1 Direct Iteration
5.4.2 Newton Raphson Iteration
5.5 Computer Implementation
5.5.1 Introduction
5.5.2 Numerical Integration
5.5.3 Element Calculations
Problems
References
6 Nonlinear Bending of Elastic Plates
6.1 Introduction
6.2 Classical Plate Theory
6.2.1 Assumptions of the Kinematics
6.2.2 Displacement Field and Strains
6.3 Variational Formulation of CPT
6.3.1 Virtual Work
6.3.2 Weak Forms
6.3.3 Equilibrium Equations
6.3.4 Boundary Conditions
6.3.5 Stress Resultant-Deflection Relations
6.4 Finite Element Models of CPT
6.4.1 General Formulation
6.4.2 Tangent Stiffness Coefficients
6.4.3 Some Plate Finite Elements
6.5 Computer Implementation Aspects and Numerical Results
of CPT Elements
6.5.1 Computer Implementation
6.5.2 Results of Linear Analysis
6.5.3 Results of Nonlinear Analysis
6.6 First-Order Shear Deformation Plate Theory
6.6.1 Introduction
6.6.2 Displacement Field
6.6.3 Weak Formulation
6.7 Finite Element Models of FSDT
6.7.1 Virtual Work Statements
6.7.2 Finite Element Model
6.7.3 Tangent Stiffness Coefficients
6.7.4 Shear and Membrane Locking
6.8 Computer Implementation and Numerical Results
of FSDT Elements
6.8.1 Computer Implementation
6.8.2 Results of Linear Analysis
6.8.3 Results of Nonlinear Analysis
6.9 Theory of Doubly-Curved Shells
6.9.1 Introduction
6.9.2 Geometric Description
6.9.3 Strain Displacement Relations
6.9.4 Stress Resultants
6.9.5 Equations of Motion
6.10 Finite Element Analysis of Shells
6.10.1 Weak Forms
6.10.2 Finite Element Model
6.10.3 Linear Results
6.10.4 Nonlinear Results
Problems
References
7 Flows of Viscous Incompressible Fluids
7.1 Introduction
7.2 Governing Equations
7.2.1 Introduction
7.2.2 Conservation of Mass
7.2.3 Conservation of Momenta
7.2.4 Conservation of Energy
7.2.5 Constitutive Equations
7.2.6 Boundary Conditions
7.3 Governing Equations in Terms of Primitive Variables
7.3.1 Vector Form
7.3.2 Cartesian Component Form
7.4 Velocity-Pressure Finite Element Model
7.4.1 Weak Form
7.4.2 Finite Element Model
7.5 Penalty Finite Element Models
7.5.1 Introduction
7.5.2 Penalty Function Method
7.5.3 Reduced Integration Penalty Model
7.5.4 Consistent Penalty Model
7.6 Computational Aspects
7.6.1 Properties of the Matrix Equations
7.6.2 Choice of Elements
7.6.3 Evaluation of Element Matrices in Penalty Models
7.6.4 Post-Computation of Stresses
7.7 Computer Implementation
7.7.1 Mixed Model
7.7.2 Penalty Model
7.8 Numerical Examples
7.8.1 Preliminary Comments
7.8.2 Fluid Squeezed Between Parallel Plates
7.8.3 Flow of a Viscous Lubricant in a Slider Bearing
7.8.4 Wall-Driven Cavity Flow
7.8.5 Backward-Facing Step
7.9 Least-Squares Finite Element Models
7.9.1 Introduction
7.9.2 Finite Element Model
7.9.3 Computational Aspects
7.9.4 Numerical Examples
Problems
References
8 Nonlinear Analysis of Time-Dependent Problems
8.1 Introduction
8.2 Time Approximations
8.2.1 Introduction
8.2.2 Parabolic Equations
8.2.3 Hyperbolic Equations
8.3 Stability and Accuracy
8.3.1 Preliminary Comments
8.3.2 Stability Criteria
8.4 Transient Analysis of Nonlinear Problems
8.4.1 Introduction
8.4.2 Heat Transfer
8.4.3 Flows of Viscous Incompressible Fluids
8.4.4 Plate Bending (FSDT)
8.5 Computer Implementation
8.6 Numerical Examples 307
8.6.1 Linear Problems 307
8.6.2 Nonlinear Problems 314
Problems 323
References 325
9 Finite Element Formulations of Solid Continua 327
9.1 Introduction 327
9.1.1 Background 327
9.1.2 Descriptions of Motion 328
9.2 Strain and Stress Measures 329
9.2.1 Deformation Gradient Tensor 329
9.2.2 Green and Almansi Strain Tensors 331
9.2.3 Polar Decomposition 334
9.2.4 Stress Tensors 334
9.2.5 Energetically-Conjugate Stresses and Strains 335
9.3 Strain and Stress Measures Between Configurations 337
9.3.1 Notation 337
9.3.2 Conservation of Mass 339
9.3.3 Green Strain Tensors for Various Configurations 340
9.3.4 Euler Strain Tensor 342
9.3.5 Relationships Between Various Stress Tensors 343
9.4 Constitutive Equations 346
9.5 Total Lagrangian and Updated Lagrangian Formulations
of Continua 347
9.5.1 Principle of Virtual Displacements 347
9.5.2 Total Lagrangian Formulation 348
9.5.3 Updated Lagrangian Formulation 350
9.6 Finite Element Models of Two-Dimensional Continua 353
9.6.1 Introduction 353
9.6.2 Total Lagrangian Formulation 353
9.6.3 Updated Lagrangian Formulation 360
9.6.4 Computer Implementation 361
9.6.5 Numerical Results 361
9.7 Shell Finite Elements 369
9.7.1 Introduction 369
9.7.2 Incremental Equations of Motion 369
9.7.3 Finite Element Models of a Continuum 370
9.7.4 Shell Finite Element 372
9.7.5 Numerical Examples 378
Problems 381
References 387XV
10 Material Nonlinearities and Coupled Problems
10.1 Introduction
10.2 Nonlinear Elastic Problems
10.3 Small Deformation Theory of Plasticity
10.3.1 Introduction
10.3.2 Ideal Plasticity
10.3.3 Strain Hardening Plasticity
10.3.4 Elastic-Plastic Analysis of a Bar
10.4 Non-Newtonian Fluids
10.4.1 Introduction
10.4.2 Governing Equations in Cylindrical Coordinates
10.4.3 Power-Law Fluids
10.4.4 White-Metzner Fluids
10.4.5 Numerical Examples
10.5 Coupled Fluid Flow and Heat Transfer
10.5.1 Finite Element Models
10.5.2 Numerical Examples
References
Appendix Al: Solution Procedures for Linear Algebraic
Equations 425
Al.l Introduction 425
A1.2 Direct Methods
Al.2.1 Preliminary Comments
Al.2.2 Symmetric Solver
A1.2.3 Unsymmetric Solver
A1.3 Iterative Methods
Al.3.1 General Comments
Al.3.2 Solution Algorithms
References
Appendix A2: Solution Procedures for Nonlinear Algebraic
Equations 439
A2.1 Introduction 439
A2.2 Picard Iteration Method
A2.3 Newton-Raphson Iteration Method
A2.4 Riks and Modified Riks Schemes
References
Subject Index
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