Finite Element Procedures

Preface 	 xiii 
CHAPTER ONE — 
An Introduction to the Use of Finite Element Procedures  1 

1.1  	Introduction  1 
1.2  	Physical Problems, Mathematical Models, and the Finite Element Solution  2 
1.3  	Finite Element Analysis as an Integral Part of Computer-Aided Design  11 
1.4  	A Proposal on How to Study Finite Element Methods  14 

CHAPTER TWO
Vectors, Matrices, and Tensors  17
2.1  	Introduction  17 
22  	Introduction to Matrices  18 
23  	Vector Spaces  34 
24  	Definition of Tensors  40  ⋅ 
2.5  	The Symmetric Eigenproblem Av = Av  51 
2.6  	The Rayleigh Quotient and the Minimax Characterization 
	of Eigenvalues  60 
2.7  	Vector and Matrix Norms  66 
2.8  	Exercises  72 

CHAPTER TRHEE
Some Basic Concepts of Engineering Analysis and an Introduction  77

to the Finite Element Method  77
3.1 	Introduction  77 
3.2 	Solution of Discrete-System Mathematical Models 78 
	3.2.1  Steady-State Problems, 78 
	3.2.2  Propagation Problems, 87 
	3.2.3  FEigenvalue Problems, 90 
	3.2.4  On the Nature of Solutions, 96 
	3.2.5  Exercises, 101 
33 	Solution of Continuous-System Mathematical Models  105 
	3.3.1  Differential Formulation, 105 
	3.3.2  Variational Formulations, 110 
	3.3.3  Weighted Residual Methods; Ritz Method, 116 
	3.3.4  An Overview: The Differential and Galerkin Formulations, the Principle of 
	Virtual Displacements, and an Introduction to the Finite Element Solution, 124 
	3.3.5  Finite Difference Differential and Energy Methods, 129 
	3.3.6  Exercises, 138 
3.4 	Imposition of Constraints  143 
	3.4.1  An Introduction to Lagrange Multiplier and Penalty Methods, 143 
	3.4.2  Exercises, 146 

CHAPTER FOUR 	
Formulation of the Finite Element Method—Linear Analysis in Solid and Structural Mechanics 	148 

4.1 	Introduction  148 
4.2 	Formulation of the Displacement-Based Finite Element Method  149 
	4.2.1  General Derivation of Finite Element Equilibrium Equations, 153 
	4.2.2  Imposition of Displacement Boundary Conditions, 187 
	4.2.3  Generalized Coordinate Models for Specific Problems, 193 
	4.2.4  Lumping of Structure Properties and Loads, 212 
	4.2.5  Exercises, 214 
4.3 	Convergence of Analysis Results  225 
	4.3.1  The Model Problem and a Definition of Convergence, 225 
	4.3.2  Criteria for Monotonic Convergence, 229 
	4.3.3  The Monotonically Convergent Finite Element Solution: A Ritz Solution, 234 
	4.3.4  Properties of the Finite Element Solution, 236 
	4.3.5  Rate of Convergence, 244 
	4.3.6  Calculation of Stresses and the Assessment of Error, 254 
	4.3.7  Exercises, 259 
4.4 	Incompatible and Mixed Finite Element Models  261 
	4.4.1  Incompatible Displacement-Based Models, 262 
	4.4.2  Mixed Formulations, 268 
	4.4.3  Mixed Interpolation— Displacement/Pressure Formulations for 
	Incompressible Analysis, 276 
	4.4.4  Exercises, 296 
4.5 	The Inf-Sup Condition for Analysis of Incompressible Media and Structural 
	Problems  300 
	4.5.1  The Inf-Sup Condition Derived from Convergence Considerations, 301 
	4.5.2  The Inf-Sup Condition Derived from the Matrix Equations, 312 
	4.5.3  The Constant (Physical) Pressure Mode, 315 
	4.5.4  Spurious Pressure Modes—The Case of Total Incompressibility, 316 
	4.5.5  Spurious Pressure Modes—The Case of Near Incompressibility, 318 
	4.5.6  The Inf-Sup Test, 322 
	4.5.7  An Application to Structural Elements: The Isoparametric Beam Elements, 330 
	4.5.8  Exercises, 335 

CHAPTER FIVE
Formulation-and Calculation of Isoparametric Finite Element Matrices 	338 

5.1 	Introduction  338 
5.2 	Isoparametric Derivation of Bar Element Stiffness Matrix  339 
53 	Formulation of Continuum Elements  341 
	5.3.1  Quadrilateral Elements, 342 
	5.3.2  Triangular Elements, 363 
	5.3.3  Convergence Considerations, 376 
	5.3.4  Element Matrices in Global Coordinate System, 386 
	5.3.5  Displacement/Pressure Based Elements for Incompressible Media, 388 
	5.3.6  Exercises, 389 
5.4 	Formulation of Structural Elements  397 
	5.4.1  Beam and Axisymmetric Shell Elements, 399 
	5.4.2  Plate and General Shell Elements, 420 
	5.4.3  Exercises, 450 
55 	Numerical Integration  455 
	5.5.1  Interpolation Using a Polynomial, 456 
	5.5.2  The Newton-Cotes Formulas (One-Dimensional Integration), 457 
	5.5.3  The Gauss Formulas (One-Dimensional Integration), 461 
	5.5.4  Integrations in Two and Three Dimensions, 464 
	5.5.5  Appropriate Order of Numerical Integration, 465 
	5.5.6  Reduced and Selective Integration, 476 
	5.5.7  Exercises, 478 
5.6 	Computer Program Implementation of Isoparametric Finite Elements 480 

CHAPTER SIX
Finite Element Nonlinear Analysis in Solid and Structural Mechanics 	485 

6.1 	Introduction to Nonlinear Analysis  485 
6.2 	Formulation of the Continuum Mechanics Incremental Equations of 
	Motion 497 
	6.2.1  The Basic Problem, 498 
	6.2.2  The Deformation Gradient, Strain, and Stress Tensors, 502 
	6.2.3  Continuum Mechanics Incremental Total and Updated Lagrangian 
	Formulations, Materially- Nonlinear-Only Analysis, 522 
	6.2.4  Exercises, 529 
6.3 	Displacement-Based Isoparametric Continuum Finite Elements  538 
	6.3.1  Linearization of the Principle of Virtual Work with Respect to Finite Element 
	Variables, 538 
	6.3.2  General Matrix Equations of Displacement-Based Continuum Elements, 540 
	6.3.3  Truss and Cable Elements, 543 
	6.3.4  Two-Dimensional Axisymmetric, Plane Strain, and Plane Stress Elements, 549 
	6.3.5  Three-Dimensional Solid Elements, 555 
	6.3.6  Exercises, 557 
6.4 	Displacement/Pressure Formulations for Large Deformations 561 
	6.4.1  Total Lagrangian Formulation, 561 
	6.4.2  Updated Lagrangian Formulation, 565 
	6.4.3  Exercises, 566 
6.5 	Structural Elements  568 
	6.5.1  Beam and Axisymmetric Shell Elements, 568 
	6.5.2  Plate and General Shell Elements, 575 
	6.5.3  Exercises, 578 
6.6 	Use of Constitutive Relations  581 
	6.6.1  Elastic Material Behavior—Generalization of Hooke’s Law, 583 
	6.6.2  Rubberlike Material Behavior, 592 
	6.6.3  Inelastic Material Behavior; Elastoplasticity, Creep, and Viscoplasticity, 595 
	6.6.4  Large Strain Elastoplasticity, 612 
	6.6.5  Exercises, 617 
6.7 	Contact Conditions  622 
	6.7.1  Continuum Mechanics Equations, 622 
	6.7.2  A Solution Approach for Contact Problems: The Constraint Function Method, 626 
	6.7.3  Exercises, 628 
6.8 	Some Practical Considerations  628 
	6.8.1  The General Approach to Nonlinear Analysis, 629 
	6.8.2  Collapse and Buckling Analyses, 630 
	6.8.3  The Effects of Element Distortions, 636 
	6.8.4  The Effects of Order of Numerical Integration, 637 
	6.8.5  Exercises, 640 

CHAPTER SEVEN 	
Finite Element Analysis of Heat Transfer, Field Problems, and Incompressible Fluid Flows 	642 

7.1  	Introduction  642 
7.2 	Heat Transfer Analysis  642 
	7.2.1  Governing Heat Transfer Equations, 642 
	7.2.2  Incremental Equations, 646 
	7.2.3  Finite Element Discretization of Heat Transfer Equations, 651 
	7.2.4  Exercises, 659 
7.3 	Analysis of Field Problems  661 
	7.3.1  Seepage, 662 
	7.3.2  Incompressible Inviscid Flow, 663 
	7.3.3  Torsion, 664 
	7.3.4  Acoustic Fluid, 666 
	7.3.5  Exercises, 670 
1.4 	Analysis of Viscous Incompressible Fluid Flows  671 
	7.4.1  Continuum Mechanics Equations, 675 
	7.4.2  Finite Element Governing Equations, 677 
	7.4.3  High Reynolds and High Peclet Number Flows, 682 
	7.4.4  Exercises, 691 

CHAPTER EIGHT 
Solution of Equilibrium Equations in Static Analysis 	695 

8.1 	Introduction  695 
8.2 	Direct Solutions Using Algorithms Based on Gauss Elimination  696 
	8.2.1  Introduction to Gauss Elimination, 697 
	8.2.2  The LDLT Solution, 705 
	8.2.3  Computer Implementation of Gauss Elimination—The Active Column Solution, 708 
	8.2.4  Cholesky Factorization, Static Condensation, Substructures, and Frontal 
	Solution,  717 
	8.2.5  Positive Definiteness, Positive Semidefiniteness, and the Sturm Sequence 
	Property,  726 
	8.2.6  Solution Errors, 734 
	8.2.7  Exercises, 741 
83 	Iterative Solution Methods  745 
	8.3.1  The Gauss-Seidel Method, 747 
	8.3.2  Conjugate Gradient Method with Preconditioning, 749 
	8.3.3  Exercises, 752 
8.4 	Solution of Nonlinear Equations  754 
	8.4.1  Newton-Raphson Schemes, 755 
	8.4.2  The BFGS Method, 759 
	8.4.3  Load-Displacement-Constraint Methods, 761 
	84.4  Convergence Criteria, 764 
	8.4.5  Exercises, 765 

CHAPTER NINE 	
Solution of Equilibrium Equations in Dynamic Analysis 	768 

9.1 	Introduction  768 
9.2 	Direct Integration Methods  769 
	9.2.1  The Central Difference Method, 770 
	9.2.2  The Houbolt Method,  774 
	9.2.3  The Wilson 0 Method, 777 
	9.2.4  The Newmark Method, 780 	
	9.2.5  The Coupling of Different Integration Operators, 782 	
	9.2.6  Exercises, 784 	
9.3 	Mode Superposition  785 	
	9.3.1  Change of Basis to Modal Generalized Displacements,  785 	
	9.3.2  Analysis with Damping Neglected, 789 	
	9.3.3  Analysis with Damping Included, 796 	
	9.3.4  Exercises, 801 	
9.4 	Analysis of Direct Integration Methods  801 	
	9.4.1  Direct Integration Approximation and Load Operators, 803 	
	9.4.2  Stability Analysis, 806 	
	9.4.3  Accuracy Analysis, 810 	
	9.4.4  Some Practical Considerations, 813 	
	9.4.5  Exercises, 822 	
9.5 	Solution of Nonlinear Equations in Dynamic Analysis  824 	
	9.5.1  Explicit Integration, 824 	
	9.5.2  Implicit Integration, 826 	
	9.5.3  Solution Using Mode Superposition, 828 	
	9.5.4  Exercises, 829 	
9.6 	Solution of Nonstructural Problems; Heat Transfer and Fluid Flows 	830 
	9.6.I The a-Method of Time Integration, 830 	
	9.6.2  Exercises, 836 	

CHAPTER TEN
Preliminaries to the Solution of Eigenproblems 	838 

10.1 	Introduction  838 
10.2 	Fundamental Facts Used in the Solution of Eigensystems  840 
	10.2.1  Properties of the Eigenvectors, 841 
	10.2.2  The Characteristic Polynomials of the Eigenproblem K&  = AMd  and of Its 
	Associated Constraint Problems, 845 
	10.2.3  Shifting, 851 
	10.2.4  Effect of Zero Mass, 852 
	10.2.5  Transformation of the Generalized Eigenproblem Kb  = AMd  1o a Standard 
	Form, 854 
	10.2.6  Exercises, 860 
10.3 	Approximate Solution Techniques  861 
	10.3.1  Static Condensation, 861 
	10.3.2  Rayleigh-Ritz Analysis, 868 
	10.3.3  Component Mode Synthesis, 875 
	10.3.4  Exercises, 879 
10.4 	Solution Errors  880  . 
	10.4.1  Error Bounds, 880 
	10.4.2  Exercises, 886 

CHAPTER ELEVEN o  —— 	
Solution Methods for Eigenproblems 	887 

11.1 	Introduction  887 
11.2 	Vector Iteration Methods  889 
	11.2.1  Inverse Iteration, 890 
	11.2.2  Forward Iteration, 897 
	11.2.3  Shifting in Vector Iteration, 899 
	11.2.4  Rayleigh Quotient Iteration, 904 
	11.2.5  Matrix Deflation and Gram-Schmidt Orthogonalization, 906 
	11.2.6  Some Practical Considerations Concerning Vector Iterations, 909 
	11.2.7  Exercises, 910 
11.3 	Transformation Methods  911 
	11.3.1  The Jacobi Method, 912 
	11.3.2  The Generalized Jacobi Method, 919 
	11.3.3  The Householder-QR-Inverse Iteration Solution, 927 
	11.3.4  Exercises, 937 
11.4 	Polynomial Iterations and Sturm Sequence Techniques - 938 
	11.4.1  Explicit Polynomial Iteration, 938 
	11.4.2  Implicit Polynomial Iteration, 939 
	11.4.3  Iteration Based on the Sturm Sequence Property, 943 
	11.4.4  Exercises, 945 
11.5 	The Lanczos Iteration Method  945 
	11.5.1  The Lanczos Transformation, 946 
	11.5.2  Iteration with Lanczos Transformations, 950 
	11.5.3  Ezxercises, 953 
11.6 	The Subspace Iteration Method  954 
	11.6.1  Preliminary Considerations, 955 
	11.6.2  Subspace lteration, 958 
	11.6.3  Starting Iteration Vectors, 960 
	11.6.4  Convergence, 963 
	11.6.5  Implementation of the Subspace Iteration Method, 964 
	11.6.6  Exercises, 978 

CHAPTER TWELVE 	
implementation of the Finite Element Method 	979 

12.1 	Introduction  979 
12.2 	Computer Program Organization for Calculation of System Matrices  980 
	12.2.1  Nodal Point and Element Information Read-in, 981 
	12.2.2  Calculation of Element Stiffness, Mass, and Equivalent Nodal Loads, 983 
	12.2.3  Assemblage of Matrices, 983 
12.3 	 Calculation of Element Stresses  987 
124  	Example Program STAP  988 
	12.4.1  Data Input to Computer Program STAP, 988 
	12.4.2  Listing of Program STAP, 995 
12.5  	Exercises and Projects  1009 

References 	1013 
Index 	1029