Finite element procedures

✍ Scribed by Klaus-Jürgen Bathe.

• Publisher: Prentice Hall, Inc.
• Year: 1996
• Tongue: English
• Leaves: 1050
• Category: Library

✦ Synopsis

Preface
Finite element procedures are now an important and frequently indispensable part of
engineering analysis and design. Finite element computer programs are now widely used in
practically all branches of engineering for the analysis of structures, solids, and fluids.
My objective in writing this book was to provide a text for upper-level undergraduate
and graduate courses on finite element analysis and to provide a book for self-study by
engineers and scientists.
With this objective in mind, I have developed this book from my earlier publication
Finite Element Procedures in Engineering Analysis (Prentice-Hall, 1982). I have kept the
same mode of presentation but have consolidated, updated, and strengthened the earlier
writing to the current state of finite element developments. Also, I have added new sections,
both to cover some important additional topics for completeness of the presentation and to
facilitate (through exercises) the teaching of the material discussed in the book.
This text does not present a survey of finite element methods. For such an endeavor,
anumber of volumes would be needed. Instead, this book concentrates only on certain finite
element procedures, namely, on techniques that I consider very useful in engineering
practice and that will probably be employed for many years to come. Also, these methods
are introduced in such a way that they can be taught effectively—and in an exciting
manner—to students.
An important aspect of a finite element procedure is its reliability, so that the method
can be used in a confident manner in computer-aided design. This book emphasizes this
point throughout the presentations and concentrates on finite element procedures that are
general and reliable for engineering analysis.
Hence, this book is clearly biased in that it presents only certain finite element
procedures and in that it presents these procedures in a certain manner. In this regard, the
book reflects my philosophy toward the teaching and the use of finite element methods.
While the basic topics of this book focus on mathematical methods, an exciting and
thorough understanding of finite element procedures for engineering applications is
achieved only if sufficient attention is given to both the physical and mathematical charac-
teristics of the procedures. The combined physical and mathematical understanding greatly
enriches our confident use and further development of finite element methods and is there-
fore emphasized in this text.
These thoughts also indicate that a collaborauon between engineers and mathemati-
cians to deepen our understanding of finite element methods and to further advance in the
fields of research can be of great benefit. Indeed, I am thankful to the mathematician Franco
Brezzi for our research collaboration in this spirit, and for his valuable suggestions regard-
ing this book.
I consider it one of the greatest achievements for an educator to write a valuable book.
In these times, all fields of engineering are rapidly changing, and new books for students are
needed in practically all areas of engineering. I am therefore grateful that the Mechanical
Engineering Department of M.L.T. has provided me with an excellent environment in which.
to pursue my interests in teaching, research, and scholarly writing. While it required an
immense effort on my part to write this book, I wanted to accomplish this task as a
commitment to my past and future students, to any educators and researchers who might
have an interest in the work, and, of course, to improve upon my teaching at M.I.T.
I have been truly fortunate to work with many outstanding students at M.I.T., for
which I am very thankful. It has been a great privilege to be their teacher and work with
them. Of much value has also been that I have been intimately involved, at my company
ADINA R & D, Inc., in the development of finite element methods for industry. This
involvement has been very beneficial in my teaching and research, and in my writing of this
book.
A text of significant depth and breadth on a subject that came to life only a few decades
ago and that has experienced tremendous advances, can be written only by an author who
has had the benefit of interacting with many people in the field. I would like to thank all my
students and friends who contributed—and will continue to contribute—to my knowledge
and understanding of finite element methods. My interaction with them has given me great
joy and satisfaction.
I also would like to thank my secretary, Kristan Raymond, for her special efforts in
typing the manuscript of this text.
Finally, truly unbounded thanks are due to my wife, Zorka, and children, Ingrid and
Mark, who, with their love and their understanding of my efforts, supported me in writing
this book.

K. J. Bathe

✦ Table of Contents

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

✦ Subjects

finite element