Enhanced descriptions from Syndetics:

This text is geared toward assisting engineering and physical science students in cultivating comprehensive skills in linear static and dynamic finite element methodology. Based on courses taught at Stanford University and the California Institute of Technology, it ranges from fundamental concepts to practical computer implementations. Additional sections touch upon the frontiers of research, making the book of potential interest to more experienced analysts and researchers working in the finite element field. In addition to its examination of numerous standard aspects of the finite element method, the volume includes many unique components, including a comprehensive presentation and analysis of algorithms of time-dependent phenomena, plus beam, plate, and shell theories derived directly from three-dimensional elasticity theory. It also contains a systematic treatment of "weak," or variational, formulations for diverse classes of initial/boundary-value problems. Directed toward students without in-depth mathematical training, the text incorporates introductory material on the mathematical theory of finite elements and many important mathematical results, making it an ideal primer for more advanced works on this subject.

Table of contents provided by Syndetics

• Preface (p. XV)
• A Brief Glossary of Notations (p. XXII)
• Part 1 Linear Static Analysis
• 1 Fundamental Concepts; A Simple One-Dimensional Boundary-Value Problem (p. 1)
• 1.1 Introductory Remarks and Preliminaries (p. 1)
• 1.2 Strong, or Classical, Form of the Problem (p. 2)
• 1.3 Weak, or Variational, Form of the Problem (p. 3)
• 1.4 Eqivalence of Strong and Weak Forms; Natural Boundary Conditions (p. 4)
• 1.5 Galerkin's Approximation Method (p. 7)
• 1.6 Matrix Equations; Stiffness Matrix K (p. 9)
• 1.7 Examples: 1 and 2 Degrees of Freedom (p. 13)
• 1.8 Piecewise Linear Finite Element Space (p. 20)
• 1.9 Properties of K (p. 22)
• 1.10 Mathematical Analysis (p. 24)
• 1.11 Interlude: Gauss Elimination; Hand-calculation Version (p. 31)
• 1.12 The Element Point of View (p. 37)
• 1.13 Element Stiffness Matrix and Force Vector (p. 40)
• 1.14 Assembly of Global Stiffness Matrix and Force Vector; LM Array (p. 42)
• 1.15 Explicit Computation of Element Stiffness Matrix and Force Vector (p. 44)
• 1.16 Exercise: Bernoulli-Euler Beam Theory and Hermite Cubics (p. 48)
• Appendix 1.I An Elementary Discussion of Continuity, Differentiability, and Smoothness (p. 52)
• References (p. 55)
• 2 Formulation of Two- and Three-Dimensional Boundary-Value Problems (p. 57)
• 2.1 Introductory Remarks (p. 57)
• 2.2 Preliminaries (p. 57)
• 2.3 Classical Linear Heat Conduction: Strong and Weak Forms; Equivalence (p. 60)
• 2.4 Heat Conduction: Galerkin Formulation; Symmetry and Positive-definiteness of K (p. 64)
• 2.5 Heat Conduction: Element Stiffness Matrix and Force Vector (p. 69)
• 2.6 Heat Conduction: Data Processing Arrays ID, IEN, and LM (p. 71)
• 2.7 Classical Linear Elastostatics: Strong and Weak Forms; Equivalence (p. 75)
• 2.8 Elastostatics: Galerkin Formulation, Symmetry, and Positive-definiteness of K (p. 84)
• 2.9 Elastostatics: Element Stiffness Matrix and Force Vector (p. 90)
• 2.10 Elastostatics: Data Processing Arrays ID, IEN, and LM (p. 92)
• 2.11 Summary of Important Equations for Problems Considered in Chapters 1 and 2 (p. 98)
• 2.12 Axisymmetric Formulations and Additional Exercises (p. 101)
• References (p. 107)
• 3 Isoparametric Elements and Elementary Programming Concepts (p. 109)
• 3.1 Preliminary Concepts (p. 109)
• 3.2 Bilinear Quadrilateral Element (p. 112)
• 3.3 Isoparametric Elements (p. 118)
• 3.4 Linear Triangular Element; An Example of "Degeneration" (p. 120)
• 3.5 Trilinear Hexahedral Element (p. 123)
• 3.6 Higher-order Elements; Lagrange Polynomials (p. 126)
• 3.7 Elements with Variable Numbers of Nodes (p. 132)
• 3.8 Numerical Integration; Gaussian Quadrature (p. 137)
• 3.9 Derivatives of Shape Functions and Shape Function Subroutines (p. 146)
• 3.10 Element Stiffness Formulation (p. 151)
• 3.11 Additional Exercises (p. 156)
• Appendix 3.I Triangular and Tetrahedral Elements (p. 164)
• Appendix 3.II Methodology for Developing Special Shape Functions with Application to Singularities (p. 175)
• References (p. 182)
• 4 Mixed and Penalty Methods, Reduced and Selective Integration, and Sundry Variational Crimes (p. 185)
• 4.1 "Best Approximation" and Error Estimates: Why the standard FEM usually works and why sometimes it does not (p. 185)
• 4.2 Incompressible Elasticity and Stokes Flow (p. 192)
• 4.2.1 Prelude to Mixed and Penalty Methods (p. 194)
• 4.3 A Mixed Formulation of Compressible Elasticity Capable of Representing the Incompressible Limit (p. 197)
• 4.3.1 Strong Form (p. 198)
• 4.3.2 Weak Form (p. 198)
• 4.3.3 Galerkin Formulation (p. 200)
• 4.3.4 Matrix Problem (p. 200)
• 4.3.5 Definition of Element Arrays (p. 204)
• 4.3.6 Illustration of a Fundamental Difficulty (p. 207)
• 4.3.7 Constraint Counts (p. 209)
• 4.3.8 Discontinuous Pressure Elements (p. 210)
• 4.3.9 Continuous Pressure Elements (p. 215)
• 4.4 Penalty Formulation: Reduced and Selective Integration Techniques; Equivalence with Mixed Methods (p. 217)
• 4.4.1 Pressure Smoothing (p. 226)
• 4.5 An Extension of Reduced and Selective Integration Techniques (p. 232)
• 4.5.1 Axisymmetry and Anisotropy: Prelude to Nonlinear Analysis (p. 232)
• 4.5.2 Strain Projection: The B-approach (p. 232)
• 4.6 The Patch Test; Rank Deficiency (p. 237)
• 4.7 Nonconforming Elements (p. 242)
• 4.8 Hourglass Stiffness (p. 251)
• 4.9 Additional Exercises and Projects (p. 254)
• Appendix 4.I Mathematical Preliminaries (p. 263)
• 4.I.1 Basic Properties of Linear Spaces (p. 263)
• 4.I.2 Sobolev Norms (p. 266)
• 4.I.3 Approximation Properties of Finite Element Spaces in Sobolev Norms (p. 268)
• 4.I.4 Hypotheses on a(.,.) (p. 273)
• Appendix 4.II Advanced Topics in the Theory of Mixed and Penalty Methods: Pressure Modes and Error Estimates (p. 276)
• 4.II.1 Pressure Modes, Spurious and Otherwise (p. 276)
• 4.II.2 Existence and Uniqueness of Solutions in the Presence of Modes (p. 278)
• 4.II.3 Two Sides of Pressure Modes (p. 281)
• 4.II.4 Pressure Modes in the Penalty Formulation (p. 289)
• 4.II.5 The Big Picture (p. 292)
• 4.II.6 Error Estimates and Pressure Smoothing (p. 297)
• References (p. 303)
• 5 The C[superscript 0]-Approach to Plates and Beams (p. 310)
• 5.1 Introduction (p. 310)
• 5.2 Reissner-Mindlin Plate Theory (p. 310)
• 5.2.1 Main Assumptions (p. 310)
• 5.2.2 Constitutive Equation (p. 313)
• 5.2.3 Strain-displacement Equations (p. 313)
• 5.2.4 Summary of Plate Theory Notations (p. 314)
• 5.2.5 Variational Equation (p. 314)
• 5.2.6 Strong Form (p. 317)
• 5.2.7 Weak Form (p. 317)
• 5.2.8 Matrix Formulation (p. 319)
• 5.2.9 Finite Element Stiffness Matrix and Load Vector (p. 320)
• 5.3 Plate-bending Elements (p. 322)
• 5.3.1 Some Convergence Criteria (p. 322)
• 5.3.2 Shear Constraints and Locking (p. 323)
• 5.3.3 Boundary Conditions (p. 324)
• 5.3.4 Reduced and Selective Integration Lagrange Plate Elements (p. 327)
• 5.3.5 Equivalence with Mixed Methods (p. 330)
• 5.3.6 Rank Deficiency (p. 332)
• 5.3.7 The Heterosis Element (p. 335)
• 5.3.8 T1: A Correct-rank, Four-node Bilinear Element (p. 342)
• 5.3.9 The Linear Triangle (p. 355)
• 5.3.10 The Discrete Kirchhoff Approach (p. 359)
• 5.3.11 Discussion of Some Quadrilateral Bending Elements (p. 362)
• 5.4 Beams and Frames (p. 363)
• 5.4.1 Main Assumptions (p. 363)
• 5.4.2 Constitutive Equation (p. 365)
• 5.4.3 Strain-displacement Equations (p. 366)
• 5.4.4 Definitions of Quantities Appearing in the Theory (p. 366)
• 5.4.5 Variational Equation (p. 368)
• 5.4.6 Strong Form (p. 371)
• 5.4.7 Weak Form (p. 372)
• 5.4.8 Matrix Formulation of the Variational Equation (p. 373)
• 5.4.9 Finite Element Stiffness Matrix and Load Vector (p. 374)
• 5.4.10 Representation of Stiffness and Load in Global Coordinates (p. 376)
• 5.5 Reduced Integration Beam Elements (p. 376)
• References (p. 379)
• The C[superscript 0]-Approach to Curved Structural Elements (p. 383)
• 6.1 Introduction (p. 383)
• 6.2 Doubly Curved Shells in Three Dimensions (p. 384)
• 6.2.1 Geometry (p. 384)
• 6.2.2 Lamina Coordinate Systems (p. 385)
• 6.2.3 Fiber Coordinate Systems (p. 387)
• 6.2.4 Kinematics (p. 388)
• 6.2.5 Reduced Constitutive Equation (p. 389)
• 6.2.6 Strain-displacement Matrix (p. 392)
• 6.2.7 Stiffness Matrix (p. 396)
• 6.2.8 External Force Vector (p. 396)
• 6.2.9 Fiber Numerical Integration (p. 398)
• 6.2.10 Stress Resultants (p. 399)
• 6.2.11 Shell Elements (p. 399)
• 6.2.12 Some References to the Recent Literature (p. 403)
• 6.2.13 Simplifications: Shells as an Assembly of Flat Elements (p. 404)
• 6.3 Shells of Revolution; Rings and Tubes in Two Dimensions (p. 405)
• 6.3.1 Geometric and Kinematic Descriptions (p. 405)
• 6.3.2 Reduced Constitutive Equations (p. 407)
• 6.3.3 Strain-displacement Matrix (p. 409)
• 6.3.4 Stiffness Matrix (p. 412)
• 6.3.5 External Force Vector (p. 412)
• 6.3.6 Stress Resultants (p. 413)
• 6.3.7 Boundary Conditions (p. 414)
• 6.3.8 Shell Elements (p. 414)
• References (p. 415)
• Part 2 Linear Dynamic Analysis
• 7 Formulation of Parabolic, Hyperbolic, and Elliptic-Elgenvalue Problems (p. 418)
• 7.1 Parabolic Case: Heat Equation (p. 418)
• 7.2 Hyperbolic Case: Elastodynamics and Structural Dynamics (p. 423)
• 7.3 Eigenvalue Problems: Frequency Analysis and Buckling (p. 429)
• 7.3.1 Standard Error Estimates (p. 433)
• 7.3.2 Alternative Definitions of the Mass Matrix; Lumped and Higher-order Mass (p. 436)
• 7.3.3 Estimation of Eigenvalues (p. 452)
• Appendix 7.I Error Estimates for Semidiscrete Galerkin Approximations (p. 456)
• References (p. 457)
• 8 Algorithms for Parabolic Problems (p. 459)
• 8.1 One-step Algorithms for the Semidiscrete Heat Equation: Generalized Trapezoidal Method (p. 459)
• 8.2 Analysis of the Generalized Trapezoidal Method (p. 462)
• 8.2.1 Modal Reduction to SDOF Form (p. 462)
• 8.2.2 Stability (p. 465)
• 8.2.3 Convergence (p. 468)
• 8.2.4 An Alternative Approach to Stability: The Energy Method (p. 471)
• 8.2.5 Additional Exercises (p. 473)
• 8.3 Elementary Finite Difference Equations for the One-dimensional Heat Equation; the von Neumann Method of Stability Analysis (p. 479)
• 8.4 Element-by-element (EBE) Implicit Methods (p. 483)
• 8.5 Modal Analysis (p. 487)
• References (p. 488)
• 9 Algorithms for Hyperbolic and Parabolic-Hyperbolic Problems (p. 490)
• 9.1 One-step Algorithms for the Semidiscrete Equation of Motion (p. 490)
• 9.1.1 The Newmark Method (p. 490)
• 9.1.2 Analysis (p. 492)
• 9.1.3 Measures of Accuracy: Numerical Dissipation and Dispersion (p. 504)
• 9.1.4 Matched Methods (p. 505)
• 9.1.5 Additional Exercises (p. 512)
• 9.2 Summary of Time-step Estimates for Some Simple Finite Elements (p. 513)
• 9.3 Linear Multistep (LMS) Methods (p. 523)
• 9.3.1 LMS Methods for First-order Equations (p. 523)
• 9.3.2 LMS Methods for Second-order Equations (p. 526)
• 9.3.3 Survey of Some Commonly Used Algorithms in Structural Dynamics (p. 529)
• 9.3.4 Some Recently Developed Algorithms for Structural Dynamics (p. 550)
• 9.4 Algorithms Based upon Operator Splitting and Mesh Partitions (p. 552)
• 9.4.1 Stability via the Energy Method (p. 556)
• 9.4.2 Predictor/Multicorrector Algorithms (p. 562)
• 9.5 Mass Matrices for Shell Elements (p. 564)
• References (p. 567)
• 10 Solution Techniques for Eigenvalue Problems (p. 570)
• 10.1 The Generalized Eigenproblem (p. 570)
• 10.2 Static Condensation (p. 573)
• 10.3 Discrete Rayleigh-Ritz Reduction (p. 574)
• 10.4 Irons-Guyan Reduction (p. 576)
• 10.5 Subspace Iteration (p. 576)
• 10.5.1 Spectrum Slicing (p. 578)
• 10.5.2 Inverse Iteration (p. 579)
• 10.6 The Lanczos Algorithm for Solution of Large Generalized Eigenproblems (p. 582)
• 10.6.1 Introduction (p. 582)
• 10.6.2 Spectral Transformation (p. 583)
• 10.6.3 Conditions for Real Eigenvalues (p. 584)
• 10.6.4 The Rayleigh-Ritz Approximation (p. 585)
• 10.6.5 Derivation of the Lanczos Algorithm (p. 586)
• 10.6.6 Reduction to Tridiagonal Form (p. 589)
• 10.6.7 Convergence Criterion for Eigenvalues (p. 592)
• 10.6.8 Loss of Orthogonality (p. 595)
• 10.6.9 Restoring Orthogonality (p. 598)
• References (p. 601)
• 11 Dlearn--A Linear Static and Dynamic Finite Element Analysis Program (p. 603)
• 11.1 Introduction (p. 603)
• 11.2 Description of Coding Techniques Used in DLEARN (p. 604)
• 11.2.1 Compacted Column Storage Scheme (p. 605)
• 11.2.2 Crout Elimination (p. 608)
• 11.2.3 Dynamic Storage Allocation (p. 616)
• 11.3 Program Structure (p. 622)
• 11.3.1 Global Control (p. 623)
• 11.3.2 Initialization Phase (p. 623)
• 11.3.3 Solution Phase (p. 625)
• 11.4 Adding an Element to DLEARN (p. 631)
• 11.5 DLEARN User's Manual (p. 634)
• 11.5.1 Remarks for the New User (p. 634)
• 11.5.2 Input Instructions (p. 635)
• 11.5.3 Examples (p. 663)
• 1. Planar Truss (p. 663)
• 2. Static Analysis of a Plane Strain Cantilever Beam (p. 666)
• 3. Dynamic Analysis of a Plane Strain Cantilever Beam (p. 666)
• 4. Implicit-explicit Dynamic Analysis of a Rod (p. 668)
• 11.5.4 Subroutine Index for Program Listing (p. 670)
• References (p. 675)
• Index (p. 676)