Enhanced descriptions from Syndetics:

Directed toward students without in-depth mathematical training, this text cultivates comprehensive skills in linear static and dynamic finite element methodology. Included are a comprehensive presentation and analysis of algorithms of time-dependent phenomena plus beam, plate, and shell theories derived directly from three-dimensional elasticity theory. Solution guide available upon request.

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)
• 2.12 Axisymmetric Formulations and Additional Exercises (p. 101)
• 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)
• 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)
• 4.5 An Extension of Reduced and Selective Integration Techniques (p. 232)
• 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)
• 4.5.1 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)
• 5.4.10 Representation of Stiffness and Load in Global Coordinates (p. 376)
• 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)
• 5.5 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)
• 8.2.3 Convergence (p. 468)
• 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)
• 8.2.4 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)
• 11 Dlearn--A Linear Static and Dynamic Finite Element Analysis Program (p. 603)
• 10.6.8 Loss of Orthogonality (p. 595)
• 10.6.9 Restoring Orthogonality (p. 598)
• References (p. 601)
• 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)