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Preface | |

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A Brief Glossary of Notations | |

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Linear Static Analysis | |

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Fundamental Concepts; A Simple One-Dimensional Boundary-Value Problem | |

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Introductory Remarks and Preliminaries | |

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Strong, or Classical, Form of the Problem | |

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Weak, or Variational, Form of the Problem | |

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Eqivalence of Strong and Weak Forms; Natural Boundary Conditions | |

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Galerkin's Approximation Method | |

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Matrix Equations; Stiffness Matrix K | |

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Examples: 1 and 2 Degrees of Freedom | |

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Piecewise Linear Finite Element Space | |

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Properties of K | |

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Mathematical Analysis | |

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Interlude: Gauss Elimination; Hand-calculation Version | |

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The Element Point of View | |

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Element Stiffness Matrix and Force Vector | |

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Assembly of Global Stiffness Matrix and Force Vector; LM Array | |

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Explicit Computation of Element Stiffness Matrix and Force Vector | |

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Exercise: Bernoulli-Euler Beam Theory and Hermite Cubics | |

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An Elementary Discussion of Continuity, Differentiability, and Smoothness | |

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References | |

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Formulation of Two- and Three-Dimensional Boundary-Value Problems | |

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Introductory Remarks | |

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Preliminaries | |

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Classical Linear Heat Conduction: Strong and Weak Forms; Equivalence | |

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Heat Conduction: Galerkin Formulation; Symmetry and Positive-definiteness of K | |

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Heat Conduction: Element Stiffness Matrix and Force Vector | |

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Heat Conduction: Data Processing Arrays ID, IEN, and LM | |

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Classical Linear Elastostatics: Strong and Weak Forms; Equivalence | |

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Elastostatics: Galerkin Formulation, Symmetry, and Positive-definiteness of K | |

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Elastostatics: Element Stiffness Matrix and Force Vector | |

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Elastostatics: Data Processing Arrays ID, IEN, and LM | |

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Summary of Important Equations for Problems Considered in Chapters 1 and 2 | |

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Axisymmetric Formulations and Additional Exercises | |

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References | |

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Isoparametric Elements and Elementary Programming Concepts | |

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Preliminary Concepts | |

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Bilinear Quadrilateral Element | |

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Isoparametric Elements | |

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Linear Triangular Element; An Example of "Degeneration" | |

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Trilinear Hexahedral Element | |

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Higher-order Elements; Lagrange Polynomials | |

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Elements with Variable Numbers of Nodes | |

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Numerical Integration; Gaussian Quadrature | |

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Derivatives of Shape Functions and Shape Function Subroutines | |

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Element Stiffness Formulation | |

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Additional Exercises | |

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Triangular and Tetrahedral Elements | |

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Methodology for Developing Special Shape Functions with Application to Singularities | |

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References | |

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Mixed and Penalty Methods, Reduced and Selective Integration, and Sundry Variational Crimes | |

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"Best Approximation" and Error Estimates: Why the standard FEM usually works and why sometimes it does not | |

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Incompressible Elasticity and Stokes Flow | |

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Prelude to Mixed and Penalty Methods | |

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A Mixed Formulation of Compressible Elasticity Capable of Representing the Incompressible Limit | |

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Strong Form | |

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Weak Form | |

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Galerkin Formulation | |

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Matrix Problem | |

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Definition of Element Arrays | |

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Illustration of a Fundamental Difficulty | |

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Constraint Counts | |

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Discontinuous Pressure Elements | |

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Continuous Pressure Elements | |

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Penalty Formulation: Reduced and Selective Integration Techniques; Equivalence with Mixed Methods | |

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Pressure Smoothing | |

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An Extension of Reduced and Selective Integration Techniques | |

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Axisymmetry and Anisotropy: Prelude to Nonlinear Analysis | |

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Strain Projection: The B-approach | |

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The Patch Test; Rank Deficiency | |

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Nonconforming Elements | |

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Hourglass Stiffness | |

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Additional Exercises and Projects | |

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Mathematical Preliminaries | |

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Basic Properties of Linear Spaces | |

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Sobolev Norms | |

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Approximation Properties of Finite Element Spaces in Sobolev Norms | |

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Hypotheses on a(.,.) | |

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Advanced Topics in the Theory of Mixed and Penalty Methods: Pressure Modes and Error Estimates | |

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Pressure Modes, Spurious and Otherwise | |

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Existence and Uniqueness of Solutions in the Presence of Modes | |

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Two Sides of Pressure Modes | |

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Pressure Modes in the Penalty Formulation | |

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The Big Picture | |

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Error Estimates and Pressure Smoothing | |

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References | |

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The C[superscript 0]-Approach to Plates and Beams | |

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Introduction | |

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Reissner-Mindlin Plate Theory | |

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Main Assumptions | |

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Constitutive Equation | |

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Strain-displacement Equations | |

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Summary of Plate Theory Notations | |

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Variational Equation | |

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Strong Form | |

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Weak Form | |

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Matrix Formulation | |

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Finite Element Stiffness Matrix and Load Vector | |

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Plate-bending Elements | |

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Some Convergence Criteria | |

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Shear Constraints and Locking | |

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Boundary Conditions | |

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Reduced and Selective Integration Lagrange Plate Elements | |

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Equivalence with Mixed Methods | |

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Rank Deficiency | |

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The Heterosis Element | |

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T1: A Correct-rank, Four-node Bilinear Element | |

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The Linear Triangle | |

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The Discrete Kirchhoff Approach | |

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Discussion of Some Quadrilateral Bending Elements | |

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Beams and Frames | |

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Main Assumptions | |

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Constitutive Equation | |

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Strain-displacement Equations | |

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Definitions of Quantities Appearing in the Theory | |

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Variational Equation | |

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Strong Form | |

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Weak Form | |

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Matrix Formulation of the Variational Equation | |

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Finite Element Stiffness Matrix and Load Vector | |

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Representation of Stiffness and Load in Global Coordinates | |

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Reduced Integration Beam Elements | |

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References | |

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The C[superscript 0]-Approach to Curved Structural Elements | |

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Introduction | |

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Doubly Curved Shells in Three Dimensions | |

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Geometry | |

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Lamina Coordinate Systems | |

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Fiber Coordinate Systems | |

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Kinematics | |

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Reduced Constitutive Equation | |

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Strain-displacement Matrix | |

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Stiffness Matrix | |

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External Force Vector | |

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Fiber Numerical Integration | |

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Stress Resultants | |

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Shell Elements | |

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Some References to the Recent Literature | |

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Simplifications: Shells as an Assembly of Flat Elements | |

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Shells of Revolution; Rings and Tubes in Two Dimensions | |

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Geometric and Kinematic Descriptions | |

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Reduced Constitutive Equations | |

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Strain-displacement Matrix | |

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Stiffness Matrix | |

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External Force Vector | |

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Stress Resultants | |

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Boundary Conditions | |

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Shell Elements | |

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References | |

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Linear Dynamic Analysis | |

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Formulation of Parabolic, Hyperbolic, and Elliptic-Elgenvalue Problems | |

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Parabolic Case: Heat Equation | |

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Hyperbolic Case: Elastodynamics and Structural Dynamics | |

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Eigenvalue Problems: Frequency Analysis and Buckling | |

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Standard Error Estimates | |

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Alternative Definitions of the Mass Matrix; Lumped and Higher-order Mass | |

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Estimation of Eigenvalues | |

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Error Estimates for Semidiscrete Galerkin Approximations | |

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References | |

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Algorithms for Parabolic Problems | |

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One-step Algorithms for the Semidiscrete Heat Equation: Generalized Trapezoidal Method | |

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Analysis of the Generalized Trapezoidal Method | |

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Modal Reduction to SDOF Form | |

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Stability | |

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Convergence | |

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An Alternative Approach to Stability: The Energy Method | |

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Additional Exercises | |

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Elementary Finite Difference Equations for the One-dimensional Heat Equation; the von Neumann Method of Stability Analysis | |

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Element-by-element (EBE) Implicit Methods | |

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Modal Analysis | |

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References | |

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Algorithms for Hyperbolic and Parabolic-Hyperbolic Problems | |

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One-step Algorithms for the Semidiscrete Equation of Motion | |

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The Newmark Method | |

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Analysis | |

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Measures of Accuracy: Numerical Dissipation and Dispersion | |

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Matched Methods | |

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Additional Exercises | |

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Summary of Time-step Estimates for Some Simple Finite Elements | |

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Linear Multistep (LMS) Methods | |

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LMS Methods for First-order Equations | |

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LMS Methods for Second-order Equations | |

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Survey of Some Commonly Used Algorithms in Structural Dynamics | |

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Some Recently Developed Algorithms for Structural Dynamics | |

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Algorithms Based upon Operator Splitting and Mesh Partitions | |

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Stability via the Energy Method | |

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Predictor/Multicorrector Algorithms | |

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Mass Matrices for Shell Elements | |

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References | |

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Solution Techniques for Eigenvalue Problems | |

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The Generalized Eigenproblem | |

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Static Condensation | |

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Discrete Rayleigh-Ritz Reduction | |

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Irons-Guyan Reduction | |

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Subspace Iteration | |

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Spectrum Slicing | |

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Inverse Iteration | |

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The Lanczos Algorithm for Solution of Large Generalized Eigenproblems | |

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Introduction | |

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Spectral Transformation | |

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Conditions for Real Eigenvalues | |

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The Rayleigh-Ritz Approximation | |

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Derivation of the Lanczos Algorithm | |

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Reduction to Tridiagonal Form | |

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Convergence Criterion for Eigenvalues | |

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Loss of Orthogonality | |

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Restoring Orthogonality | |

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References | |

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Dlearn--A Linear Static and Dynamic Finite Element Analysis Program | |

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Introduction | |

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Description of Coding Techniques Used in DLEARN | |

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Compacted Column Storage Scheme | |

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Crout Elimination | |

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Dynamic Storage Allocation | |

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Program Structure | |

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Global Control | |

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Initialization Phase | |

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Solution Phase | |

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Adding an Element to DLEARN | |

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DLEARN User's Manual | |

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Remarks for the New User | |

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Input Instructions | |

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Examples | |

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Planar Truss | |

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Static Analysis of a Plane Strain Cantilever Beam | |

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Dynamic Analysis of a Plane Strain Cantilever Beam | |

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Implicit-explicit Dynamic Analysis of a Rod | |

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Subroutine Index for Program Listing | |

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References | |

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Index | |