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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 | |