The finite element method : linear static and dynamic finite element analysis /

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 de...

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Detalles Bibliográficos
Clasificación:Libro Electrónico
Autor principal: Hughes, Thomas J. R.
Formato: Electrónico eBook
Idioma:Inglés
Publicado: Mineola, NY : Dover Publications, 2000.
Temas:
Finite element method.
Boundary value problems.
Méthode des éléments finis.
Problèmes aux limites.
TECHNOLOGY & ENGINEERING > Civil > General.
TECHNOLOGY & ENGINEERING > Engineering (General)
TECHNOLOGY & ENGINEERING > Reference.
Boundary value problems
Finite element method
Método dos elementos finitos.
Diferenças finitas.
Acceso en línea:Texto completo
Tabla de Contenidos:
  • Part 1. Linear Static Analysis
  • 1. Fundamental Concepts; A Simple One-Dimensional Boundary-Value Problem
  • 1.1. Introductory Remarks and Preliminaries
  • 1.2. Strong, or Classical, Form of the Problem
  • 1.3. Weak, or Variational, Form of the Problem
  • 1.4. Eqivalence of Strong and Weak Forms; Natural Boundary Conditions
  • 1.5. Galerkin's Approximation Method
  • 1.6. Matrix Equations; Stiffness Matrix K
  • 1.7. Examples: 1 and 2 Degrees of Freedom
  • 1.8. Piecewise Linear Finite Element Space
  • 1.9. Properties of K
  • 1.10. Mathematical Analysis
  • 1.11. Interlude: Gauss Elimination; Hand-calculation Version
  • 1.12. The Element Point of View
  • 1.13. Element Stiffness Matrix and Force Vector
  • 1.14. Assembly of Global Stiffness Matrix and Force Vector; LM Array
  • 1.15. Explicit Computation of Element Stiffness Matrix and Force Vector
  • 1.16. Exercise: Bernoulli-Euler Beam Theory and Hermite Cubics
  • Appendix 1.I. An Elementary Discussion of Continuity, Differentiability, and Smoothness
  • 2. Formulation of Two- and Three-Dimensional Boundary-Value Problems
  • 2.1. Introductory Remarks
  • 2.2. Preliminaries
  • 2.3. Classical Linear Heat Conduction: Strong and Weak Forms; Equivalence
  • 2.4. Heat Conduction: Galerkin Formulation; Symmetry and Positive-definiteness of K
  • 2.5. Heat Conduction: Element Stiffness Matrix and Force Vector
  • 2.6. Heat Conduction: Data Processing Arrays ID, IEN, and LM
  • 2.7. Classical Linear Elastostatics: Strong and Weak Forms; Equivalence
  • 2.8. Elastostatics: Galerkin Formulation, Symmetry, and Positive-definiteness of K
  • 2.9. Elastostatics: Element Stiffness Matrix and Force Vector
  • 2.10. Elastostatics: Data Processing Arrays ID, IEN, and LM
  • 2.11. Summary of Important Equations for Problems Considered in Chapters 1 and 2
  • 2.12. Axisymmetric Formulations and Additional Exercises
  • 3. Isoparametric Elements and Elementary Programming Concepts
  • 3.1. Preliminary Concepts
  • 3.2. Bilinear Quadrilateral Element
  • 3.3. Isoparametric Elements
  • 3.4. Linear Triangular Element; An Example of "Degeneration"
  • 3.5. Trilinear Hexahedral Element
  • 3.6. Higher-order Elements; Lagrange Polynomials
  • 3.7. Elements with Variable Numbers of Nodes
  • 3.8. Numerical Integration; Gaussian Quadrature
  • 3.9. Derivatives of Shape Functions and Shape Function Subroutines
  • 3.10. Element Stiffness Formulation
  • 3.11. Additional Exercises
  • Appendix 3.I. Triangular and Tetrahedral Elements
  • Appendix 3. II. Methodology for Developing Special Shape Functions with Application to Singularities
  • 4. Mixed and Penalty Methods, Reduced and Selective Integration, and Sundry Variational Crimes
  • 4.1. "Best Approximation" and Error Estimates: Why the standard FEM usually works and why sometimes it does not
  • 4.2. Incompressible Elasticity and Stokes Flow
  • 4.2.1. Prelude to Mixed and Penalty Methods
  • 4.3. A Mixed Formulation of Compressible Elasticity Capable of Representing the Incompressible Limit
  • 4.3.1. Strong Form
  • 4.3.2. Weak Form
  • 4.3.3. Galerkin Formulation
  • 4.3.4. Matrix Problem
  • 4.3.5. Definition of Element Arrays
  • 4.3.6. Illustration of a Fundamental Difficulty
  • 4.3.7. Constraint Counts
  • 4.3.8. Discontinuous Pressure Elements
  • 4.3.9. Continuous Pressure Elements
  • 4.4. Penalty Formulation: Reduced and Selective Integration Techniques; Equivalence with Mixed Methods
  • 4.4.1. Pressure Smoothing
  • 4.5. An Extension of Reduced and Selective Integration Techniques
  • 4.5.1. Axisymmetry and Anisotropy: Prelude to Nonlinear Analysis
  • 4.5.2. Strain Projection: The B-approach
  • 4.6. The Patch Test; Rank Deficiency
  • 4.7. Nonconforming Elements
  • 4.8. Hourglass Stiffness
  • 4.9. Additional Exercises and Projects
  • Appendix 4.I. Mathematical Preliminaries
  • 4.I.1. Basic Properties of Linear Spaces
  • 4.I.2. Sobolev Norms
  • 4.I.3. Approximation Properties of Finite Element Spaces in Sobolev Norms
  • 4.I.4. Hypotheses on a(., .)
  • Appendix 4. II. Advanced Topics in the Theory of Mixed and Penalty Methods: Pressure Modes and Error Estimates / David S. Malkus
  • 4. II.1. Pressure Modes, Spurious and Otherwise
  • 4. II.2. Existence and Uniqueness of Solutions in the Presence of Modes
  • 4. II.3. Two Sides of Pressure Modes
  • 4. II.4. Pressure Modes in the Penalty Formulation
  • 4. II.5. The Big Picture
  • 4. II.6. Error Estimates and Pressure Smoothing
  • 5. The C[superscript 0]-Approach to Plates and Beams
  • 5.1. Introduction
  • 5.2. Reissner-Mindlin Plate Theory
  • 5.2.1. Main Assumptions
  • 5.2.2. Constitutive Equation
  • 5.2.3. Strain-displacement Equations
  • 5.2.4. Summary of Plate Theory Notations
  • 5.2.5. Variational Equation
  • 5.2.6. Strong Form
  • 5.2.7. Weak Form
  • 5.2.8. Matrix Formulation
  • 5.2.9. Finite Element Stiffness Matrix and Load Vector
  • 5.3. Plate-bending Elements
  • 5.3.1. Some Convergence Criteria
  • 5.3.2. Shear Constraints and Locking
  • 5.3.3. Boundary Conditions
  • 5.3.4. Reduced and Selective Integration Lagrange Plate Elements
  • 5.3.5. Equivalence with Mixed Methods
  • 5.3.6. Rank Deficiency
  • 5.3.7. The Heterosis Element
  • 5.3.8. T1: A Correct-rank, Four-node Bilinear Element
  • 5.3.9. The Linear Triangle
  • 5.3.10. The Discrete Kirchhoff Approach
  • 5.3.11. Discussion of Some Quadrilateral Bending Elements
  • 5.4. Beams and Frames
  • 5.4.1. Main Assumptions
  • 5.4.2. Constitutive Equation
  • 5.4.3. Strain-displacement Equations
  • 5.4.4. Definitions of Quantities Appearing in the Theory
  • 5.4.5. Variational Equation
  • 5.4.6. Strong Form
  • 5.4.7. Weak Form
  • 5.4.8. Matrix Formulation of the Variational Equation
  • 5.4.9. Finite Element Stiffness Matrix and Load Vector
  • 5.4.10. Representation of Stiffness and Load in Global Coordinates
  • 5.5. Reduced Integration Beam Elements
  • The C[superscript 0]-Approach to Curved Structural Elements
  • 6.1. Introduction
  • 6.2. Doubly Curved Shells in Three Dimensions
  • 6.2.1. Geometry
  • 6.2.2. Lamina Coordinate Systems
  • 6.2.3. Fiber Coordinate Systems
  • 6.2.4. Kinematics
  • 6.2.5. Reduced Constitutive Equation
  • 6.2.6. Strain-displacement Matrix
  • 6.2.7. Stiffness Matrix
  • 6.2.8. External Force Vector
  • 6.2.9. Fiber Numerical Integration
  • 6.2.10. Stress Resultants
  • 6.2.11. Shell Elements
  • 6.2.12. Some References to the Recent Literature
  • 6.2.13. Simplifications: Shells as an Assembly of Flat Elements
  • 6.3. Shells of Revolution; Rings and Tubes in Two Dimensions
  • 6.3.1. Geometric and Kinematic Descriptions
  • 6.3.2. Reduced Constitutive Equations
  • 6.3.3. Strain-displacement Matrix
  • 6.3.4. Stiffness Matrix
  • 6.3.5. External Force Vector
  • 6.3.6. Stress Resultants
  • 6.3.7. Boundary Conditions
  • 6.3.8. Shell Elements
  • Part 2. Linear Dynamic Analysis
  • 7. Formulation of Parabolic, Hyperbolic, and Elliptic-Elgenvalue Problems
  • 7.1. Parabolic Case: Heat Equation
  • 7.2. Hyperbolic Case: Elastodynamics and Structural Dynamics
  • 7.3. Eigenvalue Problems: Frequency Analysis and Buckling
  • 7.3.1. Standard Error Estimates
  • 7.3.2. Alternative Definitions of the Mass Matrix; Lumped and Higher-order Mass
  • 7.3.3. Estimation of Eigenvalues
  • Appendix 7.I. Error Estimates for Semidiscrete Galerkin Approximations
  • 8. Algorithms for Parabolic Problems
  • 8.1. One-step Algorithms for the Semidiscrete Heat Equation: Generalized Trapezoidal Method
  • 8.2. Analysis of the Generalized Trapezoidal Method
  • 8.2.1. Modal Reduction to SDOF Form
  • 8.2.2. Stability
  • 8.2.3. Convergence
  • 8.2.4. An Alternative Approach to Stability: The Energy Method
  • 8.2.5. Additional Exercises
  • 8.3. Elementary Finite Difference Equations for the One-dimensional Heat Equation; the von Neumann Method of Stability Analysis
  • 8.4. Element-by-element (EBE) Implicit Methods
  • 8.5. Modal Analysis
  • 9. Algorithms for Hyperbolic and Parabolic-Hyperbolic Problems
  • 9.1. One-step Algorithms for the Semidiscrete Equation of Motion
  • 9.1.1. The Newmark Method
  • 9.1.2. Analysis
  • 9.1.3.
  • Measures of Accuracy: Numerical Dissipation and Dispersion
  • 9.1.4. Matched Methods
  • 9.1.5. Additional Exercises
  • 9.2. Summary of Time-step Estimates for Some Simple Finite Elements.
  • 9.3. Linear Multistep (LMS) Methods
  • 9.3.1. LMS Methods for First-order Equations
  • 9.3.2. LMS Methods for Second-order Equations
  • 9.3.3. Survey of Some Commonly Used Algorithms in Structural Dynamics
  • 9.3.4. Some Recently Developed Algorithms for Structural Dynamics
  • 9.4. Algorithms Based upon Operator Splitting and Mesh Partitions
  • 9.4.1. Stability via the Energy Method
  • 9.4.2. Predictor/Multicorrector Algorithms
  • 9.5. Mass Matrices for Shell Elements
  • 10. Solution Techniques for Eigenvalue Problems
  • 10.1. The Generalized Eigenproblem
  • 10.2. Static Condensation
  • 10.3. Discrete Rayleigh-Ritz Reduction
  • 10.4. Irons-Guyan Reduction
  • 10.5. Subspace Iteration
  • 10.5.1. Spectrum Slicing
  • 10.5.2. Inverse Iteration
  • 10.6. The Lanczos Algorithm for Solution of Large Generalized Eigenproblems / Bahram Nour-Omid
  • 10.6.1. Introduction
  • 10.6.2. Spectral Transformation
  • 10.6.3. Conditions for Real Eigenvalues
  • 10.6.4. The Rayleigh-Ritz Approximation
  • 10.6.5. Derivation of the Lanczos Algorithm
  • 10.6.6. Reduction to Tridiagonal Form
  • 10.6.7. Convergence Criterion for Eigenvalues
  • 10.6.8. Loss of Orthogonality
  • 10.6.9. Restoring Orthogonality
  • 11. Dlearn
  • A Linear Static and Dynamic Finite Element Analysis Program / Thomas J.R. Hughes, Robert M. Ferencz and Arthur M. Raefsky
  • 11.1. Introduction
  • 11.2. Description of Coding Techniques Used in DLEARN
  • 11.2.1. Compacted Column Storage Scheme
  • 11.2.2. Crout Elimination
  • 11.2.3. Dynamic Storage Allocation
  • 11.3. Program Structure
  • 11.3.1. Global Control
  • 11.3.2. Initialization Phase
  • 11.3.3. Solution Phase
  • 11.4. Adding an Element to DLEARN
  • 11.5. DLEARN User's Manual
  • 11.5.1. Remarks for the New User
  • 11.5.2. Input Instructions
  • 11.5.3. Examples
  • 1. Planar Truss
  • 2. Static Analysis of a Plane Strain Cantilever Beam
  • 3. Dynamic Analysis of a Plane Strain Cantilever Beam
  • 4. Implicit-explicit Dynamic Analysis of a Rod
  • 11.5.4. Subroutine Index for Program Listing.

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