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Finite Element Method Linear Static and Dynamic Finite Element Analysis

ISBN-10: 0486411818
ISBN-13: 9780486411811
Edition: 2000
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Description: Directed towards 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  More...

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Book details

List price: $32.95
Copyright year: 2000
Publisher: Dover Publications, Incorporated
Publication date: 8/16/2000
Binding: Hardcover
Pages: 672
Size: 6.25" wide x 9.00" long x 1.50" tall
Weight: 2.310

Directed towards 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.

Preface
A Brief Glossary of Notations
Linear Static Analysis
Fundamental Concepts; A Simple One-Dimensional Boundary-Value Problem
Introductory Remarks and Preliminaries
Strong, or Classical, Form of the Problem
Weak, or Variational, Form of the Problem
Eqivalence of Strong and Weak Forms; Natural Boundary Conditions
Galerkin's Approximation Method
Matrix Equations; Stiffness Matrix K
Examples: 1 and 2 Degrees of Freedom
Piecewise Linear Finite Element Space
Properties of K
Mathematical Analysis
Interlude: Gauss Elimination; Hand-calculation Version
The Element Point of View
Element Stiffness Matrix and Force Vector
Assembly of Global Stiffness Matrix and Force Vector; LM Array
Explicit Computation of Element Stiffness Matrix and Force Vector
Exercise: Bernoulli-Euler Beam Theory and Hermite Cubics
An Elementary Discussion of Continuity, Differentiability, and Smoothness
References
Formulation of Two- and Three-Dimensional Boundary-Value Problems
Introductory Remarks
Preliminaries
Classical Linear Heat Conduction: Strong and Weak Forms; Equivalence
Heat Conduction: Galerkin Formulation; Symmetry and Positive-definiteness of K
Heat Conduction: Element Stiffness Matrix and Force Vector
Heat Conduction: Data Processing Arrays ID, IEN, and LM
Classical Linear Elastostatics: Strong and Weak Forms; Equivalence
Elastostatics: Galerkin Formulation, Symmetry, and Positive-definiteness of K
Elastostatics: Element Stiffness Matrix and Force Vector
Elastostatics: Data Processing Arrays ID, IEN, and LM
Summary of Important Equations for Problems Considered in Chapters 1 and 2
Axisymmetric Formulations and Additional Exercises
References
Isoparametric Elements and Elementary Programming Concepts
Preliminary Concepts
Bilinear Quadrilateral Element
Isoparametric Elements
Linear Triangular Element; An Example of "Degeneration"
Trilinear Hexahedral Element
Higher-order Elements; Lagrange Polynomials
Elements with Variable Numbers of Nodes
Numerical Integration; Gaussian Quadrature
Derivatives of Shape Functions and Shape Function Subroutines
Element Stiffness Formulation
Additional Exercises
Triangular and Tetrahedral Elements
Methodology for Developing Special Shape Functions with Application to Singularities
References
Mixed and Penalty Methods, Reduced and Selective Integration, and Sundry Variational Crimes
"Best Approximation" and Error Estimates: Why the standard FEM usually works and why sometimes it does not
Incompressible Elasticity and Stokes Flow
Prelude to Mixed and Penalty Methods
A Mixed Formulation of Compressible Elasticity Capable of Representing the Incompressible Limit
Strong Form
Weak Form
Galerkin Formulation
Matrix Problem
Definition of Element Arrays
Illustration of a Fundamental Difficulty
Constraint Counts
Discontinuous Pressure Elements
Continuous Pressure Elements
Penalty Formulation: Reduced and Selective Integration Techniques; Equivalence with Mixed Methods
Pressure Smoothing
An Extension of Reduced and Selective Integration Techniques
Axisymmetry and Anisotropy: Prelude to Nonlinear Analysis
Strain Projection: The B-approach
The Patch Test; Rank Deficiency
Nonconforming Elements
Hourglass Stiffness
Additional Exercises and Projects
Mathematical Preliminaries
Basic Properties of Linear Spaces
Sobolev Norms
Approximation Properties of Finite Element Spaces in Sobolev Norms
Hypotheses on a(.,.)
Advanced Topics in the Theory of Mixed and Penalty Methods: Pressure Modes and Error Estimates
Pressure Modes, Spurious and Otherwise
Existence and Uniqueness of Solutions in the Presence of Modes
Two Sides of Pressure Modes
Pressure Modes in the Penalty Formulation
The Big Picture
Error Estimates and Pressure Smoothing
References
The C[superscript 0]-Approach to Plates and Beams
Introduction
Reissner-Mindlin Plate Theory
Main Assumptions
Constitutive Equation
Strain-displacement Equations
Summary of Plate Theory Notations
Variational Equation
Strong Form
Weak Form
Matrix Formulation
Finite Element Stiffness Matrix and Load Vector
Plate-bending Elements
Some Convergence Criteria
Shear Constraints and Locking
Boundary Conditions
Reduced and Selective Integration Lagrange Plate Elements
Equivalence with Mixed Methods
Rank Deficiency
The Heterosis Element
T1: A Correct-rank, Four-node Bilinear Element
The Linear Triangle
The Discrete Kirchhoff Approach
Discussion of Some Quadrilateral Bending Elements
Beams and Frames
Main Assumptions
Constitutive Equation
Strain-displacement Equations
Definitions of Quantities Appearing in the Theory
Variational Equation
Strong Form
Weak Form
Matrix Formulation of the Variational Equation
Finite Element Stiffness Matrix and Load Vector
Representation of Stiffness and Load in Global Coordinates
Reduced Integration Beam Elements
References
The C[superscript 0]-Approach to Curved Structural Elements
Introduction
Doubly Curved Shells in Three Dimensions
Geometry
Lamina Coordinate Systems
Fiber Coordinate Systems
Kinematics
Reduced Constitutive Equation
Strain-displacement Matrix
Stiffness Matrix
External Force Vector
Fiber Numerical Integration
Stress Resultants
Shell Elements
Some References to the Recent Literature
Simplifications: Shells as an Assembly of Flat Elements
Shells of Revolution; Rings and Tubes in Two Dimensions
Geometric and Kinematic Descriptions
Reduced Constitutive Equations
Strain-displacement Matrix
Stiffness Matrix
External Force Vector
Stress Resultants
Boundary Conditions
Shell Elements
References
Linear Dynamic Analysis
Formulation of Parabolic, Hyperbolic, and Elliptic-Elgenvalue Problems
Parabolic Case: Heat Equation
Hyperbolic Case: Elastodynamics and Structural Dynamics
Eigenvalue Problems: Frequency Analysis and Buckling
Standard Error Estimates
Alternative Definitions of the Mass Matrix; Lumped and Higher-order Mass
Estimation of Eigenvalues
Error Estimates for Semidiscrete Galerkin Approximations
References
Algorithms for Parabolic Problems
One-step Algorithms for the Semidiscrete Heat Equation: Generalized Trapezoidal Method
Analysis of the Generalized Trapezoidal Method
Modal Reduction to SDOF Form
Stability
Convergence
An Alternative Approach to Stability: The Energy Method
Additional Exercises
Elementary Finite Difference Equations for the One-dimensional Heat Equation; the von Neumann Method of Stability Analysis
Element-by-element (EBE) Implicit Methods
Modal Analysis
References
Algorithms for Hyperbolic and Parabolic-Hyperbolic Problems
One-step Algorithms for the Semidiscrete Equation of Motion
The Newmark Method
Analysis
Measures of Accuracy: Numerical Dissipation and Dispersion
Matched Methods
Additional Exercises
Summary of Time-step Estimates for Some Simple Finite Elements
Linear Multistep (LMS) Methods
LMS Methods for First-order Equations
LMS Methods for Second-order Equations
Survey of Some Commonly Used Algorithms in Structural Dynamics
Some Recently Developed Algorithms for Structural Dynamics
Algorithms Based upon Operator Splitting and Mesh Partitions
Stability via the Energy Method
Predictor/Multicorrector Algorithms
Mass Matrices for Shell Elements
References
Solution Techniques for Eigenvalue Problems
The Generalized Eigenproblem
Static Condensation
Discrete Rayleigh-Ritz Reduction
Irons-Guyan Reduction
Subspace Iteration
Spectrum Slicing
Inverse Iteration
The Lanczos Algorithm for Solution of Large Generalized Eigenproblems
Introduction
Spectral Transformation
Conditions for Real Eigenvalues
The Rayleigh-Ritz Approximation
Derivation of the Lanczos Algorithm
Reduction to Tridiagonal Form
Convergence Criterion for Eigenvalues
Loss of Orthogonality
Restoring Orthogonality
References
Dlearn--A Linear Static and Dynamic Finite Element Analysis Program
Introduction
Description of Coding Techniques Used in DLEARN
Compacted Column Storage Scheme
Crout Elimination
Dynamic Storage Allocation
Program Structure
Global Control
Initialization Phase
Solution Phase
Adding an Element to DLEARN
DLEARN User's Manual
Remarks for the New User
Input Instructions
Examples
Planar Truss
Static Analysis of a Plane Strain Cantilever Beam
Dynamic Analysis of a Plane Strain Cantilever Beam
Implicit-explicit Dynamic Analysis of a Rod
Subroutine Index for Program Listing
References
Index

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