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A First Course in Finite Elements.pdf

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    A First Course in Finite Elements
    Contents
    Preface
    1 Introduction
    1.1 Background
    1.2 Applications of Finite elements
    References
    2 Direct Approach for Discrete Systems
    2.1 Describing the Behavior of a Single Bar Element
    2.2 Equations for a System
    2.2.1 Equations for Assembly
    2.2.2 Boundary Conditions and System Solution
    2.3 Applications to Other Linear Systems
    2.4 Two-Dimensional Truss Systems
    2.5 Transformation Law
    2.6 Three-Dimensional Truss Systems
    References
    Problems
    3 Strong and Weak Forms for One-Dimensional Problems
    3.1 The Strong Form in One-Dimensional Problems
    3.1.1 The Strong Form for an Axially Loaded Elastic Bar
    3.1.2 The Strong Form for Heat Conduction in One Dimension
    3.1.3 Diffusion in One Dimension
    3.2 The Weak Form in One Dimension
    3.3 Continuity
    3.4 The Equivalence Between the Weak and Strong Forms
    3.5 One-Dimensional Stress Analysis with Arbitrary Boundary Conditions
    3.5.1 Strong Form for One-Dimensional Stress Analysis
    3.5.2 Weak Form for One-Dimensional Stress Analysis
    3.6 One-Dimensional Heat Conduction with Arbitrary Boundary Conditions
    3.6.1 Strong Form for Heat Conduction in One Dimension with Arbitrary Boundary Conditions
    3.6.2 Weak Form for Heat Conduction in One Dimension with Arbitrary Boundary Conditions
    3.7 Two-Point Boundary Value Problem with Generalized Boundary Conditions
    3.7.1 Strong Form for Two-Point Boundary Value Problems with Generalized Boundary Conditions
    3.7.2 Weak Form for Two-Point Boundary Value Problems with Generalized Boundary Conditions
    3.8 Advection–Diffusion
    3.8.1 Strong Form of Advection–Diffusion Equation
    3.8.2 Weak Form of Advection–Diffusion Equation
    3.9 Minimum Potential Energy
    3.10 Integrability
    References
    Problems
    4 Approximation of Trial Solutions, Weight Functions and Gauss Quadrature for One-Dimensional Problems
    4.1 Two-Node Linear Element
    4.2 Quadratic One-Dimensional Element
    4.3 Direct Construction of Shape Functions in One Dimension
    4.4 Approximation of the Weight Functions
    4.5 Global Approximation and Continuity
    4.6 Gauss Quadrature
    Reference
    Problems
    5 Finite Element Formulation for One-Dimensional Problems
    5.1 Development of Discrete Equation: Simple Case
    5.2 Element Matrices for Two-Node Element
    5.3 Application to Heat Conduction and Diffusion Problems
    5.4 Development of Discrete Equations for Arbitrary Boundary Conditions
    5.5 Two-Point Boundary Value Problem with Generalized Boundary Conditions
    5.6 Convergence of the FEM
    5.6.1 Convergence by Numerical Experiments
    5.6.2 Convergence by Analysis
    5.7 FEM for Advection–Diffusion Equation
    References
    Problems
    6 Strong and Weak Forms for Multidimensional Scalar Field Problems
    6.1 Divergence Theorem and Green’s Formula
    6.2 Strong Form
    6.3 Weak Form
    6.4 The Equivalence Between Weak and Strong Forms
    6.5 Generalization to Three-Dimensional Problems
    6.6 Strong and Weak Forms of Scalar Steady-State Advection–Diffusion in Two Dimensions
    References
    Problems
    7 Approximations of Trial Solutions, Weight Functions and Gauss Quadrature for Multidimensional Problems
    7.1 Completeness and Continuity
    7.2 Three-Node Triangular Element
    7.2.1 Global Approximation and Continuity
    7.2.2 Higher Order Triangular Elements
    7.2.3 Derivatives of Shape Functions for the Three-Node Triangular Element
    7.3 Four-Node Rectangular Elements
    7.4 Four-Node Quadrilateral Element
    7.4.1 Continuity of Isoparametric Elements
    7.4.2 Derivatives of Isoparametric Shape Functions
    7.5 Higher Order Quadrilateral Elements
    7.6 Triangular Coordinates
    7.6.1 Linear Triangular Element
    7.6.2 Isoparametric Triangular Elements
    7.6.3 Cubic Element
    7.6.4 Triangular Elements by Collapsing Quadrilateral Elements
    7.7 Completeness of Isoparametric Elements
    7.8 Gauss Quadrature in Two Dimensions
    7.8.1 Integration Over Quadrilateral Elements
    7.8.2 Integration Over Triangular Elements
    7.9 Three-Dimensional Elements
    7.9.1 Hexahedral Elements
    7.9.2 Tetrahedral Elements
    References
    Problems
    8 Finite Element Formulation for Multidimensional Scalar Field Problems
    8.1 Finite Element Formulation for Two-Dimensional Heat Conduction Problems
    8.2 Verification and Validation
    8.3 Advection–Diffusion Equation
    References
    Problems
    9 Finite Element Formulation for Vector Field Problems – Linear Elasticity
    9.1 Linear Elasticity
    9.1.1 Kinematics
    9.1.2 Stress and Traction
    9.1.3 Equilibrium
    9.1.4 Constitutive Equation
    9.2 Strong and Weak Forms
    9.3 Finite Element Discretization
    9.4 Three-Node Triangular Element
    9.4.1 Element Body Force Matrix
    9.4.2 Boundary Force Matrix
    9.5 Generalization of Boundary Conditions
    9.6 Discussion
    9.7 Linear Elasticity Equations in Three Dimensions
    Problems
    10 Finite Element Formulation for Beams
    10.1 Governing Equations of the Beam
    10.1.1 Kinematics of Beam
    10.1.2 Stress–Strain Law
    10.1.3 Equilibrium
    10.1.4 Boundary Conditions
    10.2 Strong Form to Weak Form
    10.2.1 Weak Form to Strong Form
    10.3 Finite Element Discretization
    10.3.1 Trial Solution and Weight Function Approximations
    10.3.2 Discrete Equations
    10.4 Theorem of Minimum Potential Energy
    10.5 Remarks on Shell Elements
    Reference
    Problems
    11 Commercial Finite Element Program ABAQUS Tutorials
    11.1 Introduction
    11.1.1 Steady-State Heat Flow Example
    11.2 Preliminaries
    11.3 Creating a Part
    11.4 Creating a Material Definition
    11.5 Defining and Assigning Section Properties
    11.6 Assembling the Model
    11.7 Configuring the Analysis
    11.8 Applying a Boundary Condition and a Load to the Model
    11.9 Meshing the Model
    11.10 Creating and Submitting an Analysis Job
    11.11 Viewing the Analysis Results
    11.12 Solving the Problem Using Quadrilaterals
    11.13 Refining the Mesh
    11.13.1 Bending of a Short Cantilever Beam
    11.14 Copying the Model
    11.15 Modifying the Material Definition
    11.16 Configuring the Analysis
    11.17 Applying a Boundary Condition and a Load to the Model
    11.18 Meshing the Model
    11.19 Creating and Submitting an Analysis Job
    11.20 Viewing the Analysis Results
    11.20.1 Plate with a Hole in Tension
    11.21 Creating a New Model
    11.22 Creating a Part
    11.23 Creating a Material Definition
    11.24 Defining and Assigning Section Properties
    11.25 Assembling the Model
    11.26 Configuring the Analysis
    11.27 Applying a Boundary Condition and a Load to the Model
    11.28 Meshing the Model
    11.29 Creating and Submitting an Analysis Job
    11.30 Viewing the Analysis Results
    11.31 Refining the Mesh
    Appendix
    A.1 Rotation of Coordinate System in Three Dimensions
    A.2 Scalar Product Theorem
    A.3 Taylor’s Formula with Remainder and the Mean Value Theorem
    A.4 Green’s Theorem
    A.5 Point Force (Source)
    A.6 Static Condensation
    A.7 Solution Methods
    Direct Solvers
    Iterative Solvers
    Conditioning
    References
    Problem
    Index
    Colour Section
    A First Course in Finite Elements Jacob Fish Rensselaer Polytechnic Institute, USA Ted Belytschko Northwestern University, USA John Wiley & Sons, Ltd
    A First Course in Finite Elements
    A First Course in Finite Elements Jacob Fish Rensselaer Polytechnic Institute, USA Ted Belytschko Northwestern University, USA John Wiley & Sons, Ltd
    Copyright ß 2007 John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England Telephone (þ44) 1243 779777 Email (for orders and customer service enquiries): cs-books@wiley.co.uk Visit our Home Page on www.wiley.com All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except under the terms of the Copyright, Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency Ltd, 90 Tottenham Court Road, London W1T 4LP, UK, without the permission in writing of the Publisher. Requests to the Publisher should be addressed to the Permissions Department, John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England, or emailed to permreq@wiley.co.uk, or faxed to (þ44) 1243 770620. Designations used by companies to distinguish their products are often claimed as trademarks. All brand names and product names used in this book are trade names, service marks, trademarks or registered trademarks of their respective owners. The Publisher is not associated with any product or vendor mentioned in this book. This publication is designed to provide accurate and authoritative information in regard to the subject matter covered. It is sold on the understanding that the Publisher is not engaged in rendering professional services. If professional advice or other expert assistance is required, the services of a competent professional should be sought. Other Wiley Editorial Offices John Wiley & Sons Inc., 111 River Street, Hoboken, NJ 07030, USA Jossey-Bass, 989 Market Street, San Francisco, CA 94103-1741, USA Wiley-VCH Verlag GmbH, Boschstr. 12, D-69469 Weinheim, Germany John Wiley & Sons Australia Ltd, 42 McDougall Street, Milton, Queensland 4064, Australia John Wiley & Sons (Asia) Pte Ltd, 2 Clementi Loop #02-01, Jin Xing Distripark, Singapore 129809 John Wiley & Sons Canada Ltd, 6045 Freemont Blvd, Mississauga, ONT, Canada L5R 4J3 Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN 978-0-470-03580-1 (PB) Typeset in 9/11 pt in Thomson Digital Printed and bound in Great Britain by Antony Rowe Ltd, Chippenham, Wiltshire This book is printed on acid-free paper responsibly manufactured from sustainable forestry in which at least two trees are planted for each one used for paper production.
    Contents Preface 1 Introduction 1.1 Background 1.2 Applications of Finite elements References 2 Direct Approach for Discrete Systems 2.1 Describing the Behavior of a Single Bar Element 2.2 Equations for a System 2.2.1 Equations for Assembly 2.2.2 Boundary Conditions and System Solution 2.3 Applications to Other Linear Systems 2.4 Two-Dimensional Truss Systems 2.5 Transformation Law 2.6 Three-Dimensional Truss Systems References Problems 3 Strong and Weak Forms for One-Dimensional Problems 3.1 The Strong Form in One-Dimensional Problems 3.1.1 The Strong Form for an Axially Loaded Elastic Bar 3.1.2 The Strong Form for Heat Conduction in One Dimension 3.1.3 Diffusion in One Dimension 3.2 The Weak Form in One Dimension 3.3 Continuity 3.4 The Equivalence Between the Weak and Strong Forms 3.5 One-Dimensional Stress Analysis with Arbitrary Boundary Conditions 3.5.1 Strong Form for One-Dimensional Stress Analysis 3.5.2 Weak Form for One-Dimensional Stress Analysis xi 1 1 7 9 11 11 15 18 20 24 27 30 35 36 37 41 42 42 44 46 47 50 51 58 58 59
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