A First Course in Finite Elements

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1 A First Course in Finite Elements A First Course in Finite Elements J. Fish and T. Belytschko # 2007 John Wiley & Sons, Ltd. ISBN:

2 A First Course in Finite Elements Jacob Fish Rensselaer Polytechnic Institute, USA Ted Belytschko Northwestern University, USA John Wiley & Sons, Ltd

3 Copyright ß 2007 John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England Telephone (þ44) (for orders and customer service enquiries): Visit our Home Page on 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 ed to permreq@wiley.co.uk, or faxed to (þ44) 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 , USA Wiley-VCH Verlag GmbH, Boschstr. 12, D 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 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 (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.

4 Contents Preface xi 1 Introduction Background Applications of Finite elements 7 References 9 2 Direct Approach for Discrete Systems Describing the Behavior of a Single Bar Element Equations for a System Equations for Assembly Boundary Conditions and System Solution Applications to Other Linear Systems Two-Dimensional Truss Systems Transformation Law Three-Dimensional Truss Systems 35 References 36 Problems 37 3 Strong and Weak Forms for One-Dimensional Problems The Strong Form in One-Dimensional Problems The Strong Form for an Axially Loaded Elastic Bar The Strong Form for Heat Conduction in One Dimension Diffusion in One Dimension The Weak Form in One Dimension Continuity The Equivalence Between the Weak and Strong Forms One-Dimensional Stress Analysis with Arbitrary Boundary Conditions Strong Form for One-Dimensional Stress Analysis Weak Form for One-Dimensional Stress Analysis 59

5 vi CONTENTS 3.6 One-Dimensional Heat Conduction with Arbitrary Boundary Conditions Strong Form for Heat Conduction in One Dimension with Arbitrary Boundary Conditions Weak Form for Heat Conduction in One Dimension with Arbitrary Boundary Conditions Two-Point Boundary Value Problem with Generalized Boundary Conditions Strong Form for Two-Point Boundary Value Problems with Generalized Boundary Conditions Weak Form for Two-Point Boundary Value Problems with Generalized Boundary Conditions Advection Diffusion Strong Form of Advection Diffusion Equation Weak Form of Advection Diffusion Equation Minimum Potential Energy Integrability 71 References 72 Problems 72 4 Approximation of Trial Solutions, Weight Functions and Gauss Quadrature for One-Dimensional Problems Two-Node Linear Element Quadratic One-Dimensional Element Direct Construction of Shape Functions in One Dimension Approximation of the Weight Functions Global Approximation and Continuity Gauss Quadrature 85 Reference 90 Problems 90 5 Finite Element Formulation for One-Dimensional Problems Development of Discrete Equation: Simple Case Element Matrices for Two-Node Element Application to Heat Conduction and Diffusion Problems Development of Discrete Equations for Arbitrary Boundary Conditions Two-Point Boundary Value Problem with Generalized Boundary Conditions Convergence of the FEM Convergence by Numerical Experiments Convergence by Analysis FEM for Advection Diffusion Equation 120 References 122 Problems 123

6 CONTENTS vii 6 Strong and Weak Forms for Multidimensional Scalar Field Problems Divergence Theorem and Green s Formula Strong Form Weak Form The Equivalence Between Weak and Strong Forms Generalization to Three-Dimensional Problems Strong and Weak Forms of Scalar Steady-State Advection Diffusion in Two Dimensions 146 References 148 Problems Approximations of Trial Solutions, Weight Functions and Gauss Quadrature for Multidimensional Problems Completeness and Continuity Three-Node Triangular Element Global Approximation and Continuity Higher Order Triangular Elements Derivatives of Shape Functions for the Three-Node Triangular Element Four-Node Rectangular Elements Four-Node Quadrilateral Element Continuity of Isoparametric Elements Derivatives of Isoparametric Shape Functions Higher Order Quadrilateral Elements Triangular Coordinates Linear Triangular Element Isoparametric Triangular Elements Cubic Element Triangular Elements by Collapsing Quadrilateral Elements Completeness of Isoparametric Elements Gauss Quadrature in Two Dimensions Integration Over Quadrilateral Elements Integration Over Triangular Elements Three-Dimensional Elements Hexahedral Elements Tetrahedral Elements 183 References 185 Problems Finite Element Formulation for Multidimensional Scalar Field Problems Finite Element Formulation for Two-Dimensional Heat Conduction Problems Verification and Validation 201

7 viii CONTENTS 8.3 Advection Diffusion Equation 207 References 209 Problems Finite Element Formulation for Vector Field Problems Linear Elasticity Linear Elasticity Kinematics Stress and Traction Equilibrium Constitutive Equation Strong and Weak Forms Finite Element Discretization Three-Node Triangular Element Element Body Force Matrix Boundary Force Matrix Generalization of Boundary Conditions Discussion Linear Elasticity Equations in Three Dimensions 240 Problems Finite Element Formulation for Beams Governing Equations of the Beam Kinematics of Beam Stress Strain Law Equilibrium Boundary Conditions Strong Form to Weak Form Weak Form to Strong Form Finite Element Discretization Trial Solution and Weight Function Approximations Discrete Equations Theorem of Minimum Potential Energy Remarks on Shell Elements 265 Reference 269 Problems Commercial Finite Element Program ABAQUS Tutorials Introduction Steady-State Heat Flow Example Preliminaries Creating a Part Creating a Material Definition Defining and Assigning Section Properties Assembling the Model Configuring the Analysis Applying a Boundary Condition and a Load to the Model Meshing the Model 282

8 CONTENTS ix Creating and Submitting an Analysis Job Viewing the Analysis Results Solving the Problem Using Quadrilaterals Refining the Mesh Bending of a Short Cantilever Beam Copying the Model Modifying the Material Definition Configuring the Analysis Applying a Boundary Condition and a Load to the Model Meshing the Model Creating and Submitting an Analysis Job Viewing the Analysis Results Plate with a Hole in Tension Creating a New Model Creating a Part Creating a Material Definition Defining and Assigning Section Properties Assembling the Model Configuring the Analysis Applying a Boundary Condition and a Load to the Model Meshing the Model Creating and Submitting an Analysis Job Viewing the Analysis Results Refining the Mesh 299 Appendix 303 A.1 Rotation of Coordinate System in Three Dimensions 303 A.2 Scalar Product Theorem 304 A.3 Taylor s Formula with Remainder and the Mean Value Theorem 304 A.4 Green s Theorem 305 A.5 Point Force (Source) 307 A.6 Static Condensation 308 A.7 Solution Methods 309 Direct Solvers 310 Iterative Solvers 310 Conditioning 311 References 312 Problem 312 Index 313

9 Preface This book is written to be an undergraduate and introductory graduate level textbook, depending on whether the more advanced topics appearing at the end of each chapter are covered. Without the advanced topics, the book is of a level readily comprehensible by junior and senior undergraduate students in science and engineering. With the advanced topics included, the book can serve as the textbook for the first course in finite elements at the graduate level. The text material evolved from over 50 years of combined teaching experience by the authors of graduate and undergraduate finite element courses. The book focuses on the formulation and application of the finite element method. It differs from other elementary finite element textbooks in the following three aspects: 1. It is introductory and self-contained. Only a modest background in mathematics and physics is needed, all of which is covered in engineering and science curricula in the first two years. Furthermore, many of the specific topics in mathematics, such as matrix algebra, some topics in differential equations, and mechanics and physics, such as conservation laws and constitutive equations, are reviewed prior to their application. 2. It is generic. While most introductory finite element textbooks are application specific, e.g. focusing on linear elasticity, the finite element method in this book is formulated as a general purpose numerical procedure for solving engineering problems governed by partial differential equations. The methodology for obtaining weak forms for the governing equations, a crucial step in the development and understanding of finite elements, is carefully developed. Consequently, students from various engineering and science disciplines will benefit equally from the exposition of the subject. 3. It is a hands-on experience. The book integrates finite element theory, finite element code development and the application of commercial software package. Finite element code development is introduced through MATLAB exercises and a MATLAB program, whereas ABAQUS is used for demonstrating the use of commercial finite element software. The material in the book can be covered in a single semester and a meaningful course can be constructed from a subset of the chapters in this book for a one-quarter course. The course material is organized in three chronological units of about one month each: (1) finite elements for one-dimensional problems; (2) finite elements for scalar field problems in two dimensions and (3) finite elements for vector field problems in two dimensions and beams. In each case, the weak form is developed, shape functions are described and these ingredients are synthesized to obtain the finite element equations. Moreover, in a web-base chapter, the application of general purpose finite element software using ABAQUS is given for linear heat conduction and elasticity. Each chapter contains a comprehensive set of homework problems, some of which require programming with MATLAB. Each book comes with an accompanying ABAQUS Student Edition CD, and

10 xii PREFACE MATLAB finite element programs can be downloaded from the accompanying website hosted by John Wiley & Sons: A tutorial for the ABAQUS example problems, written by ABAQUS staff, is also included in the book. Depending on the interests and background of the students, three tracks have been developed: 1. Broad Science and Engineering (SciEng) track 2. Advanced (Advanced) track 3. Structural Mechanics (StrucMech) track The SciEng track is intended for a broad audience of students in science and engineering. It is aimed at presenting FEM as a versatile tool for solving engineering design problems and as a tool for scientific discovery. Students who have successfully completed this track should be able to appreciate and apply the finite element method for the types of problems described in the book, but more importantly, the SciEng track equips them with a set of skills that will allow them to understand and develop the method for a variety of problems that have not been explicitly addressed in the book. This is our recommended track. The Advanced track is intended for graduate students as well as undergraduate students with a strong focus on applied mathematics, who are less concerned with specialized applications, such as beams and trusses, but rather with a more detailed exposition of the method. Although detailed convergence proofs in multidimensions are left out, the Advanced track is an excellent stepping stone for students interested in a comprehensive mathematical analysis of the method. The StrucMech track is intended for students in Civil, Mechanical and Aerospace Engineering whose main interests are in structural and solid mechanics. Specialized topics, such as trusses, beams and energybased principles, are emphasized in this track, while sections dealing with topics other than solid mechanics in multidimensions are classified as optional. The Table P1 gives recommended course outlines for the three tracks. The three columns on the right list are the recommended sections for each track. Table P1 Suggested outlines for Science and Engineering (SciEng) track, Advanced Track and Structural Mechanics (StrucMech) Track. Outline SciEng Advanced StrucMech Part 1: Finite element formulation for one-dimensional problems Chapter 1: Introduction All All All Chapter 2: Direct approach for discrete systems , 2.2, 2.4 Chapter 3: Strong and weak forms for All 3.1.1, , 3.9 one-dimensional problems Chapter 4: Approximation of trial solutions, All All All weight functions and Gauss quadrature for one-dimensional problems Chapter 5: Finite element formulation for , 5.6, All 5.1, 5.2, 5.4, 5.6, one-dimensional problems Part 2: Finite element formulation for scalar field problems in multidimensions Chapter 6: Strong and weak forms for All 6, 6.1 multi-dimensional scalar field problems Chapter 7: Approximation of trial solutions, , All , weight functions and Gauss quadrature for multi-dimensional problems Chapter 8: Finite element formulation for multi 8.1, 8.2 All dimensional scalar field problems

11 PREFACE xiii Table P1 (Continued) Outline SciEng Advanced StrucMech Part 3: Finite element formulation for vector field problems in two dimensions Chapter 9: Finite element formulation for vector All field problems linear elasticity Chapter 10: Finite element formulation for beams Chapter 11: Commercial finite element program All All All ABAQUS tutorial Chapter 12: Finite Element Programming with , , MATLAB (on the web only) A BRIEF GLOSSARY OF NOTATION Scalars, Vectors, Matrices a, B Scalars a, B Matrices ~a; ~B Vectors a i ; B ij Matrix or vector components Integers n np n el n gp n en e IJ Number of nodal points Number of element Number of Gauss points Number of element nodes Element number Kronecker delta Sets 8 For all [ Union \ Intersection 2 Member of Subset of Spaces, Continuity U Space of trial solutions U 0 Space of weight functions C n Functions whose j th derivatives 0 j n are continuous H s A space of functions with s square-integrable derivatives Strong Forms-General Problem domain Boundary of domain n ¼ðn x ; n y Þ Unit normal to ðn ¼1 in1dþ ðx; yþ Physical coordinates (x in 1D) =; = S Gradient and symmetric gradient matrices ~r Gradient vector Strong Form-Heat Conduction T Temperature q ¼ðq x ; q y Þ T Flux (q in 1D) T Essential boundary q Natural boundary s Heat source q; T Boundary flux and temperature D Conductivity matrix k xx ; k yy ; k xy Conductivities (k in 1D) Strong Form-Elasticity u ¼ðu x ; u y Þ T Displacements (u in 1D) ~s x ;~s y stress vectors acting on the planes normal to x and y directions e; r Strain and stress matrices (e and s in 1D) s Stress tensor e xx ; e yy ; g xy Strain components s xx ; s yy ; s xy Stress components b ¼ðb x ; b y Þ T Body forces (b in 1D) t ¼ðt x ; t y Þ T Tractions E; Young s modulus and Poisson s ratio. D Material moduli matrix t ¼ðt x ;t y Þ T Prescribed traction (t in 1D) u ¼ðu x ; u y Þ T Prescribed displacements (u in 1D) u ; t Essential (displacement) and natural (traction) boundary

12 xiv PREFACE Strong Form - Beams u M x ðxþ Displacement in x at midline m(x) Internal moment s(x) Internal shear force p(x) Distributed loading I Moment of inertia Curvature u y Vertical displacements, Rotations m; s Prescribed moments and shear forces u y ; Prescribed vertical displacements and rotations m ; s Natural boundary: moments and shear u ; Essential boundary: vertical displacements and rotations Finite Elements-General e Domain of element e (l e in 1D) A e Area of element e (cross-sectional area in 1D) x e I ; ye I Coordinates of node I in element e N e ; N Element and global shape function matrices B e ; B Element and global shape function derivative matrices L e Gather matrix L et Scatter matrix J e Jacobian matrix e ; h Element and global trial solutions w e ; w h Element and global weight functions W i Gauss quadrature weights ; ; I Parent/natural coordinate xð; Þ x - coordinate mapping yð; Þ y - coordinate mapping K E ; K F ; K EF Partition into E- and F- nodes w Global weight functions matrix R e Rotation matrix from element to global coordinate system Finite Elements-Heat Conduction T e Finite element temperature d; d e Global and element temperature matrices K; K e Global and element conductance matrices f ; f e Global and element boundary flux matrices f ; f e Global and element source matrices r Global residual matrix f Global flux matrix Finite Elements-Elasticity u e Finite element displacements u e xi ; ue yi Displacements at element node I in x and y directions, respectively d; d e Global and element displacement matrix K; K e Global and element stiffness matrices f ; f e Global and element boundary force matrix f ; f e Global and element body force matrices f; f e Global and element force matrix r Global reaction force matrix Finite Elements-Beams u e y Finite element vertical displacements d e Element displacement matrix ½u y1 ; 1 ; u y2 ; 2 Š T K; K e Global and element stiffness matrices f ; f e Global and element boundary force matrices f ; f e Global and element body force matrices f; f e Global and element force matrices r Global reaction force matrix

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