Severity: 8192
Message: Creation of dynamic property CI_URI::$config is deprecated
Filename: core/URI.php
Line Number: 102
Backtrace:
File: /home/glencovo/public_html/index.php
Line: 315
Function: require_once
Severity: 8192
Message: Creation of dynamic property CI_Router::$uri is deprecated
Filename: core/Router.php
Line Number: 128
Backtrace:
File: /home/glencovo/public_html/index.php
Line: 315
Function: require_once
Severity: 8192
Message: Creation of dynamic property Main::$benchmark is deprecated
Filename: core/Controller.php
Line Number: 83
Backtrace:
File: /home/glencovo/public_html/application/controllers/Main.php
Line: 6
Function: __construct
File: /home/glencovo/public_html/index.php
Line: 315
Function: require_once
Severity: 8192
Message: Creation of dynamic property Main::$hooks is deprecated
Filename: core/Controller.php
Line Number: 83
Backtrace:
File: /home/glencovo/public_html/application/controllers/Main.php
Line: 6
Function: __construct
File: /home/glencovo/public_html/index.php
Line: 315
Function: require_once
Severity: 8192
Message: Creation of dynamic property Main::$config is deprecated
Filename: core/Controller.php
Line Number: 83
Backtrace:
File: /home/glencovo/public_html/application/controllers/Main.php
Line: 6
Function: __construct
File: /home/glencovo/public_html/index.php
Line: 315
Function: require_once
Severity: 8192
Message: Creation of dynamic property Main::$log is deprecated
Filename: core/Controller.php
Line Number: 83
Backtrace:
File: /home/glencovo/public_html/application/controllers/Main.php
Line: 6
Function: __construct
File: /home/glencovo/public_html/index.php
Line: 315
Function: require_once
Severity: 8192
Message: Creation of dynamic property Main::$utf8 is deprecated
Filename: core/Controller.php
Line Number: 83
Backtrace:
File: /home/glencovo/public_html/application/controllers/Main.php
Line: 6
Function: __construct
File: /home/glencovo/public_html/index.php
Line: 315
Function: require_once
Severity: 8192
Message: Creation of dynamic property Main::$uri is deprecated
Filename: core/Controller.php
Line Number: 83
Backtrace:
File: /home/glencovo/public_html/application/controllers/Main.php
Line: 6
Function: __construct
File: /home/glencovo/public_html/index.php
Line: 315
Function: require_once
Severity: 8192
Message: Creation of dynamic property Main::$exceptions is deprecated
Filename: core/Controller.php
Line Number: 83
Backtrace:
File: /home/glencovo/public_html/application/controllers/Main.php
Line: 6
Function: __construct
File: /home/glencovo/public_html/index.php
Line: 315
Function: require_once
Severity: 8192
Message: Creation of dynamic property Main::$router is deprecated
Filename: core/Controller.php
Line Number: 83
Backtrace:
File: /home/glencovo/public_html/application/controllers/Main.php
Line: 6
Function: __construct
File: /home/glencovo/public_html/index.php
Line: 315
Function: require_once
Severity: 8192
Message: Creation of dynamic property Main::$output is deprecated
Filename: core/Controller.php
Line Number: 83
Backtrace:
File: /home/glencovo/public_html/application/controllers/Main.php
Line: 6
Function: __construct
File: /home/glencovo/public_html/index.php
Line: 315
Function: require_once
Severity: 8192
Message: Creation of dynamic property Main::$security is deprecated
Filename: core/Controller.php
Line Number: 83
Backtrace:
File: /home/glencovo/public_html/application/controllers/Main.php
Line: 6
Function: __construct
File: /home/glencovo/public_html/index.php
Line: 315
Function: require_once
Severity: 8192
Message: Creation of dynamic property Main::$input is deprecated
Filename: core/Controller.php
Line Number: 83
Backtrace:
File: /home/glencovo/public_html/application/controllers/Main.php
Line: 6
Function: __construct
File: /home/glencovo/public_html/index.php
Line: 315
Function: require_once
Severity: 8192
Message: Creation of dynamic property Main::$lang is deprecated
Filename: core/Controller.php
Line Number: 83
Backtrace:
File: /home/glencovo/public_html/application/controllers/Main.php
Line: 6
Function: __construct
File: /home/glencovo/public_html/index.php
Line: 315
Function: require_once
Severity: 8192
Message: Creation of dynamic property Main::$db is deprecated
Filename: core/Loader.php
Line Number: 397
Backtrace:
File: /home/glencovo/public_html/application/controllers/Main.php
Line: 6
Function: __construct
File: /home/glencovo/public_html/index.php
Line: 315
Function: require_once
Severity: 8192
Message: Creation of dynamic property CI_DB_mysqli_driver::$failover is deprecated
Filename: database/DB_driver.php
Line Number: 372
Backtrace:
File: /home/glencovo/public_html/application/controllers/Main.php
Line: 6
Function: __construct
File: /home/glencovo/public_html/index.php
Line: 315
Function: require_once
Severity: Warning
Message: ini_set(): Session ini settings cannot be changed after headers have already been sent
Filename: Session/Session.php
Line Number: 303
Backtrace:
File: /home/glencovo/public_html/application/controllers/Main.php
Line: 6
Function: __construct
File: /home/glencovo/public_html/index.php
Line: 315
Function: require_once
Severity: Warning
Message: session_set_cookie_params(): Session cookie parameters cannot be changed after headers have already been sent
Filename: Session/Session.php
Line Number: 328
Backtrace:
File: /home/glencovo/public_html/application/controllers/Main.php
Line: 6
Function: __construct
File: /home/glencovo/public_html/index.php
Line: 315
Function: require_once
Severity: Warning
Message: ini_set(): Session ini settings cannot be changed after headers have already been sent
Filename: Session/Session.php
Line Number: 355
Backtrace:
File: /home/glencovo/public_html/application/controllers/Main.php
Line: 6
Function: __construct
File: /home/glencovo/public_html/index.php
Line: 315
Function: require_once
Severity: Warning
Message: ini_set(): Session ini settings cannot be changed after headers have already been sent
Filename: Session/Session.php
Line Number: 365
Backtrace:
File: /home/glencovo/public_html/application/controllers/Main.php
Line: 6
Function: __construct
File: /home/glencovo/public_html/index.php
Line: 315
Function: require_once
Severity: Warning
Message: ini_set(): Session ini settings cannot be changed after headers have already been sent
Filename: Session/Session.php
Line Number: 366
Backtrace:
File: /home/glencovo/public_html/application/controllers/Main.php
Line: 6
Function: __construct
File: /home/glencovo/public_html/index.php
Line: 315
Function: require_once
Severity: Warning
Message: ini_set(): Session ini settings cannot be changed after headers have already been sent
Filename: Session/Session.php
Line Number: 367
Backtrace:
File: /home/glencovo/public_html/application/controllers/Main.php
Line: 6
Function: __construct
File: /home/glencovo/public_html/index.php
Line: 315
Function: require_once
Severity: Warning
Message: ini_set(): Session ini settings cannot be changed after headers have already been sent
Filename: Session/Session.php
Line Number: 368
Backtrace:
File: /home/glencovo/public_html/application/controllers/Main.php
Line: 6
Function: __construct
File: /home/glencovo/public_html/index.php
Line: 315
Function: require_once
Severity: Warning
Message: ini_set(): Session ini settings cannot be changed after headers have already been sent
Filename: Session/Session.php
Line Number: 426
Backtrace:
File: /home/glencovo/public_html/application/controllers/Main.php
Line: 6
Function: __construct
File: /home/glencovo/public_html/index.php
Line: 315
Function: require_once
Severity: Warning
Message: session_set_save_handler(): Session save handler cannot be changed after headers have already been sent
Filename: Session/Session.php
Line Number: 110
Backtrace:
File: /home/glencovo/public_html/application/controllers/Main.php
Line: 6
Function: __construct
File: /home/glencovo/public_html/index.php
Line: 315
Function: require_once
Severity: Warning
Message: session_start(): Session cannot be started after headers have already been sent
Filename: Session/Session.php
Line Number: 137
Backtrace:
File: /home/glencovo/public_html/application/controllers/Main.php
Line: 6
Function: __construct
File: /home/glencovo/public_html/index.php
Line: 315
Function: require_once
Severity: 8192
Message: Creation of dynamic property Main::$session is deprecated
Filename: core/Loader.php
Line Number: 1284
Backtrace:
File: /home/glencovo/public_html/application/controllers/Main.php
Line: 6
Function: __construct
File: /home/glencovo/public_html/index.php
Line: 315
Function: require_once
Severity: 8192
Message: Creation of dynamic property Main::$cart is deprecated
Filename: core/Loader.php
Line Number: 1284
Backtrace:
File: /home/glencovo/public_html/application/controllers/Main.php
Line: 6
Function: __construct
File: /home/glencovo/public_html/index.php
Line: 315
Function: require_once
Severity: 8192
Message: Creation of dynamic property Main::$ajax_pagination is deprecated
Filename: core/Loader.php
Line Number: 1284
Backtrace:
File: /home/glencovo/public_html/application/controllers/Main.php
Line: 10
Function: library
File: /home/glencovo/public_html/index.php
Line: 315
Function: require_once
Severity: 8192
Message: Creation of dynamic property Main::$email is deprecated
Filename: core/Loader.php
Line Number: 1284
Backtrace:
File: /home/glencovo/public_html/application/controllers/Main.php
Line: 11
Function: library
File: /home/glencovo/public_html/index.php
Line: 315
Function: require_once
Severity: 8192
Message: Creation of dynamic property Main::$main_model is deprecated
Filename: core/Loader.php
Line Number: 359
Backtrace:
File: /home/glencovo/public_html/application/controllers/Main.php
Line: 12
Function: model
File: /home/glencovo/public_html/index.php
Line: 315
Function: require_once
Severity: 8192
Message: Creation of dynamic property Main::$catalogue_model is deprecated
Filename: core/Loader.php
Line Number: 359
Backtrace:
File: /home/glencovo/public_html/application/controllers/Main.php
Line: 13
Function: model
File: /home/glencovo/public_html/index.php
Line: 315
Function: require_once
Severity: 8192
Message: Creation of dynamic property Main::$perPage is deprecated
Filename: controllers/Main.php
Line Number: 14
Backtrace:
File: /home/glencovo/public_html/application/controllers/Main.php
Line: 14
Function: _error_handler
File: /home/glencovo/public_html/index.php
Line: 315
Function: require_once
Severity: 8192
Message: Creation of dynamic property CI_Loader::$load is deprecated
Filename: core/Loader.php
Line Number: 932
Backtrace:
File: /home/glencovo/public_html/application/controllers/Main.php
Line: 360
Function: view
File: /home/glencovo/public_html/index.php
Line: 315
Function: require_once
Severity: 8192
Message: Creation of dynamic property CI_Loader::$benchmark is deprecated
Filename: core/Loader.php
Line Number: 932
Backtrace:
File: /home/glencovo/public_html/application/controllers/Main.php
Line: 360
Function: view
File: /home/glencovo/public_html/index.php
Line: 315
Function: require_once
Severity: 8192
Message: Creation of dynamic property CI_Loader::$hooks is deprecated
Filename: core/Loader.php
Line Number: 932
Backtrace:
File: /home/glencovo/public_html/application/controllers/Main.php
Line: 360
Function: view
File: /home/glencovo/public_html/index.php
Line: 315
Function: require_once
Severity: 8192
Message: Creation of dynamic property CI_Loader::$config is deprecated
Filename: core/Loader.php
Line Number: 932
Backtrace:
File: /home/glencovo/public_html/application/controllers/Main.php
Line: 360
Function: view
File: /home/glencovo/public_html/index.php
Line: 315
Function: require_once
Severity: 8192
Message: Creation of dynamic property CI_Loader::$log is deprecated
Filename: core/Loader.php
Line Number: 932
Backtrace:
File: /home/glencovo/public_html/application/controllers/Main.php
Line: 360
Function: view
File: /home/glencovo/public_html/index.php
Line: 315
Function: require_once
Severity: 8192
Message: Creation of dynamic property CI_Loader::$utf8 is deprecated
Filename: core/Loader.php
Line Number: 932
Backtrace:
File: /home/glencovo/public_html/application/controllers/Main.php
Line: 360
Function: view
File: /home/glencovo/public_html/index.php
Line: 315
Function: require_once
Severity: 8192
Message: Creation of dynamic property CI_Loader::$uri is deprecated
Filename: core/Loader.php
Line Number: 932
Backtrace:
File: /home/glencovo/public_html/application/controllers/Main.php
Line: 360
Function: view
File: /home/glencovo/public_html/index.php
Line: 315
Function: require_once
Severity: 8192
Message: Creation of dynamic property CI_Loader::$exceptions is deprecated
Filename: core/Loader.php
Line Number: 932
Backtrace:
File: /home/glencovo/public_html/application/controllers/Main.php
Line: 360
Function: view
File: /home/glencovo/public_html/index.php
Line: 315
Function: require_once
Severity: 8192
Message: Creation of dynamic property CI_Loader::$router is deprecated
Filename: core/Loader.php
Line Number: 932
Backtrace:
File: /home/glencovo/public_html/application/controllers/Main.php
Line: 360
Function: view
File: /home/glencovo/public_html/index.php
Line: 315
Function: require_once
Severity: 8192
Message: Creation of dynamic property CI_Loader::$output is deprecated
Filename: core/Loader.php
Line Number: 932
Backtrace:
File: /home/glencovo/public_html/application/controllers/Main.php
Line: 360
Function: view
File: /home/glencovo/public_html/index.php
Line: 315
Function: require_once
Severity: 8192
Message: Creation of dynamic property CI_Loader::$security is deprecated
Filename: core/Loader.php
Line Number: 932
Backtrace:
File: /home/glencovo/public_html/application/controllers/Main.php
Line: 360
Function: view
File: /home/glencovo/public_html/index.php
Line: 315
Function: require_once
Severity: 8192
Message: Creation of dynamic property CI_Loader::$input is deprecated
Filename: core/Loader.php
Line Number: 932
Backtrace:
File: /home/glencovo/public_html/application/controllers/Main.php
Line: 360
Function: view
File: /home/glencovo/public_html/index.php
Line: 315
Function: require_once
Severity: 8192
Message: Creation of dynamic property CI_Loader::$lang is deprecated
Filename: core/Loader.php
Line Number: 932
Backtrace:
File: /home/glencovo/public_html/application/controllers/Main.php
Line: 360
Function: view
File: /home/glencovo/public_html/index.php
Line: 315
Function: require_once
Severity: 8192
Message: Creation of dynamic property CI_Loader::$db is deprecated
Filename: core/Loader.php
Line Number: 932
Backtrace:
File: /home/glencovo/public_html/application/controllers/Main.php
Line: 360
Function: view
File: /home/glencovo/public_html/index.php
Line: 315
Function: require_once
Severity: 8192
Message: Creation of dynamic property CI_Loader::$session is deprecated
Filename: core/Loader.php
Line Number: 932
Backtrace:
File: /home/glencovo/public_html/application/controllers/Main.php
Line: 360
Function: view
File: /home/glencovo/public_html/index.php
Line: 315
Function: require_once
Severity: 8192
Message: Creation of dynamic property CI_Loader::$cart is deprecated
Filename: core/Loader.php
Line Number: 932
Backtrace:
File: /home/glencovo/public_html/application/controllers/Main.php
Line: 360
Function: view
File: /home/glencovo/public_html/index.php
Line: 315
Function: require_once
Severity: 8192
Message: Creation of dynamic property CI_Loader::$ajax_pagination is deprecated
Filename: core/Loader.php
Line Number: 932
Backtrace:
File: /home/glencovo/public_html/application/controllers/Main.php
Line: 360
Function: view
File: /home/glencovo/public_html/index.php
Line: 315
Function: require_once
Severity: 8192
Message: Creation of dynamic property CI_Loader::$email is deprecated
Filename: core/Loader.php
Line Number: 932
Backtrace:
File: /home/glencovo/public_html/application/controllers/Main.php
Line: 360
Function: view
File: /home/glencovo/public_html/index.php
Line: 315
Function: require_once
Severity: 8192
Message: Creation of dynamic property CI_Loader::$main_model is deprecated
Filename: core/Loader.php
Line Number: 932
Backtrace:
File: /home/glencovo/public_html/application/controllers/Main.php
Line: 360
Function: view
File: /home/glencovo/public_html/index.php
Line: 315
Function: require_once
Severity: 8192
Message: Creation of dynamic property CI_Loader::$catalogue_model is deprecated
Filename: core/Loader.php
Line Number: 932
Backtrace:
File: /home/glencovo/public_html/application/controllers/Main.php
Line: 360
Function: view
File: /home/glencovo/public_html/index.php
Line: 315
Function: require_once
Severity: 8192
Message: Creation of dynamic property CI_Loader::$perPage is deprecated
Filename: core/Loader.php
Line Number: 932
Backtrace:
File: /home/glencovo/public_html/application/controllers/Main.php
Line: 360
Function: view
File: /home/glencovo/public_html/index.php
Line: 315
Function: require_once
Preface ix
1. Overview of the Finite Element Method 1
1. Introduction 1
2. Historical Background 2
3. Basic concepts 4
4. Technical discussion 7
5. Various types of finite element methods 13
6. Link with the gradient discretization method 19
7. Comparison to the finite difference method 19
8. Numerical Implementation with Finite Element Method 21
9. Finite Element Analysis for a 1D Problem 23
10. Simulation Implementation in COMSOL Multiphysics 24
Epilogue 30
References 30
MCQs 32
Answers 33
2. Vectors, Matrices, and Tensors 34
1. Introduction 34
2. Vectors 35
3. Vector subtraction 42
4. Matrices 46
5. Operations with Matrices 49
6. Tensor 54
Epilogue 55
References 55
MCQs 58
Answers 58
3. Formulation of the Finite Element Method 59
1. Introduction 59
2. Virtual Work Method 59
The Finite Element Method
vi
3. Energy Methods 65
4. Integral Formulations for Beam Problem 67
5. Virtual Work and Variational Formulations 70
Epilogue 74
References 74
MCQs 75
Answers 76
4. Formulation and Calculation of lsoparametric Finite
Element Matrices 77
1. Introduction 77
2. Finite Element Method in One Dimension 78
3. Shape Functions 80
4. Bilinear Quadrilateral 85
5. Eight-Noded Quadratic Quadrilateral 86
6. Linear Triangle 87
7. Quadratic Triangle 90
8. Directional Cosines 90
9. The Element Matrices 91
10. Inviscid Flow Example 94
Epilogue 104
References 104
MCQs 106
Answers 106
5. Nonlinear Analysis in Solid and Structural Mechanics 107
1. Introduction 107
2. Equations for Three-Dimensional Solids 109
3. Equations for Two-Dimensional Solids 115
4. Constitutive Equations 117
5. Equations For truss Members 118
6. Stress and Strain 119
7. Dynamic Equilibrium Equations 119
8. Equations For Plates 124
9. Finite rotations 124
10. Analysis of stress and strain 125
11. Linear Elastic Materials 151
12. Boundary Conditions 151
13. Approximations for Slender Geometries 155
14. Failure Mechanisms 158
15. Components of a Numerical Solution Method 162
Contents
vii
Epilogue 163
References 163
MCQs 163
Answers 164
6. FEA of Heat Transfer, Field Problems, & Incompressible
Fluid Flows 165
1. Introduction 165
2. Heat Transfer in a Two-Dimensional Fin 166
3. Weighted Residual Approach For FEM 171
4. 2D Heat Transfer Problem 188
Epilogue 193
References 193
MCQs 194
Answers 195
7. Solution of Equilibrium Equations in Static Analysis 196
1. Introduction 196
2. Problem-Solving Strategy: Static Equilibrium 196
3. Equations of Static Equilibrium 198
4. Equilibrium Analysis for Concurrent Force Systems 200
5. Analyzing a Hanging Sign 205
Epilogue 208
References 208
MCQs 208
Answers 209
8. Solution of Eigenproblems 210
1. Introduction 210
2. Numerical Stability and Conditioning 211
3. Hermitian Eigenproblems 212
4. Generalized Hermitian Eigenproblems 215
5. Singular Value Decomposition 217
Epilogue 221
References 221
MCQs 222
Answers 222
9. Problems in Theory of Plasticity 223
1. Introduction 223
2. Static boundary value problems 224
3. Mathematical problems in the theory of the equations of state 231
The Finite Element Method
viii
4. Physical mechanisms 231
5. Computational Mechanics 238
6. Linear and angular momentum principles: stress and equations
of motion 241
7. Geometry of deformation 246
Epilogue 250
References 250
MCQs 251
Answers 251
10. Automatic Mesh Generation 252
1. Introduction 252
2. Techniques 253
3. Mathematical definition and variants 255
4. Mesh improvement 256
5. Unstructured 2-D Mesh Generation 257
6. Some Important Numerical and Implementation Issues 261
Epilogue 262
References 262
MCQs 263
Answers 263
Index 265
The finite element method (FEM) is a popular technique for numerically solving differential equations in engineering and mathematics. The classic domains of structural analysis, heat transfer, fluid flow, mass transport, and electromagnetic potential are typical issue areas of interest. The finite difference (FD), finite element (FE), finite volume (FV), spectral, and discontinuous galerkin (DG) methods are among the various numerical approaches for estimating the solution to differential equations. When the mathematical equations are too hard to solve analytically, these methods are applied. The FEM breaks a big system into smaller, simpler sections called finite elements in order to solve a problem. This is accomplished through a specific space discretization in the spatial dimensions, which is accomplished through the creation of a mesh of the object: the numerical domain for the solution, which has a finite number of points.
This book elucidates the concepts and innovative models around prospective developments with respect to finite element method. It talks in detail about the various concepts and theories related to this subject. This is one of the unique book of its type which includes comprehensive knowledge of vectors, matrices, lsoparametric finite element matrices, nonlinear analysis & incompressible fluid flows. It also gives detail of solution of equilibrium equations, eigenproblems, problems in theory of plasticity and automatic mesh generation. Most of the topics introduced in the book cover new techniques and the applications of this field. Through this book, we attempts to further enlighten the readers about the new concepts in this field. Coherent flow of topics, student-friendly language and extensive use of examples make this book an invaluable source of knowledge.
Elza Irvine holds Ph.D. in Engineering Mechanics from Technological University, Rostock. She is a Senior Science Research Specialist at Community College, Antwerp. Irvine has written several books, articles and research papers on kinematics, fluid mechanics, energy and pneumatics. Her research field includes strength of materials, finite elements, boundary elements, mechanics of machines and mechatronics. She has also been an invited speaker at various international production management conferences.