The revised and updated second edition of Finite Element Analysis: Method, Verification, and Validation offers a comprehensive review of the theoretical foundations of the finite element method and highlights the fundamentals of solution verification, validation, and uncertainty quantification. Written by noted experts on the topic, the book covers the theoretical fundamentals as well as the algorithmic structure of the finite element method. The text contains numerous examples and helpful exercises that clearly illustrate the techniques and procedures needed for accurate estimation of the quantities of interest. In addition, the authors describe the technical requirements for the formulation and application of design rules.

Designed as an accessible resource, the book has a companion website that contains a solutions manual, PowerPoint slides for instructors, and a link to finite element software. This important text:

Offers a comprehensive review of the theoretical foundations of the finite element method
Puts the focus on the fundamentals of solution verification, validation, and uncertainty quantification
Presents the techniques and procedures of quality assurance in numerical solutions of mathematical problems
Contains numerous examples and exercises

Written for students in mechanical and civil engineering, analysts seeking professional certification, and applied mathematicians, Finite Element Analysis: Method, Verification, and Validation, Second Edition includes the tools, concepts, techniques, and procedures that help with an understanding of finite element analysis.

Auteur

Barna Szabó is Senior Professor in the Department of Mechanical Engineering and Materials Science at Washington University in St. Louis, USA. He is also co-founder and chairman of Engineering Software Research and Development, Inc. Ivo Babuka is Professor Emeritus of The University of Texas at Austin, USA, Professor of Aerospace Engineering and Engineering Mechanics, Professor of Mathematics, and Senior Research Scientist of the Oden Institute of Computational Engineering and Sciences.

Texte du rabat

Finite Element Analysis An updated and comprehensive review of the theoretical foundation of the finite element method The revised and updated second edition of Finite Element Analysis: Method, Verification, and Validation offers a comprehensive review of the theoretical foundations of the finite element method and highlights the fundamentals of solution verification, validation, and uncertainty quantification. Written by noted experts on the topic, the book covers the theoretical fundamentals as well as the algorithmic structure of the finite element method. The text contains numerous examples and helpful exercises that clearly illustrate the techniques and procedures needed for accurate estimation of the quantities of interest. In addition, the authors describe the technical requirements for the formulation and application of design rules. Designed as an accessible resource, the book has a companion website that contains a solutions manual, PowerPoint slides for instructors, and a link to finite element software. This important text: Offers a comprehensive review of the theoretical foundations of the finite element method Puts the focus on the fundamentals of solution verification, validation, and uncertainty quantification Presents the techniques and procedures of quality assurance in numerical solutions of mathematical problems Contains numerous examples and exercises Written for students in mechanical and civil engineering, analysts seeking professional certification, and applied mathematicians, Finite Element Analysis: Method, Verification, and Validation, Second Edition includes the tools, concepts, techniques, and procedures that help with an understanding of finite element analysis.

Contenu

1 Introduction to FEM 3 1.1 An introductory problem 6 1.2 Generalized formulation 9 1.2.1 The exact solution 9 1.2.2 The principle of minimum potential energy 14 1.3 Approximate solutions 16 1.3.1 The standard polynomial space 17 1.3.2 Finite element spaces in one dimension 20 1.3.3 Computation of the coefficient matrices 22 1.3.4 Computation of the right hand side vector 26 1.3.5 Assembly 27 1.3.6 Condensation 30 1.3.7 Enforcement of Dirichlet boundary conditions 30 1.4 Post-solution operations 33 1.4.1 Computation of the quantities of interest 33 1.5 Estimation of error in energy norm 37 1.5.1 Regularity 38 1.5.2 A priori estimation of the rate of convergence 38 1.5.3 A posteriori estimation of error 40 1.5.4 Error in the extracted QoI 46 1.6 The choice of discretization in 1D 47 1.6.1 The exact solution lies in Hk(I), k . 1 > p 47 1.6.2 The exact solution lies in Hk(I), k . 1 p 146 4.2.2 2D model problem, uEX element of Hk(), k . 1 <= p 148 4.2.3 Computation of the flux vector in a given point 151 4.2.4 Computation of the flux intensity factors 153 4.2.5 Material interfaces 158 4.3 The Laplace equation in three dimensions 160 4.4 Planar elasticity 164 4.4.1 Problems of elasticity on an L-shaped domain 165 4.4.2 Crack tip singularities in 2D 165 4.4.3 Forcing functions acting on boundaries 170 4.5 Robustness 172 4.6 Solution verification 177 5 Simulation 185 5.1 Development of a mathematical model 186 5.1.1 The Bernoulli-Euler beam model 187 5.1.2 Historical notes 188 5.2 FE modeling vs simulation 190 5.2.1 Numerical simulation 190 5.2.2 Finite element modeling 192 5.2.3 Calibration versus tuning 195 5.2.4 Simulation governance 196 5.2.5 Milestones in numerical simulation 197 5.2.6 Example: The Girkmann problem 199 5.2.7 Example: Fastened structural connection 203 5.2.8 Finite element model 210 5.2.9 Example: Coil spring with displacement boundary conditions 215 5.2.10 Example: Coil spring segment 220 6 Calibration, Validation and Ranking 225 6.1 Fatigue data 226 6.1.1 Equivalent stress 227 6.1.2 Statistical models 227 6.1.3 The effect of notches 228 6.1.4 Formulation of predictors of fatigue life 229 6.2 The predictors of Peterson and Neuber 230 6.2.1 The effect of notches - calibration 232 6.2.2 The effect of notches - validation 235 6.2.3 Updated calibration 237 6.2.4 The fatigue limit 240 6.2.5 Discussion 242 6.3 The predictor Galpha 243 6.3.1 Calibration of ß(V, alpha) 244 6.3.2 Ranking 246 6.3.3 Comparison of Galpha with Peterson's revised predictor 246 6.4 Biaxial test data 247 6.4.1 Axial, torsional and combined in-phase loading 248 6.4.2 The domain of calibration 249 6.4.3 Out-of-phase biaxial loading 252 6.4.4 Validation 255 6.4.5 Selection of the prior 256 6.4.6 Discussion 259 7 Beams, plates and shells 261 7.1 Beams 261 7.1.1 The Timoshenko beam 263 7.1.2 The Bernoulli-Euler beam 268 7.2 Plates 273 7.2.1 The Reissner-Mindlin plate 276 7.2.2 The Kirchhoff plate 281 7.2.3 The transverse variation of displacements 283 7.3 Shells 287 7.3.1 Hierarchic thin solid models 291 7.4 Chapter summary 295 8 Aspects of multiscale models 297 8.1 Unidirectional fiber-reinforced laminae 297 8.1.1 Determination of material constants 300 8.1.2 The coefficients of thermal expansion 300 8.1.3 Examples 301 8.1.4 Localization 304 8.1.5 Prediction of failure in composite materials 305 8.1.6 Uncertainties 307 8.2 Discussion 307 9 Non-linear models 309 9.1 Heat conduction 309 9.1.1 Radiation 309 9.1.2 Nonlinear material properties 310 9.2 Solid mechanics 310 9.2.1 Large strain and rotation 311 9.2.2 Structural stability and stress stiffening 314 9.2.3 Plasticity 321 9.2.4 Mechanical contact 327 9.3 Chapter summary 335 A Definitions 337 A.1 Normed linear spaces, linear functionals and bilinear forms 338 A.1.1 Normed linear spaces 338 A.1.2 Linear forms 339 A.1.3 Bilinear forms 339 A.2 Convergence in the space X 339 A.2.1 The space of continuous functions 339 A.2.2 The space …