Finite Element Analysis
Description
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.
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Persons
Barna Szabo 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 Babu?ka 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.
Content
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
> p 47
1.6.2 The exact solution lies in Hk(I), k - 1 = p 49
1.7 Eigenvalue problems 52
1.8 Other finite element methods 57
1.8.1 The mixed method 59
1.8.2 Nitsche's method 60
2 Boundary value problems 63
2.1 Notation 63
2.2 The scalar elliptic boundary value problem 65
2.2.1 Generalized formulation 66
2.2.2 Continuity 68
2.3 Heat conduction 68
2.3.1 The differential equation 70
2.3.2 Boundary and initial conditions 71
2.3.3 Boundary conditions of convenience 73
2.3.4 Dimensional reduction 75
2.4 Linear elasticity - strong form 82
2.4.1 The Navier equations 86
2.4.2 Boundary and initial conditions 86
2.4.3 Symmetry, antisymmetry and periodicity 88
2.4.4 Dimensional reduction in linear elasticity 89
2.4.5 Incompressible elastic materials 93
2.5 Stokes flow 95
2.6 Elasticity - generalized formulation 96
2.6.1 The principle of minimum potential energy 98
2.6.2 The RMS measure of stress 100
2.6.3 The principle of virtual work 101
2.6.4 Uniqueness 102
2.7 Residual stresses 106
2.8 Chapter summary 108
3 Implementation 111
3.1 Standard elements in two dimensions 111
3.2 Standard polynomial spaces 111
3.2.1 Trunk spaces 111
3.2.2 Product spaces 112
3.3 Shape functions 112
3.3.1 Lagrange shape functions 113
3.3.2 Hierarchic shape functions 115
3.4 Mapping functions in two dimensions 118
3.4.1 Isoparametric mapping 118
3.4.2 Mapping by the blending function method 121
3.4.3 Mapping algorithms for high order elements 123
3.5 Finite element spaces in two dimensions 125
3.6 Essential boundary conditions 125
3.7 Elements in three dimensions 126
3.7.1 Mapping functions in three-dimensions 127
3.8 Integration and differentiation 129
3.8.1 Volume and area integrals 129
3.8.2 Surface and contour integrals 131
3.8.3 Differentiation 131
3.9 Stiffness matrices and load vectors 132
3.9.1 Stiffness matrices 133
3.9.2 Load vectors 134
3.10 Post-solution operations 135
3.11 Computation of the solution and its first derivatives 135
3.12 Nodal forces 137
3.12.1 Nodal forces in the h-version 137
3.12.2 Nodal forces in the p-version 140
3.12.3 Nodal forces and stress resultants 141
3.13 Chapter summary 142
4 Verification 143
4.1 Regularity in two and three dimensions 143
4.2 The Laplace equation in two dimensions 144
> p 146
4.2.2 2D model problem, uEX ¿ 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 Ga 243
6.3.1 Calibration of ß(V, a) 244
6.3.2 Ranking 246
6.3.3 Comparison of Ga 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 Lp() 340
A.2.3 Sobolev space of order 1 340
A.2.4 Sobolev spaces of fractional index 341
A.3 The Schwarz inequality for integrals 342
B Proof of convergence 343
C Convergence in 3D 345
D Legendre polynomials 349
D.1 Shape functions based on Legendre polynomials 350
E Numerical quadrature 353
E.1 Gaussian quadrature 353
E.2 Gauss-Lobatto quadrature 355
F Polynomial mapping functions 357
F.1 Interpolation on surfaces 359
F.1.1 Interpolation on the standard quadrilateral element 359
F.1.2 Interpolation on the standard triangle 359
G Corner singularities 361
G.1 The Airy stress function 361
G.2 Stress-free edges 363
G.2.1 Symmetric eigenfunctions 364
G.2.2 Antisymmetric eigenfunctions 365
G.2.3 The L-shaped domain 366
G.2.4 Corner points 367
H Stress intensity factors 369
H.1 Singularities at crack tips 369
H.2 The contour integral method 370
H.3 The energy release rate 372
H.3.1 Symmetric (Mode I) loading 372
H.3.2 Antisymmetric (Mode II) loading 373
H.3.3 Combined (Mode I and Mode II) loading 373
H.3.4 Computation by the stiffness derivative method 374
I Fundamentals of data analysis 375
I.1 Statistical foundations 375
I.2 Test data 377
I.3 Statistical models 378
I.4 Ranking 387
I.5 Confidence intervals 387
J Fastener forces 389
K Useful algorithms 393
K.1 The traction vector 393
K.2 Transformation of vectors 394
K.3 Transformation of stresses 396
K.4 Principal stresses 396
K.5 The von Mises stress 397
K.6 Statically equivalent forces and moments 398
K.6.1 Technical formulas for stress 400
1
Introduction to the finite element method
This book covers the fundamentals of the finite element method in the context of numerical simulation with specific reference to the simulation of the response of structural and mechanical components to mechanical and thermal loads.
We begin with the question: what is the meaning of the term "simulation"? By its dictionary definition, simulation is the imitative representation of the functioning of one system or process by means of the functioning of another. For instance, the membrane analogy introduced by Prandtl1 in 1903 made it possible to find the shearing stresses in bars of arbitrary cross-section, loaded by a twisting moment, through mapping the deflected shape of a thin elastic membrane. In other words, the distribution and magnitude of shearing stress in a twisted bar can be simulated by the deflected shape of an elastic membrane.
The membrane analogy exists because two unrelated phenomena can be modeled by the same partial differential equation. The physical meaning associated with the coefficients of the differential equation depends on which problem is being solved. However, the solution of one is proportional to the solution of the other: At corresponding points the shearing stress in a bar, subjected to a twisting moment, is oriented in the direction of the tangent to the contour lines of a deflected thin membrane and its magnitude is proportional to the slope of the membrane. Furthermore, the volume enclosed by the deflected membrane is proportional to the twisting moment.
In the pre-computer years the membrane analogy provided practical means for estimating shearing stresses in prismatic bars. This involved cutting the shape of the cross-section out of sheet metal or a wood panel, covering the hole with a thin elastic membrane, applying pressure to the membrane and mapping the contours of the deflected membrane. In present-day practice both problems would be formulated as mathematical problems which would then be solved by a numerical method, most likely by the finite element method.
There are many other useful analogies. For example, the same differential equations simulate the response of assemblies of mechanical components, such as linear spring-mass-viscous damper systems and assemblies of electrical components, such as capacitors, inductors and resistors. This has been exploited by the use of analogue computers. Obviously, it is much easier to build and manipulate electrical circuitry than mechanical assemblies. In present-day practice both simulation problems would be formulated as mathematical problems which would be solved by a numerical method.
At the heart of simulation of aspects of physical reality is a mathematical problem cast in a generalized form2. The solution of the mathematical problem is approximated by a numerical method, such as the finite element method, which is the subject of this book. The quantities of interest (QoI) are extracted from the approximate solution. The errors of approximation in the QoI depend on how the mathematical problem was discretized3 and how the QoI were extracted from the numerical solution. When the errors of approximation are larger than what is considered acceptable then the discretization has to be changed either by an automated adaptive process or by action of the analyst.
Estimation and control of numerical errors are fundamentally important in numerical simulation. Consider, for example, the problem of design certification. Design rules are typically stated in the form
(1.1)where (resp. ) is the maximum (resp. allowable) value of a quantity of interest, for example the first principal stress. Since in numerical simulation only an approximation to Fmax is available, denoted by , it is necessary to know the size of the numerical error t:
(1.2)In design and design certification the worst case scenario has to be considered, which is underestimation of Fmax, that is,
(1.3)Therefore it has to be shown that
(1.4)Without a reliable estimate of the size of the numerical error it is not possible to certify design and, furthermore, numerical errors penalize design by lowering the allowable value, as indicated by eq. (1.4). Generally speaking, it is far more economical to ensure that t is small than to accept the consequences of decreased allowable values.
We distinguish between finite element modeling and numerical simulation. As explained in greater detail in Chapter 5, finite element modeling evolved well before the theoretical basis of numerical simulation was developed. In finite element modeling a numerical problem is formulated by assembling elements from a library of finite elements that contains intuitively constructed beam, plate, shell, solid elements of various description. The numerical problem so created may not correspond to a well defined mathematical problem and therefore a solution may not even exist. For that reason it is not possible to speak of errors of approximation. Nevertheless, finite element modeling is widely practiced with success in some cases but with disappointing results in others. Such practice should be regarded as a practice of art, guided by intuition and experience, rather than a scientific activity. This is because practitioners of finite element modeling have to balance two kinds of very large errors: (a) conceptual errors in the formulation and (b) approximation errors in the numerical solution of an improperly posed mathematical problem.
In numerical simulation, on the other hand, the formulation of mathematical models is treated separately from their numerical solution. A mathematical model should be understood to be a precise statement of an idea of physical reality that permits the prediction of the occurrence, or probability of occurrence, of physical events, given certain data. The intuitive aspects of simulation are confined to the formulation of mathematical models whereas their numerical solution involves the application of well established procedures of applied mathematics. Separation of mathematical models from their numerical solution makes separate treatment of errors associated with the formulation of mathematical models and their numerical approximation possible. Errors associated with the formulation of mathematical models are called model form errors. Errors associated with the numerical solution of mathematical problems are called errors of approximation or errors of discretization. In the early papers and books on the finite element method no such distinction was made.
In this chapter we introduce the finite element method as a method by which the exact solution of a mathematical problem, cast in a generalized form, can be approximated. We also introduce the relevant mathematical concepts, terminology and notation in the simplest possible setting. Generalization of these concepts to two- and three-dimensional problems will be discussed in subsequent chapters.
We first consider the formulation of a second order ordinary differential equation without reference to any physical interpretation. This is to underline that once a mathematical problem was formulated, the approximation process is independent from why the mathematical problem was formulated. This important point is often missed by engineering users of legacy finite element codes because the formulation and approximation of mathematical problems is mixed in finite element libraries.
We show that the exact solution of the generalized formulation is unique. Approximation of the exact solution by the finite element method is described and various discretization strategies are explored. Efficient methods for the computation of QoIs and a posteriori error estimation are described. This chapter serves as a foundation for subsequent chapters.
We would like to assure engineering students who are not yet familiar with the concepts and notation of that branch of applied mathematics on which the finite element method is based that their investment of time and effort to master the contents of this chapter will prove to be highly rewarding.
1.1 An introductory problem
We introduce the finite element method through approximating the exact solution of the following second order ordinary differential equation
(1.5)with the boundary conditions
(1.6)where the prime indicates differentiation with respect to x. It is assumed that where a and ß are real numbers, on , and are defined such that the indicated operations are meaningful on I. For example, the indicated operations would not be meaningful if , c or f would not be finite in one or more points on the interval . The function f is called a forcing function.
We seek an approximation to u in the form:
(1.7)where are fixed functions, called basis functions, and aj are the coefficients of the basis functions to be determined. Note that the basis functions satisfy the zero boundary conditions.
Let us find aj such that the integral defined by
(1.8)is minimum. While there are other plausible criteria for selecting aj, we will see that this criterion is fundamentally important in the finite element method. Differentiating with respect to ai and letting the derivative equal to zero, we...
