Finite Element Method: Its Basis and Fundamentals (eBook)
752 Seiten
Elsevier Science (Verlag)
978-0-08-047277-5 (ISBN)
.The classic FEM text, written by the subject's leading authors
.Enhancements include more worked examples and exercises, plus a companion website with a solutions manual and downloadable algorithms
.With a new chapter on automatic mesh generation and added materials on shape function development and the use of higher order elements in solving elasticity and field problems
Active research has shaped The Finite Element Method into the pre-eminent tool for the modelling of physical systems. It maintains the comprehensive style of earlier editions, while presenting the systematic development for the solution of problems modelled by linear differential equations.
Together with the second and third self-contained volumes (0750663219 and 0750663227), The Finite Element Method Set (0750664312) provides a formidable resource covering the theory and the application of FEM, including the basis of the method, its application to advanced solid and structural mechanics and to computational fluid dynamics.
* The classic introduction to the finite element method, by two of the subject's leading authors
* Any professional or student of engineering involved in understanding the computational modelling of physical systems will inevitably use the techniques in this key text
* Enhancements include more worked examples, exercises, plus a companion website with a worked solutions manual for tutors and downloadable algorithms"
The Sixth Edition of this influential best-selling book delivers the most up-to-date and comprehensive text and reference yet on the basis of the finite element method (FEM) for all engineers and mathematicians. Since the appearance of the first edition 38 years ago, The Finite Element Method provides arguably the most authoritative introductory text to the method, covering the latest developments and approaches in this dynamic subject, and is amply supplemented by exercises, worked solutions and computer algorithms.* The classic FEM text, written by the subject's leading authors * Enhancements include more worked examples and exercises* With a new chapter on automatic mesh generation and added materials on shape function development and the use of higher order elements in solving elasticity and field problemsActive research has shaped The Finite Element Method into the pre-eminent tool for the modelling of physical systems. It maintains the comprehensive style of earlier editions, while presenting the systematic development for the solution of problems modelled by linear differential equations. Together with the second and third self-contained volumes (0750663219 and 0750663227), The Finite Element Method Set (0750664312) provides a formidable resource covering the theory and the application of FEM, including the basis of the method, its application to advanced solid and structural mechanics and to computational fluid dynamics. - The classic introduction to the finite element method, by two of the subject's leading authors- Any professional or student of engineering involved in understanding the computational modelling of physical systems will inevitably use the techniques in this key text
Front Cover 1
The Finite Element Method: Its Basis and Fundamentals 4
Copyright Page 5
Contents 8
Preface 14
Chapter 1. The standard discrete system and origins of the finite element method 16
1.1 Introduction 16
1.2 The structural element and the structural system 18
1.3 Assembly and analysis of a structure 20
1.4 The boundary conditions 21
1.5 Electrical and fluid networks 22
1.6 The general pattern 24
1.7 The standard discrete system 25
1.8 Transformation of coordinates 26
1.9 Problems 28
Chapter 2. A direct physical approach to problems in elasticity: plane stress 34
2.1 Introduction 34
2.2 Direct formulation of finite element characteristics 35
2.3 Generalization to the whole region– internal nodal force concept abandoned 46
2.4 Displacement approach as a minimization of total potential energy 49
2.5 Convergence criteria 52
2.6 Discretization error and convergence rate 53
2.7 Displacement functions with discontinuity between elements – non-conforming elements and the patch test 54
2.8 Finite element solution process 55
2.9 Numerical examples 55
2.10 Concluding remarks 61
2.11 Problems 62
Chapter 3. Generalization of the finite element concepts. Galerkin-weighted residual and variational approaches 69
3.1 Introduction 69
3.2 Integral or 'weak' statements equivalent to the differential equations 72
3.3 Approximation to integral formulations: the weighted residual-Galerkin method 75
3.4 Virtual work as the 'weak form' of equilibrium equations for analysis of solids or fluids 84
3.5 Partial discretization 86
3.6 Convergence 89
3.7 What are 'variational principles'? 91
3.8 'Natural' variational principles and their relation to governing differential equations 93
3.9 Establishment of natural variational principles for linear, self-adjoint, differential equations 96
3.10 Maximum, minimum, or a saddle point? 98
3.11 Constrained variational principles. Lagrange multipliers 99
3.12 Constrained variational principles. Penalty function and perturbed lagrangian methods 103
3.13 Least squares approximations 107
3.14 Concluding remarks – finite difference and boundary methods 110
3.15 Problems 112
Chapter 4. 'Standard' and 'hierarchical' element shape functions: some general families of C0 continuity 118
4.1 Introduction 118
4.2 Standard and hierarchical concepts 119
4.3 Rectangular elements– some preliminary considerations 122
4.4 Completeness of polynomials 124
4.5 Rectangular elements– Lagrange family 125
4.6 Rectangular elements– 'serendipity' family 127
4.7 Triangular element family 131
4.8 Line elements 134
4.9 Rectangular prisms – Lagrange family 135
4.10 Rectangular prisms – 'serendipity' family 136
4.11 Tetrahedral elements 137
4.12 Other simple three-dimensional elements 140
4.13 Hierarchic polynomials in one dimension 140
4.14 Two- and three-dimensional, hierarchical elements of the 'rectangle' or 'brick' type 143
4.15 Triangle and tetrahedron family 143
4.16 Improvement of conditioning with hierarchical forms 145
4.17 Global and local finite element approximation 146
4.18 Elimination of internal parameters before assembly – substructures 147
4.19 Concluding remarks 149
4.20 Problems 149
Chapter 5. Mapped elements and numerical integration– 'infinite' and 'singularity elements' 153
5.1 Introduction 153
5.2 Use of 'shape functions' in the establishment of coordinate transformations 154
5.3 Geometrical conformity of elements 158
5.4 Variation of the unknown function within distorted, curvilinear elements. Continuity requirements 158
5.5 Evaluation of element matrices. Transformation in .,.,. coordinates 160
5.6 Evaluation of element matrices. Transformation in area and volume coordinates 163
5.7 Order of convergence for mapped elements 166
5.8 Shape functions by degeneration 168
5.9 Numerical integration– one dimensional 175
5.10 Numerical integration– rectangular (2D) or brick regions (3D) 177
5.11 Numerical integration – triangular or tetrahedral regions 179
5.12 Required order of numerical integration 179
5.13 Generation of finite element meshes by mapping. Blending functions 184
5.14 Infinite domains and infinite elements 185
5.15 Singular elements by mapping – use in fracture mechanics, etc. 191
5.16 Computational advantage of numerically integrated finite elements 192
5.17 Problems 193
Chapter 6. Problems in linear elasticity 202
6.1 Introduction 202
6.2 Governing equations 203
6.3 Finite element approximation 216
6.4 Reporting of results: displacements, strains and stresses 222
6.5 Numerical examples 224
6.6 Problems 232
Chapter 7. Field problems – heat conduction, electric and magnetic potential and fluid flow 244
7.1 Introduction 244
7.2 General quasi-harmonic equation 245
7.3 Finite element solution process 248
7.4 Partial discretization– transient problems 252
7.5 Numerical examples– an assessment of accuracy 254
7.6 Concluding remarks 268
7.7 Problems 268
Chapter 8. Automatic mesh generation 279
8.1 Introduction 279
8.2 Two-dimensional mesh generation– advancing front method 281
8.3 Surface mesh generation 301
8.4 Three-dimensional mesh generation– Delaunay triangulation 318
8.5 Concluding remarks 338
8.6 Problems 338
Chapter 9. The patch test, reduced integration, and non-conforming elements 344
9.1 Introduction 344
9.2 Convergence requirements 345
9.3 The simple patch test (tests A and B) – a necessary condition for convergence 347
9.4 Generalized patch test (test C) and the single-element test 349
9.5 The generality of a numerical patch test 351
9.6 Higher order patch tests 351
9.7 Application of the patch test to plane elasticity elements with 'standard' and 'reduced' quadrature 352
9.8 Application of the patch test to an incompatible element 358
9.9 Higher order patch test– assessment of robustness 362
9.10 Concluding remarks 362
9.11 Problems 365
Chapter 10. Mixed formulation and constraints– complete field methods 371
10.1 Introduction 371
10.2 Discretization of mixed forms – some general remarks 373
10.3 Stability of mixed approximation. The patch test 375
10.4 Two-field mixed formulation in elasticity 378
10.5 Three-field mixed formulations in elasticity 385
10.6 Complementary forms with direct constraint 390
10.7 Concluding remarks – mixed formulation or a test of element 'robustness' 394
10.8 Problems 394
Chapter 11. Incompressible problems, mixed methods and other procedures of solution 398
11.1 Introduction 398
11.2 Deviatoric stress and strain, pressure and volume change 398
11.3 Two-field incompressible elasticity (u–p form) 399
11.4 Three-field nearly incompressible elasticity (u–p–ev form) 408
11.5 Reduced and selective integration and its equivalence to penalized mixed problems 413
11.6 A simple iterative solution process for mixed problems: Uzawa method 419
11.7 Stabilized methods for some mixed elements failing the incompressibility patch test 422
11.8 Concluding remarks 436
11.9 Problems 437
Chapter 12. Multidomain mixed approximations– domain decomposition and 'frame' methods 444
12.1 Introduction 444
12.2 Linking of two or more subdomains by Lagrange multipliers 445
12.3 Linking of two or more subdomains by perturbed lagrangian and penalty methods 451
12.4 Interface displacement 'frame' 457
12.5 Linking of boundary (or Trefftz)-type solution by the 'frame' of specified displacements 460
12.6 Subdomains with 'standard' elements and global functions 466
12.7 Concluding remarks 466
12.8 Problems 466
Chapter 13. Errors, recovery processes and error estimates 471
13.1 Definition of errors 471
13.2 Superconvergence and optimal sampling points 474
13.3 Recovery of gradients and stresses 480
13.4 Superconvergent patch recovery – SPR 482
13.5 Recovery by equilibration of patches – REP 489
13.6 Error estimates by recovery 491
13.7 Residual-based methods 493
13.8 Asymptotic behaviour and robustness of error estimators – the Babuška patch test 503
13.9 Bounds on quantities of interest 505
13.10 Which errors should concern us? 509
13.11 Problems 510
Chapter 14. Adaptive finite element refinement 515
14.1 Introduction 515
14.2 Adaptive h-refinement 518
14.3 p-refinement and hp-refinement 529
14.4 Concluding remarks 533
14.5 Problems 535
Chapter 15. Point-based and partition of unity approximations. Extended finite element methods 540
15.1 Introduction 540
15.2 Function approximation 542
15.3 Moving least squares approximations – restoration of continuity of approximation 548
15.4 Hierarchical enhancement of moving least squares expansions 553
15.5 Point collocation– finite point methods 555
15.6 Galerkin weighting and finite volume methods 561
15.7 Use of hierarchic and special functions based on standard finite elements satisfying the partition of unity requirement 564
15.8 Concluding remarks 573
15.9 Problems 573
Chapter 16. The time dimension– semi-discretization of field and dynamic problems and analytical solution procedures 578
16.1 Introduction 578
16.2 Direct formulation of time-dependent problems with spatial finite element subdivision 578
16.3 General classification 585
16.4 Free response – eigenvalues for second-order problems and dynamic vibration 586
16.5 Free response – eigenvalues for first-order problems and heat conduction, etc. 591
16.6 Free response– damped dynamic eigenvalues 593
16.7 Forced periodic response 594
16.8 Transient response by analytical procedures 594
16.9 Symmetry and repeatability 598
16.10 Problems 599
Chapter 17. The time dimension– discrete approximation in time 604
17.1 Introduction 604
17.2 Simple time-step algorithms for the first-order equation 605
17.3 General single-step algorithms for first- and second-order equations 615
17.4 Stability of general algorithms 624
17.5 Multistep recurrence algorithms 630
17.6 Some remarks on general performance of numerical algorithms 633
17.7 Time discontinuous Galerkin approximation 634
17.8 Concluding remarks 639
17.9 Problems 641
Chapter 18. Coupled systems 646
18.1 Coupled problems– definition and classification 646
18.2 Fluid-structure interaction (Class I problems) 649
18.3 Soil-pore fluid interaction (Class II problems) 660
18.4 Partitioned single-phase systems – implicit-explicit partitions (Class I problems) 668
18.5 Staggered solution processes 670
18.6 Concluding remarks 675
Chapter 19. Computer procedures for finite element analysis 679
19.1 Introduction 679
19.2 Pre-processing module: mesh creation 679
19.3 Solution module 681
19.4 Post-processor module 681
19.5 User modules 682
Appendix A: Matrix algebra 683
Appendix B: Tensor-indicial notation in the approximation of elasticity problems 689
Appendix C: Solution of simultaneous linear algebraic equations 698
Appendix D: Some integration formulae for a triangle 707
Appendix E: Some integration formulae for a tetrahedron 708
Appendix F: Some vector algebra 709
Appendix G: Integration by parts in two or three dimensions (Green's theorem) 714
Appendix H: Solutions exact at nodes 716
Appendix I: Matrix diagonalization or lumping 719
Author index 726
Subject index 734
Color Plate Section 750
The standard discrete system and origins of the finite element method
1.1 Introduction
The limitations of the human mind are such that it cannot grasp the behaviour of its complex surroundings and creations in one operation. Thus the process of subdividing all systems into their individual components or ‘elements’, whose behaviour is readily understood, and then rebuilding the original system from such components to study its behaviour is a natural way in which the engineer, the scientist, or even the economist proceeds.
In many situations an adequate model is obtained using a finite number of well-defined components. We shall term such problems discrete. In others the subdivision is continued indefinitely and the problem can only be defined using the mathematical fiction of an infinitesimal. This leads to differential equations or equivalent statements which imply an infinite number of elements. We shall term such systems continuous.
With the advent of digital computers, discrete problems can generally be solved readily even if the number of elements is very large. As the capacity of all computers is finite, continuous problems can only be solved exactly by mathematical manipulation. The available mathematical techniques for exact solutions usually limit the possibilities to oversimplified situations.
To overcome the intractability of realistic types of continuous problems (continuum), various methods of discretization have from time to time been proposed by engineers, scientists and mathematicians. All involve an approximation which, hopefully, approaches in the limit the true continuum solution as the number of discrete variables increases.
The discretization of continuous problems has been approached differently by mathematicians and engineers. Mathematicians have developed general techniques applicable directly to differential equations governing the problem, such as finite difference approximations,1–3 various weighted residual procedures,4,5 or approximate techniques for determining the stationarity of properly defined ‘functionals’.6 The engineer, on the other hand, often approaches the problem more intuitively by creating an analogy between real discrete elements and finite portions of a continuum domain. For instance, in the field of solid mechanics McHenry,7 Hrenikoff,8 Newmark,9 and Southwell2 in the 1940s, showed that reasonably good solutions to an elastic continuum problem can be obtained by replacing small portions of the continuum by an arrangement of simple elastic bars. Later, in the same context, Turner et al.10 showed that a more direct, but no less intuitive, substitution of properties can be made much more effectively by considering that small portions or ‘elements’ in a continuum behave in a simplified manner.
It is from the engineering ‘direct analogy’ view that the term ‘finite element’ was born. Clough11 appears to be the first to use this term, which implies in it a direct use of a standard methodology applicable to discrete systems (see also reference 12 for a history on early developments). Both conceptually and from the computational viewpoint this is of the utmost importance. The first allows an improved understanding to be obtained; the second offers a unified approach to the variety of problems and the development of standard computational procedures.
Since the early 1960s much progress has been made, and today the purely mathematical and ‘direct analogy’ approaches are fully reconciled. It is the object of this volume to present a view of the finite element method as a general discretization procedure of continuum problems posed by mathematically defined statements.
In the analysis of problems of a discrete nature, a standard methodology has been developed over the years. The civil engineer, dealing with structures, first calculates forcedisplacement relationships for each element of the structure and then proceeds to assemble the whole by following a well-defined procedure of establishing local equilibrium at each ‘node’ or connecting point of the structure. The resulting equations can be solved for the unknown displacements. Similarly, the electrical or hydraulic engineer, dealing with a network of electrical components (resistors, capacitances, etc.) or hydraulic conduits, first establishes a relationship between currents (fluxes) and potentials for individual elements and then proceeds to assemble the system by ensuring continuity of flows.
All such analyses follow a standard pattern which is universally adaptable to discrete systems. It is thus possible to define a standard discrete system, and this chapter will be primarily concerned with establishing the processes applicable to such systems. Much of what is presented here will be known to engineers, but some reiteration at this stage is advisable. As the treatment of elastic solid structures has been the most developed area of activity this will be introduced first, followed by examples from other fields, before attempting a complete generalization.
The existence of a unified treatment of ‘standard discrete problems’ leads us to the first definition of the finite element process as a method of approximation to continuum problems such that
(a) the continuum is divided into a finite number of parts (elements), the behaviour of which is specified by a finite number of parameters, and
(b) the solution of the complete system as an assembly of its elements follows precisely the same rules as those applicable to standard discrete problems.
The development of the standard discrete system can be followed most closely through the work done in structural engineering during the nineteenth and twentieth centuries. It appears that the ‘direct stiffness process’ was first introduced by Navier in the early part of the nineteenth century and brought to its modern form by Clebsch13 and others. In the twentieth century much use of this has been made and Southwell,14 Cross15 and others have revolutionized many aspects of structural engineering by introducing a relaxation iterative process. Just before the Second World War matrices began to play a larger part in casting the equations and it was convenient to restate the procedures in matrix form. The work of Duncan and Collar,16–18 Argyris,19 Kron20 and Turner10 should be noted. A thorough study of direct stiffness and related methods was recently conducted by Samuelsson.21
It will be found that most classical mathematical approximation procedures as well as the various direct approximations used in engineering fall into this category. It is thus difficult to determine the origins of the finite element method and the precise moment of its invention.
Table 1.1 shows the process of evolution which led to the present-day concepts of finite element analysis. A historical development of the subject of finite element methods has been presented by the first author in references 34–36. Chapter 3 will give, in more detail, the mathematical basis which emerged from these classical ideas.1,22–27,29,30,32
Table 1.1
History of approximate methods
1.2 The structural element and the structural system
To introduce the reader to the general concept of discrete systems we shall first consider a structural engineering example with linear elastic behaviour.
Figure 1.1 represents a two-dimensional structure assembled from individual components and interconnected at the nodes numbered 1 to 6. The joints at the nodes, in this case, are pinned so that moments cannot be transmitted.
As a starting point it will be assumed that by separate calculation, or for that matter from the results of an experiment, the characteristics of each element are precisely known. Thus, if a typical element labelled (1) and associated with nodes 1, 2, 3 is examined, the forces acting at the nodes are uniquely defined by the displacements of these nodes, the distributed loading acting on the element (p), and its initial strain. The last may be due to temperature, shrinkage, or simply an initial ‘lack of fit’. The forces and the corresponding displacements are defined by appropriate components (U, V and u, v) in a common coordinate system (x, y).
Listing the forces acting on all the nodes (three in the case illustrated) of the element (1) as a matrix† we have
1=q11q21q31q11=U1V1,etc.
(1.1)
and for the corresponding nodal displacements
1=u11u21u31u11=u1v1,etc.
(1.2)
Assuming linear elastic behaviour of the element, the characteristic relationship will always be of the form
1=K1u1+f1
(1.3)
in which f1 represents the nodal forces required to balance any concentrated or distributed loads...
| Erscheint lt. Verlag | 26.5.2005 |
|---|---|
| Sprache | englisch |
| Themenwelt | Sachbuch/Ratgeber |
| Mathematik / Informatik ► Mathematik ► Analysis | |
| Mathematik / Informatik ► Mathematik ► Angewandte Mathematik | |
| Naturwissenschaften | |
| Technik ► Bauwesen | |
| Technik ► Maschinenbau | |
| ISBN-10 | 0-08-047277-X / 008047277X |
| ISBN-13 | 978-0-08-047277-5 / 9780080472775 |
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