Techniques of finite elements

목차 (Table of Contents)

• CONTENTS
• Dedication = 22
• Authors' Preface and Acknowledgements = 23
• List of Symbols = 27
• Part I INTRODUCTION
• Chapter 1 OVERVIEW
• 1.1 Impact of Computation in Engineering - the supersonic flying wing project = 33
• 1.2 Computer Aided Experiment - X-ray tomography etc = 34
• 1.3 Introductory Example - a flat membrane, spring-supported and without tension = 35
• 1.4 The Structural Test-Rig and the Element Stiffness Matrix - a physical introduction to reactions from earth = 36
• 1.5 External Loads, and Element Assembly - creation of the linear equations, by considering equilibrium = 37
• 1.6 Shape Function Constraints and the Continuum - does the finite element model approach the continuum with fine mesh? = 39
• 1.7 Why Finite Elements? - a practical view : they suit modern computers, they are widely used, they are usually successful, and they are convenient = 40
• 1.8 Limitations - They fail in difficult problems, or when the physical problem is inadequately defined = 40
• 1.9 Typical Problem Sizes - These cover a very wide spectrum = 41
• 1.10 Styles of Numeracy, Ancient and Modern - There still remains a place for simple rough calculations. The computer has adverse side-effects = 41
• Chapter 2 BASIC TECHNIQUES
• 2.1 Energy - damping and plastic dissipation, also flutter. The role of symmetric matrices = 43
• 2.2 The Spring-supported Triangle - The expression for strain energy yields the stiffness matrix K with no further effort = 45
• 2.3 The Continuously Supported Triangle introduces the concept of numerical integration = 46
• 2.4 Thin Beam - The idea of shape functions = 48
• 2.5 Rectangular Elements are not useful in themselves = 51
• 2.5.1 Shape Functions - Rigid needles as constraint mechanisms. The pantograph for plane stress. A shape function is a vector, not always parallel to the controlling deflection = 51
• 2.5.2 Element Stiffness - The shape function matrix yields the B-matrix, and the modulus matrix D combines with it to give the stiffness = 53
• 2.5.3 Numerical Integration - The product-rules, based initially on the Simpson rule = 55
• 2.6 The Gauss Rules are about twice as effective as Simpson's rule etc = 58
• 2.7 The Taig Quadrilateral - the first isoparametric element = 58
• 2.7.1 Introduction - Iso-P elements are approximately integrated, but this does not really matter in practice = 58
• 2.7.2 The Shape Function Subroutine uses the local variables ξ and η, both between - 1 and 1 inside the element = 58
• 2.7.3 Construction of the Shape Function Subroutine - The chain rule leads to the Jacobian matrix : by inverting this we deduce the x - y derivatives needed for the strains = 60
• 2.7.4 The Stiffness Matrix - To integrate, we need the Jacobian determinant = 61
• Chapter 3 SHAPE FUNCTIONS
• 3.1 The Shape Function Subroutine Habit is central to our message. Life is easier. Fewer mistakes are made. Planning is straight - forward. It should be universal = 65
• 3.2 Eight-node Quadrilateral - A useful element, especially with 2×2 integration = 66
• 3.3 Eight-node Brick - Trilinear : similar to Taig's quadrilateral = 67
• 3.4 Twenty node Brick - Another useful element, given a frontal solver = 68
• 3.5 Three-node Triangle, the Turner linear triangle introduces area coordinates = 69
• 3.6 Six-node Triangle - another useful iso-P element = 70
• 3.7 Fifteen-node Wedge - mixes area coordinates with a Taig-like local coordinate. A useful element but not indispensible = 72
• 3.8 Ten-node Tetrahedron - A special purpose iso-P element = 73
• 3.9 The Iso-P Theorems demonstrate convergence if the integration is exact. It is not, and we shall rely on the patch test = 74
• Chapter 4 VARIOUS ELASTIC PROBLEMS
• 4.1 The Versatility of Iso-P Elements - Very little effort to code a new type of problem, given the shape function subroutine = 77
• 4.2 The Elastic Membrane - The only obstacle here is the differential geometry, but care is needed in using the element = 77
• 4.3 Thick Shefl Element : the Membrane Stack - A simple and versatile curved shell. The trick is to include the shear strain energy = 79
• 4.4 Thick Plate Element - The degenerate case : simple, but not competitive = 84
• 4.5 Body of Revolution with Pressure Loads etc. - A very widely used case = 85
• 4.6 Body of Revolution under Torsion - Little used, but it should be taught = 86
• 4.7 Body of Revolution with Sinusoidal Loading - Fourier analysis models any deformations : 3D approached via 2D = 89
• 4.8 Helical Body of Revolution, General Case - This is a larger problem computationally. Hermitian matrices are already worthwhile = 92
• 4.9 Beams in Torsion, approached directly via the warpingfunction, instead of St. Venant's bubble analogy, which obscures the physics = 94
• 4.10 Beams in Bending with Shear ; again warping is the nodal variable. Combining torsion with bending gives the shear centre etc = 95
• Chapter 5 NODAL LOADS FROM SHAPE FUNCTION ROUTINES
• 5.1 Terminology and Practice - We must use virtual work. With shape function constraints the nodal loads are indistinguishable from the actual loads = 99
• 5.2 Point loads for initial instruction = 99
• 5.2.1 Uniform Beam - The nodal loads from virtual work are surprisingly effective = 99
• 5.2.2 Iso-P Membrane - The nodal load from the shape function at the point of application, sometimes with a scalar product = 100
• 5.2.3 An Alternative to Fourier Analysis, which is never needed. Structural arguments suffice = 101
• 5.3 Distributed Loads are simply integrals of point loads = 102
• 5.3.1 Numerical integration - Linear pressure on a triangular membrane element = 102
• 5.3.2 The Taig Quadrilateral - In the event, this is merely a rehearsal of dimensional checks = 103
• 5.3.3 Pore-Water Pressure completely describes the action of the head of water on a dam = 104
• 5.4 Distributed Surface Forces are more difficult and involve some differential geometry = 104
• 5.5 Misfits, Prestrains and Thermal Loads are all treated as cases of forced assembly of parts whose geometry is wrong = 106
• 5.6 Internal forces - due to a stress field arising during iteration, are calculated with surprising ease in almost every case = 107
• 5.7 A Program = 108
• 5.7.1 Generalities - The program details are the major difficulty = 108
• 5.7.2 The Package the User Sees if he wants for example to add a new element = 108
• 5.7.3 Dictionary for ELFILE, NFUNC, SHELL, STRESS, and PRINPL to help the reader to understand the code = 109
• 5.7.4 The Program ELFILE etc. - in FORTRAN, with sufficient comments to understand at a first reading = 111
• Part Ⅱ ORGANIZATION
• Cbapter 6 PROBLEMS OF MANAGEMENT
• 6.1 Ugly Facts introduce the realities of being a chief programmer = 119
• 6.2 Overview of the Finite Element Process in the vernacular = 120
• 6.2.1 Input - the primitive approach, graphics terminals, and mesh generation routines = 120
• 6.2.2 Computer Diagnostics are indispensible, by consensus = 121
• 6.2.3 Element Stresses - during, or immediately after the back-substitution? = 122
• 6.2.4 Output - Editing and the problem of communicating efficiently = 122
• 6.2.5 Systems are now huge. Languages and the amateur programmer : diagnostics and the problem of inaccesible array dimensions = 123
• 6.2.6 Minicomputers - Small jobs at the right time are the most useful = 123
• 6.3 A Panacea : Numerical Integration? - Ease of programming, modularity, and superior bug-control. Other element formulators should learn = 124
• Cbapter 7 MATRIX-STRUCTURAL THEORY
• 7.1 The Two Aspects of Wisdom - It pays dividends to appreciate structural Principles from both a matrix and a physical view-point = 127
• 7.2 Nodal Variables are usually representable by arrows. If these are mutually perpendicular, they denote forces or deflections = 127
• 7.3 Reactions from Earth can confuse people. We re-emphasise the remedy in Section 1.4 = 128
• 7.4 Influence Surfaces are particularly easy in matrix form = 128
• 7.5 Perturbation Theorem - If a structure is perturbated, the increase in potential energy equals the strain energy of the perturbation = 129
• 7.6 Hierarchies of Idealisation - Matrix inequalities : a mesh is absolutely stiffer than a subdivided mesh, sometimes = 130
• 7.7 Elimination and Deletion in matrices are duals in the structural sense. Elimination in a matrix equals deletion in its inverse = 131
• 7.8 Part-Inversion means taking only certain variables across the equals sign. An important artifice preserves symmetry = 132
• 7.9 Strain Energy Theorem - Surprisingly, S due to forces and loads does not carry the complete information = 133
• 7.10 Potential Energy Theorem - However, the minimum U does = 133
• 7.11 Matrix Inequality Theorem - A > B survives any part-inversion. It is valid to speak of structure A being stiffer than B = 134
• 7.12 Re-Design Theorem - For a small change, keep the deflections the same, and the change in U is nearly correct = 134
• 7.13 Hierarchies of Elements - In particular it is valid to speak of formulation A being stiffer than B. Today, 'flexible' is usually good = 135
• 7.14 Fracture Mechanics - Van der Wahl's forces, with a clear understanding of potential energy = 135
• 7.15 Conclusions - Matrix-structural theory and the intelligent use of finite elements = 136
• Chapter 8 THE MATCHED SOLUTION
• 8.1 Motivation - Mainly it is to explain pathological cases to unsympathetic managers = 137
• 8.2 Linear Triangle Theorem - In a very general mesh, we give every node the correct displacement for an exact bending solution. This is also an exact finite element solution = 137
• 8.3 Beam Theorem - Provided the nodal forces are found by virtual work, the nodal slopes and deflections are exact in any assemblage of beams = 139
• 8.4 Stretched Cable on an Elastic Foundation - whose exact and finite element solution are compared analytically. The matched solution is a third case ; we artificially give the nodes the 'exact' values. It bounds U = 140
• 8.5 Practical Examples of Intrinsic Errors which have shocked people = 142
• 8.5.1 Bilinear Elements in Bending tend to lock because of spurious shears = 142
• 8.5.2 Linear Triangles in Bending tend to lock because of spurious hydrostatic stresses = 143
• 8.5.3 Linear Triangles : Pressurised Sphere is similar, but worse = 145
• 8.6 Barlow Points for Better Stresses - Some elements have points where the stresses are about ten times as accurate as at the corner nodes = 146
• 8.7 Strategies too numerous to summarise = 147
• Chapter 9 CONVERGENCE - THE PATCH TEST
• 9.1 The Many Traps - Mathematical vs. useful convergence = 149
• 9.2 Recommended Procedure - Neighbouring elements have sensibly equal stresses with fine mesh = 150
• 9.3 Nonstandard Beam Element, with discontinuities of deflection, passes the patch test = 151
• 9.4 The New Morality - Anything goes, but to keep out of trouble one must be careful = 154
• 9.5 The Iso-P Elements owe their commercial existence to the patch test, but the argument is tricky and slightly perverse = 157
• 9.6 Other Applications - Technical exercises, no practical value = 159
• 9.7 Interim Remarks on the Patch Test - Many developments will come. It is not trivial = 162
• Chapter 10 DEVELOPING AND IMPLEMENTING ELEMENTS
• 10.1 The Need for Discipline in programming extends far beyond the mere use of the patch test = 163
• 10.2 Progressive Testing, Debugging and Validation of elements : a thoughtfully conceived and practical sequence of computer runs = 163
• 10.3 Mechanisms : The Witches' Curse - An exhaustive account of the arguments used the phenomena observed. Very few general principles emerge = 166
• 10.4 Comparison Test by Eigenvalues - given two formulations. Easy to do but difficult to interpret = 170
• 10.5 Response of Repeating Patterns - Occasionally useful, akin to the matched solution. The patch test is a special case giving zero error = 171
• 10.6 The Programmer responds to chapter 6 = 172
• 10.6.1 The Computing Environment is typically very unsatisfactory, but if manufacturers exploit the new technology it could be nearly perfect = 172
• 10.6.2 Teamwork, Documentation and Readability are profitable = 174
• 10.6.3 Effective Problem Solving - A plan of campaign, successful in difficult programs. It also generates exceptionally good documentation = 175
• 10.7 Summary - We re-emphasise the need for self-discipline = 177
• PART Ⅲ SOLUTION TECHNIQUES
• Chapter 11 HOW NODES HANG TOGETHER : FRONT OR BAND?
• 11.1 Equation and Nodal Topology : the key for the sparse solvers widely used for finite elements = 181
• 11.2 Nodal Valency denotes the number of elements to which a node connects = 181
• 11.3 Extensive Meshes with a simple pattern enable us to estimate the number of assembled variables per element etc = 182
• 11.4 Substructures and Isolation - The 'front' insulates substructure A from a large force in substructure B = 185
• 11.5 The Use of Substructures is not generally recommended, although there are attractive possibilities = 186
• 11.6 Front Solutions - A creeping substructure engulfs each element in turn = 186
• 11.7 Band Solutions are illustrated by examples ; no reference in this chapter to banded matrices = 187
• 11.8 Estimates of the Frontwidth and Bandwidth are proposed, for simple extensive meshes. Frontal solutions are imperative when there are midside nodes = 188
• 11.9 Conclusions - Band of Front? - There are other considerations, too numerous to summarise. Mostly, the front is better = 189
• Chapter 12 ELEMENT ASSEMBLY AND EQUATION SOLVING
• 12.1 The Case for Ignorance - If a solver does everything one needs, leave well alone = 191
• 12.2 The Gauss-Jordan Solution - A good introduction, but almost useless for practical purposes = 191
• 12.3 Gauss Reduction - the norn. Positive-definitenesss helps, and hence also symmetry = 193
• 12.4 Re-solution without re-processing the coefficients = 194
• 12.5 Cholesky Decomposition as cultural background : seldom used = 195
• 12.6 Prescribed Variables - We advocate King's technique exclusively = 196
• 12.7 Housekeeping and Assembling Elements - Both are purely a question of re-addressing. The band solution = 197
• 12.8 The Sliced-Band Techniques could evolve today from the concept of virtual core = 199
• 12.9 Triangular Matrices to take advantage of symmetry = 199
• 12.10 Simultaneous Assembly and Reduction should be the norm = 201
• 12.11 Strategies of Element Creation etc, - Decisions on the solver may have wide ramifications = 202
• 12.12 Simultaneous Back-Substitution and Output is a profitable off-shoot of Section 12.10 = 202
• 12.13 The Frontal Technique = 203
• 12.13.1 Historical Roots - Who invented it? = 203
• 12.13.2 Active Variables are those whose coefficients must be in core = 204
• 12.13.3 Destinations to enter the submatrix. An example makes the addressing clear = 206
• 12.13.4 Nodes with Different Numbers of Variables - Hellen's artifice for Semiloof etc = 208
• 12.13.5 Zeros within the Front waste some time duting reduction. Selection of address by longevity = 208
• 12.13.6 Numerical Example - that of chapter 2 = 210
• Chapter 13 A FRONTAL SOLUTION PACKAGE
• 13.1 Programs and People - The computing environment, and the political system that governs it = 215
• 13.2 The Principles of Planning - Maximise safety, and recognise the germs of muddle and confusion = 215
• 13.3 Activities of the Subroutines - How the program handles its data : a verbal flow-diagram, with details = 216
• 13.4 The Problems of Housekeeping - The economics and security of dynamic core storage ; backing storage in solution and re-solution = 217
• 13.5 User Guide - Data for a small Semiloofjob = 219
• 13.6 Dictionary of TYRANT, NURSE, INPUT, MATRON, DOCTOR, and FRONT - i.e., the whole system apart from Semiloof = 221
• 13.7 The Program TYRANT etc. - with ample FORTRAN and textual comments throughout = 223
• 13.8 Diagnostic Messages - 84 error conditions for the system give the typical user much food for thought = 239
• 13.9 Fatal Errors from the Shape Function Subroutines - the Semiloof shell and beam contribute 15 more = 243
• Chapter 14 ROUNDOFF ERRORS
• 14.1 The Need for Criteria - Different bugs require different weapons. Useful insight is always physical = 245
• 14.2 Roundoff Forces are large when large numbers are encountered. Their order of magnitude is estimated from a check substitution = 246
• 14.3 The $$K^{-1}$$R Criterion interprets the roundoff forces into approximate errors of deflection, stress, etc = 247
• 14.4 The Diagonal Decay Criterion is cheap and easy, and gives an early warning = 248
• 14.5 The Diagonal Energy Criterion : the standard deviation of the strain energy. Not applicable with nonzero prescribed deflections. Most important, it leads to = 249
• 14.6 The Physical Causes of Roundoff - namely large deflections and stiff elements = 250
• Part Ⅳ TRENDS IN ELEMENT FORMULATION
• Chapter 15 FURTHER MATRIX - STRUCTURAL THEORY
• 15.1 Ends and Means - Physical insight into hybrid elements, Lagrange multipliers etc = 253
• 15.2 Levers to the Strain Energy Generators - A general pseudo-physical model is introduced via a numerically integrated element = 253
• 15.3 Surplus elastic constraints - In a typical finite element structure, there are many more strain energy generators than are needed = 257
• 15.4 The Plan Hybrids follow directly = 257
• 15.5 Lagrange Multipliers also follow = 258
• 15.6 The Right-Left Leverage System - More polynomial terms than the shape functions need = 260
• 15.7 Surplus Responses : Applications. Speculative and not very serious, but the implications are crucial = 263
• Chapter 16 PLATE BENDING
• 16.1 History of disasters, but the pioneering days are past = 265
• 16.2 The Quadratic Pyramid with midside nodes, to introduce the plate theorem = 265
• 16.3 Problems of Rotation = 267
• 16.3.1 The Bogner-Fox-Schmit rectangle with an extra nodal variable, $$W_{xy}$$ = 267
• 16.3.2 The Singularity Theorem - without $$W_{xy}$$ at the corners there must be a jump in the second derivatives : polynomials are out = 268
• 16.4 Escapes = 269
• 16.4.1 Rejection of $$W_{xy}$$ implies acceptance of a singularity. Complicated, disappointing = 269
• 16.4.2 Acceptance of $$W_{xy}$$ with Friends, i. e., $$W_{xx}$$ and $$W_{yy}$$ All the second derivatives are nodal variables = 270
• 16.4.3 Layered models with Discrete Kirchhoff avoid the theorem, and perform well = 271
• 16.4.4 Pian Hybrids - Invented to meet the challenge of plates. We predict that hybrids will be universal = 273
• 16.4.5 Stress Smoothing sometimes gives hybrids. It leads directly to polynomial shape function routines, which is helpful. P$${o \over a}$$l Bergan's triangle and Nagwa's bubble = 274
• 16.5 The Spin Off - Plane and 3D elements benefit from techniques invented for plate bending = 277
• Chapter 17 SHELLS
• 17.1 Overview - the foolish habit of regarding shells as super-plates = 281
• 17.2 Problems of Real Shells - The thin membrane is a better introduction = 281
• 17.3 Razzaque's Smooth Shell Element - A promising element, but limited = 285
• 17.4 The Problem of Corners and multiple junctions demand a different approach = 286
• 17.5 The Answer : to avoid nodal rotations at corners. An easy decision, but reinforcing beams present problems = 287
• 17.6 Further Theory on the Problem of Rotation refutes the idea of including in-plane rotation as a corner variable = 289
• Chapter 18 THE SEMILOOF BEAM AND SHELL
• 18.1 The Future of the She Element is assured, but in its present form it is too complicated = 291
• 18.2 Peculiarities of the Beam Element, which initially has excess variables : these are condensed out = 291
• 18.3 The Philosophy of the Beam is subtle. It deforms in shear : the shell does not. Most of the nodal rotations are independent of the slopes of the shell = 292
• 18.4 Dictionary for BEAM, LOFBEM, SFRBEM, SCALAR and VECTOR, comprising the whole beam package = 293
• 18.5 The Program BEAM etc. is not unduly difficult = 295
• 18.6 Haloof, etc. : Dictionary introduces the package for the shell = 301
• 18.7 The Program, HALOOF etc. is too complex for study unless the reader has special motivations = 304
• Part Ⅴ TRENDS IN SOLUTION TECHNIQUES
• Chapter 19 SYMMETRY
• 19.1 Attitudes Towards Symmetry : neither flippant nor over-serious = 317
• 19.2 The Pictorial Approach : a surprisingly rigorous way to confirm or disprove our first impressions = 317
• 19.2.1 The Cantilever Beam shows both the power and the pitfalls of the technique = 317
• 19.2.2 The Thick Plate : the need for careful drawing = 318
• 19.2.3 Finite Elements - An infinite mesh is more re-warding than a symmetric finite mesh = 319
• 19.3 Sectorial Symmetry - Less general than in chapter 20 = 320
• 19.4 Vibration of a symmetric body almost always exhibits symmetry or anti-symmetry in its normal modes = 321
• 19.5 The Staircase Problem - A difficult practical case : also a less familiar variant = 322
• 19.6 Postlude : attitudes again. Cost-effectiveness in avoiding expensive mistakes = 323
• Chapter 20 SECTORIAL SYMMETRY
• 20.1 Evolution and the Classroom - The assimilation of new techniques depends critically on effective teaching = 325
• 20.2 Extended Shape Functions - A deflection in one sector implies an equal deflection in the others = 325
• 20.3 Hermitian Forms : The General Case is derived by elementary trigonometry. The deflections at corresponding nodes in different sectors lie on a sine curve = 328
• 20.4 Worked Example - simple and artificial, to inspire confidence = 330
• 20.5 Finite Fourier Series - Which terms are needed for an exact solution with general loading? = 332
• 20.6 Further Possibilities - Following Rashid, we speculate on iterative techniques = 333
• Chapter 21 NONLINEARITY
• 21.1 Medical Aspects - Stress-induced diseases and programs that sometimes work. The two-bar paradox and other simple cases = 335
• 21.2 Residual Forces : Strain-to-Stress Approach - The most natural method = 337
• 21.3 The Newton and Modified Newton Strategies - M. N. is Preferred to N. but not with over-relaxation by guesswork = 338
• 21.4 The Stress-to-Strain Approach cannot be justified, but is widely used. Almost every finite element job is structurally redundant = 339
• 21.5 Modified Newton with Search is a very intelligent choice, given a suitable equation-solver = 341
• 21.6 Conjugate Search Directions add some complication and perform better. For some problems Rashid iteration is immensely attractive with conjugate Newton = 342
• Chapter 22 EIGENVALUES AND NUMERICAL STABILITY
• 22.1 Convergence of Iterative Procedures - Breakdown of the error vector into its component eigenvectors = 345
• 22.2 Rashid's Approach in Two Dimensions: modified Newton should always succeed = 346
• 22.3 Hermitian Matrices Made Real - The same case in different guise = 346
• 22.4 Body of Revolution with Cut-outs - A controversial and interesting development from Fourier analysis = 347
• 22.5 Bounds on the Eigenvalues - The smallest eigenvalue of the smallest element often bounds the eigenvdlues of the assemblage = 349
• 22.6 Plasticity - But often we must consider the extremes of the modulus matrix, or of some other local material Property, to predict whether modified Newton will succeed = 350
• 22.7 Time Marching, Instability - The roughest estimates of the bounds are good enough for trouble-shooting = 351
• 22.8 Time Marching and Noise is more insidious. We can estimate rough die-away rates or, guided by the scalar case, we can recommend different formulae = 353
• 22.9 Creep is a process in time that uses local bounds, i. e. material properties = 355
• 22.10 Large Deflections cannot be discussed. They are impossible with modified Newton unless a search is made in each iteration = 355
• Chapter 23 EIGENVALUES AND STRUCTURAL PROBLEMS
• 23.1 Structural Dynamics, an easy linearised theory = 357
• 23.2 Techniques of Solution - dynamic response algorithms that people use = 358
• 23.3 Design for Eigenvalues Rayleigh's principle can differentiate a natural frequency with respect to a design change. Optimization = 360
• 23.4 Calculation of Eigenvalues - Wilson's subspace technique leads to some interesting developments = 361
• Part Ⅵ SPECULATIONS
• Chapter 24 NON-STRUCTURAL PROBLEMS
• 24.1 The Two Routes ty Hydrodynamics - A wistful look at energy methods which bypass the Navier-Stokes equations = 367
• 24.2 Creeping Flow - Stream functions and vector potential = 368
• 24.3 Turbulence : Moving and Fictitious Boundaries - High Reynolds numbers present philosophical as well as numerical problems = 372
• 24.4 Green's Theorem unfortunately gives no clear physical image. However, it is easily remembered in a very general form = 374
• 24.5 Thermal Conduction is easy ;a teaching vehicle for the Galerkin approach using Green's theorem. = 375
• 24.6 Graphics and Statistics - A powerful theorem on higher-order splines = 377
• 24.7 Conclusion - A final plea for methods that yield physical insight = 379
• Chapter 25 IMPLICATIONS OF THE PATCH TEST
• 25.1 The Ivory Tower - Pure science may yield dividends eventually = 381
• 25.2 Are These Elements Possible? - There is no harm in speculating = 381
• 25.2.1 The Quadratic Hexagon - Definitely not possible = 381
• 25.2.2 The 12-Variable Plate Bending Quadrilateral has failed the cubic test. We think it will always fail = 383
• 25.2.3 The 14-Node Brick and Other Elements, all three desirable, but all fail a higher patch test, with enough variables to satisfy it. The lure of 'superpatch = 384
• 25.3 Boundary Force Theorems - The nodal tractions are independent of the formulation = 385
• 25.4 Small Element Theorems - These are tentative but appealing = 386
• 25.5 Superfluous Nodes,superfluous that is to the patch test satisfied = 386
• 25.6 Forces on a Side - These appear to be independent of the formulation, if superfluous nodes are absent = 387
• 25.7 The Mixability Theorem - If two elements pass a patch test, they can normally be mixed in another patch test = 387
• 25.8 Superpatch elements appear to be impossible, by their nature = 388
• 25.9 Valediction - The end of the exposition = 389
• Part Ⅶ THEORETICAL DETAILS
• Chapter 26 INTERPOLATION AND NUMERICAL INTEGRATION
• 26.1 Introduction - partⅦ is designed for browsing = 393
• 26.2 Interpolation - The Choices. Do present computer capabilities encourage the finite element style of interpolation? = 393
• 26.3 Lagrangian Interpolation 'flaps its wings' near the ends of the interval covered by the data = 395
• 26.4 Numerical Integration as a by-product of polynomial interpolation = 396
• 26.5 The Mechanics of Generating a Simple Rule describes the direct approach = 398
• 26.6 Segmental Integration including the Euler-Maclaurin rules = 399
• 26.7 Gaussian Integration is very effective, even for a step function = 400
• 26.8 Some Useful Integration Formulae for a simple curve, for,cubes, and for triangles = 403
• 26.9 Legendre-related Polynomials are successive integrals of $$p_n$$(x). Useful as bubble functions = 407
• 26.10 Techniques for Finding Shape Functions - Rescue missions: neutral functions. The Loof case = 408
• Chapter 27 MATRICES
• 27.1 Introduction and Definitions - suffixes and symmetry, etc. = 415
• 27.2 Ordinary Multiplication, definition only = 417
• 27.3 Extended Products and the distributive law = 417
• 27.4 Suffix Notation and the rule of repeated suffixes = 418
• 27.5 Double-Dot Product, Extended Scalar Products - All the terms in A times all in B. A useful transformation for a triple product = 418
• 27.6 Unit Matrix, alias Identity Matrix: the $$\delta _{ij}$$ of tensors = 419
• 27.7 Matrix Inverse - The pre-inverse equals the post-inverse = 419
• 27.8 Inverse Products - Remember to reverse the order = 419
• 27.9 Transposed Products - Transpose each, and reverse the order too = 419
• 27.10 Quadratic Forms are usually energies = 420
• 27.11 Derivatives of a Quadratic Form are particularly simple = 420
• 27.12 Gram-Schmit Orthogonalisation to make vectors mutually perpendicular = 421
• 27.13 Positive Definite and Semidefinite Matrices - a most important concept = 421
• 27.14 Matrix Inequalities are a natural by-product = 422
• 27.15 Orthogonal Matrices are introduced geometrically = 422
• 27.16 Rank is described : nothing is proved, or even argued = 422
• 27.17 Rank of Products : a usefull result = 423
• 27.18 Partitioning, distinguished from submatrices = 423
• 27.19 Pseudo-Reduction for Triple Products : Strain Energy - A simple and perennially usefull technique = 423
• 27.20 Cost of Equation Solving - How many multiplications are involved? = 424
• 27.21 Pivotting is needed to solve or invert, if a matrix is not symmetrical = 424
• 27.22 Inequality Theorem for Part-Inversion Inequalities are preserved - the algebraic proof = 425
• 27.23 Lagrange Multipliers are developed algebraically = 426
• 27.24 Inversion of Matrices Incorporating Leverage Matrices - Conditions that non-positive-definite matrices of chapter 16 are nonsingular 4 = 427
• 27.25 Eigenvalues, Eigenvectors are defined = 428
• 27.26 Rayleigh's Principle is proved by a simple argument = 428
• 27.27 The Fried-Tong-Treharne Inequality for eigenvalues : a useful result peculiar to finite elements = 429
• 27.28 Constraints and Interlaced Eigenvalues occasionally gives simple bounds = 429
• 27.29 The Power Method : the idea historically behind the Wilson sub- space techniques = 430
• 27.30 Normal Modes - Eigenvectors are mutually orthogonal in two different ways = 430
• 27.31 Relative Eigenvalues for Rashid in 2D This relates to the convergence of a modified Newton procedure in a special case = 431
• 27.32 Relative Eigenvalues of Semidefinite Matrices - We accept $$\lambda $$ ＝ 0/0 etc = 431
• 27.33 Other Eigenvalues Theorems - Positive, negative and zero $$\lambda $$ = 431
• 27.34 Matrix Series, e.g., for the square root of a matrix = 432
• 27.35 Determinants as the product of the pivots = 432
• 27.36 Real, Imaginary, Complex as applied to matrices = 432
• 27.37 Hermitian Matrices are effectively symmetrical, but complex = 433
• 27.38 Hermitian Forms are like quadratic forms = 433
• 27.39 Mixed Hermitian-Real Forms still allow more efficient processing = 434
• 27.40 Numerical Conditioning - A simple example = 435
• 27.41 Scaled Matrices set an identical conditioning problem when solving or inverting = 435
• 27.42 Notorious Matrices - with all terms nearly equal : and the Hilbert matrices, with a few specimen pivots = 435
• Chapter 28 VECTORS AND DIFFERENTIAL GEOMETRY
• 28.1 Vector Addition and Terminology - Displacement and deflections = 437
• 28.2 Scalar Product - The work done by a force = 437
• 28.3 Vector Product - Vector area ＝ load due to unit pressure = 437
• 28.4 Triple Scalar Product, or Box Product - The volume of a para- llelopiped : a determinant also = 438
• 28.5 Perspective by Vectors : much easier than with 4x4 matrices = 439
• 28.6 Derivatives in an iso-P Brick using the inverse Jacobian matrix to transform to x, y, z = 441
• 28.7 Elementary Volume as the integral of a triple scalar product, alias the Jacobian determinant = 442
• 28.8 Green's Theorem in the simple case = 443
• 28.9 Derivatives in an iso-P Membrane - a 2x2 Jacobian matrix in 3D space = 443
• 28.10 Nodal Axes for Layered Shells : an algorithm for erecting nodal and local axes = 444
• 28.11 Derivatives in Local Cartesian Axes - The second transformation uses an orthogonal Jacobian matrix = 445
• 28.12 Classical and Mindlin Plate Theories - Both emerge from the concept of the layered plate or shell = 446
• 28.13 The Iso-P Shell Theory for Semiloof also emerges thus. We attempt to convey these elusive ideas pictorially = 447
• 28.14 Area Coordinates = 451
• 28.14.1 General Concepts and Notation - Special techniques for first and second derivatives = 451
• 28.14.2 Taylor's Series - A simple form, as far as quadratic terms = 452
• 28.14.3 x-y Derivatives from L Derivatives including curved iso-P elements = 453
• 28.14.4 Local -Shell Derivatives involve special axes in the surface, but again a cyclic formula exists = 455
• 28.14.5 Plate Bending - A special case, yielding unfamiliar and attractive formulae = 456
• 28.14.6 Programming Techniques to economise in the length and number of statements = 458
• Chapter 29 STRESS AND STRAIN
• 29.1 Tensile Effects : an Introduction - The stretched wire = 461
• 29.2 Shear Effects : an Introduction : shown pictorially = 462
• 29.3 The isotropic Moduh in 3D : inversion of the 6x6 modulus matrix = 462
• 29.4 Plane Strain is trivial, but see Section 29.6 = 463
• 29.5 Plane Stress is less trivial. We must sometimes use the 'wrong' modulus : names are misleading = 463
• 29.6 Anisotropic Elasticity is dangerous in practice ; anistropy is common in nature. A perversion of plain strain = 464
• 29.7 The Stress Matrix - Usually written as a tensor. Our presentation is pictorial = 465
• 29.8 The Strain Matrix is argued physically = 466
• 29.9 Large Pre-Strains present a worrying philosophical problem = 468
• 29.10 Stress and Strain Vectors are a convenience denied to those who use tensors exclusively = 469
• 29.11 Strain Energy follows immediately, for small deflections = 470
• 29.12 Lagrangian Stresses use this definition for large deflections = 470
• 29.13 The Unsymmetrical Stress Matrix gives the correct virtual work in conjunction with changes in the Jacobian matrix = 471
• 29.14 The Euler Stress Matrix - Ordinary stresses, giving the expected tractions on a deformed area = 471
• 29.15 Incremental Stiffnesses - A laborious tensorial argument = 472
• 29.16 Euler Buckling is a simplifled form of 29.15 = 474
• 29.17 Unit Stress and Strain Vectors - A very useful artifice, and the difference between them is crucial = 475
• 29.18 Plastic Flow is Cumulative - A simple thought-experiment establishes that any plasticity law must be cumulative = 476
• 29.19 The Plastic Yield Surface - A concept that idealises and to some extent falsifies the response = 477
• 29.20 Incremental Strain is normal to Yield Surface - Unit stress and strain vectors. Another thought-experiment = 478
• 29.21 Perfect Plasticity, Additional Elastic Strains - A useful fiction, to introduce the incremental modulus matrix = 479
• 29.22 The Strain-Hardening Case - In theory, two methods for interpreting a test are acceptable = 479
• 29.23 Preferred matrix forms for Elasto-Plastic Modulus A Lagrange multiplier was used above : now discarded = 481
• 29.24 Principal Stresses as Eigenvalues - A method for differentiating a principal stress, by Rayleigh's principle = 481
• 29.25 Some Actual Surfaces which possess extreme symmetry if the material remains isotropic = 482
• 29.26 The Bauschinger Effect and Kinematic Hardening - An easy approach to a very complex phenomenon. Materials do not remain isotropic = 485
• SOLUTIONS = 487
• REFERENCES and Author Index = 511
• SUBJECT INDEX = 523