The Finite Element Method: Its Basis and Fundamentals
Sixth edition
O.C. Zienkiewicz, CBE, FRS UNESCO Professor of Numerical Methods in Engineering International Centre for Numerical Methods in Engineering, Barcelona Previously Director of the Institute for Numerical Methods in Engineering University of Wales, Swansea
R.L. Taylor J.Z. Zhu Professor in the Graduate School Senior Scientist Department of Civil and Environmental Engineering ESI USR&DInc. University of California at Berkeley 5850 Waterloo Road, Suite 140 Berkeley, California Columbia, Maryland
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First published in 1967 by McGraw-Hill Fifth edition published by Butterworth-Heinemann 2000 Reprinted 2002 Sixth edition 2005
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Typeset by Kolam Information Services P Ltd, Pondicherry, India Dedication
This book is dedicated to our wives Helen, Mary Lou and Song and our families for their support and patience during the preparation of this book, and also to all of our students and colleagues who over the years have contributed to our knowledge of the finite element method. In particular we would like to mention Professor Eugenio Onate˜ and his group at CIMNE for their help, encouragement and support during the preparation process. Preface
It is thirty-eight years since the The Finite Element Method in Structural and Continuum Mechanics was first published. This book, which was the first dealing with the finite element method, provided the basis from which many further developments occurred. The expanding research and field of application of finite elements led to the second edition in 1971, the third in 1977, the fourth as two volumes in 1989 and 1991 and the fifth as three volumes in 2000. The size of each of these editions expanded geometrically (from 272 pages in 1967 to the fifth edition of 1482 pages). This was necessary to do justice to a rapidly expanding field of professional application and research. Even so, much filtering of the contents was necessary to keep these editions within reasonable bounds. In the present edition we have decided not to pursue the course of having three contiguous volumes but rather we treat the whole work as an assembly of three separate works, each one capable of being used without the others and each one appealing perhaps to a different audience. Though naturally we recommend the use of the whole ensemble to people wishing to devote much of their time and study to the finite element method. In particular the first volume which was entitled The Finite Element Method: The Basis is now renamed The Finite Element Method: Its Basis and Fundamentals. This volume has been considerably reorganized from the previous one and is now, we believe, better suited for teaching fundamentals of the finite element method. The sequence of chapters has been somewhat altered and several examples of worked problems have been added to the text. A set of problems to be worked out by students has also been provided. In addition to its previous content this book has been considerably enlarged by including more emphasis on use of higher order shape functions in formulation of problems and a new chapter devoted to the subject of automatic mesh generation. A beginner in the finite element field will find very rapidly that much of the work of solving problems consists of preparing a suitable mesh to deal with the whole problem and as the size of computers has seemed to increase without limits the size of problems capable of being dealt with is also increasing. Thus, meshes containing sometimes more than several million nodes have to be prepared with details of the material interfaces, boundaries and loads being well specified. There are many books devoted exclusively to the subject of mesh generation but we feel that the essence of dealing with this difficult problem should be included here for those who wish to have a complete ‘encyclopedic’ knowledge of the subject. xiv Preface
The chapter on computational methods is much reduced by transferring the computer source program and user instructions to a web site.† This has the very substantial advantage of not only eliminating errors in program and manual but also in ensuring that the readers have the benefit of the most recent version of the program available at all times. The two further volumes form again separate books and here we feel that a completely different audience will use them. The first of these is entitled The Finite Element Method in Solid and Structural Mechanics and the second is a text entitled The Finite Element Method in Fluid Dynamics. Each of these two volumes is a standalone text which provides the full knowledge of the subject for those who have acquired an introduction to the finite element method through other texts. Of course the viewpoint of the authors introduced in this volume will be continued but it is possible to start at a different point. We emphasize here the fact that all three books stress the importance of considering the finite element method as a unique and whole basis of approach and that it contains many of the other numerical analysis methods as special cases. Thus, imagination and knowledge should be combined by the readers in their endeavours. The authors are particularly indebted to the International Center of Numerical Methods in Engineering (CIMNE) in Barcelona who have allowed their pre- and post-processing code (GiD) to be accessed from the web site. This allows such difficult tasks as mesh generation and graphic output to be dealt with efficiently. The authors are also grateful to Professors Eric Kasper and Jose Luis Perez-Aparicio for their careful scrutiny of the entire text and Drs Joaquim Peiro´ and C.K. Lee for their review of the new chapter on mesh generation. Resources to accompany this book Worked solutions to selected problems in this book are available online for teachers and lecturers who either adopt or recommend the text. Please visit http://books.elsevier.com/ manuals and follow the registration and log in instructions on screen.
OCZ, RLT and JZZ
† Complete source code and user manual for program FEAPpv may be obtained at no cost from the publisher’s web page: http://books.elsevier.com/companions or from the authors’ web page: http://www.ce.berkeley.edu/˜rlt Contents
Preface xiii
1 The standard discrete system and origins of the finite element method 1 1.1 Introduction 1 1.2 The structural element and the structural system 3 1.3 Assembly and analysis of a structure 5 1.4 The boundary conditions 6 1.5 Electrical and fluid networks 7 1.6 The general pattern 9 1.7 The standard discrete system 10 1.8 Transformation of coordinates 11 1.9 Problems 13 2 A direct physical approach to problems in elasticity: plane stress 19 2.1 Introduction 19 2.2 Direct formulation of finite element characteristics 20 2.3 Generalization to the whole region – internal nodal force concept abandoned 31 2.4 Displacement approach as a minimization of total potential energy 34 2.5 Convergence criteria 37 2.6 Discretization error and convergence rate 38 2.7 Displacement functions with discontinuity between elements – non-conforming elements and the patch test 39 2.8 Finite element solution process 40 2.9 Numerical examples 40 2.10 Concluding remarks 46 2.11 Problems 47 3 Generalization of the finite element concepts. Galerkin-weighted residual and variational approaches 54 3.1 Introduction 54 3.2 Integral or ‘weak’ statements equivalent to the differential equations 57 3.3 Approximation to integral formulations: the weighted residual- Galerkin method 60 viii Contents
3.4 Virtual work as the ‘weak form’ of equilibrium equations for analysis of solids or fluids 69 3.5 Partial discretization 71 3.6 Convergence 74 3.7 What are ‘variational principles’? 76 3.8 ‘Natural’ variational principles and their relation to governing differential equations 78 3.9 Establishment of natural variational principles for linear, self-adjoint, differential equations 81 3.10 Maximum, minimum, or a saddle point? 83 3.11 Constrained variational principles. Lagrange multipliers 84 3.12 Constrained variational principles. Penalty function and perturbed lagrangian methods 88 3.13 Least squares approximations 92 3.14 Concluding remarks – finite difference and boundary methods 95 3.15 Problems 97 4 ‘Standard’ and ‘hierarchical’ element shape functions: some general families of C0 continuity 103 4.1 Introduction 103 4.2 Standard and hierarchical concepts 104 4.3 Rectangular elements – some preliminary considerations 107 4.4 Completeness of polynomials 109 4.5 Rectangular elements – Lagrange family 110 4.6 Rectangular elements – ‘serendipity’ family 112 4.7 Triangular element family 116 4.8 Line elements 119 4.9 Rectangular prisms – Lagrange family 120 4.10 Rectangular prisms – ‘serendipity’ family 121 4.11 Tetrahedral elements 122 4.12 Other simple three-dimensional elements 125 4.13 Hierarchic polynomials in one dimension 125 4.14 Two- and three-dimensional, hierarchical elements of the ‘rectangle’ or ‘brick’ type 128 4.15 Triangle and tetrahedron family 128 4.16 Improvement of conditioning with hierarchical forms 130 4.17 Global and local finite element approximation 131 4.18 Elimination of internal parameters before assembly – substructures 132 4.19 Concluding remarks 134 4.20 Problems 134 5 Mapped elements and numerical integration – ‘infinite’ and ‘singularity elements’ 138 5.1 Introduction 138 5.2 Use of ‘shape functions’ in the establishment of coordinate transformations 139 5.3 Geometrical conformity of elements 143 5.4 Variation of the unknown function within distorted, curvilinear elements. Continuity requirements 143 Contents ix
5.5 Evaluation of element matrices. Transformation in ξ, η, ζ coordinates 145 5.6 Evaluation of element matrices. Transformation in area and volume coordinates 148 5.7 Order of convergence for mapped elements 151 5.8 Shape functions by degeneration 153 5.9 Numerical integration – one dimensional 160 5.10 Numerical integration – rectangular (2D) or brick regions (3D) 162 5.11 Numerical integration – triangular or tetrahedral regions 164 5.12 Required order of numerical integration 164 5.13 Generation of finite element meshes by mapping. Blending functions 169 5.14 Infinite domains and infinite elements 170 5.15 Singular elements by mapping – use in fracture mechanics, etc. 176 5.16 Computational advantage of numerically integrated finite elements 177 5.17 Problems 178
6 Problems in linear elasticity 187 6.1 Introduction 187 6.2 Governing equations 188 6.3 Finite element approximation 201 6.4 Reporting of results: displacements, strains and stresses 207 6.5 Numerical examples 209 6.6 Problems 217
7 Field problems – heat conduction, electric and magnetic potential and fluid flow 229 7.1 Introduction 229 7.2 General quasi-harmonic equation 230 7.3 Finite element solution process 233 7.4 Partial discretization – transient problems 237 7.5 Numerical examples – an assessment of accuracy 239 7.6 Concluding remarks 253 7.7 Problems 253
8 Automatic mesh generation 264 8.1 Introduction 264 8.2 Two-dimensional mesh generation – advancing front method 266 8.3 Surface mesh generation 286 8.4 Three-dimensional mesh generation – Delaunay triangulation 303 8.5 Concluding remarks 323 8.6 Problems 323
9 The patch test, reduced integration, and non-conforming elements 329 9.1 Introduction 329 9.2 Convergence requirements 330 9.3 The simple patch test (tests A and B) – a necessary condition for convergence 332 9.4 Generalized patch test (test C) and the single-element test 334 9.5 The generality of a numerical patch test 336 9.6 Higher order patch tests 336 x Contents
9.7 Application of the patch test to plane elasticity elements with ‘standard’ and ‘reduced’ quadrature 337 9.8 Application of the patch test to an incompatible element 343 9.9 Higher order patch test – assessment of robustness 347 9.10 Concluding remarks 347 9.11 Problems 350 10 Mixed formulation and constraints – complete field methods 356 10.1 Introduction 356 10.2 Discretization of mixed forms – some general remarks 358 10.3 Stability of mixed approximation. The patch test 360 10.4 Two-field mixed formulation in elasticity 363 10.5 Three-field mixed formulations in elasticity 370 10.6 Complementary forms with direct constraint 375 10.7 Concluding remarks – mixed formulation or a test of element ‘robustness’ 379 10.8 Problems 379 11 Incompressible problems, mixed methods and other procedures of solution 383 11.1 Introduction 383 11.2 Deviatoric stress and strain, pressure and volume change 383 11.3 Two-field incompressible elasticity (u–p form) 384 11.4 Three-field nearly incompressible elasticity (u–p–εv form) 393 11.5 Reduced and selective integration and its equivalence to penalized mixed problems 398 11.6 A simple iterative solution process for mixed problems: Uzawa method 404 11.7 Stabilized methods for some mixed elements failing the incompressibility patch test 407 11.8 Concluding remarks 421 11.9 Problems 422 12 Multidomain mixed approximations – domain decomposition and ‘frame’ methods 429 12.1 Introduction 429 12.2 Linking of two or more subdomains by Lagrange multipliers 430 12.3 Linking of two or more subdomains by perturbed lagrangian and penalty methods 436 12.4 Interface displacement ‘frame’ 442 12.5 Linking of boundary (or Trefftz)-type solution by the ‘frame’ of specified displacements 445 12.6 Subdomains with ‘standard’ elements and global functions 451 12.7 Concluding remarks 451 12.8 Problems 451 13 Errors, recovery processes and error estimates 456 13.1 Definition of errors 456 13.2 Superconvergence and optimal sampling points 459 13.3 Recovery of gradients and stresses 465 Contents xi
13.4 Superconvergent patch recovery – SPR 467 13.5 Recovery by equilibration of patches – REP 474 13.6 Error estimates by recovery 476 13.7 Residual-based methods 478 13.8 Asymptotic behaviour and robustness of error estimators – the Babuˇskapatch test 488 13.9 Bounds on quantities of interest 490 13.10 Which errors should concern us? 494 13.11 Problems 495 14 Adaptive finite element refinement 500 14.1 Introduction 500 14.2 Adaptive h-refinement 503 14.3 p-refinement and hp-refinement 514 14.4 Concluding remarks 518 14.5 Problems 520 15 Point-based and partition of unity approximations. Extended finite element methods 525 15.1 Introduction 525 15.2 Function approximation 527 15.3 Moving least squares approximations – restoration of continuity of approximation 533 15.4 Hierarchical enhancement of moving least squares expansions 538 15.5 Point collocation – finite point methods 540 15.6 Galerkin weighting and finite volume methods 546 15.7 Use of hierarchic and special functions based on standard finite elements satisfying the partition of unity requirement 549 15.8 Concluding remarks 558 15.9 Problems 558 16 The time dimension – semi-discretization of field and dynamic problems and analytical solution procedures 563 16.1 Introduction 563 16.2 Direct formulation of time-dependent problems with spatial finite element subdivision 563 16.3 General classification 570 16.4 Free response – eigenvalues for second-order problems and dynamic vibration 571 16.5 Free response – eigenvalues for first-order problems and heat conduction, etc. 576 16.6 Free response – damped dynamic eigenvalues 578 16.7 Forced periodic response 579 16.8 Transient response by analytical procedures 579 16.9 Symmetry and repeatability 583 16.10 Problems 584 17 The time dimension – discrete approximation in time 589 17.1 Introduction 589 xii Contents
17.2 Simple time-step algorithms for the first-order equation 590 17.3 General single-step algorithms for first- and second-order equations 600 17.4 Stability of general algorithms 609 17.5 Multistep recurrence algorithms 615 17.6 Some remarks on general performance of numerical algorithms 618 17.7 Time discontinuous Galerkin approximation 619 17.8 Concluding remarks 624 17.9 Problems 626 18 Coupled systems 631 18.1 Coupled problems – definition and classification 631 18.2 Fluid–structure interaction (Class I problems) 634 18.3 Soil–pore fluid interaction (Class II problems) 645 18.4 Partitioned single-phase systems – implicit–explicit partitions (Class I problems) 653 18.5 Staggered solution processes 655 18.6 Concluding remarks 660 19 Computer procedures for finite element analysis 664 19.1 Introduction 664 19.2 Pre-processing module: mesh creation 664 19.3 Solution module 666 19.4 Post-processor module 666 19.5 User modules 667 Appendix A: Matrix algebra 668 Appendix B: Tensor-indicial notation in the approximation of elasticity problems 674 Appendix C: Solution of simultaneous linear algebraic equations 683 Appendix D: Some integration formulae for a triangle 692 Appendix E: Some integration formulae for a tetrahedron 693 Appendix F: Some vector algebra 694 Appendix G: Integration by parts in two or three dimensions (Green’s theorem) 699 Appendix H: Solutions exact at nodes 701 Appendix I: Matrix diagonalization or lumping 704 Author index 711 Subject index 719 1
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 over- simplified 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, sci- entists 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 mathemati- cians 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 ap- proaches 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 por- tions 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 2 The standard discrete system and origins of the finite element method
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 force– displacement 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 The structural element and the structural system 3
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
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
Table 1.1 History of approximate methods ENGINEERING MATHEMATICS
Trial Finite functions differences Rayleigh 187022 Richardson 19101 Ritz 190823 Liebman 191824 Southwell 19462
Variational Weighted methods residuals − − − − − − − − − − − − − − − − − −
22 25 Rayleigh 1870 −−− Gauss 1795
23 26 Ritz 1908 −−− −−→ Galerkin 1915 −−→ Biezeno–Koch 192327
Structural Piecewise analogue continuous substitution trial functions −−→ Hrenikoff 19418 Courant 194329 McHenry 194328 Prager–Synge 194730 Newmark 19499 Argyris 195519 − −−→ Zienkiewicz 196431 −−−−−−−−−−
Direct Variational continuum −−−−−−−−−− −−−−−−−−−− finite elements differences −−→
10 32 Turner et al. 1956 −−→ −−→ Varga 1962 Wilkins 196433 PRESENT-DAY FINITE ELEMENT METHOD 4 The standard discrete system and origins of the finite element method
Y4
3 4 (2) X p 4 y (1) (4)
(3) 1 2 5 6
x
v3(V3)
3 u (U ) p 3 3 y (1) Nodes 2
1 x A typical element (1)
Fig. 1.1 A typical structure built up from interconnected elements.
displacements are defined by appropriate components (U, V and u, v) in a common co- ordinate 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 ⎧ ⎫ ⎨⎪q1⎬⎪ 1 U 1 = 1 1 = 1 , . q q2 q1 etc (1.1) ⎪ ⎪ V1 ⎩ 1⎭ q3 and for the corresponding nodal displacements ⎧ ⎫ ⎨⎪u1⎬⎪ 1 u 1 = 1 1 = 1 , . u u2 u1 etc (1.2) ⎪ ⎪ v1 ⎩ 1⎭ u3 Assuming linear elastic behaviour of the element, the characteristic relationship will always be of the form q1 = K1u1 + f 1 (1.3) in which f 1 represents the nodal forces required to balance any concentrated or distributed loads acting on the element. The first of the terms represents the forces induced by dis- placement of the nodes. The matrix Ke is known as the stiffness matrix for the element (e). Equation (1.3) is illustrated by an example of an element with three nodes with the interconnection points capable of transmitting only two components of force. Clearly, the
†A limited knowledge of matrix algebra will be assumed throughout this book. This is necessary for reasonable conciseness and forms a convenient book-keeping form. For readers not familiar with the subject a brief appendix (Appendix A) is included in which sufficient principles of matrix algebra are given to follow the development intelligently. Matrices and vectors will be distinguished by bold print throughout. Assembly and analysis of a structure 5 same arguments and definitions will apply generally. An element (2) of the hypothetical structure will possess only two points of interconnection; others may have quite a large number of such points. Quite generally, therefore, ⎧ e ⎫ ⎧ ⎫ ⎪q ⎪ ⎪ 1 ⎪ ⎪u1 ⎪ ⎨⎪ e ⎬⎪ ⎨⎪ ⎬⎪ q2 u2 e = e = . q ⎪ . ⎪ and u ⎪ . ⎪ (1.4) ⎪ . ⎪ ⎪ . ⎪ ⎩⎪ ⎭⎪ ⎩ ⎭ e um qm e with each qa and ua possessing the same number of components or degrees of freedom. The stiffness matrices of the element will clearly always be square and of the form ⎡ ⎤ e e ··· e K11 K12 K1m ⎢ . . ⎥ ⎢ e .. . ⎥ e ⎢K21 ⎥ K = ⎢ . . . ⎥ (1.5) ⎣ . . . ⎦ e ··· ··· e Km1 Kmm e e l × l in which K11, K12, etc., are submatrices which are again square and of the size , where l is the number of force and displacement components to be considered at each node. The element properties were assumed to follow a simple linear relationship. In principle, similar relationships could be established for non-linear materials, but discussion of such problems will be postponed at this stage. In most cases considered in this volume the element matrices Ke will be symmetric.
1.3 Assembly and analysis of a structure
Consider again the hypothetical structure of Fig. 1.1. To obtain a complete solution the two conditions of (a) displacement compatibility and (b) equilibrium have to be satisfied throughout. Any system of nodal displacements u: ⎧ ⎫ ⎨⎪u1⎬⎪ . u = . (1.6) ⎩⎪ ⎭⎪ un listed now for the whole structure in which all the elements participate, automatically satisfies the first condition. As the conditions of overall equilibrium have already been satisfied within an element, all that is necessary is to establish equilibrium conditions at the nodes (or assembly points) of the structure. The resulting equations will contain the displacements as unknowns, and once these have been solved the structural problem is determined. The internal forces in elements, or the stresses, can easily be found by using the characteristics established a priori for each element. 6 The standard discrete system and origins of the finite element method
If now the equilibrium conditions of a typical node, a, are to be established, the sum of the component forces contributed by the elements meeting at the node are simply accumulated. Thus, considering all the force components we have m e 1 2 qa = qa + qa +···=0 (1.7) e = 1
1 2 in which qa is the force contributed to node a by element 1, qa by element 2, etc. Clearly, only the elements which include point a will contribute non-zero forces, but for conciseness in notation all the elements are included in the summation. Substituting the forces contributing to node a from the definition (1.3) and noting that nodal variables ua are common (thus omitting the superscript e), we have m m m e + e +···+ e = Ka1 u1 Ka2 u2 fi 0 (1.8) e=1 e=1 e=1 The summation again only concerns the elements which contribute to node a. If all such equations are assembled we have simply
Ku + f = 0 (1.9)
in which the submatrices are m m e e Kab = Kab and fa = fa (1.10) e=1 e=1 with summations including all elements. This simple rule for assembly is very convenient because as soon as a coefficient for a particular element is found it can be put immediately into the appropriate ‘location’ specified in the computer. This general assembly process can be found to be the common and fundamental feature of all finite element calculations and should be well understood by the reader. If different types of structural elements are used and are to be coupled it must be remem- bered that at any given node the rules of matrix summation permit this to be done only if these are of identical size. The individual submatrices to be added have therefore to be built up of the same number of individual components of force or displacement.
1.4 The boundary conditions
The system of equations resulting from Eq. (1.9) can be solved once the prescribed support displacements have been substituted. In the example of Fig. 1.1, where both components of displacement of nodes 1 and 6 are zero, this will mean the substitution of 0 u = u = 1 6 0
which is equivalent to reducing the number of equilibrium equations (in this instance 12) by deleting the first and last pairs and thus reducing the total number of unknown displacement Electrical and fluid networks 7 components to eight. It is, nevertheless, often convenient to assemble the equation according to relation (1.9) so as to include all the nodes. Clearly, without substitution of a minimum number of prescribed displacements to pre- vent rigid body movements of the structure, it is impossible to solve this system, because the displacements cannot be uniquely determined by the forces in such a situation. This physically obvious fact will be interpreted mathematically as the matrix K being singular, i.e., not possessing an inverse. The prescription of appropriate displacements after the assembly stage will permit a unique solution to be obtained by deleting appropriate rows and columns of the various matrices. If all the equations of a system are assembled, their form is
K11u1 + K12u2 +···+f1 = 0
K21u1 + K22u2 +···+f2 = 0 (1.11) etc. and it will be noted that if any displacement, such as u1 = u¯ 1, is prescribed then the total ‘force’ f1 cannot be simultaneously specified and remains unknown. The first equa- tion could then be deleted and substitution of known values u¯ 1 made in the remaining equations. When all the boundary conditions are inserted the equations of the system can be solved for the unknown nodal displacements and the internal forces in each element obtained.
1.5 Electrical and fluid networks
Identical principles of deriving element characteristics and of assembly will be found in many non-structural fields. Consider, for instance, the assembly of electrical resistances shown in Fig. 1.2. If a typical resistance element, ab, is isolated from the system we can write, by Ohm’s law, the relation between the currents (J ) entering the element at the ends and the end voltages (V ) as J e = 1 (V − V ) J e = 1 (V − V ) a re a b and b re b a (1.12) or in matrix form e J − Va a = 1 1 1 e e Jb r −11Vb which in our standard form is simply
Je = KeVe (1.13)
This form clearly corresponds to the stiffness relationship (1.3); indeed if an external current were supplied along the length of the element the element ‘force’ terms could also be found. To assemble the whole network the continuity of the voltage (V ) at the nodes is assumed and a current balance imposed there. With no external input of current at node a we must 8 The standard discrete system and origins of the finite element method
b
Pa Jb,Vb b a
r e
Ja ,Va a
Fig. 1.2 A network of electrical resistances.
have, with complete analogy to Eq. (1.8),