The Finite Element Method Fifth edition Volume 1: The Basis Professor O.C Zienkiewicz, CBE, FRS, FREng is Professor Emeritus and Director of the Institute for Numerical Methods in Engineering at the University of Wales, Swansea, UK He holds the UNESCO Chair of Numerical Methods in Engineering at the Technical University of Catalunya, Barcelona, Spain He was the head of the Civil Engineering Department at the University of Wales Swansea between 1961 and 1989 He established that department as one of the primary centres of ®nite element research In 1968 he became the Founder Editor of the International Journal for Numerical Methods in Engineering which still remains today the major journal in this ®eld The recipient of 24 honorary degrees and many medals, Professor Zienkiewicz is also a member of ®ve academies ± an honour he has received for his many contributions to the fundamental developments of the ®nite element method In 1978, he became a Fellow of the Royal Society and the Royal Academy of Engineering This was followed by his election as a foreign member to the U.S Academy of Engineering (1981), the Polish Academy of Science (1985), the Chinese Academy of Sciences (1998), and the National Academy of Science, Italy (Academia dei Lincei) (1999) He published the ®rst edition of this book in 1967 and it remained the only book on the subject until 1971 Professor R.L Taylor has more than 35 years' experience in the modelling and simulation of structures and solid continua including two years in industry In 1991 he was elected to membership in the U.S National Academy of Engineering in recognition of his educational and research contributions to the ®eld of computational mechanics He was appointed as the T.Y and Margaret Lin Professor of Engineering in 1992 and, in 1994, received the Berkeley Citation, the highest honour awarded by the University of California, Berkeley In 1997, Professor Taylor was made a Fellow in the U.S Association for Computational Mechanics and recently he was elected Fellow in the International Association of Computational Mechanics, and was awarded the USACM John von Neumann Medal Professor Taylor has written several computer programs for ®nite element analysis of structural and non-structural systems, one of which, FEAP, is used world-wide in education and research environments FEAP is now incorporated more fully into the book to address non-linear and ®nite deformation problems Front cover image: A Finite Element Model of the world land speed record (765.035 mph) car THRUST SSC The analysis was done using the ®nite element method by K Morgan, O Hassan and N.P Weatherill at the Institute for Numerical Methods in Engineering, University of Wales Swansea, UK (see K Morgan, O Hassan and N.P Weatherill, `Why didn't the supersonic car ¯y?', Mathematics Today, Bulletin of the Institute of Mathematics and Its Applications, Vol 35, No 4, 110±114, Aug 1999) The Finite Element Method Fifth edition Volume 1: The Basis O.C Zienkiewicz, CBE, FRS, FREng UNESCO Professor of Numerical Methods in Engineering International Centre for Numerical Methods in Engineering, Barcelona Emeritus Professor of Civil Engineering and Director of the Institute for Numerical Methods in Engineering, University of Wales, Swansea R.L Taylor Professor in the Graduate School Department of Civil and Environmental Engineering University of California at Berkeley Berkeley, California OXFORD AUCKLAND BOSTON JOHANNESBURG MELBOURNE NEW DELHI Butterworth-Heinemann Linacre House, Jordan Hill, Oxford OX2 8DP 225 Wildwood Avenue, Woburn, MA 01801-2041 A division of Reed Educational and Professional Publishing Ltd First published in 1967 by McGraw-Hill Fifth edition published by Butterworth-Heinemann 2000 # O.C Zienkiewicz and R.L Taylor 2000 All rights reserved No part of this publication may be reproduced in any material form (including photocopying or storing in any medium by electronic means and whether or not transiently or incidentally to some other use of this publication) without the written permission of the copyright holder except in accordance with the provisions of the Copyright, Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency Ltd, 90 Tottenham Court Road, London, England W1P 9HE Applications for the copyright holder's written permission to reproduce any part of this publication should be addressed to the publishers British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloguing in Publication Data A catalogue record for this book is available from the Library of Congress ISBN 7506 5049 Published with the cooperation of CIMNE, the International Centre for Numerical Methods in Engineering, Barcelona, Spain (www.cimne.upc.es) Typeset by Academic & Technical Typesetting, Bristol Printed and bound by MPG Books Ltd Dedication This book is dedicated to our wives Helen and Mary Lou 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 ®nite element method In particular we would like to mention Professor Eugenio Onate and his group at CIMNE for Ä their help, encouragement and support during the preparation process Contents Preface xv Some 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 preliminaries: the standard discrete system Introduction The structural element and the structural system Assembly and analysis of a structure The boundary conditions Electrical and ¯uid networks The general pattern The standard discrete system Transformation of coordinates References A direct approach to problems in elasticity 2.1 Introduction 2.2 Direct formulation of ®nite element characteristics 2.3 Generalization to the whole region 2.4 Displacement approach as a minimization of total potential energy 2.5 Convergence criteria 2.6 Discretization error and convergence rate 2.7 Displacement functions with discontinuity between elements 2.8 Bound on strain energy in a displacement formulation 2.9 Direct minimization 2.10 An example 2.11 Concluding remarks References Generalization of the ®nite element concepts Galerkin-weighted residual and variational approaches 3.1 Introduction 3.2 Integral or `weak' statements equivalent to the di€erential equations 3.3 Approximation to integral formulations 3.4 Virtual work as the `weak form' of equilibrium equations for analysis of solids or ¯uids 1 10 12 14 15 16 18 18 19 26 29 31 32 33 34 35 35 37 37 39 39 42 46 53 viii Contents 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 Partial discretization Convergence What are `variational principles'? `Natural' variational principles and their relation to governing di€erential equations Establishment of natural variational principles for linear, self-adjoint di€erential equations Maximum, minimum, or a saddle point? Constrained variational principles Lagrange multipliers and adjoint functions Constrained variational principles Penalty functions and the least square method Concluding remarks References 55 58 60 stress and plane strain Introduction Element characteristics Examples ± an assessment of performance Some practical applications Special treatment of plane strain with an incompressible material Concluding remark References 87 87 87 97 100 110 111 111 62 66 69 70 76 82 84 Plane 4.1 4.2 4.3 4.4 4.5 4.6 Axisymmetric stress analysis 5.1 Introduction 5.2 Element characteristics 5.3 Some illustrative examples 5.4 Early practical applications 5.5 Non-symmetrical loading 5.6 Axisymmetry ± plane strain and plane stress References 112 112 112 121 123 124 124 126 Three-dimensional stress analysis 6.1 Introduction 6.2 Tetrahedral element characteristics 6.3 Composite elements with eight nodes 6.4 Examples and concluding remarks References 127 127 128 134 135 139 Steady-state ®eld problems ± heat conduction, electric and magnetic potential, ¯uid ¯ow, etc 7.1 Introduction 7.2 The general quasi-harmonic equation 7.3 Finite element discretization 7.4 Some economic specializations 7.5 Examples ± an assessment of accuracy 7.6 Some practical applications 140 140 141 143 144 146 149 Contents ix 7.7 Concluding remarks References `Standard' and `hierarchical' element shape functions: some general families of C0 continuity 8.1 Introduction 8.2 Standard and hierarchical concepts 8.3 Rectangular elements ± some preliminary considerations 8.4 Completeness of polynomials 8.5 Rectangular elements ± Lagrange family 8.6 Rectangular elements ± `serendipity' family 8.7 Elimination of internal variables before assembly ± substructures 8.8 Triangular element family 8.9 Line elements 8.10 Rectangular prisms ± Lagrange family 8.11 Rectangular prisms ± `serendipity' family 8.12 Tetrahedral elements 8.13 Other simple three-dimensional elements 8.14 Hierarchic polynomials in one dimension 8.15 Two- and three-dimensional, hierarchic, elements of the `rectangle' or `brick' type 8.16 Triangle and tetrahedron family 8.17 Global and local ®nite element approximation 8.18 Improvement of conditioning with hierarchic forms 8.19 Concluding remarks References Mapped elements and numerical integration ± `in®nite' and `singularity' elements 9.1 Introduction 9.2 Use of `shape functions' in the establishment of coordinate transformations 9.3 Geometrical conformability of elements 9.4 Variation of the unknown function within distorted, curvilinear elements Continuity requirements 9.5 Evaluation of element matrices (transformation in , ,  coordinates) 9.6 Element matrices Area and volume coordinates 9.7 Convergence of elements in curvilinear coordinates 9.8 Numerical integration ± one-dimensional 9.9 Numerical integration ± rectangular (2D) or right prism (3D) regions 9.10 Numerical integration ± triangular or tetrahedral regions 9.11 Required order of numerical integration 9.12 Generation of ®nite element meshes by mapping Blending functions 9.13 In®nite domains and in®nite elements 9.14 Singular elements by mapping for fracture mechanics, etc 161 161 164 164 165 168 171 172 174 177 179 183 184 185 186 190 190 193 193 196 197 198 198 200 200 203 206 206 208 211 213 217 219 221 223 226 229 234 x Contents 9.15 9.16 9.17 9.18 A computational advantage of numerically integrated ®nite elements Some practical examples of two-dimensional stress analysis Three-dimensional stress analysis Symmetry and repeatability References 10 The patch test, reduced integration, and non-conforming elements 10.1 Introduction 10.2 Convergence requirements 10.3 The simple patch test (tests A and B) ± a necessary condition for convergence 10.4 Generalized patch test (test C) and the single-element test 10.5 The generality of a numerical patch test 10.6 Higher order patch tests 10.7 Application of the patch test to plane elasticity elements with `standard' and `reduced' quadrature 10.8 Application of the patch test to an incompatible element 10.9 Generation of incompatible shape functions which satisfy the patch test 10.10 The weak patch test ± example 10.11 Higher order patch test ± assessment of robustness 10.12 Conclusion References 11 Mixed formulation and constraints± complete ®eld methods 11.1 Introduction 11.2 Discretization of mixed forms ± some general remarks 11.3 Stability of mixed approximation The patch test 11.4 Two-®eld mixed formulation in elasticity 11.5 Three-®eld mixed formulations in elasticity 11.6 An iterative method solution of mixed approximations 11.7 Complementary forms with direct constraint 11.8 Concluding remarks ± mixed formulation or a test of element `robustness' References 12 Incompressible materials, mixed methods and other procedures of solution 12.1 Introduction 12.2 Deviatoric stress and strain, pressure and volume change 12.3 Two-®eld incompressible elasticity (u±p form) 12.4 Three-®eld nearly incompressible elasticity (u±p±"v form) 12.5 Reduced and selective integration and its equivalence to penalized mixed problems 12.6 A simple iterative solution process for mixed problems: Uzawa method 236 237 238 244 246 250 250 251 253 255 257 257 258 264 268 270 271 273 274 276 276 278 280 284 291 298 301 304 304 307 307 307 308 314 318 323 Moving least square approximations ± restoration of continuity of approximation where n ˆ H…x† ˆ wk …x ÿ xk †p…xk †T p…xk † …16:19† kˆ1 and gj …x† ˆ wj …x ÿ xj †p…xj †T …16:20† In matrix form the arrays H…x† and g…x† may be written as H…x† ˆ PT w…Áx†P …16:21† g…x† ˆ w…Áx†P in which P T T w…Áx† ˆ T R w1 …x ÿ x1 † ÁÁÁ ÁÁÁ F F F w2 …x ÿ x2 † F F F FF F ÁÁÁ F F F ÁÁÁ ÁÁÁ Q U U U S …16:22† wn …x ÿ xn † The moving least square algorithm produces solutions for a which depend continuously on the point selected for each ®t The approximation for the function u…x† now may be written as ” u…x† ˆ n ˆ j ˆ1 Nj …x†~j u …16:23† where Nj …x† ˆ p…x†Hÿ1 …x†gj …x† …16:24† ~ de®ne interpolation functions for each data item uj We note that in general these `shape functions' not possess the Kronecker delta property which we noted previously for ®nite element methods ± that is Nj …xi † Tˆ ji …16:25† It must be emphasized that all least square approximations generally have values at the de®ning points xj in which ” ~ uj Tˆ u…xj † …16:26† i.e., the local values of the approximating function not ®t the nodal unknown ” values (e.g., Fig 16.2) Indeed u will be the approximation used in seeking solutions ~ to di€erential equations and boundary conditions and uj are simply the unknown parameters de®ning this approximation The main drawback of the least square approach is that the approximation rapidly deteriorates if the number of points used, n, largely exceeds that of the m polynomial terms in p This is reasonable since a least square ®t usually does not match the data points exactly A moving least square interpolation as de®ned by Eq (16.23) can approximate globally all the functions used to de®ne p…x† To show this we consider the set of 441 442 Point-based approximations approximations Uˆ n ˆ j ˆ1 ~ Nj …x†Uj …16:27† where ” U ˆ ‰ u1 …x† and ” u2 …x† F F F  ~ ~ Uj ˆ uj1 ~ uj2 ” un …x† ŠT ~ F F F ujn ÃT …16:28† …16:29† ~ Next, assign to each ujk the value of the polynomial pk …xj † (i.e., the kth entry in p) so that Uj ˆ p…xj † …16:30† Using the de®nition of the interpolation functions given by Eqs (16.23) and (16.24) we have Uˆ n ˆ jˆ1 Nj …x†p…xj † ˆ n ˆ j ˆ1 p…x†Hÿ1 …x†gj …x†p…xj † …16:31† which after substitution of the de®nition of gj …x† yields Uˆ n ˆ j ˆ1 p…x†Hÿ1 …x†wj …x ÿ xj †p…xj †T p…xj † ˆ p…x†Hÿ1 n ˆ jˆ1 wj …x ÿ xj †p…xj †T p…xj † ˆ p…x†Hÿ1 H…x† ˆ p…x† …16:32† Equation (16.32) shows that a moving least square form can exactly interpolate any function included as part of the de®nition of p…x† If polynomials are used to de®ne the functions, the interpolation always includes exact representations for each included polynomial Inclusion of the zero-order polynomial (i.e., 1), implies that n ˆ jˆ1 Nj …x† ˆ …16:33† This is called a partition of unity (provided it is true for all points, x, in the domain).22 It is easy to recognize that this is the same requirement as applies to standard ®nite element shape functions Derivatives of moving least square interpolation functions may be constructed from the representation Nj …x† ˆ p…x†vj …x† …16:34† H…x†vj …x† ˆ gj …x† …16:35† where Hierarchical enhancement of moving least square expansions For example, the ®rst derivatives with respect to x is given by @Nj @p @vj v ‡p ˆ @x j @x @x …16:36† @vj @H @gj v ˆ ‡ @x @x j @x …16:37† and H where n @H ˆ @wk …x ÿ xk † ˆ p…xk †T p…xk † @x k ˆ @x …16:38† @gj @wj …x ÿ xj † ˆ p…xj † @x @x …16:39† and Higher derivatives may be computed by repeating the above process to de®ne the higher derivatives of vj An important ®nding from higher derivatives is the order at which the interpolation becomes discontinuous between the interpolation subdomains This will be controlled by the continuity of the weight function only For weight functions which are Cq continuous in each subdomain the interpolation will be continuous for all derivatives up to order q For the truncated Gauss function given by Eq (16.10) only the approximated function will be continuous in the domain, no matter how high the order used for the p basis functions On the other hand, use of the Hermitian interpolation given by Eq (16.11) produces C1 continuous interpolation and use of Eq (16.12) produces Cn continuous interpolation This generality can be utilized to construct approximations for high order di€erential equations Nayroles et al suggest that approximations ignoring the derivatives of a may be used to de®ne the derivatives of the interpolation functions.11ÿ13 While this approximation simpli®es the construction of derivatives as it is no longer necessary to compute the derivatives for H and gj , there is little additional e€ort required to compute the derivatives of the weighting function Furthermore, for a constant in p no derivatives are available Consequently, there is little to recommend the use of this approximation 16.4 Hierarchical enhancement of moving least square expansions The moving least square approximation of the function u…x† was given in the previous section as ” u…x† ˆ n ˆ j ˆ1 Nj …x†~j u …16:40† where Nj …x† de®ned the interpolation or shape functions based on linearly independent functions prescribed by p…x† as given by Eq (16.24) Here we shall restrict 443 444 Point-based approximations attention to one-dimensional forms and employ polynomial functions to describe p…x† up to degree k Accordingly, we have  à p…x† ˆ x x2 F F F xk …16:41† For this case we will denote the resulting interpolation functions using the notation ~ Njk …x†, where j is associated with the location of the point where the parameter uj is given and k denotes the order of the polynomial approximating functions Duarte and Oden suggest using Legendre polynomials instead of the form given above;16 however, conceptually the two are equivalent and we use the above form for simplicity A hierarchical construction based on Njk …x† can be established which increases the order of the complete polynomial to degree p The hierarchical interpolation is written as V WI H ~ b bj1 b b b b bg b b f b~ b n ˆf k  Ã` bj2 ag g f ” u…x† ˆ Nj …x†~j ‡ Njk …x† xk ‡ xk ‡ F F F xp u g f b F bg f b F be j ˆ 1d b F b b b b b X~ Y bjq ˆ n ˆ j ˆ1 Njk …x† ÿ @ A n ~ uj Á ˆ k ~j ˆ ~ uj ‡ q…x†b Nj …x†‰ q…x† Š ~j b jˆ1 …16:42† ~ where q ˆ p ÿ k and bjm ; m ˆ 1; F F F ; q; are additional parameters for the approximation Derivatives of the interpolation function may be constructed using the method described by Eqs (16.34)±(16.39) The advantage of the above method lies in the reduced cost of computing the interpolation function Njk …x† compared to that required to compute the p-order interpolations Njp …x† Shepard interpolation For example, use of the functions Nj0 …x†, which are called Shepard interpolations,8 leads to a scalar matrix H which is trivial to invert to de®ne the Nj0 Speci®cally, the Shepard interpolations are Nj0 …x† ˆ H ÿ1 …x†gj …x† …16:43† where H…x† ˆ n ˆ wk …x ÿ xk † …16:44† kˆ1 and gj …x† ˆ wj …x ÿ xj † …16:45† The fact that the hierarchical interpolations include polynomials up to order p is easy to demonstrate Based on previous results from standard moving least squares the interpolation with ~j ˆ contains all the polynomials up to degree k Higher b Hierarchical enhancement of moving least square expansions degree polynomials may be constructed from H f n ˆf k  f k ” u…x† ˆ u fNj …x†~j ‡ Nj …x† xk ‡ f j ˆ 1d xk ‡ FFF V WI b bj1 b b~ b b bg b b b~ b Ã` bj2 ag g p g x b F bg b F be b F b b b b b X~ Y bjq …16:46† ~ ~ by setting all uj to zero and for each interpolation term setting one of the bjk to unity ~ with the remaining values set to zero For example, setting bj1 to unity results in the expansion ” u…x† ˆ n ˆ j ˆ1 Njk …x†xk ‡ ˆ xk ‡ …16:47† This result requires only the partition of unity property n ˆ j ˆ1 Njk …x† ˆ …16:48† ~ The remaining polynomials are obtained by setting the other values of bjk to unity one at a time We note further that the same order approximation is obtained using k ˆ 0; or p.16 The above hierarchical form has parameters which not relate to approximate values of the interpolation function For the case where k ˆ (i.e., Shepard interpolation), BabusÏ ka and Melenk23 suggest an alternate expression be used in which  à q in Eq (16.42) is taken as x x2 F F F xp and the interpolation written as p n ˆ p ˆ ” …16:49† Nj …x† l k …x†~jk u u…x† ˆ j ˆ1 kˆ0 l p …x† k In this form the are Lagrange interpolation polynomials (e.g., see Sec 8.5) and ~ ujk are parameters with dimensions of u for the jth term at point xk of the Lagrange interpolation The above result follows since Lagrange interpolation polynomials have the property & 1; if k ˆ iY lk …xi † ˆ ki ˆ …16:50† 0; otherwise We should also note that options other than polynomials may be used for the q…x† ~ Thus, for any function qi …x† we can set the associated bji to unity (with all others and ~ uj set to zero) and obtain ” u…x† ˆ n ˆ jˆ1 k NJ …x†qi …x† ˆ qi …x† …16:51† Again the only requirement is that n ˆ j ˆ1 Njk …x† ˆ …16:52† 445 446 Point-based approximations Thus, for any basic functions satisfying the partition of unity a hierarchical enrichment may be added using any type of functions For example, if one knows the structure of the solution involves exponential functions in x it is possible to include them as members of the q…x† functions and thus capture the essential part of the solution with just a few terms This is especially important for problems which involve solutions with di€erent length scales A large length scale can be included in the basic functions, Njk …x†, while other smaller length scales may be included in the functions q…x† This will be illustrated further in Volume in the chapter dealing with waves The above discussion has been limited to functions in one space variable, however, extensions to two and three dimensions can be easily constructed In the process of this extension we shall encounter some diculties which we address in more detail in the section on partition-of-unity ®nite element methods Before doing this we explore in the next section the direct use of least square methods to solve di€erential equations using collocation methods 16.5 Point collocation ± ®nite point methods Finite di€erence methods based on Taylor formula expansions on regular grids can, as explained in Chapter 3, Sec 3.13, always be considered as point collocation methods applied to the di€erential equation They have been used to solve partial di€erential equations for many decades.24ÿ26 Classical ®nite di€erence methods commonly restrict applications to regular grids This limits their use in obtaining accurate solutions to general engineering problems which have curved (irregular) boundaries and/or multiple material interfaces To overcome the boundary approximation and interface problem curvilinear mapping may be used to de®ne the ®nite di€erence operators.27 The extension of the ®nite di€erence methods from regular grids to general arbitrary and irregular grids or sets of point has received considerable attention (Girault,1 Pavlin and Perrone,2 Snell et al.3 ) An excellent summary of the current state of the art may be found in a recent paper by Orkisz27 who himself has contributed very much to the subject since the late 1970s (Liszka and Orkisz4 ) More recently such ®nite di€erence approximations on irregular grids have been proposed by Batina28 in the context of aerodynamics and by Onate et al.29ÿ31 who Ä introduced the name `®nite point method' Here both elasticity and ¯uid mechanics problems have been addressed In point collocation methods the set of di€erential equations, which here is taken in the form described in Sec 3.1, is used directly without the need to construct a weak form or perform domain integrals Accordingly, we consider A…u† ˆ …16:53† as a set of governing di€erential equations in a domain  subject to boundary conditions B…u† ˆ …16:54† applied on the boundaries ÿ An approximation to the dependent variable u may be constructed using either a weighted or moving least square approximation since at each collocation point the methods become identical In this we must ®rst describe Point collocation ± ®nite point methods 447 the (collocation) points and the weighting function The approximation is then constructed from Eq (16.23) by assuming a sucient order polynomial for p in Eq (16.14) such that all derivatives appearing in Eqs (16.53) and (16.54) may be computed Generally, it is advantageous to use the same order of interpolation to approximate both the di€erential and boundary conditions.27 The resulting discrete form for the di€erential equations at each collocation point becomes A…N…xi †~i † ˆ 0Y u i ˆ 1; 2; F F F ; ne …16:55† and the discrete form for each boundary condition is B…N…xi †~i † ˆ 0Y u i ˆ 1; 2; F F F ; nb …16:56† The total number of equations must equal the number of collocation points selected Accordingly, ne ‡ nb ˆ n …16:57† It would appear that little di€erence will exist between continuous approximations involving moving least squares and discontinuous ones as in both locally the same polynomial will be used This may well account for the convergence of standard least square approximations which we have observed in Chapter for discontinuous least square forms but in view of our previous remarks about di€erentiation, a slight di€erence will in fact exist if moving least squares are used and in the work of Onate Ä et al.29ÿ31 which we mentioned before such moving least squares are adopted In addition to the choice for p…x†, a key step in the approximation is the choice of the weighting function for the least square method and the domain over which the weighting function is applied In the work of Orkisz32 and Liszka33 two methods are used: A `cross' criterion in which the domain at a point is divided into quadrants in a cartesian coordinate system originating at the `point' where the equation is to be evaluated The domain is selected such that each quadrant contains a ®xed number of points, nq The product of nq and the number of quadrants, q, must equal or exceed the number of polynomial terms in p less one (the central node point) An example is shown in Fig 16.9(a) for a two-dimensional problem (q ˆ quadrants) and nq ˆ 2 A `Voronoi neighbour' criterion in which the closest nodes are selected as shown for a two-dimensional example in Fig 16.9(b) There are advantages and disadvantages to both approaches ± namely, the cross criterion leads to dependence on the orientation of the global coordinate axes while the Voronoi method gives results which are sometimes too few in number to get appropriate order approximations The Voronoi method is, however, e€ective for use in Galerkin solution methods or ®nite volume (subdomain collocation) methods in which only ®rst derivatives are needed The interested reader can consult reference 27 for examples of solutions obtained by this approach Additional results for ®nite point solutions may be found in work by Onate et al.29 and Batina.28 Ä One advantage of considering moving least square approximations instead of simple ®xed point weighted least squares is that approximations at points other 448 Point-based approximations 15 15 10 10 5 0 –5 –5 –10 –10 –15 –15 –15 –10 –5 10 –15 –10 15 –5 10 15 (b) (a) Fig 16.9 Methods for selecting points: (a) cross; (b) Voronoi than those used to write the di€erential equations and boundary conditions are also continuously available Thus, it is possible to perform a full post-processing to obtain the contours of the solution and its derivatives In the next part of this section we consider the application of the moving least square method to solve a second-order ordinary di€erential equation using point collocation Example: Collocation (point) solution of ordinary di€erential equations We consider the solution of ordinary di€erential equations using a point collocation method The di€erential equation in our examples is taken as ÿa d2 u du ‡ b ‡ cu ÿ f …x† ˆ dx dx …16:58† on the domain < x < L with constant coecients a, b, c, subject to the boundary conditions u…0† ˆ g1 and u…L† ˆ g2 The domain is divided into an equally spaced set of points located at xi ; i ˆ 1; F F F ; n The moving least square approximation described in Sec 16.3 is used to write di€erence equations at each of the interior points (i.e., i ˆ 2; F F F ; n ÿ 1) The boundary conditions are also written in terms of discrete approximations using the moving least square approximation Accordingly, for the approximate solution using p-order polynomials to de®ne the p…x† in the interpolations ” u…x† ˆ n ˆ j ˆ1 Nip …x†~i u …16:59† we have the set of n equations in n unknowns: n ˆ n ˆ iˆ1 iˆ1 d2 Nip d2 Nip ÿa ‡b ‡ cNip dx2 dx2 Nip …x1 †~i ˆ g1 u …16:60† ~ ui ÿ f …xj † ˆ 0Y x ˆ xj j ˆ 2; F F F ; n ÿ …16:61† Point collocation ± ®nite point methods 449 and n ˆ iˆ1 Nip …xn †ui ˆ g2 …16:62† The above equations may be written compactly as: Ku ‡ f ˆ …16:63† where K is a square coecient matrix, f is a load vector consisting of the entries from gi and f …xj †, and u is the vector of unknown parameters de®ning the approximate ” solution u…x† A unique solution to this set of equations requires K to be non-singular (i.e., rank…K† ˆ n) The rank of K depends both on the weighting function used to construct the least square approximation as well as the number of functions used to de®ne the polynomials p In order to keep the least square matrices as well conditioned as possible, a di€erent approximation is used at each node with p… j† …x† ˆ ‰ x ÿ xj …x ÿ xj †2 FFF …x ÿ xj †p Š …16:64† de®ning the interpolations associated with Njp …x† The matrix K will be of correct rank provided the weighting function can generate linearly independent equations The accurate approximation of second derivatives in the di€erential equation requires the use of quadratic or higher order polynomials in p…x†.27 In addition, the span of the weighting function must be sucient to keep the least squares matrix H non-singular at every collocation point Thus, the minimum span needed to de®ne quadratic interpolations of p…x† (i.e., p ˆ k ˆ 2) must include at least three mesh points with non-zero contributions At the problem boundaries only half of the weighting function span will be used (e.g., the right half at the left boundary) Consequently, for weighting functions which go smoothly to zero at their boundary, a span larger than four mesh spaces is required The span should not be made too large, however, since the sparse structure of K will then be lost and overdi€use solutions may result Use of hierarchical interpolations reduces the required span of the weighting function For example, use of interpolations with k ˆ requires only a span at each point for which the domain is just covered (since any span will include its de®ning point, xk , the H matrix will always be non-singular) For a uniformly spaced set of points this is any span greater than one mesh spacing For the example we use the weighting function described by Eq (16.12) with a weight span 4.4 (rm ˆ 2:2h) times the largest adjacent mesh space for the quadratic interpolations with k ˆ p ˆ and a weight 2.01 times the mesh space for the hierarchical quadratic interpolations with k ˆ 0, p ˆ We consider the example of a string on an elastic foundation with the di€erential equation ÿa d2 u ‡ cu ‡ f ˆ 0Y dx2 0