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(BQ) Part 1 book The finite element method in engineering has contents: Overview of finite element method, discretization of the domain, interpolation models, higher order and isoparametric elements, higher order and isoparametric elements, assembly of element matrices and vectors and derivation of system equations,...and other contents.
The Finite Element Method in Engineering This Page Intentionally Left Blank The Finite Element Method in Engineering FOURTH EDITION Singiresu S Rao Professor and Chairman Department of Mechanical and Aerospace Engineering University of Miami, Coral Gables, Florida, USA Amsterdam • Boston • Heidelberg • London • New York Paris • San Diego • San Francisco • Singapore • Sydney • • Oxford Tokyo Elsevier Butterworth–Heinemann 30 Corporate Drive, Suite 400, Burlington, MA 01803, USA Linacre House, Jordan Hill, Oxford OX2 8DP, UK Copyright c 2005, Elsevier Inc All rights reserved No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone: (+44) 1865 843830, fax: (+44) 1865 853333, e-mail: permissions@elsevier.com.uk You may also complete your request on-line via the Elsevier homepage (http://elsevier.com), by selecting “Customer Support” and then “Obtaining Permissions.” Recognizing the importance of preserving what has been written, Elsevier prints its books on acid-free paper whenever possible Library of Congress Cataloging-in-Publication Data APPLICATION SUBMITTED British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library ISBN: 0-7506-7828-3 For information on all Butterworth-Heinemann publications visit our Web site at www.books.elsevier.com 04 05 06 07 08 09 10 10 Printed in the United States of America This Page Intentionally Left Blank CONTENTS xiii Preface Principal Notation xv INTRODUCTION 1 Overview of Finite Element Method 1.1 Basic Concept 1.2 Historical Background 1.3 General Applicability of the Method 1.4 Engineering Applications of the Finite Element Method 1.5 General Description of the Finite Element Method 1.6 Comparison of Finite Element Method with Other Methods of Analysis 1.7 Finite Element Program Packages References Problems 3 10 10 26 43 43 45 BASIC PROCEDURE 51 Discretization of the Domain 2.1 Introduction 2.2 Basic Element Shapes 2.3 Discretization Process 2.4 Node Numbering Scheme 2.5 Automatic Mesh Generation References Problems 53 53 53 56 65 68 73 74 Interpolation Models 3.1 Introduction 3.2 Polynomial Form of Interpolation Functions 3.3 Simplex, Complex, and Multiplex Elements 3.4 Interpolation Polynomial in Terms of Nodal Degrees of Freedom 3.5 Selection of the Order of the Interpolation Polynomial 3.6 Convergence Requirements 3.7 Linear Interpolation Polynomials in Terms of Global Coordinates 3.8 Interpolation Polynomials for Vector Quantities 80 80 82 83 vii 84 85 86 91 97 viii CONTENTS 3.9 Linear Interpolation Polynomials in Terms of Local Coordinates References Problems Higher Order and Isoparametric Elements 4.1 Introduction 4.2 Higher Order One-Dimensional Elements 4.3 Higher Order Elements in Terms of Natural Coordinates 4.4 Higher Order Elements in Terms of Classical Interpolation Polynomials 4.5 One-Dimensional Elements Using Classical Interpolation Polynomials 4.6 Two-Dimensional (Rectangular) Elements Using Classical Interpolation Polynomials 4.7 Continuity Conditions 4.8 Comparative Study of Elements 4.9 Isoparametric Elements 4.10 Numerical Integration References Problems Derivation of Element Matrices and Vectors 5.1 Introduction 5.2 Direct Approach 5.3 Variational Approach 5.4 Solution of Equilibrium Problems Using Variational (Rayleigh–Ritz) Method 5.5 Solution of Eigenvalue Problems Using Variational (Rayleigh–Ritz) Method 5.6 Solution of Propagation Problems Using Variational (Rayleigh–Ritz) Method 5.7 Equivalence of Finite Element and Variational (Rayleigh–Ritz) Methods 5.8 Derivation of Finite Element Equations Using Variational (Rayleigh–Ritz) Approach 5.9 Weighted Residual Approach 5.10 Solution of Eigenvalue Problems Using Weighted Residual Method 5.11 Solution of Propagation Problems Using Weighted Residual Method 5.12 Derivation of Finite Element Equations Using Weighted Residual (Galerkin) Approach 5.13 Derivation of Finite Element Equations Using Weighted Residual (Least Squares) Approach References Problems 100 109 110 113 113 113 115 125 131 133 136 139 140 149 154 155 162 162 162 168 174 178 179 180 180 187 192 193 194 198 201 202 CONTENTS ix Assembly of Element Matrices and Vectors and Derivation of System Equations 6.1 Coordinate Transformation 6.2 Assemblage of Element Equations 6.3 Computer Implementation of the Assembly Procedure 6.4 Incorporation of Boundary Conditions 6.5 Incorporation of Boundary Conditions in the Computer Program References Problems 221 222 223 Numerical Solution of Finite Element Equations 7.1 Introduction 7.2 Solution of Equilibrium Problems 7.3 Solution of Eigenvalue Problems 7.4 Solution of Propagation Problems 7.5 Parallel Processing in Finite Element Analysis References Problems 230 230 231 242 258 269 270 272 209 209 211 215 216 APPLICATION TO SOLID MECHANICS PROBLEMS 277 Basic Equations and Solution Procedure 8.1 Introduction 8.2 Basic Equations of Solid Mechanics 8.3 Formulations of Solid and Structural Mechanics 8.4 Formulation of Finite Element Equations (Static Analysis) References Problems 279 279 279 295 300 305 306 Analysis of Trusses, Beams, and Frames 9.1 Introduction 9.2 Space Truss Element 9.3 Beam Element 9.4 Space Frame Element 9.5 Planar Frame Element 9.6 Computer Program for Frame Analysis References Problems 309 309 309 319 321 331 334 341 342 10 Analysis of Plates 10.1 Introduction 10.2 Triangular Membrane Element 10.3 Numerical Results with Membrane Element 10.4 Computer Program for Plates under Inplane Loads 10.5 Bending Behavior of Plates 10.6 Finite Element Analysis of Plate Bending 357 357 357 365 371 374 378 262 NUMERICAL SOLUTION OF FINITE ELEMENT EQUATIONS of the equations into a different form is used prior to numerical integration The direct integration methods are based on the following ideas: (a) Instead of trying to find a solution X(t) that satisfies Eq (7.68) for any time t, we can try to satisfy Eq (7.68) only at discrete time intervals Δt apart (b) Within any time interval, the nature of variation of X (displacement), ˙ ¨ X (velocity), and X (acceleration) can be assumed in a suitable manner ˙ ¨ Here, the time interval Δt and the nature of variation of X, X, and X within any Δt are chosen by considering factors such as accuracy, stability, and cost of solution The finite difference, Houbolt, Wilson, and Newmark methods fall under the category of direct methods [7.20–7.22] The finite difference method (a direct integration method) is outlined next Finite Difference Method By using central difference formulas [7.23], the velocity and acceleration at any time t can be expressed as (−Xt−Δt + Xt+Δt ) 2Δt ¨ [Xt−Δt − 2Xt + Xt+Δt ] Xt = (Δt)2 ˙ Xt = (7.73) (7.74) If Eq (7.68) is satisfied at time t, we have ˙ ¨ [A]Xt + [B]Xt + [C]Xt = Ft (7.75) By substituting Eqs (7.73) and (7.74) in Eq (7.75), we obtain 1 [A] + [A] Xt [B] Xt+Δt = Ft − [C] − (Δt) 2Δt (Δt)2 − 1 [B] Xt−Δx [A] − (Δt) 2Δt (7.76) Equation (7.76) can now be solved for Xt+Δt Thus, the solution Xt+Δt is based on the equilibrium conditions at time t Since the solution of Xt+Δt involves Xt and Xt−Δt , we need to know X−Δt for finding ˙ ¨ XΔt For this we first use the initial conditions X0 and X0 to find X0 using Eq (7.75) for t = Then we compute X−Δt using Eqs (7.73)–(7.75) as (Δt)2 ¨ ˙ X−Δt = Xo − ΔtXo + Xo (7.77) A disadvantage of the finite difference method is that it is conditionally stable—that is, the time step Δt has to be smaller than a critical time step (Δt)cri If the time step Δt is larger than (Δt)cri , the integration is unstable in the sense that any errors resulting from the numerical integration or round-off in the computations grow and makes the calculation of X meaningless in most cases SOLUTION OF PROPAGATION PROBLEMS 263 Acceleration X Xt+Δt b = 1– (Linear) b = 1– (Constant) Xt b = 1– (Stepped) Time (t) t t+Δt b = o only if Xt = Xt+Δt = constant in between t and t+Δt ¨ Figure 7.3 Values of β for Different Types of Variation of X 7.4.5 Newmark Method The basic equations of the Newmark method (or Newmark’s β method) are given by [7.20] ˙ ¨ ¨ ˙ Xt+Δt = Xt + (1 − γ)ΔtXt + Δtγ Xt+Δt (7.78) ˙ ¨ ¨ Xt+Δt = Xt + ΔtXt + ( − β)(Δt)2 Xt + β(Δt)2 Xt+Δt (7.79) where γ and β are parameters that can be determined depending on the desired accuracy and stability Newmark suggested a value of γ = 1/2 for avoiding artificial damping The ¨ value of β depends on the way in which the acceleration, X, is assumed to vary during the time interval t and t + Δt The values of β to be taken for different types of variation ¨ of X are shown in Figure 7.3 In addition to Eqs (7.78) and (7.79), Eqs (7.68) are also assumed to be satisfied at time t + Δt so that ˙ ¨ [A]Xt+Δt + [B]Xt+Δt + [C]Xt+Δt = [F ]t+Δt (7.80) ¨ To find the solution at the t + Δt, we solve Eq (7.79) to obtain Xt+Δt in terms of Xt+Δt , ˙ ¨ substitute this Xt+Δt into Eq (7.78) to obtain Xt+Δt in terms of Xt+Δt , and then use ˙ ¨ Eq (7.80) to find Xt+Δt Once Xt+Δt is known, Xt+Δt and Xt+Δt can be calculated from Eqs (7.78) and (7.79) 7.4.6 Mode Superposition Method It can be seen that the computational work involved in the direct integration methods is proportional to the number of time steps used in the analysis Hence, in general, the 264 NUMERICAL SOLUTION OF FINITE ELEMENT EQUATIONS use of direct integration methods is expected to be effective when the response over only a relatively short duration (involving few time steps) is required On the other hand, if the integration has to be carried for many time steps, it may be more effective to transform Eqs (7.68) into a form in which the step-by-step solution is less costly The mode superposition or normal mode method is a technique wherein Eq (7.68) is first transformed into a convenient form before integration is carried Thus, the vector X is transformed as X(t) = n×1 [T ] Y (t) n×r r×1 (7.81) where [T ] is a rectangular matrix of order n × r, and Y (t) is a time-dependent vector of order r(r ≤ n) The transformation matrix [T ] is still unknown and will have to be determined Although the components of X have physical meaning (like displacements), the components of Y need not have any physical meaning and hence are called generalized displacements By substituting Eq (7.81) into Eq (7.68), and premultiplying throughout by [T ]T , we obtain where ¨ ˙ [A ]Y + [ B ]Y + [ C ]Y = F ∼ ∼ ∼ ∼ (7.82) [A ] = [T ]T [A][T ], ∼ (7.83) ] = [T ]T [B][T ], [B ∼ (7.84) ] = [T ]T [C][T ], [C ∼ (7.85) = [T ]T F F ∼ and (7.86) The basic idea behind using the transformation of Eq (7.81) is to obtain the new system of equations (7.82) in which the matrices [A ], [B ], and [C ] will be of much smaller order ∼ ∼ ∼ than the original matrices [A], [B], and [C] Furthermore, the matrix [T ] can be chosen so as to obtain the matrices [A ], [B ], and [C ] in diagonal form, in which case Eq (7.82) ∼ ∼ ∼ represents a system of r uncoupled second-order differential equations The solution of these independent equations can be found by standard techniques, and the solution of the original problem can be found with the help of Eq (7.81) In the case of structural mechanics problems, the matrix [T ] denotes the modal matrix and Eqs (7.82) can be expressed in scalar form as (see Section 12.6) Y¨i (t) + 2ζi ωi Y˙ i (t) + ωi2 Yi (t) = Ni (t), i = 1, 2, , r (7.87) ], [B ], and [C ] have been expressed in diagonal form as where the matrices [A ∼ ∼ ∼ ωi2 - [C ]= ∼ - 2ζi ωi , - [B ]= ∼ - [A ] = [I], ∼ (7.88) SOLUTION OF PROPAGATION PROBLEMS 265 Figure 7.4 Arbitrary Forcing Function Ni (t) as and the vector F ∼ ⎧ ⎫ N (t)⎪ ⎪ ⎬ ⎨ = F ∼ ⎪ ⎪ ⎩ ⎭ Nr (t) (7.89) Here, ωi is the rotational frequency (square root of the eigenvalue) corresponding to the ith natural mode (eigenvector), and ζi is the modal damping ratio in the ith natural mode 7.4.7 Solution of a General Second-Order Differential Equation We consider the solution of Eq (7.87) in this section In many practical problems the forcing functions f1 (t), f2 (t), , fn (t) (components of F ) are not analytical expressions but are represented by a series of points on a diagram or a list of numbers in a table Furthermore, the forcing functions N1 (t), N2 (t), , Nr (t) of Eq (7.87) are given by premultiplying F by [T ]T as indicated in Eq (7.86) Hence, in many cases, the solution of Eq (7.87) can only be obtained numerically by using a repetitive series of calculations Let the function Ni (t) vary with time in some general manner, such as that represented by the curve in Figure 7.4 This forcing function may be approximated by a series of rectangular impulses of various magnitudes and durations as indicated in Figure 7.4 For good accuracy the magnitude 266 NUMERICAL SOLUTION OF FINITE ELEMENT EQUATIONS (j) Ni of a typical impulse should be chosen as the ordinate of the curve at the middle of the time interval Δtj as shown in Figure 7.4 In any time interval tj−1 ≤ t ≤ tj , the solution of Eq (7.87) can be computed as the sum of the effects of the initial conditions at time tj−1 and the effect of the impulse within the interval Δtj as follows [7.24]: Yi (t) = e−ζi ωi (t−tj−1 ) Yi (j−1) + (j−1) (j−1) + ζi ωi Yi Y˙ i sin ωdi (t − tj−1 ) ωdi (j) N + i2 ωi + cos ωdi (t − tj−1 ) − e−ζi ωi (t−tj−1 ) cos ωdi (t − tj−1 ) ζi ωi sin ωdi (t − tj−1 ) ωdi (7.90) ωdi = ωi (1 − ζi2 )1/2 where (7.91) At the end of the interval, Eq (7.90) becomes (j) Yi = Yi (t = tj ) = e−ζi ωi Δtj Yi (j−1) + cos ωdi Δtj (j−1) (j−1) + ζi ωi Yi Y˙ i sin ωdi Δtj ωdi (j) N + i2 ωi − e−ζi ωi Δtj cos ωdi Δtj + ζi ωi sin ωdi Δtj ωdi (7.92) By differentiating Eq (7.90) with respect to time, we obtain (j−1) (j−1) Y˙ i + ζi ωi Yi (j) (j−1) −ζi ωi Δtj ˙ ˙ Yi = Yi (t = tj ) = ωdi e −Yi sin ωdi Δtj + ωdi × cos ωdi Δtj − (j) (j−1) (j−1) Y˙ + ζi ωi Yi ζi ωi (j−1) Yi cos ωdi Δtj + i sin ωdi Δtj ωdi ωdi N ωdi + i e−ζi ωi Δtj ωi ζi2 ωi2 1+ ωdi sin ωdi Δtj (7.93) Thus, Eqs (7.92) and (7.93) represent recurrence formulas for calculating the solution at the end of the jth time step They also provide the initial conditions of Yj and Y˙ j at the beginning of step j + These formulas may be applied repetitively to obtain the time history of response for each of the normal modes i Then the results for each time station can be transformed back, using Eq (7.81), to obtain the solution of the original problem SOLUTION OF PROPAGATION PROBLEMS 267 7.4.8 Computer Implementation of Mode Superposition Method A subroutine called MØDAL is given to implement the mode superposition or normal mode method This subroutine calls a matrix multiplication subroutine called MATMUL The arguments of the subroutine MØDAL are as follows: NMØDE = number of modes to be considered in the analysis = r of Eq (7.81) = input N = number of degrees of freedom = order of the square matrices [A], [B], and [C] = input GM = array of N × N in which the (mass) matrix [A] is stored = input ØMEG = array of size NMØDE in which the natural frequencies (square root of eigenvalues) are stored = input T = array of size N × NMØDE in which the eigenvectors (modes) are stored columnwise = matrix [T] = input ZETA = array of size NMØDE in which the modal damping ratios of various modes are stored = input NSTEP = number of integration points = input XØ = array of size N in which the initial conditions x1 (0), x2 (0), , xn (0) are stored = input XDØ = array of size N in which the initial conditions dx1 dx2 dxn (0) = Y1 (0), (0) = Y2 (0), , (0) = Yn (0) dt dt dt are stored = input = array of size NMØDE × NSTEP = array of size NMØDE × N = transpose of the matrix [T] = array of size NSTEP in which the magnitudes of the force applied at coordinate M at times t1 , t2 , , tNSTEP are stored = input YØ, YDØ = arrays of size NMØDE (j) (j) U, V = arrays of size NMODE × NSTEP in which the values of Yi and Y˙ i are stored (j) X = array of size N × NSTEP in which the solution of the original problem, xi , is stored TIME = array of size NSTEP in which the times t1 , t2 , , tNSTEP are stored = input DT = array of size NSTEP in which the time intervals Δt1 , Δt2 , , ΔtNSTEP are stored M = coordinate number at which the force is applied = input TGM = array of size NMØDE × N used to store the product [T ]T [A] TGMT = array of size NMØDE × NMØDE used to store the matrix [A ] = [T ]T [A][T ] ∼ XX TT F To demonstrate the use of the subroutine MØDAL, the solution of the following problem is considered: ¨ ˙ [A]X + [B]X + [C]X = F with X(0) = Xo and ˙ X(0) = Yo Known data: n = 3, r = 3; ζi = 0.05 for i = 1, 2, 3; (E1 ) 268 NUMERICAL SOLUTION OF FINITE ELEMENT EQUATIONS ⎤ 0 [B] = [0]; [A] = ⎣0 0⎦ ; 0 ⎤ ⎡ 1.000 1.000 1.000 0.445 −1.247⎦ ; [T ] = ⎣1.802 2.247 −0.802 0.555 ⎡ ⎡ ω1 = 0.445042, −1 −1 [C] = ⎣−1 ω2 = 1.246978, ⎤ −1⎦ ; ω3 = 1.801941; Yo = 0; Xo = 0; ⎧ ⎫ ⎧ ⎫ ⎨f1 ⎬ ⎨ ⎬ F = f2 = ⎩ ⎭ ⎩ ⎭ f f3 Value of time t 10 Magnitude of f 1 1 1 1 1 Thus, in this case, NMODE = 3, N = 3, NSTEP = 10, M = 3, TIME (I) = I for I = 1–10, and F (I) = for I = 1–10 The main program for this problem and the results given by the program are given below C======================================================================== C C RESPONSE OF MULTI-DEGREE-OF-FREEDOM SYSTEM BY MODAL ANALYSIS C C======================================================================== DIMENSION GM(3,3),OMEG(3),T(3,3),ZETA(3),TT(3,3),TGMT(3,3), XO(3),XDO(3),YO(3),YDO(3),WN(3,10),F(10),U(3,10),TGM(3,3), V(3,10),X(3,10),TIME(10),DT(10) DATA NMODE,N,NSTEP,M/3,3,10,3/ DATA GM/1.0,0.0,0.0,0.0,1.0,0.0,0.0,0.0,1.0/ DATA OMEG/0.445042,1.246978,1.801941/ DATA ZETA/0.05,0.05,0.05/ DATA(T(I,1),I=1,3)/0.445042,0.8019375,1.0/ DATA(T(I,2),I=1,3)/-1.246984, -0.5549535,1.0/ DATA(T(I,3),I=1,3)/1.801909,-2.246983,1.0/ DATA X0/0.0,0.0,0.0/ DATA XD0/0.0,0.0,0.0/ DATA TIME/1.0,2.0,3.0,4.0,5.0,6.0,7.0,8.0,9.0,10.0/ DATA F/1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0/ DO 10 I=1,NMODE DO 10 J=1,NMODE 10 TT(I,J) = T(J,I) CALL-MODAL(GM,OMEG,T,ZETA,XO,XDO,YO,YDO,WN,F,U,V,X, TIME,DT,TT,M, NSTEP,N,NMODE,TGMT,TGM) PRINT 20,M 20 FORMAT(/,69H RESPONSE OF THE SYSTEM TO A TIME VARYING FORCE APPL 2IED AT COORDINATE,I2,/) PARALLEL PROCESSING IN FINITE ELEMENT ANALYSIS 30 40 269 DO 30 I=1,N PRINT 40,I,(X(I,J),J=1,NSTEP) FORMAT(/,11H COORDINATE,I5,/,1X,5E14.8,/,1X,5E14.8) STOP END RESPONSE OF THE SYSTEM TO A TIME VARYING FORCE APPLIED AT COORDINATE COORDINATE 0.30586943E-020.75574949E-010.44995812E+000.11954379E+010.18568037E+01 0.20145943E+010.18787326E+010.18112489E+010.17405561E+010.14387581E+01 COORDINATE 0.42850435E-010.45130622E+000.13615156E+010.23700531E+010.31636245E+01 0.36927490E+010.38592181E+010.36473811E+010.32258503E+010.26351447E+01 COORDINATE 0.44888544E+000.14222714E+010.24165692E+010.33399298E+010.42453833E+01 0.50193300E+010.54301090E+010.52711205E+010.45440755E+010.35589526E+01 7.5 PARALLEL PROCESSING IN FINITE ELEMENT ANALYSIS Parallel processing is defined as the exploitation of parallel or concurrent events in the computing process [7.25] Parallel processing techniques are being investigated because of the high degree of sophistication of the computational models required for future aerospace, transportation, nuclear, and microelectronic systems Most of the present-day supercomputers, such as CRAY X-MP, CRAY-2, CYBER-205, and ETA-10, achieve high performance through vectorization/parallelism Efforts have been devoted to the development of vectorized numerical algorithms for performing the matrix operations, solution of algebraic equations, and extraction of eigenvalues [7.26, 7.27] However, the progress has been slow, and no effective computational strategy exists that performs the entire finite element solution in the parallel processing mode The various phases of the finite element analysis can be identified as (a) input of problem characteristics, element and nodal data, and geometry of the system; (b) data preprocessing; (c) evaluation of element characteristics; (d) assembly of elemental contributions; (e) incorporation of boundary conditions; (f) solution of system equations; and (g) postprocessing of the solution and evaluation of secondary fields The input and preprocessing phases can be parallelized Since the element characteristics require only information pertaining to the elements in question, they can be evaluated in parallel The assembly cannot utilize the parallel operation efficiently since the element and global variables are related through a Boolean transformation The incorporation of boundary conditions, although usually not time-consuming, can be done in parallel The solution of system equations is the most critical phase For static linear problems, the numerical algorithm should be selected to take advantage of the symmetric banded structure of the equations and the type of hardware used A variety of efficient direct iterative and noniterative solution techniques have been developed for different computers by exploiting the parallelism, pipeline (or vector), and chaining capabilities [7.28] For nonlinear steady-state problems, the data structure is essentially the same as for linear problems The major difference lies in the algorithms for evaluating the nonlinear terms 270 NUMERICAL SOLUTION OF FINITE ELEMENT EQUATIONS and solving the nonlinear algebraic equations For transient problems, several parallel integration techniques have been proposed [7.29] The parallel processing techniques are still evolving and are expected to be the dominant methodologies in the computing industry in the near future Hence, it can be hoped that the full potentialities of parallel processing in finite element analysis will be realized in the next decade REFERENCES 7.1 S.S Rao: Applied Numerical Methods for Engineers and Scientists, Prentice Hall, Upper Saddle River, NJ, 2002 7.2 G Cantin: An equation solver of very large capacity, International Journal for Numerical Methods in Engineering, 3, 379–388, 1971 7.3 B.M Irons: A frontal solution problem for finite element analysis, International Journal for Numerical Methods in Engineering, 2, 5–32, 1970 7.4 A Razzaque: Automatic reduction of frontwidth for finite element analysis, International Journal for Numerical Methods in Engineering, 15, 1315–1324, 1980 7.5 G Beer and W Haas: A partitioned frontal solver for finite element analysis, International Journal for Numerical Methods in Engineering, 18, 1623–1654, 1982 7.6 R.S Varga: Matrix Iterative Analysis, Prentice Hall, Englewood Cliffs, NJ, 1962 7.7 I Fried: A gradient computational procedure for the solution of large problems arising from the finite element discretization method, International Journal for Numerical Methods in Engineering, 2, 477–494, 1970 7.8 G Gambolati: Fast solution to finite element flow equations by Newton iteration and modified conjugate gradient method, International Journal for Numerical Methods in Engineering, 15, 661–675, 1980 7.9 O.C Zienkiewicz and R Lohner: Accelerated “relaxation” or direct solution? Future prospects for finite element method, International Journal for Numerical Methods in Engineering, 21, 1–11, 1985 7.10 N Ida and W Lord: Solution of linear equations for small computer systems, International Journal for Numerical Methods in Engineering, 20, 625–641, 1984 7.11 J.H Wilkinson: The Algebraic Eigenvalue Problem, Clarendon, Oxford, UK, 1965 7.12 A.R Gourlay and G.A Watson: Computational Methods for Matrix Eigen Problems, Wiley, London, 1973 7.13 M Papadrakakis: Solution of the partial eigenproblem by iterative methods, International Journal for Numerical Methods in Engineering, 20, 2283–2301, 1984 7.14 P Roberti: The accelerated power method, International Journal for Numerical Methods in Engineering, 20, 1179–1191, 1984 7.15 K.J Bathe and E.L Wilson: Large eigenvalue problems in dynamic analysis, Journal of Engineering Mechanics Division, Proc of ASCE, 98(EM6), 1471–1485, 1972 7.16 F.A Akl, W.H Dilger, and B.M Irons: Acceleration of subspace iteration, International Journal for Numerical Methods in Engineering, 18, 583–589, 1982 7.17 T.C Cheu, C.P Johnson, and R.R Craig, Jr.: Computer algorithms for calculating efficient initial vectors for subspace iteration method, International Journal for Numerical Methods in Engineering, 24, 1841–1848, 1987 7.18 A Ralston: A First Course in Numerical Analysis, McGraw-Hill, New York, 1965 7.19 L Brusa and L Nigro: A one-step method for direct integration of structural dynamic equations, International Journal for Numerical Methods in Engineering, 15, 685–699, 1980 REFERENCES 271 7.20 S.S Rao: Mechanical Vibrations, Addison-Wesley, Reading, MA, 1986 7.21 W.L Wood, M Bossak, and O.C Zienkiewicz: An alpha modification of Newmark’s method, International Journal for Numerical Methods in Engineering, 15, 1562–1566, 1980 7.22 W.L Wood: A further look at Newmark, Houbolt, etc., time-stepping formulae, International Journal for Numerical Methods in Engineering, 20, 1009–1017, 1984 7.23 S.H Crandall: Engineering Analysis: A Survey of Numerical Procedures, McGrawHill, New York, 1956 7.24 S Timoshenko, D.H Young, and W Weaver: Vibration Problems in Engineering (4th Ed.), Wiley, New York, 1974 7.25 A.K Noor: Parallel processing in finite element structural analysis, Engineering with Computers, 3, 225–241, 1988 7.26 C Farhat and E Wilson: Concurrent iterative solution of large finite element systems, Communications in Applied Numerical Methods, 3, 319–326, 1987 7.27 S.W Bostic and R.E Fulton: Implementation of the Lanczos method for structural vibration analysis on a parallel computer, Computers and Structures, 25, 395–403, 1987 7.28 L Adams: Reordering computations for parallel execution, Communications in Applied Numerical Methods, 2, 263–271, 1986 7.29 M Ortiz and B Nour-Omid: Unconditionally stable concurrent procedures for transient finite element analysis, Computer Methods in Applied Mechanics and Engineering, 58, 151–174, 1986 272 NUMERICAL SOLUTION OF FINITE ELEMENT EQUATIONS PROBLEMS 7.1 Find the inverse of the following matrix using the decomposition [A] = [U ]T [U ]: ⎡ ⎣ [A] = −1 −1 −4 ⎤ −4⎦ 7.2 Find the inverse of the matrix [A] given in Problem 7.1 using the decomposition [A] = [L][L]T , where [L] is a lower triangular matrix Hint: If a symmetric matrix [A] of order n is decomposed as [A] = [L][L]T , the elements of [L] are given by i−1 lii = aii − (1/2) lik , i = 1, 2, , n k=1 lmi = ami − lii lij = 0, i−1 lik lmk , m = i + 1, , n and i = 1, 2, , n k=1 ij k=j λij = 0, i