WHAT EVERY ENGINEER SHOULD KNOW ABOUT COMPUTATIONAL TECHNIQUES OF FINITE ELEMENT ANALYSIS Second Edition WHAT EVERY ENGINEER SHOULD KNOW ABOUT COMPUTATIONAL TECHNIQUES OF FINITE ELEMENT ANALYSIS Second Edition LOUIS KOMZSIK Boca Raton London New York CRC Press is an imprint of the Taylor & Francis Group, an informa business CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2009 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S Government works Version Date: 20131125 International Standard Book Number-13: 978-1-4398-0295-3 (eBook - PDF) This book contains information obtained from authentic and highly regarded sources Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint Except as permitted under U.S Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers For permission to photocopy or use material electronically from this work, please access www.copyright com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400 CCC is a not-for-profit organization that provides licenses and registration for a variety of users For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com To my son, Victor Contents Preface to the second edition xiii Preface to the first edition xv Acknowledgments I xvii Numerical Model Generation 1 Finite Element Analysis 1.1 Solution of boundary value problems 1.2 Finite element shape functions 1.3 Finite element basis functions 1.4 Assembly of finite element matrices 1.5 Element matrix generation 1.6 Local to global coordinate transformation 1.7 A linear quadrilateral finite element 1.8 Quadratic finite elements References 3 12 15 19 20 26 29 31 31 37 40 43 48 50 51 54 57 Modeling of Physical Phenomena 3.1 Lagrange’s equations of motion 3.2 Continuum mechanical systems 3.3 Finite element analysis of elastic continuum 3.4 A tetrahedral finite element 3.5 Equation of motion of mechanical system 3.6 Transformation to frequency domain 59 59 61 63 65 69 71 Finite Element Model Generation 2.1 Bezier spline approximation 2.2 Bezier surfaces 2.3 B-spline technology 2.4 Computational example 2.5 NURBS objects 2.6 Geometric model discretization 2.7 Delaunay mesh generation 2.8 Model generation case study References vii viii References 74 Constraints and Boundary Conditions 4.1 The concept of multi-point constraints 4.2 The elimination of multi-point constraints 4.3 An axial bar element 4.4 The concept of single-point constraints 4.5 The elimination of single-point constraints 4.6 Rigid body motion support 4.7 Constraint augmentation approach References Singularity Detection of Finite 5.1 Local singularities 5.2 Global singularities 5.3 Massless degrees of freedom 5.4 Massless mechanisms 5.5 Industrial case studies References 75 76 79 82 85 86 88 90 92 Element Models 93 93 97 99 100 102 104 105 105 106 109 111 112 113 115 Coupling Physical Phenomena 6.1 Fluid-structure interaction 6.2 A hexahedral finite element 6.3 Fluid finite elements 6.4 Coupling structure with compressible fluid 6.5 Coupling structure with incompressible fluid 6.6 Structural acoustic case study References II Computational Reduction Techniques Matrix Factorization and Linear Systems 7.1 Finite element matrix reordering 7.2 Sparse matrix factorization 7.3 Multi-frontal factorization 7.4 Linear system solution 7.5 Distributed factorization and solution 7.6 Factorization and solution case studies 7.7 Iterative solution of linear systems 7.8 Preconditioned iterative solution technique References 117 119 119 122 124 126 127 130 134 137 139 ix Static Condensation 8.1 Single-level, single-component condensation 8.2 Computational example 8.3 Single-level, multiple-component condensation 8.4 Multiple-level static condensation 8.5 Static condensation case study References 141 141 144 147 152 155 158 Real Spectral Computations 9.1 Spectral transformation 9.2 Lanczos reduction 9.3 Generalized eigenvalue problem 9.4 Eigensolution computation 9.5 Distributed eigenvalue computation 9.6 Dense eigenvalue analysis 9.7 Householder reduction technique 9.8 Normal modes analysis case studies References 159 159 161 164 166 168 172 175 177 181 10 Complex Spectral Computations 10.1 Complex spectral transformation 10.2 Biorthogonal Lanczos reduction 10.3 Implicit operator multiplication 10.4 Recovery of physical solution 10.5 Solution evaluation 10.6 Reduction to Hessenberg form 10.7 Rotating component application 10.8 Complex modal analysis case studies References 183 183 184 186 188 190 191 192 196 199 11 Dynamic Reduction 11.1 Single-level, single-component dynamic reduction 11.2 Accuracy of dynamic reduction 11.3 Computational example 11.4 Single-level, multiple-component dynamic reduction 11.5 Multiple-level dynamic reduction 11.6 Multi-body analysis application References 201 201 203 206 208 210 212 215 12 Component Mode Synthesis 12.1 Single-level, single-component modal synthesis 12.2 Mixed boundary component mode reduction 12.3 Computational example 12.4 Single-level, multiple-component modal synthesis 12.5 Multiple-level modal synthesis 217 217 219 222 225 228 x 12.6 Component mode synthesis case study 230 References 232 III Engineering Solution Computations 13 Modal Solution Technique 13.1 Modal solution 13.2 Truncation error in modal solution 13.3 The method of residual flexibility 13.4 The method of mode acceleration 13.5 Coupled modal solution application 13.6 Modal contributions and energies References 235 237 237 239 241 245 246 247 250 14 Transient Response Analysis 14.1 The central difference method 14.2 The Newmark method 14.3 Starting conditions and time step changes 14.4 Stability of time integration techniques 14.5 Transient response case study 14.6 State-space formulation References 251 251 252 254 255 258 259 262 15 Frequency Domain Analysis 15.1 Direct and modal frequency response analysis 15.2 Reduced-order frequency response analysis 15.3 Accuracy of reduced-order solution 15.4 Frequency response case study 15.5 Enforced motion application References 263 263 264 267 268 269 271 16 Nonlinear Analysis 16.1 Introduction to nonlinear analysis 16.2 Geometric nonlinearity 16.3 Newton-Raphson methods 16.4 Quasi-Newton iteration techniques 16.5 Convergence criteria 16.6 Computational example 16.7 Nonlinear dynamics References 273 273 275 278 282 284 285 287 288 17 Sensitivity and Optimization 289 17.1 Design sensitivity 289 17.2 Design optimization 290 17.3 Planar bending of the bar 294 Contents 17.4 Computational example 17.5 Eigenfunction sensitivities 17.6 Variational analysis References xi 297 302 304 308 18 Engineering Result Computations 18.1 Displacement recovery 18.2 Stress calculation 18.3 Nodal data interpolation 18.4 Level curve computation 18.5 Engineering analysis case study References 309 309 311 312 314 316 319 Annotation 321 List of Figures 323 List of Tables 325 Index 327 Closing Remarks 331 312 Chapter 18 σ3 y σ1 σv σ2 x FIGURE 18.1 Stresses in triangle 18.3 Nodal data interpolation In industrial practice it is common to calculate nodal values out of the constant element value with least square minimalization Assume that the element value given is σv and we need corner values at a triangular element as shown in Figure 18.1 With the help of the shape functions, the stress at a point in the element is expressed in terms of the nodal values as σ = N1 σ1 + N2 σ2 + N3 σ3 , where σi are the yet unknown corner stresses and the Ni are the shape functions as defined earlier Our desire is to minimize the following squared error for this element Ee = (σv − σ)2 dA, A Engineering Result Computations 313 where σv is the computed element stress Expanding the square results in Ee = A σv2 dA − σv σdA + A σ dA A As σv is constant over the element, the first integral is simply a constant σ A = Ce v Introducing ⎡ ⎤ σ1 σe = ⎣ σ2 ⎦ , σ3 the stress is the product of two vectors σ = N σe , where N = N1 N2 N3 By substituting the second integral becomes σv N σe dA = σe σv A N dA, A as σe and σv now both are constant for the element The third integral similarly changes to T σ e N T N dA σe A The evaluation of these element integrals proceeds along the same lines as the computations shown in Chapters and We sum (assemble) all the element errors as m E = Σm e=1 Ee = Σe=1 (Ce − σv σe N dA + σeT A N T N dAσe ), A where m is the number of elements in the finite element model We then introduce ⎤ ⎡ σ1 ⎢ σ2 ⎥ ⎥ ⎢ ⎢ σ3 ⎥ ⎥ ⎢ ⎥ ⎢ S = Σm e=1 σe = ⎢ ⎥ , ⎢ σn−2 ⎥ ⎥ ⎢ ⎣ σn−1 ⎦ σn 314 Chapter 18 where n is the number of nodes in the finite element model Also introduce R = Σm e=1 N T N dA, A and T = Σm e=1 σv N dA A With these the assembled error is T S RS − S T T + C We obtain the minimum of this error when E= ∂E = 0, ∂S which occurs with RS = T, Σm e=1 Ce since C = is constant Note, that the calculation of R is essentially the same as the calculation of the mass matrix in Section 3.5, apart from the absence of the density ρ Hence, the matrix of this linear system is similar to the mass matrix in structure and as such very likely extremely sparse or strongly banded Therefore, this linear system may now be solved with the techniques shown in Chapter The resulting nodal stress values in S form the basis for the last computational technique discussed in the next section 18.4 Level curve computation The nodal values, whether they were direct solution results such as the displacements or were interpolated as the stresses, are best presented to the engineer in the form of level, or commonly called, contour curves Let us consider the example of the triangular element with nodes 1, 2, and nodal values σ1 , σ2 , σ3 shown in Figure 18.2 We are to find the location of the points R, S on the sides of the triangle that correspond to points P, Q that define a constant level of stress Let the value of the level curve be σl Introduce the ratio p= σl − σ1 σ3 − σ1 Engineering Result Computations 315 σ3 P Q σl σl R S σ1 σ2 FIGURE 18.2 Level curve computation Some arithmetic yields the coordinates of point R as xR = px3 + (1 − p)x1 , and yR = py3 + (1 − p)y1 Similarly the ratio r= σl − σ2 σ3 − σ2 yields the coordinates of S as xS = rx3 + (1 − r)x2 , and yS = ry3 + (1 − r)y2 These calculations, executed for all the elements, result in continuous level curves across elements and throughout the finite element model 316 Chapter 18 It should be noted that above method is fine when the stresses of the exact solution are smooth The method could result in large errors in cases when the exact stress is not smooth, for example, in elements located on singular edges of the model FIGURE 18.3 Physical load on bracket 18.5 Engineering analysis case study The engineering analysis process and the computed results are demonstrated by the bracket model for which the geometric and finite element models were presented in Chapter The first step is to apply the loads to the model as Engineering Result Computations 317 shown on Figure 18.3 by a pressure load applied horizontally to the left face of the model FIGURE 18.4 Constraint conditions of bracket Note, that in modern engineering environments, such as the NX CAE environment of the virtual product development suite of Siemens PLM Software [2], the loads are applied to the geometric model in accordance with the physical intentions of the engineer In this case the load is distributed on the face bounded by the highlighted edges and the direction of the load is denoted by the arrows The constraints are also applied to the geometric model In the case of the example the constraints were applied to the interior cylindrical holes where the bolts will be located and made visible by the arrows on Figure 18.4 318 Chapter 18 FIGURE 18.5 Deformed shape of bracket The computational steps of the analysis process were executed by NX NASTRAN [3] The displacements superimposed on the finite element mesh of the model result in the deformed shape of the model as shown in Figure 18-5 This is one of the most useful informations for the engineers The stress results projected to the undeformed finite element model are visualized in Figure 18.6 The lighter shades indicate the areas of higher stresses The level curve representation of the results described in the last section was used for the contours The NX environment also enable the animation of the deformed shape with a contiguous process between the undeformed and the deformed geometry, another useful tool for the engineer Engineering Result Computations 319 FIGURE 18.6 Stress contours of bracket References [1] Craig, R R Jr.; Structural dynamics, An introduction to computer methods, Wiley, New York, 1981 [2] www.plm.automation.siemens.com/en_us/products/nx/design/index.shtml [3] www.siemens.com/plm/nxnastran Annotation Notation Meaning P Pk T D D WP Ps Ni N B J qe ke Ae Ve Ee K M B F G H1 I C L U S Sj Pi R V (Pi ) Ys T Q Potential energy, permutation Householder matrix Kinetic energy, transformation matrix Dissipative function, material matrix Diagonal factor matrix Work potential Strain energy Matrix of shape functions Shape functions Strain displacement matrix Jacobian matrix Element displacement Element stiffness Element area Element volume Element energy Stiffness matrix Mass matrix Damping matrix Force matrix Static condensation matrix Hilbert space Identity matrix Cholesky factor Lower triangular factor matrix Upper triangular factor matrix Dynamic reduction transformation matrix Spline segment Point coordinates Rigid constraint matrix Voronoi polygon Vector of enforced displacements Tridiagonal matrix Permutation matrix, Lanczos vector matrix 321 322 Annotation X Y Z Linear system solution Intermediate solution Residual flexibility matrix σ α β λ Λ φ Φ Ψ ω μ λs Δt ΔF Δu κk θ Strain vector Stress vector Diagonal Lanczos coefficient Off-diagonal Lanczos coefficient Eigenvalue, Lagrange multiplier Eigenvalue matrix Eigenvector Eigenvector matrix Residual flexibility matrix Frequency Shifted eigenvalue Spectral shift Time step Nonlinear force imbalance Nonlinear displacement increment Krylov subspace Rotational degrees of freedom bi f (x) g(x) ki mi qk qi r t u v w wi Modal damping Objective function Constraint function Modal stiffness Modal mass Lanczos vectors Generalized degrees of freedom Residual vector Time Displacement in frequency domain Displacement in time domain Modal displacement Weight coefficients List of Figures 1.1 1.2 1.3 1.4 1.5 1.6 Membrane model Local coordinates of triangular element Meshing the membrane model Parametric coordinates of triangular element A planar quadrilateral element Parametric coordinates of quadrilateral element 12 16 21 22 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13 2.14 2.15 2.16 Bezier polygon The effect of weights on the shape of spline Multiple Bezier segments Continuity of spline segments Bezier patch definition Patch continuity definition B spline interpolation B spline approximation Clamped B spline approximation Closed B spline approximation Voronoi polygon Delaunay triangle Delaunay triangularization Design sketch of a bracket Geometric model of bracket Finite element model of bracket 32 34 35 36 37 39 44 46 47 48 50 52 53 55 56 57 3.1 3.2 3.3 Discrete mechanical system 60 Degrees of freedom of mechanical particle 62 Tetrahedron element 66 4.1 4.2 Rigid bar 76 Boundary conditions 87 5.1 5.2 Deformed shape of bar 96 Typical automobile body-in-white model 103 6.1 6.2 6.3 Hexahedral finite element 106 Fuel tank model 113 Truck cabin model 114 323 324 List of Figures 7.1 Crankshaft casing finite element model 131 8.1 8.2 8.3 8.4 Single-level, single-component partitioning Single-level, multiple-component partitioning Multiple-level, multiple-component partitioning Automobile crankshaft industrial example 142 148 152 156 9.1 9.2 9.3 9.4 9.5 Spectral transformation Generalized solution scheme Trimmed car body model Speedup of parallel normal modes analysis Engine block model 160 165 178 179 181 10.1 10.2 10.3 10.4 Campbell diagram Stability diagram Brake model Rotating machinery model 195 196 197 198 11.1 Steering mechanism 213 12.1 Convertible car body 230 13.1 13.2 13.3 13.4 Time dependent load The effect of residual vector Modal contributions Modal kinetic energy distribution 244 244 248 250 14.1 Transient response 258 15.1 Satellite model 268 16.1 16.2 16.3 16.4 Nonlinear stress-strain relationship Rotated bar model Newton-Raphson iteration Modified Newton iteration 274 277 280 281 17.1 Optimum condition 293 17.2 Planar bending of bar element 294 17.3 Design space of optimization example 300 18.1 18.2 18.3 18.4 18.5 18.6 Stresses in triangle Level curve computation Physical load on bracket Constraint conditions of bracket Deformed shape of bracket Stress contours of bracket 312 315 316 317 318 319 List of Tables 1.1 1.2 1.3 Basis function terms for two-dimensional elements 10 Basis function terms for three-dimensional elements 11 Gauss weights and locations 18 5.1 5.2 Element statistics of automobile model examples 103 Reduction sizes of automobile model examples 104 6.1 6.2 Local coordinates of hexahedral element 107 Acoustic response analysis matrix statistics 114 7.1 7.2 7.3 Size statistics of casing component model 132 Computational statistics of casing component model 132 Linear static analysis matrix statistics 133 8.1 8.2 Component statistics of crankshaft model 157 Performance statistics of crankshaft model 157 9.1 9.2 9.3 Model statistics of trimmed car body 178 Distributed normal modes analysis statistics 180 Normal modes analysis dense matrix statistics 180 10.1 Statistics of brake model 198 10.2 Complex eigenvalue analysis statistics 199 12.1 Element types of case study automobile model 231 12.2 Problem statistics 231 12.3 Execution statistics 232 325 Closing Remarks The book’s goal was to give a working knowledge of the main computational techniques of finite element analysis Extra effort was made to make the material accessible with the usual engineering mathematical tools Some chapters contained a simple computational example to demonstrate the details of the computation It was hoped that those help the reader to develop a good understanding of these computational steps Some chapters contained a description of an industrial application or an actual real life case study They were meant to demonstrate the awesome practical power of the technology discussed In order to produce a logically contiguous material and to increase the readability, some of the more tedious details were omitted In these areas the reader is encouraged to follow the cited references The reference sections at the end of each chapter are organized in alphabetic order by authors’ names They are all publicly available references Many original publications on topics contained in the book are given The best review references are also cited, especially those dealing with the mathematical, engineering and geometric theory of finite elements 331 .. .WHAT EVERY ENGINEER SHOULD KNOW ABOUT COMPUTATIONAL TECHNIQUES OF FINITE ELEMENT ANALYSIS Second Edition WHAT EVERY ENGINEER SHOULD KNOW ABOUT COMPUTATIONAL TECHNIQUES OF FINITE ELEMENT ANALYSIS. .. thank Professor Duc Nguyen for his proofreading of the extensions of this edition His use of the first edition in his teaching provided me with valuable feedback and confirmation of the approach of. .. the second edition xiii Preface to the first edition xv Acknowledgments I xvii Numerical Model Generation 1 Finite Element Analysis 1.1 Solution of boundary value problems 1.2 Finite element shape