WHAT EVERY ENGINEER SHOULD KNOW ABOUT COMPUTATIONAL TECHNIQUES OF FINITE ELEMENT ANALYSIS Second Edition

LOUIS KOMZSIK

Ltfi) CRC Press V V J Taylor &. Francis Croup

Boca Raton London New York

CRC Press is an imprint of the Taylor & Francis Group, an informa business

Contents

Preface to the second edition xiii

Preface to the first edition xv

Acknowledgments xvii

I Numerical Model Generation 1

1 Finite Element Analysis 3 1.1 Solution of boundary value problems 3 1.2 Finite element shape functions 6 1.3 Finite element basis functions 9 1.4 Assembly of finite element matrices 12 1.5 Element matrix generation 15 1.6 Local to global coordinate transformation 19 1.7 A linear quadrilateral finite element 20 1.8 Quadratic finite elements 26 References 29

2 Finite Element Model Generation 31 2.1 Bezier spline approximation 31 2.2 Bezier surfaces 37 2.3 B-spline technology 40 2.4 Computational example 43 2.5 NURBS objects 48 2.6 Geometric model discretization 50 2.7 Delaunay mesh generation 51 2.8 Model generation case study 54 References 57

3 Modeling of Physical Phenomena 59 3.1 Lagrange's equations of motion 59 3.2 Continuum mechanical systems 61 3.3 Finite element analysis of elastic continuum 63 3.4 A tetrahedral finite element 65 3.5 Equation of motion of mechanical system 69 3.6 Transformation to frequency domain 71

vii

V l l l

References 74

4 Constraints and Boundary Conditions 75 4.1 The concept of multi-point constraints 76 4.2 The elimination of multi-point constraints 79 4.3 An axial bar element 82 4.4 The concept of single-point constraints 85 4.5 The elimination of single-point constraints 86 4.6 Rigid body motion support 88 4.7 Constraint augmentation approach 90 References 92

5 Singularity Detection of Finite Element Models 93 5.1 Local singularities 93 5.2 Global singularities 97 5.3 Massless degrees of freedom 99 5.4 Massless mechanisms 100 5.5 Industrial case studies 102 References 104

6 Coupling Physical Phenomena 105 6.1 Fluid-structure interaction 105 6.2 A hexahedral finite element 106 6.3 Fluid finite elements 109 6.4 Coupling structure with compressible fluid I l l 6.5 Coupling structure with incompressible fluid 112 6.6 Structural acoustic case study 113 References 115

II Computational Reduction Techniques 117

7 Matrix Factorization and Linear Systems 119 7.1 Finite element matrix reordering 119 7.2 Sparse matrix factorization 122 7.3 Multi-frontal factorization 124 7.4 Linear system solution 126 7.5 Distributed factorization and solution 127 7.6 Factorization and solution case studies 130 7.7 Iterative solution of linear systems 134 7.8 Preconditioned iterative solution technique 137 References 139

IX

8 Static Condensation 141 8.1 Single-level, single-component condensation 141 8.2 Computational example 144 8.3 Single-level, multiple-component condensation 147 8.4 Multiple-level static condensation 152 8.5 Static condensation case study 155 References 158

9 Real Spectral Computations 159 9.1 Spectral transformation 159 9.2 Lanczos reduction 161 9.3 Generalized eigenvalue problem 164 9.4 Eigensolution computation 166 9.5 Distributed eigenvalue computation 168 9.6 Dense eigenvalue analysis 172 9.7 Householder reduction technique 175 9.8 Normal modes analysis case studies 177 References 181

10 Complex Spectral Computations 183 10.1 Complex spectral transformation 183 10.2 Biorthogonal Lanczos reduction 184 10.3 Implicit operator multiplication 186 10.4 Recovery of physical solution 188 10.5 Solution evaluation 190 10.6 Reduction to Hessenberg form 191 10.7 Rotating component application 192 10.8 Complex modal analysis case studies 196 References 199

11 Dynamic Reduction 201 11.1 Single-level, single-component dynamic reduction 201 11.2 Accuracy of dynamic reduction 203 11.3 Computational example 206 11.4 Single-level, multiple-component dynamic reduction 208 11.5 Multiple-level dynamic reduction 210 11.6 Multi-body analysis application 212 References 215

12 Component Mode Synthesis 217 12.1 Single-level, single-component modal synthesis 217 12.2 Mixed boundary component mode reduction 219 12.3 Computational example 222 12.4 Single-level, multiple-component modal synthesis 225 12.5 Multiple-level modal synthesis 228

X

12.6 Component mode synthesis case study 230

References 232

III Engineering Solution Computations 235

13 Modal Solution Technique 237 13.1 Modal solution 237 13.2 Truncation error in modal solution 239 13.3 The method of residual flexibility 241 13.4 The method of mode acceleration 245 13.5 Coupled modal solution application 246 13.6 Modal contributions and energies 247 References 250

14 Transient Response Analysis 251 14.1 The central difference method 251 14.2 The Newmark method 252 14.3 Starting conditions and time step changes 254 14.4 Stability of time integration techniques 255 14.5 Transient response case study 258 14.6 State-space formulation 259 References 262

15 Frequency Domain Analysis 263 15.1 Direct and modal frequency response analysis 263 15.2 Reduced-order frequency response analysis 264 15.3 Accuracy of reduced-order solution 267 15.4 Frequency response case study 268 15.5 Enforced motion application 269 References 271

16 Nonlinear Analysis 273 16.1 Introduction to nonlinear analysis 273 16.2 Geometric nonlinearity 275 16.3 Newton-Raphson methods 278 16.4 Quasi-Newton iteration techniques 282 16.5 Convergence criteria 284 16.6 Computational example 285 16.7 Nonlinear dynamics 287 References 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 xi

17.4 Computational example 297 17.5 Eigenfunction sensitivities 302 17.6 Variational analysis 304 References 308

18 Engineering Result Computations 309 18.1 Displacement recovery 309 18.2 Stress calculation 311 18.3 Nodal data interpolation 312 18.4 Level curve computation 314 18.5 Engineering analysis case study 316 References 319

Annotation 321

List of Figures 323

List of Tables 325

Index 327

Closing Remarks 331