Numerical Simulation of Mechatronic Sensors and Actuators (eBook)

Preface to the Third Edition 5
Preface to the Second Edition 8
Preface to the First Edition 10
Contents 14
Notation 21
1 Introduction 26
References 30
2 The Finite Element (FE) Method 31
2.1 Finite Element Formulation 32
2.2 Finite Element Method for a 1D Problem 37
2.3 Nodal (Lagrangian) Finite Elements 44
2.3.1 Basic Properties 45
2.3.2 Quadrilateral Element in R2 47
2.3.3 Triangular Element in R2 50
2.3.4 Tetrahedron Element in R3 51
2.3.5 Hexahedron Element in R3 52
2.3.6 Wedge Element in R3 54
2.3.7 Pyramidal Element in R3 55
2.3.8 Global/Local Derivatives 56
2.3.9 Numerical Integration 58
2.4 Finite Element Procedure 60
2.5 Time Discretization 65
2.5.1 Parabolic Differential Equation 65
2.5.2 Hyperbolic Differential Equation 69
2.6 Integration over Surfaces 72
2.7 Edge Nédélec Finite Elements 73
2.8 Discretization Error 75
2.9 Finite Elements of Higher Order 79
2.9.1 Legendre Polynomials and Hierarchical Finite Elements 81
2.9.2 Lagrange Polynomials and Spectral Elements 89
2.10 Flexible Discretization 91
2.10.1 Mortar FEM 93
2.10.2 Nitsche Type Mortaring 106
2.10.3 Numerical Example 111
References 113
3 Mechanical Field 116
3.1 Navier's Equation 116
3.2 Deformation and Displacement Gradient 120
3.3 Mechanical Strain 121
3.4 Constitutive Equations 125
3.4.1 Plane Strain State 127
3.4.2 Plane Stress State 128
3.4.3 Axisymmetric Stress--Strain Relations 129
3.5 Waves in Solid Bodies 129
3.6 Material Properties 131
3.7 Numerical Computation 133
3.7.1 Linear Elasticity 133
3.7.2 Damping Model 135
3.7.3 Geometric Nonlinear Case 137
3.7.4 Numerical Example 143
3.8 Locking and Efficient Solution Approaches 144
3.8.1 Incompatible Modes Method 147
3.8.2 Enhanced Assumed Strain Method 149
3.8.3 Balanced Reduced and Selective Integration 151
References 157
4 Flow Field 159
4.1 Spatial Reference Systems 160
4.2 Reynolds' Transport Theorem 161
4.3 Conservation Equations 162
4.3.1 Conservation of Mass 162
4.3.2 Conservation of Momentum 163
4.3.3 Conservation of Energy 166
4.3.4 Constitutive Equations 167
4.4 Navier-Stokes Equations 167
4.5 Characterization of Flows by Dimensionless Numbers 168
4.6 Finite Element Formulation 169
4.7 Numerical Examples 173
4.7.1 Steady Channel Flow 173
4.7.2 Unsteady Flow Around a Square 176
References 178
5 Acoustic Field 180
5.1 Wave Theory of Sound 180
5.1.1 Conservation of Mass (Continuity Equation) 182
5.1.2 Conservation of Momentum (Euler Equation) 182
5.1.3 Pressure-Density Relation (State Equation) 183
5.1.4 Linear Acoustic Wave Equation 185
5.1.5 Acoustic Quantities 187
5.1.6 Plane and Spherical Waves 189
5.2 Quantitative Measure of Sound 193
5.3 Nonlinear Acoustic Wave Equation 197
5.4 Numerical Computation 202
5.4.1 Linear Acoustic Wave Equation 202
5.4.2 Linear Acoustic Conservation Equations 205
5.4.3 Nonlinear Acoustics 208
5.4.4 Non-conforming Grids 211
5.4.5 Discretization Error 215
5.5 Treatment of Open Domain Problems 219
5.5.1 Absorbing Boundary Conditions 220
5.5.2 Perfectly Matched Layer (PML) Technique 222
5.6 Numerical Examples 233
5.6.1 Transient Wave Propagation in Unbounded Domains 233
5.6.2 Harmonic Wave Propagation in Unbounded Domains 237
5.6.3 Nonlinear Wave Propagation in a Channel 239
References 244
6 Electromagnetic Field 247
6.1 Maxwell's Equations 247
6.1.1 Law of Ampère 249
6.1.2 Law of Faraday 250
6.1.3 Law of Gauss 253
6.1.4 Solenoidal Magnetic Field 254
6.2 Quasistatic Electromagnetic Fields 255
6.2.1 Magnetic Vector Potential 255
6.2.2 Skin Effect 256
6.3 Electrostatic Field 258
6.4 Material Properties 259
6.4.1 Magnetic Permeability 259
6.4.2 Electrical Conductivity 262
6.4.3 Dielectric Permittivity 263
6.5 Electromagnetic Interface Conditions 264
6.5.1 Continuity Relations for Magnetic Field 264
6.5.2 Continuity Relations for Electric Field 265
6.5.3 Continuity Relations for Electric Current Density 267
6.6 Numerical Computation: Electrostatics 267
6.7 Numerical Computation: Electromagnetics 269
6.7.1 Formulation 269
6.7.2 Discretization with Edge Elements 275
6.7.3 Discretization with Nodal Finite Elements 277
6.7.4 Newton's Method for the Nonlinear Case 280
6.7.5 Approximation of BH Curve 283
6.7.6 Higher Order Edge Elements 285
6.7.7 Modeling of Current-Loaded Coil 291
6.7.8 Computation of Global Quantities 292
6.7.9 Induced Electric Voltage 295
6.7.10 Voltage-Loaded Coil 295
6.8 Numerical Examples 297
6.8.1 Thin Iron Plate 297
6.8.2 TEAM-13 Benchmark Problem 300
References 302
7 Coupled Flow-Structural Mechanical Systems 304
7.1 Fluid-Solid Interaction 304
7.2 Coupling Types and Strategies 305
7.3 Grid Adaption 308
7.4 Numerical Examples 311
7.4.1 Solid Plunger 311
7.4.2 Flag in a Flow 312
References 315
8 Coupled Mechanical-Acoustic Systems 316
8.1 Solid--Fluid Interface 316
8.2 Coupled Field Formulation 318
8.3 Numerical Computation 319
8.3.1 Finite Element Formulation 319
8.3.2 Non-conforming Grids 321
8.3.3 Numerical Examples 322
References 327
9 Computational Aeroacoustics 328
9.1 Requirements for Numerical Schemes 328
9.2 Lighthill's Analogy 331
9.3 Curle's Theory 336
9.4 Vortex Sound 341
9.5 Perturbation Equations 343
9.6 Finite Element Formulation 346
9.6.1 Lighthills' Inhomogeneous Wave Equation 346
9.6.2 Perturbation Equations 349
9.6.3 Source Term Treatment 352
9.7 Comparison of Different Aeroacoustic Analogies 353
References 356
10 Coupled Electrostatic-Mechanical Systems 358
10.1 Electrostatic Force 358
10.2 Numerical Computation 365
10.2.1 Calculation Scheme 366
10.2.2 Voltage-Driven Bar 368
References 370
11 Coupled Magnetomechanical Systems 371
11.1 General Moving/Deforming Body 371
11.2 Electromagnetic Force 373
11.3 Numerical Computation 375
11.3.1 Force Computation Via the Principle of Virtual Work 375
11.3.2 Grid Adaption Techniques 378
11.3.3 Calculation Scheme 382
11.3.4 Moving Current/Voltage-Loaded Coil 384
References 391
12 Piezoelectric Systems 393
12.1 Constitutive Equations 393
12.2 Governing Equations: Linear Piezoelectricity 396
12.3 Piezoelectric Material Properties 397
12.4 Models for Nonlinear Piezoelectricity 402
12.4.1 Macroscopic Model with Hysteresis Operators 402
12.4.2 Micro-mechanical Switching Model 410
12.5 Numerical Computation 411
12.5.1 Linear Case 412
12.5.2 Macroscopic Hysteresis Based Approach 414
12.5.3 Micro-mechanical Switching Model 418
12.6 Numerical Examples 423
12.6.1 Computation of Impedance Curve 423
12.6.2 Piezoelectric Disc Actuator 426
12.6.3 Polarization and Depolarization Process 427
References 430
13 Algebraic Solvers 432
13.1 Preconditioned Conjugate Gradient (PCG) Method 432
13.2 Multigrid (MG) Method 434
13.3 Geometric MG Method 437
13.3.1 Geometric MG for Edge Elements 437
13.3.2 Case Study 440
13.4 Algebraic MG Method 443
13.4.1 Auxiliary Matrix 444
13.4.2 Coarsening Process 444
13.4.3 Prolongation Operators 448
13.4.4 Smoother and Coarse-Grid Operator 448
13.4.5 AMG for Nodal Elements 449
13.4.6 AMG for Edge Elements 450
13.4.7 AMG for Time-Harmonic Case 453
13.4.8 Case Studies 454
13.5 Block Preconditioner for Higher Order Edge Element Discretization 460
References 467
14 Industrial Applications 470
14.1 Electrodynamic Loudspeaker 470
14.1.1 Finite Element Models 471
14.1.2 Verification of Computer Models 473
14.1.3 Numerical Analysis of the Nonlinear Loudspeaker Behavior 475
14.1.4 Computer Optimization of the Nonlinear Loudspeaker Behavior 477
14.2 Noise Computation of Power Transformers 477
14.2.1 Finite Element Models 479
14.2.2 Verification of the Computer Models 482
14.2.3 Verification of the Calculated Winding and Tank-Surface Vibrations 482
14.2.4 Verification of the Sound-Field Calculations 484
14.2.5 Influence of Tap-Changer Position 485
14.2.6 Influence of Stiffness of Winding Supports 486
14.3 Fast-Switching Electromagnetic Valves 486
14.3.1 Modeling and Solution Strategy 487
14.3.2 Actuator Characteristics 489
14.3.3 Actuator Dynamics 491
14.3.4 Dynamics Optimization I: Electrical Premagnetization 492
14.3.5 Dynamics Optimization II: Overexcitation 494
14.3.6 Switching Cycle 495
14.4 Cofired Piezoceramic Multilayer Actuators 496
14.4.1 Polarization of a Stack Actuator 497
14.4.2 Stack Actuator: Hysteresis Based Approach 500
14.5 Capacitive Micro-machined Ultrasound Transducers 502
14.5.1 Requirements to Numerical Simulation Scheme 503
14.5.2 Single CMUT Cell 505
14.5.3 CMUT Array 507
14.5.4 Controlled CMUT Array 508
14.6 High-Intensity Focused Ultrasound 512
14.6.1 Piezoelectric Transducer and Input Impedance 512
14.6.2 Pressure Pulse Computation 514
14.6.3 High-Power Pulse Sources for Lithotripsy 515
14.7 Human Phonation 520
14.7.1 Mathematical Modeling 522
14.7.2 2D Fully Coupled Simulation 522
14.7.3 3D Driven Simulation 527
14.8 Aeroacoustics of Flow Around Obstacles 532
14.8.1 Square Cylinder Geometries 532
14.8.2 Edge Tone 540
14.8.3 Airframe Noise 547
References 550
15 Summary and Outlook 553
References 554
Appendix A Norms 555
Appendix B Scalar and Vector Fields 557
Appendix C Tensors and Index Notation 573
Appendix D Appropriate Function Spaces 578
Appendix E Solution of Nonlinear Equations 583
Appendix F Hysteresis Model 589
Index 594