Numerical Simulation of Mechatronic Sensors and Actuators (eBook)

Preface to the second edition 5
Acknowledgment 6
Preface to the first edition 7
Acknowledgements 10
Notation 11
Mathematical symbols Finite Element Method 11
Acoustics 12
Electromagnetics 12
Mechanics 12
Contents 13
1 Introduction 19
2 The Finite Element (FE) Method 25
2.1 Finite Element Formulation 27
2.2 Finite Element Method for a 1D Problem 31
2.3 Nodal (Lagrangian) Finite Elements 38
2.4 Finite Element Procedure 50
2.5 Time Discretization 53
2.6 Integration over Surfaces 61
2.7 Edge (N´ ed ´ elec) Finite Elements 61
2.8 Discretization Error 63
3 Mechanical Field 68
3.1 Navier’s Equation 68
3.2 Deformation and Displacement Gradient 72
3.3 Mechanical Strain 73
3.4 Constitutive Equations 77
3.5 Waves in Solid Bodies 80
3.6 Material Properties 82
3.7 Numerical Computation 83
3.8 Locking and Efficient Solution Approaches 95
4 Electromagnetic Field 110
4.1 Maxwell’s Equations 110
4.2 Quasistatic Electromagnetic Fields 118
4.3 Electrostatic Field 121
4.4 Material Properties 121
4.5 Electromagnetic Interface Conditions 126
4.6 Numerical Computation: Electrostatics 130
4.7 Numerical Computation: Electromagnetics 131
4.8 Numerical Examples 152
5 Acoustic Field 155
5.1 Wave Theory of Sound 155
5.2 Quantitative Measure of Sound 167
5.3 Non-linear Acoustic Wave Equation 172
5.4 Numerical Computation 176
5.5 Treatment of Open Domain Problems 189
5.6 Numerical Examples 200
6 Coupled Electrostatic-Mechanical Systems 211
6.1 Electrostatic Force 212
6.2 Numerical Computation 218
7 Coupled Magnetomechanical Systems 223
7.1 General Moving/Deforming Body 223
7.2 Electromagnetic Force 225
7.3 Numerical Computation 227
8 Coupled Mechanical-Acoustic Systems 244
8.1 Solid–Fluid Interface 245
8.2 Coupled Field Formulation 246
8.3 Numerical Computation 248
9 Piezoelectric Systems 257
9.1 Constitutive Equation 257
9.2 Governing Equations 258
9.3 Piezoelectric Material Properties 259
9.4 Models for Non-linear Piezoelectricity 263
9.5 Hysteresis Model 267
9.6 Numerical Computation 271
9.7 Numerical Examples 275
10 Computational Aeroacoustics 281
10.1 Requirements for Numerical Schemes 281
10.2 Lighthill’s Analogy and its Extension 284
10.3 Finite Element Formulation 289
10.4 Validation: Co-Rotating Vortex Pair 293
11 Algebraic Solvers 297
11.1 Preconditioned Conjugate Gradient (PCG) Method 297
11.2 Multigrid (MG) Method 299
11.3 Geometric MG Method 301
11.4 Algebraic MG Method 307
12 Industrial Applications 326
12.1 Electrodynamic Loudspeaker 326
12.2 Noise Computation of Power Transformers 333
12.3 Fast-switching Electromagnetic Valves 343
12.4 Cofired Piezoceramic Multilayer Actuators 353
12.5 Capacitive Micromachined Ultrasound Transducers 358
12.6 High-Intensity Focused Ultrasound 370
12.7 Noise Generated from a Flow around a Square Cylinder 379
A Norms 394
A.1 Vector Norms 394
A.2 Matrix Norms 395
B Scalar and Vector Fields 396
B.1 The Nabla (.) Operator 399
B.2 Definition of Gradient, Divergence, and Curl 399
B.3 The Gradient 400
B.4 The Flux 402
B.5 Divergence 403
B.6 Divergence Theorem (Gauss Theorem) 405
B.7 The Circulation 405
B.8 The Curl 406
B.9 Stoke’s Theorem 408
B.10 Green’s Integral Theorems 409
B.11 Application of the Operators 410
B.12 Irrotational Vector Fields 410
B.13 Solenoidal Vector Fields 411
C Appropriate Function Spaces 412
D Solution of Nonlinear Equations 417
D.1 Fixed-point Iteration 419
D.2 Newton’s Method 420
References 423
Index 435