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