Preface	6
	Contents	9
	Contributors	16
	1 Mathematics in Laser Processing	17
	Abstract	17
	1.1 Mathematics and Its Application	17
	1.2 Formulation in Terms of Partial Differential Equations	19
	1.2.1 Length Scales	19
	1.2.2 Rectangular Cartesian Tensors	20
	1.2.3 Conservation Equations and Their Generalisations	23
	1.2.4 Governing Equations of Generalised Conservation Type	25
	1.2.4.1 Flow of a Viscous Fluid	25
	1.2.4.2 Viscous Heat Flow	27
	1.2.4.3 Conservation of Electric Charge	28
	1.2.4.4 Linear Thermo-Elasticity in a Moving Frame of Reference	28
	1.2.5 Gauss’s Law	29
	1.3 Boundary and Interface Conditions	30
	1.3.1 Generalised Conservation Conditions	30
	1.3.2 The Kinematic Condition in Fluid Dynamics	35
	1.4 Fick’s Laws	36
	1.5 Electromagnetism	37
	1.5.1 Maxwell’s Equations	37
	1.5.2 Ohm’s Law	39
	References	40
	2 Simulation of Laser Cutting	41
	2.1 Introduction	42
	2.1.1 Physical Phenomena and Experimental Observation	44
	2.2 Mathematical Formulation and Analysis	47
	2.2.1 The One-Phase Problem	49
	2.2.2 The Two-Phase Problem	62
	2.2.3 Three-Phase Problem	70
	2.3 Outlook	84
	References	85
	3 Glass Cutting	89
	Abstract	89
	3.1 Introduction	90
	3.2 Phenomenology of Glass Processing with Ultrashort Laser Radiation	90
	3.3 Modelling the Propagation of Radiation and the Dynamics of Electron Density	92
	3.4 Radiation Propagation Solved by BPM Methods	93
	3.5 The Dynamics of Electron Density Described by Rate Equations	93
	3.6 Properties of the Solution with Regard to Ablation and Damage	95
	3.7 Electronic Damage Versus Thermal Damage	98
	3.8 Glass Cutting by Direct Ablation or Filamentation?	102
	Acknowledgements	103
	References	103
	4 Keyhole Welding: The Solid and Liquid Phases	105
	Abstract	105
	4.1 Heat Generation and Heat Transfer	105
	4.1.1 Absorption	105
	4.1.2 Heat Conduction and Convection	107
	4.2 Steady State 3D-Solutions Based on Moving Point Sources of Heat	108
	4.3 Steady State 2D-Heat Conduction from a Moving Cylinder at Constant Temperature	110
	4.4 Sophisticated Quasi-3D-Model Based on the Moving Line Source of Heat	112
	4.4.1 Surface Convection and Radiation	113
	4.4.2 Phase Transformations	114
	4.4.3 Transient and Pulsed Heat Conduction	115
	4.5 Model for Initiation of Laser Spot Welding	115
	4.5.1 Geometry of the Liquid Pool	117
	4.6 Mass Balance of a Welding Joint	118
	4.7 Melt Flow	119
	4.7.1 Melt Flow Passing Around the Keyhole	120
	4.7.2 Numerical 2D-Simulation of the Melt Flow Around a Prescribed Keyhole Shape	121
	4.7.3 Marangoni Flow Driven by Surface Tension Gradients	123
	4.7.4 Flow Redirection, Inner Eddies, Spatter and Stagnation Points	124
	4.7.5 Humping Caused by Accumulating Downstream Flow	125
	4.7.6 Keyhole Front Melt Film Flow Downwards, Driven by Recoil Pressure	125
	4.8 Concluding Remarks	126
	References	127
	5 Laser Keyhole Welding: The Vapour Phase	129
	Abstract	129
	5.1 Notation	129
	5.2 The Keyhole	131
	5.3 The Keyhole Wall	135
	5.3.1 The Knudsen Layer	135
	5.3.1.1 Ablation Through the Knudsen Layer	135
	5.3.1.2 Thermal Flux and Viscous Slip in the Knudsen Layer	138
	5.3.2 Fresnel Absorption	139
	5.4 The Role of Convection in the Transfer of Energy to the Keyhole Wall	140
	5.5 Fluid Flow in the Keyhole	144
	5.5.1 General Aspects	144
	5.5.2 Turbulence in the Weld Pool and the Keyhole	145
	5.5.3 Stability of the Keyhole Wall	147
	5.5.4 Stability of Waves of Acoustic Type	147
	5.5.5 Elongation of the Keyhole	151
	5.6 Further Aspects of Fluid Flow	152
	5.6.1 Simplifying Assumptions for an Analytical Model	152
	5.6.2 Lubrication Theory Model	152
	5.6.3 Boundary Conditions	153
	5.6.4 Solution Matched to the Liquid Region	157
	5.7 Electromagnetic Effects	158
	5.7.1 Self-induced Currents in the Vapour	158
	5.7.2 The Laser Beam as a Current Guide	163
	5.7.2.1 Note on Cooling by Thermal Convection	165
	References	165
	6 Basic Concepts of Laser Drilling	168
	6.1 Introduction	169
	6.2 Technology and Laser Systems	169
	6.3 Diagnostics and Monitoring for s Pulse Drilling	171
	6.4 Phenomena of Beam-Matter Interaction	173
	6.4.1 Physical Domains---Map of Intensity and Pulse Duration	174
	6.4.2 Beam Propagation	180
	6.4.3 Refraction and Reflection	182
	6.4.4 Absorption and Scattering in the Gaseous Phase	183
	6.4.5 Kinetics and Equation of State	184
	6.5 Phenomena of the Melt Expulsion Domain	186
	6.6 Mathematical Formulation of Reduced Models	188
	6.6.1 Spectral Decomposition Applied to Dynamics in Recast Formation	189
	6.7 Analysis	190
	6.7.1 Initial Heating and Relaxation of Melt Flow	190
	6.7.2 Widening of the Drill by Convection	192
	6.7.3 Narrowing of the Drill by Recast Formation	193
	6.7.4 Melt Closure of the Drill Hole	195
	6.7.5 Drilling with Inertial Confinement---Helical Drilling	197
	6.8 Outlook	199
	References	200
	7 Arc Welding and Hybrid Laser-Arc Welding	204
	Abstract	204
	7.1 The Structure of the Welding Arc	204
	7.1.1 Macroscopic Considerations	205
	7.1.2 Arc Temperatures and the PLTE Assumption	215
	7.1.3 Multi-component Plasmas	221
	7.2 The Arc Electrodes	224
	7.2.1 The Cathode	224
	7.2.2 The Anode	226
	7.3 Fluid Flow in the Arc-Generated Weld Pool	227
	7.4 Unified Arc and Electrode Models	230
	7.5 Arc Plasma-Laser Interactions	233
	7.5.1 Absorption	234
	7.5.2 Scattering	239
	7.5.3 Absorption Measurements	241
	7.6 Laser-Arc Hybrid Welding	242
	References	250
	8 Metallurgy and Imperfections of Welding and Hardening	255
	Abstract	255
	8.1 Thermal Cycle and Cooling Rate	255
	8.2 Resolidification	258
	8.3 Metallurgy	259
	8.3.1 Diffusion	259
	8.3.2 Fe-Based Alloys	261
	8.3.2.1 Low Alloy Steel	261
	8.3.3 Model of the Metallurgy During Transformation Hardening of Low Alloy Steel	263
	8.3.4 Non-Fe-Based Alloys	265
	8.4 Imperfections	266
	8.4.1 Large Geometrical Imperfections	267
	8.4.2 Cracks	268
	8.4.3 Spatter	269
	8.4.4 Pores and Inclusions	270
	References	274
	9 Laser Cladding	276
	Abstract	276
	9.1 Introduction	276
	9.2 Beam-Particle Interaction	283
	9.2.1 Powder Mass Flow Density	283
	9.2.2 Effect of Gravity on the Mass Flow Distribution	284
	9.2.3 Beam Shadowing and Particle Heating	286
	9.3 Formation of the Weld Bead	289
	9.3.1 Particle Absorption and Dissolution	290
	9.3.2 Shape of the Cross Section of a Weld Bead	291
	9.3.3 Three-dimensional Model of the Melt Pool Surface	293
	9.3.4 Temperature Field Calculation Using Rosenthal’s Solution	294
	9.3.5 Self-consistent Calculation of the Temperature Field and Bead Geometry	296
	9.3.6 Role of the Thermocapillary Flow	297
	9.4 Thermal Stress and Distortion	300
	9.4.1 Fundamentals of Thermal Stress	300
	9.4.2 Phase Transformations	302
	9.4.3 FEM Model and Results	304
	9.4.4 Simplified Heuristic Model	305
	9.4.5 Crack Prevention by Induction Assisted Laser Cladding	311
	9.5 Conclusions and Future Work	314
	References	316
	10 Laser Forming	320
	Abstract	320
	10.1 History of Thermal Forming	322
	10.2 Forming Mechanisms	323
	10.2.1 Temperature Gradient Mechanism	324
	10.2.2 Residual Stress Point Mechanism	331
	10.2.3 Upsetting Mechanism	333
	10.2.4 Buckling Mechanism	338
	10.2.5 Residual Stress Relaxation Mechanism	342
	10.2.6 Martensite Expansion Mechanism	343
	10.2.7 Shock Wave Mechanism	344
	10.3 Applications	345
	10.3.1 Plate Bending	346
	10.3.2 Tube Bending/Forming	347
	10.3.3 High Precision Positioning Using Actuators	348
	10.3.4 Straightening of Weld Distortion	349
	10.3.5 Thermal Pre-stressing	350
	References	351
	11 Femtosecond Laser Pulse Interactions with Metals	354
	Abstract	354
	11.1 Introduction	354
	11.2 What Is Different Compared to Longer Pulses?	356
	11.2.1 The Electron-Electron Scattering Time	356
	11.2.2 The Nonequilibrium Electron Distribution	359
	11.3 Material Properties Under Exposure to Femtosecond Laser Pulses	361
	11.3.1 Optical Properties	361
	11.3.2 Thermal Properties	363
	11.3.3 Electronic Thermal Diffusivity	365
	11.4 Determination of the Electron and Phonon Temperature Distribution	366
	11.4.1 The Two-Temperature Model	366
	11.4.2 The Extended Two-Temperature Model	369
	11.5 Summary and Conclusions	372
	References	373
	12 Meta-Modelling and Visualisation of Multi-dimensional Data for Virtual Production Intelligence	375
	Abstract	375
	12.1 Introduction	375
	12.2 Implementing Virtual Production Intelligence	377
	12.3 Meta-Modelling Providing Operative Design Tools	378
	12.4 Meta-Modelling by Smart Sampling with Discontinuous Response	384
	12.5 Global Sensitivity Analysis and Variance Decomposition	389
	12.6 Reduced Models and Emulators	392
	12.7 Summary and Advances in Meta-Modelling	393
	Acknowledgements	393
	References	394
	13 Comprehensive Numerical Simulation of Laser Materials Processing	396
	13.1 Motivation---The Pursuit of Ultimate Understanding	397
	13.2 Review	398
	13.3 Correlation, The Full Picture	404
	13.4 Introduction to Numerical Techniques	405
	13.4.1 The Method of Discretisation	405
	13.4.2 Meshes	406
	13.4.3 Explicit Versus Implicit	407
	13.4.4 Discretisation of Transport pde's	408
	13.4.5 Schemes of Higher Order	411
	13.4.6 The Multi Phase Problem	413
	13.5 Solution of the Energy Equation and Phase Changes	416
	13.5.1 Gas Dynamics	419
	13.5.2 Beam Tracing and Associated Difficulties	421
	13.6 Program Development and Best Practice When Using Analysis Tools	423
	13.7 Introduction to High Performance Computing	425
	13.7.1 MPI	425
	13.7.2 openMP	427
	13.7.3 Hybrid	428
	13.7.4 Performance	429
	13.8 Visualisation Tools	430
	13.9 Summary and Concluding Remarks	431
	References	432
	Index	437