Thermomechanical Simulation of Manufacturing Processes using GPU-Accelerated Particle Methods

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(1)diss. eth no. 26833. THERMOMECHANICAL SIMULATION OF MANUFACTURING PROCESSES USING GPU-ACCELERATED PARTICLE METHODS. A dissertation submitted to eth zurich. for the degree of doctor of sciences (Dr. sc. ETH Zurich). presented by. mohamadreza afrasiabi M.Sc. from University of Tehran born on August 12, 1984. accepted on the recommendation of Prof. Dr. Eleni Chatzi, examiner Prof. Dr.-Ing. Dr. h.c. Konrad Wegener, co-examiner Prof. Dr.-Ing. Friedrich Bleicher, co-examiner Prof. Dr. Pavel Hora, co-examiner. 2020.

(2) Mohamadreza Afrasiabi: Thermomechanical Simulation of Manufacturing Processes using GPUAccelerated Particle Methods, © 2020.

(3) D E D I C AT I O N. To my parents, who instilled in me the virtues of perseverance and commitment. And ... To YM, without whom none of these would have been possible!. iii.

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(5) ABSTRACT. Due to the tremendous growth of computer technology, numerical simulations have established themselves as an integral part of most scientific and engineering advances over the past 50 years. In essence, they are best appreciated in areas where the limitations of financial resources, theoretical developments, and experimental studies are regarded as serious challenges. Metal cutting, or manufacturing processes as a whole, is entangled with all of these limitations, hence a prime candidate for numerical investigations. Unsurprisingly though, the numerical analysis of metal cutting is complicated and fraught with pitfalls since there is a wide range of diverse physical phenomena to be modeled. In this view, several grand challenges to address include thermo-mechanical coupling, severe contact/friction conditions, generation of new surfaces, very large deformations, and extremely high deformation and temperature rates. Mesh-based methods such as FEM have proven to be phenomenally successful as a numerical tool for solving such problems, according to hundreds of thousands of scientific citations. Nevertheless, they face major difficulties in handling mesh distortions and numerical instabilities without particular remedies like remeshing and ALE formulations. Not only are these solutions time consuming, but some of them (e.g., the remeshing algorithm) usually leads to the degradation of computational accuracy. Lagrangian (meshfree) particle methods, on the other hand, can handle large deformations with no theoretical limit and without the caveat of mesh distortion (since there is no mesh). They relieve the burden of remeshing procedures, thus an attractive choice for metal cutting simulations. The main technical drawbacks of these methods, however, lie in: 1. Enforcing essential boundary conditions 2. Lack of interpolation consistency 3. Some numerical and tensile instabilities 4. Lack of explicit interface representation While numerous corrections have been made so far to rectify these four deficiencies in different applications, the use of particle methods in manufacturing simulations is still in its nuclei stage and requires further development. In this thesis, a new particle-based software tool that can accurately and efficiently simulate manufacturing processes is presented. A broad array of modern algorithms and state-of-the-art enhancements are incorporated to tackle the shortcomings of particle methods outlined above. Furthermore, algorithmic and computational measures are both implemented to optimize the runtime. Special care is taken to facilitate multi-resolution simulations, where dynamic refinement and coarsening procedures are enabled. While keeping the computational times manageable, the present particle-based code is additionally accelerated by the virtue of General-Purpose computing on Graphics Processing Units (GPGPUs) using the CUDA platform. As a result of this efficiency, one can adopt the code to gain. v.

(6) valuable insights into various problems such as the modeling of complex 3D applications, performing several parametric studies, and running simulations in high resolution. Besides multiple reconstruction tests as well as structural benchmarks, the solver is applied to simulate various manufacturing problems such as laser drilling, ultra-precision machining, tribometer device, and metal cutting. The present methods are, nonetheless, more widely applicable to a range of problems in which geometrical, thermal, and mechanical aspects are substantially important. Indicatively, an enhanced Coulomb law whose coefficient of friction is a decreasing function of temperature is proposed for more realistic modeling of metal machining. Thanks to the remarkable speedup gained by GPU computing, the unknown parameters of this friction model are determined for the first time by employing an inverse identification method in iterative cutting simulations. Finally, a fully coupled geometrical-thermo-mechanical model is developed as a proof of concept to maintain the geometry of moving and/or newly generated interfaces encountered in complex multiphysics problems.. vi.

(7) K U R Z FA S S U N G. Durch den gewaltigen Anstieg an verfügbarer Rechenleistung in den letzten 50 Jahren wurden numerische Simulationen zum integralen Bestandteil von Wissenschaftlichen Erkenntnissen und Fortschritten im Ingenieurwesen. Numerische Simulationen sind insbesondere dann wichtig, wenn finanzielle Ressourcen, theoretische Entwicklungen und experimentelle Studien zu ernsthaften Herausforderungen werden. Die Metallzerspanung im Speziellen und die Fertigungstechnik im Allgemeinen unterliegen diesen genannten Herausforderungen und sind somit exzellente Kandidaten für numerische Untersuchungen. Die Vielzahl an verschiedenen physikalischen Phänomenen die in der Metallzerspanung machen ihre Analyse kompliziert und voller Hindernisse. Einige der besonderen Herausforderungen sind thermomechanische Zusammenhänge, extreme Reib- und Kontaktbedingungen, Entstehung neuer Oberflächen, sehr grosse Deformationen und Temperaturraten. Netzbasierte Methoden wie die FEM haben sich als erfolgreiches numerisches Werkzeug zur Simulation solcher Probleme bewiesen, wie man an hunderttausenden Zitaten in der Fachliteratur erkennen kann. Nichtsdestotrotz haben FEM Schwierigkeiten mit Netzverzerrung und numerischen Instabilitäten falls nicht auf bestimmte Methoden wie Neuvernetzung und oder ALE Formulierungen zurück gegriffen wird. Diese Lösungen sind erhöhen jedoch die Laufzeit oder gehen auf Kosten der Rechengenauigkeit (z.B. die Neuvernetzung). Lagransche, (netzfreie) Partikelmethoden Methoden hingegen können beliebig grosse Deformationen abbilden, ohne Neuvernetzung (zumal sie eben ohne Netz auskommen). Die Neuvernetzung wird vermieden, was diese Methoden zu einer attraktiven Wahl für die Simulation der Metallzerspanung macht. Doch auch die netzfreien Methoden haben Nachteile, insbesondere: 1. Erfüllung essentieller Randbedingungen 2. Mangelnde Interpolationskonsistenz 3. Numiersche Instabilität und Zuginstabilität 4. Keine explizite Oberflächenrepresentation Während über eine Vielzahl an Fortschritten bezüglich diesen 4 Punkten in der Literatur berichtet wird, steckt die Simulation von Fertigungsprozessen durch Partikelmethoden immer noch in den Kinderschuhen. In dieser Arbeit wird ein neues, partikelbasiertes Software Tool entwickelt, welches Fertigungstechniksprozesse akkurat und effizient abbilden kann. Die genannten Nachteile der Partikelmethoden werden mit einer breiten Auswahl an Algorithmen und Verbesserungen aus der State of the Art Literatur begegnet. Sowohl algorithmische wie auch rechnerische Massnahmen werden getroffen um die Laufzeit zu optimieren. Spezielle Sorgfalt wurde getroffen, um Simulationen mit variierender Auflösung zu ermöglichen. Diese Simulationen geniessen die Vorteile von dynamischen Verfeinerungswie auch Vergröberungsprozeduren. Weitere Laufzeitverbesserung wurde durch die Nutzung von Grafikcoprozessoren erzielt (GPGPU Computing). Dadurch wurde die Analyse von komplexen 3D Prozessen, Parameterstudien und Hochauflösungssimulationen erst möglich.. vii.

(8) Neben verschiedenen strukturellen Testrechnungen und Rekonstruktionstests wird der Löser auch verschiedenen Fertigungsprozessen wie dem Laserbohren, der Ultraprezisionsverarbeitung, Tribometertests und der Metallzerspanung unterworfen. Anzumerken ist hierbei, dass die geschaffenen Methoden für eine weit breiteres Problemfeld geeignet sind, in dem geometrische, thermische und mechanische Aspekte dominieren. In dem Kontext wird auch ein erweitertes Coulomb Modell vorgeschlagen, dessen Reibkoeffizient temperaturabhängig gemacht wird. Durch die beschleunigte Rechnung auf der GPU kann dieser Parameter zum ersten mal durch inverse Identifikationsmethoden anhand von iterativen Zerspanungssimulationen bestummen werden. Schlussendlich wird zum ersten mal ein voll geometrisch, thermisch, mechanisch gekoppeltes Metallzerspanungsmodell im netzfreien, updated Lagranschen Rahmen als Proof of Concept entwickelt.. viii.

(9) ACKNOWLEDGMENTS. Undertaking this PhD at ETH Zurich has been a truly life-changing experience for me and it would not have been possible to do without the support, corporation, guidance, and love that I received from many people. First and foremost, I would like to express my deepest sense of gratitude and profound respect to my PhD advisors, Prof. Dr.-Ing. Konrad Wegener and Prof. Dr. Eleni Chatzi, who have helped and encouraged me at all stages of my work with immense care and great patience. I enjoyed our innumerable discussions to the fullest, but also benefited from their thoughtful responses to my steady stream of emails and manuscripts. Prof. Wegener’s passion for science has been a major impetus for me to conduct this research. He was and remains my best role model for a scientist, mentor, and a teacher. Eleni is an extremely responsible, smart, and energetic professor, who always responded to my queries so promptly. I will forever be thankful to both of them! Furthermore, I would like to thank the members of my PhD committee, Prof. Dr. Pavel Hora and Prof. Dr.-Ing. Friedrich Bleicher, for reviewing this dissertation and providing me with their invaluable feedback. I would also like to say a heartfelt thank you to Prof. Dr. Soheil Mohammadi and Prof. Dr. Thomas Schumacher for their helpful suggestions and for making it possible to complete what I started back in 2010. They believed in me and encouraged me to chase my academic dreams. Thank you, Soheil and Thomas! This research was funded in part by the Swiss National Science Foundation, to whom I am extremely grateful. Particular thanks should also go to the following institute staff: Dr. Fredy Kuster, Mr. Josef Meile, and Ms. Ewa Grob for their unfailing support and assistance in all administrative, technical, and financial issues. A very special and deep appreciation goes out to my research teammates: Dr. Matthias Röthlin and Hagen Klippel. Their excellent work during this project has made a priceless contribution to my PhD. Most of all, I am grateful to them for their friendship and for making our office at Technopark ZH a fantastic environment. What a cracking place to work! I am indebted to all my friends who shared the joy, the pain, and the moments with me, who have always been so understanding and supportive in numerous ways. With a special mention to my former colleagues and current friends: Ali, Timo, Andreas, Martin, Bircan, Mohammad, Thomas, Markus, Stefan, Kevin, and Mansur for helping in whatever way they could during this challenging period. Thanks in particular to Eduard Relea for plenty of interesting lunch-time discussions about life, mechanical watches, and soccer! My thanks also go out to the support I received from the collaborative work I undertook with my co-workers at IWF and IBK. I am especially grateful to Dr. Simon Züst, Dr. Kostas Agathos, and Linus Meier for their both input and insights. And finally, last but by no means least, the most special gratitude to my beloved parents, Amir and Maryam, for their unconditional and endless love; to my siblings, Mahdieh and Alireza, for their continual encouragement through all the ups and downs of the last 5 years; and to God for making it all worth living this life! Without them, I would not have had the courage to embark on this adventurous journey in the first place. M A - Zürich, Oct 2019. ix.

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(11) CONTENTS. nomenclature & acronyms 1 introduction 1.1 Background & Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 State of the Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Simulation of Manufacturing Processes . . . . . . . . . . . . . . 1.2.2 Particle Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Simulation of Manufacturing Processes with Particle Methods 1.3 Research Gaps & Objectives . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Layout of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 mechanical framework 2.1 Solid Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Strain & Stress Measures . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Conservation of Mass . . . . . . . . . . . . . . . . . . . . . . . . 2.1.4 Conservation of Momentum . . . . . . . . . . . . . . . . . . . . 2.1.5 Conservation of Energy . . . . . . . . . . . . . . . . . . . . . . . 2.1.6 Thermal Aspects & Thermo-Mechanical Coupling . . . . . . . 2.1.7 Constitutive Laws . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.8 Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Fluid Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Conservation of Mass (Continuity) . . . . . . . . . . . . . . . . . 2.2.2 Conservation of Momentum . . . . . . . . . . . . . . . . . . . . 2.2.3 Conservation of Energy . . . . . . . . . . . . . . . . . . . . . . . 2.2.4 Surface Tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.5 Equation of State (EOS) . . . . . . . . . . . . . . . . . . . . . . . 3 computational framework 3.1 Discretization in Space with Particle Methods . . . . . . . . . . . . . . 3.1.1 Fundamentals of SPH . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Corrective Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Laplacian Approximation . . . . . . . . . . . . . . . . . . . . . . 3.2 Interface Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Level Set Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Particle Level Set Methods . . . . . . . . . . . . . . . . . . . . . 3.3 Discretization in Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Euler Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Leapfrog Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Runge-Kutta Schemes . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4 The Choice of Time-Step Size . . . . . . . . . . . . . . . . . . . . 4 algorithms & implementations 4.1 Implementation of Plasticity . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Return Mapping Algorithm . . . . . . . . . . . . . . . . . . . . . 4.1.2 Radial Return Method & Root-Finding Procedure . . . . . . . . 4.2 Contact Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. xiii 1 1 5 5 7 13 15 16 17 17 18 20 22 23 23 24 26 34 36 37 37 38 38 39 41 42 42 48 58 66 67 71 77 77 78 79 80 83 84 84 85 87. xi.

(12) contents. 4.2.1 2D Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 3D Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Implementation of Friction . . . . . . . . . . . . . . . . . . . . . . 4.3 Numerical Stabilization Schemes . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Artificial Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Artificial Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Smoothed Velocity Field . . . . . . . . . . . . . . . . . . . . . . . . 4.3.4 XSPH Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.5 Variable Smoothing Length . . . . . . . . . . . . . . . . . . . . . . 4.4 Adaptivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Refinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Coarsening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 parallel computing 5.1 GPU for Parallel Computing . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 CUDA Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Terminology & Preliminary Definitions . . . . . . . . . . . . . . . 5.2.2 Processing Flow & Execution Model . . . . . . . . . . . . . . . . . 5.3 Implementation of the Code on a GPU . . . . . . . . . . . . . . . . . . . 5.3.1 Basics & Particle Interaction . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Model Logic & Flow Diagram . . . . . . . . . . . . . . . . . . . . 5.3.3 Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 results & discussion 6.1 Geometrical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Zalesak’s Disk - 2D Rigid Body Rotation . . . . . . . . . . . . . . 6.1.2 Single Vortex - 2D Interface Stretching Flow . . . . . . . . . . . . 6.1.3 Zalesak’s Sphere - 3D Rigid Body Rotation . . . . . . . . . . . . . 6.1.4 LeVeque’s Test - 3D Deformation Field . . . . . . . . . . . . . . . 6.2 Thermal Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Validation Benchmark: 3D Heat Conduction Problem . . . . . . 6.2.2 Laser Drilling Process . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 Ultra-Precision Machining Application . . . . . . . . . . . . . . . 6.3 Mechanical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Validation Benchmark: 3D Taylor Impact Test . . . . . . . . . . . 6.3.2 Tensile Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.3 Collision of Rubber Rings . . . . . . . . . . . . . . . . . . . . . . . 6.4 Thermo-Mechanical Simulations . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 2D Cutting Simulation on CPU: Dynamic Refinement . . . . . . 6.4.2 3D Tribometer Simulation on GPU: Proof-of-Concept . . . . . . . 6.4.3 2D Cutting Simulation on GPU: Parameter Identification of µ( T ) 6.5 Geometrical Thermo-Mechanical Simulations . . . . . . . . . . . . . . . . 6.5.1 Heat Transfer in a Rising Bubble . . . . . . . . . . . . . . . . . . . 6.5.2 Interface Capturing in a Metal Cutting Test . . . . . . . . . . . . . 7 conclusions & future work 7.1 Summary & Outlook in Physical Model Directions . . . . . . . . . . . . 7.2 Summary & Outlook in Numerical Method Directions . . . . . . . . . . 7.3 Summary & Outlook in Computational Performance Directions . . . . . bibliography. xii. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 88 89 92 93 95 96 97 98 98 103 103 107 111 112 114 114 115 116 117 118 119 123 124 124 126 132 133 137 137 139 149 154 154 155 161 164 165 176 182 194 194 198 205 205 206 208 209.

(13) N O M E N C L AT U R E & A C R O N Y M S. The symbols and operations most frequently used in the text are given below followed by a list of abbreviations. Any other notation introduced will be defined in the manuscript when required.. Convention: ˙ ♦. ⇐⇒ denotes time derivative (Newton’s notation). ♦. ⇐⇒ denotes a zeroth-order tensor (scalar). ♦. ⇐⇒ denotes a first-order tensor (vector). ♦ ∼. ⇐⇒ denotes a second-order tensor. ♦. ⇐⇒ denotes a third-order tensor. ∼. ♦ ≈. h♦i. ⇐⇒ denotes a forth-order tensor ⇐⇒ approximate value of ♦ by numerical discretization. Operations: Gradient : Divergence : Laplacian :. ∇♦ ∇·♦ ∇2 ♦ = ∇ · (∇♦). H♦ = ∇ ⊗ ∇♦ = ∇(∇♦) q Norm of vector : |♦| = ♦ · ♦ Hessian :. Determinant of matrix :. det(♦ ) ∼. Trace of tensors :. tr(♦ ) ∼. Scalar/dot product :. ♦·♦. Vector/cross product :. ♦×♦. Tensor/outer product :. ♦⊗♦. Double contraction of tensors :. ♦ :♦ ∼ ∼. xiii.

(14) nomenclature & acronyms. Nomenclature: Part 1. Symbol. Definition. a AL c cp d e E G h H hc k m nc Np p PL q or Q t T Tm Tr Ts T∞ V W x, y, z. Thermal diffusivity Coefficient of laser absorption Speed of sound Specific heat capacity Distance Internal energy Young’s modulus Shear modulus Smoothing length (i.e., characteristic radius of W) Measure of distortion Heat convection coefficient Heat conductivity Mass Normalization factor Number of particles Pressure Laser power Heat source/loss term Time (and ∆t is time-step) Temperature Melting temperature Reference/room temperature Surface temperature Background temperature Volume Smoothing kernel function Spatial coordinates. xiv.

(15) nomenclature & acronyms. Nomenclature: Part 2. Symbol. Definition. α γ Γ δ (r − r 0 ) ∆x e η θ Θ κ λ µ µv ν νv π Π ρ σ σy Σ φ ϕ χ Ω0 Ω. Clearance angle Rake angle Interface Dirac delta function based at r 0 Particle spacing Emissivity coefficient Coefficient of frictional work converted into heat 1D interpolation kernel Multi-dimensional interpolation kernel Curvature Lamé’s first parameter Coefficient of friction Dynamic (shear) viscosity Poisson’s ratio Kinematic viscosity Ratio of a circle’s circumference to its diameter, approximately equal to 3.14159 Artificial viscosity term Density Stefan–Boltzmann constant Yield stress Sum Mapping function Level set function Coefficient of plastic work converted into heat or TQ factor Reference configuration/domain Current configuration/domain. xv.

(16) nomenclature & acronyms. Nomenclature: Part 3. Symbol. Definition. a f Fb n r or x u v ω. Acceleration Force Body forces Unit normal Position vector Distance vector (i.e., displacement) Velocity Vorticity. A ∼ B ≈ C ∼ D ∼ E ∼ E ∼ F ∼ F ∼ G ∼ I ∼ K ∼ L ∼ N ∼ P ∼ s ∼ S ∼ ε ∼ Λ ∼ σ ∼ τ∼ ω ∼ Υ ≈. Randles-Liberky or CSPM correction tensor ICSPM correction tensor Right Cauchy–Green stretch tensor Strain rate tensor Cauchy Green (or Green St-Venant) strain tensor FMFS left corrector Deformation gradient tensor FMFS right corrector FMFS re-normalization matrix Identity tensor Heat conductivity tensor Velocity gradient tensor Nominal stress tensor First Piola-Kirchhoff stress tensor Deviatoric stress tensor Second Piola-Kirchhoff stress tensor Strain tensor (infinitesimal) Artificial stress tensor Cauchy stress tensor Fluid stress tensor Spin tensor Elastic moduli tensor. xvi.

(17) nomenclature & acronyms. List of Abbreviations:. Acronym. Meaning. 1D 2D 3D ALE BC CAD CEL CFD CFL CPU CSPM CUDA DIN DP EFG FD FE FEA FEM FLOP FLOPS FMFS FMM FSM GPGPU GPU ICSPM IFMM IFSM ISO JC LS MC MD MJC MLPG MLS MPM MS ODE OTM PDE PIC. One Dimensional Two Dimensional Three Dimensional Arbitrary Lagrangian Eulerian Boundary Condition Computer-Aided Design Coupled Eulerian Lagrangian Computational Fluid Dynamics Courant-Friedrichs-Levy Central Processing Unit Corrected Smoothed Particle Method Compute Unified Device Architecture Deutsches Institut für Normung (i.e., German Institute of Standardization) Double Precision Element-Free Galerkin Finite Difference Finite Element Finite Element Analysis Finite Element Method FLOating Point Operation FLOating Point Operations per Second Fatehi’s Mesh-Free Scheme Fast Marching Method Fast Sweeping Method General Purpose Computing on the GPU Graphics Processing Unit Improved CSPM Improved FMM Improved FSM International Organization for Standardization Johnson-Cook Level Set Monte Carlo Molecular Dynamics Modified JC Meshless Local Petrov-Galerkin Moving Least Squares Material Point Method Multi Stencils Ordinary Differential Equation Optimal Transportation Meshfree Partial Differential Equation Particle In Cell. xvii.

(18) nomenclature & acronyms. PLS PSE RKPM RPI SHPB SIMD SP SPH TL TQ UL. xviii. Particle Level Set Particle Strength Exchange Reproducing Kernel Particle Method Radial Point Interpolation Split-Hopkinson Pressure Bar Single Instruction Multiple Data Single Precision Smoothed Particle Hydrodynamics Total Lagrangian Taylor-Quinney Updated Lagrangian.

(19) 1 INTRODUCTION. “Success seems to be largely a matter of hanging on after others have let go.” William Feather. The first chapter seeks to construct a framework for this dissertation by providing first an overview of the problem statement, its importance, and then some common difficulties faced in its modeling. It gives a motivation for research in the field of numerical simulation of manufacturing processes and to develop suitable means to tackle those very complicated problems. A research survey is carried out next, after which the knowledge gaps, as well as the tactics to bridge those gaps, are recognized. The literature review is divided into three separate groups: Simulation of manufacturing processes, particle methods, and simulation of manufacturing processes using particle methods. This survey, in turn, sets the scene to identify the scope and structure of this dissertation.. Foreword & Disclaimer This dissertation is a multidisciplinary work where state-of-the-art developments from different subjects are incorporated. Taking the metal machining simulation into account, the analysis contains a wide range of research areas such as numerical methods for mechanical modeling, thermal issues, and interface representations. Since each of these topics is a very extensive field and has a community of its own, the present discussion and literature review are confined to the most relevant and notable works fitting the application at hand.. 1.1 Background & Motivation According to the standard DIN 8580, metal machining/cutting is defined as a group of processes through which a workpiece is brought to the desired form by removing its undesired material. The unwanted material being cut away is also known as chips. Turning, milling, drilling, grooving, and grinding are some of the most widely-used machining processes. There are, however, several other methods of changing the geometry of bulk material to produce a mechanical part by either putting material together or just moving it from one region to another. The manufacturing processes with which this work is concerned are performed by removal operations and are mainly fallen into the cutting scale category. Figure 1.1 shows three pictures of a metal cutting operation taken at three different scales (see the left column), accompanied by a schematic illustration of the same system. 1.

(20) introduction. Machine scale. Cutting Tool. Workpiece. Cutting scale. Tool scale. Machine Frame. Workpiece. Figure 1.1: Left: Photographs of a metal cutting operation in 3 different system scales. Right: Schematic illustration of a metal cutting operation in these 3 scales. The lathe machine image (top left) corresponds to the computer numerical control-milling lathe by Haas Automation, Inc. downloaded from https://de.cleanpng.com/png-1vru1y/ for personal use. The middle left picture by Florian Schott / CC BY 4.0, downloaded from https://upload.wikimedia.org/wikipedia/commons/5/58/StechenDrehen.jpg.. 2.

(21) 1.1 background & motivation. Rake face. Tool holder. Chip. Tool Flank face New surface Cutting edge. 𝛼𝛼. Original surface. 𝛾𝛾 Workpiece. Figure 1.2: Geometry of an orthogonal metal cutting process and its basic definitions. scale (see the right column). In these figures, the sub-micron scale (i.e., the fine grain and crystal lattice structure) of the workpiece is excluded from the depiction. Some thermal problems of this thesis are simulated on the machine scale; nevertheless, the focus of this work is mainly on the cutting scale. Therefore, the essential terminology and areas of an orthogonal metal machining on the cutting scale are specified in Figure 1.2 with more details. In this schematic illustration, the basic definitions related to the tool and cutting geometry such as the rake angle γ, the clearance angle α (i.e., the inclination between the flank face and the new surface), and the cutting depth (also known as, feed) are demonstrated. In the late 90s, Merchant [187] proclaimed that more than 15% of the total added production value in all developed countries is allocated to machining costs. Prof. Shaw, in the second edition of his book [258], stated that more than 10% of Gross National Product (GNP) in the US is also allocated to the cost of manufacturing removal operations. The importance of studying these processes is, consequently, realized by their undeniable role in industrial technologies and their enormous economic impact on the global market. Investigation of metal cutting processes has generated a myriad of publications that address the interests of its devotees. Perhaps the groundbreaking research published by Frederick Taylor [286] in 1907 can still be regarded as a pioneering work in the context of cutting models. Excellent theoretical and experimental studies have been carried out ever since, evidenced by countless achievements from the 1900s to 1970s [70, 102, 186]. These developments have proposed numerous models for process variables (e.g., cutting forces, thermal issues, tool wear, friction, and so on), some of which are still considered as a good estimate in the current state of the art. Nevertheless, it turns out that: • The range of machining operations is very wide, leading to so many different situations with variable geometries and parameters. • Most of the experimental data cannot be recorded directly, thus involved with large measurement uncertainties. • Important zones of a metal cutting, like the chip root and tool-chip interface, are not visible from outside. Therefore, in-process data measurements become extremely difficult and challenging. • Some process variables, such as temperature near the chamfer radius, are not feasible to measure experimentally.. 3.

(22) introduction. • Some phenomena, like those occurring inside the chip, are entirely impossible to observe in experiments. • Experimental optimization of cutting conditions, tool geometries, and other process variables are far from trivial because the parameter space is extremely vast. Just as an example, even the simplest tool is defined with multiple parameters like cutting speed, chamfer radius, depth, clearance, and rake angles. Most, if not all, of these issues, can be tackled with the help of computer-based modeling. For instance, process variables that are not feasible to measure experimentally in metal cutting can be observed in numerical simulations. These computer-based models can also survey the parameter space more comprehensively, providing a basis for the optimization of cutting conditions, tool geometries, and other parameters. But again, the computer-based modeling of metal cutting is not without its challenges either: • Both strains and strain rates are extremely large. It is not unusual to observe strains exceeding 700% with strain rates up to 106 s−1 in high-speed cutting operations [22, 47]. • High temperatures (up to 1800 K [15, 50]) and extremely large temperature gradients (up to 108 K/m as can be seen in Figure 1.3) heavily affect the material and friction behavior. As a result of this, the conventional isothermal models become reliable only for qualitative evaluations. • Severe contact conditions near the cutting edge necessitate separate friction zones and complicate the modeling thereof. • Material separation, an inevitable consequence of cutting operations, is not an easy task to handle numerically. • Expensive cost of computations associated with 3D simulations brings out the majority of available models to 2D simulations and orthogonal cutting cases. • The common “orthogonal” assumption in most cutting simulations is afflicting. It limits the number of applications to a great extent, while the level of its realism can be questionable in the first place. A recurring question naturally arises: What would be an efficient approach for the computer-based modeling of cutting processes? The literature review in the next section is intended to find the research gaps by addressing this question.. 4.

(23) 1.2 state of the art. ∆𝑥𝑥 = 10 𝜇𝜇𝜇𝜇. ∆𝑇𝑇 > 1000 K. Figure 1.3: Shear bands in orthogonal cutting simulation of Ti6Al4V using the present meshfree solver. Large temperature gradients in the order of 108 K/m can be observed from this close-up in the chipping zone. For better visualization, the upper limit of colorbar in the image is set to 1300 K.. 1.2 State of the Art As pointed out at the beginning of this chapter, the overall subject is broken down into the following three subgroups for a clearer presentation. Conclusive remarks concerning each subgroup are outlined separately and given at the end of their corresponding subsection.. 1.2.1 Simulation of Manufacturing Processes When simulating a manufacturing process in general and cutting operation in particular, the majority of available publications make use of methods that are either mesh-based or particle-based. The most well-established and popular mesh-based technique in this context is the Finite Element Method (FEM). Inarguable is what FEM has achieved so far in the modeling of machining operations. The success and popularity of FEM models stem from the fact that the number of available commercial packages (e.g., ABAQUS, DEFORM, and, AdvantEdge) is remarkable, but also the algorithms and implementations of FEM have reached maturity. The following three approaches of FE analysis are briefly reviewed from the literature. • Eulerian Although not accessible over the Internet, the article [263] published by Shirakashi and Usui in 1976 seems to be the first thermo-mechanically coupled simulation of a metal cutting problem. An earlier work using the Eulerian approach was presented by [285], but they accounted for thermal simulations only. In 1990, Strenkowski and Moon [277] developed a model for predicting the chip geometry and temperature distribution in the workpiece, chip, and tool. Since the mesh resolution is a stringent constraint that affects the chip shape in Eulerian formulations, the. 5.

(24) introduction. Lagrangian. Eulerian. Figure 1.4: Lagrangian vs. Eulerian description. In a Lagrangian approach, the boundary is well defined because the mesh is attached to the body and therefore it deforms during the deformation process. Mesh distortion can happen in this case. Since each grid point is always associated with only one single material point, the Lagrangian approach is ideally suitable for modeling history-dependent materials. In an Eulerian approach, the mesh is fixed in space and the deformation process is represented by flowing the material through the overlaying grid. Mesh distortion cannot happen in this methodology. interest in this type of approach has been gradually abolished. For a graphical representation of the Eulerian and Lagrangian descriptions, see Figure 1.4. • Lagrangian A milestone on the road to FEM cutting simulations seems to be the development of re-meshed Lagrangian techniques in the 90s. It is evidenced by numerous research papers published in this decade, focused on metal cutting simulations using the Lagrangian FEM approach. Childs et al. [69] considered turning processes to study its chips flow and tool wear. The early-stage attempts in this view, however, suffered from a major shortcoming. For generating and designing the mesh elements, a separation pattern had to be known a priori. This modeling constraint was then released by the work [254] of Sekhon and Chenot on non-stationary orthogonal cutting tests, and afterward, by a high-speed machining model presented in [183]. Mackerle gives a good overview of the notable FEM cutting simulations between 1976 and 1996 in [179]. It would not be until the early 2000s that the FEM models could take a major leap forward in their proficiency. The prosperity of computational power at that period enabled the exploration of more sophisticated models and theories. A broad array of Lagrangian FEM accomplishments in the modeling of metal machining processes is elaborated by Arrazola et el. [22]. Some notable developments include the FE analysis of residual stresses by [261], and the influence of different friction models and cutting geometries on finite element simulations of machining by [224, 262] and [260]. More recently, a few researchers in the field have attempted 3D modeling of cutting [52, 308] and single grain grinding operation [17] with FEM. • Arbitrary Lagrangian Eulerian (ALE) In spite of this overwhelming amount of research dedicated to Lagrangian FEM simulations of machining processes, there are some technical issues with their applications. A deeper dive into the topic reveals that this type of FEM formulation confronts serious uncertainties in some aspects. For example, Bil et al. [43] performed an extensive comparison of orthogonal cutting. 6.

(25) 1.2 state of the art. Figure 1.5: Mesh-based and particle-based discretization of a sphericon body in space. data between three FEM solvers. They showed these solvers can produce significantly different passive forces under the same initial setups. In 2015, Zhang and his coworkers [316] made use of three different FEM formulations, namely Lagrangian, ALE, and Coupled Eulerian-Lagrangian (CEL), to evaluate the impact of constitutive material parameters on the simulation results. Comparing the measured and predicted data, they concluded that both ALE and CEL models outperform the Lagrangian formulation, especially if the chip geometry is concerned. These coupled approaches take advantage of both formulations in the spirit of frequent interpolations between a Lagrangian and Eulerian mesh. Numerous articles have discussed the robustness of this technique in metal machining simulations, e.g., [20, 176].. 1.2.2 Particle Methods Another group of machining simulations employs methods that are not mesh-dependent. A (meshfree) particle method does exactly what it says on the tin: It solves differential equations by a finite set of particles as the discretization points without requiring a mesh (i.e., a connection between nodes of the computational domain). The approximation procedure in particle methods is rather based on the interaction of each particle with its neighbors. To illustrate, Figure 1.5 shows the discretization of a sphericon with triangular mesh elements and particles. Smoothed Particle Hydrodynamics (SPH) is perhaps the earliest of the truly meshfree methods, and certainly the most popular one. Since its birth in 1977 [114, 178], SPH has progressed tremendously owing to intensive theoretical work and computational improvements. What triggered off this method was its prominent ability in solving astrophysical problems like the evolution of proto-stars and galaxies. Due to the Lagrangian nature of SPH, the interpolation points can be interpreted as discrete particles moving with the continuum in question. The material response can, therefore, be visualized by tracking these moving interpolation points. This type of methodology makes SPH geometrically flexible yet computationally efficient for some applications. Soon after its debut, and on the grounds that it has a privilege of functioning with no underlying grid, SPH has attracted much attention from the computational science community. A diverse spectrum of analyses from solid [117, 236, 237] and fluid [9, 10, 13, 193, 197] mechanics problems, to droplet spreading and solidification [103, 167, 189] can be found in the literature. The application of SPH has gone even beyond engineering problems. As examples the following articles may serve: Applications in virtual reality surgery [200], movies special effects [201] and computer graphics [204]. Excellent. 7.

(26) introduction. overviews of the SPH theory and capabilities are given by [195, 232]. All in one, SPH can be currently regarded as a numerical tool in a variety of industrial and environmental applications. It is worth noting that Molecular Dynamics (MD) [112, 239] and Monte Carlo (MC) simulations [44, 146, 188] can also be categorized as particle-based approaches since they perform without any mesh connectivity. The history of these methods is clearly older than SPH, but the discretization of a body in MD and MC methods is different than SPH. In this respect, SPH and FEM are similar and discretize a continuum in the same fashion (see Figure 1.5). There are several other techniques that can still be classified as particle methods such as Vortex Method (VM) [246], Lattice-Boltzmann Methods (LBM) [67], and Peridynamics method [268]. In contrast to SPH, the principal idea behind these techniques does not emerge from a partition of unity and their application is hence not studied in this thesis. Like any other methodology, SPH also has its pros and cons. The main technical drawbacks of SPH related to this thesis are outlined in the following. 1. 2. 3. 4. 5.. Enforcing essential boundary conditions Some numerical/tensile instabilities Lack of interpolation consistency Lack of explicit interface representation Computation cost. In order to obtain the degree of realism required of cutting applications, the nonlinear phenomena present in such processes cannot be modeled and simulated with particle methods unless these five issues are addressed. In the past 30 years, various ideas have been proposed to eliminate these sore points and improve the computational performance of SPH. A good summary of these efforts is provided by [160, 167]. In what follows, the most important of achievements in the literature for generating a functional and suitable code resolving these five issues are summarized.. 1.2.2.1. Issues with Boundary Conditions. It sounds from the title of this section that the strength of particle methods is, paradoxically, also their weakness. The treatment of boundary conditions is often cited as one of the main shortcomings of particle methods [37], while the central idea of these techniques is to provide efficient numerical solutions for PDEs with arbitrary boundary conditions [166]. In fact, there is no canonical way to impose general boundary conditions in SPH, say, and the imposition of boundary conditions in particle methods is still subject to active research. This is illustrated by the fact that the previous “Joe Monaghan Prize” winner was a work regarding boundary conditions in SPH flow simulations [77]. Within a diverse range of developments for solid boundary treatment in SPH such as [83, 197, 240, 284], the so-called ghost particle approach [162] is incorporated for solid mechanics cases in this work because of its simplicity and efficiency. Given a CFD or fluid-structure application, another more recent algorithm by Adami et al. [5] is utilized in the present dissertation. This scheme was termed as a generalized wall boundary condition by its developers, where they propose to solve a local momentum equilibrium in the vicinity of the fluid-solid interfaces for calculating the field variables such as pressure on boundaries. It was shown in the original publication that SPH models using this type of boundary formulation can provide a more accurate approximation with a lower computational cost compared to [197]. The first issue outlined above is, therefore, addressed by implementing these two efficient algorithms (i.e., [162] and [5]) for treating solid and wall boundary conditions.. 8.

(27) 1.2 state of the art. 1.2.2.2. Issues with Numerical Instabilities. Major stability issues to be considered in an SPH simulation model include the tensile instability and zero-energy modes [167, 195]. Preliminary studies show that SPH without using proper stabilizers cannot provide stable solutions to a simple tensile test of an elastic square in the update Lagrangian frame. This situation occurs due to the tensile instability, which usually results in unphysical particle clumping [282] or even complete blowup of the simulation. Thus, special care is taken about the numerical instabilities of SPH in this work. A convenient way to mitigate these circumstances in physical problems is to introduce some numerical diffusion by inserting some artificial terms into the conservation equations. In this respect, the artificial viscosity [192] and artificial stress [118] terms have proven quite effective and are widely used by the experts in the field, according to numerous citations and relevant works including [118, 167, 190, 192, 194, 282, 297]. This notion has become a central theme of SPH analyses since its early stage and will be studied in Section 4.3 with more details.. 1.2.2.3. Issues with Consistency & Higher-Order Schemes. In principle, the particle-particle interaction in SPH methods is controlled by a kernel function similar to a shape function in FEM (see Figure 1.6). Unlike the finite element shape functions, however, the particle-based kernel functions are not necessarily consistent throughout the computational domain as they lose their consistency near/on the boundaries. To address the third issue, corrective SPH schemes are thus implemented to enforce consistency conditions in interpolation. Their enhancing idea relies on some correction functions that render higher-order kernels, by which the consistency and completeness of SPH interpolant can be satisfied. This thesis takes into account a broad array of such corrections: The Brookshaw approximation [51], Reproducing Kernel Particle Method (RKPM) [145, 170], Randles-Libserky correction [163], and Corrective Smoothed Particle Method (CSPM) [61, 62]. Furthermore, two more recent schemes derived by [104] and [148] are also scrutinized. At the cost of extra computing time and programming effort, both methods can deliver results that are second-order consistent and converge quadratically (i.e., second-order accurate). This dissertation presents the first 3D implementation of these schemes, but also extends their application to problems with moving particles. A more extensive state-of-the-art review of the most popular particle techniques goes beyond the aforementioned methods. While falling within a more general class of meshfree methods, a selection of most relevant schemes is reviewed in the following. More technical investigations of these techniques can be found in [8, 37, 59, 108, 206]. • In 1992, Nayroles and his co-workers [203] introduced the Diffuse Element Method (DEM) as a generalization of the popular FEM. Coined as the new “diffusive approximation” method by the authors of [203], this technique is known to be useful for solving PDEs by releasing some of the FEM limitations regarding the mesh generation requirements and the regularity of approximated functions. • In the same year, a novel stochastic method for simulating hydrodynamic phenomena was devised by Hoogerbrugge and Koelman [135, 147]. This particle-based technique is called Dissipative Particle Dynamics (DPD), through which some interesting features of molecular dynamics and lattice-gas automata are combined to achieve higher computational efficiency and robustness.. 9.

(28) introduction. Figure 1.6: Shape function in a finite element discretization (left), and smoothing kernel function in a particle-based discretization (right). For the particles near/on the boundaries, the completeness of the smoothing kernel function is violated and the approximation results become inconsistent. • Initially applied to elasticity and heat conduction problems, the Element-Free Galerkin (EFG) method (or EFGM) was developed by Belytschko et al. [36]. It employs a weak form of the system (like FEM) but uses the shape functions of Moving Least Squares (MLS) approximation (unlike FEM) for its test and trial functions. • In 1995, Onate et al. [215] proposed another novel scheme, termed as Finite Point Method (FPM), for solving PDEs on scattered distributions of data. They demonstrated the workability of this method by first applying it to convection-diffusion and fluid flow type problems. A set of more sophisticated applications was subsequently simulated by [214, 216], and has continuously been presented, using this fully meshless procedure. • As a purely Lagrangian meshless method, the Finite Pointset Method (also abbreviated as FPM, but not to be confused with Finite Point Method) is yet another approach taken for the numerical simulation of continuum mechanical and CFD problems. Some successful applications of this method can be found in [289, 290], and the references they contain. • Motivated by a Natural Neighbor (N-N) interpolation scheme, Natural Element Method (NEM) is also a meshless technique that was developed for solving PDEs on highly irregular evolving grids [49]. It was later applied to solid mechanics problems in 2D small-displacement elastostatic by [278]. • A particularly robust meshfree scheme for the spatial discretization of multiphase interactions is the Material Point Method, i.e. MPM. Sulsky et al. [279] initially formulated this method for the impact/bouncing tests and history-dependent materials. A good overview of MPM and its computational strength in the simulation of engineering applications is given by [30, 31]. • Last but not least, another effective meshless approach was propounded for solving boundary value problems in 1998 [24]. Analogous to the EFG/EFGM methodology, the Meshless Local Petrov-Galerkin (MLPG) approach makes use of a local symmetric weak form together with shape functions from the MLS approximation [24, 25]. This overview shall give some insight into the capability and popularity of particle methods among the computational science community.. 10.

(29) 1.2 state of the art. The formulation of particle methods may be viewed from another more mathematical perspective. That is, one could categorize the family of particle methods into two principal classifications: (1) weakform; (2) strong-form. The weak-form formulation incorporates a variational principle to minimize the residual weight of the differential equations, which is prevalent in the associated physics. As a distinctly different approach, the strong-form formulation directly uses the time-fractional differential equations governing the physical model. On one hand, the weak-form meshfree methods have been adopted to numerous applications from static-dynamic fracture [39] and incompressible Navier-Stokes equations [23] using the EFG method, to more sophisticated developments in the analysis of nonlinear wave equations up to 3D with MLPG [264]. The particle methods in weak-form, nevertheless, are not examined in this work, as they manipulate the governing PDEs by reducing the order of differential equations. On the other hand, the strong-form particle methods have technically no limitation in the order of the spatial derivative that they can handle. Some recent achievements have been reported on the use of several strong and combined weak-strong formulations in the state of the art, including the Singular Boundary Method (SBM) for stress analysis of thin structures in elastic regime [122] and biharmonic problems [312], an extensive investigation of the Meshless Local Radial Point Interpolation (MLRPI) method [266] and its application in population dynamics [265], and the recent improvements of Spectral Meshless Radial Point Interpolation (SMRPI) for fractional evolution equations [267].. 1.2.2.4. Issues with Interface Representations. Generally speaking, particle methods are incapable of representing the interface explicitly. In other words, they smooth out the mass of particles, and as a result of this, their spatial discretization size determines the accuracy of the interface represented by these methods. Hence, to obtain a clearly defined interface, a separate geometrical solver is required. The computational representation of moving interfaces comprises a dedicated mechanism of its own that forms an active area of current research. Since some of the simulations in this thesis involve the generation of new surfaces as guided by fluid flows or material separation, the associated geometric evolution influences not only the accuracy of the numerical solution but also the reliability of the physical field variables. The need to capture complex topological features, therefore, may dictate the choice of method to model moving interfaces in such problems. Almost three decades ago, Osher and Sethian [220] published their seminal work introducing the Level Set (LS) method, which has been cited almost 16’000 times to date! This method presents a conceptual framework on a fixed Cartesian grid (thus an Eulerian approach) that allows for capturing complex topological changes. In addition to Eulerian [105, 218] and Lagrangian [28, 120, 177] formulations, which have been adopted in the computational solution of moving interfaces, there exists another class of hybrid Particle Level Set (PLS) approach in this context. The first hybrid PLS method, in the spirit of Enright et al. [101], employs Lagrangian marker particles to rebuild the level set function in regions which are underresolved. These marker particles improve the mass conservation properties of the level set method in an Eulerian frame, where the interface is passively advected in flow fields. In their original work [101] and later development [100], the authors demonstrated the effectiveness of this approach in various benchmark tests. Another class of PLS methods considers a mass-conservative solution of the level set equation in a Lagrangian frame. To maintain a smooth geometrical description of the interface, the Lagrangian PLS. 11.

(30) introduction. method necessitates a consistent remeshing procedure to regularize the particle locations. Hieber and Koumoutsakos [130] were the first who proposed this methodology for capturing complex interfaces. As pointed out in their article, the efficiency and accuracy of Lagrangian PLS formulation lie in a frequent particle-mesh interpolation. This strategy is, in fact, necessary because the coherence (and smoothness) of the interface representation with particles can be lost in cases where the particle map undergoes severe distortions. In a textbook published 17 years earlier than [130], Hockney [134] extensively discussed the idea of remeshing in particle simulations. This scheme was further developed and established for vortex particle methods by [81, 149]. The first application of remeshing in the context of PLS methods presented by [130] showed a remarkable performance of this Lagrangian PLS method in problems of moving interfaces. It was shown by the authors that, for example in a single vortex problem, results produced by the Lagrangian PLS scheme can be computed with only a fraction of computational elements required by the hybrid PLS method of [101]. This method was then extended in the work of Bergdorf et al. [40], where a multi-resolution Lagrangian particle method with enhanced, waveletbased adaptivity is presented for transport problems, fusing the efficiency of wavelet collocation with the inherent numerical stability of particle methods. More recently, Cottet et al. [80] offered a treatise on the definition and implementation of re-meshed particle methods without LS reinitialization, offering a consistency and stability analysis of a large class of second- and fourth-order methods. In brief, the elegance of PLS methods lies in the coexistence of particles and grids, coupling the accuracy of Lagrangian advection with the simplicity of the Eulerian level set surface representation. In Section 3.2, the relevant theories are described in some detail, including the more important of the Lagrangian particle level set formulation employed for interface representations in this work.. 1.2.2.5. Issues with Cost of Computation. Thus far, the discussion has focused on the software aspects of a solver. For example, why a particle method is targeted for the present applications instead of FEM, or how its numerical accuracy and performance can be enhanced. While the fifth issue (i.e., the cost of computations) can, and will, be addressed by multi-resolution particle simulations in this work, there is yet another aspect of computation which has been ignored: The computer hardware. That is, making the most of the available hardware is of paramount importance for paving the road towards the maximum efficiency of a computational approach. This gives rise to a more throughput-oriented design by the introduction of high-performance and parallel computing. In parallel architectures, the Graphics Processing Unit (GPU), as well as the Central Processing Unit (CPU), are the common engines available to perform computation. CPUs are traditionally the predominant devices for parallel codes employing a variety of technologies like MPI, OpenMP, and so on. GPU, on the other hand, is relatively a newer class of processors in terms of parallel computing. A GPU offers thousands of cores, hundreds of times more than what a CPU with the same (or similar) price may feature. Particularly important is also to ensure if the associated numerical method would be suitable for the parallelism strategy. In [84], the authors developed a GPU-accelerated framework for CFD applications and concluded that SPH codes are indeed appropriate for this type of parallelism. The focus of this work is consequently on GPU-accelerated particle codes. Even though GPU computing in other fields of application has achieved so much so fast (e.g., [85, 154, 231]), it is still in its infant stages for cutting simulations. Except from a very few recent publications by [107, 249, 250], a GPU-based SPH code for solid mechanics (and for metal cutting in particular) is still missing from the state of the art. Chapter 5 aims to elaborate upon the details of this topic.. 12.

(31) 1.2 state of the art. Figure 1.7: Left: Temperature distribution in a frame of Ti6Al4V cutting, simulated with LS-DYNA by [252], where heat is not transfered into the tool. Right: An exemplary SPH cutting simulation by the present work in a higher resolution computed on the GPU, where multiple corrections are applied and also heat transfer to the tool is also taken into account.. 1.2.3 Simulation of Manufacturing Processes with Particle Methods. Conceptually, Lagrangian particle methods may have a clear advantage over mesh-based techniques in problems consisting of new surface generation, large deformations, and high gradients. As mentioned, one area where such complexities are conspicuously faced is the modeling of cutting processes. Complying with the overarching theme of this dissertation, the literature survey in this section singles out the meshfree cutting models. In 1997, a new theater of computational cutting frameworks was opened up when Heinstein and Segalman [128] applied SPH to a high-speed metal cutting problem for the first time. Continuous adoptions have been investigated ever since, but an abrupt increase of research embarked on the publications once SPH was introduced into the commercial packages like LS-DYNA. A summary of some relevant contributions to the field using this solver is as follows. Between 2006 and 2008, several developments such as [164, 165] and [300] were proposed to increase the robustness of SPH models for high-speed cutting applications. While the authors of these publications could demonstrate the capability of their method to a great extent, their numerical results were not validated by experimental data in, for example, an oblique cutting test. It would not be until 2010 that Rüttimann et al. [252] published the first SPH work, in which a 3D cutting test was verified experimentally. More interestingly, they demonstrated that the computational time of an SPH single grain model is only a fraction of what FEM needs for a comparable result. The first author of this article gives a good summary of SPH capabilities for different cutting simulations in his doctoral dissertation [251]. A frame of Ti6Al4V machining simulation from [251] is shown in Figure 1.7, next to a stabilized SPH simulation in higher resolution computed by the code developed in this thesis. Concentrating on the tool wear and chip formation of SPH models, further insights were gained by the results of [56] and [26, 180]. While much less frequent than LS-DYNA, research on the use of other commercial packages like ABAQUS for SPH machining simulations still exists, e.g. [29, 313]. Although very limited in number, accounts on the development of in-house particle codes for machining simulations can also be found. For instance, the authors of [97, 273, 274] put great. 13.

(32) introduction. effort into modeling of cutting processes with their own SPH program package called Pasimodo. Meanwhile, researchers in this field have attempted to increase the efficiency of their numerical tool. About 10 years ago, Rabczuk and Samaniego [238] modeled the cutting of a 3D block with particles using a plastic material model with isotropic hardening/softening law. In their work, however, a reference result from either FEM simulations or experimental data is lacking and only a qualitative illustration is presented. In 2014, Spreng et al. [275] simulated an elementary 2D orthogonal cutting process with SPH. Both refining and coarsening procedures were employed within the course of spatial adaptation and consequently applied to their cutting test. While suffering from the same sore point as [238] in terms of results validation, the spatial coherence and quality of the chip formation in [275] are notably worse than their corresponding standard SPH single-resolution reference. In their more recent work, nonetheless, Spreng and Eberhard [272] verified the adaptive-resolution SPH result by comparing it to the experimental data. For this purpose, they took the cutting force of a 3D oblique cutting test into account. In a nutshell, the improvement gained by adaptivity is not recognizable, neither from a quality/quantity perspective nor from a computational cost point of view. Incremental improvements in the algorithmic aspects of SPH cutting models have led to substantial increases in the number and variety of other particle methods. Among these efforts, results reported by the Material Point Method (MPM) in [19, 121] and the Optimal Transportation Meshfree (OTM) method in [139] are particularly interesting and worthwhile to mention. In MPM, the Lagrangian material points are surrounded by a background mesh to facilitate the calculation of gradients terms. In OTM, two sets of particles are used to discretize the spatial domain, namely the material/integration points and the nodal points. What distinguishes between these developments and the methodology of this work is the co-existence of different discretization elements in MPM and OTM. Moreover, none of these 3 articles considers the heat transfer from the workpiece to the cutting tool, while [19] and [139] entirely neglect the heat conduction. More closely fallen into the scope of this work, Fraser [107] developed a simulation code called SPHriction-3D that can be used for finding optimal process parameters in friction stir welding. His work is based on GPU parallel computing, but: (1) does not include any applications with material separation; (2) does not implement any corrective SPH methods. Niu et al. [209] demonstrated that using corrected SPH kernels which are first-order consistent is an essential consideration for resolving the chip shape. Later, they took a modified Johnson-Cook flow rule (referred to as the TANH model) in [208] for obtaining a better prediction of chip morphology. Their simulations, however, do not benefit from parallel computing and are limited to 2D cases.. 14.

(33) 1.3 research gaps & objectives. 1.3 Research Gaps & Objectives A multiplicity of knowledge gaps can be recognized in the wake of the preceding literature survey. Among these research opportunities in different disciplines, the present thesis sets its sights on the following categories. 1. Maturity of the physical model • The impact of material constitutive models on particle-based cutting simulations was only recently investigated [208]. Nevertheless, the influence of friction models on meshfree simulations is still absent from all available works in this context. • A more realistic thermal boundary condition is a substantial concern, especially in applications dominated by thermal effects like laser drilling and metal cutting. • Transfer of heat from the workpiece to the tool has been ignored by most of the previous studies. See Figure 1.7, for instance. This issue is recognized as one of the major shortcomings of some commercial codes like LS-DYNA. 2. Maturity of the numerical method • Several stabilizations measures such as artificial viscosity [192], artificial stress [118], variable smoothing length [191], smoothed velocity field [223], and XSPH [190] have long been utilized by the particle methods community in various problems, but cannot find an extensive implementation in meshfree cutting simulations. • Adoption of corrective SPH methods in meshfree simulation of manufacturing operations is missing. At best, only one first-order consistent kernel for the gradient operator was employed by [209]. Second-order kernels [104, 148] are completely absent from manufacturing simulations, especially in the thermal modeling of such processes. • There is no thermomechanical simulation of a machining process in 3D space that incorporates corrected kernels. • Meshfree cutting models are entirely devoid of a separate geometrical tool for an explicit interface representation. 3. Efficiency of the computational performance • There is no multi-resolution simulation of laser processing with particle methods, at least not to the best of the author’s knowledge. • The improvement gained by employing multi-resolution algorithms is not clear in meshfree cutting simulations. The very limited publications available do not provide a clear comparison with the baseline. • The current state of the art lacks a GPU-accelerated code for particle simulations of solid mechanics. As a result, this work follows a combinational logic to propose an enhancement to the current state of particle-based machining simulations. A host of contemporary algorithms are borrowed from different fields to bridge the research gaps identified above. Ultimately, a robust and efficient simulation code is developed using advanced particle methods in the updated Lagrangian frame. Both GPU-acceleration and multi-resolution algorithms are utilized in order for the approach to exploit the software and hardware aspects as much as possible. Alongside a number of trial simulations and preliminary benchmarks, various manufacturing processes such as a single-phase laser drilling, ultra-precision machining, orthogonal metal cutting, and tribometer device are investigated.. 15.

(34) introduction. 1.4 Layout of the Thesis After this introductory chapter, the following subject matters are covered in Chapters 2 to 5 before presenting the numerical results in Chapter 6. • Theories and equations • Numerical methods and algorithms • Computer implementation In Chapter 2, the fundamentals of physics needed for understanding the numerical examples are reviewed. This entails an overview of the mechanics of both solids and fluids for thermo-mechanically coupled analyses. Special attention is paid to friction and material constitutive models. In this regard, a Coulomb law whose coefficient of friction is a decreasing function of the temperature is proposed. Chapter 3 is a core constituent of this work, which presents the numerical methods as applied to the mechanical problems. Various techniques borrowed from the state of the art are (re)derived and presented in consistent notation to facilitate the assessment of their differences and similarities. The concept of particle methods together with their kernel, gradient, and Laplacian corrections are introduced in this chapter. The constant, linear, and quadratic consistency of the discrete SPH equations is enforced with the addition of different correction terms to the kernel function and its gradient. Purely mathematical examples are immediately provided to benchmark the respective performance of the presented schemes. It will be shown that using higher-order kernel/gradient corrections offered by state-of-the-art SPH schemes, the second-order consistency (i.e., completeness) can be obtained. In other words, these advanced methods allow us to approximate the second derivatives of quadratic functions without error. Besides, the very vast topic of computational moving interfaces is explored with particular consideration of Lagrangian particle level set methods. Although the central focus of Chapter 3 is on the particle-based discretization of continua in space and geometry representation, a brief review of time integration methods is additionally given for the completeness of the manuscript. Through several schematic illustrations, Chapter 4 plans to create the basis for straightforward computer implementation. To that end, necessary remarks on the most important implementation details of plasticity and contact algorithms are provided. Furthermore, we outline a multiplicity of stabilization measures to address the main numerical instability issues of SPH methods. The presentation is then followed by the general principles of spatial adaptivity in the present code. A simple refinement-coarsening strategy is developed based on particle splitting-merging procedures from the literature. At the end of this chapter, a flowchart of the model logic is demonstrated to summarize the main computational blocks of the solver. Next, in Chapter 5, the basic terminology of GPU parallel computing is revisited. How to implement the working SPH code on a GPU is then explained. The remainder of this thesis is divided into two chapters. In Chapter 6, all the essential ingredients are brought together and incorporated into a versatile, multi-purpose particle code including the geometrical, thermal, and mechanical toolkits. The theoretical contents presented within the previous chapters are applied to various numerical test cases to validate the implementation. The range of these examples varies from solid mechanics to metal cutting, laser drilling, and CFD multiphase flow applications. This dissertation closes in Chapter 7 by summarizing the major contributions to the field, opening up a few avenues for future research, and recommending a way forward for better metal machining simulations.. 16.

(35) 2 MECHANICAL FRAMEWORK. “Within every desire is the mechanics of its fulfillment.” Deepak Chopra. The fundamentals of mechanics for solids and fluids are revisited in a concise and consistent manner in this chapter. Prior to discussing the computational methods, it may prove necessary to skim this chapter to become familiar with the terminology conventions used in this text. Without delving deeply into the equations and their derivations, only an overview of the relations needed to simulate the chosen examples is described. The chapter is divided into two main sections of solid and fluid mechanics. In this chapter: • The basics of continuum mechanics for solids is reviewed. • Kinematics is followed by the selected stress and strain measures. • The conservation laws are formulated for solids in the updated Lagrangian frame. • A brief discussion on the constitutive laws for engineering materials is presented. • An extra attention to friction and thermal effects is paid. • The Navier-Stokes equations are expressed as the conservation laws for fluids. • Further considerations regarding the incompressibility and equation of state are brought up. The conventions used throughout this chapter is in accordance with the terminology defined in the list of symbols.. 2.1 Solid Mechanics In principle, continuum mechanics is the mathematical description of the motions and deformations of continuous bodies under the influence of external effects. External effects could appear in different forms such as forces, displacements, thermal changes, and so on. The description starts with reviewing the basic laws of motions, in accordance with the procedure given by available textbooks [38, 133]. Illustrated in Figure 2.1 is the sketch of basic definitions in solid mechanics, wherein the reference and current configurations are displayed as Ω0 and Ω, respectively. Material points which belong to Ω0 are denoted by uppercase X. These points can be mapped to a new (i.e., current or deformed) configuration Ω by means of a mapping function φ(♦). Lowercase x represents the location of points. 17.

(36) mechanical framework. Ω0. Ω. 𝑥𝑥 = 𝜙𝜙(𝑋𝑋, 𝑡𝑡) 𝑢𝑢 = 𝑥𝑥 − 𝑋𝑋. 𝑃𝑃 𝑋𝑋. 𝑋𝑋3, 𝑥𝑥3 𝑂𝑂. 𝑋𝑋1, 𝑥𝑥1. 𝑋𝑋2, 𝑥𝑥2. 𝑝𝑝 𝑥𝑥. Figure 2.1: Motion and deformation of a continuum body. X in Ω at time t. As shown in Figure 2.1, to further clarify, a representative material point P with position vector X in Ω0 is chosen. The same material point, but, in the deformed configuration Ω will be p with position of x. Given these definitions, the movement can be formulated as x = φ( X, t). (2.1). and the distance vector u can be defined for any arbitrary points by u = x−X. (2.2). which is known as displacement. Subsequently, velocity v and acceleration a are derived as d ( X + u) = u̇ dt ẍ = a = v̇ = ü ẋ = v =. (2.3). Next, the kinematic equations as well as some useful measures for describing the movement from Ω0 to Ω will be discussed. The description is formulated to account for large deformations in general, and metal cutting processes in particular.. 2.1.1 Kinematics. Solid and structural mechanics deal with the relationships between stresses and strains, displacements and forces for given boundary conditions. It is therefore a vital task to properly formulate these relationships, as they are the backbone of any modeling, simulation, or design of engineered mechanical systems. In what follows, a number of essential quantities that will be used repeatedly in this chapter are summarized. • Deformation gradient: F ∼ By definition, F indicates the gradient of the mapping function φ, describing the motion of a ∼ continuum. It is termed as the deformation gradient because it characterizes the local deformation at a material point X. This can be realized in Figure 2.2 by transforming an arbitrary material. 18.