Os métodos numéricos convencionais, baseados em malhas, têm sido amplamente aplicados na resolução de problemas da Dinâmica dos Fluidos Computacional. Entretanto, em problemas de escoamento de fluidos que envolvem superfícies livres, grandes explosões, grandes deformações, descontinuidades, ondas de choque etc., estes métodos podem apresentar algumas dificuldades práticas quando da resolução destes problemas. Como uma alternativa viável, existem os métodos de partículas livre de malhas. Neste trabalho é feita uma introdução ao método Lagrangeano de partículas, livre de malhas, Smoothed Particle Hydrodynamics (SPH) voltado para a simulação numérica de escoamentos de fluidos newtonianos compressíveis e quase-incompressíveis. Dois códigos numéricos foram desenvolvidos, uma versão serial e outra em paralelo, empregando a linguagem de programação C/C++ e a Compute Unified Device Architecture (CUDA), que possibilita o processamento em paralelo empregando os núcleos das Graphics Processing Units (GPUs) das placas de vídeo da NVIDIA Corporation. Os resultados numéricos foram validados e a eficiência computacional avaliada considerandose a resolução dos problemas unidimensionais Shock Tube e Blast Wave e bidimensional da Cavidade (Shear Driven Cavity Problem). / The conventional mesh-based numerical methods have been widely applied to solving problems in Computational Fluid Dynamics. However, in problems involving fluid flow free surfaces, large explosions, large deformations, discontinuities, shock waves etc. these methods suffer from some inherent difficulties which limit their applications to solving these problems. Meshfree particle methods have emerged as an alternative to the conventional grid-based methods. This work introduces the Smoothed Particle Hydrodynamics (SPH), a meshfree Lagrangian particle method to solve compressible flows. Two numerical codes have been developed, serial and parallel versions, using the Programming Language C/C++ and Compute Unified Device Architecture (CUDA). CUDA is NVIDIAs parallel computing architecture that enables dramatic increasing in computing performance by harnessing the power of the Graphics Processing Units (GPUs). The numerical results were validated and the speedup evaluated for the Shock Tube and Blast Wave one-dimensional problems and Shear Driven Cavity Problem.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Neste trabalho, foi desenvolvido um simulador numérico baseado no método livre de malhas Smoothed Particle Hydrodynamics (SPH) para a resolução de escoamentos de fluidos newtonianos incompressíveis. Diferentemente da maioria das versões existentes deste método, o código numérico faz uso de uma técnica iterativa na determinação do campo de pressões. Este procedimento emprega a forma diferencial de uma equação de estado para um fluido compressível e a equação da continuidade a fim de que a correção da pressão seja determinada. Uma versão paralelizada do simulador numérico foi implementada usando a linguagem de programação C/C++ e a Compute Unified Device Architecture (CUDA) da NVIDIA Corporation. Foram simulados três problemas, o problema unidimensional do escoamento de Couette e os problemas bidimensionais do escoamento no interior de uma Cavidade (Shear Driven Cavity Problem) e da Quebra de Barragem (Dambreak). / In this work a numerical simulator was developed based on the mesh-free Smoothed Particle Hydrodynamics (SPH) method to solve incompressible newtonian fluid flows. Unlike most existing versions of this method, the numerical code uses an iterative technique in the pressure field determination. This approach employs a differential state equation for a compressible fluid and the continuity equation to calculate the pressure correction. A parallel version of the numerical code was implemented using the Programming Language C/C++ and Compute Unified Device Architecture (CUDA) from the NVIDIA Corporation. The numerical results were validated and the speed-up evaluated for an one-dimensional Couette flow and two-dimensional Shear Driven Cavity and Dambreak problems.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Em uma grande gama de problemas físicos, governados por equações diferenciais, muitas vezes é de interesse obter-se soluções para o regime transiente e, portanto, deve-se empregar técnicas de integração temporal. Uma primeira possibilidade seria a de aplicar-se métodos explícitos, devido à sua simplicidade e eficiência computacional. Entretanto, esses métodos frequentemente são somente condicionalmente estáveis e estão sujeitos a severas restrições na escolha do passo no tempo. Para problemas advectivos, governados por equações hiperbólicas, esta restrição é conhecida como a condição de Courant-Friedrichs-Lewy (CFL). Quando temse a necessidade de obter soluções numéricas para grandes períodos de tempo, ou quando o custo computacional a cada passo é elevado, esta condição torna-se um empecilho. A fim de contornar esta restrição, métodos implícitos, que são geralmente incondicionalmente estáveis, são utilizados. Neste trabalho, foram aplicadas algumas formulações implícitas para a integração temporal no método Smoothed Particle Hydrodynamics (SPH) de modo a possibilitar o uso de maiores incrementos de tempo e uma forte estabilidade no processo de marcha temporal. Devido ao alto custo computacional exigido pela busca das partículas a cada passo no tempo, esta implementação só será viável se forem aplicados algoritmos eficientes para o tipo de estrutura matricial considerada, tais como os métodos do subespaço de Krylov. Portanto, fez-se um estudo para a escolha apropriada dos métodos que mais se adequavam a este problema, sendo os escolhidos os métodos Bi-Conjugate Gradient (BiCG), o Bi-Conjugate Gradient Stabilized (BiCGSTAB) e o Quasi-Minimal Residual (QMR). Alguns problemas testes foram utilizados a fim de validar as soluções numéricas obtidas com a versão implícita do método SPH. / In a wide range of physical problems governed by differential equations, it is often of interest to obtain solutions for the unsteady state and therefore it must be employed temporal integration techniques. One possibility could be the use of an explicit methods due to its simplicity and computational efficiency. However, these methods are often only conditionally stable and are subject to severe restrictions for the time step choice. For advective problems governed by hyperbolic equations, this restriction is known as the Courant-Friedrichs-Lewy (CFL) condition. When there is the need to obtain numerical solutions for long periods of time, or when the computational cost for each time step is high, this condition becomes a handicap. In order to overcome this restriction implicit methods can be used, which are generally unconditionally stable. In this study, some implicit formulations for time integration are used in the Smoothed Particle Hydrodynamics (SPH) method to enable the use of larger time increments and obtain a strong stability in the time evolution process. Due to the high computational cost required by the particles tracking at each time step, the implementation will be feasible only if efficient algorithms were applied for this type of matrix structure such as Krylov subspace methods. Therefore, we carried out a study for the appropriate choice of methods best suited to this problem, and the methods chosen were the Bi-Conjugate Gradient (BiCG), the Bi-Conjugate Gradient Stabilized (BiCGSTAB) and the Quasi-Minimal Residual(QMR). Some test problems were used to validate the numerical solutions obtained with the implicit version of the SPH method.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Neste trabalho, foi desenvolvido um simulador numérico baseado no método livre de malhas Smoothed Particle Hydrodynamics (SPH) para a resolução de escoamentos de fluidos newtonianos incompressíveis. Diferentemente da maioria das versões existentes deste método, o código numérico faz uso de uma técnica iterativa na determinação do campo de pressões. Este procedimento emprega a forma diferencial de uma equação de estado para um fluido compressível e a equação da continuidade a fim de que a correção da pressão seja determinada. Uma versão paralelizada do simulador numérico foi implementada usando a linguagem de programação C/C++ e a Compute Unified Device Architecture (CUDA) da NVIDIA Corporation. Foram simulados três problemas, o problema unidimensional do escoamento de Couette e os problemas bidimensionais do escoamento no interior de uma Cavidade (Shear Driven Cavity Problem) e da Quebra de Barragem (Dambreak). / In this work a numerical simulator was developed based on the mesh-free Smoothed Particle Hydrodynamics (SPH) method to solve incompressible newtonian fluid flows. Unlike most existing versions of this method, the numerical code uses an iterative technique in the pressure field determination. This approach employs a differential state equation for a compressible fluid and the continuity equation to calculate the pressure correction. A parallel version of the numerical code was implemented using the Programming Language C/C++ and Compute Unified Device Architecture (CUDA) from the NVIDIA Corporation. The numerical results were validated and the speed-up evaluated for an one-dimensional Couette flow and two-dimensional Shear Driven Cavity and Dambreak problems.

Os métodos numéricos convencionais, baseados em malhas, têm sido amplamente aplicados na resolução de problemas da Dinâmica dos Fluidos Computacional. Entretanto, em problemas de escoamento de fluidos que envolvem superfícies livres, grandes explosões, grandes deformações, descontinuidades, ondas de choque etc., estes métodos podem apresentar algumas dificuldades práticas quando da resolução destes problemas. Como uma alternativa viável, existem os métodos de partículas livre de malhas. Neste trabalho é feita uma introdução ao método Lagrangeano de partículas, livre de malhas, Smoothed Particle Hydrodynamics (SPH) voltado para a simulação numérica de escoamentos de fluidos newtonianos compressíveis e quase-incompressíveis. Dois códigos numéricos foram desenvolvidos, uma versão serial e outra em paralelo, empregando a linguagem de programação C/C++ e a Compute Unified Device Architecture (CUDA), que possibilita o processamento em paralelo empregando os núcleos das Graphics Processing Units (GPUs) das placas de vídeo da NVIDIA Corporation. Os resultados numéricos foram validados e a eficiência computacional avaliada considerandose a resolução dos problemas unidimensionais Shock Tube e Blast Wave e bidimensional da Cavidade (Shear Driven Cavity Problem). / The conventional mesh-based numerical methods have been widely applied to solving problems in Computational Fluid Dynamics. However, in problems involving fluid flow free surfaces, large explosions, large deformations, discontinuities, shock waves etc. these methods suffer from some inherent difficulties which limit their applications to solving these problems. Meshfree particle methods have emerged as an alternative to the conventional grid-based methods. This work introduces the Smoothed Particle Hydrodynamics (SPH), a meshfree Lagrangian particle method to solve compressible flows. Two numerical codes have been developed, serial and parallel versions, using the Programming Language C/C++ and Compute Unified Device Architecture (CUDA). CUDA is NVIDIAs parallel computing architecture that enables dramatic increasing in computing performance by harnessing the power of the Graphics Processing Units (GPUs). The numerical results were validated and the speedup evaluated for the Shock Tube and Blast Wave one-dimensional problems and Shear Driven Cavity Problem.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Em uma grande gama de problemas físicos, governados por equações diferenciais, muitas vezes é de interesse obter-se soluções para o regime transiente e, portanto, deve-se empregar técnicas de integração temporal. Uma primeira possibilidade seria a de aplicar-se métodos explícitos, devido à sua simplicidade e eficiência computacional. Entretanto, esses métodos frequentemente são somente condicionalmente estáveis e estão sujeitos a severas restrições na escolha do passo no tempo. Para problemas advectivos, governados por equações hiperbólicas, esta restrição é conhecida como a condição de Courant-Friedrichs-Lewy (CFL). Quando temse a necessidade de obter soluções numéricas para grandes períodos de tempo, ou quando o custo computacional a cada passo é elevado, esta condição torna-se um empecilho. A fim de contornar esta restrição, métodos implícitos, que são geralmente incondicionalmente estáveis, são utilizados. Neste trabalho, foram aplicadas algumas formulações implícitas para a integração temporal no método Smoothed Particle Hydrodynamics (SPH) de modo a possibilitar o uso de maiores incrementos de tempo e uma forte estabilidade no processo de marcha temporal. Devido ao alto custo computacional exigido pela busca das partículas a cada passo no tempo, esta implementação só será viável se forem aplicados algoritmos eficientes para o tipo de estrutura matricial considerada, tais como os métodos do subespaço de Krylov. Portanto, fez-se um estudo para a escolha apropriada dos métodos que mais se adequavam a este problema, sendo os escolhidos os métodos Bi-Conjugate Gradient (BiCG), o Bi-Conjugate Gradient Stabilized (BiCGSTAB) e o Quasi-Minimal Residual (QMR). Alguns problemas testes foram utilizados a fim de validar as soluções numéricas obtidas com a versão implícita do método SPH. / In a wide range of physical problems governed by differential equations, it is often of interest to obtain solutions for the unsteady state and therefore it must be employed temporal integration techniques. One possibility could be the use of an explicit methods due to its simplicity and computational efficiency. However, these methods are often only conditionally stable and are subject to severe restrictions for the time step choice. For advective problems governed by hyperbolic equations, this restriction is known as the Courant-Friedrichs-Lewy (CFL) condition. When there is the need to obtain numerical solutions for long periods of time, or when the computational cost for each time step is high, this condition becomes a handicap. In order to overcome this restriction implicit methods can be used, which are generally unconditionally stable. In this study, some implicit formulations for time integration are used in the Smoothed Particle Hydrodynamics (SPH) method to enable the use of larger time increments and obtain a strong stability in the time evolution process. Due to the high computational cost required by the particles tracking at each time step, the implementation will be feasible only if efficient algorithms were applied for this type of matrix structure such as Krylov subspace methods. Therefore, we carried out a study for the appropriate choice of methods best suited to this problem, and the methods chosen were the Bi-Conjugate Gradient (BiCG), the Bi-Conjugate Gradient Stabilized (BiCGSTAB) and the Quasi-Minimal Residual(QMR). Some test problems were used to validate the numerical solutions obtained with the implicit version of the SPH method.

Cette thèse porte sur le développement d’une nouvelle technique de modélisation des problèmes IFS utilisant les méthodes particulaires. Ce travail s’inscrit dans la continuité des travaux de recherche de l’équipe biomécanique du LAMIH, concernant la compréhension du comportement de l’os humain dans son environnement de moelle osseuse. La méthode SPH a été utilisée pour la modélisation des travées osseuses, supposées dans une première approche comme des solides élastiques. La méthode LB a été développée pour la modélisation des écoulements de moelle considérée comme un fluide visqueux incompressible. L’efficacité et la performance de ces deux méthodes ont été démontrées grâce aux benchmarks académiques évalués et les résultats comparés à ceux de la littérature ou ceux obtenus par des logiciels commerciaux. A l’issue d’une revue de l’état de l’art des techniques de couplage fluide-structure, une approche partitionnée en temps a été choisie, permettant d’utiliser deux codes distincts basés sur des algorithmes de résolution de type dynamique explicite. La discrétisation spatiale est faite par une technique spécifique basée sur les domaines fictifs, cette technique est très efficace car elle ne nécessite pas de rediscrétisation des domaines. L’approche de couplage développée a été appliquée à des benchmarks académiques ainsi qu’à une application en biomécanique, ayant permis d’aboutir à des résultats numériques satisfaisants. Plusieurs pistes d’amélioration sont maintenant nécessaires afin d’aller vers des modélisations plus biofidèles telles que la prise en compte du contact et de l’endommagement. / The objective of this thesis is the development of a new technique for the FSI problems modelling using particulars methods. This work is in the continuity of the LAMIH biomechanics team research works, regarding the comprehension of behavior of bone in its environment of marrow. The SPH method was used for the trabeculae modelling, supposed in a first attempt as an elastic solid. The LB method was developed for the marrow flow modelling considered as a viscous incompressible liquid. The efficacy and performance of these two methods were demonstrated using academics benchmarks which were evaluated and the results were compared of those of literature and of those obtained from commercials softwares. Following a bibliographic review of FSI coupling techniques, a partitioned approach in time was chosen, allowing the use of two separates codes, both based on a dynamic explicit algorithm resolution scheme. The special discretization was done based on a specific technique of fictional domain, this technique is very efficient because it doesn’t require an additional domain discretization. The coupling approach developed was applied on academic benchmarks and on a biomechanical application, leading to satisfactory numerical results. Many Improvement track are now necessary to go towards more biofidelic modeling as taking into account the contact and the damage.

This thesis deals with the optimal control of PDEs. After a brief introduction in the theory of elliptic and parabolic PDEs, we introduce a software that solves systems of PDEs by the finite elements method. In the second chapter we derive optimality conditions in terms of function spaces, i.e. a systems of PDEs coupled by some pointwise relations. Now we present algorithms to solve the optimality systems numerically and present some numerical test cases. A further chapter deals with the so called lack of adjointness, an issue of gradient methods applied on parabolic optimal control problems. However, since optimal control problems lead to large numerical schemes, model reduction becomes popular. We analyze the proper orthogonal decomposition method and apply it to our model problems. Finally, we apply all considered techniques to a real world problem.:Introduction The state equation Optimal control and optimality conditions Algorithms The \"lack of adjointness\" Numerical examples Efficient solution of PDEs and KKT- systems A real world application Functional analytical basics Codes of the examples

In this work, the mesh-free smoothed particle hydrodynamics (SPH) method is applied in the modelling of the direct laser interference patterning (DLIP) of metal surfaces. The DLIP technique allows the fabrication of periodic microstructures on technical surfaces using nanosecond laser pulses. Here, the interference of two coherent partial beams with a sinusoidal energy density distribution of the interference pattern is concerned, which is employed to generate line-like surface structures. However, the mechanisms effective during nanosecond pulsed DLIP of metals are not yet fully understood. The physical phenomena occurring due to the interaction of laser radiation with metallic materials are first considered and the governing differential equations are stated. The fundamentals of the SPH method and the approaches to the numerical treatment of the conservation equations are presented. Physical processes relevant to the modelling of laser material processing are solved by suitable SPH techniques, i.e. the approximations are verified with respect to test problems with analytical or known numerical solutions. Consequently, the SPH method is used to devise a thermal model of the DLIP process, considering the absorption of the laser radiation, the heat conduction into the workpiece and the latent heat of involved phase changes. This model is extended to compute the melt pool convection during DLIP, which is driven by surface tension gradients due to temperature gradients. For this purpose, an incompressible SPH (ISPH) method is used, representing a novel approach to the modelling of the laser-induced melt pool flow. The numerical model is employed to perform simulations of DLIP on metal substrates. Firstly, the thermal simulation of the single pulse patterning of stainless steel is in good agreement with experimental results. The application of DLIP to stainless steel and aluminium is then simulated by the comprehensive model including the melt pool flow. Moreover, this model is further extended to consider the non-linear temperature dependence of surface tension, as in liquid steel in the presence of a surface active element. The simulation results reveal a distinct behaviour of stainless steel and aluminium substrates. A markedly deeper melt pool and considerable velocity magnitudes of the thermocapillary convection at the melt surface are computed for DLIP of aluminium. In contrast, the melt pool flow is less pronounced during DLIP of stainless steel, whereas higher surface temperatures are predicted. Hence the Marangoni convection is a conceivable effective mechanism during the structuring of aluminium at moderate energy density. The different character of the melt pool convection during DLIP of stainless steel and aluminium is corroborated by experimental observations. Furthermore, the simulations for stainless steel with different sulphur content indicate distinct melt pool flow patterns and support the explanation of the microstructures found after DLIP experiments. The role of vapourisation and the induced recoil pressure in the microstructure evolution due to DLIP on metal substrates at elevated fluences could be prospectively investigated. In this regard, the consideration of the melt pool surface deformation in the ISPH algorithm, and particularly a suitable pressure boundary condition, is required.:I The research problem 1 Motivation 2 Modelling of laser material processing 2.1 Interaction of laser radiation with materials 2.1.1 Absorption of laser radiation 2.1.2 Heat conduction and phase change 2.1.3 Molten pool convection 2.1.4 Vapourisation regime 2.2 Mathematical modelling of laser material interaction 2.2.1 Conservation equations in Lagrangian formulation 2.2.2 Influence of surface tension 3 State of the art in laser microprocessing and the SPH method 3.1 Laser microprocessing 3.2 Simulation of direct laser interference patterning 3.3 The mesh-free smoothed particle hydrodynamics method 3.3.1 Fundamental approximations and kernel function 3.3.2 Particle distribution and interaction length 3.3.3 Approximation of derivatives 3.3.4 Treatment of boundaries 3.3.5 Neighbourhood search 3.4 Numerical modelling of laser material processing by SPH II SPH model development for direct laser interference patterning 4 SPH modelling of heat transfer and fluid flow 4.1 Solution of the heat diffusion equation 4.2 Formulation of equations governing fluid flow 4.2.1 Equation of continuity 4.2.2 Approximation of pressure gradient term 4.2.3 Treatment of viscosity 4.3 Weakly compressible SPH method for solving fluid flow 4.3.1 Particle motion 4.3.2 Time integration 4.3.3 Time step criteria 4.4 Incompressible SPH method for solving fluid flow 4.4.1 Time integration 4.4.2 Discrete incompressible SPH algorithm 4.4.3 Time step criteria 4.5 Simulation of thermal fluid flow using ISPH 4.5.1 Semi-implicit time integration 4.5.2 Solution of the pressure Poisson equation 5 Verification of the SPH implementation 5.1 Transient heat conduction in laser-irradiated plate 5.1.1 Problem description 5.1.2 Dimensionless formulation 5.1.3 Numerical solution and results 5.2 Viscous flow 5.2.1 Couette flow 5.2.2 Poiseuille flow 5.3 Thermal convection 5.3.1 Natural convection in a square cavity 5.3.2 Rayleigh--Marangoni--Bénard convection in liquid aluminium 6 SPH model of direct laser interference patterning 6.1 Characteristics of the process 6.2 Thermal model 6.2.1 Non-dimensionalisation 6.2.2 Numerical solution of governing equation 6.2.3 Verification of the computation 6.2.4 Numerical test 6.3 Thermofluiddynamic model 6.3.1 Non-dimensionalisation 6.3.2 Numerical solution of governing equations 6.3.3 Discretisation 6.3.4 Resolution independence study 7 SPH simulation of direct laser interference patterning 7.1 Thermal model 7.1.1 DLIP experiments on stainless steel substrates 7.1.2 Thermal simulation of DLIP on steel substrate 7.2 Thermofluiddynamic model 7.2.1 Material properties and simulation parameters 7.2.2 Numerical results for steel substrate 7.2.3 Numerical results for aluminium substrate 7.2.4 Discussion and comparison with experiments 7.3 Extended thermofluiddynamic model 7.3.1 Model parameters 7.3.2 Influence of sulphur content on DLIP of stainless steel 8 Conclusions and outlook Bibliography / In dieser Arbeit wird die direkte Laserinterferenzstrukturierung (Direct Laser Interference Patterning, DLIP) von Metallen mit der netzfreien Smoothed Particle Hydrodynamics (SPH) Methode modelliert. Das DLIP-Verfahren ermöglicht die Fertigung periodischer Mikrostrukturen auf technischen Oberflächen mit Nanosekunden-Laserpulsen. Hier wird die Zweistrahlinterferenz mit einer sinusförmigen Energiedichteverteilung des Interferenzmusters behandelt, die linienförmige Oberflächenstrukturen erzeugt. Die bei der direkten Interferenzstrukturierung von Metallen mit Nanosekunden-Laserpuls wirksamen Mechanismen sind jedoch noch nicht verstanden. Die aufgrund der Wechselwirkung von Laserstrahlung mit metallischen Werkstoffen auftretenden physikalischen Phänomene werden zuerst betrachtet und die sie bestimmenden Differentialgleichungen angegeben. Die Grundlagen der SPH-Methode sowie deren Herangehensweisen an die numerische Behandlung der Erhaltungsgleichungen werden vorgestellt. Für die Modellierung der Lasermaterialbearbeitung relevante physikalische Vorgänge werden mittels geeigneter SPH-Ansätze gelöst, d. h. anhand von Testproblemen mit bekannter Lösung verifiziert. Das mit SPH zunächst erstellte thermische Modell des DLIP-Prozesses berücksichtigt die Absorption der Laserstrahlung, die Wärmeleitung im Werkstück und die Enthalpien der Phasenübergänge. Das Modell wird zur Berechnung der Schmelzbadströmung bei der DLIP-Anwendung, angetrieben von Oberflächenspannungsgradienten verursacht durch Temperaturgradienten, erweitert. Hierbei wird eine inkompressible SPH (ISPH) Methode eingesetzt, in der Simulation laserinduzierter Schmelzbäder ein neuartiger Ansatz. Mit dem numerischen Modell werden Simulationen des DLIP-Verfahrens für metallische Substrate durchgeführt. Die thermische Simulation der Strukturierung von Edelstahl stimmt gut mit einem Experiment überein. Weiterhin wird die Anwendung von DLIP auf Edelstahl und Aluminium mit dem thermofluiddynamischen Modell simuliert. Außerdem wird das Modell um eine nichtlinear temperaturabhängige Oberflächenspannung, wie sie für Stahlschmelze in Anwesenheit eines oberflächenaktiven Elements vorliegt, ergänzt. Die Simulationen zeigen ein verschiedenes Verhalten von Edelstahl und Aluminium. Bei der Strukturierung von Aluminium treten ein deutlich tieferes Schmelzbad und erhebliche Geschwindigkeitsbeträge der thermokapillaren Konvektion an der Schmelzeoberfläche auf. Hingegen ist die Strömung bei der DLIP-Anwendung auf Edelstahl schwächer ausgeprägt und höhere Oberflächentemperaturen werden erreicht. Die Marangoni-Konvektion ist daher ein wirksamer Schmelzeverdrängungsmechanismus bei der Strukturierung von Aluminium mit moderater Energiedichte. Die unterschiedliche Schmelzbadströmung für die beiden Werkstoffe wird durch experimentelle Beobachtungen bestätigt. In Abhängigkeit des Schwefelgehalts von Edelstahl zeigen Simulationen verschiedene Strömungsmuster im Schmelzbad und unterstützen die Erklärung experimentell festgestellter Mikrostrukturen. Die Untersuchung der Wirkung der Verdampfung und des induzierten Rückstoßdruckes auf die Strukturausbildung bei höheren Fluenzen erfordert die Berücksichtigung der Oberflächendeformation sowie eine geeignete Druckrandbedingung im ISPH-Algorithmus.:I The research problem 1 Motivation 2 Modelling of laser material processing 2.1 Interaction of laser radiation with materials 2.1.1 Absorption of laser radiation 2.1.2 Heat conduction and phase change 2.1.3 Molten pool convection 2.1.4 Vapourisation regime 2.2 Mathematical modelling of laser material interaction 2.2.1 Conservation equations in Lagrangian formulation 2.2.2 Influence of surface tension 3 State of the art in laser microprocessing and the SPH method 3.1 Laser microprocessing 3.2 Simulation of direct laser interference patterning 3.3 The mesh-free smoothed particle hydrodynamics method 3.3.1 Fundamental approximations and kernel function 3.3.2 Particle distribution and interaction length 3.3.3 Approximation of derivatives 3.3.4 Treatment of boundaries 3.3.5 Neighbourhood search 3.4 Numerical modelling of laser material processing by SPH II SPH model development for direct laser interference patterning 4 SPH modelling of heat transfer and fluid flow 4.1 Solution of the heat diffusion equation 4.2 Formulation of equations governing fluid flow 4.2.1 Equation of continuity 4.2.2 Approximation of pressure gradient term 4.2.3 Treatment of viscosity 4.3 Weakly compressible SPH method for solving fluid flow 4.3.1 Particle motion 4.3.2 Time integration 4.3.3 Time step criteria 4.4 Incompressible SPH method for solving fluid flow 4.4.1 Time integration 4.4.2 Discrete incompressible SPH algorithm 4.4.3 Time step criteria 4.5 Simulation of thermal fluid flow using ISPH 4.5.1 Semi-implicit time integration 4.5.2 Solution of the pressure Poisson equation 5 Verification of the SPH implementation 5.1 Transient heat conduction in laser-irradiated plate 5.1.1 Problem description 5.1.2 Dimensionless formulation 5.1.3 Numerical solution and results 5.2 Viscous flow 5.2.1 Couette flow 5.2.2 Poiseuille flow 5.3 Thermal convection 5.3.1 Natural convection in a square cavity 5.3.2 Rayleigh--Marangoni--Bénard convection in liquid aluminium 6 SPH model of direct laser interference patterning 6.1 Characteristics of the process 6.2 Thermal model 6.2.1 Non-dimensionalisation 6.2.2 Numerical solution of governing equation 6.2.3 Verification of the computation 6.2.4 Numerical test 6.3 Thermofluiddynamic model 6.3.1 Non-dimensionalisation 6.3.2 Numerical solution of governing equations 6.3.3 Discretisation 6.3.4 Resolution independence study 7 SPH simulation of direct laser interference patterning 7.1 Thermal model 7.1.1 DLIP experiments on stainless steel substrates 7.1.2 Thermal simulation of DLIP on steel substrate 7.2 Thermofluiddynamic model 7.2.1 Material properties and simulation parameters 7.2.2 Numerical results for steel substrate 7.2.3 Numerical results for aluminium substrate 7.2.4 Discussion and comparison with experiments 7.3 Extended thermofluiddynamic model 7.3.1 Model parameters 7.3.2 Influence of sulphur content on DLIP of stainless steel 8 Conclusions and outlook Bibliography

In meshfree methods, partial differential equations are solved on an unstructured cloud of points distributed throughout the computational domain. In collocated meshfree methods, the differential operators are directly approximated at each grid point based on a local cloud of neighboring points. The set of neighboring nodes used to construct the local approximation is determined using a variable search radius. The variable search radius establishes an implicit nodal connectivity and hence a mesh is not required. As a result, meshfree methods have the potential flexibility to handle problem sets where the computational grid may undergo large deformations as well as where the grid may need to undergo adaptive refinement. In this work we develop the sharp interface formulation of the immersed boundary method for collocated meshfree approximations. We use the framework to implement three meshfree methods: General Finite Differences (GFD), Smoothed Particle Hydrodynamics (SPH), and Moving Least Squares (MLS). We evaluate the numerical accuracy and convergence rate of these methods by solving the 2D Poisson equation. We demonstrate that GFD is computationally more efficient than MLS and show that its accuracy is superior to a popular corrected form of SPH and comparable to MLS. We then use GFD to solve several canonic steady state fluid flow problems on meshfree grids generated using uniform and variable radii Poisson disk algorithm.