Simulation of the Particle Distribution and Resulting Laser Processing of Selective Laser Melting Processes

Selective Laser Melting (SLM) is a 3D printing technology which is suited for additively manufacturing of metals and polymers. The main barriers of this process are the lack of reproducibility and the control of the influence of the process parameters, like laser power, scan rate, layer height and particle distribution for instance. A high fidelity simulation scheme for SLM processes can not only give an insight into the physical behaviour during the process, but can also help to control the whole 3D printing process in order to guarantee the reproducibility of the desired final product properties. However many challenges have to be overcome in order to guarantee a high fidelity simulation, like modelling the heat source or the phase change for instance. In this work the modelling of the heat source is addressed. A Ray Tracing algorithm should be implemented into the existing thermomechanical Optimal Transportation Meshfree (OTM) code. A Ray Tracing algorithm for the simulation of laser processes is investigated in combination with the Discrete Element Method in the literature. Therefore in a first step this algorithm should be recoded in order to benchmark the implemented Ray Tracing algorithm. In the next step this algorithm has to be implemented into the OTM code. An investigation by means of some examples should demonstrate the influence of the Ray Tracing algorithm on the fusion of two metal particles

Peridynamic Galerkin methods for nonlinear solid mechanics

Simulation-driven product development is nowadays an essential part in the industrial digitalization. Notably, there is an increasing interest in realistic high-fidelity simulation methods in the fast-growing field of additive and ablative manufacturing processes. Thanks to their flexibility, meshfree solution methods are particularly suitable for simulating the stated processes, often accompanied by large deformations, variable discontinuities, or phase changes. Furthermore, in the industrial domain, the meshing of complex geometries represents a significant workload, which is usually minor for meshfree methods. Over the years, several meshfree schemes have been developed. Nevertheless, along with their flexibility in discretization, meshfree methods often endure a decrease in accuracy, efficiency and stability or suffer from a significantly increased computation time. Peridynamics is an alternative theory to local continuum mechanics for describing partial differential equations in a non-local integro-differential form. The combination of the so-called peridynamic correspondence formulation with a particle discretization yields a flexible meshfree simulation method, though does not lead to reliable results without further treatment.\newline In order to develop a reliable, robust and still flexible meshfree simulation method, the classical correspondence formulation is generalized into the Peridynamic Galerkin (PG) methods in this work. On this basis, conditions on the meshfree shape functions of virtual and actual displacement are presented, which allow an accurate imposition of force and displacement boundary conditions and lead to stability and optimal convergence rates. Based on Taylor expansions moving with the evaluation point, special shape functions are introduced that satisfy all the previously mentioned requirements employing correction schemes. In addition to displacement-based formulations, a variety of stabilized, mixed and enriched variants are developed, which are tailored in their application to the nearly incompressible and elasto-plastic finite deformation of solids, highlighting the broad design scope within the PG methods. Extensive numerical validations and benchmark simulations are performed to show the impact of violating different shape function requirements as well as demonstrating the properties of the different PG formulations. Compared to related Finite Element formulations, the PG methods exhibit similar convergence properties. Furthermore, an increased computation time due to non-locality is counterbalanced by a considerably improved robustness against poorly meshed discretizations

On the largest component of a hyperbolic model of complex networks

We consider a model for complex networks that was introduced by Krioukov et al. In this model, N points are chosen randomly inside a disk on the hyperbolic plane and any two of them are joined by an edge if they are within a certain hyperbolic distance. The N points are distributed according to a quasi-uniform distribution, which is a distorted version of the uniform distribution. The model turns out to behave similarly to the well-known Chung-Lu model, but without the independence between the edges. Namely, it exhibits a power-law degree sequence and small distances but, unlike the Chung-Lu model and many other well-known models for complex networks, it also exhibits clustering. The model is controlled by two parameters α and ν where, roughly speaking, α controls the exponent of the power-law and ν controls the average degree. The present paper focuses on the evolution of the component structure of the random graph. We show that (a) for α > 1 and ν arbitrary, with high probability, as the number of vertices grows, the largest component of the random graph has sublinear order; (b) for α < 1 and ν arbitrary with high probability there is a "giant" component of linear order, and (c) when α = 1 then there is a non-trivial phase transition for the existence of a linear-sized component in terms of ν

Mixed peridynamic formulations for compressible and incompressible finite deformations

The large flexibility of meshfree solution schemes makes them attractive for many kinds of engineering applications, like Additive Manufacturing or cutting processes. While numerous meshfree methods were developed over the years, the accuracy and robustness are still challenging and critical issues. Stabilization techniques of various kinds are typically used to overcome these problems, but often require the tuning of unphysical parameters. The Peridynamic Petrov–Galerkin method is a generalization of the peridynamic theory of correspondence materials and offers a stable and robust alternative. In this work, the stabilization free approach is extended to three dimensional problems of finite elasticity. Locking-free mixed formulations for nearly incompressible and incompressible materials are developed and investigated in convergence studies. In general, an efficient implicit quasi-static framework based on Automatic Differentiation is presented. The numerical examples highlight the convergence properties and robustness of the proposed formulations. © 2020, The Author(s)

Entwicklung einer netzfreien Simulationsmethode auf Basis der flexiblen Elemente

Die Finite Element Methode (FEM) hat sich im Ingenieurwesen aufgrund seiner stabilen und numerisch genauen Approximation der Lösung von partiellen Differenzialgleichungen als Simulationswerkzeug bewährt. Aufgrund der Unterteilung des Lösungsgebiets in feste Elemente sind bei sehr großen Deformationen zusätzliche Algorithmen von Nöten, die meist die Stabilität und Genauigkeit negativ beeinflussen. Netzfreie Simulationsverfahren, wie die ”Optimal Transportation Meshfree“ (OTM) Methode, bestechen gerade hierbei durch Flexibilität, da keine feste Konnektivität zwischen einzelnen Knoten benötigt wird. Anstatt fester Elemente wird die Anzahl von Knoten, die den jeweiligen Materialpunkt beeinflusst, in jedem Zeitpunkt durch einen Suchalgorithmus neu bestimmt. Dieser Bereich wird dabei als Einflussgebiet des Materialpunkts bezeichnet. Auf der anderen Seite sind netzfreie Verfahren von Haus aus nicht stabil. Einer der Hauptgründe ist die Unterintegration in den Einflussgebieten der Materialpunkte. Daher soll im Rahmen dieser Studienarbeit die OTM Methode dahingehend erweitert werden, dass zusätzliche Integrationspunkte in das Einflussgebiet des jeweiligen Materialpunkts gelegt wird. Besonderer Fokus soll dabei auf die benötigte Anzahl von Integrationspunkten und die Kopplung zu ”Mean Value“ Ansatzfunktionen gelegt werden. Aufgrund der dadurch entstehenden Nähe zum Ablaufschema der FEM, soll dieses Verfahren als flexible Finite Element Methode bezeichnet werden. Nach der Implementation des neuen Algorithmus soll des Weiteren der Vorteil der neuen Methode anhand von Beispielen untersucht werden

Coexistence of charge and ferromagnetic order in fcc Fe

Phase coexistence phenomena have been intensively studied in strongly correlated materials where several ordered states simultaneously occur or compete. Material properties critically depend on external parameters and boundary conditions, where tiny changes result in qualitatively different ground states. However, up to date, phase coexistence phenomena have exclusively been reported for complex compounds composed of multiple elements. Here we show that charge- and magnetically ordered states coexist in double-layer Fe on Rh(001). Scanning tunneling microscopy and spectroscopy measurements reveal periodic charge order stripes below a temperature of 130 K. Close to liquid helium temperature, they are superimposed by ferromagnetic domains as observed by spin-polarized scanning tunneling microscopy. Temperature-dependent measurements reveal a pronounced cross-talk between charge and spin order at the ferromagnetic ordering temperature about 70 K, which is successfully modeled within an effective Landau theory including sixth-order terms. Our results show that subtle balance between structural modifications can lead to competing ordering phenomena

The Role of Differential Diffusion during Early Flame Kernel Development under Engine Conditions -- Part II: Effect of Flame Structure and Geometry

From experimental spark ignition (SI) engine studies, it is known that the slow-down of early flame kernel development caused by the ($\mathrm{Le}>1$)-property of common transportation-fuel/air mixtures tends to increase cycle-to-cycle variations (CCV). To improve the fundamental understanding of the complex phenomena inside the flame structure of developing flame kernels, an engine-relevant DNS database is investigated in this work. Conclusive analyses are enabled by considering equivalent flame kernels and turbulent planar flames computed with $\mathrm{Le}>1$ and $\mathrm{Le}=1$. In Part I of the present study (Falkenstein et al., Combust. Flame, 2019), a reduced representation of the local mixture state based on the parameters local enthalpy, local equivalence ratio, and H-radical mass fraction was proposed for the purpose of this analysis. Here, a coupling relation for the diffusion-controlled mixture parameter local enthalpy with local flame geometry and structure is derived, characterized by the key parameters $\kappa$ and ${\ |\nabla c \ t|/\ |\nabla c \ |_{\mathrm{lam}}}$. The analysis shows that the large positive global mean curvature intrinsic to the flame kernel configuration may detrimentally affect the local mixture state inside the reaction zone, particularly during the initial kernel development phase. External energy supply by spark ignition may effectively bridge over this critical stage, which causes the impact of global mean flame kernel curvature to be small under the present conditions compared to the overall effect of$\mathrm{Le}\neq1$. Once ignition effects have decayed, the mixture state inside the reaction zone locally exhibits an identical dependence on$\ |\nabla c \ |$ as in a strained laminar flame. This implies that differential diffusion effects under engine-typical Karlovitz numbers are not weakened by small-scale turbulent mixing

The Role of Differential Diffusion during Early Flame Kernel Development under Engine Conditions -- Part I: Analysis of the Heat-Release-Rate Response

Although experimental evidence for the correlation between early flame kernel development and cycle-to-cycle variations (CCV) in spark ignition (SI) engines was provided long ago, there is still a lack of fundamental understanding of early flame/turbulence interactions, and accurate models for full engine simulations do not exist. Since the flame kernel is initiated with small size, i.e. with large positive curvature, differential diffusion is expected to severely alter early flame growth in non-unity-Lewis-number (${\mathrm{Le}\neq1}$) mixtures as typically used in engines. In this work, a DNS database of developing iso-octane/air flame kernels and planar flames has been established with flame conditions representative for stoichiometric engine part-load operation. Differential diffusion effects on the global heat release rate are analyzed by relating the present findings to equivalent flames computed in the ${\mathrm{Le}=1}$ limit. It is shown that in the early kernel development phase, the normal propagation velocity is significantly reduced with detrimental consequences on the global burning rate of the flame kernel. Besides this impact on the overall mass burning rate, the initial production of flame surface area by the normal propagation term in the flame area balance equation is noticeably reduced. By using the optimal estimator concept, it is shown that strong fluctuations in local heat release rate inherent to ${\mathrm{Le}\neq1}$ flames in the thin reaction zones regime are mainly contained in the parameters local equivalence ratio, enthalpy, and H-radical mass fraction. Differential diffusion couples the evolution of these parameters to the unsteady flame geometry and structure, which is analyzed in Part II of the present study (Falkenstein et al., Combust. Flame, 2019)

The probability of connectivity in a hyperbolic model of complex networks

We consider a model for complex networks that was introduced by Krioukov et al. (Phys Rev E 82 (2010) 036106). In this model, N points are chosen randomly inside a disk on the hyperbolic plane according to a distorted version of the uniform distribution and any two of them are joined by an edge if they are within a certain hyperbolic distance. This model exhibits a power-law degree sequence, small distances and high clustering. The model is controlled by two parameters α and ν where, roughly speaking, α controls the exponent of the power-law and ν controls the average degree. In this paper we focus on the probability that the graph is connected. We show the following results. For α > 1/2 and ν arbitrary, the graph is disconnected with high probability. For α < 1/2 and ν arbitrary, the graph is connected with high probability. When α = 1/2 and ν is fixed then the probability of being connected tends to a constant f(ν) that depends only on ν, in a continuous manner. Curiously, f(ν) = 1 for ν ≥ Π while it is strictly increasing, and in particular bounded away from zero and one, for 0 < ν < Π