1,270 research outputs found

    Symmetries of Riemann surfaces and magnetic monopoles

    Get PDF
    This thesis studies, broadly, the role of symmetry in elucidating structure. In particular, I investigate the role that automorphisms of algebraic curves play in three specific contexts; determining the orbits of theta characteristics, influencing the geometry of the highly-symmetric Bring’s curve, and in constructing magnetic monopole solutions. On theta characteristics, I show how to turn questions on the existence of invariant characteristics into questions of group cohomology, compute comprehensive tables of orbit decompositions for curves of genus 9 or less, and prove results on the existence of infinite families of curves with invariant characteristics. On Bring’s curve, I identify key points with geometric significance on the curve, completely determine the structure of the quotients by subgroups of automorphisms, finding new elliptic curves in the process, and identify the unique invariant theta characteristic on the curve. With respect to monopoles, I elucidate the role that the Hitchin conditions play in determining monopole spectral curves, the relation between these conditions and the automorphism group of the curve, and I develop the theory of computing Nahm data of symmetric monopoles. As such I classify all 3-monopoles whose Nahm data may be solved for in terms of elliptic functions

    Data-assisted modeling of complex chemical and biological systems

    Get PDF
    Complex systems are abundant in chemistry and biology; they can be multiscale, possibly high-dimensional or stochastic, with nonlinear dynamics and interacting components. It is often nontrivial (and sometimes impossible), to determine and study the macroscopic quantities of interest and the equations they obey. One can only (judiciously or randomly) probe the system, gather observations and study trends. In this thesis, Machine Learning is used as a complement to traditional modeling and numerical methods to enable data-assisted (or data-driven) dynamical systems. As case studies, three complex systems are sourced from diverse fields: The first one is a high-dimensional computational neuroscience model of the Suprachiasmatic Nucleus of the human brain, where bifurcation analysis is performed by simply probing the system. Then, manifold learning is employed to discover a latent space of neuronal heterogeneity. Second, Machine Learning surrogate models are used to optimize dynamically operated catalytic reactors. An algorithmic pipeline is presented through which it is possible to program catalysts with active learning. Third, Machine Learning is employed to extract laws of Partial Differential Equations describing bacterial Chemotaxis. It is demonstrated how Machine Learning manages to capture the rules of bacterial motility in the macroscopic level, starting from diverse data sources (including real-world experimental data). More importantly, a framework is constructed though which already existing, partial knowledge of the system can be exploited. These applications showcase how Machine Learning can be used synergistically with traditional simulations in different scenarios: (i) Equations are available but the overall system is so high-dimensional that efficiency and explainability suffer, (ii) Equations are available but lead to highly nonlinear black-box responses, (iii) Only data are available (of varying source and quality) and equations need to be discovered. For such data-assisted dynamical systems, we can perform fundamental tasks, such as integration, steady-state location, continuation and optimization. This work aims to unify traditional scientific computing and Machine Learning, in an efficient, data-economical, generalizable way, where both the physical system and the algorithm matter

    Proceedings of SIRM 2023 - The 15th European Conference on Rotordynamics

    Get PDF
    It was our great honor and pleasure to host the SIRM Conference after 2003 and 2011 for the third time in Darmstadt. Rotordynamics covers a huge variety of different applications and challenges which are all in the scope of this conference. The conference was opened with a keynote lecture given by Rainer Nordmann, one of the three founders of SIRM “Schwingungen in rotierenden Maschinen”. In total 53 papers passed our strict review process and were presented. This impressively shows that rotordynamics is relevant as ever. These contributions cover a very wide spectrum of session topics: fluid bearings and seals; air foil bearings; magnetic bearings; rotor blade interaction; rotor fluid interactions; unbalance and balancing; vibrations in turbomachines; vibration control; instability; electrical machines; monitoring, identification and diagnosis; advanced numerical tools and nonlinearities as well as general rotordynamics. The international character of the conference has been significantly enhanced by the Scientific Board since the 14th SIRM resulting on one hand in an expanded Scientific Committee which meanwhile consists of 31 members from 13 different European countries and on the other hand in the new name “European Conference on Rotordynamics”. This new international profile has also been emphasized by participants of the 15th SIRM coming from 17 different countries out of three continents. We experienced a vital discussion and dialogue between industry and academia at the conference where roughly one third of the papers were presented by industry and two thirds by academia being an excellent basis to follow a bidirectional transfer what we call xchange at Technical University of Darmstadt. At this point we also want to give our special thanks to the eleven industry sponsors for their great support of the conference. On behalf of the Darmstadt Local Committee I welcome you to read the papers of the 15th SIRM giving you further insight into the topics and presentations

    Microcredentials to support PBL

    Get PDF

    Easy-to-implement hp-adaptivity for non-elliptic goal-oriented problems

    Get PDF
    The FEM has become a foundational numerical technique in computational mechanics and civil engineering since its inception by Courant in 1943 Courant1943. Originating from the Ritz method and variational calculus, the FEM was primarily employed to derive solutions for vibrational systems. A distinctive strength of the FEM is its capability to represent mathematical models through the weak variational formulation of PDE, facilitating computational feasibility even in intricate geometries. However, the search for accuracy often imposes a significant computational task. In the FEM, adaptive methods have emerged to balance the accuracy of solutions with computational costs. The hh-adaptive FEM designs more efficient meshes by reducing the mesh size hh locally while keeping the polynomial order of approximation pp fixed (usually p=1,2p=1,2). An alternative approach to the hh-adaptive FEM is the pp-adaptive FEM, which locally enriches the polynomial space pp while keeping the mesh size hh constant. By dynamically adapting hh and pp, the hphp-adaptive FEM achieves exponential convergence rates. Adaptivity is crucial for obtaining accurate solutions. However, the traditional focus on global norms, such as L2L^2 or H1H^1, might only sometimes serve the requirements of specific applications. In engineering, controlling errors in specific domains related to a particular QoI is often more critical than focusing on the overall solution. That motivated the development of GOA strategies. In this dissertation, we develop automatic GO hphp-adaptive algorithms tailored for non-elliptic problems. These algorithms shine in terms of robustness and simplicity in their implementation, attributes that make them especially suitable for industrial applications. A key advantage of our methodologies is that they do not require computing reference solutions on globally refined grids. Nevertheless, our approach is limited to anisotropic pp and isotropic hh refinements. We conduct multiple tests to validate our algorithms. We probe the convergence behavior of our GO hh- and pp-adaptive algorithms using Helmholtz and convection-diffusion equations in one-dimensional scenarios. We test our GO hphp-adaptive algorithms on Poisson, Helmholtz, and convection-diffusion equations in two dimensions. We use a Helmholtz-like scenario for three-dimensional cases to highlight the adaptability of our GO algorithms. We also create efficient ways to build large databases ideal for training DNN using hphp MAGO FEM. As a result, we efficiently generate large databases, possibly containing hundreds of thousands of synthetic datasets or measurements

    Data-Driven Exploration of Coarse-Grained Equations: Harnessing Machine Learning

    Get PDF
    In scientific research, understanding and modeling physical systems often involves working with complex equations called Partial Differential Equations (PDEs). These equations are essential for describing the relationships between variables and their derivatives, allowing us to analyze a wide range of phenomena, from fluid dynamics to quantum mechanics. Traditionally, the discovery of PDEs relied on mathematical derivations and expert knowledge. However, the advent of data-driven approaches and machine learning (ML) techniques has transformed this process. By harnessing ML techniques and data analysis methods, data-driven approaches have revolutionized the task of uncovering complex equations that describe physical systems. The primary goal in this thesis is to develop methodologies that can automatically extract simplified equations by training models using available data. ML algorithms have the ability to learn underlying patterns and relationships within the data, making it possible to extract simplified equations that capture the essential behavior of the system. This study considers three distinct learning categories: black-box, gray-box, and white-box learning. The initial phase of the research focuses on black-box learning, where no prior information about the equations is available. Three different neural network architectures are explored: multi-layer perceptron (MLP), convolutional neural network (CNN), and a hybrid architecture combining CNN and long short-term memory (CNN-LSTM). These neural networks are applied to uncover the non-linear equations of motion associated with phase-field models, which include both non-conserved and conserved order parameters. The second architecture explored in this study addresses explicit equation discovery in gray-box learning scenarios, where a portion of the equation is unknown. The framework employs eXtended Physics-Informed Neural Networks (X-PINNs) and incorporates domain decomposition in space to uncover a segment of the widely-known Allen-Cahn equation. Specifically, the Laplacian part of the equation is assumed to be known, while the objective is to discover the non-linear component of the equation. Moreover, symbolic regression techniques are applied to deduce the precise mathematical expression for the unknown segment of the equation. Furthermore, the final part of the thesis focuses on white-box learning, aiming to uncover equations that offer a detailed understanding of the studied system. Specifically, a coarse parametric ordinary differential equation (ODE) is introduced to accurately capture the spreading radius behavior of Calcium-magnesium-aluminosilicate (CMAS) droplets. Through the utilization of the Physics-Informed Neural Network (PINN) framework, the parameters of this ODE are determined, facilitating precise estimation. The architecture is employed to discover the unknown parameters of the equation, assuming that all terms of the ODE are known. This approach significantly improves our comprehension of the spreading dynamics associated with CMAS droplets

    Inkjet printing digital image generation and compensation for surface chemistry effects

    Get PDF
    Additive manufacturing (AM) of electronic materials using digital inkjet printing (DIJP) is of research interests nowadays because of its potential benefits in the semiconductor industry. Current trends in manufacturing electronics feature DIJP as a key technology to enable the production of customised and microscale functional devices. However, the fabrication of microelectronic components at large scale demands fast printing of tight features with high dimensional accuracy on substrates with varied surface topography which push inkjet printing process to its limits. To understand the DIJP droplet deposition on such substrates, generally requires computational fluid dynamics modelling which is limited in its physics approximation of surface interactions. Otherwise, a kind of “trial and error” approach to determining how the ink spreads, coalesce and solidifies over the substrate is used, often a very time-consuming process. Consequently, this thesis aims to develop new modelling techniques to predict fast and accurately the surface morphology of inkjet-printed features, enabling the optimisation of DIJP control parameters and the compensation of images for better dimensional accuracy of printed electronics devices. This investigation explored three categories of modelling techniques to predict the surface morphology of inkjet-printed features: physics-based, data-driven and hybrid physics-based and data-driven. Two physics-based numerical models were developed to reproduce the inkjet printing droplet deposition and solidification processes using a lattice Boltzmann (LB) multiphase flow model and a finite element (FE) chemo-mechanical model, respectively. The LB model was limited to the simulation of single tracks and small square films and the FE model was mainly employed for the distortion prediction of multilayer objects. Alternatively, two data-driven models were implemented to reconstruct the surface morphology of single tracks and free-form films using images from experiments: image analysis (IA) and shape from shading (SFS). IA assumed volume conservation and minimal energy drop shape to reconstruct the surface while SFS resolved the height of the image using a reflection model. Finally, a hybrid physics-based and data-driven approach was generated which incorporates the uncertainty of droplet landing position and footprint, hydrostatic analytical models, empirical correlations derived from experiments, and relationships derived from physics-based models to predict fast and accurately any free-form layer pattern as a function of physical properties, printing parameters and wetting characteristics. Depending on the selection of the modelling technique to predict the deformed geometry, further considerations were required. For the purely physics-based and data-driven models, a surrogate model using response surface equations was employed to create a transfer function between printing parameters, substrate wetting characteristics and the resulting surface morphology. The development of a transfer function significantly decreased the computational time required by purely physics-based models and enabled the parameter optimisation using a multi-objective genetic algorithm approach to attain the best film dimensional accuracy. Additionally, for multilayer printing applications, a layer compensation approach was achieved utilizing a convolutional neural network trained by the predicted (deformed) geometry to reduce the out of plane error to target shape. The optimal combination of printing parameters and input image compensation helped with the generation of fine features that are traditionally difficult for inkjet, improved resolution of edges and corners by reducing the amount of overflow from material, accounted for varied topography and capillary effects thereof on the substrate surface and considered the effect of multiple layers built up on each other. This study revealed for the first time to the best of our knowledge the role of the droplet location and footprint diameter uncertainty in the stability and uniformity of printed features. Using a droplet overlap map which was proposed as a universal technique to assess the effect of printing parameters on pattern geometry, it was shown that reliable limits for break-up and bulging of printed features were obtained. Considering droplet position and diameter size uncertainties, predicted optimal printing parameters improved the quality of printed films on substrates with different wettability. Finally, a stability diagram illustrating the onset of bulging and separation for lines and films as well as the optimal drop spacing, printing frequency and stand-off distance was generated to inform visually the results. This investigation has developed a predictive physics-based model of the surface morphology of DIJP features on heterogeneous substrates and a methodology to find the printing parameters and compensate the layer geometry required for optimum part dimensional accuracy. The simplicity of the proposed technique makes it a promising tool for model driven inkjet printing process optimization, including real time process control and paves the way for better quality devices in the printed electronics industry

    Machine learning algorithms for efficient process optimisation of variable geometries at the example of fabric forming

    Get PDF
    FĂŒr einen optimalen Betrieb erfordern moderne Produktionssysteme eine sorgfĂ€ltige Einstellung der eingesetzten Fertigungsprozesse. Physikbasierte Simulationen können die Prozessoptimierung wirksam unterstĂŒtzen, jedoch sind deren Rechenzeiten oft eine erhebliche HĂŒrde. Eine Möglichkeit, Rechenzeit einzusparen sind surrogate-gestĂŒtzte Optimierungsverfahren (SBO1). Surrogates sind recheneffiziente, datengetriebene Ersatzmodelle, die den Optimierer im Suchraum leiten. Sie verbessern in der Regel die Konvergenz, erweisen sich aber bei verĂ€nderlichen Optimierungsaufgaben, etwa hĂ€ufigen Bauteilanpassungen nach Kundenwunsch, als unhandlich. Um auch solche variablen Optimierungsaufgaben effizient zu lösen, untersucht die vorliegende Arbeit, wie jĂŒngste Fortschritte im Maschinenlernen (ML) – im Speziellen bei neuronalen Netzen – bestehende SBO-Techniken ergĂ€nzen können. Dabei werden drei Hauptaspekte betrachtet: erstens, ihr Potential als klassisches Surrogate fĂŒr SBO, zweitens, ihre Eignung zur effiziente Bewertung der Herstellbarkeit neuer BauteilentwĂŒrfe und drittens, ihre Möglichkeiten zur effizienten Prozessoptimierung fĂŒr variable Bauteilgeometrien. Diese Fragestellungen sind grundsĂ€tzlich technologieĂŒbergreifend anwendbar und werden in dieser Arbeit am Beispiel der Textilumformung untersucht. Der erste Teil dieser Arbeit (Kapitel 3) diskutiert die Eignung tiefer neuronaler Netze als Surrogates fĂŒr SBO. Hierzu werden verschiedene Netzarchitekturen untersucht und mehrere Möglichkeiten verglichen, sie in ein SBO-Framework einzubinden. Die Ergebnisse weisen ihre Eignung fĂŒr SBO nach: FĂŒr eine feste Beispielgeometrie minimieren alle Varianten erfolgreich und schneller als ein Referenzalgorithmus (genetischer Algorithmus) die Zielfunktion. Um die Herstellbarkeit variabler Bauteilgeometrien zu bewerten, untersucht Kapitel 4 anschließend, wie Geometrieinformationen in ein Prozess-Surrogate eingebracht werden können. Hierzu werden zwei ML-AnsĂ€tze verglichen, ein merkmals- und ein rasterbasierter Ansatz. Der merkmalsbasierte Ansatz scannt ein Bauteil nach einzelnen, prozessrelevanten Geometriemerkmalen, der rasterbasierte Ansatz hingegen interpretiert die Geometrie als Ganzes. Beide AnsĂ€tze können das Prozessverhalten grundsĂ€tzlich erlernen, allerdings erweist sich der rasterbasierte Ansatz als einfacher ĂŒbertragbar auf neue Geometrievarianten. Die Ergebnisse zeigen zudem, dass hauptsĂ€chlich die Vielfalt und weniger die Menge der Trainingsdaten diese Übertragbarkeit bestimmt. Abschließend verbindet Kapitel 5 die Surrogate-Techniken fĂŒr flexible Geometrien mit variablen Prozessparametern, um eine effiziente Prozessoptimierung fĂŒr variable Bauteile zu erreichen. Hierzu interagiert ein ML-Algorithmus in einer Simulationsumgebung mit generischen Geometriebeispielen und lernt, welche Geometrie, welche Umformparameter erfordert. Nach dem Training ist der Algorithmus in der Lage, auch fĂŒr nicht-generische Bauteilgeometrien brauchbare Empfehlungen auszugeben. Weiter zeigt sich, dass die Empfehlungen mit Ă€hnlicher Geschwindigkeit wie die klassische SBO zum tatsĂ€chlichen Prozessoptimum konvergieren, jedoch kein bauteilspezifisches A-priori-Sampling nötig ist. Einmal trainiert, ist der entwickelte Ansatz damit effizienter. Insgesamt zeigt diese Arbeit, wie ML-Techniken gegenwĂ€rtige SBOMethoden erweitern und so die Prozess- und Produktoptimierung zu frĂŒhen Entwicklungszeitpunkten effizient unterstĂŒtzen können. Die Ergebnisse der Untersuchungen mĂŒnden in Folgefragen zur Weiterentwicklung der Methoden, etwa die Integration physikalischer Bilanzgleichungen, um die Modellprognosen physikalisch konsistenter zu machen

    Computational modelling and optimal control of interacting particle systems: connecting dynamic density functional theory and PDE-constrained optimization

    Get PDF
    Processes that can be described by systems of interacting particles are ubiquitous in nature, society, and industry, ranging from animal flocking, the spread of diseases, and formation of opinions to nano-filtration, brewing, and printing. In real-world applications it is often relevant to not only model a process of interest, but to also optimize it in order to achieve a desired outcome with minimal resources, such as time, money, or energy. Mathematically, the dynamics of interacting particle systems can be described using Dynamic Density Functional Theory (DDFT). The resulting models are nonlinear, nonlocal partial differential equations (PDEs) that include convolution integral terms. Such terms also enter the naturally arising no-flux boundary conditions. Due to the nonlocal, nonlinear nature of such problems they are challenging both to analyse and solve numerically. In order to optimize processes that are modelled by PDEs, one can apply tools from PDE-constrained optimization. The aim here is to drive a quantity of interest towards a target state by varying a control variable. This is constrained by a PDE describing the process of interest, in which the control enters as a model parameter. Such problems can be tackled by deriving and solving the (first-order) optimality system, which couples the PDE model with a second PDE and an algebraic equation. Solving such a system numerically is challenging, since large matrices arise in its discretization, for which efficient solution strategies have to be found. Most work in PDE-constrained optimization addresses problems in which the control is applied linearly, and which are constrained by local, often linear PDEs, since introducing nonlinearity significantly increases the complexity in both the analysis and numerical solution of the optimization problem. However, in order to optimize real-world processes described by nonlinear, nonlocal DDFT models, one has to develop an optimal control framework for such models. The aim is to drive the particles to some desired distribution by applying control either linearly, through a particle source, or bilinearly, though an advective field. The optimization process is constrained by the DDFT model that describes how the particles move under the influence of advection, diffusion, external forces, and particle–particle interactions. In order to tackle this, the (first-order) optimality system is derived, which, since it involves nonlinear (integro-)PDEs that are coupled nonlocally in space and time, is significantly harder than in the standard case. Novel numerical methods are developed, effectively combining pseudospectral methods and iterative solvers, to efficiently and accurately solve such a system. In a next step this framework is extended so that it can capture and optimize industrially relevant processes, such as brewing and nano-filtration. In order to do so, extensions to both the DDFT model and the numerical method are made. Firstly, since industrial processes often involve tubes, funnels, channels, or tanks of various shapes, the PDE model itself, as well as the optimization problem, need to be solved on complicated domains. This is achieved by developing a novel spectral element approach that is compatible with both the PDE solver and the optimal control framework. Secondly, many industrial processes, such as nano-filtration, involve more than one type of particle. Therefore, the DDFT model is extended to describe multiple particle species. Finally, depending on the application of interest, additional physical effects need to be included in the model. In this thesis, to model sedimentation processes in brewing, the model is modified to capture volume exclusion effects
    • 

    corecore