459 research outputs found

    Numerical simulation of combustion instability: flame thickening and boundary conditions

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    Combustion-driven instabilities are a significant barrier for progress for many avenues of immense practical relevance in engineering devices, such as next generation gas turbines geared towards minimising pollutant emissions being susceptible to thermoacoustic instabilities. Numerical simulations of such reactive systems must try to balance a dynamic interplay between cost, complexity, and retention of system physics. As such, new computational tools of relevance to Large Eddy Simulation (LES) of compressible, reactive flows are proposed and evaluated. High order flow solvers are susceptible to spurious noise generation at boundaries which can be very detrimental for combustion simulations. Therefore Navier-Stokes Characteristic Boundary conditions are also reviewed and an extension to axisymmetric configurations proposed. Limitations and lingering open questions in the field are highlighted. A modified Artificially Thickened Flame (ATF) model coupled with a novel dynamic formulation is shown to preserve flame-turbulence interaction across a wide range of canonical configurations. The approach does not require efficiency functions which can be difficult to determine, impact accuracy and have limited regimes of validity. The method is supplemented with novel reverse transforms and scaling laws for relevant post-processing from the thickened to unthickened state. This is implemented into a wider Adaptive Mesh Refinement (AMR) context to deliver a unified LES-AMR-ATF framework. The model is validated in a range of test case showing noticeable improvements over conventional LES alternatives. The proposed modifications allow meaningful inferences about flame structure that conventionally may have been restricted to the domain of Direct Numerical Simulation. This allows studying the changes in small-scale flow and scalar topologies during flame-flame interaction. The approach is applied to a dual flame burner setup, where simulations show inclusion of a neighbouring burner increases compressive flow topologies as compared to a lone flame. This may lead to favouring convex scalar structures that are potentially responsible for the increase in counter-normal flame-flame interactions observed in experiments.Open Acces

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

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    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

    Exponential integrators: tensor structured problems and applications

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    The solution of stiff systems of Ordinary Differential Equations (ODEs), that typically arise after spatial discretization of many important evolutionary Partial Differential Equations (PDEs), constitutes a topic of wide interest in numerical analysis. A prominent way to numerically integrate such systems involves using exponential integrators. In general, these kinds of schemes do not require the solution of (non)linear systems but rather the action of the matrix exponential and of some specific exponential-like functions (known in the literature as phi-functions). In this PhD thesis we aim at presenting efficient tensor-based tools to approximate such actions, both from a theoretical and from a practical point of view, when the problem has an underlying Kronecker sum structure. Moreover, we investigate the application of exponential integrators to compute numerical solutions of important equations in various fields, such as plasma physics, mean-field optimal control and computational chemistry. In any case, we provide several numerical examples and we perform extensive simulations, eventually exploiting modern hardware architectures such as multi-core Central Processing Units (CPUs) and Graphic Processing Units (GPUs). The results globally show the effectiveness and the superiority of the different approaches proposed

    On improving the efficiency of ADER methods

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    The (modern) arbitrary derivative (ADER) approach is a popular technique for the numerical solution of differential problems based on iteratively solving an implicit discretization of their weak formulation. In this work, focusing on an ODE context, we investigate several strategies to improve this approach. Our initial emphasis is on the order of accuracy of the method in connection with the polynomial discretization of the weak formulation. We demonstrate that precise choices lead to higher-order convergences in comparison to the existing literature. Then, we put ADER methods into a Deferred Correction (DeC) formalism. This allows to determine the optimal number of iterations, which is equal to the formal order of accuracy of the method, and to introduce efficient pp-adaptive modifications. These are defined by matching the order of accuracy achieved and the degree of the polynomial reconstruction at each iteration. We provide analytical and numerical results, including the stability analysis of the new modified methods, the investigation of the computational efficiency, an application to adaptivity and an application to hyperbolic PDEs with a Spectral Difference (SD) space discretization

    On thermodynamically compatible finite volume schemes for continuum mechanics

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    In this paper we present a new family of semi-discrete and fully-discrete finite volume schemes for overdetermined, hyperbolic and thermodynamically compatible PDE systems. In the following we will denote these methods as HTC schemes. In particular, we consider the Euler equations of compressible gasdynamics, as well as the more complex Godunov-Peshkov-Romenski (GPR) model of continuum mechanics, which, at the aid of suitable relaxation source terms, is able to describe nonlinear elasto-plastic solids at large deformations as well as viscous fluids as two special cases of a more general first order hyperbolic model of continuum mechanics. The main novelty of the schemes presented in this paper lies in the fact that we solve the \textit{entropy inequality} as a primary evolution equation rather than the usual total energy conservation law. Instead, total energy conservation is achieved as a mere consequence of a thermodynamically compatible discretization of all the other equations. For this, we first construct a discrete framework for the compressible Euler equations that mimics the continuous framework of Godunov's seminal paper \textit{An interesting class of quasilinear systems} of 1961 \textit{exactly} at the discrete level. All other terms in the governing equations of the more general GPR model, including non-conservative products, are judiciously discretized in order to achieve discrete thermodynamic compatibility, with the exact conservation of total energy density as a direct consequence of all the other equations. As a result, the HTC schemes proposed in this paper are provably marginally stable in the energy norm and satisfy a discrete entropy inequality by construction. We show some computational results obtained with HTC schemes in one and two space dimensions, considering both the fluid limit as well as the solid limit of the governing partial differential equations

    Monolithic multiphysics simulation of hypersonic aerothermoelasticity using a hybridized discontinuous Galerkin method

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    This work presents implementation of a hybridized discontinuous Galerkin (DG) method for robust simulation of the hypersonic aerothermoelastic multiphysics system. Simulation of hypersonic vehicles requires accurate resolution of complex multiphysics interactions including the effects of high-speed turbulent flow, extreme heating, and vehicle deformation due to considerable pressure loads and thermal stresses. However, the state-of-the-art procedures for hypersonic aerothermoelasticity are comprised of low-fidelity approaches and partitioned coupling schemes. These approaches preclude robust design and analysis of hypersonic vehicles for a number of reasons. First, low-fidelity approaches limit their application to simple geometries and lack the ability to capture small scale flow features (e.g. turbulence, shocks, and boundary layers) which greatly degrades modeling robustness and solution accuracy. Second, partitioned coupling approaches can introduce considerable temporal and spatial inaccuracies which are not trivially remedied. In light of these barriers, we propose development of a monolithically-coupled hybridized DG approach to enable robust design and analysis of hypersonic vehicles with arbitrary geometries. Monolithic coupling methods implement a coupled multiphysics system as a single, or monolithic, equation system to be resolved by a single simulation approach. Further, monolithic approaches are free from the physical inaccuracies and instabilities imposed by partitioned approaches and enable time-accurate evolution of the coupled physics system. In this work, a DG method is considered due to its ability to accurately resolve second-order partial differential equations (PDEs) of all classes. We note that the hypersonic aerothermoelastic system is composed of PDEs of all three classes. Hybridized DG methods are specifically considered due to their exceptional computational efficiency compared to traditional DG methods. It is expected that our monolithic hybridized DG implementation of the hypersonic aerothermoelastic system will 1) provide the physical accuracy necessary to capture complex physical features, 2) be free from any spatial and temporal inaccuracies or instabilities inherent to partitioned coupling procedures, 3) represent a transition to high-fidelity simulation methods for hypersonic aerothermoelasticity, and 4) enable efficient analysis of hypersonic aerothermoelastic effects on arbitrary geometries

    Deep Learning Methods for Partial Differential Equations and Related Parameter Identification Problems

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    Recent years have witnessed a growth in mathematics for deep learning--which seeks a deeper understanding of the concepts of deep learning with mathematics and explores how to make it more robust--and deep learning for mathematics, where deep learning algorithms are used to solve problems in mathematics. The latter has popularised the field of scientific machine learning where deep learning is applied to problems in scientific computing. Specifically, more and more neural network architectures have been developed to solve specific classes of partial differential equations (PDEs). Such methods exploit properties that are inherent to PDEs and thus solve the PDEs better than standard feed-forward neural networks, recurrent neural networks, or convolutional neural networks. This has had a great impact in the area of mathematical modeling where parametric PDEs are widely used to model most natural and physical processes arising in science and engineering. In this work, we review such methods as well as their extensions for parametric studies and for solving the related inverse problems. We equally proceed to show their relevance in some industrial applications

    Wave scattering from nontrivial boundary conditions

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    Die vorliegende Arbeit untersucht numerische Verfahren zur Simulation von akustischen und elektromagnetischen Wellen im Kontext von zeitabhängingen Streuproblemen, die an eine nichttriviale Randbedingung gekoppelt werden. Eine Vielzahl solcher Randbedingungen sind in der Praxis von Interesse, insbesondere wenn mehrere physikalische Skalen involviert sind. Effektive Randbedingungen beinhalten Modelle für dünne Schichten auf reflektierenden Materialien, oder beschreiben das Verhalten eines stark absorbierenden Mediums. Motiviert durch diese Anwendungen, behandelt die vorliegende Arbeit drei Klassen von Randbedingungen: 1. akustische Streuprobleme mit einer abstrakten, linearen Randbedingung, die neben den beschriebenen Anwendungen auch akustische Randbedingungen beinhaltet; 2. elektromagnetische Streuprobleme mit einer abstrakten linearen Randbedingung; 3. elektromagnetische Streuprobleme mit einer nichtlinearen Randbedingung. Zur Bearbeitung dieser Problemstellungen werden, basierend auf Repräsentationsformeln, zeit-abhängige Randintegralgleichungen hergeleitet. Diese Gleichungen sind vollständig auf dem Rand des Streuobjekts formuliert und äquivalent zum ursprünglichen Streuproblem. Essenzielle Eigenschaften der zugrunde liegenden zeitabhängigen Randintegraloperatoren und Repräsentationsformeln werden mithilfe von Transmissionsproblemen gezeigt. Mithilfe dieser fundamentalen Resultate wird die Wohlgestelltheit der Randintegralgleichungen hergeleitet, womit die Wohlgestelltheit der jeweiligen Randwertprobleme insgesamt gezeigt wird. Die Randintegralgleichungen werden in der Zeit durch Faltungsquadraturen basierend auf den Radau IIA Runge--Kutta Methoden diskretisiert. Die Stabilität der Semi-Diskretisierungen folgen aus den fundamentalen Eigenschaften der Randintegraloperatoren und allgemeinen Eigenschaften der Faltungsquadraturen. Komplementiert wird die Zeitdiskretisierung mit der Randelementmethode im Raum, um Volldiskretisierungen zu konstruieren, deren Lösungen effektiv berechnet werden können. Die resultierenden Verfahren berechnen in einem ersten Schritt die numerischen Lösungen auf dem Rand. Anschließend können die Approximationen durch diskrete Repräsentationsformeln an beliebigen Punkten im Gebiet ausgewertet werden. Fehleranalysen leiten Konvergenzraten für die numerischen Approximationen her. Die Notation der Behandlung der linearen Randbedingungen für akustische und elektromagnetische Wellen wurde entsprechend angepasst, sodass Gemeinsamkeiten und Unterschiede herausgestellt werden. Für die nichtlinearen Randbedingungen wird eine Fehleranalyse mithilfe neuer Techniken basierend auf diskreten Transmissionsproblemen durchgeführt. Alle numerischen Verfahren wurden implementiert und mit verschiedenen Parametern und Gittern getestet. Empirische Konvergenzraten illustrieren und komplementieren die theoretischen Ergebnisse. Visualisierungen der numerischen Approximationen zeigen den Nutzen der untersuchten Verfahren.This dissertation studies the numerical approximation of time-dependent acoustic and electromagnetic wave scattering problems in the presence of non-standard boundary conditions. Of particular interest is the numerical treatment of generalized impedance boundary conditions, effective models that approximate the wave-material interaction of partially penetrable obstacles. Classical applications of such boundary conditions are the scattering of highly absorbing materials and perfectly reflecting obstacles with a thin coating. Moreover, acoustic boundary conditions are discussed in the context of the acoustic wave equation. Finally, a class of nonlinear boundary conditions is covered in the context of electromagnetic scattering. Formulated on the time domain, these boundary conditions contain surface differential operators and temporal convolution operators. The resulting boundary value problems on exterior domains are reformulated to retarded boundary integral equations, which are themselves nonlocal in time and space, but fully formulated on the boundary. Several new fundamental properties of the time-harmonic classical potential operators and boundary operators for the acoustic wave equation and the Maxwell's equations are shown, in particular in view of their temporal counterparts. These theoretical results are the necessary preparations for the subsequent numerical analysis of these problems. To derive numerical methods, the boundary integral equations are then discretized in time and space. The temporal discretization is carried out using the Runge--Kutta convolution quadrature method. Fully discrete schemes are derived by combining the time discretization with appropriate boundary element methods in space. Error bounds with specific convergence rates are shown for all boundary conditions. The presentation of the linear boundary conditions is focused on emphasizing the similarities and differences of the acoustic and the electromagnetic settings. The error analysis for the nonlinear scattering problem substantially differs from the analysis of the linear boundary conditions and several new concepts are necessary to overcome the difficulties arising through the nonlinearity of the corresponding boundary integral equation. Numerical experiments illustrate the theoretical results and investigate practical aspects of the proposed methods

    Invariant preservation in machine learned PDE solvers via error correction

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    Machine learned partial differential equation (PDE) solvers trade the reliability of standard numerical methods for potential gains in accuracy and/or speed. The only way for a solver to guarantee that it outputs the exact solution is to use a convergent method in the limit that the grid spacing Δx\Delta x and timestep Δt\Delta t approach zero. Machine learned solvers, which learn to update the solution at large Δx\Delta x and/or Δt\Delta t, can never guarantee perfect accuracy. Some amount of error is inevitable, so the question becomes: how do we constrain machine learned solvers to give us the sorts of errors that we are willing to tolerate? In this paper, we design more reliable machine learned PDE solvers by preserving discrete analogues of the continuous invariants of the underlying PDE. Examples of such invariants include conservation of mass, conservation of energy, the second law of thermodynamics, and/or non-negative density. Our key insight is simple: to preserve invariants, at each timestep apply an error-correcting algorithm to the update rule. Though this strategy is different from how standard solvers preserve invariants, it is necessary to retain the flexibility that allows machine learned solvers to be accurate at large Δx\Delta x and/or Δt\Delta t. This strategy can be applied to any autoregressive solver for any time-dependent PDE in arbitrary geometries with arbitrary boundary conditions. Although this strategy is very general, the specific error-correcting algorithms need to be tailored to the invariants of the underlying equations as well as to the solution representation and time-stepping scheme of the solver. The error-correcting algorithms we introduce have two key properties. First, by preserving the right invariants they guarantee numerical stability. Second, in closed or periodic systems they do so without degrading the accuracy of an already-accurate solver.Comment: 41 pages, 10 figure
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