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Dissertation, RWTH Aachen University, 2023; Aachen : RWTH Aachen University 1 Online-Ressource : Illustrationen, Diagramme (2023). = Dissertation, RWTH Aachen University, 2023, Solving the Boltzmann equation numerically is an area of ongoing research. There are already existing results based on the Spectral-Fourier Approach by Pareschi & Russo (2000) and Gamba & Tharkabhushanam (2009), and based on the spectral Hermite ansatz introduced by Grad (1949) and linearizing the collision operator by Cai, Torrilhon (2015).Wang and Cai were recently (2019) able to calculate the bilinear collision operator based on the spectral Hermite ansatz. However, their method is computationally expensive. In response, Cai & Fan & Wang (2020) developed and implemented a more efficient algorithm based on the spectral Burnett ansatz. This ansatz has also been looked into analytically (Kumar 1966) and numerically (Gamba 2018). Inspired by Struchtrup's (2005) tensorial approach to the collision operator in Equation 6.35 in "Macroscopic Transport Equations for Rarefied Gas Flows: Approximation Methods in Kinetic Theory", we chose a different approach for calculating the Spectral-Burnett approximation of the Boltzmann collision operator. The main difference lies in the exploitation of features of the basis set, which is adapted to the irreducible subspaces with respect to the orthogonal group of the polynomial space, as it consists of real solid spherical harmonics multiplied with Laguerre polynomials. While this thesis discusses the mostly analytical calculation of collision coefficients for a variety of potentials, the focus is on the understanding of these coefficients provided by representation theory. This understanding is used as inspiration for an algorithm, which has been implemented as c++ code to calculate the numerical solution to the space-homogeneous Boltzmann equation for any distribution. Utilizing the properties of the irreducible subspaces, compared to the previous works we are able to significantly reduce the memory and computation time required for calculating the numerical solution. We could achieve this due to the uniqueness up to a constant of linear maps between the irreducible subspaces. Representation theory allowed us to decompose the collision tensor into two and identify all bilinear maps, that evaluate to zero due to the mathematical properties of the irreducible subspaces in question., Published by RWTH Aachen University, Aachen

Dissertation, RWTH Aachen University, 2022; Aachen : RWTH Aachen University 1 Online-Ressource |b Illustrationen Diagramme (2023). = Dissertation, RWTH Aachen University, 2022, Both hyperbolic partial differential equations (PDEs) and the GENERIC (General Equation for Non-Equilibrium Reversible-Irreversible Coupling) framework are ubiquitous in continuum mechanics, being applicable to very similar problem areas. In particular first-order systems of conservation laws have always been intimately related to gas dynamics, which served as a starting point for the GENERIC framework. However, while a lot of mathematical theory has been developed for hyperbolic PDEs, the GENERIC framework is notable mainly for its application in the modeling of thermodynamic systems in non-equilibrium. There it facilitates the development of thermodynamically compatible evolution equations, while also imposing the severe restrictions of Poisson structures on the reversible part of the evolution equation. The original goal my thesis was to investigate the exact connection between these two disparate theories.For this there were initially two main questions:* When does a system being GENERIC imply hyperbolicity?* When does hyperbolicity imply the existence of a GENERIC formulation? Furthermore, inspired by current research, another question arose:* How can structure-preserving integrators facilitate the numerical solution of GENERIC systems? Consequently, the results factor into two distinct, but related groups.It is well-known that in the smooth regime many hyperbolic PDEs can be regarded as time-reversible infinite-dimensional Hamiltonian systems, while this property is generally lost in the presence of discontinuities. This immediately suggests to investigate the connection between hyperbolic PDEs and the reversible part of GENERIC systems, and indeed we were able to show that the irreversible part of GENERIC systems can generally not contribute first-order terms to the evolution equation. Consequently, we were able to exploit the extensive body of work on hydrodynamic Poisson systems, including recent classification results for low numbers of spatial dimensions and low numbers of dependent variables to classify combinations of differential operators and Hamiltonian functionals, yielding conditions on the Hamiltonian that lead to or preclude hyperbolicity of the resulting PDEs.Second, we developed a family of superconvergent, arbitrary order reversible-irreversible numerical integrators based on the discontinuous Galerkin (dG) timestepping method and tested it both for GENERIC ODEs and PDEs. The goal was to obtain a method which combines the advantages of reversible integrators for reversible problems and the favorable properties of the dG-methods for irreversible and stiff problems such as the heat equation. Based on a simple and easily computed heuristic (the local free energy dissipation rate) our method is able to recover the characteristic L-stability and superconvergence of the dG-method for purely irreversible problems, while also retaining A-stability, time-reversibility and gaining an increased-by-one order of convergence for purely reversible problems. For intermediate problems, the qualitative behavior of the solution (as measured in the free energy dissipation rate) is generally better captured for large timesteps by our method than by either the purely reversible or standard (purely irreversible) dG-method, while the quantitative behavior of the solution converges towards the higher-order purely reversible dG-method for decreasing timestep sizes., Published by RWTH Aachen University, Aachen

Dissertation, Rheinisch-Westfälische Technische Hochschule Aachen, 2021; Aachen : RWTH Aachen University 1 Online-Ressource : Illustrationen (2021). = Dissertation, Rheinisch-Westfälische Technische Hochschule Aachen, 2021, Electron probe microanalysis (EPMA) is a well-established technique to image the mass concentration of the chemical elements inside a material probe on the micro- to nanometer scale. The material image follows from solving the inverse problem of finding the mass concentration fields that best reproduce characteristic X-Rays intensities measured in an electron microscope for a given synthetic model of characteristic X-Ray emission under excitation by a focused electron beam. Within the last decades significant improvements in the spatial resolution of EPMA were obtained by instrumental enhancements, while the models of characteristic X-Ray emission andinversion procedures essentially remained unchanged. Since classical analytical models of characteristic X-Ray emission, like ZAF and φ(ρz), assume either a homogeneous or a multi-layered material structure, the spatial resolution of EPMA is essentially limited by the size of the volume where characteristic X-Rays are produced in interactions between beam electrons and X-Rays with material atoms. In this thesis we develop a deterministic model of characteristic X-Ray emission for EPMA that bases on deterministic modelling of electron transport using moment models of the Boltzmann equation of electron transport in continuous slowing down (CSD) approximation. Modelling the transport of beam electrons allows us to omit drastic a-priory assumptions on the material structure, which constitutes the first step towards high-resolution EPMA through reconstruction of material structures on scales smaller than the interaction volume. We give an introduction to classical EPMA, motivate our approach to high-resolution EPMA as well as give an overview of the physical processes involved in the emission of characteristic X-Rays before formulating our model of characteristic X-Ray emission as a function of the fluence of beam electrons. We then motivate the method of moments as a technique to derive models of electron transport that balance accuracy and computational cost of numerical solutions - an essential ingredient for the practicability of high-resolution EPMA. In this thesis we consider moment models for electron transport from two families: First and second order minimum entropy (MN) moment models, M1 and M2, and the classic spherical harmonic (PN) approximation up to moderately high moment order N as well as PN with filtering (FPN). Besides motivating and deriving the MN and PN moment equations of electron transport in CSD approximation, two important parts of this thesis are the explicit approximation of the M2 closure to avoid computationally expensive numerical minimization schemes, and the formulation of compatible and energy stable strong-form formulations of boundary conditions for PN equations. Numerical schemes tailored to M1, M2 and (F)PN equations are given in the appendix. We conduct several numerical experiments in which we verify the accuracy of the considered moment models of electron transport by comparison with Monte-Carlo simulation of electron trajectories as well as experimental data, compare our moment-based model of characteristic X-Ray generation to classical φ(ρz) models and demonstrate the efficiency of moment models in terms of accurate and noise-free computation of characteristic X-Ray generation. From a first systematic comparison of the different moment models of electron transport considered here, we found the P N approximation with appropriate filtering is the most attractive moment model for our purpose: The simple structure of P N equations provides flexibility in accuracy by adapting the moment order N at acceptable increase of computational cost and with appropriate filtering the fluence of beam electrons is captured very accurately even in challenging situations like a very narrow electron beam or when changes in material properties are very sharp. On the example of our (F)PN-based k-ratio model we furthermore give an outlook on the development of efficient iterative inversion procedures using gradient-based minimization techniques together with the adjoint-state method for efficient gradient computation. To demonstrate the iterative reconstruction of fine material structures we also present results of a first numerical high resolution EPMA test case., Published by RWTH Aachen University, Aachen

Dissertation, RWTH Aachen University, 2020; Aachen 1 Online-Ressourece (v, 151 Seiten) : Illustrationen, Diagramme (2020). = Dissertation, RWTH Aachen University, 2020, In this doctoral thesis, efficient particle methods describing phase transition and multi-scale fluid flow far from equilibrium are devised. In the first part of this thesis, the short and long-range interactions are modeled through a continuous stochastic process and a Poisson-type partial differential equation, respectively, which allows the use of efficient numerics in practice. In particular, a Fokker-Planck type equation is devised to model the transition of probability measure associated with the underlying jump process of Enskog equation. Hence, as the main advantage over the direct Monte Carlo solution algorithms, the cost of velocity evolution can be decoupled from density and temperature since the numerical cost of It\={o} process associated with Fokker-Planck equation scales only with the number of computational particles. Furthermore, an efficient solution algorithm approximating the exact Vlasov integral, i.e., a kernel that describes long-range interactions of dense gases and liquids at moderate densities, was devised. The main idea is to relate the long-range molecular potential to the fundamental solution of a well-posed elliptic partial differential equation, which allows the use of efficient Poisson solvers. Therefore, the challenges associated with the numerics of local, long-tail, and high dimensional Vlasov integral for the interior of the simulation domain is avoided by introducing an accurate global solution at a lower cost. The devised models were tested against the benchmark in several test cases, such as the lid-driven cavity, Couette flow, evaporation to vacuum, and inverted temperature gradients. Reasonable accuracy at lower cost was attained in comparison with the direct Monte Carlo solution algorithm.In the second part of this thesis, the coupling of continuum conservation laws with a kinetic description of fluid flows far from the equilibrium is investigated. The mapping from one scale to another raises several challenges. Although the moments of distribution can be estimated using samples simply by the law of large numbers subject to a statistical noise, the inverse is ill-posed, which can be resolved by restricting the solution to the one that minimizes Shannon entropy, i.e., maximum entropy distribution (MED). The numerical challenges in the constrained optimization problem of MED motivated this research. As the first step, a high-dimensional regression method based on Gaussian process is devised to provide an accurate and efficient estimate of MED for a desired space of moments. Then, an all particle solution algorithm is devised to couple approximated solution of the Navier-Stokes-Fourier system of equations, e.g., Smoothed-Particle-Hydrodynamics, with the one for the Boltzmann equation, e.g., direct simulation Monte Carlo. The trained Gaussian process was tested in predicting the exact MED solution in several cases, such as the relaxation of distribution to equilibrium governed by Boltzmann equation in a homogeneous setting. Furthermore, the devised hybrid solution algorithm was tested by simulating the Sod's shock tube. Overall, reasonable accuracy compared with the benchmark with a speedup of two orders of magnitude was obtained., Published by Aachen

Dissertation, RWTH Aachen University, 2019; Aachen 1 Online-Ressource (xvi, 194 Seiten) : Illustrationen (2019). = Dissertation, RWTH Aachen University, 2019, State of a gas, for all flow regimes, is accurately described by the Boltzmann equation (BE) which governs the evolution of a phase density functional. Presently, out of all available numerical methods, Direct Simulation Monte Carlo (DSMC) solves the BE with highest fidelity. However, DSMC method is expensive for low Mach number flows which motivates the search for other deterministic methods for solving the BE. An appealing deterministic method is a Galerkin type approach which involves approximating the BE's solution in some finite dimensional space. Present work is devoted to developing one of such Galerkin method which is based upon Grad's expansion along the velocity space and a continuous Galerkin type discretization in the physical space. For low Mach number gas flows, we linearise the BE around a local equilibrium distribution function. To approximate the solution to the linearised BE, we develop a Galerkin method which preserves: (i) mass, momentum and energy conservation, (ii) Galelian invariance and, (iii) L2-entropy stability. Many previous works have discussed techniques to preserve the first two properties and in Grad's expansion both of these properties are preserved due to the structure of Hermite polynomials. In the present work, along with preserving the first two properties, we mainly focus on preserving L2-stability for both the velocity and physical space discretization. Solution to the linearised BE lives on a seven dimensional space: three dimensional velocity space, three dimensional physical space and one dimensional temporal space. To develop our Galerkin method, we start with discretizing the velocity space through Grad's Hermite polynomials and we find that the L2-stability of such a discretization solely depends upon the entropy flux across boundaries. This allows us to ensure entropy stability through a proper design of boundary conditions. We design these boundary conditions for both inflow/outflow and solid-wall boundaries. Next, we equip our velocity space discretization with an entropy stable continuous Galerkin spatial discretization. Since a continuous Galerkin method does not allow for discontinuities across cell boundaries, entropy flux across domain's boundary remains as the only source of entropy growth. Therefore, entropy stability of spatial discretization requires a proper boundary discretization. We use a weak boundary discretization to attain entropy stability and we present three different ways of doing so. We compare all of these three approaches through numerical experiments. Stability of a numerical scheme does not only provide a robust numerical implementation but it also allows for an a-priori convergence analysis. We conduct an a-priori convergence analysis for Grad's Hermite expansion where we use its stability to develop error bounds. Under regularity assumptions on linearised BE's solution we develop explicit convergence rates. We confirm the presented convergence rates through numerical experiments involving several benchmark problems. Solving a kinetic equation with a deterministic method is computationally expensive. This motivates developing a deterministic method where both the velocity space and spatial discretization adapts such that computational resources are allocated only where they are needed. We develop such a deterministic method with the help of moment approximations where we change the order of moment approximation and the spatial grid resolution, locally. We focus on steady-state problems and we use goal oriented adjoint based a-posteriori error prediction. With the help of numerical experiments, we compare the convergence behaviour of an adaptive and a uniform deterministic method., Published by Aachen

Dissertation, RWTH Aachen University, 2018; Aachen 1 Online-Ressource (xvi, 330 Seiten) : Illustrationen (2018). = Dissertation, RWTH Aachen University, 2018, The direct simulation of the acoustic far-field generated by turbulent flows comprises very different scales, from the Kolmogorov scale for the turbulent flow to the far-field of acoustics. The disparity of these scales is further increased if porous materials are introduced as silencers. In order to allow for simulations of real world problems, a coupled approach applying appropriate solvers to the turbulent near-field and the acoustic far-field is required. This thesis is part of an effort applying a Lattice-Boltzmann solver solving the weakly compressible Navier-Stokes equations to the turbulent flow and a very-high-order Discontinuous Galerkin solver solving the linearized Euler equations to the acoustic far-field. A two-step coupling methodology, reducing the viscosity on the Lattice-Boltzmann level with subsequent coupling to macroscopic quantities on the Discontinuous Galerkin level, and corresponding building blocks are proposed. For this, consistent Lattice-Boltzmann methods for the linearized Navier-Stokes and lineraized Euler equations are developed for compressible flows with vanishing background velocity and isothermal flows with and without background velocity. A priori stability results are found for compressible and isothermal flows with vanishing background velocity and for inviscid, isothermal flows with background velocity and backed by numerical results. Additional novel Lattice-Boltzmann methods for isothermal flows with background velocity are presented. A consistent approach for gradual reduction of the viscosity on the Lattice-Boltzmann level is derived and numerically confirmed. A strategy for approximation of spatial derivatives of macroscopic from mesoscopic quantities is presented but found to suffer from problematic error constants., Published by Aachen

RWTH Aachen University, Diss., 2017; Aachen, 1 Online-Ressource (x, 210 Seiten) : Illustrationen, Diagramme(2017). = RWTH Aachen University, Diss., 2017, Hyperbolic moment equations for the simulation of rarefied gas flows are derived and solved using appropriate numerical schemes in this thesis.The numerical simulation of rarefied gases requires the solution of the Boltzmann equation as standard models like the Euler equations are invalid in the rarefied regime, which is characterized by moderate to large Knudsen numbers. An accurate direct discretization and solution of the Boltzmann equation needs many variables due to the high dimension of the phase space and is thus computationally expensive. It is the aim of this thesis to investigate efficient yet accurate moment models for the solution of the Boltzmann equation. The distribution function of the Boltzmann equation is expanded in velocity space resulting in a system of equations for the expansion coefficients, which is called moment system and can be seen as an extension of the Euler equations.However, standard approaches yield unstable solutions due to the lack of hyperbolicity, which has been a major drawback of moment models. The main contribution of this thesis is the derivation of new hyperbolic moment models and the explanation from different viewpoints that allow for a better understanding of the new models, called Quadrature-Based Moment Equations. The analytical properties of the new models are investigated, hyperbolicity is proven and explicit equations are derived.After the detailed one-dimensional derivation, an extension to the multi-dimensional case is performed in order to obtain equations for the simulation of more physically relevant test cases including a rotationally invariant version of the Quadrature-Based Moment Equations.Each hyperbolic moment model contains at least one equation that is only given in non-conservative form, which renders the whole system of equations a partially-conservative moment system. We study appropriate numerical schemes for the computation of the non-conservative products and develop an extension to a high resolution scheme on two-dimensional unstructured grids.The numerical schemes as well as the hyperbolic model equations are successfully used in various test cases that demonstrate a robust solution by the numerical schemes and accurate results of the new moment models. The new hyperbolic moment equations yield better accuracy than existing models and have a larger range of applicability due to their global hyperbolicity as shown by one-dimensional as well as two-dimensional numerical simulations., Published by Aachen

Dissertation, RWTH Aachen University, 2017; Aachen, 1 Online-Ressource (x, 163 Seiten) : Illustrationen, Diagramme (2018). = Dissertation, RWTH Aachen University, 2017, In this thesis we are interested in numerically solving conservation laws with high-order finite volume methods. Hyperbolic systems of partial differential equations are especially challenging since even smooth initial flows may develop discontinuities in finite time. Naively discretizing such flows with high-order schemes may lead to undesired oscillations at discontinuities. First-order methods do not encounter this problem since shocks are smeared out. Nevertheless, high-order schemes are in demand because they have the advantage of reaching a fixed error bound on coarser grids than low-order methods. This reduces the overall computational time and thus the total cost. Combining the advantages of several methods, limiter functions change from high- to low-order whenever necessary. This avoids oscillations at discontinuities while maintaining high-order accuracy at smooth parts of the solution. Thus, the resulting schemes are applicable to physically relevant problems which often contain smooth parts as well as large gradients, discontinuities, or shocks. The aim of this work is the development of third-order finite volume methods by improving building blocks of the method. This means, we identify main routines in the finite volume framework and present new concepts for improving their performance. We focus on two building blocks. First, the high-order reconstruction of interface values using limiter functions. Second, the numerical flux function, also referred to as Riemann solver. The former is crucial for the order of accuracy of the solution while the latter determines the amount of dissipation added to the scheme. We develop a new third-order accurate reconstruction function for the spatial approximation of hyperbolic conservation laws. This reconstruction switches between first- and third-order, resulting in a scheme which is high-order accurate in smooth parts of the solution, does not create oscillations at discontinuities, and avoids extrema clipping as encountered by total variation diminishing (TVD) methods. The novel limiter only needs information from the cell of interest and its nearest neighbors, thus keeping the stencil as compact as possible for obtaining third order accuracy. Furthermore, the reconstruction remains in the traditional second-order framework, easing the implementation of the limiter in existing codes. Finally, a decision criterion without artificial parameters is incorporated in the limiter. This decision criterion distinguishes shocks and large gradients from extrema, thus ensuring accurate shock capturing. The obtained reconstructions at each side of the cell boundaries are then inserted into the numerical flux function which solves local Riemann problems. Many numerical flux functions, also referred to as Riemann solvers, have been developed over the last decades. However, most classical solvers add too much dissipation to the scheme such that discontinuities are smeared out. On the other side of the spectrum, Riemann solvers that do not add too much dissipation need information on the eigen structure which is costly to compute for large systems. There is the need for new Riemann solvers that avoid solving for the eigen system and still reproduce all waves of the system with less dissipation than classical methods such as Rusanov and Harten-Lax-van Leer (HLL). We present a hybrid family of Riemann solvers, requiring only an estimate of the globally fastest wave speeds in both directions. Thus, the new solvers are particularly efficient for large systems of conservation laws when no explicit expression for the eigen system is available or expensive to compute. For the validation of the developed schemes we conduct a series of numerical experiments. First, we demonstrate that the novel high-order limiter function obtains the desired third-order accuracy for smooth solutions. Test cases includes mooth and discontinuous linear transport, Euler equations, and ideal magneto hydrodynamics (MHD). Problems are presented in one and two space dimensions, on uniform as well as non-uniform grids and with adaptive mesh refinement. In a second step, the hybrid family of Riemann solvers is tested in a first-order framework. Here, we show that the newly developed solvers induce less dissipation than schemes with comparable input data. This leads to sharper gradients and less smearing at discontinuities. Numerical examples contain linear transport, Euler equations, ideal MHD, as well as the regularized 13-moment equations (R13).Finally, both parts of this work are combined to obtain third-order accurate results. We reconstruct using the novel third-order limiter and insert there constructed interface values into the hybrid family of Riemann solvers. For all numerical examples, we also implement comparable methods to ascertain the quality of our schemes. The solutions obtained with the newly developed methods indeed indicate better or equally-good results and an excellent performance., Published by Aachen

RWTH Aachen University, Diss., 2016; Aachen, 1 Online-Ressource (xii, 199 Seiten) : Illustrationen, Diagramme(2017). = RWTH Aachen University, Diss., 2016, The method of moments provides a flexible mathematical framework to derive reduced-order models for the approximation of the kinetic Boltzmann equation. This thesis investigates moment equations based on the principle of entropy maximization to describe moderately rarefied gas and plasma flows in the transition regime between the collision dominated continuum and the free molecular flow regime.The maximum-entropy system has favorable mathematical features: The resulting system of partial differential equations is in conservative form, satisfies an H-theorem and is symmetric hyperbolic. However, the robust and efficient numerical solution of entropy-based moment closures is challenging: First, the maximum-entropy closure can have a singularity in the closing flux around the equilibrium distribution, rendering initial value problems with data in local thermodynamic equilibrium questionable. Second, the Hessian matrix of the Newton method used in the dual minimization problem for the Lagrange parameters is arbitrarily ill-conditioned. Third, moments of the maximum-entropy distribution function are in general not available in closed form and have to be evaluated numerically, which can result in excessive run times.The first issue can be avoided by bounding the velocity domain. Numerical examples show that enlarging the velocity domain leads to smaller sub-shocks, i.e., unphysical discontinuities in the continuous shock-structure problem. To study the effect of the singularity in the closing flux, a closed-form closure for a simplified toy model problem is considered. Numerical results demonstrate that the sub-shock in the continuous shock-structure problem can be mitigated by non-linear closures and eventually removed by a singularity in the closing flux.Several numerical examples are provided for the 35-moment system in slab geometry, showing promising results for shock-structure and Riemann problems. The use of an adaptive basis method for the dual minimization problem allows the robust simulation of strongly nonequilibrium processes.To reduce the excessive computational run times of the maximum-entropy closure, high-performance implementations of the numerical integration algorithms are developed for multi-core processors and graphics cards. Additionally, new efficient explicit and semi-implicit time-integration schemes based on a formulation in the Lagrange parameters of the dual minimization problem are presented., Published by Aachen