This thesis provides a framework to simulate non-equilibrium gas flows using the finite element method within the FEniCS computing platform. The main model equations, i.e. the R13 equations, are introduced after a motivational discussion about the classical models, given by Navier-Stokes and Fourier. The resulting system of equations is simplified to obtain a set of steady-state and linearized balance laws for two-dimensional domains. During a validation process of the numerical method with exact solutions, particular focus is put on the intuitive implementation using the tensor capabilities of FEniCS. This allows having an almost one-to-one correspondence between the mathematical formulation and the implemented source code. A documented and validated solver is developed and published. This solver allows simulating gas flows for arbitrarily shaped two-dimensional geometries using a variety of boundary conditions. In order to justify the use of extended model equations for gas flows with moderate Knudsen number, typical examples, with occurring rarefaction effects, are presented and solved. In these application cases, the Knudsen paradox and a thermal transpiration flow are observed. Masterarbeit, RWTH Aachen University, 2019; Aachen : RWTH Aachen University 1 Online-Ressource : Illustrationen (2023). doi:10.18154/RWTH-2023-12262 = Masterarbeit, RWTH Aachen University, 2019 Published by RWTH Aachen University, Aachen

This thesis presents the construction and analysis of scalable preconditioning strategies for the linear Schrödinger eigenvalue problem with periodic potentials in anisotropic structures. As only some dimensions of the computational domain expand to infinity, the resulting eigenvalue gap between the first and second eigenvalue vanishes, posing a significant challenge for iterative solvers. For these iterative eigenvalue solvers, we provide a quasi-optimal shift-and-invert preconditioning strategy such that the iterative eigenvalue algorithms converge in constant iterations for different domain sizes. In its analysis, we derive an analytic factorization of the eigenpairs and use directional homogenization to analyze the asymptotic behavior. The resulting easy-to-calculated unit cell problem can be used within a shift-and-invert preconditioning strategy. This approach leads to a uniformly bounded number of eigensolver iterations. Numerical examples illustrate the effectiveness of this quasi-optimal preconditioning strategy if direct solvers are used since the shifting strategy, by definition, leads to a smaller eigenvalue for the resulting shifted operator, which, in turn, results in a high condition number. We also provide a two-level domain decomposition preconditioner for iterative linear solvers to overcome this issue. As the calculation of the quasi-optimal shift already offered the solution to a spectral cell problem as limiting eigenfunction, it is logical to use it as a generator to construct a coarse space. Indeed, it is the case that the resulting two-level additive Schwarz preconditioner is independent of the domain's anisotropy since we obtain a condition number bound using the theory of spectral coarse spaces despite the need for only one basis function per subdomain for the coarse solver. We provide several numerical examples illustrating the effectiveness of both methods separately and combine them in the end to show their combined scalability. Dissertation, RWTH Aachen University, 2024; Aachen : RWTH Aachen University 1 Online-Ressource : Illustrationen (2024). doi:10.18154/RWTH-2024-06163 = Dissertation, RWTH Aachen University, 2024 Published by RWTH Aachen University, Aachen

All P3 basis functions and an attached WolframScript to generate the same plots for arbitrary Pm elements.

In 2008, a survey with simulation analysts through a wide range of disciplines presented an overall time percentage of 73% spent for meshing-related tasks compared to a much smaller time needed for actual simulations. For many years, automated hexahedral meshing has been considered as the holy grail within the range of mesh generation approaches, however without having definitive success for now. Compared to the relatively easy-to-obtain tetrahedral meshes, hexahedra-based meshes allow for up to ten times fewer cells and therefore decrease simulation times. This thesis focuses on the meshing of arbitrary geometries in an automatic fashion using the mesh generation software snappyHexMesh provided by the open-source licensed computational fluid dynamics simulation framework OpenFOAM. Within this thesis, an automated workflow based on geometry heuristics is proposed to generate all necessary input files for the mesh generator. The resulting meshes are then used in the context of electrothermal simulations of natural convection. The general process of heat transfer involves several mechanisms including conduction, convection and radiation. The boundary regions of the meshes, therefore, have special requirements. Luckily, snappyHexMesh provides mechanisms to also automatically insert boundary layers in these critical regions. A study of an electrical dry transformer geometry including a large air box and very thin isolation layers acts as an industrial example to test the automatic mesh generation workflow. These test cases showed the difficulty of an automatic mesh generation for arbitrary geometries with large dimension variation resulting in a large number of cells because of the isotropic octree-based refinement approach of snappyHexMesh. Bachelorarbeit, RWTH Aachen University, 2018; Aachen : RWTH Aachen University 1 Online-Ressource : Illustrationen (2023). doi:10.18154/RWTH-2023-12261 = Bachelorarbeit, RWTH Aachen University, 2018 Published by RWTH Aachen University, Aachen

Accelerating iterative eigenvalue algorithms is often achieved by employing a spectral shifting strategy. Unfortunately, improved shifting typically leads to a smaller eigenvalue for the resulting shifted operator, which in turn results in a high condition number of the underlying solution matrix, posing a major challenge for iterative linear solvers. This paper introduces a two-level domain decomposition preconditioner that addresses this issue for the linear Schrödinger eigenvalue problem, even in the presence of a vanishing eigenvalue gap in non-uniform, expanding domains. Since the quasi-optimal shift, which is already available as the solution to a spectral cell problem, is required for the eigenvalue solver, it is logical to also use its associated eigenfunction as a generator to construct a coarse space. We analyze the resulting two-level additive Schwarz preconditioner and obtain a condition number bound that is independent of the domain's anisotropy, despite the need for only one basis function per subdomain for the coarse solver. Several numerical examples are presented to illustrate its flexibility and efficiency. 26 pages, 7 figures, 2 tables

This work was supported by the German Research Foundation (DFG) under project 411724963. ddEigenLab.jl: Domain-Decomposition Eigenvalue Problem Lab

This paper provides a provably quasi-optimal preconditioning strategy of the linear Schrödinger eigenvalue problem with periodic potentials for a possibly non-uniform spatial expansion of the domain. The quasi-optimality is achieved by having the iterative eigenvalue algorithms converge in a constant number of iterations for different domain sizes. In the analysis, we derive an analytic factorization of the spectrum and asymptotically describe it using concepts from the homogenization theory. This decomposition allows us to express the eigenpair as an easy-to-calculate cell problem solution combined with an asymptotically vanishing remainder. We then prove that the easy-to-calculate limit eigenvalue can be used in a shift-and-invert preconditioning strategy to bound the number of eigensolver iterations uniformly. Several numerical examples illustrate the effectiveness of this quasi-optimal preconditioning strategy. 29 pages, 9 figures, 2 tables

We present a mixed finite element solver for the linearized regularized 13-moment equations of non-equilibrium gas dynamics. The Python implementation builds upon the software tools provided by the FEniCS computing platform. We describe a new tensorial approach utilizing the extension capabilities of FEniCS’ Unified Form Language to define required differential operators for tensors above second degree. The presented solver serves as an example for implementing tensorial variational formulations in FEniCS, for which the documentation and literature seem to be very sparse. Using the software abstraction levels provided by the Unified Form Language allows an almost one-to-one correspondence between the underlying mathematics and the resulting source code. Test cases support the correctness of the proposed method using validation with exact solutions. To justify the usage of extended gas flow models, we discuss typical application cases involving rarefaction effects. We provide the documented and validated solver publicly.