Applied and Computational Mathematics (ACoM)

RWTH Aachen University

Schinkelstr. 2

D-52062 Aachen

Germany

Room: 229 (Rogowski Building, 2nd floor)

Office Phone: +49 241 80-98671

Mobil Phone: +49 241 80-98686

Email: theisen@acom.rwth-aachen.de

**10/19 – now**: PhD Candidate at MathCCES

RWTH Aachen University, Aachen, Germany**04/18 – 09/19**: Master of Science in Computational Engineering Science (CES)

RWTH Aachen University, Aachen, Germany**10/14 – 04/18**: Bachelor of Science in Computational Engineering Science (CES)

RWTH Aachen University, Aachen, Germany

**10/17 – 03/18**: Internship: 3D CFD Simulation and Hexahedral Meshing

ABB Corporate Research Center Baden-Dättwil, Switzerland

The solution of eigenvalue problems on long (chain-like) domain is difficult due to the increased computational complexity and an often collapsing spectrum of the considered operator. We therefore want to rewrite the steps of an (possibly non-linear) eigenvalue problem solver (eigensolver) to a sequence of local linear eigenvalue problems. If the domain is the union of simpler sub-domains (say balls in 3D), then classical Schwartz-like domain decomposition algorithms can be applied to have a weakly scalable linear eigenvalue solver for some operators. However, the global solution algorithm convergence behavior also changes dramatically with an increasing domain length. Therefore, efficient preconditioning techniques have to be developed to also take these effects into account.

The possible range of applications includes electronic structure calculations in computational chemistry or quantum mechanical simulations.

**Supervision**: Benjamin Stamm

**Keywords**: Eigenvalue Problems, Schrödinger Operators, Electronic Structure Calculations, Domain Decomposition Methods, Preconditioned Iterative Eigensolvers

This thesis provides a framework to simulate non-equilibrium gas flows using the finite element method within the FEniCS computing platform [1]. 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 can be obtained from [2]. This solver allows simulating gas flows for arbitrary 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.

**Supervision**: Manuel Torrilhon

**Keywords**: R13 equations, FEniCS project, Non-equilibrium gas flows, Finite element method, CIP stabilization

**Download**: 2019_ma_lamberttheisen.pdf

**References**:

[1]: FEniCS Project: Bitbucket Repositories. FEniCS Project. url: https://bitbucket.org/fenics-project/ (visited on 08/28/2019).

[2]: Lambert Theisen and Manuel Torrilhon. fenicsR13: Solver Repository. RWTH Aachen University, 2019. url: https://git.rwth-aachen.de/lamBOO/fenicsR13 (visited on 09/15/2019).

The subject of this seminar thesis is the application of the Shear-Slip Mesh Update Method (SSMUM) for compressible flow simulations. This method allows to simulate systems of moving objects, including fluid and solid regions, in an efficient way if a regular motion is assumed. Compared to other methods, the SSMUM avoids a complete remeshing of the computational domain, that is in general very computationally intensive. This is archived by a partial update of node connectivity in a special update layer between the moving and the static mesh.

The numerical method and its underlying formulation is presented in order to introduce the SSMUM. Previous applications for incompressible flows are discussed. On that basis, a first test case for compressible flows is presented. In order to generate a suitable mesh for this test case, requirements on the mesh generation step are presented and discussed.

**Supervision**: Michel Make

**Keywords**: Shear-Slip Mesh Update Method (SSMUM), Finite Element Method, compressible flow simulation, rotating sub-domains, Deformable-Spatial-Domain/Stabilized Space-Time (DSD/SST) formulation

**Download**: ws18_ces_seminar_theisen.pdf

**References**:

[1]: CATS. url: https://www.cats.rwth-aachen.de.

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 [1] compared to a much smaller time needed for actual simulations. For many years, the automated hexahedral meshing has been considered as the holy grail [2] 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 less cells as stated in [2] and therefore decreasing 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 [3].

Within this thesis, an automated workflow based on geometry heuristics is proposed in order 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. This 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.

**Supervision**: Christoph Winkelmann

**Keywords**: OpenFOAM, snappyHexMesh, Automated Hexahedral Meshing, Natural Convection

**Download**: 2018_ba_lamberttheisen.pdf

**References**:

[1]: S. J. Owen et al. “An Immersive Topology Environment for Meshing”. In: Proceedings of the 16th International Meshing Roundtable. Springer, Berlin, Heidelberg, 2008, pp. 553–577. url: https://link.springer.com/chapter/10.1007/978-3-540-75103-8_31.

[2]: T. Blacker. “Automated Conformal Hexahedral Meshing Constraints, Challenges and Opportunities”. In: Engineering with Computers 17.3 (2001), pp. 201–210. url: https://link.springer.com/article/10.1007/PL00013384.

[3]: H. G. Weller et al. “A tensorial approach to computational continuum mechanics using object-oriented techniques”. In: Computers in Physics 12.6 (1998), pp. 620–631. url: http://powerlab.fsb.hr/ped/kturbo/OpenFOAM/docs/Foam.pdf.

- Iterative Domain Decomposition Methods for Eigenvalue Problems, Master Thesis of Hendrik Borchardt, RWTH Aachen, 2020.
- Using a Spectral Inference Network to Solve the Time-Independent Schrödinger Equation for a Two-Dimensional Hydrogen Atom, Seminar Thesis of Alexander Kristof, RWTH Aachen, 2020.
- Evaluation and Implementation of Schrödinger-Type Eigenvalue Problems in Long Rectangular Domains using the Finite Element Method, CES Project Thesis of Corinna Müller, Matthias Geratz, Celine Heger, Johanna Meyer, RWTH Aachen, 2021.

see @lamBOOO/teaching (Github)

**WS20**: Mathematische Grundlagen III (CES) @ MathCCES, RWTH Aachen, with Prof. Stamm & Prof. Hakon Hoel- Lab PDFs: 08-svd-image-compression.pdf 09a-curves.pdf 09b-shifted-inverse-power-method.pdf 10-qr-algorithm-hessenberg-givens.pdf 11-linesearch-algorithms-gradientdescent-newton.pdf 12-linesearch-stepsize-trust-regions.pdf 13-constrained-optimization-kkt-licq.pdf 14-constrained-optimization-penalty-barrier.pdf
- Lab Codes (Julia): 08-svd-image-compression.jl 09a-curves.jl 09b-shifted-inverse-power-method.jl 10-qr-algorithm-hessenberg-givens.jl 11-linesearch-algorithms-gradientdescent-newton.jl 12-linesearch-stepsize-trust-regions.jl 13-constrained-optimization-kkt-licq.jl 14-constrained-optimization-penalty-barrier.jl

**SS20**: Mathematische Grundlagen IV (CES) @ MathCCES, RWTH Aachen, with Prof. Stamm & Prof. Krumscheid**WS19/20**: Mathematische Grundlagen I (CES) @ MathCCES, RWTH Aachen, with Prof. Stamm & Prof. Krumscheid**SS19**: Mathematische Grundlagen IV (CES) @ MathCCES, RWTH Aachen**WS18/19**: Partielle Differentialgleichungen (CES) @ MathCCES, RWTH Aachen- Material: 2D Poisson FEM Tips and Tricks

Lambert Theisen and Manuel Torrilhon. 2021. **fenicsR13: A Tensorial Mixed Finite Element Solver for theLinear R13 Equations Using the FEniCS Computing Platform**. ACM Trans. Math. Softw. 47, 2, Article 17 (April 2021), 29 pages. 10.1145/3442378. Preprint @ arXiv (Free PDF) Final Version @ Journal (Free PDF through ACM Author-Izer Service)

Lambert Theisen and Benjamin Stamm. 2021. **An Optimal Factorization Preconditioner for Periodic Schrödinger Eigenstates in Anisotropically Expanding Domains**. arXiv. arXiv:2110.14982
. Preprint @ arXiv (Free PDF)

- An Optimal Factorization Preconditioner for Periodic Schrödinger Eigenstates in Non-Uniformly Expanding Domains, ACoM Lunch Seminar, RWTH Aachen, 15 June, 2021.
- Optimal Eigensolvers for Dirichlet Schrödinger Operators with Non-Negative Additively Separable Potentials in Long Domains, MathCCES Lunch Seminar, RWTH Aachen, 30 June, 2020.
- Simulation of Non-Equilibrium Gas Flows Using the FEniCS Computing Platform, Master Thesis Defense, RWTH Aachen, 15 October, 2019.
- Shear-Slip Mesh Update Method for Compressible Flow Simulations Involving Rotating Sub-Domains, CES Seminar Project, RWTH Aachen, 10 April, 2019.
- Automated Boundary Layer Mesh Generation for Simulation of Convective Cooling, Bachelor Thesis Defense, RWTH Aachen, 10 April, 2018.

- Summer school on advanced DD methods, Milano, 24-26. November 2021.

- fenicsR13: A Tensorial Mixed Finite Element Solver for the Linear R13 Equations Using the FEniCS Computing Platform
- ddEigenLab.jl: Domain-Decomposition Methods for Eigenvalue Problems for Long-Chain Domains
- P3 Finite Element Basis Functions on 2-Simplex, doi: 10.6084/m9.figshare.9767021.v1, 2019.