Mario Mally, Bernard Kapidani, Melina Merkel and Sebastian Schöps. On Dual-Primal Tearing and Interconnecting for 3D Magnetostatics.

Simulating realistic 3D magnetostatic or eddy current problems is challenging since large, possibly badly conditioned, linear equation system must be solved which follow from space discretization. An effective strategy to overcome the inherently long simulation times is to introduce parallel computations and harness HPC architectures. This, in turn, requires scalable domain decomposition approaches for the underlying partial differential equation. In this work, we focus on extending well-established Tearing and Interconnecting (TI) methods [1] to 3D magnetostatics in vector potential formulation. A significant challenge lies in the nontrivial kernel of the Curl-Curl operator. We address this by applying TreeCotree gauging similar as in [2] and [3]. However, we derive rigorous rules for constructing appropriate trees that have not been established until now. The utilization of a global graph, linked to global degrees of freedom, facilitates an explicit tree construction based on edge priorities. Combining the generated tree with a typical dual-primal procedure [2] makes concurrent computations for all subdomains available. In addition to presenting our TI method, we assess the speed and accuracy using an Isogeometric Analysis simulation framework.

[1] C. Farhat and F.-X. Roux, A Method of Finite Element Tearing and Interconnecting and Its Parallel Solution Algorithm. Int. J. Numer. Meth. Eng., Vol. 32, 1991.

[2] W. Yao, J.-M. Jin and P. T. Krein, A Highly Efficient Domain Decomposition Method Applied to 3-D Finite-Element Analysis of Electromechanical and Electric Machine Problems. IEEE Trans. Energ. Convers., Vol. 27, 2012.

[3] B. Kapidani, M. Merkel, S. Sch¨ops and R. V´azquez, Tree-Cotree Decomposition of Isogeometric Mortared Spaces in H(curl) on Multi-Patch Domains. CMAME, Vol. 395, 2022

Irina Shishkina, Florian Kummer and Martin Oberlack. Wetting simulation of the porous structure of a heat pipe using an eXtended Discontinuous Galerkin Method and a Parameterized Level-Set.

An important aspect of modeling a heat pipe involves simulating the processes within the porous structure of the wick. Understanding the transport dynamics at the liquid-vapor interface within a pore and its wetting dynamics is essential for accurately predicting the performance of heat pipes under various operating conditions and for assessing capillary dry-out limitations.

For high-order simulations, we employ the eXtended Discontinuous Galerkin (XDG) method. During simulation, singularities can arise due to the appearance of the three-phase contact line. To address this challenge and enhance the stability of our numerical method, we introduce a parameterized level-set method for interface modelling. This method involves using a level-set evolution equation to describe the interface's movement and defining the level-set function based on several time-dependent parameters. Furthermore, we assume that the interface retains its parametric shape at each time step, with only the time-dependent parameters changing over time.

In our research, the results obtained from the parameterized level-set method were compared with analytical solutions, and they demonstrated good agreement. To understand the impact of varying interface shapes and the rotation of a heat pipe on its efficiency, which is intended for our specific application, we performed calculations varying the interface shape parameters and the centrifugal force.

Muhammed Toprak. Cell agglomeration for cut cells in eXtended discontinuous Galerkin Methods.

Extended discontinuous Galerkin (XDG) methods stand as advanced high-order techniques designed for addressing multiphase problems characterized by jumps and kinks at interfaces. When dealing with interfaces, XDG methods adopt a sharp interface approach and implement a cut cell structure, in which different phases are embedded into a Cartesian grid to facilitate the mesh generation process. However, in such methods, arbitrary small cells occur due to the intersections of background cells with interfaces and lead to discretization difficulties due to their diminutive sizes. Furthermore, topological inconsistencies may arise in different time steps when the embedded phases change in size over time.

To account for these issues associated with small-cut cells and topological changes, a cell agglomeration approach is employed in the XDG method and tailored for large-scale simulations. This approach merges problematic cells with suitable neighbors and has been deemed as a simple and convenient method [1]. However, it is important to note that cell agglomeration becomes challenging in three-dimensional space where there is a high degree of neighborship between cells [2]. Furthermore, it can create cumbersome problems in large computational simulations such as forming agglomeration chains and ineffective parallel information exchange.

In our work, we provide a comprehensive strategy for the typical issues addressed to cell agglomeration in three-dimensional and multiprocessor simulations. The proposed strategy is implemented into the open-source software package BoSSS [3] and tested with large-scale simulations of immersed boundary and two-phase flows.

[1] Qin, Ruibin and Krivodonova, Lilia (2013): A discontinuous Galerkin method for solutions of the Euler equations on Cartesian grids with embedded geometries. In: Journal of Computational Science, 4(1), pp. 24–35. DOI: 10.1016/j.jocs.2012.03.008.

[2] May, Sandra and Streitb¨urger, Florian (2022): DoD Stabilization for non-linear hyperbolic conservation laws on cut cell meshes in one dimension. In: Applied Mathematics and Computation, 419, p. 126854. DOI: 10.1016/j.amc.2021.126854.

[3] Kummer, Florian; Weber, Jens; Smuda, Martin (2021): BoSSS: A package for multi grid extended discontinuous Galerkin methods. In: Computers & Mathematics with Applications, 81, pp. 237–257. DOI: 10.1016/j.camwa.2020.05.001.