arXiv subjects

Florian Kummer

Publications and source records attributed to Florian Kummer.

13 recordsLinked to original sources

arXiv2026

Comparative study of different quadrature methods for cut elements

The quadrature of cut elements is crucial for all Finite Element Methods that do not apply boundary-fitted meshes. It should be efficient, accurate, and robust. Various approaches balancing these requirements have been published, with some available as open-source implementations. This work reviews these open-sources codes and the methods used. Furthermore, benchmarking examples are developed for 2D and 3D geometries. Implicit and explicit boundary descriptions are available for all models. The different examples test the efficiency, accuracy, versatility, and robustness of the codes. Special focus is set on the influence of the input parameter, which controls the desired quadrature order, on the actual integration error. A detailed comparison of the discussed codes is carried out. The benchmarking allows a conclusive comparison and presents a valuable tool for future code development. All tests are published in an accompanying open-source repository.

cs.CE↗

arXiv2025

A Task Parallel Orthonormalization Multigrid Method For Multiphase Elliptic Problems

Multigrid methods have been a popular approach for solving linear systems arising from the discretization of partial differential equations (PDEs) for several decades. They are particularly effective for accelerating convergence rates with optimal complexity in terms of both time and space. K-cycle orthonormalization multigrid is a robust variant of the multigrid method that combines the efficiency of multigrid with the robustness of Krylov-type residual minimalizations for problems with strong anisotropies. However, traditional implementations of K-cycle orthonormalization multigrid often rely on bulk-synchronous parallelism, which can limit scalability on modern high-performance computing (HPC) systems. This paper presents a task-parallel variant of the K-cycle orthonormalization multigrid method that leverages asynchronous execution to improve scalability and performance on large-scale parallel systems.

math.NA↗

arXiv2025

Employing Continuous Integration inspired workflows for benchmarking of scientific software -- a use case on numerical cut cell quadrature

In the field of scientific computing, one often finds several alternative software packages (with open or closed source code) for solving a specific problem. These packages sometimes even use alternative methodological approaches, e.g., different numerical discretizations. If one decides to use one of these packages, it is often not clear which one is the best choice. To make an informed decision, it is necessary to measure the performance of the alternative software packages for a suitable set of test problems, i.e. to set up a benchmark. However, setting up benchmarks ad-hoc can become overwhelming as the parameter space expands rapidly. Very often, the design of the benchmark is also not fully set at the start of some project. For instance, adding new libraries, adapting metrics, or introducing new benchmark cases during the project can significantly increase complexity and necessitate laborious re-evaluation of previous results. This paper presents a proven approach that utilizes established Continuous Integration tools and practices to achieve high automation of benchmark execution and reporting. Our use case is the numerical integration (quadrature) on arbitrary domains, which are bounded by implicitly or parametrically defined curves or surfaces in 2D or 3D.

cs.SE↗

arXiv2025

Wetting simulation of the porous structure of a heat pipe using an eXtended Discontinuous Galerkin Method and a Parameterized Level-Set

We perform high-order simulations of two-phase flows in capillaries, with and without evaporation. Since a sharp-interface model is used, singularities can arise at the three-phase contact line, where the fluid-fluid interface interacts with the capillary wall. These singularities are especially challenging when a highly accurate, high-order method with very little numerical diffusion is used for the flow solver. In this work, we employ the eXtended Discontinuous Galerkin (XDG) method, which has a very high accuracy but a severe limit regarding e.g., the time-step restriction. To address this challenge and enhance the stability of our numerical method we introduce a novel approach for representing a moving interface in the case of two-phase flows. We propose a global analytical representation of the interface-describing level-set field, defined by a small set of time-dependent parameters. Noteworthy for its simplicity and efficiency, this method effectively addresses the inherent complexity of two-phase flow problems. Furthermore, it significantly improves numerical stability and enables the use of larger time steps, ensuring both reliability and computational efficiency in our simulations. We compare different analytic expressions for level-set representation, including the elliptic function and the fourth-order polynomial, and validate the method against established literature data for capillary rise, both with and without evaporation. These results highlight the effectiveness of our approach in resolving complex interfacial dynamics.

physics.flu-dyn↗

arXiv2024

Cell agglomeration strategy for cut cells in eXtended discontinuous Galerkin methods

In this work, a cell agglomeration strategy for the cut cells arising in the extended discontinuous Galerkin (XDG) method is presented. Cut cells are a fundamental aspect of unfitted mesh approaches where complex geometries or interfaces separating sub-domains are embedded into Cartesian background grids to facilitate the mesh generation process. In such methods, arbitrary small cells occur due to the intersections of background cells with embedded geometries and lead to discretization difficulties due to their diminutive sizes. Furthermore, temporal evolutions of these geometries may lead to topological changes across different time steps. Both of these issues, i.e., small-cut cells and topological changes, can be addressed with a cell agglomeration technique. In this work, a comprehensive strategy for the typical issues associated with cell agglomeration in three-dimensional and multiprocessor simulations is provided. The proposed strategy is implemented into the open-source software package BoSSS and tested with 2- and 3-dimensional simulations of immersed boundary flows.

math.NA↗

arXiv2023

An extended discontinuous Galerkin shock tracking method

In this paper, we introduce a novel high-order shock tracking method and provide a proof of concept. Our method leverages concepts from implicit shock tracking and extended discontinuous Galerkin methods, primarily designed for solving partial differential equations featuring discontinuities. To address this challenge, we solve a constrained optimization problem aiming at accurately fitting the zero iso-contour of a level set function to the discontinuities. Additionally, we discuss various robustness measures inspired by both numerical experiments and existing literature. Finally, we showcase the capabilities of our method through a series of two-dimensional problems, progressively increasing in complexity.

math.NA↗

arXiv2023

Coupling OpenFOAM(R) with BoSSS, a discontinuous Galerkin solver written in C#

In this article, we present the foam-dg project, which provides a bridge between OpenFOAM(R) and the high-order DG (discontinuous Galerkin) framework BoSSS. Thanks to the flexibility of the coupling approach, mixed calculations where some parts of the equation system are solved in OpenFOAM(R) and others are solved in BoSSS are easily possible. This is showcased using the convective Cahn-Hilliard equation, where the Cahn-Hilliard part is solved in BoSSS and the Navier-Stokes part is solved in OpenFOAM(R). The obtained results appear reasonable, though the main focus of this paper is to present and document the foam-dg project rather than on quantitative results.

math.NA↗

arXiv2023

High-Order Numerical Integration on Domains Bounded by Intersecting Level Sets

We present a high-order method that provides numerical integration on volumes, surfaces, and lines defined implicitly by two smooth intersecting level sets. To approximate the integrals, the method maps quadrature rules defined on hypercubes to the curved domains of the integrals. This enables the numerical integration of a wide range of integrands since integration on hypercubes is a well known problem. The mappings are constructed by treating the isocontours of the level sets as graphs of height functions. Numerical experiments with smooth integrands indicate a high-order of convergence for transformed Gauss quadrature rules on domains defined by polynomial, rational, and trigonometric level sets. We show that the approach we have used can be combined readily with adaptive quadrature methods. Moreover, we apply the approach to numerically integrate on difficult geometries without requiring a low-order fallback method.

math.NA↗

arXiv2021

Shear-thinning in Polymer Melts -- Molecular Origins and Hybrid Multiscale Simulations

We investigate the molecular origin of shear-thinning in melts of flexible, semiflexible and rigid oligomers with coarse-grained simulations of a sheared melt. Alignment, stretching and tumbling modes or suppression of the latter all contribute to understanding how macroscopic flow properties emerge from the molecular level. By performing simulations of single chains in a shear flow, we identify which of these phenomena are of collective nature and arise through interchain interactions and which are already present in dilute systems. Building upon these microscopic simulations we identify by means of the Irving-Kirkwood formula the corresponding macroscopic stress tensor for a non-Newtonian polymer fluid. Shear-thinning effects in oligomer melts are also demonstrated by macroscopic simulations of a channel flow. The latter have been obtained by the discontinuous Galerkin method approximating macroscopic polymer flows. Our study confirms the influence of microscopic details in the molecular structure of short polymers such as chain flexibility on macroscopic polymer flows.

cond-mat.soft↗

arXiv2020

An Extended Discontinuous Galerkin Method for High-Order Shock-Fitting

We present a sub-cell accurate shock-fitting technique using a high-order extended discontinuous Galerkin (XDG) method, where a computational cell of the background grid is cut into two cut-cells at the shock position. Our technique makes use of a sharp interface description where the shock front is implicitly defined by means of the zero iso-contour of a level-set function. A novel implicit pseudo-time-stepping procedure is employed to correct the position of the shock front inside the cut background cell by using cell-local indicators, since the position and shape of shock waves are not known a priori for the general, multi-dimensional case. This iterative correction terminates if the shock front has converged to the exact position. The procedure is demonstrated for the test case of a one-dimensional stationary normal shock wave. Furthermore, the underlying sharp interface approach drastically reduces the complexity of the grid handling, since a simple Cartesian background grid can be employed.

math.NA↗

arXiv2020

On a marching level-set method for extended discontinuous Galerkin methods for incompressible two-phase flows

In this work a solver for instationary two-phase flows on the basis of the extended Discontinuous Galerkin (extended DG/XDG) method is presented. The XDG method adapts the approximation space conformal to the position of the interface. This allows a sub-cell accurate representation of the incompressible Navier-Stokes equations in their sharp interface formulation. The interface is described as the zero set of a signed-distance level-set function and discretized by a standard DG method. For the interface, resp. level-set, evolution an extension velocity field is used and a two-staged algorithm is presented for its construction on a narrow-band. On the cut-cells a monolithic elliptic extension velocity method is adapted and a fast-marching procedure on the neighboring cells. The spatial discretization is based on a symmetric interior penalty method and for the temporal discretization a moving interface approach is adapted. A cell agglomeration technique is utilized for handling small cut-cells and topology changes during the interface motion. The method is validated against a wide range of typical two-phase surface tension driven flow phenomena including capillary waves, an oscillating droplet and the rising bubble benchmark.

math.NA↗

arXiv2020

BoSSS: a package for multigrid extended discontinuous Galerkin methods

The software package BoSSS serves the discretization of (steady-state or time-dependent) partial differential equations with discontinuous coefficients and/or time-dependent domains by means of an eXtended Discontinuous Galerkin (XDG, resp. DG) method, aka. cut-cell DG, aka. unfitted DG. This work consists of two major parts: First, the XDG method is introduced and a formal notation is developed, which captures important numerical details such as cell-agglomeration and a multigrid framework. In the second part, iterative solvers for extended DG systems are presented and their performance is evaluated.

math.NA↗

arXiv2015

Patch-recovery filters for curvature in discontinuous Galerkin-based level-set methods

In two-phase flow simulations, a difficult issue is usually the treatment of surface tension effects. These cause a pressure jump that is proportional to the curvature of the interface separating the two fluids. Since the evaluation of the curvature incorporates second derivatives, it is prone to numerical instabilities. Within this work, the interface is described by a level-set method based on a discontinuous Galerkin discretization. In order to stabilize the evaluation of the curvature, a patch-recovery operation is employed. There are numerous ways in which this filtering operation can be applied in the whole process of curvature computation. Therefore, an extensive numerical study is performed to identify optimal settings for the patch-recovery operations with respect to computational cost and accuracy.

physics.comp-ph↗