arXiv subjects

Johannes Kromer

Publications and source records attributed to Johannes Kromer.

5 recordsLinked to original sources

arXiv2023

Efficient sequential PLIC interface positioning for enhanced performance of the three-phase VoF Method

This paper presents an efficient algorithm for the sequential positioning, also called nested dissection, of two planes in an arbitrary polyhedron. Two planar interfaces are positioned such that the first plane truncates a given volume from this arbitrary polyhedron and the next plane truncates a second given volume from the residual polyhedron. This is a relevant task in the numerical simulation of three-phase flows when resorting to the geometric Volume-of-Fluid (VoF) method with a Piecewise Linear Interface Calculation (PLIC). An efficient algorithm for this task significantly speeds up the three-phase PLIC algorithm. The present study describes a method based on a recursive application of the Gaussian divergence theorem, where the fact that the truncated polyhedron shares multiple faces with the original polyhedron can be exploited to reduce the computational effort. A careful choice of the coordinate system origin for the volume computation allows for successive positioning of two planes without reestablishing polyhedron connectivity. Combined with a highly efficient root finding, this results in a significant performance gain in the reconstruction of the three-phase interface configurations. The performance of the new method is assessed in a series of carefully designed numerical experiments. Compared to a conventional decomposition-based approach, the number of iterations and, thus, of the required truncations was reduced by up to an order of magnitude. The PLIC positioning run-time was reduced by about 90% in our reference implementation. Integrated into the multi-phase flow solver Free Surface 3D (FS3D), an overall performance gain of about 20% was achieved. Allowing for simple integration into existing numerical schemes, the proposed algorithm is self-contained (example Fortran Module see https://doi.org/10.18419/darus-2488), requiring no external decomposition libraries.

math.NA↗

arXiv2021

Second-order accurate normal reconstruction from volume fractions on unstructured meshes with arbitrary polyhedral cells

This paper introduces a novel method for the efficient second-order accurate computation of normal fields from volume fractions on unstructured polyhedral meshes. Locally, i.e. in each mesh cell, an averaged normal is reconstructed by fitting a plane in a least squares sense to the volume fraction data of neighboring cells while implicitly accounting for volume conservation in the cell at hand. The resulting minimization problem is solved approximately by employing a Newton-type method. Moreover, applying the Reynolds transport theorem allows to assess the regularity of the derivatives. Since the divergence theorem implies that the volume fraction can be cast as a sum of face-based quantities, our method considerably simplifies the numerical procedure for applications in three spatial dimensions while demonstrating an inherent ability to robustly deal with unstructured meshes. We discuss the theoretical foundations, regularity and appropriate error measures, along with the details of the numerical algorithm. Finally, numerical results for convex and non-convex hypersurfaces embedded in cuboidal and tetrahedral meshes are presented, where we obtain second-order convergence for the normal alignment and symmetric volume difference. Moreover, the findings are substantiated by completely new deep insights into the minimization procedure.

math.NA↗

arXiv2021

Third-order accurate initialization of VOF volume fractions on unstructured meshes with arbitrary polyhedral cells

This paper introduces a novel method for the efficient and accurate computation of volume fractions on unstructured polyhedral meshes, where the phase boundary is an orientable hypersurface, implicitly given as the iso-contour of a sufficiently smooth level-set function. Locally, i.e. in each mesh cell, we compute a principal coordinate system in which the hypersurface can be approximated as the graph of an osculating paraboloid. A recursive application of the \textsc{Gaussian} divergence theorem then allows to analytically transform the volume integrals to curve integrals associated to the polyhedron faces, which can be easily approximated numerically by means of standard \textsc{Gauss-Legendre} quadrature. This face-based formulation enables the applicability to unstructured meshes and considerably simplifies the numerical procedure for applications in three spatial dimensions. We discuss the theoretical foundations and provide details of the numerical algorithm. Finally, we present numerical results for convex and non-convex hypersurfaces embedded in cuboidal and tetrahedral meshes, showing both high accuracy and third- to fourth-order convergence with spatial resolution.

math.NA↗

arXiv2021

Face-based Volume-of-Fluid interface positioning in arbitrary polyhedra

We introduce a fast and robust algorithm for finding a plane $Γ$ with given normal $\vec{n}_Γ$, which truncates an arbitrary polyhedron $\mathcal{P}$ such that the remaining sub-polyhedron admits a given volume $α|\mathcal{P}|$. In the literature, this is commonly referred to as Volume-of-Fluid (VoF) interface positioning problem. The novelty of our work is twofold: firstly, by recursive application of the Gaussian divergence theorem, the volume of a truncated polyhedron can be computed at high efficiency, based on summation over quantities associated to the faces of the polyhedron. One obtains a very convenient piecewise parametrization (within so-called brackets) in terms of the signed distance s to the plane $Γ$. As an implication, one can restrain from the costly necessity to establish topological connectivity, rendering the present approach most suitable for the application to unstructured computational meshes. Secondly, in the vicinity of the truncation position s, the volume can be expressed exactly, i.e. in terms of a cubic polynomial of the normal distance to the PLIC plane. The local knowledge of derivatives enables to construct a root-finding algorithm that pairs bracketing and higher-order approximation. The performance is assessed by conducting an extensive set of numerical experiments, considering convex and non-convex polyhedra of genus (i.e., number of holes) zero and one in combination with carefully selected volume fractions $α$ (including $α\approx0$ and $α\approx1$) and normal orientations $\vec{n}_Γ$. For all configurations we obtain a significant reduction of the number of (computationally costly) truncations required for the positioning: on average, our algorithm requires between one and two polyhedron truncations to find the position of the plane $Γ$, outperforming existing methods.

cs.CG↗

arXiv2018

Highly accurate numerical computation of implicitly defined volumes using the Laplace-Beltrami operator

This paper introduces a novel method for the efficient and accurate computation of the volume of a domain whose boundary is given by an orientable hypersurface which is implicitly given as the iso-contour of a sufficiently smooth level-set function. After spatial discretization, local approximation of the hypersurface and application of the Gaussian divergence theorem, the volume integrals are transformed to surface integrals. Application of the surface divergence theorem allows for a further reduction to line integrals which are advantageous for numerical quadrature. We discuss the theoretical foundations and provide details of the numerical algorithm. Finally, we present numerical results for convex and non-convex hypersurfaces embedded in cuboidal domains, showing both high accuracy and thrid- to fourth-order convergence in space.

physics.flu-dyn↗