Patch-Recovery Filters for Curvature in Discontinuous Galerkin-Based Level-Set Methods

Numerical Meth Engineering

2016

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102

This work discusses a discontinuous Galerkin (DG) discretization for two-phase flows. The fluid interface is represented by a level set, and the DG approximation space is adapted such that jumps and kinks in pressure and velocity fields can be approximated sharply. This adaption of the original DG space, which can be performed 'on-the-fly' for arbitrary interface shapes, is referred to as extended discontinuous Galerkin. By combining this ansatz with a special quadrature technique, one can regain spectral convergence properties for low-regularity solutions, which is demonstrated by numerical examples. This work focuses on the aspects of spatial discretization, and special emphasis is devoted on how to overcome problems related to quadrature, small cut cells, and condition number of linear systems. Temporal discretization will be discussed in future works.interface were investigated by Babuska [2] as well as by Barrett and Elliott [3]. For the mathematical properties, that is, convergence rate and spectral properties of the matrix, the technique of interface representation should have no effect. However, especially for moving interfaces, one typically wants to avoid repetitive remeshing, leading to so-called extended finite element methods (XFEM). A very prominent work in this field, although in the domain of solid mechanics, was presented by Moës et al. [4].The present work is based on the discontinuous Galerkin (DG) method. In cells that are cut by the fluid interface, the standard DG space is extended in order to provide separate degrees of freedom (DOF) for both phases. One difficulty is that one has to integrate accurately over an interface that is only known implicitly and can have an almost arbitrary shape. This is solved by a special procedure for numerical integration, the so-called hierarchical-moment-fitting (HMF) [5] that eliminates the need for reconstructing the interface. On this basis, we demonstrate a DG method for which we can experimentally show a convergence order of approximately k C 1 for velocity and k for pressure, where k denotes the total polynomial degree of the velocity approximation. Many names for this idea seem at hand, for example, cut-cell DG or unfitted DG [6]. In reminiscence of XFEM [4], we prefer the name extended discontinuous Galerkin (XDG). While the idea seems simple at first glance, it introduces a bunch of technical difficulties, some of which are addressed in this work.This work represents the second paper in a series of publications on the road to a fully transient, 3D XDG-multiphase solver. Within the first work [5], the HMF procedure was developed. Because an XDG method like the one proposed in this work can only be 'h kC1 -accurate' if the quadrature also is sufficiently accurate, this is a necessary prerequisite for this work. For the actual work, one simplification we make is to consider only polynomial level set functions in C 1 . /. Such a choice allows us to evaluate the curvature of the fluid interface, which is required to describe the pressure jump due to sur...

Section: Discussionmentioning

confidence: 98%

“…We propose the following discretization of the multiphase Navier-Stokes problem (3,4) with jump conditions (5,6) and boundary conditions (7) in the extended DG space: find…”

Section: The Spatial Discretization Of the Two-phase Problemmentioning

confidence: 99%

Section: Numerical Studies On Condition Numbermentioning

confidence: 99%

mentioning

confidence: 99%

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Extended discontinuous Galerkin methods for two‐phase flows: the spatial discretization

Numerical Meth Engineering

2016

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102

“…Note that a quadratic level-set formulation, for example, 𝜑 = x 2 + y 2 − r 2 , would allow an exact approximation of the interface in P 2 (𝔎 h ), so no velocities are generated, see Kummer and Warburton. 47 In order to quantify the discretization error Smolianksi 48 proposed the following setup. The droplet with radius r = 0.25 is set in the middle of the computational domain of Ω = [−0.5, 0.5] × [−0.5, 0.5], where all boundaries describe a wall boundary condition.…”

Section: Capillary Wavementioning

confidence: 99%

On a marching level‐set method for extended discontinuous Galerkin methods for incompressible two‐phase flows: Application to two‐dimensional settings

Smuda

Numerical Meth Engineering

2021

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In this work a solver for two-dimensional, 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 subcell 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 in a 2D setting including capillary waves, an oscillating droplet and the rising bubble benchmark.

The extended discontinuous Galerkin method adapted for moving contact line problems via the generalized Navier boundary condition

Smuda

Numerical Methods in Fluids

2021

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In this work, an extended discontinuous Galerkin (extended DG/XDG also called unfitted DG) solver for two‐dimensional flow problems exhibiting moving contact lines is presented. The generalized Navier boundary condition is employed within the XDG discretization for the handling of the moving contact lines. The spatial discretization is based on a symmetric interior penalty method and the numerical treatment of the surface tension force is done via the Laplace–Beltrami formulation. The XDG method adapts the approximation space conformal to the position of the interface and allows a sub‐cell accurate representation within the 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. No adaption of the level‐set evolution algorithm is needed for the extension to moving contact line problems. The developed solver is validated against typical two‐dimensional contact line driven flow phenomena including droplet simulations on a wall and the two‐phase Couette flow.