Utilization of the Brinkman Penalization to Represent Geometries in a High-Order Discontinuous Galerkin Scheme on Octree Meshes

Nikhil Anand,Neda Ebrahimi Pour,Harald Klimach,Sabine Roller

doi:10.3390/sym11091126

Nikhil Anand, Neda Ebrahimi Pour + Show 2 more

Open Access

https://doi.org/10.3390/sym11091126

Copy DOI

Journal: Symmetry	Publication Date: Sep 5, 2019
Citations: 5	License type: CC BY 4.0

Affiliation: University of Siegen

Abstract

We investigate the suitability of the Brinkman penalization method in the context of a high-order discontinuous Galerkin scheme to represent wall boundaries in compressible flow simulations. To evaluate the accuracy of the wall model in the numerical scheme, we use setups with symmetric reflections at the wall. High-order approximations are attractive as they require few degrees of freedom to represent smooth solutions. Low memory requirements are an essential property on modern computing systems with limited memory bandwidth and capability. The high-order discretization is especially useful to represent long traveling waves, due to their small dissipation and dispersion errors. An application where this is important is the direct simulation of aeroacoustic phenomena arising from the fluid motion around obstacles. A significant problem for high-order methods is the proper definition of wall boundary conditions. The description of surfaces needs to match the discretization scheme. One option to achieve a high-order boundary description is to deform elements at the boundary into curved elements. However, creating such curved elements is delicate and prone to numerical instabilities. Immersed boundaries offer an alternative that does not require a modification of the mesh. The Brinkman penalization is such a scheme that allows us to maintain cubical elements and thereby the utilization of efficient numerical algorithms exploiting symmetry properties of the multi-dimensional basis functions. We explain the Brinkman penalization method and its application in our open-source implementation of the discontinuous Galerkin scheme, Ateles. The core of this presentation is the investigation of various penalization parameters. While we investigate the fundamental properties with one-dimensional setups, a two-dimensional reflection of an acoustic pulse at a cylinder shows how the presented method can accurately represent curved walls and maintains the symmetry of the resulting wave patterns.

Highlights

In simulations of fluid motion for engineering scenarios, we generally need to deal with obstacles or containment of a non-trivial shape
In mesh-based schemes, we have two options to represent such geometries: we can try to align the mesh with the geometries, such that the walls build a boundary of the mesh or we try to embed the boundary conditions inside the mesh elements
To investigate the penalization scheme in our discontinuous Galerkin implementation, we first analyzed the fundamental behavior in two one-dimensional setups and considered the scattering at a cylinder in a two-dimensional setup

Summary

Introduction

In simulations of fluid motion for engineering scenarios, we generally need to deal with obstacles or containment of a non-trivial shape. In mesh-based schemes, we have two options to represent such geometries: we can try to align the mesh with the geometries, such that the walls build a boundary of the mesh or we try to embed the boundary conditions inside the mesh elements. The first option eases the formulation of boundary conditions and their treatment in the scheme [1]. The second option avoids the need to adapt the mesh to the, possibly complex, geometry [2]. Aligning the mesh with arbitrary geometries in the first option can become cumbersome for high-order approximations.

Methods

Results

Conclusion