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

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## Abstract

**:**

## 1. Introduction

## 2. Numerical Method

#### 2.1. The Compressible Navier–Stokes Equation

#### 2.2. The Brinkman Penalization

#### 2.3. The Discontinuous Galerkin Discretization

#### 2.4. The Implicit-Explicit Time Discretization

#### Observation for the Implicit Part

## 3. Results and Discussion

#### 3.1. One-Dimensional Acoustic Wave Reflection

#### 3.2. One-Dimensional Shock Reflection

#### 3.3. Scattering at a Cylinder

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**One-dimensional acoustic wave setup: the center of the initial pressure pulse is located at $x=0.25$ and has an amplitude of $\u03f5={10}^{-3}$. Discretization by 48 elements as denoted by grid lines, and the right half of the domain ($x>0.5$) is penalized. Note that the wall coincides with an element interface.

**Figure 2.**Plot for the pressure profile of the reflected wave at $t=0.5$ for different scaling factors $\beta $. The numerical reference is obtained with a traditional wall boundary condition and a high resolution.

**Figure 3.**Plot of the error in the wave amplitude at $t=0.5$ with decreasing porosity and different scaling factors $\beta $. The error e is given by the relative error in the pulse amplitude after the reflection at the wall.

**Figure 4.**${L}^{2}$-error for a polynomial degree of seven over an increasing number of elements (h-refinement).

**Figure 6.**${L}^{2}$-error for varying the polynomial degree. With a wall just 5% of the element length away from the element surface.

**Figure 7.**Behavior of the error in the reflected acoustic pulse with respect to computational effort. The figure on the left (

**a**) shows the error convergence for various spatial orders over the required memory in terms of degrees of freedom. The right figure (

**b**) shows the same runs, but now over the computational effort in terms of running time in seconds. All simulations were performed on a single node with 12 cores using 12 processes.

**Figure 8.**Different curves represent different discretizations using different scheme orders and a different number of elements. (

**a**) Normalized pressure of the reflected shock wave. (

**b**) Zoom of the reflected shock.

**Figure 9.**Different curves represent different locations of the porous material in the element and the solution when using a no-slip wall. (

**a**) illustrates the normalized pressure of the reflected shock wave, and (

**b**) depicts a zoom-in of the front area of the reflected shock.

**Figure 10.**Test case setup for the wave scattering; only the section containing the cylindrical obstacle, the probing points, and the initial pulse is shown, and the actual computational domain is larger. The cylindrical obstacle is represented by the black circle located at $P(10,10)$. Five observation points $(A,\dots ,E)$ around the obstacle are shown as circles. The initial pulse in pressure is indicated by the black dot with a turquoise circle around it located at $S(14,10)$.

**Figure 11.**Simulation snapshots of pressure perturbations captured at successive points in time. The cylindrical obstacle is visible as a black disk and the probe points surrounding it as white dots. The scale of the pressure perturbation is kept constant for all snapshots.

**Figure 12.**Time evolution of pressure perturbations at all five observation points surrounding the cylinder up to $t=10$. Be aware that the perturbation pressure plotted along the y axis is scaled differently from probe to probe to illustrate the pressure profile better.

Downstream speed of sound | ${c}_{1}$ | 1.0 |

Shock Mach number | $M{a}_{s}$ | 1.2 |

Shock velocity | ${u}_{s}$ | 1.2 |

Downstream density | ${\rho}_{1}$ | 1.0 |

Downstream pressure | ${p}_{1}$ | ${\gamma}^{-1}$ |

Downstream velocity | ${u}_{1}$ | 0.0 |

Isentropic coefficient | $\gamma $ | 1.4 |

Test Case | ${\mathit{p}}^{3}/{\mathit{p}}^{2}$ | Error in ${\mathit{p}}^{3}/{\mathit{p}}^{2}$ in [%] | $\mathbf{\Delta}\mathit{x}$$\phantom{\rule{0.166667em}{0ex}}\mathbf{\xb7}\phantom{\rule{0.166667em}{0ex}}{10}^{-4}$ |
---|---|---|---|

n2048, O(4) | 1.46053873 | 1.19885086 | 32.0161 |

n1024, O(8) | 1.47642541 | 0.12416375 | 13.0319 |

n512, O(16) | 1.47700446 | 0.08499256 | 8.1828 |

n256, O(32) | 1.47714175 | 0.07570497 | 7.6228 |

n128, O(64) | 1.47721414 | 0.07080803 | 6.4032 |

n2000, O(8) | 1.47740998 | 0.05755990 | 7.0317 |

$mi{n}_{error}$ | 1.47806952 | 0.012944346 | $--$ |

**Table 3.**Comparison of simulation results, when the porous material was located in the middle of the element with the exact solution.

Test Case | ${\mathit{p}}^{3}/{\mathit{p}}^{2}$ | Error in ${\mathit{p}}^{3}/{\mathit{p}}^{2}$ in [%] | $\mathbf{\Delta}\mathit{x}$$\phantom{\rule{0.166667em}{0ex}}\mathbf{\xb7}\phantom{\rule{0.166667em}{0ex}}{10}^{-4}$ |
---|---|---|---|

n2049, O(4) | 1.44333420 | 2.36268639 | 25.6373 |

n1025, O(8) | 1.47333865 | 0.33297335 | 21.5178 |

n513, O(16) | 1.47750687 | 0.05100577 | 15.4564 |

n257, O(32) | 1.47751832 | 0.05023153 | 13.0052 |

$mi{n}_{error}$ | 1.47801754 | 0.01646083 | $--$ |

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**MDPI and ACS Style**

Anand, N.; Ebrahimi Pour, N.; Klimach, H.; Roller, S.
Utilization of the Brinkman Penalization to Represent Geometries in a High-Order Discontinuous Galerkin Scheme on Octree Meshes. *Symmetry* **2019**, *11*, 1126.
https://doi.org/10.3390/sym11091126

**AMA Style**

Anand N, Ebrahimi Pour N, Klimach H, Roller S.
Utilization of the Brinkman Penalization to Represent Geometries in a High-Order Discontinuous Galerkin Scheme on Octree Meshes. *Symmetry*. 2019; 11(9):1126.
https://doi.org/10.3390/sym11091126

**Chicago/Turabian Style**

Anand, Nikhil, Neda Ebrahimi Pour, Harald Klimach, and Sabine Roller.
2019. "Utilization of the Brinkman Penalization to Represent Geometries in a High-Order Discontinuous Galerkin Scheme on Octree Meshes" *Symmetry* 11, no. 9: 1126.
https://doi.org/10.3390/sym11091126