# Integration within Fluid Dynamic Solvers of an Advanced Geometric Parameterization Based on Mesh Morphing

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## Abstract

**:**

## 1. Introduction

## 2. RBF Theory Background

- $\phi $ is the selected interpolating radial function;
- $N$ is the total number of contributing source points (also called centers);
- ${x}_{Si}=\left\{{x}_{Si},{y}_{S}{}_{i},{z}_{S}{}_{i}\right\}$ is the vector of source points positions;
- ${\gamma}_{i}={\left\{{\gamma}_{1},\dots ,{\gamma}_{N}\right\}}^{T}$ is a vector of unknown coefficients;
- $h$ is a correction polynomial.

- The value of $s\left({x}_{Si}\right)$ assumes the desired value at the point ${x}_{i}$:

- The system is completed if the orthogonality condition of the polynomial terms is verified for all polynomials p with a degree less or equal to that of polynomial $h$:

## 3. Tools and Procedure Implementation

#### 3.1. RBF Morph

**Setup**,

**Fitting**, and

**Smoothing**. The first step is taken by using the software GUI and consists of the manual definition of the RBF problem (select the source points and prescribe the parameters to drive the shape deformation). Fitting is the action in which the RBF system is solved for the prescribed values of the input parameters. The smoothing process is the morphing phase of the mesh. It is performed by applying the prescribed displacement to the grid surfaces and then smoothly propagating the deformation to the surrounding domain volume.

#### 3.2. STAR-CCM+

#### 3.3. Automatic CAE Analysis

**Reads the data stored in the morphing input files**: the coordinates and displacements of the RBF centers (referred to as the supporting mesh) are read by the code and stored in a dynamic array;**Purges nonessential source points**: the code makes a call to the RBF Morph “Purge” function, which checks the source points stored in the array and discards those that are below a certain minimum distance from the nearby points;**Solves the RBF problem**: source point coordinates and displacements are provided as arguments to the “Solve” function, which solves the RBF problem;**Reads the data stored in the baseline mesh file**: the coordinates of the vertices (referred to as the baseline mesh built inside of STAR-CCM+) are also read and memorized in a dynamic array;**Mesh smoothing**: the baseline mesh coordinates are fed to the “Morph” function, and the mesh morphing is completed;**Writes the new coordinates of the mesh into files**: the mesh nodes coordinates resulting from the morphing process are stored on a new file.

## 4. Application to Case Studies

#### 4.1. ASMO Optimization

#### 4.1.1. Shape Parameterization

#### 4.1.2. Numerical Configuration and Results

#### 4.2. Volvo Car Side-View Mirror

#### 4.2.1. Baseline Numerical Model

#### 4.2.2. Mirror Shape Parameterization

#### 4.2.3. Optimization Results

^{®}Xeon

^{®}E5-2680 v2 processors (10 cores each), operating at a base frequency of 2.80 GHz and 128 GB of RAM. A single design point evaluation required an average of 45 min. The DoE table was completed in less than 20 h.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

ID | Boat Tail (Left) [mm] | Boat Tail (Right) [mm] | Roof Drop [mm] | Front Spoiler [mm] |
---|---|---|---|---|

1 | 18.20 | −18.20 | −8.60 | 6.95 |

2 | 11.00 | −11.00 | −17.00 | 9.65 |

3 | 8.60 | −8.60 | 4.60 | 17.75 |

4 | 12.20 | −12.20 | 9.40 | 10.55 |

5 | −2.20 | 2.20 | −1.40 | 18.65 |

6 | 0.20 | −0.20 | 1.00 | −2.05 |

7 | 1.40 | −1.40 | −19.40 | 3.35 |

8 | −1.00 | 1.00 | 8.20 | 13.25 |

9 | 7.40 | −7.40 | −9.80 | 19.55 |

10 | 2.60 | −2.60 | −18.20 | 14.15 |

11 | 9.80 | −9.80 | 5.80 | 0.65 |

12 | 17.00 | −17.00 | −13.40 | 15.95 |

13 | 6.20 | −6.20 | 2.20 | 7.85 |

14 | 13.40 | −13.40 | −15.80 | 1.55 |

15 | −4.60 | 4.60 | −11.00 | −0.25 |

16 | 14.60 | −14.60 | −5.00 | −1.15 |

17 | 5.00 | −5.00 | −7.40 | 2.45 |

18 | 19.40 | −19.40 | 3.40 | 6.05 |

19 | 3.80 | −3.80 | −6.20 | 11.45 |

20 | −5.80 | 5.80 | −12.20 | 16.85 |

21 | −3.40 | 3.40 | 7.00 | 4.25 |

22 | −7.00 | 7.00 | −14.60 | 8.75 |

23 | 15.80 | −15.80 | −2.60 | 15.05 |

24 | −8.20 | 8.20 | −3.80 | 5.15 |

25 | −9.40 | 9.40 | −0.20 | 12.35 |

ID | P1 [mm] | P2 [mm] | P3 [mm] | E1 [mm] | E2 [mm] | Offset [mm] |
---|---|---|---|---|---|---|

1 | 0.70 | 14.70 | 3.90 | −0.36 | 1.72 | 0.37 |

2 | −4.10 | 2.10 | 6.90 | −0.44 | 0.52 | 0.41 |

3 | 6.10 | 5.70 | 5.70 | −0.04 | 2.92 | 0.39 |

4 | 1.30 | 8.70 | 14.70 | −0.68 | 2.68 | 0.45 |

5 | −1.70 | 1.50 | 2.10 | −1.64 | −0.20 | 0.19 |

6 | 9.70 | 12.90 | 9.30 | −0.60 | 0.28 | 0.13 |

7 | −1.10 | 7.50 | 2.70 | −0.28 | 0.76 | 0.03 |

8 | −0.50 | 14.10 | 10.50 | −0.92 | 3.16 | 0.09 |

9 | 8.50 | 10.50 | 9.90 | −1.48 | 3.64 | 0.31 |

10 | 7.30 | 12.30 | 0.30 | −1.56 | 1.00 | 0.25 |

11 | 5.50 | 3.90 | 7.50 | −1.08 | −1.88 | 0.05 |

12 | 4.90 | 2.70 | 4.50 | −1.72 | 1.24 | 0.47 |

13 | −2.90 | 13.50 | 5.10 | −1.32 | −1.16 | 0.17 |

14 | 3.10 | 5.10 | 12.90 | −0.12 | 1.96 | 0.07 |

15 | −4.70 | 4.50 | 13.50 | −1.16 | 0.04 | 0.15 |

16 | 7.90 | 0.90 | 11.70 | −0.76 | −0.44 | 0.33 |

17 | −3.50 | 9.90 | 8.10 | −1.80 | 1.48 | 0.43 |

18 | 4.30 | 9.30 | 14.10 | −1.88 | −0.68 | 0.21 |

19 | 2.50 | 8.10 | 6.30 | −1.96 | 2.20 | 0.01 |

20 | −2.30 | 6.90 | 1.50 | −1.00 | 3.88 | 0.27 |

21 | 0.10 | 11.10 | 12.30 | −0.20 | −1.40 | 0.29 |

22 | 3.70 | 6.30 | 0.90 | −0.52 | −1.64 | 0.35 |

23 | 6.70 | 11.70 | 8.70 | −1.24 | −0.92 | 0.49 |

24 | 9.10 | 3.30 | 3.30 | −0.84 | 2.44 | 0.11 |

25 | 1.90 | 0.30 | 11.10 | −1.40 | 3.40 | 0.23 |

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**Figure 9.**Shell region created on the underside of the mirror (

**a**) and inlet/outlet system through the shell region (

**b**).

**Figure 10.**Parameterization of the mirror geometry acting on the straight borders (

**a**), on the curved sides (

**b**) and on the pocket (

**c**).

RBF with global support | $\phi \left(r\right)$, $r=\Vert x-{x}_{S}{}_{i}\Vert $ |

Spline type $\left({R}_{n}\right)$ | ${r}^{n},\text{}n\text{}odd$ |

Thin plate spline $\left(TP{S}_{n}\right)$ | ${r}^{n}\mathrm{log}\left(r\right),\text{}n\text{}even$ |

Multiquadric $\left(MQ\right)$ | $\sqrt{1+{r}^{2}}$ |

Inverse multiquadric $\left(IMQ\right)$ | $\frac{1}{\sqrt{1+{r}^{2}}}$ |

Inverse quadratic $\left(IQ\right)$ | $\frac{1}{1+{r}^{2}}$ |

Gaussian $\left(GS\right)$ | ${e}^{-{r}^{2}}$ |

RBF with compact support | $\phi \left(r\right)=f\left(\xi \right),\xi \le 1,\xi =\frac{r}{{R}_{sup}}$ |

Wendland $\left({C}^{0}\right)$ | ${\left(1-\xi \right)}^{2}$ |

Wendland $\left({C}^{2}\right)$ | ${\left(1-\xi \right)}^{4}\left(4\xi +1\right)$ |

Wendland $\left({C}^{4}\right)$ | ${\left(1-\xi \right)}^{6}\left(\frac{35}{3}{\xi}^{2}+6\xi +1\right)$ |

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**MDPI and ACS Style**

Cella, U.; Patrizi, D.; Porziani, S.; Virdung, T.; Biancolini, M.E.
Integration within Fluid Dynamic Solvers of an Advanced Geometric Parameterization Based on Mesh Morphing. *Fluids* **2022**, *7*, 310.
https://doi.org/10.3390/fluids7090310

**AMA Style**

Cella U, Patrizi D, Porziani S, Virdung T, Biancolini ME.
Integration within Fluid Dynamic Solvers of an Advanced Geometric Parameterization Based on Mesh Morphing. *Fluids*. 2022; 7(9):310.
https://doi.org/10.3390/fluids7090310

**Chicago/Turabian Style**

Cella, Ubaldo, Daniele Patrizi, Stefano Porziani, Torbjörn Virdung, and Marco Evangelos Biancolini.
2022. "Integration within Fluid Dynamic Solvers of an Advanced Geometric Parameterization Based on Mesh Morphing" *Fluids* 7, no. 9: 310.
https://doi.org/10.3390/fluids7090310