#
A Novel Method for Pressure Mapping between Shell Meshes of Varying Geometries and Resolutions^{ †}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Set-Up

- Find the centroid of the triangle element, and in the case of an element being a quad, the triangle that forms half of the quad element surface
- –
- If a quad, then the first three node elements are used to calculate the area. Afterwards, the process is repeated with the first, second, and fourth node. Regardless of the node order, this should create two triangles that form the quad.
- –
- The centroid coordinates are found by simply averaging the X, Y, and Z locations of the three nodes that made up the surface triangle.

- Generate the vectors from each node to the centroid.
- Find the area formed by each of the three pairs of vectors three times.
- –
- ${R}_{1}\times {R}_{2}$, ${R}_{2}\times {R}_{3}$, and ${R}_{1}\times {R}_{3}$, where ${R}_{1}$, ${R}_{2}$, and ${R}_{3}$ represent the vector formed in between the centroid and node 1, 2, and 3, respectively.
- –
- The area is calculated as half of the absolute value of the cross product of the two vectors, to form each smaller triangle$$\begin{array}{ccc}\hfill A& =& \frac{1}{2}\xb7|{R}_{1}\times {R}_{2}|\hfill \\ & =& \frac{1}{2}\xb7\{{({R}_{1B}\xb7{R}_{2C}-{R}_{1C}\xb7{R}_{2B})}^{2}+{({R}_{1A}\xb7{R}_{2C}-{R}_{1C}\xb7{R}_{2A})}^{2}+{({R}_{1B}\xb7{R}_{2A}-{R}_{1A}\xb7{R}_{2B})}^{2}\},\hfill \end{array}$$

- The areas of these three small triangles is added up to determine the area of the larger triangle (Figure 1)
- If the element surface is a quad, this process is performed twice, treating the quad as two triangles
- –
- Different software packages use different node patterns. If the wrong pattern is used, the area could be erroneously calculated (Figure 2),
- –
- It is not an option to find the node the greatest distance away, as an oddly shaped element would give an erroneous calculation (Figure 3),
- –
- The areas of all three combinations of two possible triangles are calculated (Table 1),
- –
- The maximum of these three possible areas is selected as the area of the quad of interest.

## 3. Algorithm

- Sorting through the ${N}_{S}\xb7{N}_{X}$ array of ${W}_{P}$ for each source element,
- Adding to the target number found, the location in a ${N}_{T}\xb7{N}_{X2}$ array,
- –
- That source element, the source element number, and the pressure magnitude data ${W}_{P}$.

- Incrementing the count of sources proximate to a given target
- –
- Stored in a separate integer vector (${N}_{T}\xb71$),
- –
- Used to track how many source elements are proximate to the given target element,
- –
- Can be expected to exceed ${N}_{X}$ provided the number of source elements is greater than the number of target elements,
- –
- Should never exceed ${N}_{X2}$; if this happens, it is necessary to increase the value of ${N}_{X2}$ to avoid memory errors,

- Save the transfer matrix.

- Number of elements ${N}_{T},$
- –
- This number is saved so a future mapping algorithm knows the number of target pressures to generate,

- For ${N}_{T}$ times, list the specific target and the following information,
- –
- Target number, Number of Sources Proximate, and Sum of Pressure Impacts ${W}_{P}$
- –
- The reason for the sum of ${W}_{P}$ is because each individual value of ${W}_{P}$ is normalized by the cumulative sum of ${W}_{P}$ for all proximate targets, so that the pressure of the target element is a proportional average of these proximate source elements.

- For each target, for the number of proximate source elements,
- –
- Sequential number, source element number, and pressure impact ${W}_{P}$

## 4. Test Example

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

ALE | Arbitrary-Lagrangian-Eulerian |

FSI | Fluid Solid Interactions |

CFD | Computational Fluid Dynamics |

CSD | Computational Solid Dynamics |

FEA | Finite Element Analysis |

ANSYS | Analysis Systems, Inc. |

NAVAIR | Naval Air Systems Command |

## Appendix A. Sample Code

#### Appendix A.1. Introduction

#### Appendix A.2. Generating the Input Files

`gfortran -O3 -o makeinput Make_Input.f`

`./makeinput`

#### Appendix A.3. Generating the Transfer Matrix

- Nx = 5: This is the maximum number of targets that can be considered proximate to a given source. Matrix arrays of $Ns\xb7Nx$ will be generated, so it must be small enough that the script does not exceed the memory resources of the computer. Even if there is a lot of memory, unless the target elements are smaller than the source (definitely possible but often not the case in practice), then it should not be desirable for Nx to be greater than a few, as one would not want the source element to be mapped onto elements a significant distance away.
- Nx2 = 500: This is the maximum number of sources that can be considered proximate to a given target and used in the transfer matrix. Matrix arrays of $Nt\xb7Nx$ will be generated, so it must be small enough that the script does not exceed the memory resources of the computer, but also large enough to contain every source matrix considered proximate to a given target; in practice, having a value of Nx2 that is 100 times greater than Nx was found to work well.
- MinW0 = 0.0000000001: This is the minimum value of W that is necessary for a target particle to be considered proximate to a source. This value can be set to zero, but the computational time will be dramatically increased, and there is a risk that source elements that do not in fact overlap a target can be used for the final pressure. If a minimum value is set, then target elements that do not have any proximate source elements will not be mapped, resulting in a consistent pressure of zero. Different circumstances may prefer or oppose this possibility. This value of ${10}^{-10}$ was used in the example in the manuscript.
- hcoef = 1.0: This is a coefficient to calculate the weighting function between a source and target element centroid as a function of distance apart. The smoothing length is proportional to this arbitrary value times the square root of the average area of all of the target elements. The larger hcoef is, the more likely a source element fully separated from a target element will influence the pressure, and the less likely a target element will turn out to not be mapped.
- ct1 = 1: This value is not used in the mapping, only in the generation of the final target pressure output files. This fifth line is the number of the first of the input pressures, and should be 1 unless modified by the user.
- ct2 = 180: This value is not used in the mapping, only in the generation of the final target pressure output files. This sixth line is the number of inputs, and should be 180 to capture all 180 input pressures unless modified by the user.

`gfortran -O3 -o run make_tm.f`

`./run`

`less prompt.dat.`

#### Appendix A.4. Generating New Pressures

`gfortran -O3 -o readmap ReadMap.f`

`./readmap.`

#### Appendix A.5. Post-Processing

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**Figure 2.**An example of how picking the wrong nodes to divide a quad into two triangles (blue and yellow) can cause error due to area overlap (green), utilizing nodes 1-2-3 and 1-3-4.

**Figure 3.**Another example of how picking the wrong nodes to divide a quad into two triangles (blue and yellow) can cause error due to area overlap (green), demonstrating how an assumption cannot be made by searching for the longest vector between two nodes.

**Figure 4.**Comparison of the Total Forces in Newtons of the source versus the target, for 180 different source pressure profiles.

**Figure 5.**Comparison of the Pressures of one time step, both Source and Target, for oo = 100. The color-bar represents the pressure (Pa).

**Table 1.**Three different areas of two triangles; the maximum total area represents the correct area of the quad element face.

Quad Area | Node (First Triangle) | Node (Second Triangle) |
---|---|---|

Area 1 | Node 1, 2, 3 | Node 1, 2, 4 |

Area 2 | Node 1, 3, 2 | Node 1, 3, 4 |

Area 3 | Node 1, 4, 2 | Node 1, 4, 3 |

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**MDPI and ACS Style**

Marko, M.D.
A Novel Method for Pressure Mapping between Shell Meshes of Varying Geometries and Resolutions. *Computation* **2019**, *7*, 29.
https://doi.org/10.3390/computation7020029

**AMA Style**

Marko MD.
A Novel Method for Pressure Mapping between Shell Meshes of Varying Geometries and Resolutions. *Computation*. 2019; 7(2):29.
https://doi.org/10.3390/computation7020029

**Chicago/Turabian Style**

Marko, Matthew David.
2019. "A Novel Method for Pressure Mapping between Shell Meshes of Varying Geometries and Resolutions" *Computation* 7, no. 2: 29.
https://doi.org/10.3390/computation7020029