# Empowering Advanced Parametric Modes Clustering from Topological Data Analysis

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## Abstract

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## Featured Application

**This paper aims at proposing the use of a topological metrics based on the persistent homology, enabling efficient surfaces classification, and ordering the elastodynamics eigenmodes to construct parametric reduced bases.**

## Abstract

## 1. Introduction

- The previously referred mass lumping that transforms the so-called consistent mass matrix into its diagonal counterpart, facilitating an explicit integration;
- In the context of model order reduction—MOR—, in [7], authors proposed a Proper Orthogonal Decomposition—POD—based reduced order modeling operating in the time domain. Ladeveze and coworkers proposed an extension of their radial approximation [8] for addressing mid-frequency dynamics, the so called TVCR (variational theory of complex rays) [9]. In our former works, we considered a PGD formulation for constructing a parametric transfer function [10] that allowed efficient solutions of transient dynamics operating in the time-domain. On the other hand, the separation of variables, at the heart of PGD [11], was extensively employed in the harmonic domain for solving multi-parametric dynamics, and was successfully extended to the nonlinear case, and then combined with modal analysis [4,5,6].
- Modal analysis is one the most widely used techniques for solving dynamical problems. Other than the benefits in the time integration, due to the dynamical system decoupling, the eigenmodes benefit from a physical interpretation, of great interest for the designer or structural analyst. However, when considering parametric models, as is always the case during the design stage, when the material and geometry are not totally defined, the dynamical modes depend on those parameters as previously discussed. Having a surrogate model expressing the parametric evolution of the eigenmodes is of great interest. Constructing those surrogate models is nowadays quite mature, by using usual and advanced nonlinear regressions [12], the last making use of sparsity and appropriate regularizations for operating in high-dimensional settings, while keeping as reduced as possible the number of data (eigenproblem solution), and leading to rich enough (nonlinear) regressions while avoiding overfitting. Here, the trickiest issue is not the regression construction but the fact of ordering the different eigenmodes involved in the modal basis for each parameter choice, in order to create N clusters (or less in the reduced case), and putting in each one a mode of each modal basis, such that modes in each cluster remain close (in certain metrics). The main issue remains the metric to be used to successfully and efficiently accomplishing such clustering. In general, such clustering is performed by operating at the level of the eigenmodes, in the associated vector space, by using, for example, the Modal Assurance Criterion—MAC— [13] that proceed with comparing the modes resulting from each eigenproblem, by using the usual scalar product (modes similar to a given one should remain quite collinear).

## 2. Methods

#### 2.1. Data Description

#### 2.2. On the Surface Topology

#### 2.2.1. Geometric Features

- A vertex $\left[{\mathbf{x}}_{m}\right]$ is generated by an individual point ${\mathbf{x}}_{m}\in \mathbb{M}$;
- A segment $[{\mathbf{x}}_{m},{\mathbf{x}}_{n}]$ joins two vertex $\left[{\mathbf{x}}_{m}\right],\phantom{\rule{4pt}{0ex}}\left[{\mathbf{x}}_{n}\right]\in \mathbb{M}$$$[{\mathbf{x}}_{m},{\mathbf{x}}_{n}]:=\left(\right)open="\{"\; close="\}">\mathbf{x}\in {\mathbb{R}}^{3}:\mathbf{x}=\lambda {\mathbf{x}}_{m}+(1-\lambda ){\mathbf{x}}_{n}\phantom{\rule{4.pt}{0ex}}\mathrm{where}\phantom{\rule{4.pt}{0ex}}0\le \lambda \le 1$$
- A triangle is generated by three different vertexes $\left[{\mathbf{x}}_{m}\right],\phantom{\rule{4pt}{0ex}}\left[{\mathbf{x}}_{n}\right],\phantom{\rule{4pt}{0ex}}\left[{\mathbf{x}}_{l}\right]\in \mathbb{M}$, such that ${\mathbf{x}}_{m}-{\mathbf{x}}_{n}$ and ${\mathbf{x}}_{m}-{\mathbf{x}}_{l}$ are linearly independent, and then:$$[{\mathbf{x}}_{m},{\mathbf{x}}_{n},{\mathbf{x}}_{l}]:=\left(\right)open="\{"\; close="\}">\mathbf{x}\in {\mathbb{R}}^{3}:\mathbf{x}={\lambda}_{m}{\mathbf{x}}_{m}+{\lambda}_{n}{\mathbf{x}}_{n}+{\lambda}_{l}{\mathbf{x}}_{l}$$
- A tetrahedron is generated by four different vertices $\left[{\mathbf{x}}_{m}\right],\phantom{\rule{4pt}{0ex}}\left[{\mathbf{x}}_{n}\right],\phantom{\rule{4pt}{0ex}}\left[{\mathbf{x}}_{l}\right],\phantom{\rule{4pt}{0ex}}\left[{\mathbf{x}}_{p}\right]\in \mathbb{M}$, such that ${\mathbf{x}}_{m}-{\mathbf{x}}_{n}$, ${\mathbf{x}}_{m}-{\mathbf{x}}_{l}$, and ${\mathbf{x}}_{m}-{\mathbf{x}}_{p}$ are linearly independent, and then:$$[{\mathbf{x}}_{m},{\mathbf{x}}_{n},{\mathbf{x}}_{l},{\mathbf{x}}_{p}]:=\left(\right)open="\{"\; close="\}">\mathbf{x}\in {\mathbb{R}}^{3}:\mathbf{x}={\lambda}_{m}{\mathbf{x}}_{m}+{\lambda}_{n}{\mathbf{x}}_{n}+{\lambda}_{l}{\mathbf{x}}_{l}+{\lambda}_{p}{\mathbf{x}}_{p}$$

#### 2.2.2. Features’ Filtration

- For every simplex in ${S}_{d}\left(\mathbb{M}\right)$, the $(d-1)$-dimensional simplices forming it are in ${S}_{d-1}\left(\mathbb{M}\right)$ (e.g., a triangle is in ${S}_{2}\left(\mathbb{M}\right)$ and its three edges are in ${S}_{1}\left(\mathbb{M}\right)$);
- If two simplices in ${S}_{d}\left(\mathbb{M}\right)$ have a common element $\sigma $, then there exists $0\le l\le (d-1)$ such that $\sigma \in {S}_{l}\left(\mathbb{M}\right)$.

- First, the filtration value of each tetrahedron is computed as the circumradius of the tetrahedron if its circumsphere is empty, and as the minimum of the filtration values of the triangles that are within the circumsphere otherwise.
- Similarly, the filtration value of each triangle is computed as the circumradius of the triangle if the circumcircle is empty, and as the minimum of the filtration values of the segments that are within the circumcircle otherwise.
- Then, the filtration value of each segment is computed as its circumradius.
- Finally, the filtration value of the vertices is set to 0.

#### 2.2.3. Persistence Diagrams

- The birth scale ${b}_{\gamma}$ of the feature $\gamma $ at homology k$${b}_{\gamma}=\underset{0\le j\le m}{min}\{{\alpha}_{j}:\gamma \in {H}_{k}^{{\alpha}_{j}}\}$$
- The death scale ${d}_{\gamma}$ of the feature $\gamma $ at homology k$${d}_{\gamma}=\underset{0\le j\le m}{max}\{{\alpha}_{j}:\gamma \in {H}_{k}^{{\alpha}_{j}}\}$$

#### 2.2.4. Illustrating the Concepts on an Example

#### 2.2.5. Matching Persistence Diagrams

#### 2.2.6. Multi-Scale Topological Measure of Surface Deformation

#### 2.2.7. Comparing Tropological Descriptions of Deformed Surfaces

#### 2.3. Modal Assurance Criterion

## 3. Results

#### 3.1. Topological Modes Identification

#### 3.2. MAC Identification

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## References

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**Figure 1.**Reference undeformed structure (

**left**) and its deformed counterpart (

**right**) when a given deformation mode applied on the reference one.

**Figure 5.**Optimal matching of two persistence diagrams ${\mathcal{PD}}_{1}\left(\mathbb{M}\right)$ and ${\mathcal{PD}}_{1}\left(\mathbb{N}\right)$.

**Figure 6.**Persistence diagrams associated with the reference surface, ${\mathcal{PD}}_{k}\left({\mathbb{M}}_{0}\right)$, for $k=0$ (

**left**), $k=1$ (

**center**) and $k=2$ (

**right**).

**Figure 7.**Modal Assurance Criterion matrices when comparing the model reduced basis—RB— (of the first thickness choice) with the remaining 16 reduced modal bases for the other thickness choices.

**Figure 8.**(

**top-left**) First mode; (

**top-right**) Second mode; (

**middle-left**) Third mode; (

**middle-right**) Fourth mode; (

**bottom-left**) Fifth mode; and (

**bottom-right**) Sixth mode.

$\mathit{\alpha}$ | ${\mathit{S}}_{0}^{\mathit{\alpha}}$ | ${\mathit{S}}_{1}^{\mathit{\alpha}}$ | ${\mathit{S}}_{2}^{\mathit{\alpha}}$ | ${\mathit{S}}_{3}^{\mathit{\alpha}}$ |
---|---|---|---|---|

0.00 | $\left[{x}_{0}\right],\left[{x}_{1}\right],\left[{x}_{2}\right]$ | - | - | - |

$\left[{x}_{3}\right],\left[{x}_{4}\right],\left[{x}_{5}\right]$ | ||||

0.50 | $\left[{x}_{0}\right],\left[{x}_{1}\right],\left[{x}_{2}\right]$ | $[{x}_{1},{x}_{3}]$ | - | - |

$\left[{x}_{3}\right],\left[{x}_{4}\right],\left[{x}_{5}\right]$ | ||||

2.00 | $\left[{x}_{0}\right],\left[{x}_{1}\right],\left[{x}_{2}\right]$ | $[{x}_{1},{x}_{3}],[{x}_{0},{x}_{4}]$ | - | - |

$\left[{x}_{3}\right],\left[{x}_{4}\right],\left[{x}_{5}\right]$ | ||||

20.50 | $\left[{x}_{0}\right],\left[{x}_{1}\right],\left[{x}_{2}\right]$ | $[{x}_{1},{x}_{3}],[{x}_{0},{x}_{4}]$ | - | - |

$\left[{x}_{3}\right],\left[{x}_{4}\right],\left[{x}_{5}\right]$ | $[{x}_{0},{x}_{2}],[{x}_{4},{x}_{5}]$ | |||

45.25 | $\left[{x}_{0}\right],\left[{x}_{1}\right],\left[{x}_{2}\right]$ | $[{x}_{1},{x}_{3}],[{x}_{0},{x}_{4}]$ | - | - |

$\left[{x}_{3}\right],\left[{x}_{4}\right],\left[{x}_{5}\right]$ | $[{x}_{0},{x}_{2}],[{x}_{4},{x}_{5}]$ | |||

$[{x}_{1},{x}_{2}],[{x}_{3},{x}_{5}]$ | ||||

50.00 | $\left[{x}_{0}\right],\left[{x}_{1}\right],\left[{x}_{2}\right]$ | $[{x}_{1},{x}_{3}],[{x}_{0},{x}_{4}]$ | - | - |

$\left[{x}_{3}\right],\left[{x}_{4}\right],\left[{x}_{5}\right]$ | $[{x}_{0},{x}_{2}],[{x}_{4},{x}_{5}]$ | |||

$[{x}_{1},{x}_{2}],[{x}_{3},{x}_{5}]$ | ||||

$[{x}_{2},{x}_{5}]$ | ||||

50.02 | $\left[{x}_{0}\right],\left[{x}_{1}\right],\left[{x}_{2}\right]$ | $[{x}_{1},{x}_{3}],[{x}_{0},{x}_{4}]$ | - | - |

$\left[{x}_{3}\right],\left[{x}_{4}\right],\left[{x}_{5}\right]$ | $[{x}_{0},{x}_{2}],[{x}_{4},{x}_{5}]$ | |||

$[{x}_{1},{x}_{2}],[{x}_{3},{x}_{5}]$ | ||||

$[{x}_{2},{x}_{5}],[{x}_{2},{x}_{4}]$ | ||||

64.73 | $\left[{x}_{0}\right],\left[{x}_{1}\right],\left[{x}_{2}\right]$ | $[{x}_{1},{x}_{3}],[{x}_{0},{x}_{4}]$ | $[{x}_{1},{x}_{2},{x}_{5}],[{x}_{1},{x}_{3},{x}_{5}]$ | - |

$\left[{x}_{3}\right],\left[{x}_{4}\right],\left[{x}_{5}\right]$ | $[{x}_{0},{x}_{2}],[{x}_{4},{x}_{5}]$ | |||

$[{x}_{1},{x}_{2}],[{x}_{3},{x}_{5}]$ | ||||

$[{x}_{2},{x}_{5}],[{x}_{2},{x}_{4}]$ | ||||

$[{x}_{1},{x}_{5}]$ | ||||

70.64 | $\left[{x}_{0}\right],\left[{x}_{1}\right],\left[{x}_{2}\right]$ | $[{x}_{1},{x}_{3}],[{x}_{0},{x}_{4}]$ | $[{x}_{1},{x}_{2},{x}_{5}],[{x}_{1},{x}_{3},{x}_{5}]$ | - |

$\left[{x}_{3}\right],\left[{x}_{4}\right],\left[{x}_{5}\right]$ | $[{x}_{0},{x}_{2}],[{x}_{4},{x}_{5}]$ | $[{x}_{0},{x}_{1},{x}_{2}],[{x}_{3},{x}_{4},{x}_{5}]$ | ||

$[{x}_{1},{x}_{2}],[{x}_{3},{x}_{5}]$ | ||||

$[{x}_{2},{x}_{5}],[{x}_{2},{x}_{4}]$ | ||||

$[{x}_{1},{x}_{5}],[{x}_{0},{x}_{1}]$ | ||||

$[{x}_{3},{x}_{4}]$ | ||||

71.38 | $\left[{x}_{0}\right],\left[{x}_{1}\right],\left[{x}_{2}\right]$ | $[{x}_{1},{x}_{3}],[{x}_{0},{x}_{4}]$ | $[{x}_{1},{x}_{2},{x}_{5}],[{x}_{1},{x}_{3},{x}_{5}]$ | - |

$\left[{x}_{3}\right],\left[{x}_{4}\right],\left[{x}_{5}\right]$ | $[{x}_{0},{x}_{2}],[{x}_{4},{x}_{5}]$ | $[{x}_{0},{x}_{1},{x}_{2}],[{x}_{3},{x}_{4},{x}_{5}]$ | ||

$[{x}_{1},{x}_{2}],[{x}_{3},{x}_{5}]$ | $[{x}_{0},{x}_{1},{x}_{4}],[{x}_{1},{x}_{3},{x}_{4}]$ | |||

$[{x}_{2},{x}_{5}],[{x}_{2},{x}_{4}]$ | ||||

$[{x}_{1},{x}_{5}],[{x}_{0},{x}_{1}]$ | ||||

$[{x}_{3},{x}_{4}],[{x}_{1},{x}_{4}]$ | ||||

71.55 | $\left[{x}_{0}\right],\left[{x}_{1}\right],\left[{x}_{2}\right]$ | $[{x}_{1},{x}_{3}],[{x}_{0},{x}_{4}]$ | $[{x}_{1},{x}_{2},{x}_{5}],[{x}_{1},{x}_{3},{x}_{5}]$ | $[{x}_{0},{x}_{1},{x}_{2},{x}_{4}]$ |

$\left[{x}_{3}\right],\left[{x}_{4}\right],\left[{x}_{5}\right]$ | $[{x}_{0},{x}_{2}],[{x}_{4},{x}_{5}]$ | $[{x}_{0},{x}_{1},{x}_{2}],[{x}_{3},{x}_{4},{x}_{5}]$ | $[{x}_{1},{x}_{2},{x}_{4},{x}_{5}]$ | |

$[{x}_{1},{x}_{2}],[{x}_{3},{x}_{5}]$ | $[{x}_{0},{x}_{1},{x}_{4}],[{x}_{1},{x}_{3},{x}_{4}]$ | $[{x}_{1},{x}_{3},{x}_{4},{x}_{5}]$ | ||

$[{x}_{2},{x}_{5}],[{x}_{2},{x}_{4}]$ | $[{x}_{1},{x}_{2},{x}_{4}],[{x}_{1},{x}_{4},{x}_{5}]$ | |||

$[{x}_{1},{x}_{5}],[{x}_{0},{x}_{1}]$ | ||||

$[{x}_{3},{x}_{4}],[{x}_{1},{x}_{4}]$ |

**Table 2.**Multi-scale topological distance of the six deformed surfaces related to the six most significant deformation modes, of the 17 choices of the structure thickness with respect to the reference undeformed surface.

Case | 1st surf. | 2nd surf. | 3rd surf. | 4th surf. | 5th surf. | 6th surf. |
---|---|---|---|---|---|---|

01 | 2828 | 3742 | 6012 | 6281 | 7070 | 7314 |

02 | 3839 | 4341 | 5160 | 5269 | 8299 | 9530 |

03 | 3540 | 4003 | 4702 | 5536 | 6436 | 8023 |

04 | 2971 | 3762 | 5883 | 7481 | 7429 | 9314 |

05 | 3062 | 3762 | 5437 | 6156 | 9865 | 10,307 |

06 | 4852 | 5239 | 7294 | 8336 | 8237 | 9585 |

07 | 3482 | 4411 | 6095 | 6882 | 9319 | 10,627 |

08 | 3392 | 3684 | 5903 | 6710 | 9273 | 9438 |

09 | 4648 | 5436 | 7986 | 7707 | 10,415 | 9406 |

10 | 4425 | 4267 | 5583 | 5811 | 9620 | 9163 |

11 | 3256 | 3722 | 4782 | 4888 | 5840 | 8064 |

12 | 2993 | 3750 | 5551 | 6885 | 7135 | 8474 |

13 | 4281 | 4862 | 7127 | 8230 | 8170 | 9687 |

14 | 4367 | 5004 | 7140 | 7036 | 8285 | 8008 |

15 | 4937 | 5396 | 6484 | 6446 | 9323 | 10,031 |

16 | 3184 | 3941 | 4907 | 4965 | 7205 | 8869 |

17 | 2957 | 3652 | 5652 | 6021 | 7659 | 7571 |

**Table 3.**Surface ordering. Numbers in red indicate the permutation that must be performed in order to align surfaces with respect to its shape.

Case | 1st surf. | 2nd surf. | 3rd surf. | 4th surf. | 5th surf. | 6th surf. |
---|---|---|---|---|---|---|

01 | 1 | 2 | 3 | 4 | 5 | 6 |

02 | 1 | 2 | 3 | 4 | 5 | 6 |

03 | 1 | 2 | 3 | 4 | 5 | 6 |

04 | 1 | 2 | 3 | 5 | 4 | 6 |

05 | 1 | 2 | 3 | 4 | 5 | 6 |

06 | 1 | 2 | 3 | 5 | 4 | 6 |

07 | 1 | 2 | 3 | 4 | 5 | 6 |

08 | 1 | 2 | 3 | 4 | 5 | 6 |

09 | 1 | 2 | 4 | 3 | 6 | 5 |

10 | 2 | 1 | 3 | 4 | 6 | 5 |

11 | 1 | 2 | 3 | 4 | 5 | 6 |

12 | 1 | 2 | 3 | 4 | 5 | 6 |

13 | 1 | 2 | 3 | 5 | 4 | 6 |

14 | 1 | 2 | 4 | 3 | 6 | 5 |

15 | 1 | 2 | 4 | 3 | 5 | 6 |

16 | 1 | 2 | 3 | 4 | 5 | 6 |

17 | 1 | 2 | 3 | 4 | 6 | 5 |

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**MDPI and ACS Style**

Frahi, T.; Falco, A.; Mau, B.V.; Duval, J.L.; Chinesta, F.
Empowering Advanced Parametric Modes Clustering from Topological Data Analysis. *Appl. Sci.* **2021**, *11*, 6554.
https://doi.org/10.3390/app11146554

**AMA Style**

Frahi T, Falco A, Mau BV, Duval JL, Chinesta F.
Empowering Advanced Parametric Modes Clustering from Topological Data Analysis. *Applied Sciences*. 2021; 11(14):6554.
https://doi.org/10.3390/app11146554

**Chicago/Turabian Style**

Frahi, Tarek, Antonio Falco, Baptiste Vinh Mau, Jean Louis Duval, and Francisco Chinesta.
2021. "Empowering Advanced Parametric Modes Clustering from Topological Data Analysis" *Applied Sciences* 11, no. 14: 6554.
https://doi.org/10.3390/app11146554