# Empowering Advanced Parametric Modes Clustering from Topological Data Analysis
## 2 Methods  
As just indicated, the present paper aims at addressing the classification of a series of modal basis related to the eigenmodes of a thin structure equipped with a mesh consisting of shell elements, with displacement and rotation degrees of freedom at each node of the mesh.  
The thickness of the structural part varies, with its consequent effect on the mass and stiffness matrices (damping is assumed proportional) and consequently on the eigenvalues and eigenvectors, the former defining the number of modes to be retained in the reduced basis. In the present study, the six rigid modes representing the whole structure translation (three modes) and rotation (three modes) will be discarded and among the remaining pairs eigenvalue-eigenvector, the most relevant six eigenvectors (corresponding to the six highest eigenvalues) retained in the reduced basis related to each choice of the model parameter (the thickness).  
These six modes related to each structure (related to a thickness value) define a reduced basis that one would like render parametric. However, prior to constructing a regression able to define the reduced basis for each possible choice of the parameter (thickness), one should classify the six eigenmodes of each reduced basis associated with each structure into six clusters.  
This task is compulsory to facilitate the interpolation in the parametric space and also to attach a physical sense to those modes. One could imagine that, for a given thickness, the most relevant deformation mode could be related to the extension, whereas, for another choice of the thickness, the most relevant deformation mode could be the bending. In such a case, one prefers to create a cluster grouping similar deformation modes, to evaluate how each of them depends on the parameter from one side, and on the other to facilitate the subsequent construction of the parametric modal reduced basis.  
To preform such a clustering, we must employ an appropriate metric to compare those modes. In general, this comparison was traditionally performed by comparing the eigenmodes within the vector space to which they belong. In the present work, as announced previously, we prefer applying the deformation mode to the reference (undeformed) structure, that is, applying the eigenmode at the nodes location in the reference structure for obtaining the deformed structure related to each mode of each structure configuration (thickness) and then clusterizing the resulting deformed structures with respect to their shape.  
Thus, we are employing a metric able to compare shapes, more than a metric for comparing the vectors (eigenmodes) that produced those shapes, the last being more intrinsic and inheriting invariance features. Moreover, in the present case study, eigenmodes are heterogeneous in the sense that they involve displacements and rotations, whereas the associated deformed surfaces are purely geometrical. For that purpose, the TDA will be applied.  
In the remaining part of the present section, we will describe the available data and its organization, and then all the concepts enabling the use of a metric applying on the shapes, based on the employment of persistent homology, at the heart of the TDA, employed in our former works [15; 16].