# Empowering Advanced Parametric Modes Clustering from Topological Data Analysis
## 1 Introduction
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. [PERSON] 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]. 3. 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). The present paper focuses on the techniques based on modal analysis. As just described, usual techniques operate at the eigenmodes level, defined in a vector space. When operating in high dimensional parametric spaces, sparsely sampled, the matrices involved in the resulting eigenproblem can vary a lot from one choice of the parameters to another, and consequently the scalar product criterion at the basis of the MAC could fail. On the other hand, the fact of proceeding in a vector space needs to carefully address the expression of the different modes by considering the same frame for all the analyzed mechanical systems. For alleviating those difficulties, this paper proposes an alternative technique that instead of operating at the eigenmodes level, classifies the modes with respect to the deformed structures shape that the eigenmodes produce, by invoking the so-called Topological Data Analysis--TDA-- that benefits from the invariance property of topology. The present paper does not aim at addressing the interpolation between the original and deformed structures that could be addressed by using optimal transport, or the parametrization of the manifold as in the parallel transport [14]. The present work aims only at addressing the deformed structures comparison using an appropriate metric, in particular the one based on the employment of TDA.