# Empowering Advanced Parametric Modes Clustering from Topological Data Analysis
## 1 Introduction  
Linear structural solid dynamics [1] expressed in the time domain results in the linear system of second order ordinary differential equations  
\[\mathbf{M}\frac{d^{2}\mathbf{U}(t)}{dt^{2}}+\mathbf{C}\frac{d\mathbf{U}(t)}{ dt}+\mathbf{K}\mathbf{U}(t)=\mathbf{F}(t), \tag{1}\]  
with the mass, damping, and stiffness matrices given by \(\mathbf{M}\), \(\mathbf{C}\), and \(\mathbf{K}\) respectively, \(\mathbf{U}\) the vector that contains the nodal displacements, and \(\mathbf{F}\) the applied nodal forces. Its timeintegration can be performed by using any well experienced state of the art discretization technique, as [2] or [3]. In what follows, we will omit the damping term that results from the fact of assuming a proportional damping that expresses it as a combination of the mass and stiffness contributions. To enhance the integration efficiency, mass lumping is usually considered, leading to a mass diagonal matrix. Model analysis looks also for enhancing the solution efficiency by decoupling the motion equation. For that purpose, the last extracts the basis \(\{\mathbf{\phi}_{1},\ \mathbf{\phi}_{2},\dots,\ \mathbf{\phi}_{N}\}\) (\(N\) being the problem size, i.e., the number of degrees of freedom) by solving the eigenproblem  
\[\left(-\omega^{2}\mathbf{M}+\mathbf{K}\right)\mathbf{\phi}=\mathbf{0}, \tag{2}\]  
associated with the dynamical problem expressed in the Fourier space,  
\[\left(-\omega^{2}\mathbf{M}+\mathbf{K}\right)\mathcal{U}=\mathcal{F}, \tag{3}\]  
where \(\mathcal{U}\) and \(\mathcal{F}\) refer to the Fourier transform of the nodal displacement \(\mathbf{U}\) and forces \(\mathbf{F}\). The eigenmodes \(\mathbf{\phi}_{i}\), \(i=1,\cdots,N\) define an orthogonal basis, normalized with respect to the mass matrix, i.e.,  
\[\mathbf{\phi}_{i}^{T}\mathbf{M}\mathbf{\phi}_{j}=\delta_{ij}, \tag{4}\]  
with \(\delta\) the Kroenecker delta, and  
\[\mathbf{\phi}_{i}^{T}\mathbf{K}\mathbf{\phi}_{j}=\kappa_{i}\delta_{ij}. \tag{5}\]  
With \(\mathbf{P}\) the matrix composed by the eigenmodes, i.e., \(\mathbf{P}=(\mathbf{\phi}_{1},\cdots,\mathbf{\phi}_{N})\), the matrix form of the previous expressions reads \(\mathbf{P}^{T}\mathbf{M}\mathbf{P}=\mathbf{I}\) and \(\mathbf{P}^{T}\mathbf{K}\mathbf{P}=\mathbb{K}\), with \(\mathbf{I}\) the identity matrix and \(\mathbb{K}\) the diagonal matrix with entries \(\mathbb{K}_{ii}=\kappa_{i}\). In the modal basis \(\mathbf{U}=\mathbf{P}\mathbf{\varphi}\), the dynamical problem reads  
\[\mathbf{I}\frac{d^{2}\mathbf{\varphi}(t)}{dt^{2}}+\mathbb{K}\mathbf{\varphi}(t)= \mathbf{P}^{T}\mathbf{F}(t), \tag{6}\]  
which constitutes a system of N uncoupled second order ordinary differential equations. The main limitation of modal analysis is the lack of validity of such basis in the case of parametric models. In the case of parametrized dynamical systems, with the model parameters grouped in vector \(\mathbf{\mu}\), the model matrices will depend on those parameters, i.e., \(\mathbf{M}(\mathbf{\mu})\) and \(\mathbf{K}(\mathbf{\mu})\). The solution of parametric eigenproblems remains a tricky issue. When one is not really interested in the transient regime, but much more in the forced regime, harmonic analysis represents a valuable route as considered in our former works where the so-called Proper Generalized Decomposition--PGD--enabled considering the frequency as a model extra-parameter as well as addressing general (non-proportional) damping and nonlinear dynamics, under the stringent real-time constraint, with even the inclusion of model parameters as extra-coordinates [4; 5; 6]. However, certain applications need accurate transient responses, and in that case the formulation and solution of the dynamical problem in the time domain is retained. Three routes are usually considered:  
1. The previously referred mass lumping that transforms the so-called consistent mass matrix into its diagonal counterpart, facilitating an explicit integration;
2.