A reduced-order matrices fitting scheme with Log-Euclidean metrics for fast approximation of dynamic response of parametric structural systems

Structural engineering practice often involves tasks, such as design, optimization or statistical analysis, where iterative solutions of dynamical systems are required varying some parameter which affect the system matrices in a generally non-linear way. The approach of interpolating among matrices of reduced-order models (ROMs) is not new in the literature and it is very promising in order to speed up the calculations. In particular, the method shows a great potential since it is applicable to any kind of FE model and of geometry, with no restrictions regarding the element type constituting the full-order model or the range of parameter variability. Yet, dealing with symmetric positive definite (SPD) full-order matrices, the SPD nature of the reduced matrices must be preserved. To this end, a special mathematical framework must be used, which allows to linearize the curved space of SPD matrices. In the literature of ROMs interpolation, the common choice is to rely on the principles of Riemannian geometry through the concept of the tangent plane to the SPD manifold. Although mathematically robust, this approach brings a marked distortion in the space of SPD matrices. Also, it shows numerical instabilities when dealing with strongly anisotropic matrices. In this work, a polynomial fitting of the reduced matrices is proposed with Log-Euclidean metrics. These metrics were obtained by giving the SPD manifold the structure of a Lie group. While maintaining the already proposed approach of interpolating between a number of sampling ROMs, the advantages of using Log-Euclidean metrics are illustrated in details. In particular, a great improvement is shown in case of a unique reduction matrix whose columns span the solution of the parameter space of interest in an accurate enough way, and in case of substructuring reduction techniques. The reported examples of applications show excellent accuracy, while maintaining the dramatic decrease in the computational time of the interpolatory scheme.