Adaptive Rational Interpolation and Higher-order SVD for Low-rank Tensor Approximation in Structural Dynamics Simulations

Simulations and data analysis in structural dynamics are challenged by large multi-dimensional data. Apart from two or three spatial directions, the problem coordinates include the dimension in frequency actuation and, possibly, dimensions to model uncertainties. If stored as a multidimensional array, these data quickly exceeds all storage capacities so that efficient approximative representations are needed for the data handling and, respectively, for simulations as a reduced-order model. Although the solution exhibits wave patterns, it is smooth and hence, well accessible to tensorized proper orthogonal decomposition (POD) approaches.The frequency dimension, however, shows a number of characteristic poles which should be preserved so that the least-squares-based approach of POD may not well suited. Therefore, in this work, we call on recently developed methods for rational interpolation of matrix-valued functions, extend them to higher-dimensional arrays to approximately represent one specific mode of these tensors, and investigate the interplay with higher-order singular value decomposition of the remaining modes. We illustrate the findings for tensorized data and simulations of a two-dimensional plate under excitation.