Abstract Summary
Finding the response of a large dynamical system demands considerable computational resources. This demand further amplifies where repeated solutions are sought, such as uncertainty quantification, sensitivity analysis, optimization. Besides this, for a complex problem such as soil-structure interaction, access to the source code is not readily available. To address these issues, recently, several non-intrusive reduced order models (ROMs) have been developed that significantly bring down the computational cost and do not require access to the source code. However, all the existing non-intrusive ROMs address uncertainty in system parameters but not in the excitation. The key challenge in developing a non-intrusive ROM for random excitation is performing interpolation or regression in a very high dimension. This step is practically infeasible in the state-of-the-art methods. This high dimension corresponds to the large number of parameters needed to represent or approximate random excitations. This paper addresses this issue. To the authors’ knowledge, this is the first reported study on the non- intrusive ROM for random excitation. A novel proper orthogonal decomposition-based non-intrusive ROM is proposed here using a neural network. In various applications neural networks have been found to be very effective in high dimensional regression. In the proposed method, first, a principal component analysis (PCA) is performed on a set of random excitations generated from a given power spectral den- sity spectrum. The PCA extracts the dominant features of the random excitation, and thus helps in reducing the number of parameters. Then, these dominant features are used to train a fully connected deep feed-forward neural network. Upon completion of the training phase, vibration response for a new excitation is found from this network. The accuracy of the proposed method is tested numerically for a soil-structure interaction problem modeled as a beam on Winkler foundation. A stationary and Gaussian excitation, generated from the Clough-Penzien spectrum, is chosen here for the numerical studies. The results show that the proposed method is accurate. The accuracy depends on the number of training points used for training the network, which, in turn, depends on the range of the frequency content. A wider frequency range demands a larger training sample size. Finally, the efficiency of the proposed method in performing uncertainty quantification is tested by comparing it with the direct Monte Carlo simulation. The results show a significant speed-up. Similar to most machine learning applications, the architecture of the network is problem specific. However, numerical experiments on various vibration problems showed a generic nature of this network and the training scheme. Applicability of the proposed method for non-stationary excitations remains to be explored.
