A non-intrusive reduced basis (RB) method is proposed for parametrized nonlinear structural analysis undergoing large deformations and with elasto-plastic constitutive relations. In this method, a reduced basis is constructed from a set of full-order snapshots by the proper orthogonal decomposition (POD), and the Gaussian process regression (GPR) is used to approximate the projection coefficients. The GPR is carried out in the offline stage with active data selection, and the outputs for new parameter values can be obtained rapidly as probabilistic distributions during the online stage. Due to the complete decoupling of the offline and online stages, the proposed non-intrusive RB method provides a powerful tool to efficiently solve parametrized nonlinear problems with various engineering applications requiring multi-query or real-time evaluations. With both geometric and material nonlinearities taken into account, numerical results are presented for typical 1D and 3D examples, illustrating the accuracy and efficiency of the proposed method.

A data-driven reduced basis (RB) method is proposed for parametrized time-dependent problems. This method requires the offline preparation of a database comprising the full-order solutions of time history at parameter locations. Based on the full-order data, a reduced basis is constructed by the proper orthogonal decomposition (POD), and the maps between the time-parameter values and the projection coefficients onto the RB are approximated as a regression model. With a natural tensor grid between the inputs of time and parameters in the database, the singular-value decomposition (SVD) is used to extract the principal components in the data of projection coefficients, and the regression functions are represented as the linear combinations of several tensor products of two Gaussian processes, one of time and the other of parameters. During the online stage, the solutions at new time-parameter locations in the considered domain can be recovered rapidly via direct outputs from the regression models. Featuring a non-intrusive nature and the complete decoupling of the offline and online stages, the proposed approach provides a reliable and efficient tool for solving parametrized time-dependent problems, and its effectiveness is illustrated by non-trivial numerical examples.

Despite tremendous progress seen in the computational fluid dynamics community for the past few decades, numerical tools are still too slow for the simulation of practical flow problems, consuming thousands or even millions of computational core-hours. To enable feasible multi-disciplinary analysis and design, the numerical techniques need to be accelerated by orders of magnitude. Reduced-order modeling has been considered one promising approach for such purposes. Recently, non-intrusive reduced-order modeling has drawn great interest in the scientific computing community due to its flexibility and efficiency and undergoes rapid development at present with different approaches emerging from various perspectives. In this paper, a brief review of non-intrusive reduced-order modeling in the context of fluid problems is performed involving three key aspects: i.e. dimension reduction of the solution space, surrogate models, and sampling strategies. Furthermore, non-intrusive reduced-order modelings regarding to some interesting topics such as unsteady flows, shock-dominating flows are also discussed. Finally, discussions on future development of non-intrusive reduced-order modeling for fluid problems are presented.

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