A Training Set Subsampling Strategy for the Reduced Basis Method

We present a subsampling strategy for the offline stage of the Reduced Basis Method. The approach is aimed at bringing down the considerable offline costs associated with using a finely-sampled training set. The proposed algorithm exploits the potential of the pivoted QR decomposition and the discrete empirical interpolation method to identify important parameter samples. It consists of two stages. In the first stage, we construct a low-fidelity approximation to the solution manifold over a fine training set. Then, for the available low-fidelity snapshots of the output variable, we apply the pivoted QR decomposition or the discrete empirical interpolation method to identify a set of sparse sampling locations in the parameter domain. These points reveal the structure of the parametric dependence of the output variable. The second stage proceeds with a subsampled training set containing a by far smaller number of parameters than the initial training set. Different subsampling strategies inspired from recent variants of the empirical interpolation method are also considered. Tests on benchmark examples justify the new approach and show its potential to substantially speed up the offline stage of the Reduced Basis Method, while generating reliable reduced-order models.

Reduced-order models for radiative heat transfer of hypersonic vehicles

Aerothermoelastic analysis of hypersonic vehicles is a complex multidisciplinary coupling problem. Thus, accurate modeling of varying disciplines with low computational cost is necessary. This work developed a tractable approach-based reduced-order modeling technology to solve the radiative thermal transfer problem in a hypersonic simulation. A method that combines proper orthogonal decomposition and unassembled discrete empirical interpolation method is developed to construct the reduced-order modeling. First, high-dimensional original systems are projected on the optional basis generated by proper orthogonal decomposition, and the nonlinear term is further approximated by unassembled discrete empirical interpolation method. Then, a numerical integration method for the solution of the reduced system of nonlinear differential equations is provided. Case studies that use a classical hypersonic control surface model, in which the time history and spatial distribution of the thermal load are known a priori, are conducted to validate the accuracy and efficiency of the reduced-order modeling methodology and to assess the robustness of the reduced-order modeling for thermal solution. Results indicate the ability of reduced-order modeling to reduce the nonlinear system size with reasonable accuracy.

Mesh-Based and Meshfree Reduced Order Phase-Field Models for Brittle Fracture: One Dimensional Problems

Modelling brittle fracture by a phase-field fracture formulation has now been widely accepted. However, the full-order phase-field fracture model implemented using finite elements results in a nonlinear coupled system for which simulations are very computationally demanding, particularly for parametrized problems when the randomness and uncertainty of material properties are considered. To tackle this issue, we present two reduced-order phase-field models for parametrized brittle fracture problems in this work. The first one is a mesh-based Proper Orthogonal Decomposition (POD) method. Both the Discrete Empirical Interpolation Method (DEIM) and the Matrix Discrete Empirical Interpolation Method ((M)DEIM) are adopted to approximate the nonlinear vectors and matrices. The second one is a meshfree Krigingmodel. For one-dimensional problems, served as proof-of-concept demonstrations, in which Young’s modulus and the fracture energy vary, the POD-based model can speed up the online computations eight-times, and for the Kriging model, the speed-up factor is 1100, albeit with a slightly lower accuracy. Another merit of the Kriging’s model is its non-intrusive nature, as one does not need to modify the full-order model code.