In this paper, we propose a general framework for projection-based model order reduction assisted by deep neural networks. The proposed methodology, called ROM-net, consists in using deep learning techniques to adapt the reduced-order model to a stochastic input tensor whose nonparametrized variabilities strongly influence the quantities of interest for a given physics problem. In particular, we introduce the concept of dictionary-based ROM-nets, where deep neural networks recommend a suitable local reduced-order model from a dictionary. The dictionary of local reduced-order models is constructed from a clustering of simplified simulations enabling the identification of the subspaces in which the solutions evolve for different input tensors. The training examples are represented by points on a Grassmann manifold, on which distances are computed for clustering. This methodology is applied to an anisothermal elastoplastic problem in structural mechanics, where the damage field depends on a random temperature field. When using deep neural networks, the selection of the best reduced-order model for a given thermal loading is 60 times faster than when following the clustering procedure used in the training phase.

Summary In this work, we propose a framework that constructs reduced‐order models for nonlinear structural mechanics in a nonintrusive fashion and can handle large‐scale simulations. Three steps are carried out: (i) the production of high‐fidelity solutions by commercial software, (ii) the offline stage of the model reduction, and (iii) the online stage where the reduced‐order model is exploited. The nonintrusivity assumes that only the displacement field solution is known, and the proposed framework carries out operations on these simulation data during the offline phase. The compatibility with a new commercial code only needs the implementation of a routine converting the discretized solution into our in‐house data format. The nonintrusive capabilities of the framework are demonstrated on numerical experiments using commercial versions of Z‐set and Ansys Mechanical. The nonlinear constitutive equations are evaluated by using an external plugin. The large‐scale simulations are handled using domain decomposition and parallel computing with distributed memory. The features and performances of the framework are evaluated on two numerical applications involving elastoviscoplastic materials: the second one involves a model of high‐pressure blade, where the framework is used to extrapolate cyclic loadings in 6.5 hours, whereas the reference high‐fidelity computation would take 9.5 days.

We consider the dictionary-based ROM-net (Reduced Order Model) framework [Daniel et al., Adv. Model. Simul. Eng. Sci. 7 (2020) https://doi.org/10.1186/s40323-020-00153-6] and summarize the underlying methodologies and their recent improvements. The object of interest is a real-life industrial model of an elastoviscoplastic high-pressure turbine blade subjected to thermal, centrifugal and pressure loadings. The main contribution of this work is the application of the complete ROM-net workflow to the quantification of the uncertainty of dual quantities on this blade (such as the accumulated plastic strain and the stress tensor), generated by the uncertainty of the temperature loading field. The dictionary-based ROM-net computes predictions of dual quantities of interest for 1008 Monte Carlo draws of the temperature loading field in 2 h and 48 min, which corresponds to a speedup greater than 600 with respect to a reference parallel solver using domain decomposition, with a relative error in the order of 2%. Another contribution of this work consists in the derivation of a meta-model to reconstruct the dual quantities of interest over the complete mesh from their values on the reduced integration points.

In the following paper, we consider the problem of constructing a time stable reduced order model of the 3D turbulent and incompressible Navier–Stokes equations. The lack of stability associated with the order reduction methods of the Navier–Stokes equations is a well-known problem and, in general, it is very difficult to account for different scales of a turbulent flow in the same reduced space. To remedy this problem, we propose a new stabilization technique based on an a priori enrichment of the classical proper orthogonal decomposition (POD) modes with dissipative modes associated with the gradient of the velocity fields. The main idea is to be able to do an a priori analysis of different modes in order to arrange a POD basis in a different way, which is defined by the enforcement of the energetic dissipative modes within the first orders of the reduced order basis. This enables us to model the production and the dissipation of the turbulent kinetic energy (TKE) in a separate fashion within the high ranked new velocity modes, hence to ensure good stability of the reduced order model. We show the importance of this a priori enrichment of the reduced basis, on a typical aeronautical injector with Reynolds number of 45,000. We demonstrate the capacity of this order reduction technique to recover large scale features for very long integration times (25 ms in our case). Moreover, the reduced order modeling (ROM) exhibits periodic fluctuations with a period of 2 . 2 ms corresponding to the time scale of the precessing vortex core (PVC) associated with this test case. We will end this paper by giving some prospects on the use of this stable reduced model in order to perform time extrapolation, that could be a strategy to study the limit cycle of the PVC.