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Flow inside a lid-driven cavity (LDC) is studied here to elucidate bifurcation sequences of the flow at super-critical Reynolds numbers (Re cr1 ) with the help of analyzing the time series at most energetic points in the flow domain. The implication of Re cr1 in the context of direct simulation of Navier-Stokes equation is presented here for LDC, with or without explicit excitation inside the LDC. This is aided further by performing detailed enstrophy-based proper orthogonal decomposition (POD) of the flow field. The flow has been computed by an accurate numerical method for two different uniform grids. POD of results of these two grids help us understand the receptivity aspects of the flow field, which give rise to the computed bifurcation sequences by understanding the similarity and differences of these two sets of computations. We show that POD modes help one understand the primary and secondary instabilities noted during the bifurcation sequences.

In the last decades, numerical simulation has experienced tremendous improvements driven by massive growth of computing power. Exascale computing has been achieved this year and will allow solving ever more complex problems. But such large systems produce colossal amounts of data which leads to its own difficulties. Moreover, many engineering problems such as multiphysics or optimisation and control, require far more power that any computer architecture could achieve within the current scientific computing paradigm. In this chapter, we propose to shift the paradigm in order to break the curse of dimensionality by introducing decomposition to reduced data. We present an extended review of data reduction techniques and intends to bridge between applied mathematics community and the computational mechanics one.The chapter is organized into two parts. In the first one bivariate separation is studied, including discussions on the equivalence of proper orthogonal decomposition (POD, continuous framework) and singular value decomposition (SVD, discrete matrices). Then, in the second part, a wide review of tensor formats and their approximation is proposed. Such work has already been provided in the literature but either on separate papers or into a pure applied mathematics framework. Here, we offer to the data enthusiast scientist a description of Canonical, Tucker, Hierarchical and Tensor train formats including their approximation algorithms. When it is possible, a careful analysis of the link between continuous and discrete methods will be performed.

Computational Intelligence (CI) techniques have shown great potential as a surrogate model of expensive physics simulation, with demonstrated ability to make fast predictions, albeit at the expense of accuracy in some cases. For many scientific and engineering problems involving geometrical design, it is desirable for the surrogate models to precisely describe the change in geometry and predict the consequences. In that context, we develop graph neural networks (GNNs) as fast surrogate models for physics simulation, which allow us to directly train the models on 2/3D geometry designs that are represented by an unstructured mesh or point cloud, without the need for any explicit or hand-crafted parameterization. We utilize an encoder-processor-decoder-type architecture which can flexibly make prediction at both node level and graph level. The performance of our proposed GNN-based surrogate model is demonstrated on 2 example applications: feature designs in the domain of additive engineering and airfoil design in the domain of aerodynamics. The models show good accuracy in their predictions on a separate set of test geometries after training, with almost instant prediction speeds, as compared to O(hour) for the high-fidelity simulations required otherwise.

We propose a parametric sampling strategy for reduction of large scale PDE systems with multidimensional input parametric spaces by leveraging models of different fidelity. The design of this methodology allows a user to adaptively sample points ad hoc from a discrete training set with no prior requirement of error estimators. It is achieved by exploiting low‐fidelity models throughout the parametric space to sample points using an efficient sampling strategy, and at the sampled parametric points, high‐fidelity models are evaluated to recover the reduced basis functions. The low‐fidelity models are then adapted with the reduced order models built by projection onto the subspace spanned by the recovered basis functions. The process continues until the low‐fidelity model can represent the high‐fidelity model adequately for all the parameters in the parametric space. Since the proposed methodology leverages the use of low‐fidelity models to assimilate the solution database, it significantly reduces the computational cost in the offline stage. The highlight of this article is to present the construction of the initial low‐fidelity model, and a sampling strategy based on the discrete empirical interpolation method. We test this approach on a 2D steady‐state heat conduction problem for two different input parameters and make a qualitative comparison with the classical greedy reduced basis method and with random selection of points. Further, we test the efficacy of the proposed method on a 9‐dimensional parametric non‐coercive elliptic problem and analyze the computational performance based on different tuning of greedy selection of points.

We propose a parametric sampling strategy for the reduction of large-scale PDE systems with multidimensional input parametric spaces by leveraging models of different fidelity. The design of this methodology allows a user to adaptively sample points ad hoc from a discrete training set with no prior requirement of error estimators.It is achieved by exploiting low-fidelity models throughout the parametric space to sample points using an efficient sampling strategy, and at the sampled parametric points, high-fidelity models are evaluated to recover the reduced basis functions. The low-fidelity models are then adapted with the reduced order models ( ROMs) built by projection onto the subspace spanned by the recovered basis functions. The process continues until the low-fidelity model can represent the high-fidelity model adequately for all the parameters in the parametric space. Since the proposed methodology leverages the use of low-fidelity models to assimilate the solution database, it significantly reduces the computational cost in the offline stage. The highlight of this article is to present the construction of the initial low-fidelity model, and a sampling strategy based on the discrete empirical interpolation method (DEIM). We test this approach on a 2D steady-state heat conduction problem for two different input parameters and make a qualitative comparison with the classical greedy reduced basis method (RBM), and further test on a 9-dimensional parametric non-coercive elliptic problem and analyze the computational performance based on different tuning of greedy selection of points.

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