Conference Agenda

Reduced Order Models (ROMs) have been widely used to efficiently solve large-scale problems in many fields including computational fluid dynamics (CFD) [1]. ROMs techniques allow to replace the expansive Full Order Model (FOM), by a ROM that captures the essential features of the system while significantly reducing the computational cost. In this work, we draw inspiration from [2] to implement a reduced basis (RB) method for model reduction of the Shallow Water Equations (SWEs) using Proper Orthogonal Decomposition (POD) and Artificial Neural Networks (ANNs). This method, referred to as POD-NN, starts with the POD technique to construct a reduced basis, and then makes us of an ANN to learn the associated coefficients in the reduced basis. It follows an offline-online strategy: the POD reduced basis along with the training of the ANN are performed in an offline stage, and then the surrogate model can be used for hyper-fast predictions. The process is non-intrusive since it does not require opening the black box of the FOM. The developed method is tested [3] on a real data set aiming at simulating an inundation of the Aude river (Southern France). The results show that the proposed method can achieve significant computational savings while maintaining satisfactory accuracy on the hydraulic variables of interest compared to the full-order hydraulic model. The proposed method is able to capture the key features of the SWEs in particular the wave propagation. Overall, the proposed non-intrusive POD-NN method offers a promising approach for ROM of SWEs while being affordable in view of fast real time inundation simulations.

[1] Benner, P., Schilders, W., Grivet-Talocia, S., Quarteroni, A., Rozza, G., & Miguel Silveira, L. (2020). Model Order Reduction: Volume 2: Snapshot-Based Methods and Algorithms (p. 348). De Gruyter.

[2] Hesthaven, J. S., & Ubbiali, S. (2018). Non-intrusive reduced order modeling of nonlinear problems using neural networks. Journal of Computational Physics, 363, 55-78.

[3] IMT-INSA, INRAE et al., “DassFlow: Data Assimilation for Free Surface Flows”, Open source computational software. https://www.math.univ-toulouse.fr/DassFlow

Let us consider wave propagation problems over 2-dimensional domains with piecewise-linear boundaries, possibly including scatterers. Under some assumptions on the initial conditions and forcing terms, we have proposed an approximation of the propagating wave as the sum of some special nonlinear space-time functions. Each term in this sum identifies a particular ray, modeling the result of a single reflection or diffraction effect. In this talk I will describe an algorithm for identifying such rays automatically, based on the domain geometry.

To showcase our proposed method, I will present several numerical examples, such as waves scattering off wedges and waves propagating through a room in presence of obstacles.

This contribution is motivated by the combined advantages of an integrated framework from CAD geometries to simulation in real time. In a typical workflow for design and shape optimization, multiple simulations are required for all possible designs represented by different geometrical parameters. This might entail a high computational cost in particular for real world, engineering applications. The development of efficient reduced order models that enable fast parametric analysis is essential for such applications. At the same time, the capabilities of splines and isogeometric analysis allow for flexible geometric design and higher-order continuity in the analysis. In CAD design, trimmed multi-patch geometries are widely used to represent complex shapes. The presence of geometric parameters introduces challenges for efficient reduced order modeling of problems formulated on such unfitted geometries. We propose a localized reduced basis method to circumvent the shortcomings of standard reduced order models in this context [1]. In this talk we present the developed strategy and address fast parametric analysis of problems in structural mechanics. The construction of efficient reduced order models for geometries described by multiple trimmed patches as well as their use in parametric shape optimization will be discussed. Numerical examples will be presented to demonstrate the accuracy and computational efficiency of the method.

[1] M. Chasapi, P. Antolin, A. Buffa, A localized reduced basis approach for unfitted domain methods on parameterized geometries, Comput. Methods Appl. Mech. Engrg. 410 (2023) 115997.

In many applications in Computer Aided Engineering, like parametric studies, structural optimization or virtual material design, a large number of almost similar models have to be simulated. Although the individual scenarios may differ only slightly in both space and time, the same effort is invested for every single new simulation with no account for experience and knowledge from previous simulations. Therefore, we have developed a method that combines data-based Model Order Reduction (MOR) and reanalysis, thus exploiting knowledge from previous simulation runs to accelerate computations in multi-query contexts. While MOR allows reducing model fidelity in space and time without significantly deteriorating accuracy, reanalysis uses results from previous computations as a predictor or preconditioner.

The workflow of our method, named Reduced Model Reanalysis (RMR), is divided into an offline and online phase. In the offline phase, data are generated to cover a wide range of the parameter space. From this data a surrogate model is learned in a reduced space using regression algorithms from the field of machine learning. Depending on the requirements of the system, different regression algorithms are favorable, e.g. linear regression, a k-nearest neighbor algorithm, a neural network, or a Gaussian process. The models are learned in the reduced space due to the prohibitively large number of degrees of freedom of the full finite element model. The reduced subspaces are obtained via a snapshot POD (proper orthogonal decomposition). In the online phase, approximations of all relevant solution quantities are obtained from the surrogate model. Their projection to the full space provides predictors that allow for an accelerated solution of the system in comparison to a standard structural mechanics computation.

In the case of nonlinear stability analysis this method can for example be used to accelerate the exact computation of critical points by the method of extended systems. Data generation in the offline phase is also accelerated by a newly developed adaptive time stepping scheme. With this scheme the number of steps to approach critical points with a path following scheme can be significantly reduced. Further potential fields of application of RMR are general nonlinear static and transient problems, with particular challenges as soon as path-dependence comes into play.