Serge Prudhomme

Clément Vella, Serge Prudhomme

We introduce an adjoint-based reduced-order model framework for fast and accurate estimation of quantities of interest for many-query linear problems. The method builds a reduced-order model with respect to the adjoint problem, thus bypassing the solution of the primal problem and drastically reducing computational cost. It creates a surrogate model that is independent of the loading configurations. It enables fast evaluation across multiple load cases and the generation of virtual charts to support decision-making. Numerical experiments on the Poisson equation and a plane-stress elasticity problem demonstrate that the adjoint reduced-order model converges rapidly, outperforms its primal counterpart, and provides reliable estimates of the quantities of interest. Importantly, it is often more practical to parameterize a kernel function than an entire set of external loads, making the method generic and particularly suited for early-stage prototyping and design optimization.

Noujoud Nader, Patrick Diehl, Marta D'Elia et al.

Local-nonlocal coupling approaches combine the computational efficiency of local models and the accuracy of nonlocal models. However, the coupling process is challenging, requiring expertise to identify the interface between local and nonlocal regions. This study introduces a machine learning-based approach to automatically detect the regions in which the local and nonlocal models should be used in a coupling approach. This identification process uses the loading functions and provides as output the selected model at the grid points. Training is based on datasets of loading functions for which reference coupling configurations are computed using accurate coupled solutions, where accuracy is measured in terms of the relative error between the solution to the coupling approach and the solution to the nonlocal model. We study two approaches that differ from one another in terms of the data structure. The first approach, referred to as the full-domain input data approach, inputs the full load vector and outputs a full label vector. In this case, the classification process is carried out globally. The second approach consists of a window-based approach, where loads are preprocessed and partitioned into windows and the problem is formulated as a node-wise classification approach in which the central point of each window is treated individually. The classification problems are solved via deep learning algorithms based on convolutional neural networks. The performance of these approaches is studied on one-dimensional numerical examples using F1-scores and accuracy metrics. In particular, it is shown that the windowing approach provides promising results, achieving an accuracy of 0.96 and an F1-score of 0.97. These results underscore the potential of the approach to automate coupling processes, leading to more accurate and computationally efficient solutions for material science applications.