arXiv subjects

Jean Ruel

Publications and source records attributed to Jean Ruel.

4 recordsLinked to original sources

arXiv2026

Certification of PGD reduced-order models with separated spatial variables

Model order reduction techniques have become an attractive approach for obtaining fast approximations of multidimensional problems. Besides computational efficiency, ensuring the reliability of the resulting approximations is of primary importance. This work focuses on the certification of PGD-based reduced-order models based on the separation of spatial variables, which are particularly well suited to plate and shell geometries. Considering diffusion problems defined in plate-like domains, we introduce a guaranteed global error estimate associated with the PGD approximation. To this end, the error bounds are derived from the Constitutive Relation Error (CRE) method. The main difficulty of this approach lies in the construction of equilibrated fluxes, for which a dedicated procedure is proposed. Based on the resulting estimator, an adaptive strategy is developed to control both the discretization error and the number of PGD modes. This certification procedure is further extended to the error control in quantities of interest. We provide several numerical examples illustrating the reliability and efficiency of our procedure.

math.NA↗

arXiv2026

Towards a new PGD strategy for the simulation of slender structures

Effective models for slender structures derived from well-known plate (or shell) theories are justified within the limit of a small thickness, and may therefore prove limited for intermediate slenderness. On the other hand, direct 3D simulation of such structures is sub-optimal because it does not take advantage of the presence of small dimensions in some directions and is sometimes too costly and ill-conditioned. In this context, the Proper Generalized Decomposition (PGD) method, a model order reduction method based on a modal representation of the solution with separation of variables, makes it possible to obtain a 3D solution with 2D resolution complexity. In this work, an analysis of the links between the PGD reduced order model and the solution provided by plate theory is carried out using asymptotic expansion. It is shown that, in the limit of large slenderness, the first mode of the PGD exhibits Kirchhoff-Love type kinematics, but only corresponds to the asymptotic solution in very special cases of loading and boundary conditions. To capture the asymptotic solution, a new PGD strategy is introduced consisting of computing the first two modes simultaneously. We also demonstrate that the PGD is subject to shear locking, and we show how to deal with it. Numerical experiments are provided, demonstrating the interest of this approach and confirming the theoretical analysis.

math.NA↗

arXiv2025

Quantum remeshing and efficient encoding for fracture mechanics

We present a variational quantum algorithm for structural mechanical problems, specifically addressing crack opening simulations that traditionally require extensive computational resources. Our approach provides an alternative solution for a relevant 2D case by implementing a parametrized quantum circuit that stores nodal displacements as quantum amplitudes and efficiently extracts critical observables. The algorithm achieves optimal nodal displacements by minimizing the elastic energy obtained from finite element method. The energy is computed with only a polylogarithmic number of measurements. Extracting relevant scalar observables such as the stress intensity factor is then done efficiently on the converged solution. To validate the scalability of our approach, we develop a warm start strategy based on a remeshing technique that uses coarse solutions to circumvent barren plateaus in the optimization landscape of the more refined problems. Our method has been experimentally validated on Quandela's photonic quantum processor Ascella and comprehensive numerical simulations demonstrate its scalability across increasingly complex quantum systems.

quant-ph↗

arXiv2023

Maximum principle preserving time implicit DGSEM for linear scalar hyperbolic conservation laws

We investigate the properties of the high-order discontinuous Galerkin spectral element method (DGSEM) with implicit backward-Euler time stepping for the approximation of hyperbolic linear scalar conservation equation in multiple space dimensions. We first prove that the DGSEM scheme in one space dimension preserves a maximum principle for the cell-averaged solution when the time step is large enough. This property however no longer holds in multiple space dimensions and we propose to use the flux-corrected transport limiting [Boris and Book, J. Comput. Phys., 11 (1973)] based on a low-order approximation using graph viscosity to impose a maximum principle on the cell-averaged solution. These results allow to use a linear scaling limiter [Zhang and Shu, J. Comput. Phys., 229 (2010)] in order to impose a maximum principle at nodal values within elements. Then, we investigate the inversion of the linear systems resulting from the time implicit discretization at each time step. We prove that the diagonal blocks are invertible and provide efficient algorithms for their inversion. Numerical experiments in one and two space dimensions are presented to illustrate the conclusions of the present analyses.

math.NA↗