A posteriori error estimation for PGD reduced order models

Abstract:

Although numerical simulation has become a common tool for various engineering activities, standard numerical methods can prove limited when it comes to simulating complex multi-dimensional models in real time. In this context, model reduction methods have been widely developed in recent decades, providing an effective solution when fast predictions are required. If the efficiency of simulation techniques is an important objective, so is the reliability of the results obtained.

The Proper Generalized Decomposition (PGD) is a model reduction method based on a modal representation of the solution with separation of variables, and has proven its efficiency in many applications. The approximation error in PGD comes mainly from two sources: the truncation of the modal representation and the discretization error linked to the underlying numerical method used to compute modes. In order to assess the accuracy of PGD solutions, some a posteriori error estimation tools have been developed, using extensions of classical verification procedures in finite element analysis, which will be briefly reviewed in this presentation. These tools lead to error evaluation strategies for parametric PGD solutions or PGD solutions based on the separation of space and time.

In this seminar, I will present applications where space variables are separated one from another in the PGD decomposition. This strategy is particularly relevant for applications related to plate or shell geometries. The present work introduces an a posteriori error estimator, taking all error sources into account, with guaranteed bounds for the verification of PGD reduced models with separation of space variables. The estimator that we introduce make it possible to certify the reduced model as well as to drive an adaptive strategy. I will also show numerical examples illustrating the efficiency of our procedure.

Short bio:

Jean Ruel is a PhD candidate at the Navier laboratory since October 2023, under the supervision of Frédéric Legoll, Arthur Lebée and Ludovic Chamoin (LMPS). He graduated from ENS Paris-Saclay in mechanical engineering and is particularly interested in computational mechanics. His research focuses on the development of robust and certified numerical methods for the simulation of slender structures, using model order reduction approaches.