Model order reduction for numerical simulation of slender structures

Abstract:

Slender structures such as plates or shells are common in mechanical systems. This is particularly true in the automotive (metal sheets) and aerospace (composite panels) sectors. Numerical simulation of these three-dimensional (3D) structures, with at least one dimension smaller than the others, is therefore of current practical interest in engineering. Simplifying this problem is useful, if not necessary, to keep computational costs down.
On the one hand, effective models for slender structures derived from well-known theories (e.g. the Kirchhoff-Love theory for plates) are justified within the limit of a small thickness, and may therefore prove limited for intermediate thicknesses. On the other hand, direct 3D simulation of such structures is sub-optimal because it does not take advantage of small dimensions and is sometimes too costly. Model order reduction methods developed over the last few decades are therefore attractive in this context. In particular, the Proper Generalized Decomposition (PGD) technique, based on a modal representation of the solution with separation of variables, makes it possible to separate the plane coordinates from the transverse coordinate according to the usual approach in this field. A 3D solution is thus obtained with 2D resolution complexity, taking advantage of the particular geometry of the structure.
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 this model in very special cases of loading and boundary conditions.
To capture the asymptotic solution from the first PGD computation sequence, a new PGD strategy is introduced, which consists of computing the first two modes simultaneously. Numerical experiments show the interest of this approach both in terms of accuracy and computational cost, and confirm the theoretical analysis.

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.