Titles and Abstracts:
Luca Heltai (Pisa):Non-matching coupling in fluid-structure interaction: constraints, singular forcing, and dimensional reduction
Abstract: Fluid–structure interaction problems often involve coupling conditions imposed on interfaces whose geometry is complex, thin, or evolving in time. In many relevant regimes, enforcing such conditions on matching discretizations is either impractical or conceptually inadequate, motivating the use of non-matching and immersed formulations.
In this talk, we present an analytical overview of non-matching coupling methods for fluid–structure interaction, interpreting interface conditions as functional constraints, imposed on sets of varying dimensions. We focus on Lagrange multiplier formulations (classical, distributed, and reduced) and discuss their well-posedness and stability properties at both the continuous and discrete levels.
A central theme of the talk is the analysis of singular forcing terms generated by non-matching constraints.
On the one hand, variational formulations of non-matching coupling typically induce a loss of global regularity in the solution. We show that this loss is in fact a purely local phenomenon, and we characterize it using weighted Sobolev spaces, thereby recovering optimal approximation properties away from the interface.
On the other hand, non-matching fluid–structure interaction formulations can be reinterpreted as Navier–Stokes-type problems with singular forcing terms. While classical Immersed Boundary Methods employ regularized Dirac distributions primarily for implementation purposes, we provide a general framework showing that regularization and analysis are not separate issues. In particular, we establish a rigorous connection between regularized and variational formulations, proving that suitably chosen regularizations lead to equivalent finite element approximations, with comparable accuracy and computational cost.
We then address dimensional reduction strategies for slender or thin structures, showing how mixed-dimensional models naturally arise as asymptotic limits of non-matching coupling formulations. We discuss how reduced Lagrange multiplier spaces can be designed to preserve well-posedness and stability uniformly with respect to vanishing geometric scales.
The presented results provide a unified functional-analytic perspective on a class of non-matching methods used in fluid–structure interaction and related problems, highlighting the mechanisms underlying their stability, accuracy, and robustness across dimensions.
Carlo Mantegazza (Napoli): Quasi-convexity in the Riemannian setting
Abstract:We introduce a notion of quasiconvexity for integrands defined on the tangent bundle of a Riemannian manifold. We prove that this condition characterizes the sequential lower semicontinuity of the associated integral functionals with respect to the weak$^*$ topology of $W^{1,\infty}$, generalizing the classical Euclidean results by Morrey and Acerbi--Fusco.
Moreover, we also extend the notions of polyconvexity and rank--one convexity to this context and establish the hierarchy between polyconvexity, quasiconvexity, and rank--one convexity, as in the Euclidean setting.
Joint work with Aurora Corbisiero and Chiara Leone.
Guofang Wang (Freiburg): Analysis of spinor fields
Last modified:Tue May 28 17:47:49 CEST 2024