Prague, Czech Republic, 22nd–27th September 2024
The conference starts on 22nd September in the evening (registration/welcome drink), the first lecture takes place on Monday 23rd September, the last lecture takes place on Friday 27th September 2024 at noon. The preliminary program might help you with your travel arrangements. (Last update on 9th May 2024.).
The plate bending or Babuska paradox refers to the failure of convergence when a linear bending problem with simple support boundary conditions is approximated using polygonal domain approximations. We provide an explanation based on a variational viewpoint and identify sufficient conditions that avoid the paradox and which show that boundary conditions have to be suitably modified. We show that the paradox also matters in nonlinear thin-sheet folding problems and devise approximations that correctly converge to the original problem.
To control instabilities in architected materials is a new challenge in which solid and structural mechanics become fully complementary. Homogenization techniques will be presented for periodic elastic structures subject to prestress, to give evidence to the emergence of material instabilities such as shear bands, occurring for both for compressive [1] and tensile [2] prestress. Moreover, the architecture of the analyzed structures leads to the emergence of multiple band gaps, flat bands, and Dirac cones [3]. The experience gained on structural flutter [4, 5] is exploited to implement a new concept, namely, the possibility of producing a Hopf bifurcation in a continuous medium. This possibility is proven through a rigorous application of Floquet-Bloch wave asymptotics, which yields an unsymmetric acoustic tensor governing the incremental dynamics of the effective material [6]. The latter represents the incremental response of a hypo-elastic solid, which does not follow from a strain potential and thus apparently breaks the wall of hyperelasticity, leading to non-Hermitian mechanics. The discovery of elastic materials capable of collecting or releasing energy in closed strain cycles through interactions with the environment introduces new micro and nano technologies and finds definite applications, for example, in the field of energy harvesting.
Acknowledgements: Financial support from ERC-ADG-2021-101052956-BEYOND is gratefully acknowledged.
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In this talk, I will focus on some models for the motion of rigid bodies immersed in a fluid. The different variants of this model have motivated many references in the past 20 years. Whatever the properties of the fluid (incompressible vs compressible, viscous vs inviscid, ...) a specific difficulty arises in the mathematical treatment because of collisions between the bodies or between one body and the container boundary. Such collisions are difficult to handle because they induce a geometrical singularity in the fluid domain and they require to introduce an advanced physical description of solid/solid contacts in the modeling. It is then mandatory to discuss the possibility of a collision and more generally to obtain an analytical description of the flow when solid bodies are close to contact. Such a description is also of high importance to the description of dense suspensions. In this talk, I will review results on this topic discussing the importance of the fluid and body properties and discuss related open problems.
In this talk, we will explore the optimal approximation rates for the critical constants associated with fundamental inequalities in partial differential equations (PDE) analysis, specifically the Hardy and Sobolev inequalities. We will outline a systematic methodology to achieve sharp convergence rates.
Additionally, we will present several applications within the context of numerical approximation for parabolic evolution problems.
We present a mechanically consistent model for the numerical simulation of fluid-structure interactions (FSI) with contact [2]. The main novelty compared to previous works is the consideration of seepage through a porous layer of co-dimension one during contact. For the latter, a Darcy model is considered in a thin porous layer attached to a solid boundary in the limit of infinitesimal thickness. To obtain a stable and efficient numerical algorithm, we use a relaxation of the contact conditions and a weak imposition of the FSI coupling and contact conditions by means of a unified Nitsche approach [1]. To test the approach, we present a benchmark setting of a falling elastic ball including experimental results [3].
Joint work with E. Burman and M. A. Fernández.
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