## Dispersive effective behavior of high-contrast periodic media

Abstract: I will discuss my recent work with Y. Ershova and A. Kiselev, demonstrating that spectral problems for quantum graphs with rapidly oscillating high-contrast weights are asymptotically equivalent to “homogenised” models with energy-dependent interface conditions. We show that these asymptotically equivalent models are directly related (in the sense of Schur-Frobenius duality) to models for time-dispersive media, which in the time domain involve memory, and we characterize the corresponding time convolution kernels explicitly.

Read More## Junction of models of different dimension: applications in biology and engineering

ABSTRACT: The partial differential equations (PDEs) set in thin structures are considered. These domains are some finite unions of thin rectangles (in 2D settings) or cylinders (in 3D settings) depending on small parameter epsilon << 1 that is, the ratio of the thickness of the rectangle (cylinder) to its length. An asymptotic analysis of the solutions of these PDEs is applied for justification of the method of asymptotic partial decomposition of domain (MAPDD) introduced in [1] and developed in [2-3]. This method reduces the 2D or 3D model of a structure to some model of hybrid...

Read More## Robust shape optimisation with small uncertainties

Abstract: In this talk we propose two approaches for dealing with small uncertainties in geometry and topology optimisation of structures. Uncertainties occur in the loadings, the material properties, the geometry or the imposed vibration frequency. A first approach, in a worst-case scenario, amounts to linearise the considered cost function with respect to the uncertain parameters, then to consider the supremum function of the obtained linear approximation, whichcan be rewritten as a more `classical’ function of the design, owing to standardadjoint techniques from optimal...

Read More## From the fluid flow through a partially perforated domain to Darcy’s law: the homogenization of the natural Stokes problem

Abstract: One considers the Stokes problem in the case of a partially perforated domain (the perforated part models a porous medium, the boundary of which is just assumed Lipschitz). A single fluid is considered and the external force in the porous medium is properly scaled. The purpose of the talk is to present a proof of the homogenization of this problem using the periodic unfolding method. In the homogenized limit, the problem become an uncoupled pair of problems, one is the standard Stokes problem in the normal part of the domain, the other an elliptic equation for the limit of...

Read More## Hierarchy of pre-strained behaviors for thin materials

ABSTRACT: A hierarchy of four models for thin structures was first obtained by means of a syste- matic, but formal, approach in [5]. Namely, the nonlinear membrane model, the nonlinear bending model, the slightly nonlinear von Ka`rm`an model and the linear model were reco- vered. Emphasis was put on the loading order of magnitude and on the induced magnitude of the deformations. A rigorous variational derivation of the nonlinear membrane model was then given in [7]. In particular, the degeneracy under compression was exhibited. Then, [6] provided a rigidity result that allowed to rigorously...

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