Forthcoming Seminars › Seminar Series

OxPDE Lunchtime Seminar

Thu, 22/01/2009
12:30 		Mason Porter (University of Oxford) OxPDE Lunchtime Seminar Add to calendar Gibson 1st Floor SR
Wave Propagation in One-Dimensional Granular Lattices
I will discuss the investigatation of highly nonlinear solitary waves in heterogeneous one-dimensional granular crystals using numerical computations, asymptotics, and experiments. I will focus primarily on periodic arrangements of particles in experiments in which stiffer/heavier stainless stee are alternated with softer/lighter ones. The governing model, which is reminiscent of the Fermi-Pasta-Ulam lattice, consists of a set of coupled ordinary differential equations that incorporate Hertzian interactions between adjacent particles. My collaborators and I find good agreement between experiments and numerics and gain additional insight by constructing an exact compaction solution to a nonlinear partial differential equation derived using long-wavelength asymptotics. This research encompasses previously-studied examples as special cases and provides key insights into the influence of heterogeneous, periodic lattice on the properties of the solitary waves. I will briefly discuss more recent work on lattices consisting of randomized arrangements of particles, optical versus acoustic modes, and the incorporation of dissipation.
Mon, 26/01/2009
13:30 		Pierluigi Cesana (SISSA, Trieste, Italy) OxPDE Lunchtime Seminar Add to calendar Gibson 1st Floor SR
Analysis of variational models for nematic liquid crystal elastomers
The relaxation of a free-energy functional which describes the order-strain interaction in nematic liquid crystal elastomers is obtained explicitly. We work in the regime of small strains (linearized kinematics). Adopting the uniaxial order tensor theory or Frank model to describe the liquid crystal order, we prove that the minima of the relaxed functional exhibit an effective biaxial microstructure, as in de Gennes tensor model. In particular, this implies that the response of the material is soft even if the order of the system is assumed to be fixed. The relaxed energy density satisfies a solenoidal quasiconvexification formula.
Thu, 29/01/2009
12:30 		Richard Norton (University of Oxford) OxPDE Lunchtime Seminar Add to calendar Gibson 1st Floor SR
Convergence analysis of the planewave expansion method for band gap calculations in photonic crystal fibres
Modelling the behaviour of light in photonic crystal fibres requires solving 2nd-order elliptic eigenvalue problems with discontinuous coefficients. The eigenfunctions of these problems have limited regularity. Therefore, the planewave expansion method would appear to be an unusual choice of method for such problems. In this talk I examine the convergence properties of the planewave expansion method as well as demonstrate that smoothing the coefficients in the problem (to get more regularity) introduces another error and this cancels any benefit that smoothing may have.
Thu, 05/02/2009
12:30 		Duvan Henao (University of Oxford) OxPDE Lunchtime Seminar Add to calendar Gibson 1st Floor SR
Sequential weak continuity of the determinant and the modelling of cavitation and fracture in nonlinear elasticity
Motivated by the tensile experiments on titanium alloys of Petrinic et al (2006), which show the formation of cracks through the formation and coalescence of voids in ductile fracture, we consider the problem of formulating a variational model in nonlinear elasticity compatible both with cavitation and with the appearance of discontinuities across two-dimensional surfaces. As in the model for cavitation of Müller and Spector (1995) we address this problem, which is connected to the sequential weak continuity of the determinant of the deformation gradient in spaces of functions having low regularity, by means of adding an appropriate surface energy term to the elastic energy. Based upon considerations of invertibility we are led to an expression for the surface energy that admits a physical and a geometrical interpretation, and that allows for the formulation of a model with better analytical properties. We obtain, in particular, important regularity properites of the inverses of deformations, as well as the weak continuity of the determinants and the existence of minimizers. We show further that the creation of surface can be modelled by carefully analyzing the jump set of the inverses, and we point out some connections between the analysis of cavitation and fracture, the theory of SBV functions, and the theory of cartesian currents of Giaquinta, Modica and Soucek. (Joint work with Carlos Mora-Corral, Basque Center for Applied Mathematics).
Thu, 26/02/2009
12:30 		Frédéric de Gournay (Université Versailles-Saint-Quentin) OxPDE Lunchtime Seminar Add to calendar Gibson 1st Floor SR
Robust shape optimization via the level-set method
We are interested in optimizing the compliance of an elastic structure when the applied forces are partially unknown or submitted to perturbations, the so-called "robust compliance". For linear elasticity,the compliance is a solution to a minimizing problem of the energy. The robust compliance is then a min-max, the minimum beeing taken amongst the possible displacements and the maximum amongst the perturbations. We show that this problem is well-posed and easy to compute. We then show that the problem is relatively easy to differentiate with respect to the domain and to compute the steepest direction of descent. The levelset algorithm is then applied and many examples will explain the different mathematical and technical difficulties one faces when one tries to tackle this problem.
Tue, 03/03/2009
15:30 		V. A. Solonnikov (Steklov Institute of Mathematics) OxPDE Lunchtime Seminar Add to calendar Gibson 1st Floor SR
Linearisation principle for a system of equations of mixed type
Wed, 11/03/2009
13:00 		François Genoud (OxPDE, University of Oxford) OxPDE Lunchtime Seminar Add to calendar Gibson 1st Floor SR
Bifurcation and orbital stability of standing waves for some nonlinear Schrödinger equations
The aim of my talk is to present the work of my PhD Thesis and my current research. It is concerned with local/global bifurcation of standing wave solutions to some nonlinear Schrödinger equations in $ \mathbb{R}^N \ (N\geq1) $ and with stability properties of these solutions. The equations considered have a nonlinearity of the form $ V(x)|\psi|^{p-1}\psi $, where $ V:\mathbb{R}^N\to\mathbb{R} $ decays at infinity and is subject to various assumptions. In particular, $ V $ could be singular at the origin. Local/global smooth branches of solutions are obtained for the stationary equation by combining variational techniques and the implicit function theorem. The orbital stability of the corresponding standing waves is studied by means of the abstract theory of Grillakis, Shatah and Strauss.
Thu, 12/03/2009
12:30 		Habib Ammari (CNRS) OxPDE Lunchtime Seminar Add to calendar Gibson 1st Floor SR
Multi-scale elastic imaging

For any corrections/updates to these listings, please contact seminars [-at-] maths [dot] ox [dot] ac [dot] uk.

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