Poster session / Buffet lunch

Vinamra Agrawal
California Institute of Technology, USA

Shock waves in composites and ferroic materials

Behnam Aminahmadi
University of Antwerp, Belgium

Dislocation-mediated relaxation in nanograinedcolumnar palladium films revealed by on-chiptime-resolved HRTEM testing

Pd creep-abstract pire workshop final.pdf

Paul Cazeaux
University of Minnesota, USA

Cauchy-Born rule for incommensurate systems of coupled atomic chains

One of the many exciting perspectives opened by the recent experimental discovery of 2D materials, such as graphene or boron nitride, is the possibility of stacking a few layers of such materials. In principle, one can tune the properties of these heterostructures [Geim et. al., 2013], such as elasticity, conductivity, electronic and optical properties... To explore the possibilities of this atomic-scale Lego game, it is necessary to develop multiscale models and adapted numerical methods.
A particular difficulty encountered in this study is the incommensurate character resulting from the coupling of periodic lattices associated with each monolayer. The difference between cristalline structure of the materials, or the rotation of one layer relative to another results in a quasi-periodic structure, sometimes presenting a Moire pattern or a super-lattice.

In this poster, we present a first step towards the mathematical understanding and analysis of such structures and methods. We study the derivation of a Cauchy-Born rule for the elastic response to stretching a system of two periodic coupled atomic chains. We will in particular focus on the case where the respective periods are incommensurate. The two chains interact through an interatomic potential such as Lennard-Jones. When the length of the system grows to infinity, we show that the elastic response converges uniformly to a homogenized response. We further study the impact of the incommensurate character on the convergence rate of numerical approaches using the tools of discrepancy theory.

Two cases are studied in particular: first, two rigid, parallel chains, and next a relaxed system allowing in-plane ripples. In this second case, the equations of the model can be seen as an innovating application of the generalized Frenkel-Kontorova model. In particular, the minimum energy configuration can be analytically and numerically studied by the tools of Aubry-Mather theory.

Lukas Döring
RWTH Aachen University, Germany

Reduced models for domain walls in soft ferromagnetic films

In soft ferromagnetic films of moderate thickness, the transition layers ("domain walls") that separate domains of constant magnetization may favor a two scale-structure, combining a stray-field free domain wall in the core of the transition with logarithmically decaying tails that complete the rotation between neighboring domains.

On this poster, reduced models for an isolated domain wall in an infinitely extended film (joint work with R. Ignat and F. Otto) and for periodic systems of parallel walls with potentially interacting tails are presented that quantify the above-mentioned splitting of domain walls into a stray-field free core and logarithmic tails (both are obtained from the Landau-Lifshitz energy via Gamma convergence).

In the case of periodic, interacting walls, also a prediction of the average hard-axis magnetization in the domains (as a function of the domain width and the strength of an external hard-axis magnetic field) is available.

For CoFeB films, provided the film thickness is not too large, the predicted hard-axis magnetization agrees well with corresponding experimental data (joint work with C. Hengst, R. Schäfer and F. Otto).

John Gemmer
Brown University, USA

Isometric Immersions and Self Similar Buckling in Non-Euclidean Elastic Sheets

The edge of torn elastic sheets and growing leaves often form a hierarchical buckling pattern. Within non-Euclidean plate theory this complex morphology can be understood as low bending energy isometric immersions of Riemannian metrics. In this poster we show that for a large class of growth profiles there exist periodic and self-similar deformations of the sheet with vanishing in-plane strain. The construction of these surfaces consists of gluing together local solutions to the isometric immersion problem along "lines of inflection" and "branch points" in such a manner that the resulting surface has finite bending energy. For hyperbolic non-Euclidean sheet, complex wrinkling patters are thus possible and our results identify the key the regularity of the isometric immersion plays in determining the global structure of a non-Euclidean elastic sheet.

Arianna Giunti
Max Planck Insitute for Mathematics in the Sciences, Germany

Quantitative stochastic homogenization: Green's function estimates for elliptic systems

Carolin Kreisbeck
Universität Regensburg, Germany

Homogenization of layered materials with rigid components in single-slip finite plasticity

Abstract_Kreisbeck.pdf

Tim Laux
Max Planck Institute for Mathematics in the Sciences, Germany

Thresholding schemes for geometric flows

The thresholding scheme by Merriman, Bence and Osher is an algorithm to construct solutions to the mean-curvature flow equation. We prove convergence of two variants of the scheme, one for multi-phase mean-curvature flow and one for volume-preserving mean-curvature flow. The results establish convergence towards a weak formulation in the
BV-framework. The proofs are based on the interpretation of the schemes as minimizing movement schemes by Esedoglu and Otto. This interpretation means that the schemes preserve the gradient flow structure. In this sense, the proofs are similar to the convergence results of Almgren, Taylor and Wang and Luckhaus and Sturzenhecker.
This is based on joint works with Felix Otto and Drew Swartz.

Faizan Nazar
University of Warwick, United Kingdom

Locality of the TFW equations

In this poster I will discuss the existence and uniqueness of a coupled system of partial differential equations that arises from minimising the Thomas-Fermi-von Weizsäcker energy functional for general infinite nuclear arrangements. This gives rise to stability estimates, which give pointwise control of the electron density in terms of a local nuclear defect. We then discuss the applications of this result, including the neutrality of local defects in TFW theory and the lattice relaxation problem.​

Grigor Nika
Worcester Polytechnic Institute, USA

RATE OF CONVERGENCE AND CORRECTORS FOR A MULTI-SCALE MODEL OF DILUTE EMULSIONS

In this study we are interested in a problem of dilute emulsions of two immiscible viscous fluids, in which one is distributed in the other in the form of droplets of arbitrary shape, with non-uniform surface tension due to surfactants. The problem includes an essential kinematic condition on the droplets. In the periodic homogenization framework, it can be shown using Mosco-convergence that, as the size of the droplets converges to zero faster than the distance between the droplets, the emulsion behaves in the limit like the continuous phase. Here we determine the rate of convergence of the velocity field for the emulsion to that of the velocity for the one fluid problem and in addition, we determine the corrector in terms of the bulk and surface polarization tensors.

Ethan O'Brien
Courant Institute (NYU), USA

Twisting of an Elastic Ribbon

An elastic ribbon, when twisted and held with low tension, will form wrinkles in the center. We analyze this by proving an upper and lower bound for an energy functional in terms of the ribbon's thickness. We also find some information about the low energy states (an in particular the minimizer): we get a rigidity result for the outer edges of the ribbon and estimates on the amplitude of the wrinkles in the interior. This work uses an energy functional reminiscent of von Kármán plate theory, derived formally by considering small deflections from a helicoid. Chopin, Démery and Davidovitch studied the relaxed problem using a closely related energy.

This is joint work with Robert V. Kohn.

Ananya Renuka Balakrishna
University of Oxford, United Kingdom

Study of polarisation patterns in ferroelectrics using a phase-field model

Matthias Ruf
Technical University Munich, Germany

Homogenization of interfacial energies defined on random lattices

Thilo Simon
MPI MIS, Germany

Rigidity of Shape Memory Alloys

We study a geometrically linear model for shape memory alloys undergoing cubic-to-tetragonal transformations. The aim is to explain the alignment of habit planes by energy minimization.

Jake Steiner
University of Maryland, College Park, USA

Microstructural Origin of Magnetostriction in FeGa and FePd

Both FePd and FeGa show large magnetostriction in the cubic state: ≈200 and ≈400 ppm, respectively. These large values are difficult to reconcile with classical Joulian magnetostriction. We used magnetostriction and magnetic torque measurements as well as microstructural imaging to show that both large values are of microstructural origin. In FePd, the magnetostriction is a property of the well-known tweed structure and hence isotropic; in FeGa, coherent magnetically and elastically self-sufficient microcells form whose reorientation under applied field dictates magnetostrictive behavior. The unique microstructures also lead to linear and completely reversible magnetization characters for both alloys.

Drew Swartz
Purdue University, USA

Dynamics of a Second Order Gradient Model for Phase Transitions

In 2000 Fonseca and Mantegazza introduced a second order gradient theory for phase transitions. The model is similar in spirit to the Ginzburg-Landau model, which is a first order model. In their work, Fonseca and Mantegazza showed that their energy Gamma-converges to a perimeter functional. The current project examines the gradient flow dynamics for the Fonseca-Mantegazza energy. The corresponding evolution equation is fourth order, thus creating some interesting difficulties in its analysis. We analyze properties of the optimal transition profile through a combination of analytical and numerical techniques. Then in the radially symmetric setting, we use this to demonstrate that the gradient flow dynamics converges to motion by mean curvature. This is joint work with Prof. Aaron Yip.

Sergey Tikhomirov
Max Planck Institute for Mathematics in the Science, Deutschland

Reaction-diffusion equations with hysteresis

We consider reaction-diffusion equations with discontinuous hysteretic nonlinearities. We connect these equations with free boundary problems and introduce a related notion of spatial transversality for initial data and solutions. We assert that the equation with transverse initial data possesses a unique solution, which remains transverse for some time. At a moment when the solution becomes nontransverse, we discretize the spatial variable and analyze the resulting lattice dynamical system with hysteresis. In particular,
we discuss a new pattern formation mechanism — rattling, which indicates how one should reset the continuous model to make it well posed.

Ian Tobasco
Courant Institute of Mathematical Sciences, USA

Energy scaling laws for an axially compressed thin elastic cylinder

A longstanding open problem in elasticity is to identify the minimum energy scaling law of a crumpled sheet of paper whose thickness tends to zero. Though much is known about scaling laws for thin sheets in tensile settings, the compressive regime is mostly unexplored. In the poster, we discuss an axially confined thin elastic cylinder which is prevented from inward displacement by a hard mandrel core. Our focus is on the dependence of the minimum energy on the thickness and confinement of the cylinder in the Foppl-von Karman model. We prove upper and lower bounds for this scaling.

Huan Wu
University of Warwick, United Kingdom

Atomistic-to-continuum coupling: the quasi-non-local approach

Atomistic-to-continuum (a/c) coupling methods are a class of computational multi-scale schemes for simulating crystal defects. It combines the accuracy of atomistic models and the efficiency of continuum models. Among a few popular a/c methods, the quasi-non-local coupling exhibits the advantage of eliminating ghost-force which otherwise presents great difficulty in error analysis. My work focuses on the rigorous error analysis of qnl-type models in 2D with high-order finite elements.

Anna Zubkova
University of Graz, Austria

On generalized Poisson-Nernst-Planck equations

A time-dependent Poisson-Nernst-Planck system of nonlinear partial differential equations is considered. It is modeled in terms of the Fickian multiphase diffusion law coupled with electrostatic and quasi-Fermi electrochemical potentials. The model describes a plenty of electrokinetic phenomena in physical and biological sciences. The generalized model is supplemented by a positivity and volume constraints, by quasi-Fermi electrochemical potentials depending on the pressure, and by inhomogeneous transmission boundary conditions representing reactions at the micro-scale level. We aim at a proper variational modeling, optimization, and asymptotic analysis as well as homogenization of the model at the macro-scale level. The work is supported by the Austrian Science Fund (FWF) in the framework of the research project P26147-N26: PION.