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Abstracts and Talk Materials
Mathematics at the Interface of Partial Differential Equations, the Calculus of Variations, and Materials Science
May 21 - 23, 2014

Andrew Bernoff (Harvey Mudd College)

Energy Driven Pattern Formation in Planar Dipole–Dipole Systems
May 21, 2014

Many two-dimensional fluid-like systems are mediated by a dipole-dipole interactions. We show that the microscopic details of any such system are irrelevant in the macroscopic limit and contribute only a constant offset to the system's energy. We develop a numeric model, track harmonic bifurcations from a circular domain and characterize some stable domain morphologies. Curiously, the stable domains are highly symmetric and bear little resemblance to experimental observation. By adding a random energy background we recover a smörgåsbord of diverse morphologies that were previously unstable and that show strong similarities to physical systems. Finally, we develop a method for recovering information about the microscopic parameters of any system from the perimeter and topology of their shape.

Joint work with Jaron P. Kent-Dobias.

Daniel Thomas Brinkman (Arizona State University)

Grain Boundary Diffusion
May 21, 2014

We consider the problem of understanding defect diffusion in Cadmium Telluride (CdTe) solar cells. Of particular interest is the motion of Copper and Chlorine. The modeling is complicated by the grain structure of CdTe. The boundaries between the grains are responsible for the primary motion of the defects, but the width of such boundaries is below experimental tolerance. We seek to understand how the geometry of the grain boundaries changes the distribution of defects throughout the device.

In this poster, we specifically consider numerical simulation of the problem with several models. We consider simplified two-species modeling, the classical method of Fisher, and a simplified model which assumes a finite width boundary. We demonstrate numerical agreement of the models in several regimes. Possible extensions to a 1D Network embedded in a 2D domain are also discussed.

Jeff Calder (University of Michigan)

Hamilton-Jacobi Equations for Sorting and Percolation Problems
May 21, 2014

We show that two related combinatorial problems have continuum limits that correspond to solving Hamilton-Jacobi equations. The first problem is non-dominated sorting, which is fundamental in multi-objective optimization, and the second is directed last passage percolation (DLPP), which is an important stochastic growth model closely related to directed polymers and the totally asymmetric simple exclusion process (TASEP). We give convergent numerical schemes for both Hamilton-Jacobi equations and explore some applications.

Xian Chen (University of Minnesota, Twin Cities)
Richard D. James (University of Minnesota, Twin Cities)

From Compatibility Conditions to Stress-Free Microstructure
May 21, 2014

The cofactor conditions (CC) are the conditions of super compatibility between phases for martensitic transformation. By satisfying CC, austenite and variants of martensite can fit together without elastic transition layers for any twinning volume fraction between 0 and 1. Here we discuss different forms of CC in Type I/II, Compound twins and domains, followed by the prediction of their possible microstructures. Then we calculate the geometric linear case of CC. Finally, we show real examples whose lattice parameters were tuned to satisfy CC closely for both Type I and II twin system and their bizarre microstructure.

(with Yintao Song)

Rustum Choksi (McGill University)

Self-Assembly: Variational Models and Energy Landscapes
May 21, 2014

Self-assembly, a process in which a disordered system of preexisting components forms an organized structure or pattern, is both ubiquitous in nature and important for the synthesis of many designer materials. In this talk, we will address two variational paradigms for self-assembly from the point of view of analysis and computation.

The first variational model is a nonlocal perturbation (of Coulombic-type) to the well-known Ginzburg-Landau/Cahn-Hilliard free energy. The functional has a rich and complex energy landscape with many metastable states. We present recent joint work with Dave Shirokoff and J.C. Nave at McGill on developing a method for assessing whether or not a particular (computed) metastable state is a global minimizer. Our method is based upon a very simple idea of using a ``suitable" global convex envelope of the energy. We present full details for global minimality of the constant state, and then present a few partial results on the application to non-constant, computed metastable states.

The second variational model is purely geometric and finite-dimensional: Centroidal Voronoi Tessellations (CVT) of rigid bodies. Using a level set formulation, we a priori fix the geometry for the structures and consider self-assembly entirely dictated by distance functions. We introduce a novel fast algorithm for simulating CVTs of rigid bodies in any space dimension. The method allows us to empirically explore the CVT energy landscape. This is joint work with Lisa Larsson and J.C. Nave at McGill.

Matt Elsey (New York University)
Benedikt Wirth (Westfälische Wilhelms-Universität Münster)

Robust Multilabel Segmentation via Redistancing Dynamics: Grain Identification in Polycrystal Images
May 21, 2014

A novel numerical method for multilabel segmentation of vector-valued images is presented. The algorithm seeks minimizers for a generalization of the piecewise-constant Mumford-Shah energy and is particularly appropriate for energies with a fitting (or fidelity) term that is computationally expensive to evaluate. The framework for the algorithm is the standard alternating-minimization scheme in which the update of the partition is alternated with the update of the vector-valued constants associated with each part of the segmentation. The update of the partition is based on the distance function-based diffusion-generated motion algorithms for mean curvature flow. The update of the vector-valued constants is based on an Augmented Lagrangian method. The scheme automatically chooses the appropriate number of segments in the partition. It is initialized with a partition of many more segments than are expected to be necessary. Adjacent segmentations of the partition are merged when energetically advantageous. The utility of the algorithm is demonstrated in the context of atomic-resolution polycrystalline image segmentation.

Selim Esedoglu (University of Michigan)

Algorithms for Computing Grain Boundary Motion in Polycrystalline Materials
May 21, 2014

We study a class of algorithms known as threshold dynamics that was originally proposed by Merriman, Bence, and Osher for generating the motion by mean curvature of a network of surfaces. Our new, variational formulation of this algorithm allows us to extend it to the setting in which each interface in the network can have a different surface tension. Preliminary results on the anisotropic case, where the interfacial energy can depend on the direction of the normal, will also be discussed. Based on joint works with Matt Elsey and Felix Otto.

Adriana Garroni (Università di Roma "La Sapienza")

Energies Concentrated on Lines and the Modeling of Dislocations in 3D
May 22, 2014

Dislocations are topological singularities in crystals, which may be described by lines to which a lattice-valued vector, called Burgers vector, is associated. They may be identified with divergence-free matrix-valued measures supported on curves or with 1-currents with multiplicity in a lattice. In the modeling of dislocations one is thus often lead to energies concentrated on lines, where the integrand depends on the orientation and on the Burgers vector of the dislocation. In this talk I will present the theory of relaxation for such energies and I will show how they may arise in a multiscale analysis of dislocations, starting from discrete and semi-discrete models.

Yoshikazu Giga (University of Tokyo)

On a Level-set Crystalline Curvature Flow of Surfaces
May 23, 2014

Keywords of the presentation: Crystalline curvature flow, viscosity solution, comparison principle.

It is by now standard that a level-set approach provides a global unique generalized solution (up to fattening) for mean curvature flow equations [G]. Even from the early stage of the theory, it is known that the method is very flexible to apply anisotropic curvature flow equations which correspond to anisotropic interfacial energy.

The anisotropy is very important in materials sciences. However, if the anisotropy is very singular for example a crystalline curvature flow corresponding to crystalline interfacial energy a level set approach was not available except evolution of curves to which a foundation of the theory was established by M.-H. Giga and Y. Giga more than ten years ago.

In this talk we push forward a level-set approach to surface evolution by crystalline curvature. The main difficulty is that crystalline curvature is a nonlocal quantity and it may not be a constant on each flat potion of a surface. (In the case of curve evolution it is always a constant over a segment.) We overcome this difficulty by introducing a suitable notion of viscosity solutions so that a comparison principle holds. We further construct a global-in-time solution as a limit of smoother problem. A delicate analysis is necessary to achieve the goal. A similar but a simpler problem was studied in [MGP1], [MGP2]. We elaborate these approaches for our purpose.

[G] Y. Giga, Surface evolution equations. A level set approach. Monographs in Mathematics, 99. Birkhäuser Verlag, Basel, 2006. xii+264 pp.

[MGP1] M.-H. Giga, Y. Giga and N. Požár, Periodic total variation flow of non-divergence type in $R^n$, J. Math. Pures Appl., to appear.

[MGP2] M.-H. Giga, Y. Giga and N. Požár, Anisotropic total variation flow of non-divergence type on a higher dimensional torus. Adv. Math. Sci. Appl. 23 (2013), no. 1, 235–266.

Yury Grabovsky (Temple University)

Scaling Instability of the Buckling Load in Axially Compressed Circular Cylindrical Shells
May 23, 2014

The classical formula for the buckling load of axially compressed cylindrical shells predicts 4-5 times higher critical load than observed in experiment. The discrepancy is explained by high sensitivity of the buckling load to imperfections. However, the exact mechanism of this sensitivity remains elusive. Our rigorous theory of "near-flip" buckling of slender bodies reveals a scaling instability of the critical load caused by imperfections of load. A possible mechanism of sensitivity to imperfections of shape can also be seen in our analysis. This is a joint work with Davit Harutyunyan.

Davit Harutyunyan (University of Utah)

Examples of Extremal Quasiconvex Quadratic Forms that are Not Polyconvex
May 21, 2014

We prove that if the associated fourth order tensor of a quadratic form has a linear elastic cubic symmetry then it is a quasiconvex form if and only if it is polyconvex, i.e. a sum of convex and null-Lagrangian quadratic forms. We prove that allowing for slightly less symmetry, namely only cyclic and axis-reflection symmetry, gives rise to a class of extremal quasiconvex quadratic forms, that also turn out to be non-polyconvex.

This a joint work with Graeme W. Milton

Richard D. James (University of Minnesota, Twin Cities)

Twisted X-rays: New Methods for the Determination of Noncrystalline Atomic Structure
May 23, 2014

Keywords of the presentation: Materials science, structure determination, x-rays, isometry groups, objective structures, Maxwell's equations

A central problem of materials science is to determine atomic structure from macroscopic measurements. Von Laue developed a theoretical method that was put into practice and popularized by Bragg, based on the scattering of plane waves by a crystal lattice. Recently, new structures have emerged like buckyballs (Nobel Prize, Chemistry, 1996) and graphene (Nobel Prize, Physics, 2010), and the third fascinating form of carbon, the carbon nanotube (no Nobel prize yet). These have a regular structure but are not crystalline. Regular but noncrystalline structures are also quite common in biology: examples of medical interest include many parts of viruses, and amyloid protein fibrils that cause diseases like Alzheimer's, Parkinson's and Creutzfeldt-Jakob disease. Structures like buckyballs, graphene and carbon nanotubes were not discovered — they are believed to have been present on earth since nearly its beginning — but the recent interest lies in the fact that they were isolated and studied, and exhibited interesting properties. There may well be related, maybe equally interesting but perhaps less common, structures around us. Thus, the determination of the atomic structure of noncrystalline structures by macroscopic methods is a central problem. After clearing up some fallacies about ordinary x-ray crystallography (Bragg’s incorrect picture, the intensity of scattered radiation is not the squared norm of the Fourier transform of the lattice), we propose a new method, which involves exploiting the relationship between structure and the invariance group of Maxwell’s equations. We work out the details for helical structures like carbon nanotubes and amyloid protein fibrils. This is joint work with Dominik Juestel and Gero Friesecke, TU Munich.

David Kinderlehrer (Carnegie-Mellon University)

Coarsening in Microstructure
May 22, 2014

Cellular networks are ubiquitous in nature. Most technologically useful materials arise as polycrystalline microstructures, composed of a myriad of small crystallites, the grains, separated by interfaces, the grain boundaries. The coarsening of these networks is of obvious concern for applications and has been since pre-history. Any order in the system must be conferred by the evolving boundary network. We discuss how this arises and some implications for future research. We still have much to learn about these very ancient questions.

Xingjie Helen Li (Brown University)

Theory-based Benchmarking of the Blended Force-based Quasicontinuum Method
May 21, 2014

We formulate an atomistic-to-continuum coupling method based on blending atomistic and continuum forces. We present a comprehensive error analysis that is valid in two and three dimensions, for finite many-body interactions (e.g., EAM type), and in the presence of lattice defects (point defects and dislocations). Based on a precise choice of blending mechanism, the error estimates are considered in terms of degrees of freedom. The numerical experiments confirm and extend the theoretical predictions, and demonstrate a superior accuracy of B-QCF over energy-based blending schemes.

Fanghua Lin (New York University)

Some Mathematical Consequences From Liquid Crystal Flows
May 22, 2014

Keywords of the presentation: Liquid Crystals, Heat Flow of Harmonic Maps, Navier-Stokes Equations

The nematic (uniaxial) liquid crystal flows are often described by the classical Ericksen-Leslie theory. Even in the (over) simplified model case, it is given by a highly nonlinear coupling of the Navier-Stokes system and the system for the (heat) flow of harmonic maps. Both of these two systems were studied by numerous authors, and much of the earlier theory developed for the dynamics of liquid crystals were based on. In this talk, I am going to describe some recent studies on the flows of liquid crystals and their consequences in the heat flow of harmonic maps.

Graeme Walter Milton (University of Utah)

The Searchlight Effect in Hyperbolic Media
May 21, 2014

Keywords of the presentation: Hyperbolic media, Superlensing, Sub-wavelength Resolution

The study of hyperbolic media, where the dielectric tensor has eigenvalues of mixed signs, has attracted considerable attention as a way of achieving subwavelength resolution. The quasistatic field around a circular hole in a two-dimensional hyperbolic medium is studied. As the loss parameter goes to zero, it is found that the electric field diverges along four lines each tangent to the hole. In this limit, the power dissipated by the field in the vicinity of these lines, per unit length of the line, goes to zero but extends further and further out so that the net power dissipated remains finite. Additionally the interaction between polarizable dipoles in a hyperbolic medium is studied. It is shown that a dipole with small polarizability can dramatically influence the dipole moment of a distant polarizable dipole, if it is appropriately placed. We call this the searchlight effect, as the enhancement depends on the orientation of the line joining the polarizable dipoles and can be varied by changing the frequency. For some particular polarizabilities the enhancement can actually increase the further the polarizable dipoles are apart, a bit like the interaction between quarks.

Barbara Niethammer (Rheinische Friedrich-Wilhelms-Universität Bonn)

Kramers- and Non-Kramers Phase Transitions in a Nonlocal Fokker-Planck Equation
May 21, 2014

Keywords of the presentation: Nonlocal Fokker Planck equation, double well potential, time-dependent consrtaint, rate independent model for hysteresis, phase transitions

We discuss a nonlocal Fokker-Planck equation that describes energy minimisation in a double well-potential and is driven by a time-dependent constraint. Via formal asymptotic analysis we identify different small parameter regimes that correspond to hysteretic and non-hysteretic phase transitions respectively. For the fast reaction regime that is related to Kramers-type phase transitions we also indicate how can rigorously derive a rate-independent evolution equation in a small parameter limit.

This is joint work with Michael Herrmann and Juan Velazquez.

Felix Otto (Max-Planck-Institut für Mathematik in den Naturwissenschaften)

Branching, Coarsening, and Nucleation
May 21, 2014

Keywords of the presentation: Branching, coarsening, nucleation, shape memory alloys, ferromagnets, interpolation inequalities

In his seminal work on twin splitting (with S. Mueller), Bob Kohn taught us how the presence of an interfacial energy selects certain types of microstructures, i. e. removes the high degree of degeneracy of an unrelaxed non-convex ariational problem. Later, he realized that the same analysis applies to domain branching in ferromagnets (with R. Choksi) and to flux tube branching in superconductors (with S. Conti). The use of suitable interpolation inequalities freed the analysis and motivated the connection to the physically unrelated subject of coarsening in spinodal decomposition. In recent work (with H. Knuepfer), Bob Kohn showed how the above insights into twin branching allow to understand even hierarchical microstructures like Martensitic inclusions in Austenite.

Sylvia Serfaty (New York University)

Systems with Coulomb Interacting Particles
May 21, 2014

Keywords of the presentation: Ginzburg-Landau, Ohta-Kawasaki, Coulomb gases, Abrikosov lattices, crystallization

I will review some results concerning vortices in the Ginzburg-Landau model, droplets in the two-dimensional Ohta-Kawasaki model, and Coulomb gases, which all have in common that they reduce to systems of points with Coulomb interaction. I will discuss the derivation of a "renormalized energy" for the limits of such systems, and the question of crystallization.

This is based on joint works with Etienne Sandier, Nicolas Rougerie, Dorian Goldman, Cyrill Muratov, Simona Rota-Nodari.

Valeriy Slastikov (University of Bristol)

Profiles and Stability of Point Defects in Landau-de Gennes Theory of Liquid Crystals
May 23, 2014

Keywords of the presentation: stability, minimizer, liquid crystal, defect

We use the framework of Landau-de Gennes theory to investigate various defect profiles and their stability in 2D and 3D liquid crystal systems. In 2D we show that a defect profile can be characterized by a system of ODEs. We show that in deep nematic regime this defect profile is a ground state of the liquid crystal system. In 3D we investigate the stability of radially symmetric profile ("melting hedgehog") corresponding to the point defect. We show its stability in the mathematically challenging temperature regime near the supercooling temperature.

Alexey B Stepanov (University of Maryland)

Equilibrium States of Nonlinearly Elastic Annuli and Spherical Shells
May 21, 2014

Within the linear theory of elasticity, the problems for the equilibria of circular annuli and spherical shells composed of homogeneous, transversely isotropic materials were solved by Gabriel Lamé. The radially symmetric equilibria of an isotropic nonlinearly elastic disk or ball is elementary. If, however, the disk or ball is aeolotropic, even for a homogeneous linearly elastic material, the solution can exhibit a rich range of singular behavior at the origin.

We show that BVPs for the equilibria of circular annuli and spherical shells composed of transversely isotropic nonlinearly elastic materials are far from elementary within the framework of geometrically exact theories. We employ a variety of mathematical approaches, discussing the virtues and idiosyncracies of each.

Michael Vogelius (Rutgers, The State University of New Jersey)

Cloaking by Mapping
May 22, 2014

Keywords of the presentation: loaking, Transformation Optics

I will give an overview of some recent mathematical results in the field of approximate Transformation Optics. The physical models discussed will range from the (single frequency) Helmholtz Equation to the full (non-local) Wave Equation. Time permitting I will also discuss optimization of approximate cloaks and the associated improved "invisibility".

Michael I. Weinstein (Columbia University)

Homogenization, and Bifurcation of Localized from Extended States
May 22, 2014

I will discuss recent results on the bifurcation of spatially localized states from a continuum of extended states. This phenomenon plays an important role in the mathematical study of wave propagation in ordered microstructures, which are perturbed by spatially compact or non-compact defects.

Near the bifurcation point, there is strong spatial scale separation and one expects the "natural'' homogenized equation to govern.

Our first example is one in which this intuition does not apply, and an appropriate effective equation must be derived. (Joint work with V. Duchene and I. Vukicevic)

Our second example concerns the bifurcation of “topologically protected edge states” in a class of periodic structures, perturbed by a (non-compact) domain wall. Such states play a central role in many recently studied systems in condensed matter physics and photonics. (Joint work with C.L. Fefferman and J.P. Lee-Thorp)

Maria G Westdickenberg (RWTH Aachen University)

Optimal Convergence Rates via Relationships among Distance, Energy, and Dissipation
May 22, 2014

We present a method developed jointly with Felix Otto to capture optimal convergence rates for a gradient flow via natural algebraic and differential relationships among distance, energy, and dissipation. The method is developed and applied in the context of relaxation to a kink profile in the one-dimensional Cahn-Hilliard equation on the line. Application to other models is discussed.

Benedikt Wirth (Westfälische Wilhelms-Universität Münster)

Energy Scaling Law for 2D Elastic Compliance Minimization Via Refined Hashin-Shtrikman Bounds
May 21, 2014

We consider the optimization of the topology and geometry of an elastic structure subjected to a fixed boundary load, i. e. we aim to minimize a weighted sum of material volume, structure perimeter, and structure compliance (which is the work done by the load). If the weight in front of the perimeter is small, optimal geometries exhibit very fine-scale structure which cannot be resolved by numerical optimization. Instead, we prove how the minimum energy scales in the perimeter weight, which involves the construction of a family of near-optimal geometries and thus provides qualitative insights. The proof of the energy scaling also requires an ansatz-independent lower bound, which we derive via a Fourier-based refinement of the Hashin-Shtrikman bounds for the effective elastic moduli of composite materials.

(Joint work with Robert Kohn)