# Reception and Poster Session

Wednesday, May 21, 2014 - 3:40pm - 5:30pm

Lind 400

**From Compatibility Conditions to Stress-Free Microstructure**

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

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)**Hamilton-Jacobi Equations for Sorting and Percolation Problems**

Jeff Calder (University of Michigan)

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.**Theory-based Benchmarking of the Blended Force-based Quasicontinuum**

Method

Xingjie Li (Brown University)

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.**Equilibrium States of Nonlinearly Elastic Annuli and Spherical Shells**

Alexey Stepanov (University of Maryland)

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.**Examples of Extremal Quasiconvex Quadratic Forms that are Not**

Polyconvex

Davit Harutyunyan (The University of Utah)

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**Energy Scaling Law for 2D Elastic Compliance Minimization Via Refined**

Hashin-Shtrikman Bounds

Benedikt Wirth (Westfälische Wilhelms-Universität Münster)

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)**Grain Boundary Diffusion**

Daniel Brinkman (Arizona State University)

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.**Energy Driven Pattern Formation in Planar Dipole–Dipole Systems**

Andrew Bernoff (Harvey Mudd College)

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.**Robust Multilabel Segmentation via Redistancing Dynamics: Grain Identification in Polycrystal Images**

Matt Elsey (Courant Institute of Mathematical Sciences)Benedikt Wirth (Westfälische Wilhelms-Universität Münster)

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.