### Center for Nonlinear Analysis

# MATHEMATICAL MODELS FOR PATTERN FORMATION

## March 8-10, 2019

#### Venue: Giant Eagle Auditorium (A51) of Baker Hall

## Program

Friday, March 8 |
Saturday, March 9 |
Sunday, March 10 |
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8:30-9:00 |
Breakfast |
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9:00-9:50 |
James | Kohn | López Jiménez | |

10:00:10:50 |
Audoly | Bella | Zwicknagl | |

11:00-11:30 |
Coffee Break |
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11:30-12:20 |
Muratov | Rüland | Truskinovsky | |

12:30-2:00 |
Lunch |
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2:00-2:50 |
Koumatos | Choksi | ||

3:00-3:50 |
Plucinsky | Morini | ||

4:00-4:30 |
Coffee Break |
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4:30-5:20 |
Poster Blitz |
Weber | ||

5:30-6:30 |
Poster Session |
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6:30-8:00 |
Reception |
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8:30-9:30 |
Community Talk: Golden |

## Abstracts

**Title:** *Asymptotically exact strain-gradient models for slender elastic structures*

**Abstract:** We propose a general method for deriving one-dimensional models for non-linear structures, that capture the strain energy associated not only with the macroscopic strain as in traditional structural models, but also its gradient. The method is applied to various types of non-linear structures featuring localization, such as a hyper-elastic cylinder that necks, an axisymmetric membrane that produces bulges, or a tape spring that snaps. By contrast with traditional models, the one-dimensional models obtained in this way can account for significant and non-uniform deformations of the cross-sections, and therefore capture the localization phenomena accurately. Work in collaboration with Claire Lestringant (ETH, Switzerland).

**Title:** *Wrinkling of a thin elastic sheet on a compliant sphere*

**Abstract:** Wrinkling of thin elastic sheets can be viewed as a way how they avoid compressive stresses. While the question of where the wrinkles appear is (mostly) well-understood, understanding properties of wrinkling is not trivial. Considering a variational viewpoint, the problem amounts to minimization of an elastic energy, which can be viewed as a non-convex membrane energy singularly perturbed by a higher-order bending term. To understand the global minimizer (ground state), the first step is to identify its energy, in particular how it depends on the small physical parameter (thickness). I will focus on one specific example – a disk-shaped thin elastic sheet bonded to a compliant sphere. There the leading-order behavior of the energy determines the macroscopic deformation of the sheet and provides insight about the length scale of the wrinkling. The next-order correction then provide insight about how the wrinkling pattern should vary across the film, and is in particular related to the form of transition between different wrinkling patterns.

**Title:** *Optimal Centroidal Voronoi Tessellations: Navigating the Energy Landscape and Gersho's Conjecture in 3D*

**Abstract:** Nonconvex and nonlocal variational problems are pervasive in energy-driven pattern formation. Two central issues are:

(Q1) can one develop hybrid numerical algorithms to navigate (or probe) the energy landscape and access low energy states whose basin of attraction might be “tiny”;

(Q2) can one conjecture and prove asymptotic statements on the (geometric) nature of global minimizers.

In this talk, we explore both (Q1) and (Q2) in the context of the simple, yet rich, paradigm of optimal quantization and centroidal Voronoi tessellations (CVT). We begin with (Q1) by presenting a new geometric hybrid algorithm which alternates gradient descent (or Lloyd's method) with movement away from the closest generator. This is ongoing work with Ivan Gonzales and JC Nave (McGill University). We then address (Q2) via the 3D Gersho's conjecture. Gersho's conjecture may be viewed as a crystallization conjecture and asserts the periodic structure, as the number of generators tends to infinity, of the optimal CVT. In joint work with Xin Yang Lu (Lakehead University), we present certain bounds which, combined with a 2D approach introduced by P. Gruber, reduce the resolution of the 3D Gersho's conjecture to a finite (albeit very large) computation of an explicit convex problem in finitely many variables.

**Title:** *Pattern Formation in Melting Arctic Sea Ice*

**Abstract:** Sea ice is a critical indicator of Earth's changing climate, with the rapid decline of the summer Arctic ice pack outpacing expert predictions. As a material, frozen sea water is a composite of pure ice with millimeter scale brine inclusions, which is quite different in structure than an ice cube in a drink. In fact, the freezing, dynamics, and melting of sea ice form beautiful patterns over a vast range of length scales. From the brine and polycrystalline microstructure of first year ice and meter scale pancakes that form in a wave field, to the complex mosaics of snow and melt ponds atop sea ice floes that stretch for kilometers, we will explore mathematical models of these patterns and the physical properties they determine. Our models are developed in conjunction with experiments we have conducted in both the Arctic and Antarctic. This work is helping to advance how sea ice is represented in climate models and to improve projections of the fate of Earth's sea ice packs and the ecosystems they support. The lecture is intended for a wide, interdisciplinary audience, and will conclude with a short video on a recent Antarctic expedition where we measured sea ice properties.

**Title:** *Atomistically inspired origami*

**Abstract:** World population is growing approximately linearly at about 80 million per year. As time goes by, there is necessarily less space per person. Perhaps this is why the scientific community seems to be obsessed with folding things. We present a mathematical approach to “rigid folding” inspired by the way atomistic structures form naturally. Their characteristic features in molecular science imply desirable features for macroscopic structures, especially 4D structures that deform. Origami structures, in turn, suggest an unusual way to look at the Periodic Table.

**Title:** *Non-uniform (zig-zag) microstructures mixing two variants of martensite*

**Abstract:** In martensitic phase transformation, the formation of microstructure is due to minimization of elastic plus surface energy. There is by now a lot of work on macroscopically-uniform microstructures. In the presence of stress or a thermal gradient, however, one expects microstructures whose volume fractions and other features vary macroscopically. I'll discuss some model problems of this kind, involving non-uniform twinning of two martensite variants. In these problems the loads or boundary conditions require the volume fractions to vary; this forces the twin boundaries to tilt away from their stress-free orientations, and leads to what might be called a "zig-zag microstructure." The goals for mathematical analysis are (a) to show that the zig-zag microstructure achieves an optimal energy scaling law, (b) to explore the local properties of energy-minimizing patterns, and (c) to understand the robust periodicity observed in experimental realizations of such microstructures. I'll discuss some recent progress toward (a) and (b) (joint work with Alex Misiats and Stefan Mueller).

**Title:** *Quasiconvex elastodynamics: weak-strong uniqueness for measure-valued solutions *

**Abstract:** A weak-strong uniqueness result is presented for dissipative measure-valued solutions to the system of conservation laws arising in elastodynamics. The main novelty of this work is that the underlying stored-energy function is assumed strongly quasiconvex. The proof borrows tools from the calculus of variations to prove a Garding type inequality involving quasiconvex functions, which allows to adapt the relative entropy method to quasiconvex energies.

**Title:** *Pattern formation in bilayer systems through substrate pre-stretching*

**Abstract:** Compressing a thin film bonded to an elastic substrate beyond a critical stress results in an elastic instability, often referred to as wrinkling or ruga, that generates complex surface deformation. Although the best-known example is the sinusoidal wrinkling that appears under uni-axial compression of the film, other loading conditions and actuation mechanisms result in a diverse range of self-organized patterns. In this talk we will present experiments in which the loading mechanisms is equi-biaxial pre-stretching of the substrate prior to the adhesion of the film, which when released results in film compression. We will show that with only moderate variations in the system and control parameters we can transition between a wide range of patterns, with a well-defined phase diagram spanning periodic wrinkles, creases, folds, and high aspect ratio ridges. We will also explore the dynamics that control the pattern formation.

**Title:** *Nonlinear stability results for nonlocal gradient flows*

**Abstract:** We consider two different nonlocal flows. We first investigate the long-time behavior of the so-called nonlocal Mullins-Sekerka flow, that is the gradient flow, with respect to a suitable Riemannian structure, of the Ohta-Kawasaki energy. In the second part of the talk we consider the evolution of interfaces of elastic materials driven by surface diffusion and elastic stress, establishing new short-time existence and asymptotic stability results.

**Title:** *The mathematics of charged liquid drops*

**Abstract:** In this talk, I will present an overview of recent analytical developments in the studies of equilibrium configurations of liquid drops in the presence of repulsive Coulombic forces. Due to the fundamental nature of Coulombic interaction, these problems arise in systems of very different physical nature and on vastly different scales: from femtometer scale of a single atomic nucleus to micrometer scale of droplets in electrosprays to kilometer scale of neutron stars. Mathematically, these problems all share a common feature that the equilibrium shape of a charged drop is determined by an interplay of the cohesive action of surface tension and the repulsive effect of long-range forces that favor drop fragmentation. More generally, these problems present a prime example of problems of energy driven pattern formation via a competition of long-range attraction and long-range repulsion. In the talk, I will focus on two classical models - Gamow's liquid drop model of an atomic nucleus and Rayleigh's model of perfectly conducting liquid drops. Surprisingly, despite a very similar physical background these two models exhibit drastically different mathematical properties. I will discuss the basic questions of existence vs. non-existence, as well as some qualitative properties of global energy minimizers in these models, and present the current state of the art for this class of geometric problems of calculus of variations.

**Title:** *Pattern formation and design in active and architectured sheets*

**Abstract:** Thin sheets exhibit a broad range of mechanical responses as the competition between stretching and bending in these structures can result in buckling, localized deformations like folding, and tension wrinkling. Active and architectured materials also exhibit a broad range of mechanical responses as pattern formation and/or design at the micro and mesoscale in these materials results in mechanical couplings at the engineering scale (thermal/ electrical/ dissipative/ ...) with novel function (the shape memory effect/ ferroelectricity/enhanced fracture toughness/ ...). Given this richness in behaviors, it is natural to wonder what happens when active and architectured materials are incorporated into thin sheets? Do phenomena inherent to these materials compete with or enhance those inherent to these structures? Does this interplay result in entirely new and unexpected phenomena? And can all this be exploited to design new functionality?

In this talk, I will explore these questions in the context of thin sheets of an active material in nematic elastomer as well as architectured sheets designed to fold continuously as origami. For the latter, I will completely characterize all rigidly and flat-foldable origami, and describe an efficient algorithm to compute their designs and deformations. For the former, I will show that a material instability inherent to nematic elastomers at the micron scale is capable of suppressing a structural instability (wrinkling) at the engineering scale. These results provide novel, yet concrete, design guidance for membrane structures (where wrinkling can diminish functionality), as well as tools to efficiently investigate robust and elegant concepts for deployable space structures, reconfigurable antennas, and soft robotics using origami.

**Title:** *Microstructures in Shape-Memory Alloys: Rigidity, Flexibility and Some Numerical Exeriments*

**Abstract:** In this talk I will discuss a striking dichotomy which occurs in the mathematical analysis of microstructures in shape-memory alloys: On the one hand, some models for these materials display a very rigid structure with only very specific microstructures, if one assumes that surface energies are penalised. On the other hand, without this penalisation, for the same models a plethora of very wild solutions exists. Motivated by this observation, we seek to further understand and analyse the underlying mechanisms. By discussing a two-dimensional toy model and by constructing explicit solutions, we show that adding only little regularity to the model does not suffice to exclude the wild solutions. We illustrate these constructions by presenting numerical simulations of them. The talk is based on joint work with J. M. Taylor, Ch. Zillinger and B. Zwicknagl.

**Title:** *A mechanical perspective on vertebral segmentation *

**Abstract:** Segmentation is a characteristic feature of the vertebrate body plan. We argue that the periodicity along the embryonic body axis, anticipating such segmentation, is controlled by mechanical rather than bio-chemical signaling. Using a prototypical model we show that regular patterning can result from a mechanical instability induced by differential strains developing between the segmenting and the surrounding tissues. We consider a uniaxially stretched hyper-elastic block with one surface free and another one subjected to mixed boundary conditions imitating differential pre-stress. We find a rich and complex bifurcation diagram containing both diffuse (necking) and localized (wrinkling) instability modes. Necking is related to the non-convexity of the energy, while wrinkling is triggered by the violation of the complementing condition. The crossover between the two is found to be remarkably sensitive to small variations of the geometrical and material parameters, reminiscent of what is observed at fluid turbulence.

**Title:** *Pattern Formation in Compressible Fluids*

**Abstract:** Motivated by recent non-uniqueness results for multi-dimensional Euler equations and numerical results, Fjordholm, Lanthaler and Mishra (2017) suggested statistical solutions as a notion of solutions for multidimensional systems of nonlinear conservation laws. In the scalar case, well-posedness of such solutions can be shown. We present numerical algorithms to approximate statistical solutions of systems of conservation laws and formulate conditions under which they converge. The conditions are inspired by Kolmogorov’s theory of turbulence (1941) and numerical experiments for the 2d compressible Euler equations show that the conditions might be satisfied at least in some cases. In addition, in the case of the incompressible Navier-Stokes equations, the statistical solutions introduced by Fjordholm, Lanthaler and Mishra can be shown to be equivalent to the statistical solutions as in the sense of Foias and Temam.

**Title:** *Scaling regimes and needle-like microstructures in shape memory alloys*

**Abstract:** I shall discuss recent results on variational models for microstructures in martensites. We consider singularly perturbed multiwell elastic energy functionals and the associated nonconvex vectorial minimization problems. In this talk, I shall focus on scaling regimes for geometrically linearized models for martensitic nuclei in the limit of low volume fraction, and on needle-like microstructures. This talk is based on joint work with S. Conti, J. Diermeier, M. Lenz, N. Lüthen, D. Melching, and M. Rumpf.

This conference is sponsored by the Department of Mathematical Sciences at Carnegie Mellon University.