Applied Mathematics Seminar, Fall 2019

## Mondays 4:10 PM - 5:10 PM, LCB 215

• This seminar can be taken for credit: Students can get 1-3 credits by registering to the Applied Math Seminar class Math 7875 Section 010 for Fall 2019. Students should talk to the seminar organizer before taking it for a credit. Grading is based on attendance and giving a talk by presenting an applied-mathematics paper (not necessarily your own). Student talks will be appropriately labeled to distinguish them from visitor talks. The seminar organizer is available to review your slides, for dry-runs etc.
• Please direct questions or comments about the seminar (or its class) to Yekaterina Epshteyn (epshteyn (at) math.utah.edu)
• Talks are announced through the applied-math mailing list. Please ask the seminar organizer for information about how to subscribe to this list.

August 19
Welcome Back Meeting

September 9
Speaker: Masashi Mizuno, Department of Mathematics, Nihon University, Japan
Title: Grain boundary motion with time-dependent misorientation and mobility effects
Abstract: In my talk, I will present a model of grain boundary motion with time-dependent misorientation and mobility effects. Dynamics of grain boundaries play an important role in defining the materials properties of polycrystals. The model, considered in the talk, is given by the system of partial differential equations, and is derived using energetic variational principle. Next, I will present the analysis of the model for the grain boundary motion. The key ingredient of the analysis is the so-called monotonicity formula of Huisken's type. Moreover, to treat the time-dependent mobility, we use the modified Gauss kernel to the monotonicity formula.

September 23
Speaker: Tom Alberts, Department of Mathematics, University of Utah
Title: Random Matrix Theory for Homogenization of Composites on Graphs
Abstract: I will discuss the random matrix theory behind two-component random resistor networks on general graphs. This involves random submatrices of the graph's Gamma projection operator, with the particular realization of the submatrix determined by the disorder of the conductances. Certain combinations of graph symmetries together with different models for the random conductances lead to exactly computable spectral statistics. Our recent results lead to exact spectral statistics for the uniform spanning tree model on a diamond hierarchical lattice. Joint work with Ken Golden, Elena Cherkaev, Ben Murphy, Han Le.

September 30
Speaker: Andrejs Treibergs, Department of Mathematics, University of Utah
Title: Compatibility Conditions for Discrete Planar Structures
Abstract: The nonlinear and linearized equations for the state of a planar material with prescribed strain are overdetermined PDE's. To be solvable, the strains must satisfy the necessary compatibility conditions. Approximations of the material by discrete network satisfy overdetermined algebraic equations whose compatibility conditions are less well understood. For generic networks, the number of compatibility conditions is given by the Maxwell number which is a measure of the resilience of a network to damage. It can be easily computed for triangulated networks. We will give a geometric description for the compatibility conditions, discuss which networks are generic, explain why the discrete problems approximate the continuous ones and consider boundary integrals over regions satisfying the compatibility conditions.

October 21
Speaker: Graeme Milton, Department of Mathematics, University of Utah
Title: The spider web problem: guiding stress with discrete Networks
Abstract: Pentamode materials are a class of materials that are useful for guiding stress. In particular, they have been proposed for acoustic cloaking by guiding stress around objects, and have been physically constructed. A key feature of pentamode materials is that each vertex in the material is the junction of 4 double cone elements. Thus the tension in one element determines the tension in the other elements, and by extension uniquely determines the stress in the entire metamaterial. Here we show how this key feature can be extended to discrete wire networks, supporting forces at the terminal nodes and which may have internal nodes where no forces are applied. In usual wire or cable networks, such as in a bridge or bicycle wheel, one distributes the forces by adjusting the tension in the wires. Here our discrete networks provide an alternative way of distributing the forces through the geometry of the network. In particular the network can be chosen so it is uniloadable, i.e. supports only one set of forces at the terminal nodes. Such uniloadable networks provide the natural generalization of pentamode materials to discrete networks.

This is joint work with G.Bouchitte, O. Mattei, and Pierre Seppecher, and is summarized in the paper "On the forces that cable webs under tension can support and how to design cable webs to channel stresses", Proceedings of the Royal Society A: 475 (2019), 20180781.

October 29. Note day is Tuesday and Time is 3pm - 3:50pm. Room is LCB 225.
Speaker: Jeffrey Rickman, Department of Materials Science and Engineering, Lehigh University
Title: Using Materials Informatics to Quantify Complex Correlations Linking Structure, Properties and Processing in Materials
Abstract: I will present several examples in which materials informatics can be used to elucidate and quantify complex correlations linking structure, properties and processing of materials. In the first example, I consider the case of high-entropy (HE) (or multi-principal element) alloys, typically comprising five or more elements. The study of these alloys is a relatively new area of materials research that has attracted intense interest in recent years as, in many cases, these systems possess unexpected and superior mechanical (and other) properties relative to those of conventional alloys. However, the identification of promising HE alloys presents a daunting challenge given the associated vastness of the chemistry/composition space. I will describe a supervised learning strategy for the efficient screening of HE alloys that combines two complementary tools, namely: (1) a canonical-correlation analysis (CCA) and (2) a genetic algorithm (GA) with a CCA-inspired fitness function. In the second example, I consider the ubiquitous phenomenon of grain abnormality in a microstructure that involves the unusually rapid growth of a minority of constituent grains, with the resulting bimodal structure often having a deleterious impact on the thermomechanical properties of a system. In this context, I will describe the use of CCA in conjunction with the formalism of extreme-value statistics to correlate abnormal grain growth with powder processing and chemistry for specialty ceramics. Finally, I will outline the use of detrended correlation analyses to interpret time series data associated with ceramic powder processing.

November 4
Speaker (Invited by Graeme Milton): Yekaterina Epshteyn, Department of Mathematics, University of Utah
Title: Grain Structure, Grain Growth and Evolution of the Grain Boundary Network
Abstract: Cellular networks are ubiquitous in nature. Most technologically useful materials arise as polycrystalline microstructures, composed of a myriad of small monocrystalline cells or grains, separated by interfaces, or grain boundaries. Grain boundaries play an essential role in determining the properties of materials across a wide range of scales. During grain growth (also termed coarsening), an initially random grain boundary arrangement reaches a steady state that is strongly correlated to the interfacial energy density. In this presentation, we will discuss open questions and recent progress on modeling, simulation, experiments and analysis of the evolution of the grain boundary network in polycrystalline materials.

Some of the most recent results that will be discussed as a part of the talk are the joint work with Patrick Bardsley, Katayun Barmak (Columbia University), Elias Clark (U of U), Lajos Horvath (U of U), Chun Liu (IIT) and Masashi Mizuno (Nihon University).

November 7
Speaker: David A. Kopriva, Department of Mathematics, Florida State University/Computational Science Research Center, San Diego State University
Title: Summation by Parts, a Discrete Integral Calculus and How We Design Spectral Element Methods that Work
Abstract: Spectral element methods (SEMs) including Discontinuous Galerkin versions (DGSEMs) have advanced to the point of having open source solvers available that can compute solutions to industrial grade problems. For advection or advection dominated problems (e.g. compressible flows, shallow water flows, electromagnetic wave propagation, etc.) it is also known that the DGSEMs can go unstable, especially for under-resolved simulations like those that occur in turbulence computations. We have developed a discrete integral calculus framework for the analysis of DGSEMs that starts with summation by parts. The framework allows us to identify when approximations are stable or not, and has enabled us to derive provably stable methods for both linear and nonlinear systems of equations approximated in the open source solvers.

November 20. Time 4pm - 5pm, Room LCB 219. Special Joint Statistics/Stochastics/Applied Math Seminar.
Speaker: Mireille Boutin, School of Electrical and Computer Engineering, Purdue University
Title: Reconstructing a Room from Echoes and other Unlabeled Distance Geometry Problems
Abstract: Suppose that some microphones are placed on a drone inside a room with planar walls/floors/ceilings. A loudspeaker emits a sound impulse and the microphones receive several delayed responses corresponding to the sound bouncing back from each planar surface. These are the first-order echoes. We are interested in reconstructing the shape of the room from the first-order echoes. The time delay for each echo determines the distance from the microphone to a mirror image of the source reflected across a wall. Since we do not know which echo corresponds to which wall, the distances are unlabeled. The problem is to figure out under which circumstances, and how, one can find out the correct distance-wall assignments and reconstruct the wall positions. This is one example of an unlabeled distance geometry problem. Another example is the problem of reconstructing a point configuration, up to a rigid motion, from the multi-set of its pairwise (unlabeled) distances. We show that under some mild genericity assumptions, these problems are well-posed. Extensions to the case where the measurements are noisy will also be discussed. This is joint work with Gregor Kemper (TU Munich).

November 25
Speaker: Junshan Lin, Department of Mathematics, Auburn University
Title: Resonances Through Subwavelength Holes and Their Applications in Sensing and Imaging
Abstract: The so-called extraordinary optical transmission (EOT) through metallic nanoholes has triggered extensive research in modern plasmonics, due to its significant applications in bio-sensing, imaging, etc. The mechanisms contributing to the EOT phenomenon can be complicated due to the multiscale nature of the underlying structure. In this talk, I will focus on mechanisms induced by scattering resonances. In the first part of the talk, based upon the layer potential technique, asymptotic analysis and the homogenization theory, I will present rigorous mathematical analysis to investigate the scattering resonances for several typical two-dimensional structures, these include Fabry-Perot resonance, Fano resonance, spoof surface plasmon, etc. In the second part of the talk, preliminary mathematical studies for their applications in sensing and super-resolution imaging will be given. I will focus on the resonance frequency sensitivity analysis and how one can achieve super-resolution by using plasmonic nanohole structures.

Seminar organizer: Yekaterina Epshteyn (epshteyn (at) math.utah.edu).

155 South 1400 East, Room 233, Salt Lake City, UT 84112-0090, T:+1 801 581 6851, F:+1 801 581 4148