Department of Mathematics
Applied Mathematics Seminar, Fall 2013

Mondays 12:55 - 1:55PM, LCB 215

July 22
Speaker: Grzegorz Dzierzanowski , Department of Civil Engineering, Warsaw University of Technology, Poland
Title: Three-phase plane composites of minimal elastic stress energy
Abstract: In this talk we discuss two issues: (i) a lower estimate for effective elastic stress energy of two-dimensional three-phase anisotropic composites; (ii) attainability of the proposed estimate on certain microstructures. It is assumed that the materials are mixed with fixed volume fractions and that one of the phases is degenerated to void, i.e. the effective material is porous. A combination of the translation method and additional inequalities on the stress fields in materials is used to derive the energy estimate and to establish sufficient conditions of optimality of stress fields in each phase of a microstructure. Next, optimal layouts of materials in the form of laminates of a rank are determined for (i) high-porosity composites; (ii) low-porosity composites subjected to isotropic or moderately anisotropic stress. In the remaining region, optimal high-rank laminates are conjectured.
This is joint work with Andrej Cherkaev.

September 9
Speaker: Davit Harutyunyan, Department of Mathematics, University of Utah
Title: Reversal process in magnetic nanowires
Abstract: We study static 180 degree domain walls in infinite magnetic wires with a bounded convex and centrally symmetric cross sections. We prove an existence of global minimizers for the energy of micromagnetics. We prove that the energies of micromagnetics Gamma-converge to a one dimensional energy as the thickness goes to zero. We show as well, that under some asymmetry condition of the cross sections the whole sequence of almost minimizers converge to a minimizer of the limiting energy up to a rotation and a translation. Combining that results, we show that a Neel wall occurs in sufficiently thin wires. We prove as well a rate of convergence for the minimal energies

September 23 (Student Talk)
Speaker: Andy Thaler, Department of Mathematics, University of Utah
Title: The near-cloak defeats the anticloak
Abstract: Recently the idea of cloaking has attracted a lot of attention. Although several methods of cloaking have been proposed in the literature, this talk will focus on a cloaking scheme known as transformation-based cloaking. In this method one surrounds an object by a suitably designed ``cloak'' that will make both the object and the cloak itself almost invisible to observers outside of the cloak. However, an anticloak has been proposed which significantly reduces the effectiveness of the transformation-based cloak. In this talk we will introduce the transformation-based cloak and the anticloak; we will then discuss a counter-measure to this anticloak and prove that the transformation-based cloak can still be very effective, even in the presence of the anticloak.

September 30
Speaker: Andrej Cherkaev, Department of Mathematics, University of Utah
Title: Optimal three-material design and structure of three-well quasiconvex envelope
Abstract: We consider an optimal layout of three elastic materials (one of them is void) in a given domain, loaded from the boundary; the amounts of materials are fixed. It is well-known that the minimizing sequences of layouts alternate in an infinitesimal scale; a limiting optimal layout is a multimaterial composite of optimal micro-geometry. This nonconvex multivariable variational problem is reduced to determination of the quasiconvex envelope of a multiwell Lagrangian, where the wells represent materials' energies plus their costs; the quasiconvex envelope represents the energy and the cost of an optimal composite. In the paper, the structure of minimizing sequences and the optimal energy density is demonstrated, the techniques are briefly discussed, and the computed three-material designs are shown. We mention that the corresponding results for two-well energy and two-material optimal composites have been known and widely used for more that two decades, but the transition to multiwell Lagrangian is done only now.
The talk is based on a joint project with Grzegorz Dzierzanowski (Warsaw Polytechnic University) and Nathan Briggs (University of Utah).

October 7
Speaker: Davit Harutyunyan, Department of Mathematics, University of Utah
Title: Korn inequality for thin walled cylinders
Abstract: Korn inequalities are known to play a central role in the theory of linear (also nonlinear) elasticity. In their recent theory of buckling of slender structures Y. Grabovsky and L. Truskinovsky realized that under some appropriate conditions (when the theory is applicable) Korn's constant gives a lower bound and in some cases the scaling for the critical buckling load of a compressed slender structure. In this work we prove sharp Korn and Korn-like inequalities for the displacement gradient componentes for perfect cylindrical shells. Using those inequalities we solve the problem of buckling of axially compressed perfect cylindrical shells. This is joint work with Yury Grabovsky.

October 21
Speaker: Hyundae Lee, Department of Mathematics, Inha University
Title: Near Cloaking Structures and Generalized Polarization Tensors
Abstract: We construct very effective near-cloaking structures for the conductivity problem. These new structures are before using the transformation optics, layered structures and are designed so that their first Generalized Polarization Tensors vanish. We show that this in particular significantly enhances the cloaking effect. This method can be extended to the Helmholtz equation. We may design a structure that any target inside some region has near-zero scattering cross section for a band of frequencies. Some numerical examples are also presented.

October 28 (Student Talk)
Speaker: Varun Shankar, School of Computing and Department of Mathematics, University of Utah
Title: Radial Basis Function-based numerical methods for the simulation of platelet aggregation
Abstract: In this talk, I will present some of my work with Aaron Fogelson, Mike Kirby (SCI Institute) and Grady Wright (Boise State University) on developing numerical methods for the simulation of platelet aggregation. In this context, I will discuss three ways in which we use Radial Basis Functions (RBFs): first, as parametric interpolants for the geometric modeling of platelets within the Immersed Boundary method; second, as generalized Hermite interpolants for enforcing boundary conditions on PDEs describing the evolution of chemical species during platelet aggregation; and third, as the basic component within a high-order meshfree method for solving reaction-diffusion equations on surfaces embedded in 2D and 3D domains.

November 4
Speaker: Sergey Yakovlev, SCI Institute, University of Utah
Title: H-to-P Efficiently: A progress report on High-Order FEM on Manifolds with Applications in Electrophysiology
Abstract: In this talk, I will describe the numerical discretization of an embedded two-dimensional manifold using high-order spectral/hp elements. Such methods provide an exponential reduction of the error with increasing polynomial order, while retaining geometric flexibility of the domain. Embedded manifolds are considered a valid approximation for many scientific problems ranging from the shallow water equations to geophysics. The Description and validation of our discretization technique as well as the motivation of its application to modeling electrical propagation on the surface of the human left atrium will be provided. In the beginning of the talk I will give a brief outline of the finite element framework, Nektar++, that was used as an implementation platform for this project.
This a joint work with Mike Kirby (SCI Institute) and our collaborators from Imperial College London: Chris Cantwell, Spencer Sherwin and Nicholas Peters.

November 8 (note, Friday at 3:50 pm in LCB 215)
Speaker: Dennis Kochmann , Graduate Aerospace Laboratories, Caltech
Title: Can negative-stiffness phases in composite materials lead to extreme effective mechanical properties?
Abstract: Engineering design is undergoing a paradigm shift in dealing with mechanical instabilities in solids and structures, away from avoiding instability and towards making positive use thereof. About a decade ago, it was first proposed to use so-called negative-stiffness phases (phases with non-positive-definite elastic moduli) in a composite to achieve extremely-high overall stiffness and damping of the composite. In this seminar, we will discuss if and under what circumstances negative stiffness (which commonly arises from a microscale instability) can indeed lead to extreme mechanical behavior in composite materials. To this end, we will review the classical conditions of elastic stability and show that negative-stiffness phases are indeed permissible if sufficiently constrained by a matrix or coating. Next, we show that composite theory predicts such negative incremental moduli to result in extreme overall mechanical properties, and we correlate composite stability and elastic and viscoelastic performance. Finally, we present experimental confirmation of the exceptional performance of such composite systems.

November 11 (Student Talk)
Speaker: Christian Sampson, Department of Mathematics, University of Utah
Title: Wave-Ice Interactions in the Marginal Ice Zone
Abstract: The Marginal Ice Zone (MIZ) in the Southern Ocean around the continent of Antarctica can be defined as the part of the ice cover that is close enough to the open ocean boundary such that it is significantly affected by ocean swell. In this context the MIZ is an area of enhanced ice drift, deformation and divergence. One way in which the open ocean affects the MIZ is through wave-ice interaction. Waves coming in from the turbulent Southern Ocean break up the ice. The ice floes in turn damp out and limit the wavelengths that can exist in the MIZ. This wave-ice interaction is important for determining the thickness, floe sizes and extent of ice in the MIZ, all of which are important factors in the interaction of the ice cover with the global climate system. In this talk I will present a model for gravity waves in the MIZ in which the ice is viewed as a suspension with an effective viscosity much greater than that of water. The Navier-Stokes equations for an incompressible fluid are analyzed yielding a dispersion relation for the waves. I will also highlight possible applications of this model aimed at gaining a better understanding of ice dynamics in the MIZ.

November 18
Speaker: Qinghai Zhang , Department of Mathematics, University of Utah
Title: Algebraic Geometry Tools for Handling Irregular and Moving Boundaries with Fourth- and Higher-order Accuracies in the Context of Numerical PDEs
Abstract: Irregular and moving boundaries are ubiquitous in real-world problem, yet current methods for representing and tracking them are at best second-order accurate. After motivating the need of high-order methods, I will first introduce the analytic solutions of two algebraic geometry problems: the classification of fluxing particles through a fixed curve and the Boolean algebra of regular open sets via bounded distributive lattice of Jordan curves. Then I will apply these tools to (i) derive an optimal algorithm of set intersection and union of polygons with spline boundaries, (ii) analyze a family of interface tracking methods including volume-of-fluid methods and the polygonal area mapping (PAM) method, (iii) propose the first fourth-order interface tracking method based on PAM, and (iv) design a general higher-order finite-volume framework that can handle multiple phases and $C^1$ discontinuities. The above analysis and algorithms are verified and validated by extensive numerical tests. Although the analysis part involves only elementary results in classical ODE, graph theory, complex analysis, algebraic topology, and fluid mechanics; the application part is at the frontier of computational geometry, computer aided geometric design, interface tracking methods, and multiphase flow simulation.

November 25
Speaker: Ilya Zaliapin , Department of Mathematics and Statistics, University of Nevada, Reno
Title: Random self-similar trees: models, statistical inference, and applications
Abstract: Hierarchical branching organization is ubiquitous in nature. It is readily seen in river basins, drainage networks, bronchial passages, botanical trees, lightening, and snowflakes, to mention but a few. Notably, empirical evidence reveals a surprising similarity among natural hierarchies - many of them are closely approximated by so-called self-similar trees (SSTs). This talk will focus on the Horton and Tokunaga self-similarity that provide easily parameterized constraints on random tree graphs. The Horton self-similarity is a weaker property that addresses the principal branching in a tree; it is a counterpart of the power-law size distribution for system's elements. The stronger Tokunaga self-similarity addresses so-called side branching; it ensures that different levels of a hierarchy have the same probabilistic structure (in a sense that can be rigorously defined). The Horton and Tokunaga self-similarity have been empirically established in numerous observed and modeled systems. This hints at the existence of a universal underlying self-similarity mechanism and prompts the question: What basic probability models can generate Horton/Tokunaga self-similar trees with a range of parameters? This talk reviews the existing results and presents recent findings on self-similarity for tree representation of branching and coalescent processes, random walks, and white noises. In particular, we establish the equivalence of tree representation for selected coalescent processes and time series models. We also describe a statistical framework for testing self-similarity in a finite tree and estimating the related parameters. Our results suggest at least a partial explanation for the omnipresence of Tokunaga self-similar structures in natural branching systems. The results are illustrated using applications in hydrology and billiard dynamics.
This is a joint work with Yevgeniy Kovchegov (Oregon State U) and Alejandro Tejedor (U of Minnesota).

December 2
Speaker: Bart Raeymaekers , Department of Mechanical Engineering, University of Utah
Title: A patterned microtexture to improve longevity of prosthetic hip joints
Abstract: Approximately 285,000 total hip replacement (THR) surgeries are performed in the US each year. Most prosthetic hip joints consist of a cobalt-chromium (CoCr) femoral head that articulates against a polyethylene acetabular component, lubricated by joint fluid. The statistical survivorship of these metal-on-polyethylene prosthetic hip joints declines significantly after 15 years of use, primarily due to wear and wear debris incited disease. The current engineering paradigm aims to manufacture ultra-smooth articulating surfaces to increase longevity of prosthetic hip joints. In contrast, we show that adding a patterned microtexture to the ultra- smooth CoCr femoral head reduces friction when articulating against the polyethylene acetabular liner. The microtexture increases the lubricant film thickness, and reduces contact, friction, and eventually wear. We have optimized the microtexture geometry to maximize the lubrication film thickness between the articulating surfaces of the prosthetic joint, and experimentally demonstrate reduced friction for the microtextured compared to the smooth articulating surfaces lubricated with joint fluid.

December 9
Speaker: Jichun Li , Department of Mathematical Sciences, University of Nevada, Las Vegas
Title: Mathematical study and computational modeling of invisibility cloak with metamaterials
Abstract: In the June 23, 2006's issue of Science magazine, Pendry et al and Leonhardt independently published their works on electromagnetic cloaking. In Nov 2006's Science, Pendry et al demonstrated the first practical realization of such a cloak with the use of artificially structured metamaterials. In 2006, Graeme Milton also showed cloaking effects with anomalous localized resonance. Since 2006, there is a growing interest in using metamaterials to construct invisibility cloaks. In this talk, I'll focus on the mathematical study and numerical simulation of some cloak models. Numerical simulations using finite element methods in both frequency domain and time domain will be presented.

Seminar organizer: Yekaterina Epshteyn (epshteyn (at)

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