feldman (at) math.utah.edu),
Victoria Kala (victoria.kala (at) utah.edu),
or Akil Narayan (akil (at) sci.utah.edu)Monday, August 26 at 4pm. In-person LCB 219
Speaker: Frederic Marazzato,
Department of Mathematics, University of Arizona
Title: Computation of origami-inspired structures and mechanical metamaterials
Abstract: Origami folds have found a large range of applications in Engineering as solar panels for satellites or to produce inexpensive mechanical metamaterials. This talk will first focus on the direct problem of computing the deformation of periodic origami surfaces. A homogenization process for origami folds proposed in [Nassar et al, 2017] and then extended in [Xu, Tobasco and Plucinsky, 2023], is first discussed. Then, its application to two specific folds is presented alongside the PDEs characterizing the associated smooth surfaces. The talk will then focus on the PDEs describing Miura surfaces by studying existence of solutions and by proposing a numerical method to approximate them.
This talk will then present the inverse problem of trying to determine an optimal fold set approximating a given target surface. The folding of a thin elastic sheet is modeled as a two-dimensional nonlinear Kirchhoff plate with an isometry constraint. We formulate the problem in the framework of special functions of bounded variation SBV, and propose to use a phase-field damage model and a Discontinuous Galerkin method to approximate the solutions of this problem. We prove that the approximation $\Gamma$-converges to the sharp interface model. Finally, some numerical examples are presented.
Monday, September 16 at 4pm. In-person LCB 219
Speaker: Victoria Kala,
Department of Mathematics, University of Utah
Title: Thermomechanical material point methods
Abstract: In this talk, we will present a computational framework for modeling the deformation of solids under thermomechanical effects such as melting and combustion. When solids burn, they often undergo significant transformations, including fracturing, shrinking, or conversion into granular forms like ash. Melting solids experience a phase transition from solid to liquid. These processes are further influenced by the surrounding air, smoke, and fire. Our approach leverages the material point method (MPM) combined with incompressible fluid dynamics to simulate these phenomena.
We will begin with an overview of the MPM and its application to melting solids. Our method is an MPM discretization of surface tension forces that arise from spatially varying energies. This method also includes an implicit treatment of thermomechanical coupling and a particle-based enforcement of Robin boundary conditions to handle convective heating. Our framework effectively simulates materials with diverse surface tension and thermomechanical properties, such as melting wax.
Next, we will review incompressible flow and introduce a hybrid solver that integrates incompressible flow and MPM techniques to simulate the combustion of flammable solids. This solver allows for artistic control over flame front propagation and solid combustion rates. Solid combustion is modeled using temperature-dependent elastoplastic constitutive laws. We demonstrate the effectiveness of our approach through a series of real-world 3D simulations, including a burning match, incense sticks, and a wood log in a fireplace.
Monday, September 23 at 4pm. In-person LCB 219
Speaker: Kerrek Stinson,
Department of Mathematics, University of Utah
Title: Variational models for fracture and regularity properties
Abstract: In this talk, I will discuss some recent results with M. Friedrich and C. Labourie. We looked at minimizers of the Griffith energy describing fracture in a linearly elastic material. In dimension 2, we proved that at most points of the crack it is actually described by a Hoelder regular surface. This result requires a novel synthesis of ideas from regularity theory for the Mumford-Shah functional (coming from image segmentation) and rigidity estimates for linear elasticity. Depending on audience preference, I will discuss our ideas behind topological separation or the generic strategy of contradiction compactness used to obtain `strong' solutions.
Monday, October 14 at 4pm. In-person, LCB 219
Speaker: Yury Grabovsky,
Department of Mathematics, Temple University
Title: Convex duality for least squares problem in Hilbert spaces
Abstract: In many applications, such as electrochemical impedance spectroscopy, remote sensing, analysis of composite materials, brain MRI, and many others the response of materials can be described in terms of special classes of functions, such as Stieltjes class, in the case of composites, or completely monotone class, in the case of diffusion MRI. These response function can be measured experimentally at a number of points on the real line or in the complex plane. The problem of their reconstruction from the noisy experimental measurements is of central importance in these applications. In this talk I will analyze this problem by placing it in a general Hilbert space framework, deriving optimality conditions, error estimates, and proposing an algorithm for its solution that comes with a certificate of optimality. This is a joint work with my Ph.D. student Henry J. Brown.
Wednesday, October 16 at 4pm. In-person, JWB 335
Speaker: Kirill Cherednichenko,
Department of Mathematical Sciences, University of Bath
Title: Operator-norm homogenisation for Maxwell equations on periodic singular structures
Abstract:
I will discuss a new approach to obtaining uniform operator asymptotic estimates in periodic homogenisation. Based on a novel uniform Poincaré-type inequality, it bears similarities to the techniques I developed with Cooper (ARMA, 2016) and Velčić (JLMS, 2022). In the context of the Maxwell system, the analytic framework I will present leads to a new representation for the asymptotics obtained by Birman and Suslina in 2007 for the full system and by Suslina in 2004 for the electric field in the presence of currents. As part of the new asymptotic construction, I will link the leading-order approximation to a family of ``homogenised" problems, which was not possible using the earlier method. The analysis presented applies to a class of inhomogeneous structures modelled by arbitrary periodic Borel measures. However, the results are new even for the particular case of the Lebesgue measure. This is joint work with Serena D'Onofrio.
Monday, October 21 at 4pm. In-person, LCB 219
Speaker: Aaron Welters,
Department of Mathematics and Systems Engineering, Florida Institute of Technology
Title: Effective operators in Electrical Network Theory: New Perspectives and Representations
Abstract: In this talk, we introduce a class of problems, called Z-problems, that arose initially from the theory of composites [1,2] in the study of the properties of effective operators (a quintessential example of this is the conductivity equation of a 3D periodic composite and the effective conductivity). Then we will discuss a new and unified approach we've developed to treat these problems, based on block operator methods, for obtaining the associated effective operator in terms of the Schur complement. This approach allows for a relaxation of the usual coercivity hypotheses required the Z-problem for defining the effective operator. And, at the same time, we derive the dual variational principles for the effective operator under weaker hypotheses. Using this we give both upper and lower bounds on the effective operator as well as operator monotonicity results in the usual Loewner ordering defined on the convex cone of positive semidefinite operators. To show these results have interesting applications, we consider several examples from electrical network (circuit) theory that are new from this perspective including the Dirichlet-to-Neumann map and the effective conductivity for finite linear graphs. This is joint work with Kenneth Beard (LSU), Anthony Stefan (Florida Tech), and Robert Viator, Jr (Denison Univ.) based on [3].
[1] G. W. Milton, The Theory of Composites, SIAM, Philadelphia, PA, 2022. Reprint of book originally published by Cambridge Univ. Press, 2002.
[2] G. W. Milton (editor), Extending the Theory of Composites to Other Areas of Science, Milton-Patton Publishers, 2016.
[3] K. Beard, A. Stefan, J. Viator, R., and A. Welters, Effective operators and their variational principles for discrete electrical network problems, Journal of Mathematical Physics, 64 (2023). https://doi.org/10.1063/5.0149861
Monday, October 28 at 4pm. In-person, LCB 219
Speaker: Mark Allen,
Department of Mathematics, BYU
Title:
Higher free boundary regularity for semilinear obstacle-type problems
Abstract: We consider free boundary problems of semilinear type. Typically, the hodograph transform is avoided in such situations; however, we show how the hodograph transform together with Sobolev estimates can be used to bootstrap Lipschitz regularity of the free boundary to $C^{\infty}$ regularity.
Monday, November 4 at 4pm. In-person, LCB 219
Speaker: Hong Zhang,
Mathematics and Computer Science Division, Argonne National Laboratory
Title: Stable and Memory-Efficient Neural ODEs with Universal Gradient Accuracy
Abstract: Neural Ordinary Differential Equations (ODEs) have been a crucial tool for data-driven modelling, time series prediction, and generative modelling; however, most existing neural ODE methods struggle to balance numerical stability, memory consumption, and gradient accuracy. In this talk, I introduce a new neural ODE framework based on high-level discrete adjoint algorithmic differentiation. This framework employs discrete adjoint time integrators and advanced checkpointing strategies tailored for these integrators, implemented in the PETSc library, to significantly improve memory efficiency while maintaining accurate and consistent gradient computations. Our approach also integrates off-the-shelf linear solvers, including both direct and matrix-free iterative solvers, for effective mini-batch training.
Furthermore, I will present our semi-implicit neural ODE strategy, which builds upon this framework, using linear-nonlinear partitioning for implicit-explicit (IMEX) time integration and gradient calculation. The excellent stability property of IMEX methods enables the use of large time steps for ODE integration, which is critical for tackling stiff problems where traditional methods are inefficient. Our approach demonstrates superior performance across a range of applications, including graph classification and learning complex dynamical systems.
Wednesday, November 20 at 4pm. In-person, LCB 323
Speaker: Davit Harutyunyan,
Department of Mathematics, UC Santa Barbara
Title: On the extreme rays of the cone of 3 times 3 quasiconvex quadratic forms
Abstract: This talk is concerned with the study of the extreme rays (strong extremals) of the convex cone of 3 times 3 quasiconvex quadratic forms. Those are interesting from both quasiconvex analysis and theory of composites perspective. We characterize the forms, the determinant of the acoustic tensor of which is an extremal polynomial, proving that the extremality of the determinant implies either strong extremality or polyconvexity of the form. We show that similar results do not hold in higher dimensions. These results improve the ones on weak extremals previously obtained by the author and G. Milton.
Monday, December 2 at 4pm. In-person, LCB 219
Speaker: Bob Palais,
Department of Mathematics, University of Utah
Title:
Abstract:
feldman (at) math.utah.edu),
Victoria Kala (victoria.kala (at) utah.edu),
and
Akil Narayan (akil (at) sci.utah.edu).
Past lectures: Spring 2024, Fall 2023, Spring 2023, Fall 2022, Spring 2022, Fall 2021, Spring 2021, Fall 2020, Spring 2020, Fall 2019, Spring 2019, Fall 2018, Spring 2018, Fall 2017, Spring 2017, Fall 2016, Spring 2016, Fall 2015, Spring 2015, Fall 2014, Spring 2014, Fall 2013, Spring 2013, Fall 2012, Spring 2012, Fall 2011, Spring 2011, Fall 2010, Spring 2010, Fall 2009, Spring 2009, Fall 2008, Spring 2008, Fall 2007, Spring 2007, Fall 2006, Spring 2006, Fall 2005, Spring 2005, Fall 2004, Spring 2004, Fall 2003, Spring 2003, Fall 2002, Spring 2002, Fall 2001, Spring 2001, Fall 2000, Spring 2000, Fall 1999, Spring 1999, Fall 1998, Spring 1998, Winter 1998, Fall 1997, Spring 1997, Winter 1997, Fall 1996, Spring 1996, Winter 1996, Fall 1995.