
Date Time 
Location  Speaker 
Title – click for abstract 

09/07 2:00pm 
BLOC 302 
Alexander Watson University of Minnesota 
Moiréscale PDE models of twisted bilayer graphene
2D materials are materials consisting of a single sheet of atoms. The first 2D material, graphene, a single sheet of carbon atoms, was isolated in 2005. In recent years, attention has shifted to materials created by stacking 2D materials with a relative twist. Such materials are known as moiré materials because of the approximate periodicity of their atomic structures over long distances, known as the moiré pattern. In 2018, experiments showed that, when twisted to the first socalled magic angle (approximately 1 degree), twisted bilayer graphene exhibits exotic quantum phenomena such as superconductivity. I will present the first rigorous justification of the BistritzerMacDonald moiréscale PDE model of twisted bilayer graphene, which played a critical role in identifying twisted bilayer graphene’s magic angles, from a microscopic tightbinding model. If time permits, I will discuss the chiral model, a simplification of the BistritzerMacDonald model with a number of remarkable properties. 

09/21 3:00pm 
BLOC 302 
Frederic Marazzato Louisiana State University 
Homogenized Origami surfaces
Origami folds have found a large range of applications in Engineering as, for instance, solar panels for satellites, or the folding of airbags for optimal deployment or metamaterials. A homogenization process turning origami folds into smooth surfaces, developed in [Nassar et al, 2017], 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 and uniqueness of solutions and by proposing a numerical method to approximate them. Finally, some numerical examples are presented. 

09/28 3:00pm 
BLOC 302 
Loic Cappanera 
Robust numerical methods for incompressible flows with variable density
The modeling and approximation of incompressible flows with variable density are important for a large range of applications in biology, engineering, geophysics and magnetohydrodynamics. Our main goal here is to develop and analyze robust numerical methods that can be used with high order finite element and spectral methods. We first discuss the main challenges we face before introducing a semiimplicit scheme based on projection methods and the use of the momentum, equal to the density times the velocity, as primary unknown. We present an analysis of the stability and convergence properties of the method and obtain a priori error estimates. A fully explicit version of the scheme is then proposed. Its robustness and convergence are studied with a pseudo spectral code over various setups involving large ratio of density, gravity and surface tension effects, or manufactured solutions. Applications to magnetohydrodynamics instabilities in industrial setups such as aluminum production cells, and liquid metal batteries will be presented. Eventually, a novel method based on artificial compressibility techniques is introduced and its performances are compared to our projectionbased method. 

10/05 3:00pm 
BLOC 302 
Andrea Bonito Texas A&M University 
Paper Folding and Curved Origami: Modeling, Analysis and Simulation
The unfolding of a ladybird's wings, the trapping mechanism used by a flytrap, the design of selfdeployable space shades, and the constructions of curved origami are diverse examples where strategically placed material defects are leveraged to generate large and robust deformations. With these applications in mind, we derive plate models incorporating the possibly of curved folds as the limit of thin threedimensional hyperelastic materials with defects. This results in a fourth order geometric partial differential equation for the plate deformations further restricted to be isometries. The latter nonconvex constraint encodes the plates inability to undergo shear nor stretch and is critical to justify large deformations.
We explore the rigidity of the folding process by taking advantage of the natural moving frames induced by piecewise isometries along the creases. We then deduce relations between the crease geodesic curvature, normal curvature, torsion, and folding angle.
Regarding the numerical approximation, we propose a locally discontinuous Galerkin method. The second order derivatives present in the energy are replaced by weakly converging discrete reconstructions. Furthermore, the isometry constraint is linearized and incorporated within a gradient flow. We show that the sequence of resulting equilibrium deformations converges to a minimizer of the exact energy (and, in particular, to an isometry) as the discretization parameters tend to infinity. This theory does not require additional smoothness on the plate deformations besides having a finite energy. The capabilities and efficiency of the proposed algorithm is documented throughout the presentation by illustrating the behavior of the model on relevant examples.


10/12 3:00pm 
BLOC 302 
David Nicholls University of Illinois Chicago 
A Stable HighOrder Perturbation of Surfaces/Asymptotic Waveform Evaluation Method for the Numerical Solution of Grating Scattering Problems
The rapid and robust simulation of linear waves interacting with
layered periodic media is a crucial capability in many areas of
scientific and engineering interest. HighOrder Perturbation of
Surfaces (HOPS) algorithms are interfacial methods which recursively
estimate scattering quantities via perturbation in the interface shape
heights/slopes. For a single incidence wavelength such methods are the
most efficient available in the parameterized setting we consider
here. In this talk we describe a generalization of one of these HOPS
schemes by incorporating a further expansion in the wavelength about a
base configuration which constitutes an "Asymptotic Waveform
Evaluation" (AWE). We not only provide a detailed specification of the
algorithm, but also verify the scheme and point out its benefits and
shortcomings. With numerical experiments we show the remarkable
efficiency, fidelity, and highorder accuracy one can achieve with an
implementation of this algorithm. 

10/19 3:00pm 
BLOC 302 
LiseMarie ImbertGerard University of Arizona 


10/26 3:00pm 
BLOC 302 
Vladimir Yushutin Clemson University 
TRex FEM: an abstract analysis framework for unfitted methods
Unfitted, nonconforming finite element methods have the following in common: there is a drastic difference between the space of solutions and the finite element space. This difference manifests on the discrete level where one needs to employ a discrete stabilization form to guarantee the wellposedness of linear problems. Convergence analysis for such methods often follows the second Strang lemma, conditions of which may be hard to verify in some situations.
Instead, we study the strong convergence of unfitted continuousintime approximations via compactness. With this goal in mind, we develop an analysis framework, called TRex FEM, that involves notions of abstract TRace and EXtension operators. We build this analysis framework sequentially starting from abstract linear elliptic, parabolic, saddle problems and applying it to NavierStokes and AllenCahn equations. The key ingredient is a problemdependent modification of the abstract discrete stabilization form that makes the scheme amenable to a proof by compactness.
We test numerically the modified scheme suggested by the TRex FEM when it is applied to the surface heat equation being solved by the Trace FEM  an unfitted method for surface PDEs which uses a bulk mesh surrounding the surface. In addition to the advantage of TRex FEM from the analysis standpoint, the new scheme restores the conditioning of linear problems, known in the fitted case for the heat equation, despite the presence of the stabilization form. 

11/02 3:00pm 
BLOC 302 
Celine Torres University of Maryland 


11/16 3:00pm 
BLOC 306 
Shawn Walker Louisiana State University 
