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Texas A&M University
Mathematics

Numerical Analysis Seminar

Fall 2022

 

Date:September 7, 2022
Time:2:00pm
Location:BLOC 302
Speaker:Alexander Watson, University of Minnesota
Title:Moiré-scale PDE models of twisted bilayer graphene
Abstract: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 so-called magic angle (approximately 1 degree), twisted bilayer graphene exhibits exotic quantum phenomena such as superconductivity. I will present the first rigorous justification of the Bistritzer-MacDonald moiré-scale PDE model of twisted bilayer graphene, which played a critical role in identifying twisted bilayer graphene’s magic angles, from a microscopic tight-binding model. If time permits, I will discuss the chiral model, a simplification of the Bistritzer-MacDonald model with a number of remarkable properties.

Date:September 21, 2022
Time:3:00pm
Location:BLOC 302
Speaker:Frederic Marazzato, Louisiana State University
Title:Homogenized Origami surfaces
Abstract: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.

Date:September 28, 2022
Time:3:00pm
Location:BLOC 302
Speaker:Loic Cappanera, University of Houston
Title:Robust numerical methods for incompressible flows with variable density
Abstract: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 semi-implicit 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 projection-based method.

Date:October 5, 2022
Time:3:00pm
Location:BLOC 302
Speaker:Andrea Bonito, Texas A&M University
Title:Paper Folding and Curved Origami: Modeling, Analysis and Simulation
Abstract:The unfolding of a ladybird's wings, the trapping mechanism used by a flytrap, the design of self-deployable 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 three-dimensional hyper-elastic 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.

Date:October 12, 2022
Time:3:00pm
Location:BLOC 302
Speaker:David Nicholls, University of Illinois Chicago
Title:A Stable High-Order Perturbation of Surfaces/Asymptotic Waveform Evaluation Method for the Numerical Solution of Grating Scattering Problems
Abstract: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. High-Order 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 high-order accuracy one can achieve with an implementation of this algorithm.

Date:October 19, 2022
Time:3:00pm
Location:BLOC 302
Speaker:Lise-Marie Imbert-Gerard, University of Arizona
Title:Wave propagation in inhomogeneous media: An introduction to quasi-Trefftz methods
Abstract:Trefftz methods rely, in broad terms, on the idea of approximating solutions to Partial Differential Equation (PDEs) using basis functions which are exact solutions of the PDE, making explicit use of information about the ambient medium. But wave propagation in inhomogeneous media is modeled by PDEs with variable coefficients, and in general no exact solutions are available.

Quasi-Trefftz methods have been introduced, in the case of the Helmholtz equation with variable coefficients, to address this problem: they rely not on exact solutions to the PDE but instead of high order approximate solutions constructed locally. We will discuss the origin, the construction, and the properties of these so-called quasi-Trefftz functions. We will also discuss the consistency error introduced by this construction process.

Date:October 26, 2022
Time:3:00pm
Location:BLOC 302
Speaker:Jean-Luc Guermond, Texas A&M University
Title:Invariant-domain preserving high-order implicit explicit time stepping for nonlinear conservation equations
Abstract:I consider high-order discretizations of a Cauchy problem where the evolution operator comprises a hyperbolic part and a parabolic part with diffusion and stiff relaxation terms. Assuming that this problem admits non-trivial invariant domains, in the talk I will discuss approximation techniques in time that preserve these invariant domains. Before going into the details, I am going to give an overview of the literature on the topic. Emphasis will be put on explicit and explicit Runge Kutta techniques using Butcher's formalism. Then I am going to describe techniques that make every implicit-explicit time stepping scheme invariant-domain preserving and mass conservative. The proposed methodology is agnostic to the space discretization and allows to optimize the time step restrictions induced by the hyperbolic sub-step.

Date:November 2, 2022
Time:3:00pm
Location:BLOC 302
Speaker:Céline Torres, University of Maryland
Title:A two-scale method for the Integral Fractional Laplace equation and the Fractional Obstacle problem
Abstract:In this talk, we are interested in the approximation of integral Fractional Laplace equation. The continuous problem satisfies a maximum principle, which is essential for the analysis of the Fractional Obstacle problem. Motivated by this application, we propose a two-scale method for the linear problem, which inherits the maximum principle naturally and leads to L-estimates for the error. We present the discretization of the Fractional Obstacle Problem as a possible application of the method and its error analysis. The work presented is in collaboration with R.H. Nochetto, J.P. Borthagaray and A. Salgado.

Date:November 9, 2022
Time:3:00pm
Location:BLOC 302
Speaker:Vladimir Yushutin, Clemson University
Title:T-Rex FEM: an abstract analysis framework for unfitted methods
Abstract:Unfitted, non-conforming 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 well-posedness 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 continuous-in-time approximations via compactness. With this goal in mind, we develop an analysis framework, called T-Rex 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 Navier--Stokes and Allen--Cahn equations.

The key ingredient is a problem-dependent 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 T-Rex 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 T-Rex 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.

Date:November 16, 2022
Time:3:00pm
Location:BLOC 306
Speaker:Shawn Walker, Louisiana State University
Title:Curvature and the HHJ Method
Abstract:This talk shows how the classic Hellan--Herrmann--Johnson (HHJ) method can be extended to surfaces, as well as approximate curvature. We start by showing how HHJ can be extended to surfaces embedded in ℝ3 to solve the surface version of the Kirchhoff plate equation. The surface Hessian of the "displacement" variable is discretized by an HHJ finite element function. Convergence is established for all possible combinations of mixed boundary conditions, e.g. clamped, simply-supported, free, and the "4th" condition. Numerical examples are shown, some which use "point" boundary conditions as well as solving the surface biharmonic equation.

We also show how the surface HHJ method can be used to post-process a discrete Lagrange function on a given surface triangulation to yield an approximation of its surface Hessian, i.e. a kind of Hessian recovery. Moreover, we demonstrate that this scheme can be used to give convergent approximations of the *full shape operator* of the underlying surface using only the known discrete surface, even for piecewise linear triangulations. Several numerical examples are given on non-trivial surfaces that demonstrate the scheme.