The goals of this seminar are to expose students, postdocs, faculty and visitors to research about waves broadly speaking in a interdisciplinary setting. Topics include non linear waves, applications in acoustics, optics, etc. This seminar also provides opportunities to share interesting findings, and to potentially contribute with research ideas/projects.

Our group meetings and seminar are led by Professors Chrysoula Tsogka and Camille Carvalho and consist of graduate students, postdocs and faculty.

## Spring 2021

For the Spring 2021 semester, seminars will take place on **Thursdays** from **15:00 to 16:00am** **via Zoom**.

**February 4: Adar Kahana (Tel Aviv University)**

Title: Solving PDE related problems using deep-learning

Abstract: In this talk I will present our work on PDE related problems and how integrating deep-learning can help achieve better results. We discuss the accelerating topic of physically-informed neural networks: a method for solving PDEs with neural networks. With this in mind, we present our work in this field and present two interesting variants of this method. Motivated by the physical experiment of acoustic waves propagating in an underwater homogeneous domain, we discuss the inverse problem: Simulating the physical experiment, we solve the acoustic wave equation and save the data at a small number of sensors over many time steps and given these sensor measurements we aim to find and identify the shape of an obstacle inside the domain. We cast it to a data-driven problem by building an image segmentation of the domain where the segment is an arbitrary polygon (the obstacle). We improve the model using a physically-informed loss term designed based on the wave equation. After that we switch to a completely different area - we discuss an explicit nonlinear numerical scheme for the 1D wave equation that remains stable when violating the CFL condition. We create a data-set based on a stable wave propagation process and train a network to infer a non-stable process. We incorporate a physically-informed loss term here as well to achieve better accuracy (lower deviation from the analytic solution) for our scheme.

**February 18: Remi Cornaggia (Sorbonne University) **

Title: Modeling and controlling dispersive waves in architected materials : second-order homogenization and topological optimization

We are interested in waves in two-phase periodic materials, whose phase distribution is to be optimized to obtain specific dispersive properties (typically, to maximize the dispersion in given directions of wave propagation).

The two-scale asymptotic homogenization procedure will first be recalled. In particular, the second-order asymptotic expansion enables to model the low-frequency dispersive behavior of waves in these media. Illustratrations will be given for bilaminates in 1D, for which we designed correctors for boundary and transmission conditions, that complement the wave model to obtain an overall second-order approximation in bounded domains [1].

The topological optimization algorithm [2] will then be presented for scalar waves in 2D media (e.g. acoustic or antiplane shear waves). First, simple dispersion indicators are extracted from the homogenized model. Cost functionals to be minimized to achieve certain goals are then defined using these indicators. The minimization is then performed thanks to an iterative algorithm, which relies on the concept of topological derivative (TD) of the cost functional. The TD quantifies the sensitivity of the functional to a localized phase change in the unit cell, and therefore indicates optimal locations where to perform these phase changes. The TD of the cost functionnal can be computed from the TDs of the coefficients of the homogenized model, whose expressions were determined in a previous work [3]. At each step, the cell problems underlying the homogenized model, whose solutions are needed to compute the TDs, are solved thanks to FFT-accelerated solvers [4].

Two applications of the method will be presented: maximizing the dispersion in given directions, and determining the microstructure of an architected material from phase velocity measurements.

[1] Second-order homogenization of boundary and transmission conditions for one-dimensional waves in periodic media,

Remi Cornaggia and Bojan B. Guzina, International Journal of Solids and Structures, 2020

[2] Tuning effective dynamical properties of periodic media by FFT-accelerated topological optimization,

Rémi Cornaggia, Cédric Bellis, International Journal for Numerical Methods in Engineering, 2020

[3] Microstructural topological sensitivities of the second-order macroscopic model for waves in periodic media.

Marc Bonnet, Rémi Cornaggia and Bojan B. Guzina, SIAM Journal on Applied Mathematics, 2018

[4] A numerical method for computing the overall response of nonlinear composites with complex microstructure,

Hervé Moulinec et Pierre Suquet, Computer Methods in Applied Mechanics and Engineering, 1998

(files of [1,2,3] available on HAL : https://cv.archives-ouvertes.fr/remi-cornaggia )

## Fall 2020

For the Fall 2020 semester, seminars will take place on **Thursdays** from **10:30am to 11:30am** **via Zoom**.

**September 3: Sean Horan (UCM)**

Title: Double Spherical Harmonics And The Radiative Transport Equation

**September 17: Arnold Kim (UCM)**

Title: Direct and inverse scattering of extended objects

Abstract:

Both the direct and inverse scattering problems for extended objects are challenging problems. In this talk, I discuss the method of fundamental solutions for the direct scattering problem and signal subspace methods for the inverse scattering problem. The key to both of these methods is knowing the fundamental solution of the governing PDE. I will show some recent results for propagating waves and diffuse waves and discuss some ongoing research going on in the Sensing and Imaging SMaRT team.

Slides Video**October 1: Boaz Ilan (UCM)**

Title: Green’s function for nonimaging optics

Abstract: Nonimaging optics has application to designing solar energy concentrators and illumination engineering. In this talk, I will discuss how nonimaging optics can be described in the context of radiative transfer theory. The radiative transport equation reduces to a local transport equation, whose solution can be expressed in terms of a Green’s function. This new formalism may be useful for nonimaging designs.

**October 15: Stéphanie Chaillat-Loseille (UMA, ENSTA ParisTech)**

Title: Recent advances on the preconditioning of 3D fast Boundary Element Solvers for 3D acoustics and elastodynamics

Abstract: Recent works in the Boundary Element Method (BEM) community have been devoted to the derivation of fast techniques to perform the matrix vector product needed in the iterative solver. Fast BEMs are now very mature. However, it has been shown that the number of iterations can significantly hinder the overall efficiency of fast BEMs. The derivation of robust preconditioners is now inevitable to increase the size of the problems that can be considered. I will present some recent works on analytic and algebraic preconditioners for fast BEMs.

**October 29: Ornella Mattei**

Title: Wave propagation in space-time microstructures: the theory of field patterns

Abstract: Field patterns are a new type of wave propagating in one-dimensional linear media with moduli that vary both in space and time. Specifically, the geometry of these space-time materials is commensurate with the slope of the characteristic lines so that a disturbance does not generate a complicate cascade of subsequent disturbances, but rather concentrates on a periodic space-time pattern, that we call field pattern. Field patterns present spectacularly novel features. One of the most interesting ones is the appearance of a wave generated from an instantaneous source, whose amplitude, unlike a conventional wake, does not tend to zero away from the wave front. Furthermore, very interestingly, the band structure associated with these special space-time geometries is infinitely degenerate: associated with each point on the dispersion diagram is an infinite space of Bloch functions, a basis for which are generalized functions each concentrated on a field pattern.

**November 12: Tomas Virgen (UCM)**

Title: Time-Dependent Scattering By A Sound-Hard Sphere

Abstract: Time-dependent scattering in three dimensions is a difficult mathematical problem to solve analytically and numerically. Analytical solutions obtained using the method of separation of variables are complicated, and numerical methods to solve this problem have challenges for computing solutions at long distances and long times. We seek a simple, effective, and efficient method to solve this problem based on the method of fundamental solutions. The specific case of a sound-hard sphere is considered in the context of acoustics.

**December 3: Camille Carvalho**

Title: Limiting Amplitude Principle for plasmonic structures with corners

## Spring 2020

**Seminars are on Thursdays, 2pm-3pm in ACS 362B**.

### February 6: Dr. Nicki Boardman, 2pm-3pm in Granite Pass 120-125

### February 20: Matthias Bussonnier

Title: Python overview

### March 5:

### Camille Carvalho

Title: Subtraction tecnhiques for the close evaluation of layer potentials

Abstract: Close evaluation of layer potentials reffer to large errors occured at evaluation points near (but not on) the boundary. This is due to peaked behaviors of the integrands near the boundary. There exist subtraction techniques to smooth out the peaked behavior for Laplace's problems. In this talk we present how to extend those ideas to waves problems.

### Arnold D. Kim

Title: Optical imaging of colloids

Abstract: A colloidal suspension is a collection of nanometer to micron scaled particles in a fluid used to study self-assembly. A key to studying colloids lies in imaging these particles accurately and efficiently within a standard microscope setup. In this talk, we introduce this imaging problem, propose an imaging method that uses space, angle or wavelength diversity at the source to compensate for the intensity-only measurements, and demonstrate its effectiveness through numerical simulations.

### March 19: Cancelled

### April 9: Lori Lewis

Title: Asymptotic approximations for boundary integral equations with regions of high curvature

### April 23:

### Ben Latham

Title: Introduction to Surface Plasmons on a Planar Interface

### Zoïs Moitier

Title: Where are the plasmons: asymptotics for the cavity case

## Fall 2019

The first meeting will be on **Thursday, September the 12th, at 10-11am in ACS 362B**. The first meeting will be an organizational

meeting and introductions. The focus of this meeting will be on setting up the schedule, hence we'll need volunteers to give talks. If you can't make this meeting but would like to give a talk, please send me your title and preferred date as soon as possible.

**September 26th:**

Arnold D. Kim, (Full Professor, Applied Mathematics, University of California, Merced)

Title Talk: Research projects about the multiple scattering of light

Abstract: In this talk I give an overview of my research projects about the multiple scattering of light. Then I discuss two specific projects in detail. First, I discuss modeling nano cloaking structures in collaboration with Prof. Ghosh's lab. Then I discuss some recent work in the diffuse optical imaging of tissues using spatially modulated light.

**October 10th:**

Chrysoula Tsogka, (Full Professor, Applied Mathematics, University of California, Merced)

Title Talk: Imaging in waveguides

Boaz Ilan, (Full Professor, Applied Mathematics, University of California, Merced)

Title Talk: Nonlinear eigenvalue problems in nonlinear waves

**October 24th: **

Jay Sharping, (Professor, Physics, University of California, Merced)

Title Talk: Tunable SRF cavities for 3D optomechanics

Abstract: Quantum mechanics is real! If you took a physics class on Quantum Mechanics, they (should have) told you to consider a mass on a spring and (should have) showed that the energy spectrum for that thing is quantized. Over the past 10 years or so we have finally been able to do just that in the lab. One of my research goals is to combine low loss oscillators such as optical or microwave cavities with mechanical oscillators in hopes of one day recognizing quantum behavior in large oscillators. In this talk I will (hopefully) get you excited about the prospects of this effort and then share designs, simulations and experiments with high-Q 3-dimensional cavities. I’ll give you a status report on my group’s journey to coupling these cavities to mechanical oscillators.

**November 7th: **

Zoïs Moitier, (Postdoc, Applied Mathematics, University of California, Merced)

Title : Asymptotic expansions of Whispering Gallery Modes optical micro-cavities

Abstract: In this talk we study the resonance frequencies of bidimensional optical cavities. More specifically, we are interested in whispering-gallery modes (modes localized along the cavity boundary with a large number of oscillations). The first part deals with the numerical computation of resonances by the finite element method using perfectly matched layers, and with a sensibility analysis in the three following situations: an unidimensional problem, a reduction of the rotationally invariant bidimensional case, and the general case. The second part focuses on the construction of asymptotic expansions of whispering-gallery modes as the number of oscillations along of boundary goes to infinity. We start by considering the case of a rotationally invariant problem for which the number of oscillations can be interpreted as a semiclassical parameter by means of an angular Fourier transform. Next, for the general case, the construction uses a phase-amplitude ansatz of WKB type which leads to a generalized Schrödinger operator. Finally, the numerically computed resonances obtained in the first part are compared to the asymptotic expansions made explicit by the use of a computer algebra software.

**November 21st: **

Cory MacCullough, (graduate student, Applied Mathematics, University of California, Merced)

Title Talk: TBA

Imran Khan, (graduate student, Physics, University of California, Merced)

Title Talk: TBA

**December 12th: **

TBA

Title Talk: TBA