Seminar 10 -S4: Multimode multiple scattering in disordered systems of spherical scatterers
Speaker: Valerie Pinfield (Chemical Engineering Department, Loughborough University, UK)
Abstract: We will consider the propagation of ultrasound in nano/microparticle suspensions consisting of disordered (randomly distributed) solid particles in a viscous liquid or other soft viscoelastic material. Understanding the complex wave propagation in such systems is important, both from the point of view of characterising soft materials – for biological tissues, or for industrial process monitoring e.g. in pharma and food – and for the design of metamaterials where enhanced absorption may be desirable – for example for stealth coatings. An important and interesting characteristic of these systems is the existence of two wavenumbers of very different magnitudes in the continuous phase – for the compressional and shear modes. For typical liquids at MHz frequencies, and with colloidal particles, the dimensionless compressional wavenumber (based on particle size) may be small whereas the dimensionless shear wavenumber can range from small to large and is always much larger than the dimensionless compressional wavenumber. In addition, we have another length scale that affects the multiple scattering, i.e. the interparticle separation; its effects are usually accounted for through the concentration or number density but it may be helpful to consider these effects in terms of additional dimensionless wavenumbers relating to the interparticle distance. Recently, a new formulation for the effective wavenumber arising from multiple scattering in a multi-mode media was published (Luppe, Conoir & Valier-Brasier, Wave Motion 115 (2022) 103082). In this presentation, we will show analytical asymptotic results obtained from the model at long compressional wavelength (low frequency), but arbitrary shear wavelength for these soft media, highlighting the dominant wave-conversion and correlation contributions. We will show numerical results to validate the approximations made for this system and compare with experimental data. Although it is common to consider that structure has minimal effect at long wavelength, in these dual-wave-mode systems the particle correlations are seen to have a significant effect because of the influence of the small shear wavelength.
Biography: Valerie Pinfield is Professor of Ultrasonics and Complex Materials at Loughborough University with a research programme focussing on the mathematical modelling of ultrasonic wave scattering in heterogeneous media with associated computational and experimental investigations. She is also developing the application of machine learning techniques to material design, surrogate models and optimised control for electrochemical systems for energy applications and for more sustainable chemical production without fossil fuel feedstocks.
Valerie studied Natural Sciences (theoretical physics) at the University of Cambridge, graduating in 1990 (MA, 1994) and completed her PhD in 1996 at the Food Science Department of The University of Leeds. She has experience in industry (The Welding Institute and Cadbury Ltd) as well as academia and developed her academic research working on ultrasonic characterisation of materials from aerospace composites suffering from porosity, to liquid-based particulate formulations found in pharma and foods. Her wave scattering research has latterly encompassed a broad range of interests including aggregated particle suspensions, acoustofluidics and metamaterials. Valerie was recently a Co-organiser of a six-month programme at the Isaac Newton Institute for Mathematical Sciences on the Theory and Applications of Multiple Wave Scattering (Jan-Jun 2023) which brought together researchers from all over the world working on wave scattering in optics, electromagnetics, acoustics, water waves and more.
Seminar 9 -S4: Valery Smyshlyaev (Department of Mathematics, University College London, UK)
Title: Two-scale homogenisation of high-contrast subwavelength resonances and error analysis
Abstract: High-contrast two-scale homogenization is a field with several decades of history. In the context of wave propagation, the critical scaling between the two small parameters of the spatial scale separation and contrast appears to be a micro-resonant scaling. Renewed interest to this area in the context of metamaterial modelling seems to be due to the two-scale asymptotic models’ ability to display often unusual macroscopic physical effects in an asymptotically explicit way, clarifying the nature and the microscopic mechanism of the observable effects such as band gap opening due to the subwavelength resonances. We give a brief background overview and discuss various scenarios displaying interesting effects. Finally, we report on most recent progress on constructing improved approximations of a two-scale type with controllably small errors for a broad and growing set of models of physical interest, see , joint work with Shane Cooper and Ilia Kamotski.
Seminar 8 -S4: The plasmonic eigenvalue problem beyond the quasi-static limit
Speaker: Matias Ruiz (School of Computing and Mathematical Sciences, University of Leicester, UK)
Abstract: Plasmonic resonances in nanoparticles can be understood in the quasi-static limit as solutions to the plasmonic eigenvalue problem, i.e. solutions to the quasi-static homogeneous Maxwell’s equations when considering the permittivity as the eigenvalue; this formulation makes the modes material-independent. In this talk I will consider analogous versions of the plasmonic eigenvalue problem in two scenarios. First, the full wave regime in 2D where the subwavelength assumption on the size of the nanoparticle is abandoned and the wave scattering problem has to be modelled by the Helmholtz equation. Second (time permitting), the nonlocal hydrodynamic Drude model, which describes, qualitatively, the light-matter interactions at scales where the quantum nature of matter becomes apparent. I will present a rigorous spectral analysis of the plasmonic eigenvalue problem in these two scenarios. The main results are the completeness of the material-independent modes for the Helmholtz equation, and the regularizing properties of nonlocality in the nonlocal hydrodynamic Drude model.
Biography: Matias is an applied mathematician working in the field of wave propagation in complex media. He uses a wide range of mathematical and computational techniques, including spectral theory, PDE analysis, and asymptotic methods, to tackle topical problems in electromagnetic theory and related wave phenomena.
Seminar 7 -S4: Diffraction of acoustic waves by edges and corners of a simple periodic material
Speaker: Anastasia Kisil (Department of Mathematics, University of Manchester, UK)
Abstract: In diffraction theory, it is well known that the interfaces between two materials (points, lines or surfaces) play a fundamental role in wave scattering. In this talk I will explain the progress that is made into semi-analytical method to compute the wave scattering by a simple periodic material. The study of wave propagation in a periodic medium is a classical and important research direction but it is usually assumed that the material is infinite in all directions. The novelty of this approach is that the periodic material is semi-infinite and hence has edges and corners. In particular, I will consider semi-infinite arrays and wedges made of point scatters. The method will be a based on the generalisation of Wiener–Hopf techniques which has classically being used to solve scattering by one array.
This is joint work with M. A. Nethercote and R. C. Assier
Biography: Dr Kisil is a Royal Society Dorothy Hodgkin Research Fellow at the University of Manchester. Previously she was held a stipendiary three-year Research Fellowship at Corpus Christi College Cambridge. She has obtained her degree from University of Cambridge, Trinity College. She is currently a leader and initiator of a Specialist Interest Group in the UK Acoustics Network “Mathematical methods in acoustics”.
Seminar 6 -S4: Wave propagation in quasiperiodic media
Speaker: Pierre Amenoagbadji (Department of Applied Physics and Applied Mathematics, Columbia University, USA)
Abstract: A quasiperiodic medium is an ordered medium without being periodic. A fairly well-known example since the 2011 Nobel Prize in Chemistry is the quasicrystal. The notion of quasi-periodic and more generally almost periodic function is a very well-defined notion in the mathematical literature. To give an idea, a 1D quasiperiodic function is the trace along a line of a periodic function of nn variables. PDEs with quasi-periodic coefficients have been the subject of a number of theoretical studies, but it seems that there has been much less work on the numerical resolution of these equations.
The objective of this work is to develop original numerical methods for the resolution of the time-harmonic wave equation in quasiperiodic media, in the spirit of the methods previously developed for periodic media. The idea is to use the fact that the study of an elliptic PDE with quasiperiodic coefficients comes down to the study of an “augmented” non-elliptic PDE in higher dimensions, but whose coefficients are periodic. This so-called lifting approach allows to use tools that are adapted for periodic media, but comes with the price that the augmented PDE is non-elliptic, in the sense of its principal part.
In this talk, I will first present the lifting method for the 1D Helmholtz equation with dissipation. I will then explain how this method can be used to solve the 2D Helmholtz equation in a junction of periodic media cut in an arbitrary direction. The method will be illustrated by numerical results.
Biography: Since January 2024, I am a postdoctoral fellow in the Department of Applied Physics and Applied Mathematics at Columbia University, under the supervision of Pr. Michael Weinstein. In December 2023, I completed my PhD titled “Wave propagation in quasiperiodic media” at POEMS (UMR CNRS-ENSTA Paris-INRIA) under the supervision of Sonia Fliss and Patrick Joly. I am interested in the analysis and simulation of wave propagation phenomena in heterogeneous media.
Seminar 5 -S4: Twinning cavities – Making a garage sound identical a cathedral
Speaker: Richard Craster (Department of Mathematics, Imperial College London, UK)
Abstract: Bounded domains have discrete eigenfrequencies/spectra, and cavities with different boundaries and areas have different spectra. A general methodology for isospectral twinning, whereby the spectra of different cavities are made to coincide, is created by combining ideas from across physics including transformation optics, inverse problems and metamaterial cloaking. We extend this to open cavities where the spectrum is no longer purely discrete and real, and we pay special attention to twinning of leaky modes in 2D open cavities associated with complex valued eigenfrequencies with an imaginary part orders of magnitude lower than the real part. Open cavities are often an essential component in the design of ultra-thin subwavelength metasurfaces and a typical requirement is that cavities have precise, often low frequency, resonances whilst simultaneously being physically compact. To aid this design challenge we develop a methodology to allow isospectral twinning of reference cavities with either smaller or larger ones, enforcing their spectra to coincide so that open resonators are identical in terms of their complex eigenfrequencies.
Biography: Richard Craster is the Dean of the Faculty of Natural Sciences at Imperial College and formerly Head of Department of Mathematics at Imperial College a role he held for six years (2011-2017). He is also Director of the CNRS-Imperial ‘Abraham de Moivre’ International Research Laboratory in Mathematics. He has been at Imperial College as an academic since 1998 apart from holding a distinguished professorship in Alberta, Canada, 2008-2010 returning to become Head of the Mathematics Department at Imperial. In addition to being a Professor of Applied Mathematics he is also a member of the Mechanical Engineering Department at Imperial.
Seminar 4 -S4: Graded arrays for spatial frequency separation and amplification of water waves
Speaker: Malte A. Peter (Institute of Mathematics, University of Augsburg)
Abstract: Wave-energy converters extracting energy from ocean waves are known to suffer from poor efficiency. We propose structures capable of substantially amplifying water waves over a broad range of frequencies at selected locations, with the idea of enhanced energy extraction. The structures consist of full or C-shaped bottom-mounted cylinders arranged in one-dimensional or two-dimensional arrays, with the cylinder properties or the array spacing graded along the array. Using linear potential-flow theory, it is shown that the energy carried by a plane incident wave is amplified within specified locations, for wavelengths comparable to the array length, and for a range of incident directions. Transfer-matrix analysis is used to analyse the large amplifications and we also show results from recent wave-flume experiments confirming the amplification phenomenon in practice.
Biography: Malte A. Peter is the professor of applied analysis at the University of Augsburg in Germany. His research is in mathematical modelling and simulation as well as analysis of partial differential equations with a focus on multiscale problems. He regularly collaborates with scientists from physics, chemistry and engineering on problems of continuum mechanics (fluids and solids), biophysics and the geosciences in particular. He is an expert in (periodic) homogenisation methods and their applications as well as (water-)wave–structure interaction problems. Professor Peter received a master’s degree in mathematics with first class honours from Massey University, Auckland, New Zealand in 2003 with a thesis on water-wave interactions with floating elastic objects (modelling sea ice). The next year, he graduated as Diplom-Mathematiker with distinction from the University of Bremen, Germany, with a thesis on homogenisation of reaction–diffusion problems and seamlessly continued this work for his PhD with a scholarship of the German Academic Scholarship Foundation there, which he obtained at the end of 2006 with the highest possible mark summa cum laude. After his PhD, he filled post-doc positions, first in Bremen and then at the University of Auckland, before he took up a tenure-track professorship at the University of Augsburg in Germany in 2009. Ever since, he has been based at Augsburg, obtaining his habilitation degree in 2011, and he was tenured to full professor in 2013. Recent projects include mathematical aspects of wave propagation through heterogeneous media, mechanical damage of materials with microstructure, degradation mechanisms in porous materials as well as physico-chemical processes in biological materials. Besides being head of the Research Unit Applied Analysis at Augsburg, he is an associate editor of the SIAM Journal on Applied Mathematics and of Springer Nature Scientific Reports.
Seminar 3 -S4: On Micropolar Elastic Foundations and Architected Interfaces
Speaker: Marcelo Dias (School of Engineering, University of Edinburgh, UK)
Abstract: We will tackle the task of modelling heterogeneous and architected materials, necessitating advanced homogenisation techniques. In simplifying this challenge, we leverage micropolar elasticity. Simultaneously, elastic foundation theory is widely applied in fracture mechanics and delamination analysis of composite materials. The objective is to seamlessly integrate these frameworks, refining elastic foundation theory to accommodate materials exhibiting micropolar behaviour. Our elastic foundation theory for micropolar materials employs a stress potentials formulation, leading to closed-form solutions for stress and couple stress reactions, including the associated restoring stiffness. Additionally, we delve into the mechanical properties of conceptual structural adhesive joints, where the adhesive function is assumed by an architected interface. Diverging from isotropic interfaces, architected interfaces exert control over properties through tailored microstructures. To augment existing theoretical frameworks, we introduce our elastic foundation theory, encompassing emerging micromechanical effects. Illustrating how characteristic lengths govern the Mode I fracture behaviour of architected interfaces, we assert control over the fracture process zone size. Our findings, validated through numerical simulations, underscore the effectiveness of the proposed method.
Biography: Dr Dias obtained his bachelor’s in physics at the State University of São Paulo, Brazil. Four years later, he commenced a MSc in theoretical physics from his alma mater. In 2012, he obtained his PhD degree from the University of Massachusetts, USA, where he researched on the mechanics of origami structures and growth mechanisms. Dr Dias has worked as a researcher on a broad range of topics in structural engineering and applied mathematics at Brown University School of Engineering (USA), Aalto University (Finland), and the Nordic Institute for Theoretical Physics at KTH (Sweden). Before joining the University of Edinburgh, Dr Dias was an Associate Professor of mechanical engineering at Aarhus University in Denmark, where he lead his research group ‘Mechanical Metamaterials and Soft Matter’.
Seminar 2 -S4: An overview of waves in random media with comparisons to periodic
Speaker: Artur Gower (the Dynamics group of University of Sheffield, UK)
Abstract: Describing exactly how waves multiply scatter between a complex arrangement of particles is challenging. There exist accurate numerical methods, but they lack intuition, and can be slow for a large quantity of particles. To resolve this, we use theoretical methods to deduce dispersion equations, usually for an infinite media, which leads to a basis for the solution. The solution can then be completely determined with some boundary conditions or matching. This is true for periodic materials, homogeneous, and also holds true for waves in material with a random arrangement of particles (averaged over all configurations) for any frequency. In this talk, I will give an overview of results for waves in a random particulate, while making some light comparisons with periodic materials. There are many opportunities and applications of this theory in both developing new sensors and designing materials, which I will mention.
Biography: My background is in applying mathematics (BSc, MSc, PhD) to understand the microstructure of complex solids. I mostly develop code and mathematical models for waves (like sound and radio). We still do not fully understand how waves (like sound, radio, light, and vibrations) behave in many materials. How well can these waves propagate, and how much information can they carry in different materials.
I work within a department of engineering, where I help propose new sensors for ultrasonic inspection, monitoring, and medical ultrasound.