Previous Webinars:



Semmianr 20 -S4: Non-reciprocal and Non-Newtonian Mechanical Metamaterials

Speaker: Lianchao WANG (Department of Mechanical Engineering, City University of Hong Kong)

Abstract: Non-Newtonian liquids are characterized by stress and velocity-dependent dynamical response. In elasticity, and in particular, in the field of phononics, reciprocity in the equations acts against obtaining a directional response for passive media. Active stimuli-responsive materials have been conceived to overcome it. Significantly, Milton and Willis have shown theoretically in 2007 that quasi-rigid bodies containing masses at resonance can display a very rich dynamical behavior, hence opening a route toward the design of non-reciprocal and non-Newtonian metamaterials. In this paper, we design a solid structure that displays unidirectional shock resistance, thus going beyond Newton’s second law in analogy to non-Newtonian fluids. We design the mechanical metamaterial with finite element analysis and fabricate it using three-dimensional printing at the centimetric scale (with fused deposition modeling) and at the micrometric scale (with two-photon lithography). The non-Newtonian elastic response is measured via dynamical velocity-dependent experiments. Reversing the direction of the impact, we further highlight the intrinsic non-reciprocal response.

Biography: Lianchao Wang is a Ph.D. in the mechanics of solids. Lianchao was awarded his first Ph.D. diploma at the University Bourgogne Franche-Comté (France) and his second one at Harbin Institute of Technology (China) in July and September 2023, respectively. Currently, he works as a postdoc researcher at the Department of Engineering, City University of Hong Kong. His research is on the static and/or dynamic mechanical properties of porous structures or metamaterials.

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Seminar 18 – S4: Weak scattering in phononic crystals

Speaker: Mario Lázaro (Universitat Politècnica de València, Spain)

DOI: doi.org/10.52843/meta-mat

Abstract: In recent decades, there has been a significant advancement in the field of phononic crystals, thanks to the development of innovative techniques for controlling waves in complex media. One area of particular interest involves materials comprising a host medium with an array of point scatterers. Hyperuniform and stealth disordered materials are especially noteworthy, as they enable the identification of point scatterers that mitigate scattering within a certain frequency range while exhibiting heterogeneity in another range. The Born hypothesis plays a crucial role in connecting the design of hyperuniform materials to wave propagation characteristics. In this seminar, we will delve into a comprehensive methodology for modeling various types of scatterers in 1D waveguides. Furthermore, we will establish the mathematical correlation between the point distribution and propagation properties to ensure that we are dealing with weak scattering phenomena. To validate this methodology, we will present numerical examples and provide physical interpretations for the mathematical findings.

Biography: Dr Mario Lázaro is Associate Professor of the Dept. of Continuum Mechanics and Theory of Structures (UPV) teaching subjects of Aeroelasticiy, Modal Analysis and Aircraft Structures for Aerospace Engineers. He is researcher for in the IUMPA (Instituto de Matemática Pura y Aplicada). With an extensive research experience in the fields of structural dynamics, mechanical vibrations, and wave propagation in elastic media, he has been part of a total of 6 competitive research projects, being principal investigator of one of them. He has about 25 JCR research publications covering different fields: structural analysis, numerical methods in continuous media, wave phenomena in complex media and energy dissipation in dynamic systems. He has been a visiting researcher at the University of Calgary and at Imperial College London (UK), where he currently collaborates with the Wave Phenomena and Metamaterials research group. In 2023 he was recipient of the prestigious Doak Award, to the most successful paper in the Journal of Sound and Vibration in 2022.

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Seminar 17 -S4: Double Floquet-Bloch transforms: complex deformation of integration surfaces and far-field asymptotics of periodic structures Green’s functions

Speaker: Raphael Assier (Department of Mathematics, University of Manchester, UK)

DOI: doi.org/10.52843/meta-mat.p0dj1n

Abstract: We propose a general procedure to study double integrals arising when considering wave propagation in periodic structures. This method, based on a complex deformation of the integration surface to bypass the integrands’ singularities, is particularly efficient to estimate the Green’s functions of such structures in the far field. We provide several illustrative examples and explicit asymptotic formulae. Special attention is devoted to the pathological cases of degeneracies, such as Dirac conical points for instance.

Biography: I am a Reader in Applied Mathematics, within the Department of Mathematics at the University of Manchester. I graduated from the Ecole Centrale de Lyon, and then worked as an engineer for Rolls-Royce (2006-2007, Derby), before returning to studying mathematics. Having obtained a MAST (formerly part III of the mathematical tripos) and a PhD in Applied Mathematics from the University of Cambridge (with Nigel Peake), I was a Postdoctoral Research Associate (2011-2013, with Xuesong Wu) and then a Junior Research Fellow (2013) at Imperial College London. I was appointed to a lectureship in Manchester in December 2013, promoted to Senior Lecturer in 2017 and to Reader in 2021.

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Seminar 16 -S4: Graded Elastic Metamaterials: Rainbow Reflection, Topological Trapping, and Zero Group Velocity Modes

Speaker: Gregory James Chaplain (Centre for Metamaterial Research and Innovation, University of Exeter, UK)

 DOI: doi.org/10.52843/meta-mat.tqpx7g

Abstract: Graded elastic metamaterials have been identified as attractive candidates for the manipulation of waves on elastic beams and half-spaces with a variety of applications ranging from vibration isolation to energy harvesting. This talk will cover a review of the elastic metawedge (proposed by Colombi et al [Sci Rep, 6, 27717 (2016)]), highlighting several modalities through interpretation of local dispersion curves which underpin the phenomena of rainbow reflection, trapping, and mode conversion. Applications of each of these effects shall be outlined: rainbow trapping and rainbow reflection are delineated, providing a route to enhanced energy harvesting by inspection of zero group velocity modes; topological interfaces are incorporated within a grading structure, motivating robust energy harvesters; and finally designer diffraction for ‘focussing’ is achieved through mode conversion along a graded structure and at abrupt interfaces.

Biography: Gregory is from Dundee, Scotland, where he lived until moving to Glasgow in 2012 to undertake an MSci in physics at the University of Glasgow. Afterwards he moved to Imperial College London where he obtained a PhD in applied mathematics under Prof. Richard Craster, winning the Yael Dowker Prize for best mathematics PhD thesis for his research into wave propagation effects in structured materials. He continued at Imperial as a doctoral prize research fellow before being awarded a Research Fellowship from the Royal Commission for the Exhibition of 1851 at the University of Exeter in 2021. He was appointed as a proleptic Lecturer in Metamaterials Physics in 2022.

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Seminar 15 -S4: Analytical solutions for Bloch waves in resonant phononic crystals

Speaker: Richard Wiltshaw (Department of Mathematics, Imperial College London, UK)

 DOI: doi.org/10.52843/meta-mat.vt1dvd

Abstract: I will discuss the canonical problem of wave scattering by arrays of Neumann inclusions. Firstly, we consider the wavefield to be governed by the Helmholtz equation. We apply the method of matched asymptotic expansions to show how small scatterers can be modelled as singular perturbations to the free space. Analytical expressions then follow in terms of singular Green’s functions (and their derivatives), from which we can construct an eigenvalue problem to consider Floquet-Bloch waves, or we can consider scattering problems as an extension to Foldy’s method. The methods presented allow for efficient, rapid, and accurate computations.

These methods will then be applied in an elastic setting, to consider waves propagating through an elastic plate, whose surface is patterned by periodic arrays of elastic beams. Our methodology is versatile and allows us to solve a range of problems regarding arrangements of multiple beams per primitive cell, over Bragg to deep-subwavelength scales. We cross-verify against finite element numerical simulations to gain further confidence in our approach. The accuracy and flexibility of our solutions are demonstrated by engineering topologically non-trivial states, from primitive cells with broken spatial symmetries, following the phononic analogue of the Quantum Valley Hall Effect. These topologically non-trivial states exist near flexural resonances of the constituent beams of the phononic crystal and hence can be tuned into a deep-subwavelength regime.

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Seminar 14 -S4: Disordered acoustic materials with target scattering properties

Speaker: Vicent Romero-García (Applied Math Department, Universitat Politècnica de València)

DOI: doi.org/10.52843/meta-mat.9yp3bf

Abstract: The ability to manipulate waves has long been one of the main goals in various areas of physics and engineering. Many-body scattering systems [1] and metamaterials [2] offer promising prospects to deal with this challenge due to their ability to be tuned and reconfigured. Properly designed highly disordered many-body systems have recently attracted attention as a tool for scattering manipulation. The introduction of local correlations between the positions of the scatterers constituting the disordered system allows to control the scattering of an incident radiation [3, 4, 5, 6, 7]. In particular, stealth materials consist of multiple scatterers distributed a way that completely suppress the scattering of the sample over a broadband frequency range [8, 9].

In this work, we develop a route to engineer acoustic materials consisting of multiple scatterers, which possess the desired scattering properties under the incidence of a plane wave. We characterize the scattering pattern of a set of scatterers under the approach of weak scattering by its structure factor. We validate this hypothesis calculating the scattered far-field amplitude using the multiple scattering theory that considers all scattering orders. We develop an optimization technique, which optimizes the positions of scatterers that lead to a chosen value of the structure factor over a given frequency range.

Acknowledgements: Author is grateful for the partial support under Grant No. PID2020-112759GB-I00 funded by MCIN/AEI/10.13039/501100011033 and from Grant No. CIAICO/2022/052 of the “Programa para la promoción de la investigación científica, el desarrollo tecnológico y la innovación en la Comunitat Valenciana” funded by Generalitat Valenciana.

Biography: Dr. Romero-García (born in 1981, with European PhD diploma with honors (Cum Laude) and extraordinary award in Applied Physics from the Universitat Politècnica de València obtained in November 2010) has strongly contributed to the progress in the field of Complex Media. Particularly in the topic of wave propagation in matematerials, periodic, and correlated disorder systems. His work has contributed to the knowledge of new control scattering processes for absorbing and diffusing waves. His ideas and results have been published in journals of high impact factor (more than 80% of the publications are in Q1 of JCR) with an output of more than 100 publications (h-index is 30, citation count 3500+).

He has been PI of several research projects: A coordinated European project with the European Space Agency to develop noise reduction systems for launch pads, one French project to generate systems to control waves with the concept of Hyperuniformity. Nowadays is PI in a regional project, METASUP, to develop metasurfaces for medical applications and Primary Coordinator of a MSCA Postdoctoral Fellowships (Horizon-MSCA-2022-PF-01). He has participated in several national projects with strong implications in several tasks. Thanks to the previous funding, he has acquired strong scientific and technical skills. In that moment, he has several setups to experimentally characterize acoustic and elastic materials.

His broad background places him in a privileged position to undertake major problems of current interest and it has a confirmed relevance in this communities as the several invitations to European and International events confirm. He has a large network of highly influential collaborators who are major players in wave Physics at local and international level. Particularly, due to his past as a researcher in the CNRS in France he has a strong collaboration with France as well as with Researchers in the rest of Europe and UK.

Due to his twofold carrier in France and in Spain, he has faced to the technological development and innovation in both countries, allowing him a deep understanding of this paradigm. He has been PI of contracts with French (NAVAL Group, SNCF) and Spanish (COMET) industries. At that moment, he has 4 Granted Patents and Applications (2 in Spain and 2 in France). He has participated in several dissemination activities as the “La nuit des chercheurs Européens”, “La Semana de la Ciencia”. He has organized several art expositions with scientific links to his results as well as scientific concerts.

He participates actively in the direction of Degree and Master students, with an output of 15 students. He has supervised 10 PhDs, all of them have secured postdoctoral positions. Moreover, he has supervised 12 PostDocs, all of them nowadays with permanent academic/industry positions.

He actively participates in the evaluation of research projects from national agencies. He is nowadays Editorial Board of New Journal of Physics, which is an open access peer review journal in the Q1 of the JCR.

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Seminar 13 -S4: Plug-and-play paradigm for the analysis of scattering by “layer-cake” periodic systems

 Speaker: Bojan Guzina (Civil, Environmental, and Geo-Engineering, University of Minnesota)

DOI: doi.org/10.52843/meta-mat.srpp24

Abstract: We investigate scattering of scalar plane waves by a heterogeneous layer that is periodic in the direction parallel to its boundary. On describing the layer as a union of periodic sub-laminae, we develop a solution of the scattering problem by combining the concept of propagator matrices and that of Bloch eigenstates featured by the unit cell of each sub-lamina. The Bloch eigenstates are obtained by solving the quadratic eigenvalue problem (QEP) that seeks a complex-valued wavenumber normal to the layer boundary given (i) the excitation frequency, and (ii) real-valued wavenumber parallel to the boundary — which is preserved throughout the system. Spectral analysis of the QEP reveals sufficient conditions for discreteness of the eigenvalue spectrum and the fact that all eigenvalues come in complex-conjugate pairs. By deploying the factorization afforded by the propagator matrix approach, we demonstrate explicitly that the contribution of individual eigenvalues (and affiliated eigenmodes) to the solution diminishes exponentially with absolute value of their imaginary part, which then forms a rational basis for truncation of the factorized Bloch-wave solution. The proposed developments cater for optimal design of the rainbow traps and metasurfaces, whose potency to manipulate waves is controlled not only by the individual dispersion and impedance characteristics of the component sub-laminae, but also by ordering and generally “fitting” of the latter into the composite layer. By deploying the Bloch propagator approach, evaluation of the trial configurations — as generated by (a) permutation and (b) window translation/stretching of the component sub-laminae — can be accelerated by decades.

Biography: Dr. Guzina is Shimizu Professor in the Department of Civil, Environmental, and Geo- Engineering at the University of Minnesota. His research focuses on direct and inverse problems involving wave motion. Example topics include inverse scattering, waves in periodic and random media, and nonlinear waves in soft solids. He tackles the scientific and engineering challenges using analytical, computational, and experimental tools. Target applications of his work include nondestructive evaluation of materials and structures, seismic imaging, medical diagnosis, wave manipulation by architected materials, and most recently wave-based characterization of mineral carbon storage. His research has been supported by the National Science Foundation, Department of Energy, and the National Institutes of Health. Dr. Guzina serves an Associate Editor for the ASCE Journal of Engineering Mechanics. He is the 2019 recipient of the Nathan M. Newmark Medal awarded by the American Society of Civil Engineers for seminal work on theoretical and computational methods related to wave propagation and inverse scattering problems, with applications to the imaging of biological and engineered systems.

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Seminar 12 -S4: Elastic wave propagation in cubic non-centrosymmetric and chiral architectured materials: insights from strain gradient elasticity

 Speaker: Giuseppe Rosi (Université Paris-Est Créteil)

DOI: doi.org/10.52843/meta-mat.9cc3qt

Abstract: The study of elastic wave propagation is a fundamental tool in different fields, from Non-destructive Damage Evaluation (NDE) to ultrasonic imaging. Usually NDE and characterisation techniques rely on inversion methods based on homogenised theories, that are valid only when the wavelength of the perturbation is considerably larger than the characteristic size of the heterogeneities of the materials. When the wavelength approaches this characteristic size, an upscaling occurs and mesoscopic effects can be transferred to the macro-scale. In this case, classic models used in the aforementioned inversion procedures can fail to predict the correct response [1] and they need to be improved [2]. In this work, we address those architectures for which the unit cell does not have any centre of inversion (non-centrosymmetric) nor symmetry plane (chiral). It will be shown that unconventional effects, in terms of dispersion and polarisation, can be observed even for large wavelengths. We will also prove that for describing these materials using an equivalent homogeneous continuum, the use of an enriched or generalized theory, such as strain gradient elasticity, is mandatory. Moreover, the analysis of the generalised acoustic (or Christoffel) tensor defined in this framework can give a useful insight on the dynamic features of the architectured material. Among others, the example of the gyroid unit cell will be detailed.

References

  • 1.G. Rosi et al. (2020) On the Failure of Classic Elasticity in Predicting Elastic Wave Propagation in Gyroid Lattices for Very Long Wavelengths. Symmetry, 12(8) 
  • 2.G. Rosi and N. Auffray (2019) Continuum modelling of frequency dependent acoustic beam focussing and steering in hexagonal lattices. European Journal of Mechanics – A/Solids, 77 
  • 3.G. Rosi and N. Auffray (2016) Anisotropic and dispersive wave propagation within strain-gradient framework. Wave Motion, 63 
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Biography: Giuseppe Rosi is an associate professor (Maître de Conference, MCF) at the Université Paris-Est Créteil. He is an expert in generalised Continua and orthopaedic implant stability assessment. He develops his research in biomechanics at the Multi Scale Modelling and Simulation (MSME) laboratory, focusing on generalised continua applied to the characterisation of architectured materials, bio inspired meta-materials and bone substitutes.

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Seminar 11 -S4: Analysis of in-plane wave propagation in elastic structured systems

Speaker: Michael Nieves (School of Computer Science and Mathematics, Keele University, UK)

DOI: doi.org/10.52843/meta-mat.hx6bhn

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Abstract: Discrete media are a paradigm of metamaterials research that have applications in understanding and designing civil engineering structures, as well as materials with novel waveguiding properties. Often these materials are modelled as being infinite in extent and periodic. There, the corresponding modal analysis can provide a good indication of the structure’s overall behaviour, especially when the medium of interest is large. However, there exists many scenarios where one needs to determine the response of stratified systems subjected to complex loads or possessing inhomogeneities, non-periodic microstructures and/or having multiple boundaries. This webinar will focus on analytical techniques that help to address such problems and enable one to characterise the in-plane waveguiding and scattering properties of two-dimensional elastic periodic systems.

In the first part of the talk, we will discuss Lamb wave propagation in microstructured elastic triangular strips attached to an array of gyroscopes. The latter makes the system non-reciprocal and this allows the medium to support uni-directional Lamb waves when subjected to forcing [1]. The solution to this problem is solved using the discrete Fourier transform and we demonstrate how this solution can be exploited to novel waveguides. Namely, we illustrate how we can create networks of structured strips that can channel waves that propagate from one point in the system and along any predefined controllable path in the system to any other point in the network.

In the second part of the talk, we investigate the scattering of waves in a triangular elastic lattice by a penetrable inertial line defect [2]. Through the application of the discrete Fourier transform, this problem can be reduced to the analysis of two scalar Wiener-Hopf equations. From there, essential information about all dynamic modes of the system and their symmetry properties can be extracted. This includes all dynamic regimes where localised modes are supported by the defect when interacting with incoming waves. The solution of the Wiener-Hopf equations is represented as a contour integral that can be used to investigate unusual scattering responses of the defect encountered outside of the low-frequency regime.

If time permits, we will then discuss a method that allows one to model vibration in finite or infinite non-periodic discrete flexural systems and that allows for a new dynamic homogenisation method that provides an accurate broadband description of the vibration response for infinite structured flexural waveguides [3].

All analytical results presented are accompanied by numerical illustrations that demonstrate their effectiveness.

Acknowledgement: MJN gratefully acknowledges the support of the EU H2020 grant MSCA-RISE-2020-101008140-EffectFact. MJN would also like to thank the Isaac Newton Institute for Mathematical Sciences (INI) for their support and hospitality during the programme “Mathematical theory and applications of multiple wave scattering” (MWS), where work on some topics from the talk was undertaken and supported by EPSRC grant no. EP/R014604/1. Additionally, MJN is grateful for the funding received from the Simons Foundation that supported his visit to INI during January-June 2023 and participation in MWS programme.

Biography:  Dr Nieves is a Marie Skłodowska-Curie Fellow and a Senior Lecturer in Applied Mathematics at Keele University. His research interests include asymptotic methods for singularly perturbed boundary value problems in mathematical physics, the dynamics of continuous and discrete multiscale systems and developing the Wiener-Hopf technique in tackling problems concerning dynamic fracture in stratified media.

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Seminar 10 -S4: Multimode multiple scattering in disordered systems of spherical scatterers

Speaker: Valerie Pinfield (Chemical Engineering Department, Loughborough University, UK)

DOI: doi.org/10.52843/meta-mat.xfkhy1

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.

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Seminar 9 -S4: Valery Smyshlyaev (Department of Mathematics, University College London, UK) 

Title: Two-scale homogenisation of high-contrast subwavelength resonances and error analysis 

DOI: doi.org/10.52843/meta-mat.nnr8x3

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 [1], joint work with Shane Cooper and Ilia Kamotski.

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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)

DOI: doi.org/10.52843/meta-mat.44g91r

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.

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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)

DOI: doi.org/10.52843/meta-mat.jn13mm

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

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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”.

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Seminar 6 -S4: Wave propagation in quasiperiodic media

Speaker: Pierre Amenoagbadji (Department of Applied Physics and Applied Mathematics, Columbia University, USA)

DOI: doi.org/10.52843/meta-mat.6p5jt8

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.

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Seminar 5 -S4: Twinning cavities – Making a garage sound identical a cathedral

Speaker: Richard Craster (Department of Mathematics, Imperial College London, UK)

DOI: doi.org/10.52843/meta-mat.rlsxv9

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.

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Seminar 4 -S4: Graded arrays for spatial frequency separation and amplification of water waves

Speaker: Malte A. Peter (Institute of Mathematics, University of Augsburg)

DOI: doi.org/10.52843/meta-mat.8dlgyy

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.

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Seminar 3 -S4: On Micropolar Elastic Foundations and Architected Interfaces

Speaker: Marcelo Dias (School of Engineering, University of Edinburgh, UK)

DOI: doi.org/10.52843/meta-mat.nlpjqp

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’.

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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)

DOI: doi.org/10.52843/meta-mat.rxb2h9

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.

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Seminar 1 – S4: Band structure and Dirac points of real-space quantum optics in periodic media

Speaker: Erik O. Hiltunen (Yale University, USA)

DOI: doi.org/10.52843/meta-mat.4bblf3

Abstract: Elastic The field of photonic crystals is almost exclusively based on a Maxwell model of light. To capture light-matter interactions, it natural to study such systems under a quantum-mechanical photon model instead. In the real-space parametrization, interacting photon-atom systems are governed by a system of nonlocal partial differential equations. In this talk, we study resonant phenomena of such systems. Using integral equations, we phrase the resonant problem as a nonlinear eigenvalue problem. In a setting of high-contrast atom inclusions, we obtain fully explicit characterizations of resonances, band structure, and Dirac cones. Additionally, we present a strikingly simple relation between the Green’s function of the nonlocal equation and that of the local (Helmholtz) equation. In particular, we generalize existing lattice-summation methods to the nonlocal case. Based on this, we are able to achieve efficient numerical calculations of band structures of interacting photon-atom systems.

Erik Orvehed Hiltunen: 2021 ECCOMAS award – Department of Mathematics | ETH  Zurich

Biography: Dr. Erik O. Hiltunen is a Gibbs Assistant Professor at Yale University, USA. Dr. Hiltunen’s research focuses on developing the mathematical understanding of wave propagation in materials governed by local or non-local PDEs, using tools from PDE theory, harmonic analysis, and solid-state physics. Before moving to Yale, Hiltunen earned his PhD from ETH Zurich under the supervision of prof. Habib Ammari, where his dissertation was recently awarded the ECCOMAS award for best PhD theses on Computational Methods in Applied Sciences.

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