## Delivered study plan 2022/2023

Curriculum in

ELETTROMAGNETISMO/ELECTROMAGNETISM

PhD courses

Title: Analytical Techniques for Wave Phenomena

Instructor: Paolo Burghignoli

Duration: 6 CFU

Tutte le lezioni si svolgeranno in italiano presso la sede della Facoltà di Ingegneria dalla Sapienza Università di Roma in Via Eudossiana 18, Roma (Fermata Metro B Colosseo o Cavour). Sono previste pause caffè (10:45-11:00, 15:45-16:00) e pause pranzo (13:00-14:00).

Seminario 1. martedì 7 febbraio 2023 ore 14:00-18:00 Aula 33

Seminario 2. mercoledì 8 febbraio 2023 ore 9:00-13:00 e 14:00-18:00 Aula 33

Seminario 3. martedì 14 febbraio 2023 ore 14:00-18:00 Aula 33

Seminario 4. mercoledì 15 febbraio 2023 ore 9:00-13:00 e 14:00-18:00 Aula 8 (pom. 33)

Abstract: The course aims at providing Ph.D. students with analytical tools useful in applied research on general wave phenomena. The unifying theme is that of complex analysis, of which a compact, self‐contained introduction is presented. Fundamental techniques asymptotic techniques are then illustrated, including ray optics and the Laplace and saddle‐point methods for the asymptotic evaluation of integrals. Applications are focused on the analysis of both time‐harmonic and transient waves excited in planar layered structures by canonical sources. As concerns the former, different wave species will be defined and physically discussed (space, surface, leaky, lateral waves). As concerns the latter, the Cagniard‐de Hoop method will be introduced.

Title: Nanophotonics and Plasmonics

Instructor: Concita Sibilia

Duration: 2 CFU

Period: Second semester

Abstract: The part of seminars related to Nanophotonics aims to introduce to students some exciting concepts that differ from conventional wave optics, with particular emphasis to the role of the evanescent fields in many practical applications, such as near field optical microscopy. The field of plasmonics (interaction of light with electrons in metals) has attracted a great deal of interest over the past two decades, but despite the many fundamental breakthroughs and exciting science it has produced, it is yet to deliver on the applications that were initially targeted as most promising. The seminars proposed examine the primary fundamental hurdles in the physics of plasmons that have been hampering practical applications and highlights some of the promising areas in which the field of plasmonics can realistically deliver.

Title: Basics of Nonlinear Optics

Instructor: Concita Sibilia

Duration: 2 CFU

Period: Second semester

Abstract: Nonlinear Optics (NLO) is the study of phenomena that occur as a consequence of the modification of the optical properties of a material system by the presence of light. Basics and more recent applications of NLO to new light sources and devices will be presented in a series of seminars.

Title: Radiazione Termica e Segnatura Infrarossa

Instructor: Roberto Li Voti

Duration: 4 CFU

Period: Second semester

Abstract: Il corso tratterà i seguenti argomenti: emissione spontanea, radiazione termica, legge di Planck, legge si Stephan -Bolzmann. Emissività, teoria del corpo nero e radiatori selettivi. Spettro elettromagnetico, radiazione luminosa, radiazione nell'infrarosso. Parametri caratteristici di un elemento radiante: radianza, radianza spettrale, potenza, potenza spettrale. Propagazione del segnale luminoso in aria, bande di assorbimento atmosferico. Effetti atmosferici: assorbimento, autoemissione, diffusione, deflessione, turbolenza. Tecniche numeriche per la valutazione della emissività variabile dei materiali; e tecniche nondistruttive fotoacustiche e fototermiche per la caratterizzazione di materiali.

Title: Laboratorio Di Applicazioni Industriali Delle Tecnologie Ottiche Fototermiche E Fotoacustiche

Instructor: Roberto Li Voti

Duration: 3 CFU

Period: Second semester

Abstract:The course will provide the theoretical bases of the photothermal, photoacoustic, radiometric, and infrared techniques for nondestructive evaluation and testing of materials (nanomaterials and metamaterials). Many applications will be introduced in different fields: industry, environment, energy, but also biology, medicine, agrifood. Final comparisons will be introduced among the diagnostic techniques. The course contains also some experimental activities in laboratory and the relative data analysis and data processing.

List of useful courses from the Master degree

Title: Optics

Instructor: Eugenio Fazio

Duration: 6 CFU

Period: 2nd semester

Title: Molecular dynamics and atomistic simulations

Instructors: Giuseppe Zollo

Duration: 6 CFU

Period: First semester

Title: Laser Fundamentals

Instructors: Concita Sibilia

Duration: 6 CFU

Period: Second semester

Title: Quantum Information, I e II

Instructor: Fabio Bovino (Leonardo SpA)

Duration: 5+5 CFU

Period: Second semester

Title: Esperienze di Laboratorio per il modulo di Ottica/Optics

Instructors: Alessandro Belardini

Duration: 2 CFU

Period: Second semester

Title: Advanced Electromagnetics and Scattering

Instructors: Fabrizio Frezza

Duration: 6 CFU

Period: Second semester

Title: Microwaves

Instructors:Marta Cavagnaro, Fabrizio Frezza, Alessandro Galli

Duration: 9 CFU

Period: First semester

Title: Artificial materials, metamaterials and plasmonics for electromagnetic applications

Instructors: Fabrizio Frezza

Duration: 6 CFU

Period: First semester

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Curriculum in

MATEMATICA PER L'INGEGNERIA /MATHEMATICS FOR ENGINEERING

PhD courses

Numerical Methods for Boundary Integral Equations PhD course

Chiara Sorgentone (Department of Basic and Applied Sciences for Engineering, Sapienza)

Program: In this course we will introduce numerical methods for boundary integral equations, mainly for the Laplace equation and Stokes flow. The main topics to be discussed include:

- Theory, derivation and main mathematical properties of boundary integral equations. Starting

with the Laplace equation, then moving on to Stokes equations. Single layer and double layer

formulations;

- Numerical discretization of boundary integral equations;

- Quadrature rules, including singularity and quasi-singularity treatments;

- Error estimates for layer potentials in 2D and 3D.

Part of the course will be devoted to numerical implementation in MATLAB. This will include discretization of geometries in 2D and 3D, and numerical resolution of simple problems.

Preliminary timetable:

Tuesday 29/11/2022, 14-17 Aula Seminari RM004, Via Scarpa 16

Thursday 01/12/2022, 14-17 Aula Seminari RM004, Via Scarpa 16

Thursday 15/12/2022, 14-17 Aula Seminari RM004, Via Scarpa 16

Tuesday 20/12/2022, 14-17 Aula Seminari RM004, Via Scarpa 16

Registration:

All the interested students are required to send an email to chiara.sorgentone@uniroma1.it

If there are problems with the timetable it can be adjusted based on the requests and the room availability.

Title: On the connection between non-local operators and probability

Instructor: Mirko D’Ovidio (Sapienza - SBAI)

Duration: 15 hours

Planned period: February 2023

Abstract:

We discuss some basic and advanced facts about initial and boundary value problems involving non-local operators. In particular, we show some stimulating connections between non-local Cauchy problems, non-local boundary value problems and stochastic processes. Non-local boundary value problems also include non-local dynamic boundary conditions. We discuss the probabilistic representation of the solutions together with the associated functionals.

First lesson: January 30, 10:30 - 12:30, Room 7, Building RM018 (via del Castro Laurenziano)

Title: Energetic relaxation of structured deformations, a multiscale geometrical basis for variational problems in continuum mechanics

Instructor: José Matias (Univ. Of Lisbon)

Duration: 12 hours

Plan:

29/3 ore 11-12 aula 4 (via Del Castro Laurenziano)

5/4, 12/4, 19/4, 26/4, 3/5 ore 17-19 aula 4 (via Del Castro Laurenziano)

Abstract:

This course will cover the material on the book [12]. Broadly speaking, first order structured deforma- tions introduced in [9] provide a mathematical framework to capture the effects at the macroscopic level of geometrical changes at submacroscopic levels. This theory was broadened by [17] in order to allow for geo- metrical changes at the level of second order derivatives (second order structured deformations). The theory of structured deformations was further enriched in [11] in order to consider different levels of microstructure, that is, hierarchical structural deformations.

Starting from this mechanical formulation of the theory, and upon describing the needed mathematical framework, namely recalling some basic properties of spaces of bounded variation and spaces of bounded hessian, the variational formulation for first order structured deformations in [7] is presented as well as two different variational formulations for second order structured deformations, in [3] and in [10]. A variational formulation for hierarchical structured deformations in [5] is also presented. Different applications in this context will be discussed, namely:

(1) Dimension reduction in the context of structured deformations [14] and [8]; (2) Derivation of explicit formulae for the relaxed energy densities [4] and [16]; (3) Optimal design in the context of structured deformations [15];

(4) Homogenization in the context of structured deformations [1];

(5) Upscaling and spatial localization of non-local energies [13].

Finally, some open problems and possible generalizations of the theory will be discussed. Program:

(1) Mechanical framework: L∞ theory of first order structured deformations [9], second order struc- tured defomations [17], and hierarchical systems of structured defomtations [11]. Examples and approximation theorems.

(2) Mathematical framework: some results on measure theory, BV spaces and Γ-convergence that will be needed for the mathematical formulation. Later on, some other mathematical preliminaries will be needed, namely Reshetnyak -type continuity theorems, BH space and the Global method for relaxation [2].

(3) The variational formulation of Choksi-Fonseca for first order structured defomations [7]. Results and sketch of the proofs.

(4) Some applications:

(a) Relaxation of purely interfacial energies;

(b) Optimal design of fractured media;

(c) Relaxation of non-local energies;

(d) Hierarchical systems of first order structured deformations. (e) Homogenization in the context of structured deformations.

(5) Variational settings for second order structured deformations [3] and [10] .

(6) Outlook for future research.

References:

[1] M. Amar, J. Matias, M. Morandotti, and E. Zappale: Periodic homogenization in the context of structured deformations. ZAMP, 73, 173 (2022)

[2] G. Bouchitté, I. Fonseca, and L. Mascarenhas: A global method for relaxation. Arch. Rational Mech. Anal., 145 (1998), 51–98.

[3] A. C. Barroso, J. Matias, M. Morandotti and D. R. Owen: Second-order structured deformations: relaxation, integral representation and examples. Arch. Rational Mech. Anal., 225 (2017), 1025–1072.

[4] A. C. Barroso, J. Matias, M. Morandotti, and D. R. Owen: Explicit formulas for relaxed energy densities arising from structured deformations. Math. Mech. Complex Syst., 5(2) (2017),

[5] A. C. Barroso, J. Matias, M. Morandotti, D. R. Owen and E. Zappale The variational modeling of hierarchical structured deformations. Submitted to J. Elasticity (2022).

[6] R. Choksi, G. Del Piero, I. Fonseca, and D. R. Owen: Structured deformations as energy minimizers in models of fracture and hysteresis. Mathematics and Mechanics of Solids 4 (1999), 321–356.

[7] R. Choksi and I. Fonseca: Bulk and interfacial energy densities for structured deformations of continua. Arch. Rational Mech. Anal., 138 (1997), 37–103.

[8] G. Carita, J. Matias, M. Morandotti, and D. R. Owen: Dimension reduction in the context of structured deformations. J. Elast. 133 Issue 1 (2018), 1–35.

[9] G. Del Piero and D. R. Owen: Structured deformations of continua. Arch. Rational Mech. Anal., 124 (1993), 99–155.

[10] I. Fonseca, A. Hagerty, and R. Paroni: Second-order structured deformations in the space of functions of bounded hessian.

J. Nonlinear Sci., 29(6) (2019), 2699–2734.

[11] L. Deseri and D. R. Owen: Elasticity with hierarchical disarrangements: a field theory that admits slips and separations

at multiple submacroscopic levels. J. Elasticity, 135 (2019), 149–182.

[12] J. Matias, M. Morandotti and D. R. Owen Energetic relaxation of structured deformations. A Multiscale Geometrical Basis

for Variational Problems in Continuum Mechanics. Book to be published by SpringerBriefs on PDEs and Data Science.

[13] J. Matias, M. Morandotti, D. R. Owen, and E. Zappale: Upscaling and spatial localization of non-local energies with

applications to crystal plasticity, Math. Mech. Solids, 26 (2021), 963–997.

[14] J. Matias and P. M. Santos: A dimension reduction result in the framework of structured deformations. Appl. Math.

Optim. 69 (2014), 459–485.

[15] J. Matias, M. Morandotti, and E. Zappale: Optimal design of fractured media with prescribed macroscopic strain. Journal

of Mathematical Analysis and Applications 449 (2017), 1094–1132.

[16] M. Šilhavý: The general form of the relaxation of a purely interfacial energy for structured deformations. Math. Mech. Com-

plex Syst., 5(2) (2017), 191–215.

[17] D. R. Owen and R. Paroni: Second-order structured deformations. Arch. Rational Mech. Anal. 155 (2000), 215–235.

Title: Convexity notions arising in the supremal setting

Instructor: Elvira Zappale

Duration: 10/12 hours

Plan:

31/3 ore 9-10 AULA 5 (Palazzina RM0018)

14/4, 21/4, 28/4, 5/5, 12/5, AULA 5 (Palazzina RM0018)

Abstract:

The course will consist of 5 or 6 lectures where I will first introduce supremal functionals, basic concepts of direct methods of Calculus of variations, with a particular emphasis on relaxation and then I will present several notions necessary and/or sufficient for the lower semicontinuity of supremal functionals and related results ensuring a power law approximation using integral energies. Also I will compare the classical notios of convexity used in the integral setting with their counterparts in the supremal context and other possible notions.

I will conclude with a quick overview of the nonlocal setting.

List of useful courses from the Master degree

Title: Metodi Numerici per l'Ingegneria Biomedica (in Italian)

Instructor: Francesca Pitolli (Sapienza -SBAI)

Duration : 20 hours (first part) + 40 (second part) hours

Planned period di erogazione: September-December 2022

Schedule: martedì h. 9:15-10:45 (Aula 25, via Eudossiana); giovedì h. 15:00-18:30 (Aula 15, via Eudossiana)

Abstract:

Prima parte: Metodi numerici per la soluzione di problemi differenziali, metodi di Runge-Kutta, metodi alle differenze finite (2CFU)

Seconda parte: Approssimazione ai minimi quadrati per l'identificazione di un modello e la stima dei parametri. Soluzione di sistemi lineari sovradeterminati. Decomposizione ai valori singolari e sue applicazioni. Problemi inversi mal posti e tecniche di regolarizzazione. Soluzione di sistemi lineari sottodeterminati. Analisi delle componenti principali e sue applicazioni (4CFU)

Per ogni argomento verranno svolte delle esercitazioni in cui si utilizzeranno i metodi numerici illustrati a lezione per risolvere alcuni problemi applicativi.

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Curriculum in

SCIENZA DEI MATERIALI / MATERIALS SCIENCE

PhD courses

Title: Radiation-Matter Interaction, Photoemission and Photoabsorption Spectroscopy

Instructors: Carlo Mariani, Settimio Mobilio, Francesco Offi, Alessandro Ruocco

Duration: 32 hours

Period: February-May 2023

Abstract: Introduction to the photoelectron spectroscopy: theoretical background, the three-step model, atoms and molecules, low-dimensional solid systems, experiments with angular resolution, time-resolved experiments. Instrumentation: charged particles, Auger electron spectroscopy and resonant photoemission. Theoretical background of absorption. Multiple scattering theory: a method for the observation of the electronic states and spectroscopy measurements. Surfaces and low-dimensional systems, electronic properties. Core-level photoemission and surface core-level shifts. Angular resolved photoemission, electronic band structure. Band structure of exemplary 1D and 2D systems. Electromagnetic radiation sources, synchrotron radiation. Introduction to the free-electron laser: a coherent source of radiation from UV to X rays. X ray absorption spectroscopy, EXAFS and XANES: fundamentals and applications. X ray elastic and anelastic scattering.

List of useful courses from the Master degree

Title: Surface Physics and Nanostructures (6 CFU) (Corso di Laurea in Fisica)

Instructor: Carlo Mariani

Planned period: October-December 2022

Title: Chimica Fisica dello Stato Solido e dei Materiali Nanostrutturati (6 CFU) (Corso di Laurea in Chimica Industriale)

Instructor: Danilo Dini

Planned period: October-December 2022

Title: Sistemi di produzione ed accumulo dell'energia (9 CFU) (Corso di Laurea in Chimica Industriale)

Instructor: Maria Assunta Navarra

Planned period: October-December 2022

Title: Fabbricazione e caratterizzazione di nanostrutture (6 CFU) (Corso di Laurea in Ingegneria delle Nanotecnologie)

Instructor: Carlo Mariani - Ernesto Placidi

Planned period: March-June 2023

Title: Laboratorio Macromolecole (9 CFU) (Corso di Laurea in Chimica Industriale)

Instructor: Andrea Martinelli

Planned period: March-June 2023

Title: Physics Laboratory II (9 CFU) (Corso di Laurea in Fisica)

Instructor: Carlo Mariani

Planned period: March-June 2023

Title: Microscopie e tecniche di nanocaratterizzazione (9 CFU) (Corso di Laurea in Ingegneria delle Nanotecnologie)

Instructor: Marco Rossi

Planned period: March-June 2023

Title: Chimica dei materiali polimerici (6 CFU) (Corso di Laurea in Chimica Analitica)

Instructor: Ilaria Fratoddi

Planned period: March-June 2022

Title: Scanning Probe Microscopy (3 CFU) (Corso di Laurea in Nanotechnology Engineering)

Instructor: Daniele Passeri

Planned period: March-June 2023

ELETTROMAGNETISMO/ELECTROMAGNETISM

PhD courses

Title: Analytical Techniques for Wave Phenomena

Instructor: Paolo Burghignoli

Duration: 6 CFU

Tutte le lezioni si svolgeranno in italiano presso la sede della Facoltà di Ingegneria dalla Sapienza Università di Roma in Via Eudossiana 18, Roma (Fermata Metro B Colosseo o Cavour). Sono previste pause caffè (10:45-11:00, 15:45-16:00) e pause pranzo (13:00-14:00).

Seminario 1. martedì 7 febbraio 2023 ore 14:00-18:00 Aula 33

Seminario 2. mercoledì 8 febbraio 2023 ore 9:00-13:00 e 14:00-18:00 Aula 33

Seminario 3. martedì 14 febbraio 2023 ore 14:00-18:00 Aula 33

Seminario 4. mercoledì 15 febbraio 2023 ore 9:00-13:00 e 14:00-18:00 Aula 8 (pom. 33)

Abstract: The course aims at providing Ph.D. students with analytical tools useful in applied research on general wave phenomena. The unifying theme is that of complex analysis, of which a compact, self‐contained introduction is presented. Fundamental techniques asymptotic techniques are then illustrated, including ray optics and the Laplace and saddle‐point methods for the asymptotic evaluation of integrals. Applications are focused on the analysis of both time‐harmonic and transient waves excited in planar layered structures by canonical sources. As concerns the former, different wave species will be defined and physically discussed (space, surface, leaky, lateral waves). As concerns the latter, the Cagniard‐de Hoop method will be introduced.

Title: Nanophotonics and Plasmonics

Instructor: Concita Sibilia

Duration: 2 CFU

Period: Second semester

Abstract: The part of seminars related to Nanophotonics aims to introduce to students some exciting concepts that differ from conventional wave optics, with particular emphasis to the role of the evanescent fields in many practical applications, such as near field optical microscopy. The field of plasmonics (interaction of light with electrons in metals) has attracted a great deal of interest over the past two decades, but despite the many fundamental breakthroughs and exciting science it has produced, it is yet to deliver on the applications that were initially targeted as most promising. The seminars proposed examine the primary fundamental hurdles in the physics of plasmons that have been hampering practical applications and highlights some of the promising areas in which the field of plasmonics can realistically deliver.

Title: Basics of Nonlinear Optics

Instructor: Concita Sibilia

Duration: 2 CFU

Period: Second semester

Abstract: Nonlinear Optics (NLO) is the study of phenomena that occur as a consequence of the modification of the optical properties of a material system by the presence of light. Basics and more recent applications of NLO to new light sources and devices will be presented in a series of seminars.

Title: Radiazione Termica e Segnatura Infrarossa

Instructor: Roberto Li Voti

Duration: 4 CFU

Period: Second semester

Abstract: Il corso tratterà i seguenti argomenti: emissione spontanea, radiazione termica, legge di Planck, legge si Stephan -Bolzmann. Emissività, teoria del corpo nero e radiatori selettivi. Spettro elettromagnetico, radiazione luminosa, radiazione nell'infrarosso. Parametri caratteristici di un elemento radiante: radianza, radianza spettrale, potenza, potenza spettrale. Propagazione del segnale luminoso in aria, bande di assorbimento atmosferico. Effetti atmosferici: assorbimento, autoemissione, diffusione, deflessione, turbolenza. Tecniche numeriche per la valutazione della emissività variabile dei materiali; e tecniche nondistruttive fotoacustiche e fototermiche per la caratterizzazione di materiali.

Title: Laboratorio Di Applicazioni Industriali Delle Tecnologie Ottiche Fototermiche E Fotoacustiche

Instructor: Roberto Li Voti

Duration: 3 CFU

Period: Second semester

Abstract:The course will provide the theoretical bases of the photothermal, photoacoustic, radiometric, and infrared techniques for nondestructive evaluation and testing of materials (nanomaterials and metamaterials). Many applications will be introduced in different fields: industry, environment, energy, but also biology, medicine, agrifood. Final comparisons will be introduced among the diagnostic techniques. The course contains also some experimental activities in laboratory and the relative data analysis and data processing.

List of useful courses from the Master degree

Title: Optics

Instructor: Eugenio Fazio

Duration: 6 CFU

Period: 2nd semester

Title: Molecular dynamics and atomistic simulations

Instructors: Giuseppe Zollo

Duration: 6 CFU

Period: First semester

Title: Laser Fundamentals

Instructors: Concita Sibilia

Duration: 6 CFU

Period: Second semester

Title: Quantum Information, I e II

Instructor: Fabio Bovino (Leonardo SpA)

Duration: 5+5 CFU

Period: Second semester

Title: Esperienze di Laboratorio per il modulo di Ottica/Optics

Instructors: Alessandro Belardini

Duration: 2 CFU

Period: Second semester

Title: Advanced Electromagnetics and Scattering

Instructors: Fabrizio Frezza

Duration: 6 CFU

Period: Second semester

Title: Microwaves

Instructors:Marta Cavagnaro, Fabrizio Frezza, Alessandro Galli

Duration: 9 CFU

Period: First semester

Title: Artificial materials, metamaterials and plasmonics for electromagnetic applications

Instructors: Fabrizio Frezza

Duration: 6 CFU

Period: First semester

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Curriculum in

MATEMATICA PER L'INGEGNERIA /MATHEMATICS FOR ENGINEERING

PhD courses

Numerical Methods for Boundary Integral Equations PhD course

Chiara Sorgentone (Department of Basic and Applied Sciences for Engineering, Sapienza)

Program: In this course we will introduce numerical methods for boundary integral equations, mainly for the Laplace equation and Stokes flow. The main topics to be discussed include:

- Theory, derivation and main mathematical properties of boundary integral equations. Starting

with the Laplace equation, then moving on to Stokes equations. Single layer and double layer

formulations;

- Numerical discretization of boundary integral equations;

- Quadrature rules, including singularity and quasi-singularity treatments;

- Error estimates for layer potentials in 2D and 3D.

Part of the course will be devoted to numerical implementation in MATLAB. This will include discretization of geometries in 2D and 3D, and numerical resolution of simple problems.

Preliminary timetable:

Tuesday 29/11/2022, 14-17 Aula Seminari RM004, Via Scarpa 16

Thursday 01/12/2022, 14-17 Aula Seminari RM004, Via Scarpa 16

Thursday 15/12/2022, 14-17 Aula Seminari RM004, Via Scarpa 16

Tuesday 20/12/2022, 14-17 Aula Seminari RM004, Via Scarpa 16

Registration:

All the interested students are required to send an email to chiara.sorgentone@uniroma1.it

If there are problems with the timetable it can be adjusted based on the requests and the room availability.

Title: On the connection between non-local operators and probability

Instructor: Mirko D’Ovidio (Sapienza - SBAI)

Duration: 15 hours

Planned period: February 2023

Abstract:

We discuss some basic and advanced facts about initial and boundary value problems involving non-local operators. In particular, we show some stimulating connections between non-local Cauchy problems, non-local boundary value problems and stochastic processes. Non-local boundary value problems also include non-local dynamic boundary conditions. We discuss the probabilistic representation of the solutions together with the associated functionals.

First lesson: January 30, 10:30 - 12:30, Room 7, Building RM018 (via del Castro Laurenziano)

Title: Energetic relaxation of structured deformations, a multiscale geometrical basis for variational problems in continuum mechanics

Instructor: José Matias (Univ. Of Lisbon)

Duration: 12 hours

Plan:

29/3 ore 11-12 aula 4 (via Del Castro Laurenziano)

5/4, 12/4, 19/4, 26/4, 3/5 ore 17-19 aula 4 (via Del Castro Laurenziano)

Abstract:

This course will cover the material on the book [12]. Broadly speaking, first order structured deforma- tions introduced in [9] provide a mathematical framework to capture the effects at the macroscopic level of geometrical changes at submacroscopic levels. This theory was broadened by [17] in order to allow for geo- metrical changes at the level of second order derivatives (second order structured deformations). The theory of structured deformations was further enriched in [11] in order to consider different levels of microstructure, that is, hierarchical structural deformations.

Starting from this mechanical formulation of the theory, and upon describing the needed mathematical framework, namely recalling some basic properties of spaces of bounded variation and spaces of bounded hessian, the variational formulation for first order structured deformations in [7] is presented as well as two different variational formulations for second order structured deformations, in [3] and in [10]. A variational formulation for hierarchical structured deformations in [5] is also presented. Different applications in this context will be discussed, namely:

(1) Dimension reduction in the context of structured deformations [14] and [8]; (2) Derivation of explicit formulae for the relaxed energy densities [4] and [16]; (3) Optimal design in the context of structured deformations [15];

(4) Homogenization in the context of structured deformations [1];

(5) Upscaling and spatial localization of non-local energies [13].

Finally, some open problems and possible generalizations of the theory will be discussed. Program:

(1) Mechanical framework: L∞ theory of first order structured deformations [9], second order struc- tured defomations [17], and hierarchical systems of structured defomtations [11]. Examples and approximation theorems.

(2) Mathematical framework: some results on measure theory, BV spaces and Γ-convergence that will be needed for the mathematical formulation. Later on, some other mathematical preliminaries will be needed, namely Reshetnyak -type continuity theorems, BH space and the Global method for relaxation [2].

(3) The variational formulation of Choksi-Fonseca for first order structured defomations [7]. Results and sketch of the proofs.

(4) Some applications:

(a) Relaxation of purely interfacial energies;

(b) Optimal design of fractured media;

(c) Relaxation of non-local energies;

(d) Hierarchical systems of first order structured deformations. (e) Homogenization in the context of structured deformations.

(5) Variational settings for second order structured deformations [3] and [10] .

(6) Outlook for future research.

References:

[1] M. Amar, J. Matias, M. Morandotti, and E. Zappale: Periodic homogenization in the context of structured deformations. ZAMP, 73, 173 (2022)

[2] G. Bouchitté, I. Fonseca, and L. Mascarenhas: A global method for relaxation. Arch. Rational Mech. Anal., 145 (1998), 51–98.

[3] A. C. Barroso, J. Matias, M. Morandotti and D. R. Owen: Second-order structured deformations: relaxation, integral representation and examples. Arch. Rational Mech. Anal., 225 (2017), 1025–1072.

[4] A. C. Barroso, J. Matias, M. Morandotti, and D. R. Owen: Explicit formulas for relaxed energy densities arising from structured deformations. Math. Mech. Complex Syst., 5(2) (2017),

[5] A. C. Barroso, J. Matias, M. Morandotti, D. R. Owen and E. Zappale The variational modeling of hierarchical structured deformations. Submitted to J. Elasticity (2022).

[6] R. Choksi, G. Del Piero, I. Fonseca, and D. R. Owen: Structured deformations as energy minimizers in models of fracture and hysteresis. Mathematics and Mechanics of Solids 4 (1999), 321–356.

[7] R. Choksi and I. Fonseca: Bulk and interfacial energy densities for structured deformations of continua. Arch. Rational Mech. Anal., 138 (1997), 37–103.

[8] G. Carita, J. Matias, M. Morandotti, and D. R. Owen: Dimension reduction in the context of structured deformations. J. Elast. 133 Issue 1 (2018), 1–35.

[9] G. Del Piero and D. R. Owen: Structured deformations of continua. Arch. Rational Mech. Anal., 124 (1993), 99–155.

[10] I. Fonseca, A. Hagerty, and R. Paroni: Second-order structured deformations in the space of functions of bounded hessian.

J. Nonlinear Sci., 29(6) (2019), 2699–2734.

[11] L. Deseri and D. R. Owen: Elasticity with hierarchical disarrangements: a field theory that admits slips and separations

at multiple submacroscopic levels. J. Elasticity, 135 (2019), 149–182.

[12] J. Matias, M. Morandotti and D. R. Owen Energetic relaxation of structured deformations. A Multiscale Geometrical Basis

for Variational Problems in Continuum Mechanics. Book to be published by SpringerBriefs on PDEs and Data Science.

[13] J. Matias, M. Morandotti, D. R. Owen, and E. Zappale: Upscaling and spatial localization of non-local energies with

applications to crystal plasticity, Math. Mech. Solids, 26 (2021), 963–997.

[14] J. Matias and P. M. Santos: A dimension reduction result in the framework of structured deformations. Appl. Math.

Optim. 69 (2014), 459–485.

[15] J. Matias, M. Morandotti, and E. Zappale: Optimal design of fractured media with prescribed macroscopic strain. Journal

of Mathematical Analysis and Applications 449 (2017), 1094–1132.

[16] M. Šilhavý: The general form of the relaxation of a purely interfacial energy for structured deformations. Math. Mech. Com-

plex Syst., 5(2) (2017), 191–215.

[17] D. R. Owen and R. Paroni: Second-order structured deformations. Arch. Rational Mech. Anal. 155 (2000), 215–235.

Title: Convexity notions arising in the supremal setting

Instructor: Elvira Zappale

Duration: 10/12 hours

Plan:

31/3 ore 9-10 AULA 5 (Palazzina RM0018)

14/4, 21/4, 28/4, 5/5, 12/5, AULA 5 (Palazzina RM0018)

Abstract:

The course will consist of 5 or 6 lectures where I will first introduce supremal functionals, basic concepts of direct methods of Calculus of variations, with a particular emphasis on relaxation and then I will present several notions necessary and/or sufficient for the lower semicontinuity of supremal functionals and related results ensuring a power law approximation using integral energies. Also I will compare the classical notios of convexity used in the integral setting with their counterparts in the supremal context and other possible notions.

I will conclude with a quick overview of the nonlocal setting.

List of useful courses from the Master degree

Title: Metodi Numerici per l'Ingegneria Biomedica (in Italian)

Instructor: Francesca Pitolli (Sapienza -SBAI)

Duration : 20 hours (first part) + 40 (second part) hours

Planned period di erogazione: September-December 2022

Schedule: martedì h. 9:15-10:45 (Aula 25, via Eudossiana); giovedì h. 15:00-18:30 (Aula 15, via Eudossiana)

Abstract:

Prima parte: Metodi numerici per la soluzione di problemi differenziali, metodi di Runge-Kutta, metodi alle differenze finite (2CFU)

Seconda parte: Approssimazione ai minimi quadrati per l'identificazione di un modello e la stima dei parametri. Soluzione di sistemi lineari sovradeterminati. Decomposizione ai valori singolari e sue applicazioni. Problemi inversi mal posti e tecniche di regolarizzazione. Soluzione di sistemi lineari sottodeterminati. Analisi delle componenti principali e sue applicazioni (4CFU)

Per ogni argomento verranno svolte delle esercitazioni in cui si utilizzeranno i metodi numerici illustrati a lezione per risolvere alcuni problemi applicativi.

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Curriculum in

SCIENZA DEI MATERIALI / MATERIALS SCIENCE

PhD courses

Title: Radiation-Matter Interaction, Photoemission and Photoabsorption Spectroscopy

Instructors: Carlo Mariani, Settimio Mobilio, Francesco Offi, Alessandro Ruocco

Duration: 32 hours

Period: February-May 2023

Abstract: Introduction to the photoelectron spectroscopy: theoretical background, the three-step model, atoms and molecules, low-dimensional solid systems, experiments with angular resolution, time-resolved experiments. Instrumentation: charged particles, Auger electron spectroscopy and resonant photoemission. Theoretical background of absorption. Multiple scattering theory: a method for the observation of the electronic states and spectroscopy measurements. Surfaces and low-dimensional systems, electronic properties. Core-level photoemission and surface core-level shifts. Angular resolved photoemission, electronic band structure. Band structure of exemplary 1D and 2D systems. Electromagnetic radiation sources, synchrotron radiation. Introduction to the free-electron laser: a coherent source of radiation from UV to X rays. X ray absorption spectroscopy, EXAFS and XANES: fundamentals and applications. X ray elastic and anelastic scattering.

List of useful courses from the Master degree

Title: Surface Physics and Nanostructures (6 CFU) (Corso di Laurea in Fisica)

Instructor: Carlo Mariani

Planned period: October-December 2022

Title: Chimica Fisica dello Stato Solido e dei Materiali Nanostrutturati (6 CFU) (Corso di Laurea in Chimica Industriale)

Instructor: Danilo Dini

Planned period: October-December 2022

Title: Sistemi di produzione ed accumulo dell'energia (9 CFU) (Corso di Laurea in Chimica Industriale)

Instructor: Maria Assunta Navarra

Planned period: October-December 2022

Title: Fabbricazione e caratterizzazione di nanostrutture (6 CFU) (Corso di Laurea in Ingegneria delle Nanotecnologie)

Instructor: Carlo Mariani - Ernesto Placidi

Planned period: March-June 2023

Title: Laboratorio Macromolecole (9 CFU) (Corso di Laurea in Chimica Industriale)

Instructor: Andrea Martinelli

Planned period: March-June 2023

Title: Physics Laboratory II (9 CFU) (Corso di Laurea in Fisica)

Instructor: Carlo Mariani

Planned period: March-June 2023

Title: Microscopie e tecniche di nanocaratterizzazione (9 CFU) (Corso di Laurea in Ingegneria delle Nanotecnologie)

Instructor: Marco Rossi

Planned period: March-June 2023

Title: Chimica dei materiali polimerici (6 CFU) (Corso di Laurea in Chimica Analitica)

Instructor: Ilaria Fratoddi

Planned period: March-June 2022

Title: Scanning Probe Microscopy (3 CFU) (Corso di Laurea in Nanotechnology Engineering)

Instructor: Daniele Passeri

Planned period: March-June 2023