## Mathematics for Engineering 2022/23

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:

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

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

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

4. 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 (Palazzina RM0018)

April 13/20/27, 2023,  ore 16-18 Aula SEMINARI
May 4/11, 2023, ore 16-18, Aula SEMINARI

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

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

6. (6)  Outlook for future research.

References:
1. [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.

2. [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.

3. [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),

4. [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).

5. [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 (1999), 321–356.

6. [7]  R. Choksi and I. Fonseca: Bulk and interfacial energy densities for structured deformations of continua. Arch. Rational Mech. Anal., 138 (1997), 37–103.

7. [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.

8. [9]  G. Del Piero and D. R. Owen: Structured deformations of continua. Arch. Rational Mech. Anal., 124 (1993), 99–155.

9. [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.

10. [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.

11. [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.

12. [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.

13. [14]  J. Matias and P. M. Santos: A dimension reduction result in the framework of structured deformations. Appl. Math.

Optim. 69 (2014), 459–485.

14. [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.

15. [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.

16. [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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