Dipartimento di Matematica

Abstract

Numerical simulations are used with growing popularity in diverse sectors of engineering. The most important applications are those which attempt to replace expensive experiments on real structures that involve material mechanical behavior beyond their elastic limit. Such circumstance makes strong requirement for formulating material constitutive models with appropriate numerical implementation and for defining protocols for their calibration. Both problems are rather challenging when dealing with advanced materials.
In order to describe mechanical behavior of materials through an appropriate constitute model experiments are needed, but the transition from measurable quantities to sought parameters cannot always be directly established. Additional difficulty is encountered when dealing with complex constitutive models which tend to capture most of the physical processes taking place during material deforming, resulting in constitutive models with elevated number of parameters. Calibration of such models on the basis of too simple experiments, risks to identify particular solutions only, managing to fit a single experiment, thus not to be treated as material representative properties. A systematic way of resolving these difficulty is through the application of inverse analysis, centered on the minimization of a discrepancy function designed to quantify the difference between measured quantities and their counter parts, computed as a function of sought material parameters. This approach is!
more and more frequently adopted despite common mathematical difficulties, such as ill-posedness, non-uniqueness of the solution and non-convex function minimization.
Within this lecture some recent research contributions achieved by our team to the methodology of inverse analysis apt for calibration of complex constitutive models will be given. Results are presented with reference to real life engineering problems, related to diverse industrial environments. The first group of problems considers diagnostic analysis of structures based on instrumented indentation test. Results concern the development of reduced basis model for the acceleration of non-linear elasto-plastic simulations. The second group of problems concerns compaction of ceramic powders and the development of phenomenological constitutive models together with protocols for their calibration. The last group of problems, discussed within the lecture, is related to applications of porous ceramics for diesel particulate filters and catalytic substrates. Some innovative modeling techniques regarding thermally induced cracking and crack healing, observed in these materials will b!e shown.

Abstract

I will present a series of works in collaboration with Henri Berestycki (PSL), dealing with systems of predators interacting with a single prey. The system is linked to the Lotka-Volterra model of interaction with diffusion, but unlike more classic works, we are interested in studying the case where competition between predators is very strong: in this context, the original domain is partitioned in different sub-territories occupied by different predators. The question that we ask is under which conditions the predators segregate in packs and whether there is a benefit to the hostility between the packs. More specifically, we study the stationary states of the system, the stability of the solutions and the bifurcation diagram, and the asymptotic properties of the system when the intensity of the competition becomes infinite.

Abstract

Network Science is a rapidly growing interdisciplinary area at the intersection of mathematics, computer science, and a multitude of disciplines from the natural and life sciences to the social sciences and even the humanities. Network analysis methods are now widely used in proteomics, in the study of social networks (both human and animal), in finance, in ecology, in bibliometric studies, in archeology, and in a host of other fields. In this talk I will introduce the audience to some of the mathematical and computational problems and methods of complex networks, with an emphasis on the basic notions of centrality and communicability. More specifically, I will describe some of the problems in large-scale numerical linear algebra arising in this area, and how they differ from the corresponding problems encountered in more traditional applications of numerical analysis. The talk will be accessible to students, requiring only a modest background in linear algebra, numerical analysis and graph theory.

Contacts: paola.antonietti@polimi.it, paolo.zunino@polimi.it