Dipartimento di Matematica

Abstract

Modeling and computational analysis play an increasingly important role in bioengineering, particularly in the design of implantable ventricular assist devices (VAD) and other blood-handling devices. Numerical simulation of blood flow and associated physiological phenomena has the potential to shorten the design cycle and give the designers important insights into causes of blood damage and suboptimal performance. A set of modeling techniques is presented which are based on stabilized space-time finite element formulation of the Navier-Stokes equations. Alternate methods that represent the rotating components in an averaged sense using a rotating frame of reference will be discussed. In order to obtain quantitative hemolysis prediction, cumulative tensor-based measures of strain experienced by individual blood cells must be developed; red blood cells under shear can be modeled as deforming droplets, and their deformation tracked throughout the flow volume. The methods are applied to a simplified rotary blood pump, which is currently a subject of an inter-laboratory round-robin study.

Contact: luca.dede@polimi.it

Abstract

Consider the Caputo evolution equation (EE) $\partial_t^\beta u =\Delta u$ with initial condition $\phi$ on $\{0\}\times\mathbb R^d$, $\beta\in(0,1)$. As it is well known, the solution reads $u(t,x)=\mathbf E_x[\phi(B_{E_t})]$. Here $B_t$ is a Brownian motion and the independent time-change $E_t$ is an inverse $\beta$-stable subordinator. The fractional kinetic $B_{E_t}$ is a popular model for subdiffusion \cite{Meerschaert2012}, with remarkable universality properties \cite{BC11,Hai18}.\
We substitute the Caputo fractional derivative $\partial_t^\beta$ with the Marchaud derivative. This results in a natural extension of the Caputo EE featuring a \emph{time-nonlocal initial condition} $\phi$ on $(-\infty,0]\times\mathbb R^d$. We derive the new stochastic representation for the solution, namely $u(t,x)=\mathbf E_x[\phi(-W_t,B_{E_t})]$. This stochastic representation has a pleasing interpretation due to the non-obvious presence of $W_t$, elucidating the notion of time-nonlocal initial conditions. Here $W_t$ denotes the waiting/trapping time of the fractional kinetic $B_{E_t}$. We discuss classical-wellposedness \cite{T18}, and time permitting weak-wellposedness \cite{DYZ17,DTZ18} for the respective extensions of Caputo-type EEs (such as in \cite{chen,HKT17}).

Bibliography:


Barlow, \u Cern\’y (2011). Probability theory and related fields, 149.3-4: 639-673.


Chen, Kim, Kumagai, Wang (2017). arXiv:1708.05863.


Du, Toniazzi, Zhou (2018). Preprint. Submitted in Sept. 2018.


Du, Yang, Zhou (2017). Discrete and continuous dynamical systems series B, Vol 22, n. 2.


Hairer, Iyer, Koralov, Novikov, Pajor-Gyulai (2018). The Annals of Probability, 46(2), 897-955.


Hern\’andez-Hern\’andez, Kolokoltsov, Toniazzi (2017). Chaos, Solitons \& Fractals, 102, 184-196.


Meerschaert, Sikorskii (2012). De Gruyter Studies in Mathematics, Book 43.


Toniazzi (2018). To appear in: Journal of Mathematical Analysis and Applications. arXiv:1805.02464.

Abstract

Additive Manufacturing (AM), commonly known as three-dimensional printing, is widely recognized as a disruptive technology, and has the potential to fundamentally change the nature of future manufacturing. Building products layer-by-layer, AM represents a paradigm shift in manufacturing with many industrial applications. It enables production of huge varieties of customized products with considerable geometric complexity, and the same time, with extended capabilities and functional performances.
Despite tremendous enthusiasm, AM faces major research challenges for widespread adoption of this innovative technology. Specifically, addressing the unique challenges associated with quality engineering of AM processes is crucial to the eventual success of AM. This talk presents an overview of quality-related issues for AM processes and products, focusing on opportunities and challenges in quality inspection, monitoring, control.

Contact: piercesare.secchi@polimi.it

Abstract

Biological soft tissues and soft gels are difficult to study and model mathematically. Bioengineers often see them as engineering materials and try to evaluate their mechanical properties with standard testing protocols, such as tensile testing, simple shear, torsion, etc. These processes are destructive for tissues, as a sample is taken out of the body and placed in a device. The resulting measured parameters and models are expected to be very different from their in vivo counterparts. To test soft tissues properly, non¬-destructively, and non¬-invasively, we can rely on elastic waves. We can study the influence of pre¬stress on their speed and obtain the nonlinear elastic parameter by inverse analysis. This idea forms the basis of the theory of acousto-¬elasticity, which can be dated back to early works of Brillouin, and has been used successfully in the past for “hard” elastic solids such as rocks and metals. With this talk, we will explore the extension of acousto-¬elasticity to “soft” elastic solids, which can be subjected to large deformations in service. We will look at theoretical, numerical, experimental, and even clinical results, generated in particular on gels, brain, breast, and skin.

Contact: pasquale.ciarletta@polimi.it

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