Applied Mathematics and Numerical Analysis Seminar 

In this talk the computational bio-mechanics (CBM) lines of BSC will be described. They go in the direction of the "Homo Computationalis", where complex CBM problems are solved using efficient parallel solvers, capable of running in supercomputing facilities, in order to get closer to realistic simulations of human organs treated as Physical systems. Three projects will be described in detail: Cerebral Artherial System, Superior Respiratory Airways and Electromechanical Model of the Heart.

We build a family of nonempty closed convex sets using the data of the Generalized Nash equilibrium problem (GNEP), and select the sets iteratively such that the intersection of the selected sets contains solutions of the GNEP. We adapt the algorithm introduced by Iusem-Sosa (2003) to obtain solutions of the GNEP. Finally we give some numerical experiments to illustrate the numerical behavior of the algorithm.

Bifurcations in stochastic delay differential equations are quite difficult to detect. In this talk, we consider how numerical methods can be used to detect changes in the underlying behaviour of the exact solution of these problems. The talk brings together ideas from deterministic delay differential equations, from stochastic ordinary differential equations and from statistical curve-fitting to provide some interesting new insights into the problem.

Modelling and simulation of fluid-structure interaction problems (FSI) has many applications in engineering, biomechanics and medical sciences. For example, modelling elasto-plastic material deformations covered with lubricant or simulation of blood flow in veins and diseases of the cardiovascular system. The mathematical approach for investigating monolithic FSI problems via the 'arbitrary Lagrangian-Eulerian' method (ALE) will be presented. The resulting equations are nonlinear and have been solved by the Newton's method. After that, some results derived so far in my PhD project will be discussed. The focus is on stationary problems (e.g. an obstacle problem) and Benchmark configurations verifying the programming code.

The general thinking is that ocean eddies are Gaussian-like instead of Rankine-like. We will show however that in several oceanic regions, such as around the Canary Islands or in the Gulf of Tanhuentepec (Mexico), eddies are initially Rankine-like and evolve towards Gaussian type. Gaussian-like vortices have a Gaussian distribution of vorticity, smooth shear zones at their periphery, their azimuthal velocity does not vary linearly and anticyclones are unstable. On the contrary, Rankine-like vortices are on solid body rotation, have strong shears at their periphery, the azimuthal velocity field varies linearly and anticyclones are stable. A simple way to detect both vortices is through hydrographic field anomalies. In Rankine-like vortices, anomalies do not have a well-defined center and gradients increase towards the periphery. In Gaussian type vortices, anomalies have a clear center with gradients increasing toward the vortex center. In Rankine vortices there is a strong diapycnal mixing at their periphery that enhances primary production.

In this talk a finite-volume discretisation of a Lattice Boltzmann equation over unstructured grids is presented. The new scheme is based on placing the unknown fields at the nodes of the mesh and evolve them based on the fluxes crossing the surfaces of the corresponding control volumes. The method, named unstructured Lattice Boltzmann equation (ULBE) is applied here to the problem of blood flow over the endothelium in small arteries. The study shows a significant variation of wall shear stress to the corrugation degree of the endothelium.

I will discuss Mathematical Immunology, concentrating on T cells. In the first part of the talk, I will consider how a diverse repertoire of T-cell clonotypes is maintained. Understanding the dynamics of "homeostasis" leads to a continuous-time Markov chain model. In the second part of the talk I will discuss timescales for the interaction of T cells with Dendritic cells in a lymph node, which leads to a continuous-space Brownian motion model

The unsteady Poiseuille flow, describing the motion of a viscous incompressible fluid in an infinite straight pipe of constant cross-section $\sigma$, is defined as a solution of the inverse problem for the heat equation on $\sigma$. The existence and uniqueness of such flow with the prescribed flow rate $F(t)$ is proved for Newtonian and second grade fluids. It is shown that the flow rate $F(t)$ and the axial pressure drop $q(t)$ are related, at each time $t$, by a linear Volterra integral equation of the second type, where the kernel depends only upon $t$ and $\sigma$. One significant consequence of this result is that it allows us to prove that the inverse parabolic problem of finding a Poiseuille flow corresponding to a given $F(t)$ is equivalent to the resolution of the classical initial-boundary value problem for the heat equation. The behavior as $t\to\infty$ of this unsteady Poiseuille solution is studied. In particular, it is proved that in the case, where the flow rate $F(t)$ exponentially tends to a constant $F_*$, the non-stationary Poiseuille solution tends as $t\to\infty$ to the steady Poiseuille flow corresponding to the flow rate $F_*$. The unsteady Navier-Stokes system is studied in a two-dimensional domain with strip-like outlets to infinity in weighted Sobolev function spaces. It is proved that under natural compatibility conditions there exists a solution with prescribed flow rates over cross-sections of outlets to infinity and that this solution tends in each outlet to the corresponding unsteady Poiseuille flow. The decay rate of the solution is conditioned only by the decay rate of an external force and initial data. The obtained results are true for arbitrary values of norms of the data (in particular, for arbitrary fluxes) and globally in time. For the three-dimensional domain with cylindrical outlets to infinity the analogous results are obtained either for small data or for a small time interval.

In this work numerical methods for one-dimensional diffusion problems are discussed. The differential equation considered, takes into account the variation of the relaxation time of the mass flux and the existence of a potential field. Consequently, according to which values of the relaxation parameter or the potential field we assume, the equation can have properties similar to a hyperbolic equation or to a parabolic equation. The numerical schemes consist of using an inverse Laplace transform algorithm to remove the time-dependent terms in the governing equation and boundary conditions. For the spatial discretization, three different approaches are discussed and we show their advantages and disadvantages according to which values of the potential field and relaxation time parameters we choose.

This talk presents some of the main mathematical and computational challenges encountered in the numerical simulation of ocean flows. These challenges and some possible solutions will be presented in the context of oceanic gravity currents. Oceanic gravity currents are cold (dense) water masses that are released into the large-scale ocean circulation from high-latitude and marginal seas. The entrainment of ambient waters into oceanic gravity currents is recognized as being a prominent oceanic process with significant impact on the ocean general circulation and climate. The numerical simulation of oceanic gravity currents at realistic parameters represents a grand challenge. Recent developments in this area, including new mathematical models and computational methodologies for stratified flows will be presented.

Lipid bilayers are currently being used for the development of many bioinspired microsystems ranging from portable and fast biosensors for detecting biological agents to biocompatible and biodegradable drug delivery carriers. Many of these microsystems work as proof-of-concept in laboratory environments but their application in real-world scenarios remains to be demonstrated due to the poor stability of lipid bilayers to mechanical stresses. An accurate characterization of the material properties of lipid bilayers, which is needed to enhance their performance, is limited by the challenges encountered in experimental in vestigations (e.g. measurement of stress and strain in the nanoscale range). Therefore, the formulation of new mathematical models is essential in making a big leap forward in the development of the next generation of bio-inspired microsystems that include lipid bilayers. We will present novel continuum models for the description of equilibrium configurations of planar lipid bilayers by accounting for their smectic A liquid crystallinity. These models represent a major improvement over existing continuum models since they incorporate significant molecular features of lipid bilayers (positional and orientational order) without requiring detailed molecular information. Unlike previous models, the proposed models capture the misalignment of the lipid molecules with the normal to the smectic layers and are derived within a new nonlinear theoretical framework for smectic A liquid crystals (IW Stewart 2007 Contin. Mech. Thermodyn. 18:343). The total energy of lipid bilayers consists of an elastic splay term, smectic layer bending and compression terms, a coupling term between the director and layer normal, a surface tension term, and a surface anchoring term. Nonlinear equilibrium equations are obtained by using variational methods and are then solved by analytical and numerical methods. The solutions illustrate the nonlinear deformations of lipid bilayers including the misalignment of lipid molecules at their interface with other media such as, for example, surface substrates.

With all budgets contractions all over the science world has to look for applied problems to attract more possibilities to infiltrate numerical analysis in a more eye-catching way. I will give a short view of my work in the theory of NA and open applied problems to be modeled and studied.

Multiple zeros of complex analytic functions can be localized and counted using certain level curves. This approach provides the setting for a computational-positional system which we have called double newtonization. This process enables the computation of high precision approximations of simple or multiple zeros regardless of their multiplicity.

The problem on water-waves is described, within the linearized theory, by a boundary-value problem for the Laplace equation with a spectral boundary condition of Steklov type. The spectrum of the problem is known to be be continuous in infinite channels and layers. In this talk, we will demonstrate that the spectrum can have a nonempty continuous component also in a pond with a gently sloped bottom topography due to the boundary singularity of cuspidal edge type.

In this talk we consider the application of the Method of Fundamental Solutions (MFS) to calculate eigenvalues and eigenfunctions of the Laplacian (2D and 3D domains). We show that a particular choice of the point-sources allows to obtain very good results for a fairly general class of domains. The case of regions with corners and cracks is also addressed enriching the MFS basis of functions with some particular solutions adapted to these domains. We also present results of the application of the MFS to the eigenvalue problem associated to the Bilaplacian operator and to the Lamé operator, in the elastic case.

One-dimensional (1D) models are exploited for the representation of the complex system of cerebral arteries, featuring a peculiar structure called the circle of Willis. These models, based on the Euler equations, are unable to capture the local details of the blood flow but are suitable for the description of the pressure wave propagation in large vascular networks. The propagative phenomenon is driven by the mechanical interaction of the blood and the vessel wall, and is therefore affected by the mechanical features of the wall. Our 1D model takes into account the wall viscoelasticity, whose effects on the wave propagation phenomena are qualitatively studied in some numerical experiments representative of realistic conditions in the cardiovascular and cerebral arterial systems. The details of the blood flow can be studied by means of three-dimensional (3D) models, based on the Navier-Stokes equations for incompressible Newtonian fluids. These models can correctly describe blood flow patterns in medium and large arteries, and in particular allow the evaluation of the stress field in the fluid. Thus, it is possible to estimate the traction exerted by the blood flow on the vessel wall (wall shear stress, WSS). We also show that by exploiting the representation of the vascular tree in terms of centerlines, it is possible to easily identify regions of interest in the computational domain, in which to restrict the fluid dynamics analysis, and to study the correlation between fluid dynamics features and the location along the arterial tree. Cerebral aneurysms are a disease of the vascular wall causing a local dilation, which tends to grow and can rupture, leading to severe damage to the brain. The mechanisms of initiation, growth and rupture have not been completely explained yet, but the effects of blood flow on the vascular wall are generally accepted as risk factors. In the context of the Aneurisk project (www2.mate.polimi.it:9080/aneurisk) it was found that certain spatial patterns of radius and curvature are associated to the presence and to the position of an aneurysm in the cerebral vasculature. Starting from this observation, a classification strategy for vascular geometries has been devised. Blood flow has been simulated in patient-specific vascular geometries reconstructed in the context of the Aneurisk project, and an index of the mechanical load exerted by the blood on the vascular wall near the aneurysm has been defined. Moreover, we show that certain values of the mechanical load are associated to the presence and the location of an aneurysm in the cerebral circulation: adding this hemodynamic parameter in the classification technique improves its efficacy. The interaction between local and global phenomena is a typical feature of the circulatory system. It is believed to be crucial in the context of the cerebral circulation, since defects or diseases at the level of the circle of Willis can induce local flow conditions associated to the initiation of an aneurysm. Geometrical multiscale models are a promising tool for the modeling of this interaction. We present a geometrical multiscale model of the cerebral circulation, based on the coupling of a 1D representation of the circle of Willis and the 3D representation of a carotid artery (T. Passerini, M. de Luca, L. Formaggia, A. Quarteroni, and A. Veneziani, 2009). Moreover, we discuss a novel method to describe the interface between the two models. The number of potential applications of numerical models, due to their proven effectiveness in the study of vascular networks, calls for the design of efficient and robust software tools. The software specifically written in the context of this work for the simulation of the circulatory system is based on the C++ LifeV (www.lifev.org) library.

Cell population can be considered as a continuous medium and described by partial differential equations. Another approach relies on the individual based modelling with soft spheres or elastic cells models. We will discuss the relation between these two approaches and some applications to morphogenesis and hematopoiesis.

We present a 1D model for a viscous fluid with axial symmetric swirling motion flowing in a circular straight tube with variable radius. Integrating the equation of conservation of linear momentum over the tube cross-section, with the velocity field approximated by the Cosserat theory, we obtain a one-dimensional system depending only on time and on a single spatial variable. The velocity field approximation satisfies both the incompressibility condition and the kinematic boundary condition exactly. From this new system, we derive the equation for the wall shear stress and the relationship between mean pressure gradient, volume flow rate and swirling scalar function over a finite section of the tube. Also, we obtain the corresponding partial differential equation for the swirling scalar function.

A behavior-oriented diffusive model, which describes the long-term evolution of coastal profiles is considered. The inverse problem of determining two functional coefficients in the aforementioned model equation has been solved, and prediction on the coastal evolution made, referring to two available realistic dataset.