Applied Mathematics and Numerical Analysis Seminar 

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.

In this presentation we address the parallel implementation of the Multi-Power Defect-Correction method for the computation of eigenpairs of compact integral operators. The integral operator is discretized by a projection method on a subspace of moderate dimension and the eigenelements of its representing matrix are computed. These are then iteratively refined, by a defect correction type procedure accelerated by a few power iterations, to yield a better approximation to the spectral elements of the operator. The algorithm is rich in matrix-vector multiplications involving matrices whose construction is distributed among the available processors. The defect correction phase requires the solution of a distributed linear system of moderate dimension. Computational approaches of parallelization using state-of-the-art packages will be discussed, and numerical results on an astrophysics application, whose mathematical model involves a weakly singular integral operator, will be presented. Co-authors: Filomena Dias d'Almeida (Faculdade de Engenharia da Universidade do Porto) e José E. Roman (Instituto ITACA, Universidad Politécnica de Valencia, Spain)

This talk is motivated by the modeling and simulation of fluid-structure interaction phenomena in the vicinity of heart valves. On the one hand, the interaction of the vessel wall is dealt with an Arbitrary Lagrangian Eulerian (ALE) formulation. On the other hand the interaction of the valves is treated with the help of Lagrange multipliers in a Fictitious Domains-like (FD) formulation. After a synthetic presentation of the several methods available for the fluid-structure interaction in blood flows, a method that permits capture the dynamics of a valve immersed in an incompressible fluid is described. The coupling algorithm is partitioned which allows the fluid and structure solvers to remain independent. In order to follow the vessel walls, the fluid mesh is mobile, but it remains none the less independent of the valve mesh. In this way we allow large displacements without the need to perform remeshing. We propose a strategy to manage contact between several immersed structures. The algorithm is completely independent of the structure solver and is well adapted to the partitioned fluid-structure coupling. The methods considered are followed with several numerical tests in 2D and 3D.

In this talk, we consider Navier-Stokes equations with Navier's slip boundary conditions. We show their weak solvability. A remarkable observation says that Navier's boundary conditions enable us to accept a class of possible collisions, in contrast with the no-slip Dirichlet boundary conditions.

Balloon angioplasty followed by stent implantation has been of great benefit in coronary applications, whereas in peripheral applications, success rates remain low. Analysis of healing patterns in successful deployments shows that six months after implantation the artery has reorganized itself and there is no purpose for the stent to remain, potentially provoking inflammation and foreign body reaction. Thus, a fully absorbable stent that fulfills the mission and steps away is of great benefit. Biodegradable polymers have a widespread usage in the biomedical field, such as sutures, scaffolds and implants. Aliphatic polyesters (the main class of biodegradable polymers used in biomedical applications) degrade by random scission due to passive hydrolysis which results in molecular weight reduction, loss of strength, and ultimately, loss of mass. A constitutive model involving degradation and its impact on mechanical properties was developed through an extension of a material which response depends on the history of the motion and on a scalar parameter reflecting the local extent of degradation. The material properties decrease with degradation and a rate equation describing the chain scission process confers characteristics of stress relaxation, creep and hysteresis. These phenomena arise due to the entropy-producing nature of degradation and are markedly different from their viscoelastic counterparts.

In this paper we discuss Nordsieck formulas applied to ordinary differential equations. We focus on local and global error evaluation techniques in the mentioned numerical schemes. The error estimators are derived for both consistent Nordsieck methods and quasi-consistent ones. It is also shown how quasi-consistent Nordsieck formulas, which suffer, on variable grids, from the order reduction phenomenon, can be modified in an optimal way to avoid the order reduction. A special task here is to study advantages of numerical integration by quasi-consistent Nordsieck formulas. All quasi-consistent numerical methods possess at least one attractive property for practical use. The global error of a quasi-consistent method has the same order as its local error. This means that the usual local error control will produce a numerical solution for the prescribed accuracy requirement if the principal term of the local error dominates strongly over remaining terms. In other words, the global error control can be as cheap as the local error control in the methods under discussion. We apply this idea to Nordsieck Adams-Moulton methods, which are known to be quasi-consistent. Moreover, some Nordsieck Adams-Moulton methods are even super-quasi-consistent. The latter property means that their propagation matrices annihilate two leading terms in the defect expansion of such methods. In turn, this can impose a strong relation between the local and global errors of the numerical solution and allow the global error to be controlled effectively by a local error control. We also introduce Implicitly Extended Nordsieck methods such that in some sense they form pairs of embedded formulas with their source Nordsieck counterparts. This facilitates the local error control in quasi-consistent Nordsieck schemes. As the most promising result, we study double quasi-consistency of numerical schemes. This property means that the principal terms of the global and local errors of a doubly quasi-consistent numerical technique coincide. This benefits the global error control, significantly. We show that doubly quasi-consistent schemes do exist and belong to the class of general linear methods. The notion of double quasi-consistency creates potentially a new area of research in numerical methods for differential equations. Numerical examples presented in this paper confirm clearly the power of the above-mentioned global error estimators in practice.

In this talk, two types of precondtioned iterative methods and SOR-like methods are proposed to solving saddle point problems. The upper bound of the condition number of the preconditioned systems are estimated. Semi-conjugate direction methods (SCD) are introduced as well. The properties of the SCD method are studied. The equivalence of the SCD methods and the LU decomposition is established. Some remedies for breakdown problem are presented. Some applications and open problems are mentioned for future research.