Applied Mathematics and Numerical Analysis Seminar 

The human airways have typically stimulated a smaller portion of research compared to the circulatory system, and in particular the arterious system. The result is that much still needs to be understood about the nasal cavity and its geometric properties. The interest in studying the nasal cavity is to increase the understanding in respiratory physiology for further possible applications of surgery, drug delivery and toxicology. The nasal cavity has the complex physiological function of conditioning (heat transfer and humidification) and filtering the inspired air, as well as to give a sense of olfaction. The result is a complex geometry that is seen to vary greatly between subjects. The work presented here is the result of recent studies in coming to terms with the topology and characterising the nasal airway. Starting from CT medical image data, anatomically accurate virtual models of three subjects are obtained for this study. By means of skeletonisation techniques, Fourier descriptors and implicit functions we are able to deconstruct the topology of these patients as a set of signals. From these we formulate an average nasal airway geometry and describe the individual subjects as a deviation from the mean. The underlying interest in studying the topology is to learn about the flow field. CFD results of these subjects are also presented for a quiet restful breathing rate, showing particle tracks and structures in the flow field.

Whole blood is a concentrated suspension of formed cellular elements that includes red blood cells (RBCs), white blood cells and platelets, representing approximately 46% of the volume in human blood. These cellular elements are suspended in plasma, an aqueous ionic solution of low viscosity. Plasma behaves as a Newtonian fluid but whole blood has non-Newtonian properties. In the large vessels, where shear rates are high enough, it is reasonable to assume that blood has a constant viscosity and a Newtonian behavior. However, in smaller vessels, or in some disease conditions, the presence of the cells induces low shear rate and whole blood exhibits remarkable non-Newtonian characteristics, like shear-thinning viscosity, thixotropy, viscoelasticity and possibly a yield stress. In this talk we discuss some mathematical models well suited to describe blood rheology and present numerical results comparing Newtonian and non-Newtonian simulations of blood flow in several geometries.

The pulsatile flow in a curved elastic pipe of circular cross section is investigated. The unsteady flow of a viscous fluid and the wall motion equations are written in a toroidal coordinate system, superimposed and linearized over a steady state solution. Being the main application relative to the vascular system, the radius of the pipe is assumed small compared with the radius of curvature. This allows an asymptotic analysis over the curvature parameter. The model results in an extension of the Womersley's model for the straight elastic tube. A numerical solution is found for the first order approximation and computational results are finally presented, demonstrating the role of curvature in the wave propagation and in the development of a secondary flow.

In this talk we focus on the modeling and numerical simulation of the fluid structure interaction mechanism in vascular dynamics. We will first present a simplified coupled fluid-structure problem where the artery is modeled as an elastic membrane deforming only in the normal direction. This model generalizes the well known "independent rings" model to an arbitrary geometry and can be embedded into the fluid equations as a Robin boundary condition, thus leading to "cheap" algorithms to simulate the blood/artery interaction. We will then consider the more complex case of a thick structure, modeled by 3D elasticity equations and explore the possibility to use the reduced model as a preconditioner of the full one. In particular we propose partitioned fluid-structure algorithms based on Robin transmission conditions. We will present some theoretical results obtained on a linear model problem as well as numerical tests, showing the robustness of the new approach with respect to the added mass effect and its superiority with respect to more traditional partitioned procedures.

The modelling of the cardiovascular system is still a computational challenge. To reduce complexity one resorts to the interplay of different models. The most complex ones resolve the full three-dimensional flow and are used only in the region of interest. Simpler models are used instead to take account of the global circulatory system. In this talk we will discuss effective numerical schemes to couple these different techniques.

The study of atmosphere phenomena is important and challenging field of research in different areas of sciences such as meteorology, geophysics, physics and mathematics. This presentation addresses some recent results obtained in mathematical and computational simulation of the atmosphere. Four problems are considered. The first one is the study of mathematical properties of atmospheric balance equations. The study of the mathematical structure of these equations revealed that certain balance relations lead to ill-posed mathematical problems. Comparison of the well-posedness conditions for balance theories of different level of complexity, made possible to conclude that the ill-posed problems arise due to applied physical simplifications, which can be inappropriate for certain atmospheric motions. The second problem is related to generation of structured grids for numerical models in spherical geometry. If the accuracy of numerical solution is required over some "limited" area of a sphere, then the most uniform flattening of a sphere can provide better properties of numerical scheme. Comparison of conformal mappings from a sphere onto a plane showed that for different "limited" area models the best stereographic projection is the stereographic one centered at the centerpoint of the chosen spherical domain. Evaluations of the advantage of the "best" stereographic grid with respect to other conformal mappings are given. The third problem is correctness of the vertical discretization in the hydrostatic atmospheric models. The positivity of spetra of different matrices arising in vertical discretization of atmospheric models was proved analytically, which assures well-posedness of initial-value problem for vertically discretized atmospheric equations. The last problem is related to application of splitting techniques in numerical atmospheric models in order to design a more efficient algorithm. Different types of physical and geometrical splitting were employed in the context of the semi-implicit scheme for shallow water and hydrostatic atmospheric models. Some techniques of reducing the additional splitting errors were also applied. Developed numerical schemes were tested with actual atmospheric data and obtained results of experiments showed high order of accuracy of forecasting fields and computational efficiency of numerical algorithms.

We provide necessary and sufficient for the exact observability of systems governed by the Schrödinger equation in a rectangle with Dirichlet or Neumann boundary conditions. Generalizing results from a previuos work with Ramdani, Takahashi and Tenenbaum, we prove that the corresponding criterion is that the observation region has non empty interior in the case of Dirichlet observation and, in the case of Neumann observation, that it has an open intersection with an edge of each direction. Thus, in both circumstances, observation regions may have arbitrarily small measures. We complement these results by proving that the above mentioned properties hold in arbitrarily small time. We also show that similar results hold for the Euler-Bernoulli plate equation. Finally, we give explicit estimates for the blow-up rate of the observability constants as the time and (or) the size of the observation region tend to zero. The main ingredients of the proofs are an effective version of a theorem of Beurling on non harmonic Fourier series and an estimate for the number of lattice points in a neighbourhood of an ellipse.

This is a joint work with G. Tenenbaum (Institut Élie Cartan).

1. Information dynamical models; 2. Heat cnoidal waves and solitons induced by electromagnetic radiation; 3. Stochastic resonance in disperse media.

We consider the wave analogue of the classical Newton problem of the body of minimal resistance. In the wave setting, instead of a homogenuous parallel flow of particles and classical resistance, one should consider a plane wave and the transport cross section. Trying to solve the problem, we found some new natural effects of scattering theory unknown even for physicists.

A mathematical model for the diffusion-transport of a substance between two porous homogeneous media of different properties and dimensions is presented. A strong analogy with the 1D transient heat conduction process across two-layered slabs is shown and a similar methodology of solution is proposed. Separation of variables leads to a Sturm-Liouville problem with discontinuous coefficients and an exact analytical solution is given in the form of an infinite series expansion. The drug-eluting stent constitutes the main application of the present model. The talk points out the role of nondimensional parameters which control the complex transfer mechanism across the two layers.

A detecção não-destrutiva de obstáculos por propagação de ondas harmónicas de baixa frequência e as suas aplicações em engenharia e medicina é um interessante problema matemático. A resolução deste problema inverso específico está presente nas tecnologias de imagiologia médica (ecografias) assim como nas tecnologias para radares, sonares e detecção de minas. Apresentamos um método híbrido para a resolução numérica do problema inverso de difracção acústica por ondas harmónicas de baixa frequência. Este método combina ideias de métodos iterativos tipo Newton e de métodos de decomposição, adquirindo vantagens de cada um deles. A fiabilidade e robustez do método são demonstradas por exemplos numéricos.

In processes driven by nonlinear diffusion a signal from a concentrated source remains confined in a finite region in space during the dynamics. Such kind of solutions appears in numerous applied problems such as gas filtration and turbulent flows. Due to their confined shape the solutions are convenient to seek in the form of power series in spatial coordinate. The original set of PDEs is converted to a dynamical system with respect to time-dependent series coefficients to analyze. We use this approach in problems involving coupled agents. To test the method we consider a single equation with linear and then quadratic diffusivity to recover the known results. Then we apply the approach to modeling of expanding free turbulent jet. Some example trajectories for the respective dynamical system are presented. The structure of the system hints an existence of an attracting centre manifold. The attractor is explicitly found for a reduced version of the system.

In this talk we present and analyse a numerical method for the solution of a distributed order differential equation of the general form $$ \int_0^m \mathcal{A}(r, D^r_*u(t)) dr = f(t) $$ where the derivative $D^r_*$ is taken to be a fractional derivative of Caputo type of order $r$. We give a convergence theory for our method and conclude with some numerical examples.

As is well-known, the Navier-Stokes equations are at the foundations of many branches of applied sciences, including Meteorology, Oceanography, Oil Industry, Airplane, Ship and Car Industries, etc. In each of the above areas, these equations have collected many undisputed successes, which definitely place them among the most accurate, simple and beautiful models of mathematical physics. However, in spite of these successes, to date, a number of unresolved basic questions — mostly, for the physically relevant case of three-dimensional (3D) motions — remain open. Among them, certainly, the most famous is that of proving or disproving existence of 3D regular solutions for all times and for data of arbitrary ‘size’, no matter how smooth. This notorious question has challenged several generations of mathematicians since the beginning of the 20th century who, yet, have not been able two furnish a complete answer. The problem has become so obsessing and intriguing that, as is known, mathematicians have decided to put a generous bounty on it. In fact, properly formulated, it is listed as the third of the seven $1M Millennium Prize Problems of the Clay Mathematical Institute. It should be observed that the analogous question in the two-dimensional (2D) case received a positive answer about half a century ago. In this talk I shall present the main known results of existence, uniqueness and regularity of solutions to the corresponding initial-boundary value problem in a way that should be accessible also to non-specialists. Moreover, I will furnish a number of significant open questions and explain why the current mathematical approaches fail to answer them. In some cases, I shall also point out possible strategies of resolution.

The problem of shape reconstruction of an unknown characteristic source inside a domain is analyzed. We consider a conductivity problem where the heat source is defined as a characteristic function. Restrictions to star-shaped sets arise from a uniqueness theorem by Novikov and are discussed in the context of a Fourier problem. A numerical method based on the reciprocity gap functional for harmonic polynomials and series truncation is proposed to recover the unknown shape from Cauchy noisy data. Extensions to other problems will be discussed. (Joint work with C J S Alves)

This work is concerned with the development of a numerical method capable of simulating viscoelastic free surface flows governed by the non-linear constitutive equation PTT (Phan-Thien-Tanner). In particular, we are interested in flows possessing moving free surfaces. The fluid is modelled by a Marker-and-Cell type method and employs an accurate representation of the fluid surface. Boundary conditions are described in detail and the full free surface stress conditions are considered. The PTT equation is solved by a high order method which requires the calculation of the extra-stress tensor on the mesh contour. The equations describing the numerical technique are solved by the finite difference method on a staggered grid. The numerical method was incorporated into the codes Freeflow2D and Freeflow3D, extending these codes to viscoelastic flows described by the non-linear constitutive equation PTT. To validate the numerical method fully developed flow in a two-dimensional channel was simulated and the numerical solutions were compared with known analytic solutions. The 3D-case was validated by simulating fully developed flow in a 3D-pipe. Convergence results were obtained throughout by using mesh refinement. To demonstrate that complex free surface flows using the PTT model can be computed, extrudate swell and a jet flowing onto a rigid plate were simulated. A short video will be shown. This is joint work with Gilcilene Paulo.

The numerical solution of linear weakly singular Volterra integro-differential equations is discussed. Using an integral equation reformulation of the initial-value problem, we apply to it a smoothing transformation so that the exact solution of the resulting equation does not contain any singularities in its derivatives up to a certain order. After that the regularized equation is solved by a piecewise polynomial collocation method on a mildly graded or uniform grid.

The aim of this work is to extend to compressible and heat conducting flows the well known concept of weak solutions to the incompressible Navier-Stokes equations due to J. Leray in 1933 and extended by P.-L. Lions in 1995 and E. Feireisl in 2001 to barotropic flows. Global existence and stability properties are obained in dimension 2 and 3 for general equation of state including polytropic gas law except close to vacuum. The main idea is to use a new mathematical entropy expressed as some additional Lyapunov function on the whole system which arises when the viscosity coefficients depend in a suitable way on the density. We will compare the results to the one by E. Feireisl where an inequality is obtained on the temperature but viscosity coefficients may depend on temperature. We will also explain why our result help to provide existence of global weak solutions for viscous shallow water systems. This is a joint work with Benoit Desjardins.

An asymptotic analysis of solutions to elliptic problems in domains with rapidly oscillating boundaries (rough and damaged surfaces) will be presented. The construction of several asymptotic terms, including boundary layers, permit for modelling the problem by a much simpler problem, solution of which, nevertheless, gives the two-term asymptotics of the original solution. Two approaches for this type of modelling will be discussed. For the sake of simplicity, boundary value problems for a scalar equation will be considered, although the investigation is directed to the elasticity theory.