Applied Mathematics and Numerical Analysis Seminar 

The complexity of the cardiovascular system makes it unfeasible to perform 3D simulations in large vascular regions. Thus, truncated 3D regions of interest must be considered, originating artificial sections. Commonly these simulations are carried out neglecting the remaining cardiovascular system. However there is a strong relation between local and global hemodynamics that must be taken into account in order to perform realistic simulations. The global circulation can be approximated through reduced (1D and 0D) models. The reduced models have lower accuracy and complexity, yet they provide the useful information to be coupled to the 3D model. On the other hand, experimental results demonstrate that blood can exhibit non-Newtonian characteristics such as shear-thinning viscosity, viscoelasticity or yield stress, which should be captured by the mathematical model. We address both issues of the geometrical multiscale modelling of the cardiovascular system and blood rheology. We describe the different models, including 3D generalized Newtonian fluids and fluid-structure interaction. We also focus on the coupling between them, detailing 1D-3D, 0D-3D and 1D-0D couplings. Several numerical results are presented to illustrate the geometrical multiscale approach.

The object of this talk is to introduce the pipeline of studying physiological flows. We start by highlighting the pipeline of going from a stack of medical images to numerical simulation. Difficulties are identified and an example demonstrating the importance of accurate virtual model boundaries is presented. Morphological variations between different patients and their correlation to disease-associated fluid mechanics properties is alsointroduced. However identifying these fluid mechanics properties is also non-trivial since the full physiological complexity and responses are too complex. The idea is to identify flow characteristics and as well as when these become abnormal. Results for both a peripheral bypass graft and nasal cavity are presented.

Efficient numerical simulation of fluid-structure interaction problems involving a viscous incompressible fluid with a moderate fluid-structure density ratio is a difficult task. Blood flow simulations in deformable arteries is a popular example. Indeed, in such situations, weak (or explicit) coupling schemes, i.e. that only involve the solution of the fluid and the structure once (or just a few times) per time step, are known to give rise to numerical instabilities. Recently, we have proposed a (stabilized) explicit coupling scheme, based on Nitsche’s method, whose stability properties are independent of the fluid and structure density ratio. Stability is obtained thanks to the dissipative structure of the Nitsche coupling and a stabilization term giving control of the time fluctuations of the interface fluid load. We will discuss some theoretical and numerical results (in 2D and 3D) illustrating the features of the method.

The talk addresses the numerical solution of singularly perturbed differential equations in one and two dimensions. Because solutions of such problems exhibit sharp boundary and interior layers (which are narrow regions where solutions change rapidly), a significant economy of computer memory and time can be attained by using special layer-adapted meshes. These meshes are fine in layer-regions and standard outside; in two dimensions they have extremely high maximum aspect ratios. Ideally, mesh layer adaptation is automated by exploiting sharp a posteriori error estimates. However, the known a posteriori error estimates are typically under the minimum angle condition, equivalent to the bounded-mesh-aspect-ratio condition, which is rather restrictive and makes a posteriori error estimates less practical for layer solutions. In contrast, we present certain new a posteriori error estimates that hold true under no mesh aspect ratio condition. These estimates are in the maximum norm, which is sufficiently strong to capture layers. Furthermore, our error estimates are uniform in the singular perturbation parameter, which is significant since in general the error constant might blow up as the perturbation parameter becomes small.

This talk is concerned with the analysis of a linear mixed type (forward-backward) functional equation, that is, a linear functional equation which has a delay and an advance term. We search for a solution of such equation which has a given form at the initial interval $[-1,0]$ and at the final interval $[k-1,k]$. This problem has been studied both analytically and numerically (see[1]). One of the most common approaches for the analysis of this problem is based on its reduction to an initial value problem for a delay differential equation (DDE). Following this approach, we search for an approximate solution in the form of a linear combination of a given set ob basis functions. The basis functions can be extended to the interval $[1,k]$ either numerically (using the finite difference method) or analytically (using recurrence formulae — “method of steps”). Finally, the coefficients of the linear combination are computed by the collocation or least squares method, so that the numerical solution fits the data at the interval $[k-1,k]$. A different approach consists on the transformation of the considered problem into a boundary value problem for a ODE. In this case, standard numerical methods for ODEs can be applied. Numerical results obtained by these methods are presented and compared with the ones, presented in previous works. The advantages and weaknesses of the introduced computational methods are discussed. This is a joint work with P. Lima, P. Lumb and N. Ford.

Reference

1. N. Ford and P. Lumb, Mixed type functional differential equations: a numerical approach (submitted).

This work is concerned with modelling the evolution of competitive chemical reactions within a small cell with a labelled and unlabelled antigen reacting with a specific antibody on the side wall. A model consisting of coupled heat conduction equations with nonlinear and nonlocal boundary conditions is considered and shown to be equivalent to a system of Volterra integral equations with weakly singular kernel. This work generalizes some previous work done on the case of the single heat equation ([1], [2]). We prove the existence and uniqueness of the nonlinear system on $[0, 1)$. The asymptotic behavior of the solution as t tends to $0$ and $t$ tends to infinity is obtained and other properties of the solution, e.g., monotonicity, are investigated. In order to obtain a numerical solution of the system of VIES we use the technique of subtracting out singularities to derive explicit and implicit Euler schemes with order one convergence and a product trapezoidal scheme with order two convergence. Numerical results are presented. This is a joint work with T. Diogo and S. McKee.

References

1. S. Jones, B. Jumarhon, S. McKee, J. A. Scott. A mathematical model of a biosensor, Journal of Engineering Mathematics 30, Netherlands, (1996) 321-337.
2. B. Jumarhon, S. McKee. On the heat equation with nonlinear an nonlocal boundary conditions, Journal of Mathematical Analysis and Applications 190, (1995) 806-820.

In this talk we address the question of identifying a source function from boundary measurements, for the Laplace equation. This is an inverse problem that models (among others) the reconstruction of heat sources from boundary measurements of temperature and heat flux, in linear diffusion problems. It is a well known inverse problem that lacks uniqueness and some extra information concerning the source must be considered. One way is to consider intrusive measurements (knowledge of the source at some domain points). In non-intrusive evaluation (which is the approach of this work) the extra source information is an indirect one. We present several classes where this indirect information is sufficient to obtain uniqueness and show how to recover the harmonic part of a source from the boundary data. In particular, a one to one relation between the Cauchy data and the harmonic part of the source is established. Several numerical simulations will be presented. This is an joint work that results from a cooperation between engineering and mathematics departments of UFRJ (Brazil) and IST.

The general framework of our contribution consists in recovering lacking data on some part of the boundary of a domain from overspecified data on the remaining part of the boundary. This kind of problem occurs in the reconstruction of cardiac activ ity. In fact, non invasive imaging of heart’s electrical activity from ElectroCardioGram becomes a standard diagnostis tool in clinical application. The reconstruction of the spread of electrical excitation in the human heart of each single beat shall facilitate cardiologists to discriminate normal from abnormal activity, localize the origin of ar rhythmias, ischemie or infarcted regions . We solve this inverse ElectroCardioGraphy (ECG) problem by the mean of an energy-like error functional.

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.