Applied Mathematics and Numerical Analysis Seminar 

A fluorescent capillary-fill device is a biosensor based on antibody-antigen technology for determining whether a patient is suffering from a particular pathogen. The specific antibody is affixed to a side wall in a small container. On the other side is the antigen with a fluorescent label which dissolves upon entry of the bulk fluid (usually urine) containing the antigen. A competitive reaction then takes place on the side wall for the antibody sites. In this talk we consider the development of a mathematical model of this competitive chemical reaction within a small cell which occurs in a biosensor. The model consists of two coupled diffusion equations with nonlinear boundary conditions which can be expressed equivalently as a system of two integro- differential equations. Through this reformulation an asymptotic result is derived, a perturbation solution is developed and a product integration method is presented. Finally, an alternative formulation is presented in the form of a system of four Volterra integral equations, which provides local existence and uniqueness of the solution to the original diffusion problem. Two product integration methods are applied to this second reformulation. We present several numerical results, using real data, that illustrate the performance of the methods.

In this talk we are concerned about singular boundary value problems arising in hydrodynamics and cosmology. In the case of spherical simmetry, the orginal partial differential equation may be reduced to a second order nonlinear ordinary differential equation (ODE). This is the case, for example, of the formation of spherical bubbles or droplets in a mixture gas-liquid. We are interested on solutions of the resulting ODE which are strictly increasing on the positive semi-axis and have finite limits at 0 and infinity (bubble-type solutions). Necessary and sufficient conditions for the existence of such solutions are obtained in the form of a restriction on the equation coefficients. The asymptotic behavior of certain solutions of this equation is analysed near the two singularities (when r tends to 0 and r tends to infinity), where the considered boundary conditions define one-parameter families of solutions. Based on the analytical study, an efficient numerical method is proposed to compute approximately the needed solutions of the above problem. Some results of the numerical experiments are displayed and their physical interpretation is discussed.

Functional differential equations are used in the modelling of nerve conduction. In the seminar we will consider the questions: a) Which equations are used? b) What modelling assumptions are made in the derivation of the models? c) Why and how are mixed type functional equations involved? We will also consider mixed type functional equations with multiple delay and advance terms which, in some applications, may lead to a more realistic mathematical model of a problem. We will consider some aspects of, and issues relating to, our current work in this area.

Cardiovascular diseases represent the major causes of death in developed countries. This family of diseases involve both dysfunctions (or illnesses) of the heart and the circulatory system. Intracranical Saccular Aneurysm is a specific type of arterial aneurysm, that is a focal dilatation of the vascular wall. It usually develops at the apex of a bifurcation in or near the circle of Willis. The goal of this research is to find a suitable model to describe the early stage development of an ICA and to implement it in a finite element library. Moreover, the effects of hemodynamics will also be studied using appropriate models for the blood and fluid-structure interaction (FSI) algorithms. This talk focuses on the mathematical models developed in literature to describe the mechanical response of the arterial wall and in particular on the cerebral arterial wall models. The finite element library used to run FSI simulations and to implement the models will also be described.

Asymmetric RBF collocation (where RBF stands for radial basis function) is a relatively new alternative to finite elements and volumes for the numerical solution of PDEs in irregular domains. It is a meshless method -requires no connectivity between the discretization nodes-, which approximates the strong form of the PDE in an interpolation space made up of shifted radial functions. These features confer appealing properties on it, such as geometric flexibility, exponential convergence, and great ease of implementation. There are some practical drawbacks as well, most notably, ill-conditioning and the lack of sound theoretical foundations. In this talk, the RBF setting will be first introduced. Then, the performance of this meshless scheme in two non-standard applications will be discussed: 1) elliptic problems with boundary singularities, and 2) quasilinear elliptic PDEs. It will be shown that, with minor modifications, the properties of the method carry over to these kind of problems.

In this seminar, we begin with a brief explanation of the significance of blood flow modelling to CoW aneurysms and an overview of the aspects of CFD blood flow modelling. We then proceed with an introduction to the latest finite volume method derivation and, following on, outline the addition of a pressure boundary condition. It represents a significant improvement with respect to the ability to manage the spatial discretization error. Wall shear stress results are shown from a two-dimensional, rigid CoW model, with pulsatile inlet pressure conditions and resistance outlet conditions. Three simulation set-ups cover a physiologically representative range of flows: symmetric inlet conditions and configuration; internal carotid artery inlet pressure phase difference and symmetric configuration; symmetric inlet conditions and a narrow proximal (A1) section of the left anterior cerebral artery. Following on from the two-dimensional CFD model, we show some results of a similar model in three dimensions which uses the ANSYS CFX solver. An evaluation of the accuracy of the three-dimensional model is shown along with some simulation results for the flow in the CoW and ACoA.

Adaptive finite elements vary element size h or/and polynomial order p to deliver approximation properties superior to standard discretization methods. The best approximation error may converge even exponentially fast to zero as a function of problem size (CPU time, memory). The adaptive methods are thus a natural candidate for singularly perturbed problems like convection-dominated diffusion, compressible gas dynamics, nearly incompressible materials, elastic deformation of structures with thin-walled components, etc. Depending upon the problem, diffusion constant, Poisson ratio or beam (plate, shell) thickness, define the small parameter. This is the good news. The bad news is that only a small number of variational formulations is stable for adaptive meshes By the stability we mean a situation where the discretization error can be bounded by the best approximation error times a constant that is independent of the mesh. To this class belong classical elliptic problems (linear and non-linear), and a large class of wave propagation problems whose discretization is based on hp spaces reproducing the classical exact grad-curl-div sequence. Examples include acoustics, Maxwell equations, elastodynamics, poroelasticity and various coupled and multiphysics problems. For singularly perturbed problems, the method should also be robust, i.e. the stability constant should be independent of the perturbation parameter. This is also the dream for wave propagation problems in the frequency domain where the (inverse of) frequency can be identified as the perturbation parameter. In this context, robustness implies a method whose stability properties do not deteriorate with the frequency (method free of pollution (phase) error). We will present a new paradigm for constructing discretization schemes for virtually arbitrary systems of linear PDE's that remain stable for arbitrary hp meshes, extending thus dramatically the applicability of hp approximations. The DPG methods build on two fundamental ideas: - a Petrov-Galerkin method with optimal test functions for which continuous stability automatically implies discrete stability, - a discontinuous Petrov-Galerkin formulation based on the so-called ultra-weak variational hybrid formulation. We will use linear acoustics and convection-dominated diffusion as model problems to present the main concepts and then review a number of other applications for which we have collected some numerical experience including: 1D and 2D convection-dominated diffusion (boundary layers) 1D Burgers and compressible Navier-Stokes equations (shocks) Timoshenko beam and axisymmetric shells (locking, boundary layers) 2D linear elasticity (mixed formulation, singularities) 1D and 2D wave propagation (pollution error control) 2D convection and 2D compressible Euler equations (contact discontinuities and shocks) The presented methodology incorporates the following features: The problem of interest is formulated as a system of first order PDE's in the distributional (weak) form, i.e. all derivatives are moved to test functions. We use the DG setting, i.e. the integration by parts is done over individual elements. As a consequence, the unknowns include not only field variables within elements but also fluxes on interelement boundaries. We do not use the concept of a numerical flux but, instead, treat the fluxes as independent, additional unknowns (a hybrid method). For each trial function corresponding to either field or flux variable, we determine a corresponding optimal test function by solving an auxiliary local problem on one element. The use of optimal test functions guarantees attaining the supremum in the famous inf-sup condition from Babuska-Brezzi theory. The resulting stiffness matrix is always hermitian and positive- definite. In fact, the method can be interpreted as a least-squares applied to a preconditioned version of the problem. By selecting right norms for test functions, we can obtain stability properties uniform not only with respect to discretization parameters but also with respect to the perturbation parameter (diffusion constant, Reynolds number, beam or shell thickness, wave number) In other words, the resulting discretization is robust.

Atherosclerosis is the gradual build up of plaque material in the arterial wall, composed primarily of lipid-rich fatty deposit, inflammatory cells, and fibrous tissue. A vast majority of heart attacks occur when there is a sudden rupture in the atherosclerotic plaque, exposing its prothrombotic core materials to the coronary blood flow, forming blood clots that can cause blockage of the arterial lumen. The diseased arteries can be treated with drugs delivered locally to these rupture–prone plaques termed “vulnerable plaques”. In designing these local drug delivery devices, important issues regarding drug distribution and targeting need to be addressed to ensure device design optimization. For example, a drug delivery implant adjacent to a target tissue may not produce the desired safe and efficacious therapy. Therefore, the main objective of this work was to develop a computational tool-set that provides predictive local pharmacokinetic insight into the design and analysis of a catheter-based drug delivery system that uses nanoparticles as drug carriers to treat vulnerable plaques and diffuse atherosclerosis. A three-dimensional mathematical model of coupled mass transport of drug and drug-encapsulated nanoparticles was developed and solved numerically by applying finite element based isogeometric analysis that uses NURBS. Simulations were run on a patient-specific multilayered coronary artery wall segment with a typical vulnerable plaque and the effect of artery and plaque inhomogeneity was analyzed. The method successfully captured trends observed in local drug delivery and demonstrated its potential utility in optimizing design parameters, including delivery location, nanoparticle surface properties, and drug release rate.

The seminar will focus on mixed-type functional dierential equations (MTFDEs) of the form $x\prime (t)=a(t)x(t)+b(t)x(t-1)+c(t)x(t+1)$. The emphasis of our work (to date) with MTFDEs (also referred to as forward-backward equations) has been on the development of numerical approaches to the solution of this type of equation. Alongside our continuing interest in this area we are now seeking to apply our approach in modelling applications. These equations are known to have application in nerve conduction theory, one of our current areas of investigation. Parameter estimation is a technique of interest to mathematical modellers. If the equation being used by the modeller admits small solutions (solutions that approach zero faster than any exponential) then no information is obtained about the parameters. We have successfully detected the presence of small solutions to several classes of delay dierential equations (DDEs) using a numerical approach. We discuss the potential for our work with DDEs to provide insight into the detection of small solutions to MTFDEs and present illustrative examples.

The development of high-resolution models is needed to reproduce a reliable cross-shelf transport which governs the heat and the salinity budgets, the dispersion of pollutants and the redistribution of nutrient-rich coastal waters toward the oligotrophic open sea. Indeed, such chaotic mixing and stirring is very sensitive to the resolution and the accuracy of the dynamical fields. The forecasts quality is then related to the precise description of unstable coastal processes and the wide variety of vortices evolving along or across the steep shelf topography. We study the impact of smooth to steep shelf topography on the destabilization of a frontal coastal current by means of numerical simulations and laboratory experiments. We used an idealized configuration of the primitive equation model NEMO to reproduce the unstable evolution of a coastal current and the eddy generation in a circular basin. The sensitivity to the spatial resolution and the dissipative scheme were tested. High and low Ekman numbers cases were studied in order to perform a quantitative comparison with laboratory experiments or the inviscid stability theory. The laboratory experiments were conducted on the UME/ENSTA turntable at Palaiseau. Non-intrusive PIV measurements using high resolution camera provide the surface velocity and vorticity fields for quantitative comparisons. Both the numerical and the laboratory studies have shown that the ratio of the shelf slope over the isopycnal slope of the coastal current is a main control parameter of the baroclinic instability. When this topographic parameter increases the most unstable wavelength decreases leading to the formation of smaller vortices. In a second stage theses surface vortices which transport large fluid parcel are affected by the deep shelf topography which prohibit cross shelf trajectories. For a steep enough shelf slope the coastal current could be completely stabilized.

We consider reaction-diffusion equations in fractured domains, with a high dimensional gap between the dimension of the domain and the dimension of the fractures. An example is provided by fluid flow problems in three-dimensional porous media with a network of thin fractures. If the fractures are reduced to one-dimensional manifolds, the 3D equation of the asymptotic model features a measure term to account for mass conservation, which makes the considered problem non-standard. The main advantage of such reduced models is that the computational grid does not need to be adapted or conformal to the fractures. The 3D and 1D mesh are independent; moreover, for time dependent problems, the characteristic time/length scales of the 3D domain and of the fracture network can also be resolved independently. These features make this class of 3D-1D multiscale models attractive for applications to bioengineering problems, such as the analysis of tissue perfusion and of drug release from thin implantable devices, or for applications in geophysics. In this talk, we consider 3D-1D coupled reaction-diffusion problems and describe the main difficulties related to non-standard measure terms that govern the coupling conditions. We introduce special analytical and numerical tools and present some useful results about the analysis of the model and the numerical techniques for the approximate solution. We also describe the accuracy of the reduced model, and present a numerical convergence analysis. Finally, we show some of the computational results that have been obtained using the 3D-1D approach, with applications to blood flow simulation and to the analysis of drug release by biomedical devices.

Consideram-se as soluções $\nu$ das equações de Navier-Stokes, com condições de escorregamento na fronteira, e com viscosidade $\nu$ convergente para $0$. Os dados iniciais $a_\nu$ convergem num espaço funcional conveniente $X$ para um elemento $a$. Se o espaço $X$ for suficientemente “regular”, o tempo de existência $T \gt 0 $ é independente de $\nu$. No caso, por exemplo, do semi-espaço demonstramos a convergência das soluções $u_\nu$ para a solução $u$ das equações de Euler, ao fazer tender para zero a viscosidade $\nu$. A convergência é forte e uniforme, isto é, obtém-se no espaço $C([ 0, T]; X )$ . Mas no caso de uma fronteira arbitrariamente regular, mas não plana, o problema encontra-se totalmente em aberto.

In this paper we shall study a numerical method for computing the spectrum of some elliptic operators. It will be shown that the essential spectrum of an elliptic operator in an infinite periodic waveguide may contain gaps. Moreover, we construct examples where the number of band-gaps can be made arbitrarily large. Finally, we present some numerical examples by using the finite element method. The theoretical tool to accomplish this task is provided by the Krein-Birman-Vishik theory. Using the Gelfand transform we formulate the spectral problem for a quadratic pencil of formally self-adjoint operators on the periodicity cell. For fixed Gelfand parameter the model problem in the periodicity cell admits an increasing sequence of real and non-negative eigenvalues. We approximate these eigenvalues using the mixed finite element method using the Raviart-Thomas elements for the approximation of the eigenfunctions.

T.B.A