Applied Mathematics and Numerical Analysis Seminar 

We aim to provide an accurate representation of industrial CFD activities in Portugal. For this purpose we review some work undertaken at blueCAPE. The presentation is thus comprised of three sections: basic description of physical models and numerical discretization techniques; application examples from industrial applications; hardware developments and their implications in CFD codes. The first section sets the context and the nomenclature for the examples that follow in broad technical and scientific terms. The second section puts forward some examples taken from a number of industrial problems that blueCAPE has addressed through CFD analysis. The third and final section discusses some of the implications that the widening gap between fine and coarse-grained parallelism imposed by hardware developments have on the current generation of commercial CFD codes. Questions will be taken.

We consider the Navier-Stokes equations in a 2D-bounded domain with general non-homogeneous Navier slip boundary conditions prescribed on permeable boundaries, and study the vanishing viscosity limit. We prove that solutions of the Navier-Stokes equations converge to solutions of the Euler equations satisfying the same Navier slip boundary condition on the inflow region of the boundary.

The communication is concerned with maximum regularity estimates in anisotropic Sobolev spaces $W_p^{2;1}$ for solutions of linear problems arising in hydrodynamics and magnetohydrodynamics. The proof is based on the Marcinkiewicz-Mikhlin-Lizorkin theorem on multipliers in the Fourier integrals.

In the recent years mathematical models and numerical simulations applied to the vascular system have been increasing due to their applicability in different physiological cases and pathologies such as aneurysms, thrombosis, atherosclerosis, etc. The presentation focuses on the modelling of the coupled fluid-structure interaction (FSI) problem which arises in haemodynamics using different constitutive laws to describe the vessel wall dynamics. The mechanical behaviour of the tissues composing the vessel wall is highly nonlinear. Moreover, it is known that it shows a non-homogeneous composition and anisotropic behaviour. In this work the arterial tissue has been described as an homogeneous isotropic non-linear material. Numerical simulations of the coupled fluid-structure interaction problem have been carried out using the different constitutive laws and analysing the effects of the different arterial modelling on the numerical results. In particular the attention of this work is referred to a anatomically realistic geometry of cerebral aneurysm that developed on the internal carotid artery.

We develop a computational method for the simulation of the flow of blood through cerebral aneurysms, which may occur in the human brain. These aneurysms are weak regions in the vessel, presenting a serious risk of rupture to the patient. The goal is to understand the flow in these diseased parts of the vessel system. An immersed boundary method, based on volume penalization, is developed to compute the pulsatile flow. The raw medical imagery representing the patient-specific geometry is processed to extract the ‘masking function’, which is needed to simulate flow patterns and obtain wall shear stresses under realistic physiological conditions. We illustrate the numerical method for several model and realistic aneurysms. In each case we observe a transition to complex time-dependent flow in case the flow speed and/or the aneurysm size become sufficiently large. High frequency variations appear in the flow, which may be an easy method for monitoring the progress of a developing aneurysm and the risk it represents.

Aneurysm, embolisms and atherosclerosis are, among different diseases affecting the cardiovascular system, the most studied. These pathologies include a variety of disorders and conditions that affect the heart and the blood and are usually associated with factors like biochemistry, haemodynamics and genetic predisposition. These factors are specific to each individual and it is important to represent accurately patient-specific information to evaluate correctly clinical state either at diagnosis and prognosis stages. Taking an example of a configuration of the Aorto-Iliac bifurcation, we examine the effects of image filtering and contrast enhancement on the computational reconstructed geometry. Methods to quantify the differences resulting in the images from the different filtering methods are based on the Signal Noise Ration, pixel intensity variance. Finally all the methods are applied to a synthetic image to assure the most accurate sequence of images. Comparison of the images and reconstructed geometries after different pre-processing methods identify a possible uncertainty range for this patient specific study that should be considered when discussing prognosis and diagnosis in a clinically relevant context, mainly when studying the measures of wall shear stress, wall shear stress gradient, and oscillatory shear index which have been largely used in the literature to correlate to disease. In this study we focus on the effects of uncertainty in clinically acquired medical imaging to variability in the reconstructed vessel geometry.

Blood coagulation is a biological process of fundamental importance and extreme complexity. It consists on the formation of blood clots at the site of vascular injury, preventing the blood loss. This process involves complex interactions among multiple molecular and cellular components in the blood and vessel wall, and it is also influenced by the flow of blood.

Mathematical modeling of the blood coagulation and fibrinolysis processes is a way of conceptualizing and understanding this complicated system, helping to optimize design of artificial devices and also to identify the regions of the arterial tree susceptible to the formation of thrombotic plaques and possible rupture in stenosed arteries. A good model should be simple enough in order to be applied in numerical simulations, and at the same time should be able to capture the process complexity, so to allow its better understanding.

The blood coagulation model we are working on consists of a system of convection-reaction-diffusion equations, describing the cascade of biochemical reactions, coupled with rheological models for the blood flow (Newtonian, shear-thinning and viscoelastic models). We introduce the effect of blood slip at the vessel wall emphasizing an extra supply of activated platelets to the clotting site. We expect that such contribution could be dominant, resulting in the acceleration of thrombin production and eventually of the whole clot progression. Such model will have the capacity to predict effects of specific perturbations in the hemostatic system that can't be done by laboratory tests, and will assist in clinical diagnosis and therapies of blood coagulation diseases.

Numerical results for 1D case will be presented, based on the solution of a system of reaction-diffusion equations, using the Finite Element Method. Evolution of concentration of biochemical species and clot formation and growth will be investigated in the injury site of the vessel wall.

We study the Nehari manifold $N$ associated to the boundary value problem \[-\Delta u=f(u)\,,\quad u\in H^1_0(\Omega)\] where $\Omega$ is a bounded regular domain in $\mathbb{R}^n$. Using elementary tools from Differential Geometry, we provide a local description of $N$ as an hypersurface of the Sobolev space $H^1_0(\Omega)$. We prove that, at any point $u\in N$, there exists an exterior tangent sphere whose curvature is the limit of the increasing sequence of principal curvatures of $N$. Also, the $H^1$-norm of $u\in N$ depends on the number of principal negative curvatures. Finally, we study basic properties of an angle decreasing flow on the Nehari manifold associated to homogeneous non-linearities.

Polynomial sequences generated by integral powers of first and second order differential operators conveniently chosen will be the issue. More precisely, the focus will lie on their connection with well known orthogonal polynomial sequences along with their foremost structural properties. This talk will be split in two parts. We will start by analysing the cases in which the aforementioned differential operator is of first order, bringing into analysis polynomial sequences associated to the classical linear functionals of Hermite, Laguerre, Bessel and Jacobi. Afterwards, the discussion will proceed towards the analysis of polynomial sequences generated by second order differential operators, which brings up the open problem of characterizing orthogonal polynomial sequences with respect to certain positive definite linear functionals. The Kontorovich-Lebedev transform and the central factorial numbers will be an asset to attain our goals.

References

1. Ana F. Loureiro, New results on the Bochner condition about classical orthogonal polynomials, J. Math An. Appl., 364 (2010) 307-323.
2. Ana F. Loureiro, P. Maroni, S. Yakubovich, On a nonorthogonal polynomial sequence associated with Bessel operator, Pre-Print CMUP 2011-10 (ArXiv:1104.4055v1)
3. Ana F. Loureiro, S. Yakubovich, On a polynomial sequence related to the Ditkin-Prudnikov problem, Pre-Print CMUP 2011-23 (arXiv:1110.6015v1)

The fully Lagrangian Smoothed Particle Hydrodynamics (SPH) method was originally invented to deal with non axisymmetric problems in astrophysics (Lucy 1977, Gingold & Monaghan 1977). Since then the use of SPH has expanded in many areas of solid and fluid dynamics (involving large deformations, impacts, free-surface and multiphase flows). For example, collision of rubber cylinders (Swegle et al., 1995) in solid mechanics, dam breaking and free-surface waves (Monaghan, 1994) and two-phase flows such as Rayleigh-Benard instability (Violeau, 1999) in fluid mechanics. A major advantage of SPH over Eulerian methods is that the method does not need a grid to calculate spatial derivatives. Instead, they are found by summation of analytical differentiated interpolation formulae (Monaghan, 1992). The momentum and energy equations become sets of ordinary differential equations which are easy to understand in mechanical and thermodynamical terms. For example, the pressure gradient becomes a force between pairs of particles. While Eulerian methods have difficulties to construct a mesh for the simulation domain when it has very complex interfaces, SPH is able to do it without any special front tracking treatment. Nevertheless, despite good agreements in general, some limitations are found in the SPH method such as very small time step, which lead to very expensive CPU cost, and pressure fluctuation.

Neste trabalho propõe-se uma nova metodologia baseada em técnicas de procura directa (DMS - Direct MultiSearch), que não agrega nenhuma das funções objectivo do problema. A estrutura algorítmica baseia-se no paradigma dos passos de procura/sondagem dos métodos de procura directa direccional, recorrendo ao conceito de dominância de Pareto para manter uma lista de pontos não dominados (de onde são seleccionadas as novas iteradas a utilizar no passo de sondagem). Um dos objectivos do método é gerar o maior námero possível de pontos na frente de Pareto unicamente a partir do passo de sondagem. Pretende-se, também, manter a estrutura algorítmica o mais geral possível, possibilitando, em particular, a incorporação de estratégias de disseminação no passo de procura (que, como se sabe, é opcional). A metodologia DMS é uma generalização para optimização multiobjectivo (OMO) de todos os métodos do tipo procura directa direccional. Dois outros subprodutos desta contribuição são (i) o desenvolvimento de uma colecção de problemas para a OMO; (ii) a extensão dos perfis de desempenho e de dados para OMO, permitindo a comparação de diferentes solvers num conjunto grande de problemas teste, em termos de eficiência e robustez na determinação da frente de Pareto. Relativamente à análise de convergência desta classe de algoritmos, supondo válidas as hipóteses habitualmente consideradas na análise de convergência dos métodos de procura directa direccional em optimização uni-objectivo e recorrendo à análise não suave de Clarke, foi demonstrado que uma subsucessão da sucessão de iteradas gerada pelo DMS converge para um ponto crítico de Pareto-Clarke.

d’Angelo, Zunino and Quarteroni have developed a multiscale methodology to handle numerically complex problems that arise naturally in microcirculation and perfusion of biological tissues through networks of vascular beds. The problem is composed by two scales: (i) at the macroscale, the tissue is treated as a homogenized porous media, usually saturated with plasma where relevant chemical species diffuse, advect and react; and (ii) the microscale is a finite and discrete network of small vessels (arterioles and capillaries), from which plasma permeates, and are naturally considered as a network of one dimensional models of fluid flow in circular tubes. Starling’s filtration law, a widely established description of the permeation of plasma from the arteriole into the tissue, couples both scales in a nonlinear fashion, i.e. the amount of fluid delivered from the vessel into the tissue (or vice versa) generally depends locally on the dependent variables of both scales. Currently, the mathematical modeling of angiogenesis is generally treated with two concurrent modeling approaches: (i) on one hand, continuum models of angiogenesis are based on reaction-diffusion-advection equations that describe endothelial cell conservation and quantify their motility accordingly to the distribution of other important fields, such as the concentration of angiogenic factor or presence of fibronectin, and fall short in the discrete identification and description of the vascular network; and (ii) stochastic discrete models based on lattice descriptions of the tissue together with a set of phenomenologically based cellular automata rules of network behavior and evolution have provided interesting qualitative results to describe a multitude of different patterns of angiogenesis. The multiscale methodology developed by d’Angelo and co-workers seems to be a promising strategy to develop quite a fundamental model of angiogenesis, based on evolution of the microscale (i.e. with an evolving network in response to biological stimuli provided by the tissue) or even on the evolution of both scales (with a growing tissue in response to improved conditions provided by the enhancement of the microscale). Several challenges naturally arise with the mathematical description of this complex problem and are currently a work in progress, such as: (i) the ability of the method to handle very complex arterial networks downstream after multiple rounds of capillary sprouting, (ii) anastomosis (when two capillary tips join and form a loop, changing the topology of the microscale domain), and among others (iii) the choice of proper one dimensional models for fluid flow in one-ended tubes and the treatment of anisotropic (directional) plasma sources at the macroscale.

On the nonnegative semi-axis, a second order linear integro-differential equation with a Volterra integral operator and strong singularities at zero and infinity is considered. Limit conditions at singular points are posed. Under some natural assumptions, it is a singular initial problem with limit normalizing conditions at infinity. An existence and uniqueness theorem is proved and asymptotic representations of the solution are given. A numerical algorithm for evaluating the solution is proposed, calculations and their interpretation are given. The singular problem under study describes the survival (non-ruin) probability of an insurance company on an infinite time interval (as a function of initial surplus) in the Cram'er-Lundberg dynamic insurance model with exponential distributions of claims and certain company's strategy at the financial market assuming investment of a fixed part of a surplus (capital) into risky assets (shares) and the rest of it into a risk-free asset (bank deposit).

To date, solving shortest path problems inside simple polygons has usually relied on triangulation of the polygons and graph theory. The question: "Can one devise a simple $O(n)$ time algorithm for computing the shortest path between two points in a simple polygon (with $n$ vertices), without resorting to a (complicated) linear-time triangulation algorithm?" raised by J. S.B. Mitchell in Handbook of Computational Geometry (J. Sack and J. Urrutia, eds., Elsevier Science B.V., 2000), is still open. The aim of this talk is to show that convexity contributes to the design of efficient algorithms for solving some versions of shortest path problems (namely, computing the convex hull of a finite set of points in 2D and 3D, computing the convex rope/shortest path between two points outside/inside a simple polygon) without triangulation of polygons or graph theory. Numerical examples are presented.

We present some classical ideas to stabilize infinite dimensional systems by means of a feedback law. Then we consider the problem of stabilization when only boundary observations are available which requires to couple an estimating problem with the feedback law. Using such ideas we introduce a numerical approach for estimation and stabilization of the nonlinear Burgers equation. Simulations including the filtering of observation and dynamic noise are also considered.

Abstract: The Perona-Malik equation (PME) is a forward-backward nonlinear diffusion equation which was proposed in the context of image processing as an image enhancement tool capable of preserving sharp features such as edges. To this day its mathematical nature has not been fully understood in spite of many attempts. After a brief historical overview of the mathematical results available for the equation and its many regularizations/relaxations, the talk will introduce and analyze a novel, rather natural, regularization which will shed light on the nature of PME. The regularization is quite mild in that PME is regularized by a family of forward-backward equations, the solutions of which are, however, better behaved.

We present a fluid-structure interaction approach with applications in medicine. Fluid structure interaction problems are nowadays a subject of intensive research. The combination of mathematics and computer science, on the one hand, with bio-mechanics, chemical processes and physics, on the other hand, becomes more and more important. In this talk, we present numerical algorithms to model and simulate two-dimensional aortic heart valve dynamics. Data have been obtained by a medical doctor expert in heart valve surgery. It is known that the door from the left ventricle to the aorta consists of three valves, which implicates that a 2D simulation is not appropriate to study all physical processes. However, we are able to investigate numerical algorithms and basic effects for these kind of problems. The mathematical treatment of fluid-structure interaction is based on the so-called “arbitrary Lagrangian-Eulerian” (ALE) method, where fluid equations are rewritten in a fixed arbitrary reference domain. Afterwards, all equations for the fluid and the structure are treated in a common framework leading to an implicit solution process. We present results recently derived but also discuss numerical and mathematical challenges, for instance, how to impose appropriate boundary conditions at the outlet of the blood vessel.

Some basic aspects of the mathematical models used to describe the cardiac tissue will be presented considering, in particular, the electrophysiology and the myocardium mechanics, as well as the methods used for their coupling. The origins of the electrical activity in the cardiomyocytes will be shown, entering into the cell physiology. During a normal heart beat, the electrical signal propagates throughout the tissue and it generates internal forces which are the main responsible of the muscle contraction. The presentation will focus on the constitutive relations that are used to describe cardiac mechanics and on two possible approaches for the description of the active forces induced by the ${\left[\mathrm{Ca}\right]}^{2+}$ ions during a contraction cycle. Some simple simulations will be shown for a better understanding of the subject.

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.