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08/02/2006, 15:00 — 16:00 — Room P3.10, Mathematics Building
, University of Chester, UK

Super-exponential solutions: their numerical modelling and detection

Certain functional differential equations may have exact solutions that either grow or decay at a rate that is faster than exponential. This provides a challenge for conventional mathematical analysis of the equations because ideas based on characteristic functions need to be revisited and generalised. It turns out that it is not always straightforward to solve equations with super-exponential solutions, nor is it usually possible to detect in advance whether or not they are present. The purpose of this talk is to describe approaches for the numerical solution of delay differential equations whose solutions decay at a rate that is faster than exponential. We show that we can find good approximations to the exact solutions in this way and that we can also use our methods to predict when super-exponential solutions are present. In conclusion we are able to show that there are situations where we have been able to predict the existence of super-exponential solutions by our methods and that these predictions have been subsequently confirmed analytically.

27/01/2006, 16:00 — 17:00 — Room P3.31, Mathematics Building
K. R. Rajagopal, Texas A&M University, College Station, USA

New perspectives in Fluid Mechanics

11/01/2006, 15:00 — 16:00 — Room P3.10, Mathematics Building
Jean-François Babadjian, SISSA - Trieste, Italy

A multiscale approach to the Neumann Sieve problem in dimensional reduction

This is a joint work with N. Ansini and C. I. Zeppieri. This talk is concerned with the characterization of the effective energy of weakly connected thin structures through a periodically distributed contact zone. We highlight the presence of three different regimes (depending on the mutual rate of convergence of the radii of the connecting zones and the thickness of the domain) and for each of them we derive the limit energy by a Gamma-convergence procedure. For each regime an interfacial energy term, depending on the jump of the deformation at the interface, appears in the limit representing the asymptotic memory of the sieve. We completely describe the interfacial energy densities by nonlinear capacitary type formulas.

07/12/2005, 11:30 — 12:30 — Room P3.31, Mathematics Building
Nilson C. Roberty, Univ. Federal do Rio de Janeiro, Brasil

Coefficient determination for the stationary anisotropic Boltzmann transport equation

The problem of simultaneous spatial determination of the absorption and scattering coeficients in the stationary linear one velocity Boltzmann transport equation via boundary measurements is investigated. The original first-order problem is shown to be equivalent to a second order self-adjoint problem. Then, I introduce an a priori operator K that can be different from the scattering but gives compactness to the problem. The associated eigenvalue problem generates a dense and complete set of eigenfunctions in the Hilbert space where the problem is defined. It is shown that the traces of eigenfunctions form a minimal system in the trace boundary space and that appropriate boundary values may be chosen in order to establish a bi-orthogonal set. The identifiability for the extinction and scattering coefficients is suggested. Numerical experiments with the original first order problem are presented.

30/11/2005, 15:00 — 16:00 — Room P3.10, Mathematics Building
Luísa Morgado, Departamento de Matemática, Universidade de Trás-os-Montes e Alto Douro

Análise e Tratamento Numérico de Problemas de Valores de Fronteira Singulares

Singular boundary value problems will be analysed for a nonlinear ordinary differential equation. Numerical methods are introduced, based on the asymptotic behavior of the solution.

16/11/2005, 17:00 — 18:00 — Room P3.10, Mathematics Building
Lionel Nadau, CEMAT - Instituto Superior Técnico

Numerical simulations of shear dependent viscoelastic flows with a combined finite element - finite volume method

In this talk we present a hybrid combined finite element - finite volume method that has been developed for the numerical simulation of shear-dependent viscoelastic flow problems, governed by a generalized Oldroyd-B model with a non-constant viscosity function. The method is applied to the 4:1 planar contraction benchmark problem, to investigate the influence of the viscosity effects on the flow and results are compared with those found in the literature for creeping Oldroyd-B flows, for a range of Weissenberg numbers. The method is also applied to flow in a smooth stenosed channel. It is shown that the qualitative behavior of the flow is influenced by the rheological properties of the fluid, namely its viscoelastic and inertial effects, as well as the shear-thinning viscosity. These results appear in the framework of a preliminary study of the numerical simulation of steady and pulsatile blood flows in two-dimensional stenotic vessels, using this hybrid finite element - finite volume method.

15/11/2005, 17:00 — 18:00 — Room P3.31, Mathematics Building
Euripides Sellountos, University of Patras, Greece

Two meshless methods for solving fluid problems in two dimensions

Two meshless methods, namely the Meshless Local Petrov Galerkin (MLPG) method and the Local Boundary Integral Equation (LBIE) method, for solving two dimensional fluid flow problems are presented. A cloud of distributed points without any connectivity requirement is employed for the approximation of the unknown fluid velocity $u(x)$. In both methodologies the interpolation of $u(x)$ is accomplished with the aid of a Moving Least Squares Approximation scheme. The weak integral formulation of MLPG and LBIE methodologies is presented in detail. The treatment of terms involving possible nonlinearities and time derivatives is explained and the numerical implementation of both techniques is addressed. Some representative examples that demonstrate the potentiality of using the aforementioned meshless methods to flow problems are shown.

09/11/2005, 17:00 — 18:00 — Room P3.10, Mathematics Building
Abdel Artoli, CEMAT - Instituto Superior Técnico, Lisboa

Lattice Boltzmann models for blood flow simulations

We will review the Lattice Boltzmann methods as numerical solvers for the based Boltzmann transport equation used in kinetic theory to describe transport phenomena. Benefits over Navier-Stokes solvers are highlighted. The method is adapted to model steady and unsteady non-newtonian blood flow in benchmarks and realistic arteries. Non-Newtonian blood flow in a tube is investigated using te Carreau-Yasuda model for the shear thinning behavior. Results on velocity and shear stress are presented and compared to the unsteady Newtonian flows. Further comparison with other numerical methods is planned as future work.

07/11/2005, 17:00 — 18:00 — Room P3.10, Mathematics Building
Claude Tadonki, University of Geneva, Switzerland

Integer programming heuristic for the dual power selection problem in wireless network.

We seek an integer programming based heuristic for solving the dual power management problem in wireless sensor networks. For a given network with two possible transmission powers (low and high), the problem is to find a minimum size subset of nodes such that if they are assigned high transmission power while the others are assigned low transmission power, the network will be strongly connected. The main purpose behind this efficient setting is to minimize the total communication power consumption while maintaining the network connectivity. In a theoretical point of view, the problem is known to be difficult to solve exactly. An approach to approximate the solution is to work with a spanning tree of clusters. In our model, a cluster is a strongly connected component in the transmission graph where only the low transmission power is considered. We follow the same approach, and we formulate the node selection problem inside clusters as an integer programming problem which is solved exactly using specialized codes. Experimental results show that our algorithm is efficient both in execution time and solution quality.

29/07/2005, 15:00 — 16:00 — Room P3.31, Mathematics Building
Giovanni Paolo Galdi, University of Pittsburgh, PA, USA

The relation between flow rate and axial pressure gradient for time-periodic Poiseuille flow in a pipe

Consider a fully developed, time-periodic motion of a Navier-Stokes fluid in an infinite straight pipe of constant cross section $\Omega$ (time-periodic Poiseuille flow). In this talk we show that the axial pressure gradient and the flow rate associated to this motion are uniquely connected through a very simple relation involving parameters depending only on $\Omega$ and, therefore, independent of the particular velocity field. One immediate and important consequence of this property is that it allows for a very elementary proof of existence of time-periodic Poiseuille flow under a given flow rate.

08/07/2005, 17:00 — 18:00 — Room P3.31, Mathematics Building
Anne M. Robertson, University of Pittsburgh, USA

On steady flows of viscoelastic fluids in curved pipes

In this talk, we will discuss results for steady, fully developed flows of viscoelastic fluids in curved pipes and contrast this behavior with flows of Newtonian fluids. Following the approach of W. R. Dean and other authors, we have used regular perturbation methods to study flows of viscoelastic fluids in curved pipes. The perturbation parameter is the curvature ratio: the cross sectional radius of the pipe divided by the radius of curvature of the pipe centerline. We have obtained explicit solutions to the perturbation equations at first order for second order fluids and a modified Oldroyd-B fluid. In the absence of inertial effects, flows of Newtonian fluids in curved pipes display a secondary flow, rather a uniaxial flow exists which differs only slightly from the straight pipe solution. In contrast, even in the absence of inertial effects, the class of viscoelastic fluids studied display a secondary motion (see, e.g. Thomas 1963, Bowen et al. 1991, Robertson and Muller 1996). Significantly, for a countable number of combinations of material parameters and Reynolds numbers, there is a loss of uniqueness of the solution to the perturbation equations. For other values of material parameters and Reynolds number, a solution does not even exist. There is a region in parameter space which is free of such singularities. It is interesting that these singularities do not arise when the second normal stress coefficient is zero. This lack of existence to the perturbation equations regardless of the magnitude of the curvature ratio, implies a lack of existence of a solution which is a steady, fully developed perturbation of the straight pipe solution. The implications of this result are under investigation.

(Short Course: Part I).

20/05/2005, 17:00 — 18:00 — Room P3.31, Mathematics Building
Manuel Guerra, Departamento de Matemática, ISEG/UTL

Generalized synthesis and computation of nonclosed accessible sets in control theory

Accessible sets are important invariants of control systems, and computation of boundary points of accessible sets plays important roles in the solution of optimal control problems, and other classical problems like motion planning or trajectory tracking. Pontryagin's Maximum Principle is an important tool to compute boundary points of accessible sets. However, it's usefulness is severely reduced when the accessible set is not closed. We discuss an approach to overcome these difficulties in the class of control-affine systems and give a geometric characterization of so-called "generalized extremals". The computations required to obtain generalized extremals by this method are considerably simplified with respect to alternative approaches.

18/05/2005, 16:00 — 17:00 — Room P3.10, Mathematics Building
Pedro Serranho, University of Goettingen (Germany) and CEMAT - IST(Portugal)

Um método híbrido para a reconstrução de obstáculos 2D

We are interested in solving the inverse problem of acoustic wave scattering for the position and the shape of sound hard obstacles in two-dimensions, given one incident field and the corresponding far-field pattern of the scattered wave. It can be seen as a hybrid between a regularized Newton method applied to a nonlinear operator equation with the operator that maps the unknown boundary on the solution of the direct problem and a decomposition method, in the spirit of the Kirsch and Kress method. The method does not require a forward solver for each Newton step. Numerical results show the feasibility of the method.

09/03/2005, 17:00 — 18:00 — Room P3.10, Mathematics Building
Tomás Bodnár, Czech Technical University of Prague and CEMAT/IST

Selected applications of incompressible viscous flows models

Some of the applications of the Navier-Stokes and consequently also Reynolds-Averaged Navier-Stokes models and their simplifications will be addressed. The range of applications will be divided into three main parts: 1) Atmospheric Boundary Layer flows Several more or less complex test cases have been solved. Starting from wind-tunnel tests and validations the flow complexity increase up to a solution of real-sized problems. Together with the flow solution, the problem of pollution dispersion will be addressed. Examples of computations including some environmental and industrial applications will be discussed. 2) Internal turbulent flows The mathematical and numerical model of 3D curved pipe flow problem will be discussed. Advanced turbulence model will be described together with the details of the finite-volume model implementation. 3) Variable-density and free-surface flows A specific class of the variable-density incompressible flows will be discussed. An example of model computation will be given to show the resolution of water/air interface in the square-sectioned pipe elbow by variable-density approach.

03/03/2005, 17:00 — 18:00 — Room P3.10, Mathematics Building
Helcio R. B. Orlande e Paulo Couto, Universidade Federal do Rio de Janeiro

Identificação de parâmetros e funções em transferência de calor e massa

Problemas diretos em transferência de calor/massa têm por objetivo a determinação do campo de temperaturas/concentrações, sendo conhecidas a geometria em questão, as propriedades físicas que aparecem na formulação do problema, bem como as condições de contorno e condição inicial. Por outro lado, os problemas inversos em transferência de calor/massa visam determinar uma ou mais de uma das seguintes características: (i) propriedades físicas, por exemplo, condutividade térmica e coeficiente de difusão molecular; (ii) condições de contorno, tal como o fluxo de calor em uma superfície na qual sensores não podem ser localizados, por exemplo, em um pistão em motor de combustão interna; (iii) a geometria do problema, por exemplo, na otimização de perfis aerodinâmicos ou aletas; e (iv) condição inicial. Para tanto, é necessário conhecer-se medidas experimentais de temperatura/concentração. Sendo assim, os problemas inversos podem ser genericamente definidos como aqueles que têm por objetivo determinar "causas", à partir de medidas experimentais dos "efeitos" (por exemplo, temperatura). Na verdade, o estudo de problemas inversos faz parte de um novo paradigma de pesquisa em engenharia, onde as simulações computacional e experimental não são realizadas isoladamente, mas sim de forma interativa, a fim de que o máximo de informação sobre o problema físico em questão seja obtido com as duas análises. Neste seminário são apresentados os resultados obtidos para a estimativa do fluxo de calor proveniente da combustão em um maçarico oxi-acetilênico. Apresenta-se a solução deste problema inverso através de técnicas de estimativa de parâmetros e de estimativa de funções, utilizando-se medidas experimentais de temperatura obtidas em um material com propriedades termofísicas conhecidas. Para a obtenção das propriedades do material, requeridas na formulação matemática do problema, foi utilizado um equipamento baseado no método Flash. Detalhes deste método também são apresentados no seminário.

25/02/2005, 17:00 — 18:00 — Room P4.35, Mathematics Building
Andreas Kirsch, University of Karlsruhe

The Factorization Method in Inverse Problems

The Factorization method is a relatively new method to characterize the support of the shape of a “defect” for a number of inverse problems. In this talk we explain the method for the problem to determine a crack or an inclusion for an elastostatic problem from the knowledge of the tractions and displacements on the surface of the body.

25/02/2005, 16:00 — 17:00 — Room P4.35, Mathematics Building
K. R. Rajagopal, Texas A&M University, College Station, USA

The Undiscovered Stokes: the Navier-Stokes equations and beyond

Stokes has made important contributions to several aspects of mathematical physics. In his seminal paper in 1845, Stokes in addition to deriving the equations which bear the name Navier-Stokes Equations, suggested generalizations that have not been studied with the detail that they deserve. He also had a lot to say about the nature of boundary conditions between two continua. This early work of Stokes has profound implications in the developments of constitutive theories for both constrained and unconstrained continua. These issues will be discussed.
Apoio: CEMAT/IST e POCTI/MAT/41898/2001.

09/02/2005, 17:00 — 16:00 — Room P3.10, Mathematics Building
Hugo Beirão da Veiga, Università di Pisa, Italy

Soluções periódicas das equações de Navier-Stokes em tubos infinitos. Parte I

Considere-se um tubo infinito, cilíndrico, $\Lambda=\Omega \times \mathbb{R}$ com secção $\Omega \subset\mathbb{R}^n$ regular e arbitrária. Seja dada uma função $g(t)$, periódica de período $T$ e arbitrária. Demonstra-se a existência de uma e uma só solução das equações de Navier-Stokes (com a condição de aderência na fronteira) cuja velocidade é paralela ao eixo do cilindro e cujo fluxo total é exactamente igual a $g(t)$. No primeiro seminário introduz-se o problema físico, deduzem-se as equaçõoes que o descrevem e enunciam-se os principais resultados. No segundo seminário dar-se-ão alguns detalhes da demonstração. Resultados e demonstrações encontram-se num artigo em publicação nos Arch. Rat. Mech. Analysis.

Um preprint está disponível no CMAF, preprint UL-MAT-2004-21.

Seminrio em colaboraço com o CMAF.
Parte II : Quinta-Feira, 10 Fevereiro, 14h15 - CMAF.

02/02/2005, 16:15 — 17:15 — Room P3.10, Mathematics Building
Neville J. Ford, University College Chester, UK

An algorithm for detecting small solutions for delay differential equations

This talk begins by introducing delay differential equations and (in particular) the need to detect small solutions. We discuss how small solutions influence the behaviour and analysis of solutions of the underlying equation and explain why they are difficult to detect. Next the talk focuses on a methodology that can be proved to show the existence of small solutions for certain linear equations and we show how the method can be automated using a computer. The talk concludes with examples of how the automated system can detect small solutions in the equations, and details of how the method can be extended to a wider class of equations.

02/02/2005, 15:00 — 16:00 — Room P3.10, Mathematics Building
Hedia Chaker, LAMSIN-ENIT, Tunisia

Modélisation d'injection de bulles dans un lac eutrophe

Apoio: Projecto Bilateral Portugal-Tunísia, CEMAT - LAMSIN / ENIT.

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