Applied Mathematics and Numerical Analysis Seminar 

Turbulence is ubiquitous in nature and central to many applications important to our life. Obtaining an accurate prediction of turbulent flow is a central difficulty in such diverse problems as global change estimation, improving the energy efficiency of engines, controlling dispersal of contaminants and designing biomedical devices. It is absolutely fundamental to understanding physical processes of geophysics, combustion, forces of fluids upon elastic bodies, drag, lift and mixing. In these lectures we introduce one of the most promising numerical methods for the study of turbulent flows: Large Eddy Simulation (LES). LES seeks to calculate the large, energetic structures (the large eddies) in a turbulent flow. The aim of LES is to do this with complexity independent of the Reynolds number and dependent only on the resolution sought. The first lecture is devoted to an introduction to the problem of modeling and to the analysis of “eddy viscosity models” originated by the work of Smagorinsky and Ladyzhenskaya. In the second lecture we present advanced methods that are based on wavenumber asymptotics. Results of numerical experiments are also shown. In the third lecture we make an overview of recent advances as: filtering on bounded domains, near wall modeling, and variational multiscale methods.

Defeitos em materiais compostos de microfibras são muitas vezes originários da concentração e orientação descontrolada durante o processo de fabricação dos mesmos. Estes defeitos podem reduzir significantemente a funcionalidade de tais materiais. Ainda, em muitas aplicações é extremamente desejável se obter materiais com dependência espacial de suas propriedades físicas, ou seja, obter-se materiais fortemente não-isotrópicos. Seria, portanto, interessante efetuar o processo de fabricação de tal forma que a concentração e orientação local das fibras pudesse ser controlada. Durante o processo de solidificação é importante se compreender o processo de formação da fase sólida. O acúmulo de sólido efetivamente reduz e deforma a seção transversal do molde e causa significantes variações de pressão e tensões cisalhantes. Este processo não pode ser efetivamente controlado no caso de forte transferência de calor, exceto se influenciado por uma força externa, a qual pode ser uma força eletromagnética que é criada em um fluido eletricamente condutor quando um campo magnético ou elétrico é aplicado. Desta forma, se um campo magnético externo é aplicado, o escoamento dentro do molde irá se modificar e a interface sólido-líquido pode ser manipulada não-intrusivamente. O problema direto do escoamento em regime transiente bidimensinonal, sujeito à forças eletromagnéticas é subdividido em dois problemas: o primeiro envolvendo somente a eletrohidrodinâmica (EHD), ou seja, o estudo de escoamentos contendo partículas carregadas sob a influência de um campo elétrico externo e com um campo magnético desprezível e, o segundo envolvendo somente a magnetohidrodinâmica (MHD), ou seja, o estudo de escoamentos influenciados por campos magnéticos externos sem partículas eletricamente carregadas. A metodologia de solução consiste em se transformar as equações dos modelos EHD e MHD para coordenadas generalizadas, as quais são discretizadas usando-se o método dos volumes finitos. O processo de solidificação / fusão é abordando através do método da entalpia. O problema de acoplamento pressão-velocidade é abordado pelo método SIMPLEC e as funções de interpolação para os termos convectivos são baseadas no método WUDS. O problema inverso é abordado através da utilização de um software de otimização híbrido, envolvendo vários métodos de otimização. A principal vantagem de um software híbrido de otimização reside no fato do mesmo poder escapar de mínimos locais. Desta forma, métodos estocásticos podem ser utilizados no início do processo de otimização de forma a localizar a região onde se encontra um possível mínimo global e, a partir daí, métodos determinísticos são utilizados de forma a encontrar rapidamente o valor do mínimo, dentro da região delimitada pelos métodos estocásticos.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.