Applied Mathematics and Numerical Analysis Seminar 

It is general consensus that most of the observed turbulence behavior in the continuum regime can be explained using the Navier-Stokes equation. It is also well known that the Boltzmann equation forms a basis of the Navier-Stokes equation. Yet, to date, little effort has been made to describe or model turbulence using the Boltzmann equation. In this talk, I will present results from direct numerical simulations (DNS) of isotropic and homogeneous turbulence performed with Boltzmann equation using the lattice Boltzmann method (LBM). Comparisons with Navier-Stokes DNS show excellent agreement. Then, I proceed to demonstrate how the Boltzmann equation can be filtered or averaged for developing closure models. It will be shown that straight-forward averaging or filtering the Boltzmann equation will not produce the desired results. A crucial transformation of the dependent and independent variables in the velocity phase-space must precede the averaging/filtering process. The similarities and difference between closure modeling in Navier-Stokes and Boltzmann contexts will also be described.

A two dimensional convection-dominated elliptic problem with discontinuous coefficients is considered. The problem is discretized using an inverse-monotone finite volume method on piecewise uniform (Shiskin) meshes, condensed near the boundary and the interior layers. A first-order global pointwise convergence uniform with respect to the perturbation parameter is established. Numerical experiments that support the theoretical results are presented. An inverse-monotone collocated finite volume method is applied for the numerical approximation of the generalized Navier-Stokes problem modeling unsteady non-Newtonian blood flow. The consistent splitting method is applied for time discretization. A Carreau-Yasuda model is used to describe the shear-thinning behavior of blood.

The Local Boundary Integral Equation (LBIE) method, for solving two dimensional incompressible Navier-Stokes equations is presented. A cloud of distributed points without any connectivity requirement is employed for the approximation of the unknown fluid velocity $u(x)$. The interpolation of $u(x)$ is accomplished with the aid of a Moving Least Squares Approximation scheme. The weak integral formulation of LBIE methodology which involves the velocity-vorticity scheme is presented in details. The treatment of terms involving possible non-linearities and time derivatives is explained and the numerical implementation is addressed. Some representative fluid flow examples that demonstrate the accuracy and efficiency of the aforementioned meshless method are solved and the numerical results are discussed.

In our talk we discuss some basic issues of the theory of nonlinear boundary value problems (BVP), such as the existence of a solution, multiplicity of solutions and properties of solutions. In the first part we consider the second order BVP. Brief historical report and overview of the method of a priori bounds and the method of the upper and lower solutions (functions) are given. We devote special attention to applicability of the second method to problems which have oscillatory solutions. Two alternative approaches are discussed, the first one, based on the analysis of a phase plane of similar autonomous equations and then generalizing the results to non-autonomous equations; and the second one, based on the so called quasi-linearization of an equation and distinguishing solutions by their types. Examples and applications are provided. In the second part we pass to the third and fourth order equations. We start with fundamentals of the linear theory and introduce some basic concepts, such as classes of equations, oscillation, conjugate points. Then we pass to the existence and multiplicity results. The existence results are formulated in terms of the upper and lower solutions (functions), where the boundary conditions may be essentially nonlinear (fully nonlinear, in terminology of some authors). Multiplicity results are provided of the similar nature, as in the case of the second order equations.

In a first case, the energy is provided by propagating elastic waves through visco-poroelastic layers. To investigate how seismic waves interact at the interface between sediment layers we introduce Biot theory of poroelasticity. On the way we clarify the long standing problem of reflection-transmission coefficients at the interface of poroelastic materials with applications to oil exploration and glaciology. In a second case, energy is provided by heat source at the boundary. To investigate modern solutions for orthopaedic surgeries, we look at heat dissipation during bone cutting. Finally, we will suggest use of poroelasticity in modelling blood flow in micro-vessels.

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.