Seminário Matemática Aplicada e Análise Numérica 

An asymptotic analysis of solutions to elliptic problems in domains with rapidly oscillating boundaries (rough and damaged surfaces) will be presented. The construction of several asymptotic terms, including boundary layers, permit for modelling the problem by a much simpler problem, solution of which, nevertheless, gives the two-term asymptotics of the original solution. Two approaches for this type of modelling will be discussed. For the sake of simplicity, boundary value problems for a scalar equation will be considered, although the investigation is directed to the elasticity theory.

Three-dimensional ($3$D) simulations of blood flow provide detailed information important for the comprehension of the cardiovascular system. These consist in coupling the $3$D Navier-Stokes equations with a model for the vessel wall structure. Due to the computational cost of fully $3$D fluid-structure interaction (FSI) problems and to the complexity of the cardiovascular system, these models can be applied only to truncated regions of interest. Following the geometrical multiscale modelling of the cardiovascular system, the remaining parts can be accounting resorting to reduced models, one-dimensional ($1$D) or zero-dimensional ($0$D). In particular, the $1$D models describe very well the wave propagation nature of blood flow and coupled with the $3$D model act as proper absorbing boundary conditions. We address the coupling of $3$D and $1$D FSI models. The difficulty of this coupling lays in putting together such different models, namely it is not evident which conditions to impose at the coupling interface. We study the stability of such coupling, bringing forth proper matching conditions between the models. Different coupling strategies are discussed and several numerical results presented.

Inverse problems for the one-speed equation, transport theory and tomography, simultaneous reconstruction of the absortion and scattering coefficients, source detector methodology.

Variational formulation, transformation of the problem to the self-adjoint form, methods treatement of the angular and spatial dependence, the concept of natural basis for the transport problem, domain partition consistent with the propagation of radiation.

Functional formulation, related Hilbert and Sobolev's spaces, operators, trace, albedo, compactness with the scattering kernel, related eigenvalue problems.

Linear Boltzmann equation, neutral particles transport, some physical concepts, scattering, absortion and fission, particles position, direction of propagation and energy, boundaries in the phase space, boundary values problems.

The solution of realistic problems in Hemodynamics generally involves the solution of very large systems of linear and non-linear equations. These systems arise from the discretization of non-linear partial differential equations, coupling the conservation of mass and momentum with some constitutive equation expressing the relation between stresses and the history of deformation. Analytical solutions are usually not available, either because of the complex geometries of blood vessels or because of the complexity of the constitutive equations. Even considering a simple geometry like a straight tube, exact solutions are only known in the Newtonian case or for very simple shear-dependent fluids like the power law fluids. We present a general method for obtaining solutions for the flow of generalized Newtonian fluids in straigth tubes (2D and 3D), with arbitrary precision, and use these solutions to create benchmarks for numerical solvers. Some numerical examples are shown for the Carreau-Yasuda viscosity model, analyzing the performance of two numerical solvers (Adina and FreeFem++), using different finite element spaces.

Leukocyte recruitment towards and rolling at inflammatory regions on activated endothelial cells has been a subject of intensive experimental and computational investigations. The mechanism of rolling is commonly attributed to active formation and breakage of bonds between selectin adhesion molecules and their ligands. The induced cells first rotate with an angular velocity of about 25 sec-1, before they migrate towards the nearest wall where they roll, deform, adhere and migrate through the endothelial cells. The rolling process may take about 20 seconds during which leukocytes are exposed to chemoattractants. The rolling velocity can be as slow as 2.73 mm/s and as large as 60 mm/s in a typical venule in the rat muscle of 23.5 mm in diameter. Aiming at a better understanding of the complexity of the rolling process and of the influence of hydrodynamic forces, we present here updated experimental and simulation results conducted in collaboration with A. S. Silva and C. Saldanha of FML.

We consider a second order elliptic problem posed on an interval or on a connected polygonal domain in a two dimensional space. We introduce an admissible mesh in the sense of [1]. The convergence order of the finite volume approximate solution (called Basic Finite Volume Solution) is in general $O(h)$ in both discrete norms $L^2$ and $H^1$. We suggest a technique, based on the so-called Fox's Difference Correction [2], which allows us to obtain a new Finite Volume Approximation of order $O(h^{\alpha })$, where $\alpha$ equal to $2$ or $\frac{3}{2}$. In addition, this new Finite Volume Approximation can be computed using the same matrix used to compute the basic finite volume solution. The computational cost is comparable to that of the new Finite Volume Approximation. If the domain problem is an interval or a rectangle, we obtain finite volume approximations of arbitrary order and these approximations can be computed using the same matrix used to compute the Basic Finite Volume Solution. We give an application of our approach to improve the convergence order of finite element solutions defined in non uniform meshes.

References

1. R. Eymard, T. Gallouet and R. Herbin: Finite Volume Methods. Handbook of Numerical Analysis. P. G. Ciarlet and J. L. Lions (eds.), vol. VII, 723-1020, 2000.
2. L. Fox: Some Improvements in the Use of Relaxation Methods for the Solution of Ordinary and Partial Defferential Equations. Proc. Roy. Soc. Lon Ser. A, 190, 31-59, 1947.

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.