Raúl A. Feijóo

An iterative algorithm for limit analysis with nonlinear yield functions

Abstract A mathematical programming algorithm is proposed for the general limit analysis problem. Plastic behavior is described by a set of linear or nonlinear yield functions. Abstract formulations of limit analysis are first considered and a Quasi-Newton strategy for solving the optimality conditions is then sketched. The structure of the problem arising from a finite element discretization is taken into account in order that the algorithm should be able to solve large scale problems.

THE TOPOLOGICAL DERIVATIVE FOR THE POISSON'S PROBLEM

The Topological Derivative has been recognized as a powerful tool in obtaining the optimal topology of several engineering problems. This derivative provides the sensitivity of a problem when a small hole is created at each point of the domain under consideration. In the present work the Topological Derivative for Poissons problem is calculated using two different approaches: the Domain Truncation Method and a new method based on Shape Sensitivity Analysis concepts. By comparing both approaches it will be shown that the novel approach, which we call Topological-Shape Sensitivity Method, leads to a simpler and more general methodology. To point out the general applicability of this new methodology, the most general set of boundary conditions for Poissons problem, Dirichlet, Neumann (both homogeneous and nonhomogeneous) and Robin boundary conditions, is considered. Finally, a comparative analysis of these two methodologies will also show that the Topological-Shape Sensitivity Method has an additional advantage of being easily extended to other types of problems.

Numerical Experiments in Complex Hæmodynamic Flows. Non-Newtonian Effects

Numerical experiments for non-trivial flows, close to realistic situations in hæmodynamics, are described and interpreted. Two geometries have been selected: an axisymmetric corrugated tube (with periodic boundary conditions) and a 3D bifurcation with an obstructed end (anastomosis). Results concern sensitivity of errors associated to the time-step size and mesh refinement, but essentially consist of the quantitative estimation of non-Newtonian effects based on Cassons rheological model, treated in retarded form. The time-step lag of such effects is the main reason for evaluating the sensitivity of errors. Due to the high computational cost characterizing the problems to be faced, we expect that the present results will be useful when real geometries should be modeled. The main conclusions are that non-Newtonian effects may be relevant (especially for secondary flows) and that, in most cases, for the same level of errors the use of Cassons law does not generate excessive additional computational costs. Thus, within this strategy, the user can accurately solve the problem using this rheological model without having to worry if the non-Newtonian effects are important or not.

A variational principle for the laplace's operator with application in the torsion of composite rods

Abstract The Ritz method with relaxed coordinate functions is used here to establish a minimizing sequence for an extended functional. The integrand of the functional involves discontinuous functions, which implies that the gradient of the related extremal curve or surface also admits discontinuities. It is discussed a systematic way to form an extended functional allowing the minimizing sequence to approach the exact solution displaying the discontinuities in the gradient of the extremal surface. The extra terms added to the classical functional are shown to be related to the Erdman-Weierstrass corner conditions. Finally the method is applied to the torsion of a composite rod, and the results compared with a solution obtained through the finite elements method.

On the search of more stable second-order lattice-Boltzmann schemes in confined flows

The von Neumann linear analysis, restricted by a heuristic selection of wave-number vectors was applied to the search of explicit lattice Boltzmann schemes which exhibit more stability than existing methods. The relative stability of the family members of quasi-incompressible collision kernels, for the Navier-Stokes equations in confined flows, was analyzed. The linear stability analysis was simplified by assuming a uniform velocity level over the whole domain, where only the wave numbers of the first harmonic normal to the flow direction were permitted. A singular equilibrium function that maximizes the critical velocity level was identified, which was afterwards tested in particular cases of confined flows of interest, validating the resulting procedure.

Configurational Derivative as a Tool for Image Segmentation

The introduction to medicine of techniques coming from areas like Computational Fluid Dynamics, Structural Analysis, and Inverse Problems, made the use of imaging data such us Computed Tomography (CT), Magnetic Resonance Imaging (MRI), Single Photon Emission Tomography (SPECT), Positron Emission Tomography (PET) and Ultrasound (US) mandatory in order to apply this techniques to patient specific data. The process of identification of different tissues and organs, called segmentation, is a maior concern in this analysis. This process can be tedious and time consuming when done by hand, so its been an early concern in image processing to automatize it. Many contributions have been made to the area since the introduction of the Mumford and Shah functional. This functional is endowed to quantify the cost associated to a specific segmentation.

An efficient method for the numerical solution of blood flow in 3D bifurcated regions

From the point of view of the potential clinical use of computational hemodynamic, it is mandatory to get the computational time of simulation each time closer to real clinical needs. Spending hours and even days to solve accurately one single cardiac cycle of the whole cardiovascular system is unfeasible on daily practice. In this sense, in this work we study the transversally enriched pipe element method (TEPEM) as an effective alternative to solve the Navier-Stokes equations in bifurcated domains with enough accuracy to provide clinically relevant information but at a significantly reduced time.

RVE-based multiscale modeling for the Navier-Stokes equations: linking continuum and Lattice-Boltzmann models

Abstract. This work addresses the multiscale modeling of fluid flow in highly involved media based on the concept of Representative Volume Element. Between scales we consider as basic principles for downscaling information the conservation of the velocity vector field and the conservation of the strain rate tensor field. In this context we formulate (i) the problem to be solved in the representative volume element (at fine scale), and (ii) the homogenization formulae for the force-like and the stress-like fields (at coarse scale) which stand for the procedure of upscaling data. Flow equations corresponding to simple fluids are considered at the coarse scale, and the classical Navier-Stokes equations are approximated in the fine scale by using the Lattice-Boltzmann method. Examples of application of flow in permeable media with small obstacles are presented to show the potentialities of the present approach.

Image Restoration via Topological Derivative

The problem of image restoration has deserved considerable attention in resent years. For the visual analysis of images, clarity and visibility of details are important factors, but for advanced processing, a high signal-to-noise ratio (SNR) is essential, as further processing steps (such as segmentation and classification) are sensitive to noise. Though the years, different techniques have been studied to improve the SNR or a degraded image. Techniques based on post-processing have the advantage of not affecting the acquisition process (Gonzalez & Woods (2001); Jain (1989); Weickert (1995)). More recently, the work of Tschumperle (2006) has explored the more extensive use of curve-preserving PDE’s for restoration of images. The calculation of mean intensities over neighboring pixels, equivalent to isotropic diffusion, considerably increases the SNR, but degrades the quality of image features (edges, lines and dots). This effect can be reduced with non-linear filters. The median filter has the characteristic of maintaining these features, but details are lost, degrading the image resolution. Perhaps the most popular technique introduced in the last couple of years is anisotropic diffusion, initially proposed by Perona and Malik (Black et al. (1998); Perona & Malik (1990)).

Topological Derivative Applied to Image Enhancement

Medical imaging techniques like Computed Tomography (CT), Magnetic Resonance Imaging (MRI), Single Photon Emission Tomography (SPECT), Positron Emission Tomography (PET) and Ultrasound (US) have introduced a formidably powerful tool in medicine. The poor quality or high signal-to-noise ratio (SNR) of this kind of images is maior limitation for image analysis. For this reason image enhancement takes an important roll in the segmentation and analysis process. Although imaging techniques (e.g., contrast agents, biological markers) should improve the image quality this is not always enough to give a good result. Much effort has been put into the area of image enhancement. Our aim in this paper is to present a method for medical image enhancement based on the well established concept of topological sensitivity analysis and borrowing image processing techniques like anisotropic diffusion. More specifically, an appropriate functional F is associated to the image indicating the cost endowed to an specific image. Let us assume that the image being segmented is characterized by a scalar field u representing the image data. Then, the segmentation algorithm can be cast as: given the evolution equation,