Topology Optimization for Loads with Multiple Points of Application

The optimal design for loads with multiple points of application is herein investigated by using a formulation of displacement-constrained minimum volume topology optimization. For each one of the several points in which a moving force may be applied, a static load case is introduced, and a local enforcement is implemented to control the relevant displacement. Inspired by some recent contributions in stress-based topology optimization of large-scale structures, an Augmented Lagrangian approach, is adopted to handle efficiently the arising multi-constrained problem, in conjunction with mathematical programming. The results of some numerical simulations are shown to comment on optimal shapes for loads with multiple points of application, as compared to classical solutions for fixed loads

Numerical Simulation of Heat Transport in Dispersed Gas-Liquid Two-Phase Flow using a Front Tracking Approach

In this paper a simulation model is presented for the Direct Numerical Simulation (DNS) of heat transport in dispersed gas-liquid two-phase flow using the Front Tracking (FT) approach. Our model extends the FT model developed by van Sint Annaland et al. (2006) to non-isothermal conditions. In FT an unstructured dynamic mesh is used to represent and track the interface explicitly by a number of interconnected marker points. The Lagrangian representation of the interface avoids the necessity to reconstruct the interface from the local distribution of the fractions of the phases and, moreover, allows a direct and accurate calculation of the surface tension force circumventing the (problematic) computation of the interface curvature. The extended model is applied to predict the heat exchange rate between the liquid and a hot wall kept at a fixed temperature. It is found that the wall-to-liquid heat transfer coefficient exhibits a maximum in the vicinity of the bubble that can be attributed to the locally decreased thickness of the thermal boundary layer

Direct numerical simulation of heat transport in dispersed gas-liquid two-phase flow using a front tracking approach

In this paper a simulation model is presented for the Direct Numerical Simulation (DNS) of heat transport in dispersed gas-liquid two-phase flow using the Front Tracking (FT) approach. Our model extends the FT model developed by van Sint Annaland et al. (2006) to non-isothermal conditions. In FT an unstructured dynamic mesh is used to represent and track the interface explicitly by a number of interconnected marker points. The Lagrangian representation of the interface avoids the necessity to reconstruct the interface from the local distribution of the fractions of the phases and, moreover, allows a direct and accurate calculation of the surface tension force circumventing the (problematic) computation of the interface curvature. The extended model is applied to predict the heat exchange rate between the liquid and a hot wall kept at a fixed temperature. It is found that the wall-to-liquid heat transfer coefficient exhibits a maximum in the vicinity of the bubble that can be attributed to the locally decreased thickness of the thermal boundary layer

WALL SHEAR STRESS TOPOLOGICAL SKELETON IDENTIFICATION IN CARDIOVASCULAR FLOWS: A PRACTICAL APPROACH

The observed co-localization of “disturbed” hemodynamics and atherosclerotic lesion prevalence has led to the identification of low and oscillatory Wall Shear Stress (WSS) as a biomechanical localizing factor for vascular dysfunction. However, recent evidences have underlined how consideration of only “low and oscillatory” WSS may oversimplify the complex hemodynamic milieu to which the endothelium is exposed. In this context, recent studies have highlighted the relevance of WSS fixed points, and the stable and unstable manifolds that connect them. These WSS topological features have a strong link with flow features like flow stagnation, separation, and recirculation, which are usually classified as “disturbed” flow. Technically, a fixed point of a vector field is a point where the vector field vanishes, while unstable/stable vector field manifolds identify contraction/expansion regions linking the fixed points. The set of fixed points and their connections form the topological skeleton of a vector field. The presence of WSS fixed points and of WSS contraction/expansion regions, highlighted by WSS manifolds, might induce focal vascular responses relevant for, e.g., early atherosclerosis, or, aneurysm rupture. For these reasons, the topological skeleton analysis of the WSS vector field is of great interest and motivates the study present herein. Lagrangian techniques have been recently proposed to identify WSS manifolds but have certain practical limitations. A Eulerian approach has also been suggested, but only for 2D analytical fields. Here we propose and demonstrate the use of a simple Eulerian approach for identifying WSS topological skeleton on 3D surfaces

An Analytical Approach to Inhomogeneous Structure Formation

We develop an analytical formalism that is suitable for studying inhomogeneous structure formation, by studying the joint statistics of dark matter halos forming at two points. Extending the Bond et al. (1991) derivation of the mass function of virialized halos, based on excursion sets, we derive an approximate analytical expression for the ``bivariate'' mass function of halos forming at two redshifts and separated by a fixed comoving Lagrangian distance. Our approach also leads to a self-consistent expression for the nonlinear biasing and correlation function of halos, generalizing a number of previous results including those by Kaiser (1984) and Mo & White (1996). We compare our approximate solutions to exact numerical results within the excursion-set framework and find them to be consistent to within 2% over a wide range of parameters. Our formalism can be used to study various feedback effects during galaxy formation analytically, as well as to simply construct observable quantities dependent on the spatial distribution of objects. A code that implements our method is publicly available at http://www.arcetri.astro.it/~evan/GeminiComment: 41 Pages, 11 figures, published in ApJ, 571, 585. Reference added, Figure 2 axis relabele

The Lax-Oleinik semi-group: a Hamiltonian point of view

The Weak KAM theory was developed by Fathi in order to study the dynamics of convex Hamiltonian systems. It somehow makes a bridge between viscosity solutions of the Hamilton-Jacobi equation and Mather invariant sets of Hamiltonian systems, although this was fully understood only a posteriori. These theories converge under the hypothesis of convexity, and the richness of applications mostly comes from this remarkable convergence. In the present course, we provide an elementary exposition of some of the basic concepts of weak KAM theory. In a companion lecture, Albert Fathi exposes the aspects of his theory which are more directly related to viscosity solutions. Here on the contrary, we focus on dynamical applications, even if we also discuss some viscosity aspects to underline the connections with Fathi's lecture. The fundamental reference on Weak KAM theory is the still unpublished book of Albert Fathi \textit{Weak KAM theorem in Lagrangian dynamics}. Although we do not offer new results, our exposition is original in several aspects. We only work with the Hamiltonian and do not rely on the Lagrangian, even if some proofs are directly inspired from the classical Lagrangian proofs. This approach is made easier by the choice of a somewhat specific setting. We work on \Rm^d and make uniform hypotheses on the Hamiltonian. This allows us to replace some compactness arguments by explicit estimates. For the most interesting dynamical applications however, the compactness of the configuration space remains a useful hypothesis and we retrieve it by considering periodic (in space) Hamiltonians. Our exposition is centered on the Cauchy problem for the Hamilton-Jacobi equation and the Lax-Oleinik evolution operators associated to it. Dynamical applications are reached by considering fixed points of these evolution operators, the Weak KAM solutions. The evolution operators can also be used for their regularizing properties, this opens a second way to dynamical applications.Comment: Proceedings of the Royal Society of Edinburgh, Section: A Mathematics (2012) to appea