Flexible shape extraction for micro/nano scale structured surfaces.

Surface feature is the one of the most important factors affecting the functionality and reliability of micro scale patterned surfaces. For micro scale patterned surface characterisation, it’s important to extract the surface feature effectively and accurately. The active contours, known as “snakes”, have been successfully used to segment, match and track the objects of interest. The active contours have been applied to facial boundary detection, medical image processing, motion correction, etc. In this paper, surface feature extraction techniques based on active contours have been investigated. Parametric active contour models and geometric active contour models have been presented. Also, a group of examples has been selected here to demonstrate the feasibility and applicability of the surface pattern extraction techniques based on active contours. At last, experimental results will be given and discussed

Non-negative mixed finite element formulations for a tensorial diffusion equation

We consider the tensorial diffusion equation, and address the discrete maximum-minimum principle of mixed finite element formulations. In particular, we address non-negative solutions (which is a special case of the maximum-minimum principle) of mixed finite element formulations. The discrete maximum-minimum principle is the discrete version of the maximum-minimum principle. In this paper we present two non-negative mixed finite element formulations for tensorial diffusion equations based on constrained optimization techniques (in particular, quadratic programming). These proposed mixed formulations produce non-negative numerical solutions on arbitrary meshes for low-order (i.e., linear, bilinear and trilinear) finite elements. The first formulation is based on the Raviart-Thomas spaces, and is obtained by adding a non-negative constraint to the variational statement of the Raviart-Thomas formulation. The second non-negative formulation based on the variational multiscale formulation. For the former formulation we comment on the affect of adding the non-negative constraint on the local mass balance property of the Raviart-Thomas formulation. We also study the performance of the active set strategy for solving the resulting constrained optimization problems. The overall performance of the proposed formulation is illustrated on three canonical test problems.Comment: 40 pages using amsart style file, and 15 figure

Development of an unstructured solution adaptive method for the quasi-three-dimensional Euler and Navier-Stokes equations

A general solution adaptive scheme based on a remeshing technique is developed for solving the two-dimensional and quasi-three-dimensional Euler and Favre-averaged Navier-Stokes equations. The numerical scheme is formulated on an unstructured triangular mesh utilizing an edge-based pointer system which defines the edge connectivity of the mesh structure. Jameson's four-stage hybrid Runge-Kutta scheme is used to march the solution in time. The convergence rate is enhanced through the use of local time stepping and implicit residual averaging. As the solution evolves, the mesh is regenerated adaptively using flow field information. Mesh adaptation parameters are evaluated such that an estimated local numerical error is equally distributed over the whole domain. For inviscid flows, the present approach generates a complete unstructured triangular mesh using the advancing front method. For turbulent flows, the approach combines a local highly stretched structured triangular mesh in the boundary layer region with an unstructured mesh in the remaining regions to efficiently resolve the important flow features. One-equation and two-equation turbulence models are incorporated into the present unstructured approach. Results are presented for a wide range of flow problems including two-dimensional multi-element airfoils, two-dimensional cascades, and quasi-three-dimensional cascades. This approach is shown to gain flow resolution in the refined regions while achieving a great reduction in the computational effort and storage requirements since solution points are not wasted in regions where they are not required

Relaminarisation of Re_{\tau} = 100 channel flow with globally stabilising linear feedback control

The problems of nonlinearity and high dimension have so far prevented a complete solution of the control of turbulent flow. Addressing the problem of nonlinearity, we propose a flow control strategy which ensures that the energy of any perturbation to the target profile decays monotonically. The controller's estimate of the flow state is similarly guaranteed to converge to the true value. We present a one-time off-line synthesis procedure, which generalises to accommodate more restrictive actuation and sensing arrangements, with conditions for existence for the controller given in this case. The control is tested in turbulent channel flow ($Re_\tau=100$) using full-domain sensing and actuation on the wall-normal velocity. Concentrated at the point of maximum inflection in the mean profile, the control directly counters the supply of turbulence energy arising from the interaction of the wall-normal perturbations with the flow shear. It is found that the control is only required for the larger-scale motions, specifically those above the scale of the mean streak spacing. Minimal control effort is required once laminar flow is achieved. The response of the near-wall flow is examined in detail, with particular emphasis on the pressure and wall-normal velocity fields, in the context of Landahl's theory of sheared turbulence

Contour Dynamics Methods

In an early paper on the stability of fluid layers with uniform vorticity Rayleigh (1880) remarks: "... In such cases, the velocity curve is composed of portions of straight lines which meet each other at finite angles. This state of things may be supposed to be slightly disturbed by bending the surfaces of transition, and the determination of the subsequent motion depends upon that of the form of these surfaces. For co retains its constant value throughout each layer unchanged in the absence of friction, and by a well-known theorem the whole motion depends upon [omega]." We can now recognize this as essentially the principal of contour dynamics (CD), where [omega] is the uniform vorticity. The theorem referred to is the Biot-Savart law. Nearly a century later Zabusky et al (1979) presented numerical CD calculations of nonlinear vortex patch evolution. Subsequently, owing to its compact form conferring a deceptive simplicity, CD has become a widely used method for the investigation of two-dimensional rotational flow of an incompressible inviscid fluid. The aim of this article is to survey the development, technical details, and vortex-dynamic applications of the CD method in an effort to assess its impact on our understanding of the mechanics of rotational flow in two dimensions at ultrahigh Reynolds numbers. The study of the dynamics of two- and three-dimensional vortex mechanics by computational methods has been an active research area for more than two decades. Quite apart from many practical applications in the aerodynamics of separated flows, the theoretical and numerical study of vortices in incompressible fluids has been stimulated by the idea that turbulent fluid motion may be viewed as comprising ensembles of more or less coherent laminar vortex structures that interact via relatively simple dynamics and by the appeal of the vorticity equation, which does not contain the fluid pressure. Two-dimensional vortex interactions have been perceived as supposedly relevant to the origins of coherent structures observed experimentally in mixing layers, jets, and wakes, and for models of large-scale atmospheric and oceanic turbulence. Interest has often focused on the limit of infinite Reynolds number, where in the absence of boundaries, the inviscid Euler equations are assumed to properly describe the flow dynamics. The numerous surveys of progress in the study of vorticity and the use of numerical methods applied to vortex mechanics include articles by Saffman & Baker (1979) and Saffman (1981) on inviscid vortex interactions and Aref (1983) on two-dimensional flows. Numerical methods have been surveyed by Chorin (1980), and Leonard (1980, 1985). Caflisch (1988) describes work on the mathematical aspects of the subject. Zabusky (1981), Aref (1983), and Melander et al (1987b) discuss various aspects of CD. The review of Dritschel (1989) gives emphasis to numerical issues in CD and to recent computations with contour surgery. This article is confined to a discussion of vortices on a two-dimensional surface. We generally follow Saffman & Baker (1979) in matters of definition. In two dimensions a vortex sheet is a line of discontinuity in velocity while a vortex jump is a line of discontinuity in vorticity. We shall, however, use filament to denote a two-dimensional ribbon of vorticity surrounded by fluid with vorticity of different magnitude (which may be zero), rather than the more usual three-dimensional idea of a vortex tube. The ambiguity is unfortunate but is already in the literature. Additionally, a vortex patch is a finite, singly connected area of uniform vorticity while a vortex strip is an infinite strip of uniform vorticity with finite thickness, or equivalently, an infinite filament. Contour Dynamics will refer to the numerical solution of initial value problems for piecewise constant vorticity distributions by the Lagrangian method of calculating the evolution of the vorticity jumps. Such flows are often related to corresponding solutions of the Euler equations that are steady in some translating or rotating frame of reference. These solutions will be called vortex equilibria, and the numerical technique for computing their shapes based on CD is often referred to as contour statics. The mathematical foundation for the study of vorticity was laid primarily by the well-known investigations of Helmholtz, Kelvin, J. J. Thomson, Love, and others. In our century, efforts to produce numerical simulations of flows governed by the Euler equations have utilized a variety of Eulerian, Lagrangian, and hybrid methods. Among the former are the class of spectral methods that now comprise the prevailing tool for large-scale two- and three-dimensional calculations (see Hussaini & Zang 1987). The Lagrangian methods for two-dimensional flows have been predominantly vortex tracking techniques based on the Helmholtz vorticity laws. The first initial value calculations were those of Rosenhead (193l) and Westwater (1935) who attempted to calculate vortex sheet evolution by the motion of O(10) point vortices. Subsequent efforts by Moore (1974) (see also Moore 1983, 1985) and others to produce more refined computations for vortex sheets have failed for reasons related to the tendency for initially smooth vortex sheet data to produce singularities (Moore 1979). Discrete vortex methods used to study the nonlinear dynamics of vortex patches and layers have included the evolution of assemblies of point vortices by direct summation (e.g. Acton 1976) and the cloud in cell method (Roberts & Christiansen 1972, Christiansen & Zabusky 1973, Aref & Siggia 1980, 1981). For reviews see Leonard (1980) and Aref (1983). These techniques have often been criticized for their lack of accuracy and numerical convergence and because they may be subject to grid scale dispersion. However, many qualitative vortex phenomena observed in nature and in experiments, such as amalgamation events and others still under active investigation (e.g. filamentation) were first simulated numerically with discrete vortices. The contour dynamics approach is attractive because it appears to allow direct access, at least for small times, to the inviscid dynamics for vorticity distributions smoother than those of either point vortices or vortex sheets, while at the same time enabling the mapping of the two-dimensional Euler equations to a one-dimensional Lagrangian form. In Section 2 we discuss the formulation and numerical implementation of contour dynamics for the Euler equations in two dimensions. Section 3 is concerned with applications to isolated and multiple vortex systems and to vortex layers. An attempt is made to relate this work to calculations of the relevant vortex equilibria and to results obtained with other methods. Axisymmetric contour dynamics and the treatment of the multi-layer model of quasigeostrophic flows are described in Section 4 while Section 5 is devoted to a discussion of the tendency shown by vorticity jumps to undergo the strange and subtle phenomenon of filamentation

Secular diffusion in discrete self-gravitating tepid discs I : analytic solution in the tightly wound limit

The secular evolution of an infinitely thin tepid isolated galactic disc made of a finite number of particles is described using the inhomogeneous Balescu-Lenard equation. Assuming that only tightly wound transient spirals are present in the disc, a WKB approximation provides a simple and tractable quadrature for the corresponding drift and diffusion coefficients. It provides insight into the physical processes at work during the secular diffusion of a self-gravitating discrete disc and makes quantitative predictions on the initial variations of the distribution function in action space. When applied to the secular evolution of an isolated stationary self-gravitating Mestel disc, this formalism predicts initially the importance of the corotation resonance in the inner regions of the disc leading to a regime involving radial migration and heating. It predicts in particular the formation of a "ridge like" feature in action space, in agreement with simulations, but over-estimates the timescale involved in its appearance. Swing amplification is likely to resolve this discrepancy. In astrophysics, the inhomogeneous Balescu-Lenard equation and its WKB limit may also describe the secular diffusion of giant molecular clouds in galactic discs, the secular migration and segregation of planetesimals in proto-planetary discs, or even the long-term evolution of population of stars within the Galactic center.Comment: 22 pages, 12 figure

Secular resonant dressed orbital diffusion II : application to an isolated self similar tepid galactic disc

The main orbital signatures of the secular evolution of an isolated self-gravitating stellar Mestel disc are recovered using a dressed Fokker-Planck formalism in angle-action variables. The shot-noise-driven formation of narrow ridges of resonant orbits is recovered in the WKB limit of tightly wound transient spirals, for a tepid Toomre-stable tapered disc. The relative effect of the bulge, the halo, the disc temperature and the spectral properties of the shot noise are investigated in turn. For such galactic discs all elements seem to impact the locus and direction of the ridge. For instance, when the halo mass is decreased, we observe a transition between a regime of heating in the inner regions of the disc through the inner Lindblad resonance to a regime of radial migration of quasi-circular orbits via the corotation resonance in the outer part of the disc. The dressed secular formalism captures both the nature of collisionless systems (via their natural frequencies and susceptibility), and their nurture via the structure of the external perturbing power spectrum. Hence it provides the ideal framework in which to study their long term evolution.Comment: 15 pages, 11 figure