Polynomial-based non-uniform interpolatory subdivision with features control

Starting from a well-known construction of polynomial-based interpolatory 4-point schemes, in this paper we present an original affine combination of quadratic polynomial samples that leads to a non-uniform 4-point scheme with edge parameters. This blending-type formulation is then further generalized to provide a powerful subdivision algorithm that combines the fairing curve of a non-uniform refinement with the advantages of a shape-controlled interpolation method and an arbitrary point insertion rule. The result is a non-uniform interpolatory 4-point scheme that is unique in combining a number of distinctive properties. In fact it generates visually-pleasing limit curves where special features ranging from cusps and flat edges to point/edge tension effects may be included without creating undesired undulations. Moreover such a scheme is capable of inserting new points at any positions of existing intervals, so that the most convenient parameter values may be chosen as well as the intervals for insertion. Such a fully flexible curve scheme is a fundamental step towards the construction of high-quality interpolatory subdivision surfaces with features control

Shock formation in the collapse of a vapor nano-bubble

In this paper a diffuse-interface model featuring phase change, transition to supercritical conditions, thermal conduction, compressibility effects and shock wave propagation is exploited to deal with the dynamics of a cavitation bubble. At variance with previous descriptions, the model is uniformly valid for all phases (liquid, vapor and supercritical) and phase transitions involved, allowing to describe the non-equilibrium processes ongoing during the collapse. As consequence of this unitary description, rather unexpectedly for pure vapor bubbles, the numerical experiments show that the collapse is accompanied by the emission of a strong shock wave in the liquid and by the oscillation of the bubble that periodically disappears and reappears, due to transition to super/sub critical conditions. The mechanism of shock wave formation is strongly related to the transition of the vapor to supercritical state, with a progressive steepening of the compression wave to form the shock which is eventually reflected as an outward propagating wave in the liquid

Vapor nucleation paths in lyophobic nanopores

Abstract.: In recent years, technologies revolving around the use of lyophobic nanopores gained considerable attention in both fundamental and applied research. Owing to the enormous internal surface area, heterogeneous lyophobic systems (HLS), constituted by a nanoporous lyophobic material and a non-wetting liquid, are promising candidates for the efficient storage or dissipation of mechanical energy. These diverse applications both rely on the forced intrusion and extrusion of the non-wetting liquid inside the pores; the behavior of HLS for storage or dissipation depends on the hysteresis between these two processes, which, in turn, are determined by the microscopic details of the system. It is easy to understand that molecular simulations provide an unmatched tool for understanding phenomena at these scales. In this contribution we use advanced atomistic simulation techniques in order to study the nucleation of vapor bubbles inside lyophobic mesopores. The use of the string method in collective variables allows us to overcome the computational challenges associated with the activated nature of the phenomenon, rendering a detailed picture of nucleation in confinement. In particular, this rare event method efficiently searches for the most probable nucleation path(s) in otherwise intractable, high-dimensional free-energy landscapes. Results reveal the existence of several independent nucleation paths associated with different free-energy barriers. In particular, there is a family of asymmetric transition paths, in which a bubble forms at one of the walls; the other family involves the formation of axisymmetric bubbles with an annulus shape. The computed free-energy profiles reveal that the asymmetric path is significantly more probable than the symmetric one, while the exact position where the asymmetric bubble forms is less relevant for the free energetics of the process. A comparison of the atomistic results with continuum models is also presented, showing how, for simple liquids in mesoporous materials of characteristic size of ca. 4nm, the nanoscale effects reported for smaller pores have a minor role. The atomistic estimates for the nucleation free-energy barrier are in qualitative accord with those that can be obtained using a macroscopic, capillary-based nucleation theory. Graphical abstract: [Figure not available: see fulltext.]

On multi-degree splines

Multi-degree splines are piecewise polynomial functions having sections of different degrees. For these splines, we discuss the construction of a B-spline basis by means of integral recurrence relations, extending the class of multi-degree splines that can be derived by existing approaches. We then propose a new alternative method for constructing and evaluating the B-spline basis, based on the use of so-called transition functions. Using the transition functions we develop general algorithms for knot-insertion, degree elevation and conversion to B\'ezier form, essential tools for applications in geometric modeling. We present numerical examples and briefly discuss how the same idea can be used in order to construct geometrically continuous multi-degree splines

Thermally activated vapor bubble nucleation: the Landau-Lifshitz/Van der Waals approach

Vapor bubbles are formed in liquids by two mechanisms: evaporation (temperature above the boiling threshold) and cavitation (pressure below the vapor pressure). The liquid resists in these metastable (overheating and tensile, respectively) states for a long time since bubble nucleation is an activated process that needs to surmount the free energy barrier separating the liquid and the vapor states. The bubble nucleation rate is difficult to assess and, typically, only for extremely small systems treated at atomistic level of detail. In this work a powerful approach, based on a continuum diffuse interface modeling of the two-phase fluid embedded with thermal fluctuations (Fluctuating Hydrodynamics) is exploited to study the nucleation process in homogeneous conditions, evaluating the bubble nucleation rates and following the long term dynamics of the metastable system, up to the bubble coalescence and expansion stages. In comparison with more classical approaches, this methodology allows on the one hand to deal with much larger systems observed for a much longer times than possible with even the most advanced atomistic models. On the other it extends contin- uum formulations to thermally activated processes, impossible to deal with in a purely determinist setting

Diffuse interface modeling of a radial vapor bubble collapse

A diffuse interface model is exploited to study in details the dynamics of a cavitation vapor bubble, by including phase change, transition to supercritical conditions, shock wave propagation and thermal conduction. The numerical experiments show that the actual dynamic is a sequence of collapses and rebounds demonstrating the importance of nonequilibrium phase changes. In particular the transition to supercritical conditions avoids the full condensation and leads to shockwave emission after the collapse and to successive bubble rebound

Piecewise Extended Chebyshev Spaces: a numerical test for design

Given a number of Extended Chebyshev (EC) spaces on adjacent intervals, all of the same dimension, we join them via convenient connection matrices without increasing the dimension. The global space is called a Piecewise Extended Chebyshev (PEC) Space. In such a space one can count the total number of zeroes of any non-zero element, exactly as in each EC-section-space. When this number is bounded above in the global space the same way as in its section-spaces, we say that it is an Extended Chebyshev Piecewise (ECP) space. A thorough study of ECP-spaces has been developed in the last two decades in relation to blossoms, with a view to design. In particular, extending a classical procedure for EC-spaces, ECP-spaces were recently proved to all be obtained by means of piecewise generalised derivatives. This yields an interesting constructive characterisation of ECP-spaces. Unfortunately, except for low dimensions and for very few adjacent intervals, this characterisation proved to be rather difficult to handle in practice. To try to overcome this difficulty, in the present article we show how to reinterpret the constructive characterisation as a theoretical procedure to determine whether or not a given PEC-space is an ECP-space. This procedure is then translated into a numerical test, whose usefulness is illustrated by relevant examples

Anisotropic fluctuations in turbulent sheared flows

An experimental analysis of small-scales anisotropic turbulent fluctuations has been performed in two different flows. We analyzed anisotropic properties of an homogeneous shear flows and of a turbulent boundary layer by means of two cross-wire probes to obtain multi-point multi-component measurements. Data are analyzed at changing inter-probe separation without the use of Taylor hypothesis. The results are consistent with the ``exponent-only'' scenario for universality, i.e. all experimental data can be fit by fixing the same set of anisotropic scaling exponents at changing only prefactors, for different shear intensities and boundary conditions.Comment: 11 pages, 8 figure