Analytical Validation of a Continuum Model for Epitaxial Growth with Elasticity on Vicinal Surfaces

Within the context of heteroepitaxial growth of a film onto a substrate, terraces and steps self-organize according to misfit elasticity forces. Discrete models of this behavior were developed by Duport et al. (J Phys I 5:1317–1350, 1995) and Tersoff et al. (Phys Rev Lett 75:2730–2733, 1995). A continuum limit of these was in turn derived by Xiang (SIAM J Appl Math 63:241–258, 2002) (see also the work of Xiang and Weinan Phys Rev B 69:035409-1–035409-16, 2004; Xu and Xiang SIAM J Appl Math 69:1393–1414, 2009). In this paper we formulate a notion of weak solution to Xiang’s continuum model in terms of a variational inequality that is satisfied by strong solutions. Then we prove the existence of a weak solution.     

Accounting for Polydispersion in the Eulerian Large Eddy Simulation of the Two-Phase Flow in an Aeronautical-type Burner

A major issue for the simulation of two-phase flows in engines concerns the modeling of the liquid disperse phase, either in the Lagrangian or the Eulerian approach. In the perspective of massively parallel computing, the Eulerian approach seems to be more suitable, as it uses the same algorithms as the gaseous phase solver. However taking into account the whole physics of a turbulent spray, especially in terms of polydispersity, requires an additional modeling effort. The Mesoscopic Eulerian Formalism (MEF) [13] accounts for the effect of turbulence on the disperse phase, and was extended to the Large Eddy Simulation framework [41], but is limited to monodisperse flows. In [38], the influence of polydispersity on resolved and unresolved turbulent motions of the disperse phase was highlighted, and a first model was proposed, based on size-conditioned statistics. Starting from this idea, a coupling between the MEF and the Multifluid Approach (MA) [30] is proposed. The MA decomposes the Eulerian phase into several fluid classes called sections, and corresponding to size intervals. Each section uses then size-conditioned closures. The original idea of this work is to use the MEF closures independently in each section, taking into account the mean droplet size of this section. This new approach, called Multifluid Mesoscopic Eulerian Formalism (MMEF), is then able to capture polydispersion with associated size-conditioned turbulent dynamics. First, the importance of polydispersity and the ability of MMEF to capture it are highlighted with a 0D evaporation case and a 2D vortex case, showing its impact on dynamics in both size and physical spaces. Then, the MMEF is applied to the MERCATO configuration of ONERA [18]. Results are compared to monodisperse Eulerian, Lagrangian and experimental results.     

Scale effect on unsteady cloud cavitation

No experiment was conducted, yet, to investigate the scale effects on the dynamics of developed cavitating flow with periodical cloud shedding. The present study was motivated by the unclear results obtained from the experiments in a Venturi-type section that was scaled down 10 times for the purpose of measurements by ultra-fast X-ray imaging (Coutier-Delgosha et al. 2009). Cavitation in the original size scale section (Stutz and Reboud in Exp Fluids 23:191–198, 1997, Exp Fluids 29:545–552 2000) always displays unsteady cloud separation. However, when the geometry was scaled down, the cavitation became quasi steady although some oscillations still existed. To investigate this phenomenon more in detail, experiments were conducted in six geometrically similar Venturi test sections where either width or height or both were scaled. Various types of instabilities are obtained, from simple oscillations of the sheet cavity length to large vapor cloud shedding when the size of the test section is increased. It confirms that small scale has a significant influence on cavitation. Especially the height of the test section plays a major role in the dynamics of the re-entrant jet that drives the periodical shedding observed at large scale. Results suggest that the sheet cavity becomes stabile when the section is scaled down to a certain point because re-entrant jet cannot fully develop.     

Comparison of dynamical cores for NWP models: comparison of COSMO and Dune

We present a range of numerical tests comparing the dynamical cores of the operationally used numerical weather prediction (NWP) model COSMO and the university code Dune, focusing on their efficiency and accuracy for solving benchmark test cases for NWP. The dynamical core of COSMO is based on a finite difference method whereas the Dune core is based on a Discontinuous Galerkin method. Both dynamical cores are briefly introduced stating possible advantages and pitfalls of the different approaches. Their efficiency and effectiveness is investigated, based on three numerical test cases, which require solving the compressible viscous and non-viscous Euler equations. The test cases include the density current (Straka et al. in Int J Numer Methods Fluids 17:1–22, 1993), the inertia gravity (Skamarock and Klemp in Mon Weather Rev 122:2623–2630, 1994), and the linear hydrostatic mountain waves of (Bonaventura in J Comput Phys 158:186–213, 2000).     

Wave fronts in second-order elasticity determined by perturbation method applied to the eikonal equation

In Marasco and Romano (Math Comput Model 49(7–8)1504–1518, 2009), Marasco (Math Comput Model 49(7–8):1644–1652, 2009; Int J Eng Sci 47(4):499–511, 2009), we have proposed a perturbation method to determine the speed and the amplitude of the acceleration waves in a second-order elastic body. In this paper, using the above results, we apply a perturbation procedure to analyze the evolution of the wave front of an acceleration wave in the same class of elastic materials. In particular, a second-order approximate solution of the eikonal equation is determined introducing a suitable system of coordinates. The general results are applied to an infinitesimal deformation, and the analytical solution of the eikonal equation is compared with the exact numerical one.     

Quasi-linear versus potential-based formulations of force–flux relations and the GENERIC for irreversible processes: comparisons and examples

An essential part in modeling out-of-equilibrium dynamics is the formulation of irreversible dynamics. In the latter, the major task consists in specifying the relations between thermodynamic forces and fluxes. In the literature, mainly two distinct approaches are used for the specification of force–flux relations. On the one hand, quasi-linear relations are employed, which are based on the physics of transport processes and fluctuation–dissipation theorems (de Groot and Mazur in Non-equilibrium thermodynamics, North Holland, Amsterdam, 1962, Lifshitz and Pitaevskii in Physical kinetics. Volume 10, Landau and Lifshitz series on theoretical physics, Pergamon Press, Oxford, 1981). On the other hand, force–flux relations are also often represented in potential form with the help of a dissipation potential (?ilhavý in The mechanics and thermodynamics of continuous media, Springer, Berlin, 1997). We address the question of how these two approaches are related. The main result of this presentation states that the class of models formulated by quasi-linear relations is larger than what can be described in a potential-based formulation. While the relation between the two methods is shown in general terms, it is demonstrated also with the help of three examples. The finding that quasi-linear force–flux relations are more general than dissipation-based ones also has ramifications for the general equation for non-equilibrium reversible–irreversible coupling (GENERIC: e.g., Grmela and Öttinger in Phys Rev E 56:6620–6632, 6633–6655, 1997, Öttinger in Beyond equilibrium thermodynamics, Wiley Interscience Publishers, Hoboken, 2005). This framework has been formulated and used in two different forms, namely a quasi-linear (Öttinger and Grmela in Phys Rev E 56:6633–6655, 1997, Öttinger in Beyond equilibrium thermodynamics, Wiley Interscience Publishers, Hoboken, 2005) and a dissipation potential–based (Grmela in Adv Chem Eng 39:75–129, 2010, Grmela in J Non-Newton Fluid Mech 165:980–986, 2010, Mielke in Continuum Mech Therm 23:233–256, 2011) form, respectively, relating the irreversible evolution to the entropy gradient. It is found that also in the case of GENERIC, the quasi-linear representation encompasses a wider class of phenomena as compared to the dissipation-based formulation. Furthermore, it is found that a potential exists for the irreversible part of the GENERIC if and only if one does for the underlying force–flux relations.     

A New Divergence Free Synthetic Eddy Method for the Reproduction of Inlet Flow Conditions for LES

This paper describes a recent development of the Synthetic Eddy Method (SEM) proposed by Jarrin et al. (Int J Heat Fluid Flow 30(3):435–442, 2009) for generation of synthetic turbulence. The present scheme is designed to produce a divergence-free turbulence field that can reproduce almost all possible states of Reynolds stress anisotropy. This improved representation, when used to provide inlet conditions for an LES, leads to reduced near-inlet pressure fluctuations in the LES and to a reduced development length, both of which lead to lower computer resource requirements. An advantage of this method with respect to forcing approaches (which require an iterative approach) is the suitability for direct usage with embedded LES. Results for a turbulent channel flow are reported here and compared to those from the original SEM, and other direct approaches such as the VORTEX method of Sergent (2002) and the Synthesized Turbulence approach of Davidson and Billson (Int J Heat Fluid Flow 27(6):1028–1042, 2006), showing overall improved performance and a more accurate representation of turbulence structures immediately downstream of the inlet.     

A phenomenological model for microstructure-dependent inelasticity in shape-memory alloys

Degradation in shape-memory alloy response is a crucial concern for a variety of innovative applications. Under cyclic loadings, these materials generally experience permanent inelastic deformations. The onset of plasticization is known to be very sensitive to the microstructure of the polycrystalline specimen. Moving from recent experimental findings (Malard et al. in Funct Mater Lett 2:45–54, 2009; Acta Mater 59:1542–1556, 2011), we present a phenomenological model for permanent inelastic effects in shape-memory alloys taking into account the polycrystalline microstructure. In particular, the mechanical response under cyclic loadings is investigated in connection with the mean crystal grain size. Formulated within the variational frame of generalized standard materials, the model consists in an extension of the model in Auricchio et al. (Int J Plast 23:207–226, 2007) to the case of microstructure-dependent parameters. The mathematical setting is discussed and numerical simulations showing the capability of the model to reproduce experiments are presented.     

On the Formulation of Continuum Thermodynamic Models for Solids as General Equations for Non-equilibrium Reversible-Irreversible Coupling

The purpose of this work is the comparison of some aspects of the formulation of material models in the context of continuum thermodynamics (e.g., ?ilhavý in The mechanics and thermodynamics of continuous media, Springer, Berlin, 1997) with their formulation in the form of a General Equation for Non-Equilibrium Reversible-Irreversible Coupling (GENERIC: e.g., Grmela and Öttinger in Phys. Rev. E 56: 6620–6632, 1997; Öttinger and Grmela in Phys. Rev. E 56: 6633–6655, 1997; Öttinger in Beyond equilibrium thermodynamics, Wiley, New York, 2005; Grmela in J. Non-Newton. Fluid Mech. 165: 980–998, 2010). A GENERIC represents a generalization of the Ginzburg-Landau model for the approach of non-equilibrium systems to thermodynamic equilibrium. Originally developed to formulate non-equilibrium thermodynamic models for complex fluids, it has recently been applied to anisotropic inelastic solids in a Eulerian setting (Hütter and Tervoort in J. Non-Newton. Fluid Mech. 152: 45–52, 2008; 53–65, 2008; Adv. Appl. Mech. 42: 254–317, 2009) as well as to damage mechanics (Hütter and Tervoort in Acta Mech. 201: 297–312, 2008). In the current work, attention is focused for simplicity on the case of thermoelastic solids with heat conduction and viscosity in a Lagrangian setting (e.g., ?ilhavý in The mechanics and thermodynamics of continuous media, Springer, Berlin, 1997, Chaps. 9–12). In the process, the relation of the two formulations to each other is investigated in detail. A particular point in this regard is the concept of dissipation and its model representation in both contexts.     

Failure mode transitions in RC beams: A cohesive/overlapping crack model application

The analysis of reinforced concrete beams in flexure taking into account the nonlinear behaviour of concrete is addressed by a numerical approach based on the Cohesive-Overlapping Crack Model. An extensive experimental research has been proposed by Bosco and Carpinteri (Scale effects and transitional phenomena of reinforced concrete beams in flexure. ESIS Technical Committée 9 Round Robin proposal, 1993), Bosco et al. (Scale effects and transitional failure phenomena of reinforced concrete beams in flexure. Report to ESIS Technical Committée 9, 1996) and El-Khatieb (Transizione di scala duttile-fragile per le travi in calcestruzzo armato. PhD Thesis, 1997) in order to obtain a rational explanation for failure transitional phenomena of RC beams by varying steel percentage and/or beam slenderness and/or beam size-scale. In the present paper, collapse mechanisms due to concrete tensile cracking, concrete compressive crushing and steel yielding and/or slippage are analysed and a numerical vs. experimental comparison is presented in order to validate the proposed model.     

Dynamics and Control at Feedback Vertex Sets. I: Informative and Determining Nodes in Regulatory Networks

We consider systems of differential equations which model complex regulatory networks by a graph structure of dependencies. We show that the concepts of informative nodes (Mochizuki and Saito, J Theor Biol 266:323–335, 2010) and determining nodes (Foias and Temam, Math Comput 43:117–133, 1984) coincide with the notion of feedback vertex sets from graph theory. As a result we can determine the long-time dynamics of the entire network from observations on only a feedback vertex set. We also indicate how open loop control at a feedback vertex set, only, forces the remaining network to stably follow prescribed stable or unstable trajectories. We present three examples of biological networks which motivated this work: a specific gene regulatory network of ascidian cell differentiation (Imai et al., Science 312:1183–1187, 2006), a signal transduction network involving the epidermal growth factor in mammalian cells (Oda et al., Mol Syst Biol 1:1–17, 2005), and a mammalian gene regulatory network of circadian rhythms (Mirsky et al., Proc Natl Acad Sci USA 106:11107–11112, 2009). In each example the required observation set is much smaller than the entire network. For further details on biological aspects see the companion paper (Mochizuki et al., J Theor Biol, 2013, in press). The mathematical scope of our approach is not limited to biology. Therefore we also include many further examples to illustrate and discuss the broader mathematical aspects.     

Traveling Wave Solutions for a Predator?CPrey System With Sigmoidal Response Function

We study the existence of traveling wave solutions for a diffusive predator?Cprey system. The system considered in this paper is governed by a Sigmoidal response function which in some applications is more realistic than the Holling type I, II responses, and more general than a simplified form of the Holling type III response considered before. Our method is an improvement to the original method introduced in the work of Dunbar (J Math Biol 17:11?C32, 1983; SIAM J Appl Math 46:1057?C1078, 1986). A bounded Wazewski set is used in this work while unbounded Wazewski sets were used in Dunbar (1983, 1986). The existence of traveling wave solutions connecting two equilibria is established by using the original Wazewski??s theorem which is much simpler than the extended version in Dunbar??s work.     

Type-I intermittency with discontinuous reinjection probability density in a truncation model of the derivative nonlinear Schrödinger equation

In previous papers, the type-I intermittent phenomenon with continuous reinjection probability density (RPD) has been extensively studied. However, in this paper type-I intermittency considering discontinuous RPD function in one-dimensional maps is analyzed. To carry out the present study the analytic approximation presented by del Río and Elaskar (Int. J. Bifurc. Chaos 20:1185–1191, 2010) and Elaskar et al. (Physica A. 390:2759–2768, 2011) is extended to consider discontinuous RPD functions. The results of this analysis show that the characteristic relation only depends on the position of the lower bound of reinjection (LBR), therefore for the LBR below the tangent point the relation \(\left\langle l \right\rangle \propto \varepsilon ^{-1/2}\) , where \(\varepsilon \) is the control parameter, remains robust regardless the form of the RPD, although the average of the laminar phases \(\left\langle l \right\rangle \) can change. Finally, the study of discontinuous RPD for type-I intermittency which occurs in a three-wave truncation model for the derivative nonlinear Schrodinger equation is presented. In all tests the theoretical results properly verify the numerical data.     

Graphical exploration of the connectivity sets of alternated Julia sets

Using computer graphics and visualization algorithms, we extend in this work the results obtained analytically in Danca et al. (Int. J. Bifurc. Chaos, 19:2123–2129, 2009), on the connectivity domains of alternated Julia sets, defined by switching the dynamics of two quadratic Julia sets. As proved in Danca et al. (Int. J. Bifurc. Chaos, 19:2123–2129, 2009), the alternated Julia sets exhibit, as for polynomials of degree greater than two, the disconnectivity property in addition to the known dichotomy property (connectedness and totally disconnectedness), which characterizes the standard Julia sets. Via experimental mathematics, we unveil these connectivity domains, which are four-dimensional fractals. The computer graphics results show here, without substituting the proof but serving as a research guide, that for the alternated Julia sets, the Mandelbrot set consists of the set of all parameter values, for which each alternated Julia set is not only connected, but also disconnected.     

Microscale Fracture Toughness of Bismuth Doped Copper Bicrystals Using Double Edge Notched Microtensile Tests

The availability of focused ion beam (FIB) milling, nanoindentation, and microelectromechanical systems (MEMS) based test platforms has enabled small-scale mechanical testing to become an increasingly popular approach for measuring material properties. While great emphasis has been placed on measuring plastic properties at the micro- and nanoscale [1, 2], an area that has received significantly less consideration is the measurement of fracture toughness. A technique for performing small-scale, in situ fracture toughness tests using double edge notched tensile (DENT) specimens has been developed and used to measure a nearly 40 % reduction in toughness associated with the addition of Bi to the grain boundary of a Cu bicrystal. That Bi embrittles Cu grain boundaries is well known [3–10], however, as shown herein, the DENT technique offers certain advantages over existing boundary fracture tests, especially when used with ductile materials.     

Experimental Measurement of Local Burning Velocity Within a Rotating Flow

The work presented in this paper details the implementation of a new technique for the measurement of local burning velocity using asynchronous particle image velocimetry. This technique uses the local flow velocity ahead of the flame front to measure the movement of the flame by the surrounding fluid. This information is then used to quantify the local burning velocity by taking into account the translation of the flame via convection. In this paper the developed technique is used to study the interaction between a flame front and a single toroidal vortex for the case of premixed stoichiometric methane and air combustion. This data is then used to assess the impact of vortex structure on flame propagation rates. The burning velocity data demonstrates that there is a significant enhancement to the rate of flame propagation where the flame directly interacts with the rotating vortex. The increases found were significantly higher than expected but are supported by burning velocities (Filatyev et al, Combust Flame 141:1?C21, 2005; Kobayashi et al, Proc Combust Inst 29:1793?C1800, 2002; Shepherd et al. 1998) found in turbulent flames of the same mixture composition. Away from this interaction with the main vortex core, the flame exhibits propagation rates around the value recorded in literature for unperturbed laminar combustion (Tahtouh et al, Combust Flame 159:1735?C1743, 2009; Hassan et al, Combust Flame 115:539?C550, 1998); Halter et al, Proc Combust Inst 30:201?C208, 2005; Coppens et al, Exp Therm Fluid Sci 31:437?C444, 2007).     

Estimation of the largest Lyapunov exponent-like (LLEL) stability measure parameter from the perturbation vector and its derivative dot product (part 2) experiment simulation

Controlling system dynamics with use of the Largest Lyapunov Exponent (LLE) is employed in many different areas of the scientific research. Thus, there is still need to elaborate fast and simple methods of LLE calculation. This article is the second part of the one presented in Dabrowski (Nonlinear Dyn 67:283–291, 2012). It develops method LLEDP of the LLE estimation and shows that from the time series of two identical systems, one can simply extract value of the stability parameter which value can be treated as largest LLE. Unlike the method presented in part, one developed method (LLEDPT) can be applied to the dynamical systems of any type, continuous, with discontinuities, with time delay and others. The theoretical improvement shows simplicity of the method and its obvious physical background. The proofs for the method effectiveness are based on results of the simulations of the experiments for Duffing and Van der Pole oscillators. These results were compared with ones obtained with use of the Stefanski method (Stefanski in Chaos Soliton Fract 11(15):2443–2451, 2000; Chaos Soliton Fract 15:233–244, 2003; Chaos Soliton Fract 23:1651–1659, 2005; J Theor Appl Mech 46(3):665–678, 2008) and LLEDP method. LLEDPT can be used also as the criterion of stability of the control system, where desired behavior of controlled system is explicitly known (Balcerzak et al. in Mech Mech Eng 17(4):325–339, 2013). The next step of development of the method can be considered in direction that allows estimation of LLE from the real time series, systems with discontinuities, with time delay and others.