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Magnetic helicity plays a tremendously important role when it is different from zero on average. Most notably, it leads to the phenomenon of an inverse cascade. Here, we consider decaying magnetohydrodynamic (MHD) turbulence as well as some less common examples of magnetic evolution under the Hall effect and ambipolar diffusion, as well as cases in which the magnetic field evolution is constrained by the presence of an asymmetry in the number density of chiral fermions, whose spin is systematically either aligned or anti-aligned with its momentum. In all those cases, there is a new conserved quantity: the Hosking integral. We present quantitative scaling results for the magnetic integral scale as well as the magnetic energy density and its spectrum. We also compare with cases were a magnetic version of the Saffman integral is initially finite. Rotation in MHD turbulence tends to suppress nonlinearity and thereby also inverse cascading. Finally, the role of the Hosking and magnetic Saffman integrals in shell models of turbulence is examined. [ABSTRACT FROM AUTHOR]

Shell models of turbulence are representation of turbulence equations in Fourier domain. Various shell models are studied for their mathematical relevance and the numerical simulations that exhibit at most resemblance with turbulent flows. One of the mathematically well‐studied shell model of turbulence is called sabra shell model. This work concerns with two important issues related to shell model, namely, feedback stabilization and robust stabilization. We first address stabilization problem related to sabra shell model of turbulence and prove that the system can be stabilized via finite dimensional controller. Thus, only finitely many modes of the shell model would suffice to stabilize the system. Later, we study robust stabilization in the presence of the unknown disturbance and corresponding control problem by solving an infinite time horizon max‐min control problem. We first prove the H∞$$ {H}^{\infty } $$ stabilization of the associated linearized system and characterize the optimal control in terms of a feedback operator by solving an algebraic Riccati equation. Using the same Riccati operator, we establish asymptotic stability of the nonlinear system. [ABSTRACT FROM AUTHOR]

The present work demonstrates the use of the node-dependent kinematics method to derive and compare several two-dimensional shell theories. The three dimensional displacement field is expressed in terms of generalized coordinates, which are subsequently expanded along the shell thickness using arbitrary functions. The in-plane unknowns, are then discretized through classical finite element approximation. Based on the Carrera Unified Formulation, the proposed method combines in a unique manner the theory of structures and the finite element method; thickness interpolation functions are defined node-wise. As a consequence, the resulting finite element model represents diverse approximation theories at each single node. In this work Taylor-based kinematics (including the Murakami Zig-Zag function) and Legendre-type nodal kinematics are incorporated at the element level without adopting mathematical artifices leading to the global–local strategy, where refined theories are selectively employed in specific areas, while maintaining acceptable computational costs. Numerical cases from the existing literature are employed to establish the effectiveness of node-dependent models in bridging a locally refined theories to global kinematics when local effects need to be considered. The analyses focus on localized loads for both homogeneous and multi-layered structures. • Global to local, geometrically exact analysis of layered shell structures is shown. • The formulation has node-dependent kinematics, allowing refinement of critical zones. • The model combines finite elements with HOTs, using LW to zig-zag ESL kinematics. • The governing equations are expressed in a unified form. • They are independent of the nodal theory expansion, allowing simulation optimization. • Several benchmarks are analyzed, including laminated shells with localized loadings. [ABSTRACT FROM AUTHOR]

Abstract Towards improving the numerical efficiency in the analysis of multi-layered shell structures with the finite element (FE) method, an adaptable two-level mathematical refinement approach is proposed for refined curvilinear shell elements. Based on Carrera Unified Formulation (CUF), the approximation of displacement functions of shell elements can be improved by refining the through-the-thickness assumptions and enriching the shape functions. By using the hierarchical Legendre polynomial expansions (HLE) as shape functions, the element capabilities can be enhanced conveniently without re-meshing. To further increase the numerical efficiency of shell FE models, Node-Dependent Kinematics (NDK) is utilized to implement local kinematic refinements on the selected FE nodes within the domain of interest. The conjunction of NDK with the two-level refinements of the shell FE models leads to an adaptable refinement approach in the analysis of shell structures, which can be used to build FE models with optimal efficiency and high fidelity. The competence of the proposed approach is investigated through numerical studies on laminated shells. [ABSTRACT FROM AUTHOR]

This paper exploits the stress recovery technique to evaluate the out-of-plane stress components in the static analysis of composite beams, plates and shells. This technique is implemented in the framework of the Carrera Unified Formulation, an approach allowing the implementation of the theories of structures in a compact way. This work uses Taylor, Legendre and Jacobi polynomials with equivalent single-layer and layer-wise approaches. The finite element method is applied to provide numerical solutions. Multi-layered beams, plates and shells subjected to different loading and boundary conditions are studied to validate and assess the proposed technique. The results are compared with those from the literature and show that the stress recovery technique provides reasonable accuracy for the shear stresses, even with lower-order models. Furthermore, results confirm that, when dealing with thick structures, the adoption of layer-wise models is mandatory to obtain accurate results. [ABSTRACT FROM AUTHOR]

We establish the anomalous mean dissipation rate of energy in the inviscid limit for a stochastic shell model of turbulent fluid flow. The proof relies on viscosity independent bounds for stationary solutions and on establishing ergodic and mixing properties for the viscous model. The shell model is subject to a degenerate stochastic forcing in the sense that noise acts directly only through one wavenumber. We show that it is hypo-elliptic (in the sense of Hörmander) and use this property to prove a gradient bound on the Markov semigroup. [ABSTRACT FROM AUTHOR]

In this paper, theories of structures based on hierarchical Jacobi expansions are explored for the static analysis of multilayered beams, plates and shells. They belong to the family of classical orthogonal polynomials. This expansion is employed in the framework of the Carrera Unified Formulation (CUF), which allows to generate finite element stiffness matrices in a straightforward way. CUF allows also to employ both layer-wise and equivalent single layer approaches in order to obtain the desired degree of precision and computational cost. In this work, CUF is exploited for the analysis of one-dimensional beams and two-dimensional plates and shells, and several case studies from the literature are analysed. Displacements, in-plane, transverse and shear stresses are shown. In particular, for some benchmarks, the shear stresses are calculated using the constitutive relations and the stress recovery technique. The obtained results clearly show the convenience of using equivalent single layer models when calculating displacements, in-plane stresses and shear stresses recovered by three-dimensional indefinite equilibrium equations. On the other hand, layer-wise models are able to accurately predict the structural behaviour, even though higher degrees of freedom are needed. [ABSTRACT FROM AUTHOR]

In this paper, theories of structures based on hierarchical Jacobi expansions are explored for the static analysis of multilayered beams, plates and shells. They belong to the family of classical orthogonal polynomials. This expansion is employed in the framework of the Carrera Unified Formulation (CUF), which allows to generate finite element stiffness matrices in a straightforward way. CUF allows also to employ both layer-wise and equivalent single layer approaches in order to obtain the desired degree of precision and computational cost. In this work, CUF is exploited for the analysis of one-dimensional beams and two-dimensional plates and shells, and several case studies from the literature are analysed. Displacements, in-plane, transverse and shear stresses are shown. In particular, for some benchmarks, the shear stresses are calculated using the constitutive relations and the stress recovery technique. The obtained results clearly show the convenience of using equivalent single layer models when calculating displacements, in-plane stresses and shear stresses recovered by three-dimensional indefinite equilibrium equations. On the other hand, layer-wise models are able to accurately predict the structural behaviour, even though higher degrees of freedom are needed. [ABSTRACT FROM AUTHOR]

The use of transfer entropy has proven to be helpful in detecting which is the verse of dynamical driving in the interaction of two processes, X and Y . In this paper, we present a different normalization for the transfer entropy, which is capable of better detecting the information transfer direction. This new normalized transfer entropy is applied to the detection of the verse of energy flux transfer in a synthetic model of fluid turbulence, namely the Gledzer-Ohkitana-Yamada shell model. Indeed, this is a fully well-known model able to model the fully developed turbulence in the Fourier space, which is characterized by an energy cascade towards the small scales (large wavenumbers k), so that the application of the information-theory analysis to its outcome tests the reliability of the analysis tool rather than exploring the model physics. As a result, the presence of a direct cascade along the scales in the shell model and the locality of the interactions in the space of wavenumbers come out as expected, indicating the validity of this data analysis tool. In this context, the use of a normalized version of transfer entropy, able to account for the difference of the intrinsic randomness of the interacting processes, appears to perform better, being able to discriminate the wrong conclusions to which the "traditional" transfer entropy would drive. [ABSTRACT FROM AUTHOR]

Abstract: This paper presents a study on the suitability of shell models to assess the buckling behaviour of single-walled carbon nanotubes (CNTs) under torsion. It is shown that the simultaneous use of both well known (i) Donnell shell model and (ii) uniform helix deflected shape (HDS) of CNTs, leads to incorrect values of the critical angle of twist per unit of length, as they do not match the results obtained from molecular dynamic simulations. Conversely, more sophisticated models, like the Sanders shell model (SSM) with non-uniform HDS is found to lead to correct results of critical angle of twist. It is established that there is a transitional aspect ratio (length-to-diameter ratio) that separates the group of short CNTs, mostly influenced by end conditions, from the group of long CNTs, mostly influenced by warping deformation. Based on the SSM with non-uniform HDS, straightforward analytical expressions to calculate the critical angle of twist are proposed for each of these groups. Despite its simplicity, the procedure presented is shown to give rather accurate results for a wide range of CNT lengths, diameter and chirality. [Copyright &y& Elsevier]

Abstract: This paper presents a molecular dynamics (MDs) study on the linear, buckling and post-buckling behaviour of carbon nanotubes (CNTs) under pure shortening and pure twisting. Its objectives are (i) to clarify the issue about the most correct thickness value to adopt in the simulation of CNTs using shell models and (ii) to evaluate their post-critical strength. Three CNTs with similar length-to-diameter ratio but different atomic structures (zig-zag, armchair and chiral) are selected for this study. Then, MD simulations are performed to investigate the pre-critical, critical buckling and post-critical behaviour of CNTs under pure shortening and pure twisting. Using available analytical formulae derived from shell models, the influence of CNT thickness on their critical strain and critical angle of twist is investigated. Some conclusions are drawn regarding (i) the most appropriate choice of the thickness value to use in shell models and (ii) the effectiveness of post-critical stiffness and strength of CNTs. [Copyright &y& Elsevier]

Abstract: There is a large body of experimental evidence which shows that the jet noise spectrum is composed of two sources. Our aim here is to prove, mathematically, that the two-source paradigm can be derived using a minimum number of self-consistent approximations based on our current knowledge of jet turbulence in cold flows. The starting point of the paper is Goldstein’s (2003)  exact re-arrangement of the Navier–Stokes equations, which shows that turbulence enters the acoustic spectrum formula through the Reynolds stress auto-covariance tensor. We extend the shell model of turbulence using a more general symmetry approximation that amounts to assuming that the Reynolds stress auto-covariance is, firstly, axi-symmetric, and secondly is equivalent to the same tensor only after it has been averaged (point-wise) over the azimuthal separation. As a consequence of these two assumptions, the space–time Fourier transform of the Reynolds stress auto-covariance (which we refer to as the spectral tensor) depends on the transverse wave vector only through the square of its magnitude and, moreover, is also an axi-symmetric tensor. This defines the generalized shell model (or GSM) and we apply it to the jet noise problem. The final result shows that the acoustic spectrum can be written as the sum of two groups of terms, one of which corresponds to the peak jet noise in the weakly non-parallel flow limit. [Copyright &y& Elsevier]

Abstract: Based on the Riemann and Caputo definition of the fractional derivative we use the fractional extensions of the standard rotation group to construct a higher dimensional representation of a fractional rotation group with mixed derivative types. An analytic extended symmetric rotor model is derived, which correctly predicts the sequence of magic numbers in metal clusters. It is demonstrated, that experimental data may be described by assuming a sudden change in the fractional derivative parameter which is interpreted as a second order phase transition in the region of cluster size with . [Copyright &y& Elsevier]

We consider the free vibration problem of thin shells of revolution of constant type of geometry, focusing on the asymptotic behaviour of the lowest eigenfrequency, as the thickness tends to zero. Numerical experiments are computed using two discretization methods, collocation and finite elements, each corresponding to a different shell model. Our results are in agreement with theoretical results obtained using interpolation theory and cited in literature. [ABSTRACT FROM AUTHOR]

Abstract: Different scaling behavior has been reported in various shell models proposed for turbulent thermal convection. In this paper, we show that buoyancy is not always relevant to the statistical properties of these shell models even though there is an explicit coupling between velocity and temperature in the equations of motion. When buoyancy is relevant (irrelevant) to the statistical properties, the scaling behavior is Bolgiano–Obukhov (Kolmogorov) plus intermittency corrections. We show that the intermittency corrections of temperature could be solely attributed to fluctuations in the entropy transfer rate when buoyancy is relevant but due to fluctuations in both energy and entropy transfer rates when buoyancy is irrelevant. This difference can be used as a criterion to distinguish whether temperature is behaving as an active or a passive scalar. [Copyright &y& Elsevier]

The nuclear shell model predicts that the next doubly magic shell-closure beyond 208Pb is at a proton number Z = 114, 120, or 126 and at a neutron number N = 172 or 184. The outstanding aim of experimental investigations is the exploration of this region of spherical 'SuperHeavy Elements' (SHEs). Experimental methods are described, which allowed for the identification of elements produced on a cross-section level of about 1 pb. The decay data reveal that for the heaviest elements, the dominant decay mode is alpha emission, not fission. Decay properties as well as reaction cross-sections are compared with results of theoretical investigations. [ABSTRACT FROM AUTHOR]

In this work we derive lower bounds for the Hausdorff and fractal dimensions of the global attractor of the Sabra shell model of turbulence in different regimes of parameters. We show that for a particular choice of the forcing term and for sufficiently small viscosity term ν, the Sabra shell model has a global attractor of large Hausdorff and fractal dimensions proportional to log ν −1 for all values of the governing parameter ε, except for ε =1. The obtained lower bounds are sharp, matching the upper bounds for the dimension of the global attractor obtained in our previous work. Moreover, the complexity of the dynamics of the shell model increases as the viscosity ν tends to zero, and we describe a precise scenario of successive bifurcations for different parameters regimes. In the “three-dimensional” regime of parameters this scenario changes when the parameter ε becomes sufficiently close to 0 or to 1. We also show that in the “two-dimensional” regime of parameters, for a certain non-zero forcing term, the long-term dynamics of the model becomes trivial for every value of the viscosity. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we study analytically the viscous “sabra” shell model of the energy turbulent cascade. We prove the global regularity of solutions and show that the shell model has finitely many asymptotic degrees of freedom, specifically: a finite dimensional global attractor and globally invariant inertial manifolds. Moreover, we establish the existence of an exponentially decaying energy dissipation range for sufficiently smooth forcing. [Copyright &y& Elsevier]

A lattice Boltzmann scheme simulating the dynamics of shell models of turbulence is developed. The influence of high-order kinetic modes (ghosts) on the dissipative properties of turbulence dynamics is studied. It is analytically found that when ghost fields relax on the same timescale as the hydrodynamic ones, their major effect is a net enhancement of the fluid viscosity. The bare fluid viscosity is recovered by letting ghost fields evolve on a much longer timescale. Analytical results are borne out by high-resolution numerical simulations. These simulations indicate that the hydrodynamic manifold is very robust towards large fluctuations of non-hydrodynamic fields. [Copyright &y& Elsevier]

A Gibbs-like approach for simultaneous multi-scale correlation functions in random, time-dependent, multiplicative processes for the turbulent energy cascade is investigated. We study the optimal log-normal Gibbs-like distribution able to describe the subtle effects induced by non-trivial time dependency on both single-scale (structure functions) and multi-scale correlation functions. We provide analytical expression for the general multi-scale correlation functions in terms of the two-point correlations between multipliers and we show that the log-normal distribution is already accurate enough to reproduce quantitatively many of the observed behavior. The main result is that non-trivial time effects renormalize the Gibbs-like effective potential necessary to describe single-time statistics. We also present a generalization of this approach to more general, non log-normal, potentials. In the latter case one obtains a formal expansion of both structure functions and multi-scale correlations in terms of cumulants of all orders. [ABSTRACT FROM AUTHOR]

We show that Kolmogorov multipliers in turbulence cannot be statistically independent of others at adjacent scales (or even a finite range apart) by numerical simulation of a shell model and by theory. As the simplest generalization of independent distributions, we suppose that the steady-state statistics of multipliers in the shell model are given by a translation-invariant Gibbs measure with a short-range potential, when expressed in terms of suitable “spin” variables: real-valued spins that are logarithms of multipliers and XY-spins defined by local dynamical phases. Numerical evidence is presented in favor of the hypothesis for the shell model, in particular novel scaling laws and derivative relations predicted by the existence of a thermodynamic limit. The Gibbs measure appears to be in a high-temperature, unique-phase regime with “paramagnetic” spin order. [ABSTRACT FROM AUTHOR]

Cascade models based on dynamical complex networks are proposed as models of the turbulent energy cascade. Taking a simple shell model as the initial regular lattice with only nearest neighbor interactions, small world network models are constructed by adding or replacing some of the existing local interactions by nonlocal ones. The models are then evolved over time, both by solving for the shell velocity variable using an arbitrary network generalization of the shell model evolution and by rewiring the network each time from the original lattice in regular time intervals. This results in a more intermittent time evolution with larger variations of the wavenumber spectrum. It also results in an increase in intermittency, computed from the exponents of structure functions obtained from these models. It is observed that the intermittency increases as the ratio of random nonlocal connections to local nearest-neighbor connections increases. The possibility of extraction of such models from direct numerical simulations is discussed and a detailed example for two dimensional turbulence is given. • Turbulent cascade can be seen as a percolation in an evolving network in wave-number domain. • Shell models can be modified to form complex network models of the turbulent cascade. • Connections in a complex network model can be extracted from direct numerical simulations. • Introduction of random non-local interactions increase intermittency in turbulence. [ABSTRACT FROM AUTHOR]

The mechanical response of pressure vessels to an applied internal pressure is essentially controlled by a few key parameters, related both to the overall geometry of the structure and to the orientations and thicknesses of the composite layers. The role of each parameter, and the way they interact to determine the structural response, can be apprehended at a very early design stage by using simple material and structural models, which enable to explore a wide range of designs with minimal computational cost. More complex models can then be called into play to predict the detailed structural response, including crucial information such as the burst pressure and failure mode. This paper discusses a hierarchy of models with increasing levels of details and complexity, which are useful to gain increasing insight on the pressure vessel response all along the design process. [ABSTRACT FROM AUTHOR]

Micrometer-scale encapsulated gas bubbles, known as ultrasound contrast agents, are used in ultrasound medical diagnostics for enhancing blood-tissue contrast during an ultrasonic examination. They are also employed in therapy as an activator of drug incorporation or extravasation. Adequate modeling of the effect of encapsulation is of primary importance because it is the encapsulating shell that determines many of the functional properties of contrast agents. In this review, existing approaches to the modeling of the radial motion of an encapsulated bubble are discussed and comparative analysis of available shell models is conducted. The capabilities of the shell models are evaluated in the context of recent experimental observations, such as compression-only behavior and the dependence of shell material properties on initial bubble radius. It is shown that for early shell models, the main problem is that the behavior of encapsulation is described by linear elastic and viscous laws, whereas recent experimental data attest to complicated rheological properties inherent in shell materials. Currently, a trend toward models involving nonlinear laws for shell elasticity and viscosity is observed. In particular, nonlinear models have been proposed that allow one to reproduce compression-only behavior. However, the problem of the radius dependence of shell material parameters remains unsolved. [ABSTRACT FROM AUTHOR]