Higher-order Terms Research Articles

Best Theory Diagrams for Laminated Composite Shells Based on Failure Indexes

This paper presents a novel approach to developing 2D structural theories for composite shells. The proposed approach uses the capabilities of the Carrera Unified Formulation (CUF) in conjunction with the Axiomatic/Asymptotic Method (AAM) to obtain the best theories for given structural layouts. Different structural cases are considered to analyze the influence of factors such as boundary conditions, lamination, and thickness on the accuracy of a model. The parameter chosen to evaluate a model’s performance is based on the Failure Indexes (FI) defined by the Hashin Failure criteria for unidirectional fiber composites. The outcome of this procedure is the Best Theory Diagram (BTD), containing the graphical representation of the highest accuracy as a function of the number of adopted unknowns. The results show the importance of higher-order terms to capture stress fields and the influence of thickness on the definition of the best theories.

A Generalized Two-Component Camassa-Holm System with Complex Nonlinear Terms and Waltzing Peakons

In this paper, we deal with the Cauchy problem for a generalized two-component Camassa-Holm system with waltzing peakons and complex higher-order nonlinear terms. By the classical Friedrichs regularization method and the transport equation theory, the local well-posedness of solutions for the generalized coupled Camassa-Holm system in nonhomogeneous Besov spaces and the critical Besov space B^{5/2}_{2,1}times B^{5/2}_{2,1} was obtained. Besides the propagation behaviors of compactly supported solutions, the global existence and precise blow-up mechanism for the strong solutions of this system are determined. In addition to wave breaking, the another one of the most essential property of this equation is the existence of waltzing peakons and multi-peaked solitray was also obtained.

A general leading-order asymptotic theory of thin microstructural plates and uncertainty quantification of the elastic performance of composite laminates

We propose an asymptotic framework that may be used as a platform to model general thin microstructural plates, and our initiative is twofold. First, it provides an explanation to why so many tailored plate theories have been introduced for modelling plates bearing microstructures. In hypotheses-based plate theories, the microstructural details within each ply are usually smeared out. But it is shown that with such in-ply details properly taken into account, one manages to address, even only with leading-order terms, several key issues, such as transverse shear strain effects, shear locking, etc., whose rationalisation is believed to necessitate higher-order terms. To a certain degree, the proposed formulation can be used as a target for tailoring kinematic shape functions in both Reissner-Mindlin types of plate theories and zig-zag theories. Second, being slender in nature, a microstructural plate should accommodate a stress state that is close to the state of plane stress. This agrees with our findings that the (leading-order) mechanical units of a microstructural plate should be two-dimensional slices parameterised with the thickness coordinate, distinguishing the proposed framework from other plate theories based on asymptotic analysis. To demonstrate what the present framework can predict beyond existing theories, the effects on the behaviour of fibrous laminates due to manufacturing uncertainties are investigated.

Multiple mixed state variable incremental integration for reconstructing extreme multistability in a novel memristive hyperchaotic jerk system with multiple cubic nonlinearity

Memristor-based chaotic systems with infinite equilibria are interesting because they generate extreme multistability. Their initial state-dependent dynamics can be explained in a reduced-dimension model by converting the incremental integration of the state variables into system parameters. However, this approach cannot solve memristive systems in the presence of nonlinear terms other than the memristor term. In addition, the converted state variables may suffer from a degree of divergence. To allow simpler mechanistic analysis and physical implementation of extreme multistability phenomena, this paper uses a multiple mixed state variable incremental integration (MMSVII) method, which successfully reconstructs a four-dimensional hyperchaotic jerk system with multiple cubic nonlinearities except for the memristor term in a three-dimensional model using a clever linear state variable mapping that eliminates the divergence of the state variables. Finally, the simulation circuit of the reduced-dimension system is constructed using Multisim simulation software and the simulation results are consistent with the MATLAB numerical simulation results. The results show that the method of MMSVII proposed in this paper is useful for analyzing extreme multistable systems with multiple higher-order nonlinear terms.

A Linear Theory of Local Thermal Circulations

The dynamics of local thermal circulations (LTCs) are examined by constructing a linear theory based on Boussinesq equations in the planetary boundary layer (PBL). Linear theory arranges LTCs into a sixth-order partial differential equation of temperature, which can be solved by using the Fourier transform and inverse Fourier transform. The analytic solution suggests that the horizontal distribution of the temperature anomaly is basically determined by surface heating, while the vertical distribution of the temperature anomaly is a combination of exponential decay and an Ekman spiral. For shallow PBL cases where the Ekman elevation is much smaller than the vertical scale of motion, the higher-order partial differential terms that represent the Ekman spiral structure can be ignored so that the equations reduce to a second-order partial differential equation. Compared with the numerical results, this so-called low-order approximation does not induce dramatic errors in the temperature distribution. However, to avoid distortion in the forced atmospheric circulation, the eddy viscosity in the motion equations should be replaced with the Rayleigh form, which is the common practice in LTCs. For deep PBL cases where the Ekman elevation is comparable to the vertical scale of motion, both the exponential decay and Ekman spiral structure play roles in the forced atmospheric circulation. The most significant influence is that there exist three additional compensating forced circulation cells that surround the direct forced circulation cell.

Shadows of new physics on Dirac materials, analog GUPs and other amusements

We discuss here how, when higher-order effects in the parameter , related to the lattice spacing ℓ, are considered, pristine graphene, and other Dirac materials, can be used as tabletop systems where generalized commutation relations are naturally realized. Such generalized algebras of quantization, which lead to generalized versions of the Heisenberg uncertainty principle, are under intense scrutiny these days, as they could manifest a fundamental length scale of spacetime. Despite the efforts and the many intriguing results, there are no experimental signatures of any generalized uncertainty principle (GUP). Therefore, our results here, which tell how to use tabletop physical systems to test certain GUPs in analog experiments, should be of interest to practitioners of quantum gravity. We identify three different energy regimes that we call “layers”, where the physics is still of a Dirac type but within precisely described limits. The higher the energy, the more sensitive the Dirac system becomes to the effects of the lattice. Here such lattice plays the role of a discrete space where the Dirac quasi-particles live. With the goals just illustrated, we had to identify the mapping between the high-energy coordinates, X i , and the low-energy ones, x i , i.e., those measured in the lab. We then obtained three generalized Heisenberg algebras. For two of them we have the noticeable result that X i = x i , and for the third one we obtained an improvement with respect to an earlier work: the generalized coordinates expressed in terms of the standard phase space variables, X i (x, p), and higher order terms.

Stabilization of Highly Nonlinear Stochastic Time-Varying Coupled Systems via Aperiodically Intermittent Control

Highly non-linear stochastic time-varying coupled systems are first considered in which the coupling structure is time-variant. It is worth pointing out that aperiodically intermittent control (AIC) is considered to achieve the stability of highly non-linear stochastic time-varying coupled systems. Since the existing research methods for processing AIC are not suitable for stochastic highly non-linear systems, a new Halanay-type differential inequality with higher-order nonlinear terms that extends the existing Halanay-type differential inequalities is established. Then, with the help of the Lyapunov method, some techniques of inequalities and the graph theory, two stabilization criteria are presented to guarantee the exponential stabilization for highly non-linear stochastic time-varying coupled systems. Finally, as an application of our results, the modified time-varying coupled Van der Pol-Duffing oscillators are studied via AIC with numerical simulations provided.

On the well-posed variational principle in degenerate point particle systems using embeddings of the symplectic manifold

Abstract A methodology on making the variational principle well-posed in degenerate systems is constructed. In the systems including higher-order time derivative terms being compatible with Newtonian dynamics, we show that a set of position variables of a coordinate system of a given system has to be fixed on the boundaries and that such systems are always Ostrogradski stable. For these systems, Frobenius integrability conditions are derived in explicit form. Relationships between integral constants indicated from the conditions and boundary conditions in a given coordinate system are also investigated by introducing three fundamental correspondences between Lagrange and Hamilton formulation. Based on these ingredients, we formulate problems that have to be resolved to realize the well-posedness in the degenerate systems. To resolve the problems, we compose a set of embeddings that extract a subspace holding the symplectic structure of the entire phase space in which the variational principle should be well-posed. Using these embeddings, we establish a methodology to set appropriate boundary conditions that the well-posed variational principle demands. Finally, we apply the methodology to examples and summarize this work as a three-step procedure such that one can use just by following it.

Bayesian uncertainty quantification of perturbative QCD input to the neutron-star equation of state

The equation of state of neutron-star cores can be constrained by requiring a consistent connection to the perturbative Quantum Chromodynamics (QCD) calculations at high densities. The constraining power of the QCD input depends on uncertainties from missing higher-order terms, the choice of the unphysical renormalization scale, and the reference density where QCD calculations are performed. Within a Bayesian approach, we discuss the convergence of the perturbative QCD series, quantify its uncertainties at high densities, and present a framework to systematically propagate the uncertainties down to neutron-star densities. We find that the effect of the QCD input on the neutron-star inference is insensitive to the various unphysical choices made in the uncertainty estimation.

Stability analysis of hybrid high-order nonlinear multiple time-delayed coupled systems via aperiodically intermittent control

New results on the stability of hybrid high-order nonlinear multiple time-delayed coupled systems (HHNCSs) are presented by aperiodically intermittent control (AIC). The model considered in this paper includes Markovian switching and multiple time delays, which make the high-order nonlinear coupled systems more accurately simulate the actual models. In addition, Halanay-type differential inequalities are powerful tools when investigating the stability of time-delayed systems with AIC. However, existing Halanay-type differential inequalities are not applicable for HHNCSs, since high-order nonlinear terms exist. Therefore, a novel Halanay-type differential inequality is established which not only generalizes the classic Halanay inequality but also develops the applications of AIC under the condition of high-order nonlinearity. On the foundation of this innovative Halanay-type differential inequality, sufficient conditions are obtained by employing the graph theory and the Lyapunov method. Finally, the obtained theoretical results can be applied to modified coupled Van Pol–Duffing oscillators and some simulation results are given to demonstrate the feasibility and validity of our results.

Tackling Modeling and Kinematic Inconsistencies by Fixed Point Iteration-Based Adaptive Control

The Fixed Point Iteration-based Adaptive Control design methodology is an alternative to the Lyapunov function-based technology. It contains higher-order feedback terms than the standard resolved acceleration rate control. This design approach strictly separates the kinematic and dynamic issues. At first, a purely kinematic prescription is formulated for driving the components of the tracking error to zero. Then an available approximate dynamic model is used to calculate the approximated necessary control forces. Before exerting on the controlled system, these forces are adaptively deformed in order to precisely obtain the prescribed kinematic behavior. The necessary deformation is iteratively found by the use of a contractive map that results in a sequence that converges to the unique fixed point of this map. In the case of underactuated systems, when the relative order of the control task also increases, the highest-order time-derivative depends on the lower-order ones according to the dynamic model of the system. This makes it impossible to realize the arbitrarily constructed kinematic design. In the paper, a resolution to this discrepancy is proposed. The method is demonstrated using two non-linear paradigms, a three-degree-of-freedom robot arm, and a two-degree-of-freedom system, i.e., two coupled non-linear springs. The operation of the method was investigated via simulations made by the use of Julia language and simple sequential programs. It was found that the suggested solution could be considered as a new variant of the fixed point iteration-based model reference adaptive control that is applicable for underactuated systems even if the relative order of the task is increased.

Identification of a soil water flow equation using a two-step data-driven method

Governing equations of soil water flow are traditionally derived from the conservation of mass and the Buckingham-Darcy law. However, due to uncertainties in many aspects, such as physical processes and constitutive relationships, a thorough understanding of soil water flow in a theory-driven way often poses many challenges. With the fast improvement of data acquisition ability, data-driven approaches have gained considerable scientific interest in discovering the governing equations of physical problems. However, most existing studies focused on linear or weakly nonlinear problems with constant coefficients in their partial differential equations. This study attempted to propose a modified two-step data-driven method to identify the strongly nonlinear equation of unsaturated flow. The Richardson-Richards equation and coefficients were jointly identified by hybridizing sparse regression and iterative smoother. The automatic differentiation scheme was introduced to reduce the risk of performance collapse from data noise common in sparse regression methods. We demonstrated the ability, and the challenge as well, of identifying the soil water flow governing equations in a data-driven way with the aid of a series of synthetic cases. Our results showed that the modified method was capable of accurately identifying the governing equation of unsaturated flow, whose performance was almost independent of prior knowledge of the equation form and constitutive relationships. If the underlying physical processes are poorly understood, then a more relaxed equation form was recommended. The reduction in the spatiotemporal resolution of sampling may cause higher-order terms to be difficult to derive accurately or some terms that were not originally present to appear in the identified equation. Although not completely eliminated, the detrimental effect of data noise on the identification of the governing equations can be greatly mitigated by introducing the automatic differentiation scheme.

Nonlinear internal wave reflection and transmission at an interface

The reflection and transmission of internal waves at a horizontal interface between layers of constant buoyancy frequency are treated. This interface is a simple model of the atmospheric tropopause. The waves are weakly nonlinear and obey the nonlinear Schrödinger equation away from the interface. The waves are horizontally periodic and vertically confined to a long packet. The interfacial conditions are formally taken to third order, and include higher-order linear as well as nonlinear effects. The higher-order linear affects show an instability at the interface for steep waves. Numerical results for shorter wave packets show that the higher-order linear terms result in non-physical wave generation, and are ultimately neglected for such cases. Nonlinear effects result in a decrease in the reflected wave amplitudes and an increase in the transmitted wave amplitudes when compared to linear theory. Overall, the nonlinear effects make the interface more transparent to upwardly propagating internal waves.

Hydrodynamic gauge fixing and higher order hydrodynamic expansion

Hydrodynamics is a powerful emergent theory for the large-scale behaviours in many-body systems, quantum or classical. It is a gradient series expansion, where different orders of spatial derivatives provide an effective description on different length scales. We report the first fully general derivation of third-order, or ‘dispersive’, terms in the hydrodynamic expansion. Our derivation is based on general principles of statistical mechanics, along with the assumption that the complete set of local and quasi-local conserved densities constitutes a good set of emergent degrees of freedom. We obtain fully general Kubo-like expressions for the associated hydrodynamic coefficients (also known as Burnett coefficients), and we determine their exact form in quantum integrable models, introducing in this way purely quantum higher-order terms into generalised hydrodynamics. We emphasise the importance of hydrodynamic gauge fixing at diffusive order, where we claim that it is parity-time-reversal, and not time-reversal, invariance that is at the source of Einstein’s relation, Onsager’s reciprocal relations, the Kubo formula and entropy production. At higher hydrodynamic orders we introduce a more general, nth order ‘symmetric’ gauge, which we show implies the validity of the higher-order hydrodynamic description.

Effect of annealing on the magnetic microstructure of high-pressure torsion iron: the relevance of higher-order contributions to the magnetic small-angle neutron scattering cross section.

The development of higher-order micromagnetic small-angle neutron scattering theory in nanocrystalline materials is still in its infancy. One key challenge remaining in this field is understanding the role played by the microstructure on the magnitude and sign of the higher-order scattering contribution recently observed in nanocrystalline materials prepared by high-pressure torsion. By combining structural and magnetic characterization techniques, namely X-ray diffraction, electron backscattered diffraction and magnetometry with magnetic small-angle neutron scattering, this work discusses the relevance of higher-order terms in the magnetic small-angle neutron scattering cross section of pure iron prepared by high-pressure torsion associated with a post-annealing process. The structural analysis confirms: (i) the preparation of ultra-fine-grained pure iron with a crystallite size below 100 nm and (ii) rapid grain growth with increasing annealing temperature. The analysis of neutron data based on the micromagnetic small-angle neutron scattering theory extended to textured ferromagnets yields uniaxial magnetic anisotropy values that are larger than the magnetocrystalline value reported for bulk iron, supporting the existence of induced magnetoelastic anisotropy in the mechanically deformed samples. Furthermore, the neutron data analysis revealed unambiguously the presence of non-negligible higher-order scattering contributions in high-pressure torsion iron. Though the sign of the higher-order contribution might be related to the amplitude of the anisotropy inhomogeneities, its magnitude appears to be clearly correlated to the changes in the microstructure (density and/or shape of the defects) induced by combining high-pressure torsion and a post-annealing treatment.

Precision measurement of the matrix elements for η→π+π−π0 and η→π0π0π0 decays

A precision measurement of the matrix elements for $\eta\to\pi^+\pi^-\pi^0$ and $\eta\to\pi^0\pi^0\pi^0$ decays is performed using a sample of $(10087\pm44)\times10^6$ $J/\psi$ decays collected with the BESIII detector. The decay $J/\psi \to \gamma \eta$ is used to select clean samples of 631,686 $\eta\to\pi^+\pi^-\pi^0$ decays and 272,322 $\eta\to\pi^0\pi^0\pi^0$ decays. The matrix elements for both channels are in reasonable agreement with previous measurements. The non-zero $gX^2Y$ term for the decay mode $\eta\to\pi^+\pi^-\pi^0$ is confirmed, as reported by the KLOE Collaboration, while the other higher-order terms are found to be insignificant. Dalitz plot asymmetries in the $\eta\to\pi^+\pi^-\pi^0$ decay are also explored and are found to be consistent with charge conjugation invariance. In addition, a cusp effect is investigated in the $\eta\to\pi^0\pi^0\pi^0$ decay, and no obvious structure around the $\pi^+\pi^-$ mass threshold is observed.

The generalized Clapeyron equation and its application to confined ice growth

AbstractMost theoretical descriptions of stresses induced by freezing are rooted in the (generalized) Clapeyron equation, which predicts the pressure that a solid can exert as it cools below its melting temperature. This equation is central for topics ranging beyond glaciology to geomorphology, civil engineering, food storage and cryopreservation. However, it has inherent limitations, requiring isotropic solid stresses and conditions near bulk equilibrium. Here, we examine when the Clapeyron equation is applicable by providing a rigorous derivation that details all assumptions. We demonstrate the natural extension for anisotropic stress states, and we show how the temperature and pressure ranges for validity depend on well-defined material properties. Finally, we demonstrate how the range of applicability of the (linear) Clapeyron equation can be extended by adding higher-order terms, yielding results that are in good agreement with experimental data for the pressure melting of ice.

Stochastic resonance of multi-stable energy harvesting system with high-order stiffness from rotational environment

This paper aims to study the stochastic resonance of multi-stable energy harvesting system and enhance the performance of energy harvesting from random rotational environment. The influences of high-order stiffness terms and multi-stable characteristics are analyzed. The spectral amplification is derived to quantify the stochastic resonance for the energy harvesting system with penta-stable configuration driven by noise and periodic forcing. The non-monotonic dependence of spectral amplification on noise intensity indicates the existence of stochastic resonance. In particular, a critical high-order stiffness coefficient is revealed, for which the stochastic resonance is optimum. Meanwhile, the maximum of spectral amplification can be optimized by choosing appropriate unstable equilibrium positions. In addition, it is found that both the average input energy and the overall efficiency in power conversion can be further enhanced under conditions of stochastic resonance. Finally, the penta-stable energy harvester demonstrates a good performance in output power for different noise intensities and damping coefficients.

Single monkey-saddle singularity of a Fermi surface and its instabilities

Fermi surfaces can undergo sharp transitions under smooth changes of parameters. Such transitions can have a topological character, as is the case when a higher-order singularity, one that requires cubic or higher-order terms to describe the electronic dispersion near the singularity, develops at the transition. When time-reversal and inversion symmetries are present, odd singularities can only appear in pairs within the Brillouin zone. In this case, the combination of the enhanced density of states that accompanies these singularities and the nesting between the pairs of singularities leads to interaction-driven instabilities. We present examples of single $n=3$ (monkey-saddle) singularities when time-reversal and inversion symmetries are broken. We then turn to the question of what instabilities are possible when the singularities are isolated. For spinful electrons, we find that the inclusion of repulsive interactions destroys any isolated monkey-saddle singularity present in the noninteracting spectrum by developing Stoner or Lifshitz instabilities. In contrast, for spinless electrons and at the mean-field level, we show that an isolated monkey-saddle singularity can be stabilized in the presence of short-range repulsive interactions.

Quantum decay of an optical soliton

Optical solitons are known to be classically stable objects which are robust to perturbations. In this work, we show that due to quantum mechanical effects, an optical soliton that is initially in a classical soliton coherent state will shed photons into the continuum and hence decay. The standard formulation of the quantized soliton uses the linearized version of the quantum nonlinear Schrodinger equation in the background of the classical soliton, and the quantized soliton remains stable in this approximation. We show that if higher-order interaction terms are taken into account, the soliton is no longer stable, and its photon number decreases quadratically as a function of the number of soliton cycles. We compute the power spectrum for the continuum radiation and find a narrow band that is localized about the initial soliton momentum with a cut-off that is inversely proportional to the initial soliton width.