On iterative learning control design for tracking iteration-varying trajectories with high-order internal model

In this paper, iterative learning control （ILC） design is studied for an iteration-varying tracking problem in which reference trajectories are generated by high-order internal models （HOLM）. An HOlM formulated as a polynomial operator between consecutive iterations describes the changes of desired trajectories in the iteration domain and makes the iterative learning problem become iteration varying. The classical ILC for tracking iteration-invariant reference trajectories, on the other hand, is a special case of HOlM where the polynomial renders to a unity coefficient or a special first-order internal model. By inserting the HOlM into P-type ILC, the tracking performance along the iteration axis is investigated for a class of continuous-time nonlinear systems. Time-weighted norm method is utilized to guarantee validity of proposed algorithm in a sense of data-driven control.

Numerical modeling of wave equation by a truncated high-order finite-difference method

Finite-difference methods with high-order accuracy have been utilized to improve the precision of numerical solution for partial differential equations. However, the computation cost generally increases linearly with increased order of accuracy. Upon examination of the finite-difference formulas for the first-order and second-order derivatives, and the staggered finite-difference formulas for the first-order derivative, we examine the variation of finite-difference coefficients with accuracy order and note that there exist some very small coefficients. With the order increasing, the number of these small coefficients increases, however, the values decrease sharply. An error analysis demonstrates that omitting these small coefficients not only maintain approximately the same level of accuracy of finite difference but also reduce computational cost significantly. Moreover, it is easier to truncate for the high-order finite-difference formulas than for the pseudospectral for- mulas. Thus this study proposes a truncated high-order finite-difference method, and then demonstrates the efficiency and applicability of the method with some numerical examples.

High-order harmonic generation of ZnO crystals in chirped and static electric fields

High harmonic generation in ZnO crystals under chirped single-color field and static electric field are investigated by solving the semiconductor Bloch equation(SBE). It is found that when the chirp pulse is introduced, the interference structure becomes obvious while the harmonic cutoff is not extended. Furthermore, the harmonic efficiency is improved when the static electric field is included. These phenomena are demonstrated by the classical recollision model in real space affected by the waveform of laser field and inversion symmetry. Specifically, the electron motion in k-space shows that the change of waveform and the destruction of the symmetry of the laser field lead to the incomplete X-structure of the crystal-momentum-resolved(k-resolved) inter-band harmonic spectrum. Furthermore, a pre-acceleration process in the solid four-step model is confirmed.

High-order targeted essentially non-oscillatory scheme for two-fluid plasma model

The weakly ionized plasma flows in aerospace are commonly simulated by the single-fluid model,which cannot describe certain nonequilibrium phenomena by finite collisions of particles,decreasing the fidelity of the solution.Based on an alternative formulation of the targeted essentially non-oscillatory(TENO)scheme,a novel high-order numerical scheme is proposed to simulate the two-fluid plasmas problems.The numerical flux is constructed by the TENO interpolation of the solution and its derivatives,instead of being reconstructed from the physical flux.The present scheme is used to solve the two sets of Euler equations coupled with Maxwell's equations.The numerical methods are verified by several classical plasma problems.The results show that compared with the original TENO scheme,the present scheme can suppress the non-physical oscillations and reduce the numerical dissipation.

Multi-factor high-order intuitionistic fuzzy timeseries forecasting model

Fuzzy sets theory cannot describe the neutrality degreeof data, which has largely limited the objectivity of fuzzy time seriesin uncertain data forecasting. With this regard, a multi-factor highorderintuitionistic fuzzy time series forecasting model is built. Inthe new model, a fuzzy clustering algorithm is used to get unequalintervals, and a more objective technique for ascertaining membershipand non-membership functions of the intuitionistic fuzzy setis proposed. On these bases, forecast rules based on multidimensionalintuitionistic fuzzy modus ponens inference are established.Finally, contrast experiments on the daily mean temperature ofBeijing are carried out, which show that the novel model has aclear advantage of improving the forecast accuracy.

Calibration of quantitative rescattering model for simulating vortex high-order harmonic generation driven by Laguerre–Gaussian beam with nonzero orbital angular momentum

We calibrate the macroscopic vortex high-order harmonic generation(HHG)obtained by the quantitative rescattering(QRS)model to compute single-atom induced dipoles against that by solving the time-dependent Schr?dinger equation(TDSE).We show that the QRS perfectly agrees with the TDSE under the favorable phase-matching condition,and the QRS can accurately predict the main features in the spatial profiles of vortex HHG if the phase-matching condition is not good.We uncover that harmonic emissions from short and long trajectories are adjusted by the phase-matching condition through the time-frequency analysis and the QRS can simulate the vortex HHG accurately only when the interference between two trajectories is absent.This work confirms that it is an efficient way to employ the QRS model in the single-atom response for precisely simulating the macroscopic vortex HHG.

A High-Order Conservative Numerical Method for Gross-Pitaevskii Equation with Time-Varying Coefficients in Modeling BEC

We propose a high-order conservative method for the nonlinear Sehodinger/Gross-Pitaevskii equation with time- varying coefficients in modeling Bose Einstein condensation （BEC）. This scheme combined with the sixth-order compact finite difference method and the fourth-order average vector field method, finely describes the condensate wave function and physical characteristics in some small potential wells. Numerical experiments are presented to demonstrate that our numerical scheme is efficient by the comparison with the Fourier pseudo-spectral method. Moreover, it preserves several conservation laws well and even exactly under some specific conditions.

A truncated implicit high-order finite-difference scheme combined with boundary conditions

In this paper, first we calculate finite-difference coefficients of implicit finite- difference methods （IFDM） for the first and second-order derivatives on normal grids and first- order derivatives on staggered grids and find that small coefficients of high-order IFDMs exist. Dispersion analysis demonstrates that omitting these small coefficients can retain approximately the same order accuracy but greatly reduce computational costs. Then, we introduce a mirrorimage symmetric boundary condition to improve IFDMs accuracy and stability and adopt the hybrid absorbing boundary condition （ABC） to reduce unwanted reflections from the model boundary. Last, we give elastic wave modeling examples for homogeneous and heterogeneous models to demonstrate the advantages of the proposed scheme.

A high-order splitting scheme for the advection diffusion equation

A high-order splitting scheme for the advection-diffusion equation of pollutants is proposed in this paper. The multidimensional advection-diffusion equation is splitted into several one-dimensional equations that are solved by the scheme. Only three spatial grid points are needed in each direction and the scheme has fourth-order spatial accuracy. Several typically pure advection and advection-diffusion problems are simulated. Numerical results show that the accuracy of the scheme is much higher than that of the classical schemes and the scheme can he efficiently solved with little programming effort.

A corrected particle method with high-order Taylor expansion for solving the viscoelastic fluid flow

In this paper, a corrected particle method based on the smoothed particle hydrodynamics (SPH) method with high-order Taylor expansion (CSPH-HT) for solving the viscoelastic flow is proposed and investigated. The validity and merits of the CSPH-HT method are first tested by solving the nonlinear high order Kuramoto-Sivishinsky equation and simulating the drop stretching, respectively. Then the flow behaviors behind two stationary tangential cylinders of polymer melt, which have been received little attention, are investigated by the CSPH-HT method. Finally, the CSPH-HT method is extended to the simulation of the filling process of the viscoelastic fluid. The numerical results show that the CSPH-HT method possesses higher accuracy and stability than other corrected SPH methods and is more reliable than other corrected SPH methods.

Retrieval of Electron Return Time from High-order Harmonics Generated in a Mixture of He and Ne Gases

In terms of single-atom induced dipole moment by Lewenstein model, we present the macroscopic high-order harmonic generation from mixed He and Ne gases with different mixture ratios by solving three-dimensional Maxwell＇s equation of harmonic field. And then we show the validity of mixture formulation by Wagner et al. [Phys. Rev. A 76 （2007） 061403（R）] in macroscopic response level. Finally, using/east squares fitting we retrieve the electron return time of short trajectory by formulation in Kanai et al. [Phys. Rev. Lett. 98 （2007） 153904] when the gas jet is put after the laser focus.

A force control high-order single-step-β method (HSM) for substructure pseudo-dynamic testing

This paper proposes a new technique which introduces the high-order single-step-β method(HSM)into the experimental study on the substructure pseudo-dynamic testing(SPDT).The technique is based on the proposed concept of equivalent shear stiffness which can meet the requirement of the HSM algorithm.A study is done to theoretically validate the technique by the numerical analysis of two-storey shear building structure,in comparison of the proposed substructure pseudo-dynamic testing algorithm with the central difference method(CDM).Then,a full-scale SPDT model,the three-storey frame-supported reinforced concrete short-limb masonry shear wall structure,is designed and tested to simulate the seismic response of the corresponding six-storey structure and verify the proposed force control HSM technique.Meanwhile,the techniques of both stiffness correction and force control are suggested to control algorithmic error,control error and measurement error.The results indicate that the force control HSM can be used in the full-scale multi-degree-of-freedom(MDOF)substructure pseudo-dynamic testing before descent segment of structure restoring force properties.

Control of high-order harmonic generation with two-colour laser field

We numerically investigate the high-order harmonic generation with two-colour optical field, taking into consideration the propagation effects. Some harmonics can be dramatically enhanced at a certain delay between the fundamental pulse and its second harmonics. Choice of the enhanced harmonics can be realised by changing the time delay between the two laser pulses.

High-order harmonic generation of Li^+ with combined infrared and extreme ultraviolet fields

We investigate high-order harmonic generation （HHG） ofLi＋ ion driven by an intense infrared （IR） laser field in corn-bination with a weak XUV pulse. To achieve this, we first construct an accurate single-active electron angular-momentumdependent model potential of Li＋ ion, by which the accurate singlet energy levels of Li＋ for the ground state and excited states with higher quantum numbers can be obtained. Then, we solve numerically the three dimensional time-dependent Schr/Sdinger equation of Li＋ ion by means of the generalized pseudospectral method to obtain HHG. Our results show that the strength of assisted XUV is not amplified during the harmonic generation process, but the yield of HHG power spectrum in the whole plateau has a significant enhancement. Furthermore, the optimal phase delay between the IR and XUV pulses allows the production of ultrabroadband supercontinuum spectra. By superposing some harmonics, a strong new single 27-attosecond ultrashort pulse can be obtained.

Lattice Boltzmann simulations of high-order statistics in isotropic turbulent flows

The lattice Boltzmann method （LBM） is coupled with the multiple-relaxation- time （MRT） collision model and the three-dimensional 19-discrete-velocity （D3Q19） model to resolve intermittent behaviors on small scales in isotropic turbulent flows. The high- order scaling exponents of the velocity structure functions, the probability distribution functions of Lagrangian accelerations, and the local energy dissipation rates are investi- gated. The self-similarity of the space-time velocity structure functions is explored using the extended self-similarity （ESS） method, which was originally developed for velocity spatial structure functions. The scaling exponents of spatial structure functions at up to ten orders are consistent with the experimental measurements and theoretical results, implying that the LBM can accurately resolve the intermittent behaviors. This valida~ tion provides a solid basis for using the LBM to study more complex processes that are sensitive to small scales in turbulent flows, such as the relative dispersion of pollutants and mesoscale structures of preferential concentration of heavy particles suspended in turbulent flows.

A High-Order Conservative Semi-Lagrangian Solver for 3D Free Surface Flows with Sediment Transport on Voronoi Meshes

In this paper,we present a conservative semi-Lagrangian scheme designed for the numeri-cal solution of 3D hydrostatic free surface flows involving sediment transport on unstruc-tured Voronoi meshes.A high-order reconstruction procedure is employed for obtaining a piecewise polynomial representation of the velocity field and sediment concentration within each control volume.This is subsequently exploited for the numerical integration of the Lagrangian trajectories needed for the discretization of the nonlinear convective and viscous terms.The presented method is fully conservative by construction,since the transported quantity or the vector field is integrated for each cell over the deformed vol-ume obtained at the foot of the characteristics that arises from all the vertexes defining the computational element.The semi-Lagrangian approach allows the numerical scheme to be unconditionally stable for what concerns the advection part of the governing equations.Furthermore,a semi-implicit discretization permits to relax the time step restriction due to the acoustic impedance,hence yielding a stability condition which depends only on the explicit discretization of the viscous terms.A decoupled approach is then employed for the hydrostatic fluid solver and the transport of suspended sediment,which is assumed to be passive.The accuracy and the robustness of the resulting conservative semi-Lagrangian scheme are assessed through a suite of test cases and compared against the analytical solu-tion whenever is known.The new numerical scheme can reach up to fourth order of accu-racy on general orthogonal meshes composed by Voronoi polygons.

Stability of High-Order Linear ItôEquations with Delays

A novel general stability analysis scheme based on a non-Lyapunov framework is explored. Several easy-to-check sufficient conditions for exponential p-stability are formulated in terms of M-matrices. Stability analysis of applied second-order It? equations with delay is provided as well. The linearization technique, in combination with the tests obtained in this paper, can be used for local stability analysis of a wide class of nonlinear stochastic differential equations.

Properties of the phase diagram from the Nambu-Jona-Lasino model with a scalar-vector interaction

We investigated the properties of the phase diagram of high-order susceptibilities,speed of sound,and polytropic index based on an extended Nambu-Jona-Lasinio model with an eight-quark scalar-vector interaction.Non-monotonic behavior was observed in all these quantities around the phase transition boundary,which also revealed the properties of the critical point.Further,this study indicated that the chiral phase transition boundary and critical point could vary depending on the scalarvector coupling constant G_(SV).At finite densities and temperatures,the negative G_(SV)term exhibited attractive interactions,which enhanced the critical point temperature and reduced the chemical potential.The G_(SV)term also affected the properties of the high-order susceptibilities,speed of sound,and polytropic index near the critical point.The non-monotonic(peak or dip)structures of these quantities shifted to a low baryon chemical potential(and high temperature)with a negative G_(SV).G_(SV)also changed the amplitude and range of the nonmonotonic regions.Therefore,the scalar-vector interaction was useful for locating the phase boundary and critical point in QCD phase diagram by comparing the experimental data.The study of the non-monotonic behavior of high-order susceptibilities,speed of sound,and polytropic index is of great interest,and further observations related to high-order susceptibilities,speed of sound,and polytropic index being found and applied to the search for critical points in heavy-ion collisions and the study of compact stars are eagerly awaited.

SimHOEPI:A resampling simulator for generating single nucleotide polymorphism data with a high-order epistasis model

Epistasis is a ubiquitous phenomenon in genetics,and is considered to be one of main factors in current efforts to unveil missing heritability of complex diseases.Simulation data is crucial for evaluating epistasis detection tools in genome-wide association studies(GWAS).Existing simulators normally suffer from two limitations:absence of support for high-order epistasis models containing multiple single nucleotide polymorphisms(SNPs),and inability to generate simulation SNP data independently.In this study,we proposed a simulator SimHOEPI,which is capable of calculating penetrance tables of high-order epistasis models depending on either prevalence or heritability,and uses a resampling strategy to generate simulation data independently.Highlights of SimHOEPI are the preservation of realistic minor allele frequencies in sampling data,the accurate calculation and embedding of high-order epistasis models,and acceptable simulation time.A series of experiments were carried out to verify these properties from different aspects.Experimental results show that SimHOEPI can generate simulation SNP data independently with high-order epistasis models,implying that it might be an alternative simulator for GWAS.

Global Climate Internal Variability in a 2000-year Control Simulation with Community Earth System Model(CESM)

Using the low-resolution （T31, equivalent to 3.75°× 3.75°） version of the Community Earth System Model （CESM） from the National Center for Atmospheric Research （NCAR）, a global climate simulation was carried out with fixed external forcing factors （1850 Common Era. （C.E.） conditions） for the past 2000 years. Based on the simulated results, spatio-temporal structures of surface air temperature, precipitation and internal variability, such as the E1 Nifio-Southem Oscillation （ENSO）, the Atlantic Multi-decadal Oscilla- tion （AMO）, the Pacific Decadal Oscillation （PDO）, and the North Atlantic Oscillation （NAO）, were compared with reanalysis datasets to evaluate the model performance. The results are as follows： 1） CESM showed a good performance in the long-term simulation and no significant climate drift over the past 2000 years; 2） climatological patterns of global and regional climate changes simulated by the CESM were reasonable compared with the reanalysis datasets; and 3） the CESM simulated internal natural variability of the climate system performs very well. The model not only reproduced the periodicity of ENSO, AMO and PDO events but also the 3-8 years vari- ability of the ENSO. The spatial distribution of the CESM-simulated NAO was also similar to the observed. However, because of weaker total irradiation and greenhouse gas concentration forcing in the simulation than the present, the model performances had some differences from the observations. Generally, the CESM showed a good performance in simulating the global climate and internal natu- ral variability of the climate system. This paves the way for other forced climate simulations for the past 2000 years by using the CESM.