Nonlinear Flap-Wise Vibration Characteristics ofWind Turbine Blades Based onMulti-Scale AnalysisMethod

This work presents a novel approach to achieve nonlinear vibration response based on the Hamilton principle.We chose the 5-MW reference wind turbine which was established by the National Renewable Energy Laboratory(NREL),to research the effects of the nonlinear flap-wise vibration characteristics.The turbine wheel is simplified by treating the blade of a wind turbine as an Euler-Bernoulli beam,and the nonlinear flap-wise vibration characteristics of the wind turbine blades are discussed based on the simplification first.Then,the blade’s large-deflection flap-wise vibration governing equation is established by considering the nonlinear term involving the centrifugal force.Lastly,it is truncated by the Galerkin method and analyzed semi-analytically using the multi-scale analysis method,and numerical simulations are carried out to compare the simulation results of finite elements with the numerical simulation results using Campbell diagram analysis of blade vibration.The results indicated that the rotational speed of the impeller has a significant impact on blade vibration.When the wheel speed of 12.1 rpm and excitation amplitude of 1.23 the maximum displacement amplitude of the blade has increased from 0.72 to 3.16.From the amplitude-frequency curve,it can be seen that the multi-peak characteristic of blade amplitude frequency is under centrifugal nonlinearity.Closed phase trajectories in blade nonlinear vibration,exhibiting periodic motion characteristics,are found through phase diagrams and Poincare section diagrams.

2D multi-scale hybrid optimization method for geophysical inversion and its application

Local and global optimization methods are widely used in geophysical inversion but each has its own advantages and disadvantages. The combination of the two methods will make it possible to overcome their weaknesses. Based on the simulated annealing genetic algorithm （SAGA） and the simplex algorithm, an efficient and robust 2-D nonlinear method for seismic travel-time inversion is presented in this paper. First we do a global search over a large range by SAGA and then do a rapid local search using the simplex method. A multi-scale tomography method is adopted in order to reduce non-uniqueness. The velocity field is divided into different spatial scales and velocities at the grid nodes are taken as unknown parameters. The model is parameterized by a bi-cubic spline function. The finite-difference method is used to solve the forward problem while the hybrid method combining multi-scale SAGA and simplex algorithms is applied to the inverse problem. The algorithm has been applied to a numerical test and a travel-time perturbation test using an anomalous low-velocity body. For a practical example, it is used in the study of upper crustal velocity structure of the A＇nyemaqen suture zone at the north-east edge of the Qinghai-Tibet Plateau. The model test and practical application both prove that the method is effective and robust.

Approximate analytical solution in slow-fast system based on modified multi-scale method

A simple,yet accurate modi?ed multi-scale method(MMSM)for an approximately analytical solution in nonlinear oscillators with two time scales under forced harmonic excitation is proposed.This method depends on the classical multi-scale method(MSM)and the method of variation of parameters.Assuming that the forced excitation is a constant,one could easily obtain the approximate analytical solution of the simpli?ed system based on the traditional MSM.Then,this solution for the oscillator under forced harmonic excitation could be established after replacing the harmonic excitation by the constant excitation.To certify the correctness and precision of the proposed analytical method,the van der Pol system with two scales subject to slowly periodic excitation is investigated;this system presents rich dynamical phenomena such as spiking(SP),spiking-quiescence(SP-QS),and quiescence(QS)responses.The approximate analytical expressions of the three types of responses are given by the MMSM,and it can be found that the precision of the new analytical method is higher than that of the classical MSM and better than that of the harmonic balance method(HBM).The results obtained by the present method are considerably better than those obtained by traditional methods,quantitatively and qualitatively,particularly when the excitation frequency is far less than the natural frequency of the system.

Integrated multi-scale approach combining global homogenization and local refinement for multi-field analysis of high-temperature superconducting composite magnets

Second-generation high-temperature superconducting(HTS)conductors,specifically rare earth-barium-copper-oxide(REBCO)coated conductor(CC)tapes,are promising candidates for high-energy and high-field superconducting applications.With respect to epoxy-impregnated REBCO composite magnets that comprise multilayer components,the thermomechanical characteristics of each component differ considerably under extremely low temperatures and strong electromagnetic fields.Traditional numerical models include homogenized orthotropic models,which simplify overall field calculation but miss detailed multi-physics aspects,and full refinement(FR)ones that are thorough but computationally demanding.Herein,we propose an extended multi-scale approach for analyzing the multi-field characteristics of an epoxy-impregnated composite magnet assembled by HTS pancake coils.This approach combines a global homogenization(GH)scheme based on the homogenized electromagnetic T-A model,a method for solving Maxwell's equations for superconducting materials based on the current vector potential T and the magnetic field vector potential A,and a homogenized orthotropic thermoelastic model to assess the electromagnetic and thermoelastic properties at the macroscopic scale.We then identify“dangerous regions”at the macroscopic scale and obtain finer details using a local refinement(LR)scheme to capture the responses of each component material in the HTS composite tapes at the mesoscopic scale.The results of the present GH-LR multi-scale approach agree well with those of the FR scheme and the experimental data in the literature,indicating that the present approach is accurate and efficient.The proposed GH-LR multi-scale approach can serve as a valuable tool for evaluating the risk of failure in large-scale HTS composite magnets.

Inexact Newton method via Lanczos decomposed technique for solving box-constrained nonlinear systems

This paper proposes an inexact Newton method via the Lanczos decomposed technique for solving the box-constrained nonlinear systems. An iterative direction is obtained by solving an affine scaling quadratic model with the Lanczos decomposed technique. By using the interior backtracking line search technique, an acceptable trial step length is found along this direction. The global convergence and the fast local convergence rate of the proposed algorithm are established under some reasonable conditions. Furthermore, the results of the numerical experiments show the effectiveness of the pro- posed algorithm.

STRAIGHTFORWARD MULTI-SCALE BOUNDARY ELEMENT METHOD FOR GLOBAL/LOCAL MECHANICAL ANALYSIS OF ELASTIC HETEROGENEOUS MATERIAL

A straightforward multi-scale boundary element method is proposed for global and local mechanical analysis of heterogeneous material.The method is more accurate and convenient than finite element based multi-scale method.The formulations of this method are derived by combining the homogenization approach and the fundamental equations of boundary element method.The solution gives the convenient formulations to compute global elastic constants and the local stress field.Finally,two numerical examples of porous material are presented to prove the accuracy and the efficiency of the proposed method.The results show that the method does not require the iteration to obtain the solution of the displacement in micro level.

Study on spline wavelet finite-element method in multi-scale analysis for foundation

A new finite element method （FEM） of B-spline wavelet on the interval （BSWI） is proposed. Through analyzing the scaling functions of BSWI in one dimension, the basic formula for 2D FEM of BSWI is deduced. The 2D FEM of 7 nodes and 10 nodes are constructed based on the basic formula. Using these proposed elements, the multiscale numerical model for foundation subjected to harmonic periodic load, the foundation model excited by external and internal dynamic load are studied. The results show the pro- posed finite elements have higher precision than the tradi- tional elements with 4 nodes. The proposed finite elements can describe the propagation of stress waves well whenever the foundation model excited by extemal or intemal dynamic load. The proposed finite elements can be also used to con- nect the multi-scale elements. And the proposed finite elements also have high precision to make multi-scale analysis for structure.

The Multi-scale Method for Solving Nonlinear Time Space Fractional Partial Differential Equations

In this paper, we present a new algorithm to solve a kind of nonlinear time space-fractional partial differential equations on a finite domain. The method is based on B-spline wavelets approximations, some of these functions are reshaped to satisfy on boundary conditions exactly. The Adams fractional method is used to reduce the problem to a system of equations. By multiscale method this system is divided into some smaller systems which have less computations. We get an approximated solution which is more accurate on some subdomains by combining the solutions of these systems. Illustrative examples are included to demonstrate the validity and applicability of our proposed technique, also the stability of the method is discussed.

Dynamic response of multi-scale structure in flexible pavement to moving load

In order to study the dynamic responses in the microstructures of the pavement structure, the multi-scale modeling subjected to moving load is analyzed using the discrete element method （DEM）. The macro-scale discrete element model of the flexible pavement structure is established. The stress and strain at the bottom of the asphalt concrete layer under moving load are calculated. The DEM model is validated through comparison between DEM predictions and the results from the classical program. Based on the validated macro-scale DEM model, the distribution and the volumetric fraction of coarse aggregate, mastics and air voids at the bottom of the asphalt layer are modeled, and then the multi-scale model is constructed. The dynamic response in the microstructures of the multi-scale model are calculated and compared with the results from the macro model. The influence of mastic stiffness on the distribution of dynamic response in the microstructures is also analyzed. Results show that the average values and the variation coefficient of the tensile stress at the aggregate-mastic interface are far more than those within the mastics. The dynamic response including stress and strain distributes non-uniformly in both mastics and the interface. An increase in mastic stiffness tends to a uniform distribution of tensile stress in asphalt concrete.

Development and application of a multi-physics and multi-scale coupling program for lead-cooled fast reactor

In this study,a multi-physics and multi-scale coupling program,Fluent/KMC-sub/NDK,was developed based on the user-defined functions(UDF)of Fluent,in which the KMC-sub-code is a sub-channel thermal-hydraulic code and the NDK code is a neutron diffusion code.The coupling program framework adopts the"master-slave"mode,in which Fluent is the master program while NDK and KMC-sub are coupled internally and compiled into the dynamic link library(DLL)as slave codes.The domain decomposition method was adopted,in which the reactor core was simulated by NDK and KMC-sub,while the rest of the primary loop was simulated using Fluent.A simulation of the reactor shutdown process of M2LFR-1000 was carried out using the coupling program,and the code-to-code verification was performed with ATHLET,demonstrating a good agreement,with absolute deviation was smaller than 0.2%.The results show an obvious thermal stratification phenomenon during the shutdown process,which occurs 10 s after shutdown,and the change in thermal stratification phenomena is also captured by the coupling program.At the same time,the change in the neutron flux density distribution of the reactor was also obtained.

Formula for calculating spatial similarity degrees between point clouds on multi-scale maps taking map scale change as the only independent variable

The degree of spatial similarity plays an important role in map generalization, yet there has been no quantitative research into it. To fill this gap, this study first defines map scale change and spatial similarity degree/relation in multi-scale map spaces and then proposes a model for calculating the degree of spatial similarity between a point cloud at one scale and its gener- alized counterpart at another scale. After validation, the new model features 16 points with map scale change as the x coordinate and the degree of spatial similarity as the y coordinate. Finally, using an application for curve fitting, the model achieves an empirical formula that can calculate the degree of spatial similarity using map scale change as the sole independent variable, and vice versa. This formula can be used to automate algorithms for point feature generalization and to determine when to terminate them during the generalization.

Seismic energy dispersion compensation by multi-scale morphology

Seismic energy decays while propagating subsurface, which may reduce the resolution of seismic data. This paper studies the method of seismic energy dispersion compensation which provides the basic principles for multi-scale morphology and the spectrum simulation method. These methods are applied in seismic energy compensation. First of all, the seismic data is decomposed into multiple scales and the effective frequency bandwidth is selectively broadened for some scales by using a spectrum simulation method. In this process, according to the amplitude spectrum of each scale, the best simulation range is selected to simulate the middle and low frequency components to ensure the authenticity of the simulation curve which is calculated by the median method, and the high frequency component is broadened. Finally, these scales are reconstructed with reasonable coefficients, and the compensated seismic data can be obtained. Examples are shown to illustrate the feasibility of the energy compensation method.

Particle Modeling Based on Interatomic Potential and Crystal Structure: A Multi-Scale Simulation of Elastic-Plastic Deformation of Metallic Material

We formulate a macroscopic particle modeling analysis of metallic materials (aluminum and copper, etc.) based on theoretical energy and atomic geometries derivable from their interatomic potential. In fact, particles in this framework are presenting a large mass composed of huge collection of atoms and are interacting with each other. We can start from cohesive energy of metallic atoms and basic crystalline unit (e.g. face-centered cubic). Then, we can reach to interparticle (macroscopic) potential function which is presented by the analytical equation with terms of exponent of inter-particle distance, like a Lennard-Jones potential usually used in molecular dynamics simulation. Equation of motion for these macroscopic particles has dissipative term and fluctuation term, as well as the conservative term above, in order to express finite temperature condition. First, we determine the parameters needed in macroscopic potential function and check the reproduction of mechanical behavior in elastic regime. By using the present framework, we are able to carry out uniaxial loading simulation of aluminum rod. The method can also reproduce Young’s modulus and Poisson’s ratio as elastic behavior, though the result shows the dependency on division number of particles. Then, we proceed to try to include plasticity in this multi-scale framework. As a result, a realistic curve of stress-strain relation can be obtained for tensile and compressive loading and this new and simple framework of materials modeling has been confirmed to have certain effectiveness to be used in materials simulations. We also assess the effect of the order of loadings in opposite directions including yield and plastic states and find that an irreversible behavior depends on different response of the particle system between tensile and compressive loadings.

Mechanical response of a tunnel subjected to strike-slip faulting processes,based on a multi-scale modeling method

The stick-slip action of strike-slip faults poses a significant threat to the safety and stability of underground structures.In this study,the north-east area of the Longmenshan fault,Sichuan,provides the geological background;the rheological characteristics of the crustal lithosphere and the nonlinear interactions between plates are described by Burger’s viscoelastic constitutive model and the friction constitutive model,respectively.A large-scale global numerical model for plate squeezing analysis is established,and the seemingly periodic stick-slip action of faults at different crust depths is simulated.For a second model at a smaller scale,a local finite element model(sub-model),the time history of displacement at a ground level location on the Longmenshan fault plane in a stick-slip action is considered as the displacement loading.The integration of these models,creating a multi-scale modeling method,is used to evaluate the crack propagation and mechanical response of a tunnel subjected to strike-slip faulting.The determinations of the recurrence interval of stick-slip action and the cracking characteristics of the tunnel are in substantial agreement with the previous field investigation and experimental results,validating the multi-scale modeling method.It can be concluded that,regardless of stratum stiffness,initial cracks first occur at the inverted arch of the tunnel in the footwall,on the squeezed side under strike-slip faulting.The smaller the stratum stiffness is,the smaller the included angle between the crack expansion and longitudinal direction of the tunnel,and the more extensive the crack expansion range.For the tunnel in a high stiffness stratum,both shear and bending failures occur on the lining under strike-slip faulting,while for that in the low stiffness stratum,only bending failure occurs on the lining.

Precise integration methods based on the Chebyshev polynomial of the first kind

This paper introduces two new types of precise integration methods based on Chebyshev polynomial of the first kind for dynamic response analysis of structures, namely the integral formula method （IFM） and the homogenized initial system method （HISM）. In both methods, nonlinear variable loadings within time intervals are simulated using Chebyshev polynomials of the first kind before a direct integration is performed. Developed on the basis of the integral formula, the recurrence relationship of the integral computation suggested in this paper is combined with the Crout decomposed method to solve linear algebraic equations. In this way, the IFM based on Chebyshev polynomial of the first kind is constructed. Transforming the non-homogenous initial system to the homogeneous dynamic system, and developing a special scheme without dimensional expansion, the HISM based on Chebyshev polynomial of the first kind is able to avoid the matrix inversion operation. The accuracy of the time integration schemes is examined and compared with other commonly used schemes, and it is shown that a greater accuracy as well as less time consuming can be achieved. Two numerical examples are presented to demonstrate the applicability of these new methods.

Precise integration method without inverse matrix calculation for structural dynamic equations

The precise integration method proposed for linear time-invariant homogeneous dynamic systems can provide accurate numerical results that approach an exact solution at integration points. However, difficulties arise when the algorithm is used for non-homogeneous dynamic systems due to the inverse matrix calculation required. In this paper, the structural dynamic equalibrium equations are converted into a special form, the inverse matrix calculation is replaced by the Crout decomposition method to solve the dynamic equilibrium equations, and the precise integration method without the inverse matrix calculation is obtained. The new algorithm enhances the present precise integration method by improving both the computational accuracy and efficiency. Two numerical examples are given to demonstrate the validity and efficiency of the proposed algorithm.

THE APPLICATION OF MULTI－SCALE PERTURBATION METHOD TO THE STABILITY ANALYSIS OF PLANE COUETTE FLOW

This paper applies the multi-scale perturbation method suggested by Ref [3] toinvestigate the linear stability behavior of distorted plane Couette .flow. Using thismethod, the unstable Tollmien-Schlichting wave in plane Couette flow can be found,but not the most unstable mode. By comparing the results of this paper with those ofRef. [3], the effectiveness of this method is investigated.

Renormalization group methods for a Mathieu equation with delayed feedback

This paper presents the application of the renormalization group （RG） methods to the delayed differential equation. By analyzing the Mathieu equation with time delay feedback, we get the amplitude and phase equations, and then obtain the approximate solutions by solving the corresponding RG equations. It shows that the approximate solutions obtained from the RG method are superior to those from the conventionally perturbation methods.

Stability analysis of a liquid crystal elastomer self-oscillator under a linear temperature field

Self-oscillating systems abound in the natural world and offer substantial potential for applications in controllers,micro-motors,medical equipments,and so on.Currently,numerical methods have been widely utilized for obtaining the characteristics of self-oscillation including amplitude and frequency.However,numerical methods are burdened by intricate computations and limited precision,hindering comprehensive investigations into self-oscillating systems.In this paper,the stability of a liquid crystal elastomer fiber self-oscillating system under a linear temperature field is studied,and analytical solutions for the amplitude and frequency are determined.Initially,we establish the governing equations of self-oscillation,elucidate two motion regimes,and reveal the underlying mechanism.Subsequently,we conduct a stability analysis and employ a multi-scale method to obtain the analytical solutions for the amplitude and frequency.The results show agreement between the multi-scale and numerical methods.This research contributes to the examination of diverse self-oscillating systems and advances the theoretical analysis of self-oscillating systems rooted in active materials.

Parametric resonance of axially functionally graded pipes conveying pulsating fluid

Based on the generalized Hamilton's principle,the nonlinear governing equation of an axially functionally graded(AFG)pipe is established.The non-trivial equilibrium configuration is superposed by the modal functions of a simply supported beam.Via the direct multi-scale method,the response and stability boundary to the pulsating fluid velocity are solved analytically and verified by the differential quadrature element method(DQEM).The influence of Young's modulus gradient on the parametric resonance is investigated in the subcritical and supercritical regions.In general,the pipe in the supercritical region is more sensitive to the pulsating excitation.The nonlinearity changes from hard to soft,and the non-trivial equilibrium configuration introduces more frequency components to the vibration.Besides,the increasing Young's modulus gradient improves the critical pulsating flow velocity of the parametric resonance,and further enhances the stability of the system.In addition,when the temperature increases along the axial direction,reducing the gradient parameter can enhance the response asymmetry.This work further complements the theoretical analysis of pipes conveying pulsating fluid.