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Made available in DSpace on 2022-04-29T08:31:12Z (GMT). No. of bitstreams: 0 Previous issue date: 2021-10-01 Organization and planning of reverse logistics networks make sustainable processes more efficient. Thus, an important sector for connecting the collection and waste paper recycling echelons is the intermediate center. In this study, a Mixed Integer Linear Programming model, which is multi-objective, multi-product, multi-level and multi-period, was developed to optimize the waste paper logistics processes of intermediate centers. The formulation includes the following echelons: collection, considering vehicle routing with different capacities; inventory of non-baling materials; baling sorted waste and inventory of bales; selling bales; disposal of non-recyclable waste. The aims of the model include: maximizing the collected waste; minimizing distances; maximizing both the production and sale of bales, and minimizing costs. According to research carried out in the literature, the developed model is a new proposal and to implement it, the Weighted Goal Programming and Revised Multi-Choice Goal Programming approaches were used to deal with multiple objectives and incorporate uncertainty into the quantity of waste available for collection. To analyze the proposed model, computational tests were executed with instances based on real data from a Brazilian company in the sector. For all performed tests, General Algebraic Modeling System 23.6.5 modeling language and CPLEX 12.2.0.2 solver were used for modeling and optimization. The results show that this study presents formulation and technological approaches that represent real situations and provide competitive solutions to the problem. Sáo Paulo State University “Júlio de Mesquita Filho” [Universidade Estadual Paulista Júlio de Mesquita Filho] School of Engineering of Guaratinguetá [Faculdade de Engenharia de Guaratinguetá] Production Department, Av. Ariberto Pereira da Cunha, 333, Portal das Colinas Agulhas Negras Military Academy [Academia Militar das Agulhas Negras] Rodovia Presidente Dutra, km 306, s/n Escola Preparatória de Cadetes do Exército Avenida Papa Pio XII 350 Jardim Chapadão Campinas - SP Brazil, Avenida Papa Pio XII, 350, Jardim Chapadão, Campinas - SP Sáo Paulo State University “Júlio de Mesquita Filho” [Universidade Estadual Paulista Júlio de Mesquita Filho] School of Engineering of Guaratinguetá [Faculdade de Engenharia de Guaratinguetá] Production Department, Av. Ariberto Pereira da Cunha, 333, Portal das Colinas

We propose a new concept which is a generalization of fuzzy soft subset and fuzzy soft equal. Using such notions, we will be able to consider the distributive law of fuzzy soft sets. Using the distributive law of fuzzy soft sets, we point out that the distributive law of trapezoidal fuzzy soft sets as proposed by Xiao et al. (2012) is not true. The correction will further improve further extensions of the results of Xiao et al. (2012). We will also establish the generalized distributive law of trapezoidal fuzzy soft sets along with illustrative examples.

Prediction models for specific energy consumption and throughput of palm nut–pulp separator were developed using mechanistic modeling technique. These models facilitate the easy production of different sizes of this machine in line with the budgets of the oil palm fruit processors. It also accounts for all the operational parameters of this machine as well as relevant properties of processed materials, digested palm fruit mash, palm nut and pulp. Experimental results indicated over 99% prediction accuracy of the models, while the prediction error of the specific energy consumption and throughput models are within the range of −0.014 to 0.076% and −0.005 to 0.003% respectively.

This paper summarises the development, implementation and typical results from a new full-scale three-dimensional numerical model of the multiphase chemically reacting flow in a continuous digester. The model is based on continuity and momentum equations for wood chips and free liquid. A new sub-model describing the tangential stresses for the solid phase has been developed and incorporated into the digester model. The solution procedure that takes into account the high inter-phase friction terms and the high values of the solid pressure has been introduced. In order to illustrate the model’s performance, simulations have been carried on for the industrial digester. The results are in quantitative agreement with available field measurements and in qualitative agreement with operating personnel observations. The model utilises curvilinear body fitted coordinates and can be used to simulate the operation of various pulp digesters or any other chemical reactors working in a similar regime.

International audience; In this paper we study a single machine scheduling problem with the objective of minimizing the sum of completion times. Each of the given jobs is either short or long. However the processing times are initially hidden to the algorithm, but can be tested. This is done by executing a processing time oracle, which reveals the processing time of a given job. Each test occupies a time unit in the schedule, therefore the algorithm must decide for which jobs it will call the processing time oracle. The objective value of the resulting schedule is compared with the objective value of an optimal schedule, which is computed using full information. The resulting competitive ratio measures the price of hidden processing times, and the goal is to design an algorithm with minimal competitive ratio. Two models are studied in this paper. In the non-adaptive model, the algorithm needs to decide beforehand which jobs to test, and which jobs to execute untested. However in the adaptive model, the algorithm can make these decisions adaptively depending on the outcomes of the job tests. In both models we provide optimal polynomial time algorithms following a two-phase strategy, which consist of a first phase where jobs are tested, and a second phase where jobs are executed obliviously. Experiments give strong evidence that optimal algorithms have this structure. Proving this property is left as an open problem

The post-buckling of a clamped column made of nonlinear elastic material and subject to axial compression is investigated in this paper. The Ramberg−Osgood type constitutive relation is adopted and it is expanded by using Taylor series. Based on Euler−Bernoulli beam theory, the exact governing equations are expressed in terms of rotation angle. Because the equations contain strongly nonlinear terms that are functions of the rotation angle, analytical solutions are virtually impossible. In this respect, this paper is focused on presenting an alternative method to construct concise yet accurate analytical approximate solutions for post-buckling of the Ramberg−Osgood column that is related to large rotation amplitude. The improved harmonic balance method is used to solve the nonlinear governing equations which are simplified via the Maclaurin series expansion and orthogonal Chebyshev polynomials. In addition, numerical solutions by applying the shooting method on the governing equations are obtained for comparison. The second-order analytical approximate solutions presented in this paper show excellent accuracy by comparing with numerical solutions. The analytical approximate method and numerical result presented in this paper can be applied as design guidelines for designing engineering structures that sustain large deformation, such as slender nonlinear compression of aluminum alloy columns, rods or braces.

The use of constant order differential equations to describe the evolution of complex systems is often unable to describe some of the changing characteristics of the systems accurately. Variable order fractional derivatives provide us with new tools to solve such problems. In this paper, the accumulation and derivative orders of the classic grey model are expanded from constants to functions, and a variable order fractional grey model is established to describe the evolution process of complex systems. Firstly, this paper defines the variable order fractional accumulation generation sequence. On the basis of this sequence, a variable order fractional derivative grey model is established, the parameters of the model are estimated using the least square method, and the quantum particle swarm optimization algorithm is used to solve the order of fractional derivative and accumulation. Sadik transform and Laplace transform are adopted to obtain the analytical solution of the new model. Lastly, the effectiveness of the new model is verified through four cases. Compared with other models, the variable order fractional model can describe the development process of complex systems more accurately.

The nonlinearity of bending stiffness caused by inter-wire slippage has become the major challenge in the prediction of strand bending behavior. This paper proposed an analytical approach to predict the hysteretic bending behavior of helical strands. As an extension of classical models, this paper additionally considered the wire slippage delayed by bending load. Moreover, the method of integral for discrete contact regions extended the always assumed continuous contact region into the discontinuous, which enables this model to analyze the realistic conditions of multi-layered strands having alternating algebraic signs of lay angles, such as ACSR conductors and S32 cable. The present model is compared with FEM and experimental data, and results demonstrated the effectiveness of this model in strand bending prediction.

This paper presents a new hybrid identification algorithm called the Average Multi-Verse Optimizer and Sine Cosine Algorithm for identifying the continuous-time Hammerstein system. In this paper, two modifications were employed on the conventional Multi-Verse Optimizer. Our first modification was an average design parameter updating mechanism to solve the local optima issue. The second modification was the hybridization of Multi-Verse Optimizer with Sine Cosine Algorithm that will balance the exploration and exploitation processes and thus improve the poor searching capability. The proposed hybrid method was used for identifying the parameters of linear and nonlinear subsystems in the Hammerstein model using the given input and output data. A continuous-time linear subsystem was considered in this study, while there were a few methods that utilize such models. Furthermore, various nonlinear subsystems such as the quadratic and hyperbolic functions had been used in those experiments. The efficiency of the novel technique is illustrated using a numerical example and two real-world applications, which are a twin rotor system and a flexible manipulator system. The numerical and experimental results analysis were observed with respect to the convergence curve of the fitness function, the parameter deviation index, time-domain and frequency-domain responses of the identified model, and the Wilcoxon's rank test. The results showed that the proposed method was efficient in identifying both the Hammerstein model subsystems in terms of the quadratic output estimation error and parameter deviation index. The proposed hybrid method also achieved better performance in modeling of the twin-rotor system as well as the flexible manipulator system and provided better solutions compared to other optimization methods including Particle Swarm Optimizer, Grey Wolf Optimizer, Multi-Verse Optimizer and Sine Cosine Algorithm.

As an attempt to investigate the torsional-longitudinal vibrations of marine propeller shafting systems, this paper develops an integrated mathematical formulation to consider different aspects of the problem as clearly as possible. The previous works in this field mostly deal with the lumped-parameter or finite element simulations of the propeller and the main shaft while this paper employs a non-FEM distributed-parameter modeling. The Newton–Euler method is used to derive dynamic equations of the cantilever blading and the rotating main shaft. The lumped effects on the main shaft such as the thrust block, the rigid coupling and the propeller loadings are considered together with the variation of cross section and pretwist angle along the blades. Galerkin method is used to discretize the equations and find the lowest number of dominant modes for each degree of freedom. The coupling effect is then explained by classifying the mode shapes into three groups. It is found that taking blade deformations into account is advantageous to the vibration analysis of the problem.

In the mechanical design industry, it is very time-consuming to convert actual engineering design problems into finite element models, and it is repeatedly modified during the product design process. Each time the structure is modified, computer simulation analysis is repeated, and the product development cycle becomes longer. Therefore, in this paper, a re-analysis theoretical method based on the inversion of the block matrix based on the change threshold is proposed to quickly obtain responses with different accuracy after the structure is modified according to the initial analysis. By modifying the size of the change threshold, the accuracy and calculation time of the reanalysis results are controlled. This reanalysis method shortens a lot of calculation time under the premise of controllable response accuracy. In this paper, numerical calculations are given to show the accuracy of the response under different thresholds, and the efficiency of this method is proved by the finite element model under the same accuracy.

The paper presents a fully Lagrangian meshfree projection-based hydroelastic FSI (Fluid-Structure Interaction) solver for simulation of incompressible fluid flows interacting with laminated composite elastic structures. The presented computational method is configured within the SPH (Smoothed Particle Hydrodynamics) framework. An enhanced version of Incompressible SPH (ISPH) is adopted as the fluid model and a structure model in the context of Hamiltonian SPH (HSPH) is coded for stable, accurate and conservative simulations of laminated composite elastic structures. The fluid-structure coupling is conducted by either the Fluid-Structure Acceleration-based (FSA) or the Pressure Integration (PI) coupling scheme with careful considerations of interface boundary conditions and numerical stability with respect to the discontinuities in densities of constituent laminates of considered composite structures. A distinct superiority of the proposed solver corresponds to absence of any artificial stabilizers or artificial smoothing of material properties at material interfaces. Verifications are coherently performed by conducting several benchmark test cases with exact analytical solutions, first for structures and then for fluid-structure systems. Finally, to portray the potential applicability of the solver for practical engineering applications, a challenging simulation of water slamming on a sandwich hull, comprising of large discontinuities in structural material properties is conducted. To the best knowledge of authors, this paper presents the first hydroelastic FSI solver corresponding to composite structures in the context of SPH.

This paper tries to analyze the inverse kinematics problem of the inextensible continuum robot from a new perspective. In the analysis, the generatrix of the robot's workspace is studied and replaced by the Kepler oval. Then the complicated inverse kinematics problem of the multi-section continuum robot is transformed into the easy problem in solving the binary equation set consist of the oval equations. Based on the analysis, the inverse kinematics algorithm for the single-section and multi-section continuum robot is designed. The accuracy and the computational efficiency of the proposed algorithm are evaluated by the positioning error and the computational time, respectively. Two simulations are implemented to verify the validation of the novel algorithm, and the results show the algorithm has a very good performance in accuracy and computational efficiency. The novel inverse kinematics algorithm using the Kepler oval proposed in this paper are generic, which can be extended for analyzing such flexible robots.

In this paper an ( r , Q ) stock policy is used to control an inventory system with deterministic demand and defective items. The fraction of defective items in a batch is a random variable whose variance maybe dependent and independent of the order quantity. Considering the numerous benefits of selling high quality items (noting the vital importance for firms like medical device companies), a full inspection of the order quantity is conducted. Although full inspection reduces returns, ensures high quality products and improves customer satisfaction, it requires a substantial inspection time. Consequently, during this time a non-zero probability of undesirable (unplanned) shortages exists. The dominant suggestion in the literature, is to set the reorder level equal to the total demand during inpection period. This paper suggests an alternative reorder level to prevent shortages during this period and to formulate the backlogging cost in the case that unplanned shortages are permitted. Then, theoretical results ensure the determination of the reorder level and the order quantity that minimize the average total cost of the inventory system. Comparisons, through numerical examples and simulation, demonstrate that the proposed model leads to significant reduction both in cost and probability of shortages (if allowed), in relation to other existing models.

In this paper, nonlinear vibrations of fractional viscoelastic functionally graded micro-beam are studied based on the fractional calculus. The Modified Couple Stress Theory (MCST) is utilized for considering the effect of the micro-scale. The micro-beam is modeled based on the Euler-Bernoulli theory and the Von Karman’s nonlinear strain relations. The viscoelastic part of the micro-beam is considered using the fractional Kelvin-Voigt viscoelastic model. Functionally graded properties vary along the thickness based on a power-law function. The small-scale effects of the micro-beam are modeled using the MCST. Hamiltons principle is used to derive the governing equation of motion. For the solution, the Finite Difference Method (FDM) and the Finite Element Method (FEM) are employed. The FDM is used for discretizing the time domain, and the FEM is used for discretizing the space domain. Effects of the fractional-order, microstructure parameters, and Functionally Graded Material (FGM) properties on the time response of the viscoelastic micro-beam are analyzed numerically. Numerical results show that the fractional-order modeling of the viscoelastic micro-beam can change the behavior of the structure, and increasing the fractional-order, can increase the damping of the system. Numerical simulations also suggest that the effect of the fractional derivative order on the responses of free and forced vibrations is different because there is an important correlation between the fractional-order derivative and the excitation frequency. The results of this paper can be used for modeling the damping of the viscoelastic structures.

The paper reports the new forms of the fundamental solutions for 3D magnetoelectroelasticity equations. The state-of-art presented in the paper shows that the fundamental solutions for magnetoelectroelasticity equations take complex forms, especially considering their applications in the boundary element method. In the paper, the magnetoelectroelastic continuum serves as a homogenized model of the piezoelectric-piezomagnetic composites. The model is obtained by the Mori-Tanaka micromechanical approach. The so-called quasi-static approximation of dynamics is applied to obtain the set of the partial differential equations consists of the hyperbolic equation of motion and two elliptic equations for the conservation laws of the electric and magnetic charge. The motivation example shows the analogy between the impulse response of the linear system of ordinary differential equations, known from the classical linear control theory, and the fundamental solution of the linear system of partial differential equations. The spatial Fourier transform changes the coupled system of hyperbolic-elliptic equations of magnetoelectroelasticity into differential-algebraic equations in the k-space and the time domain. Instead of the classic approach eliminating the constraint equations for the electromagnetic static potentials, the semi-state vector is introduced and the descriptor system with the Kronecker index equals 1 is obtained. The structural analysis is performed for the resultant system and the regular and singular matrix pencils were analyzed. The slowness surfaces of the mainly acoustic modes in the quasi-static approximation are constructed for the 3-phase composite. Further, the shifting method is applied and the general solution of the semi-state equations is obtained as an analogon of the impulse response solution. The obtained result serves as a base for obtaining the new form of the fundamental solution of dynamic magnetoelectroelasticity in 3-dimensional space and also for the step response and steady-state fundamental solution for magnetoelectroelastic continuum.

Imperfect production processes are frequently investigated in supply chain literature. Many recent articles assume that items that do not meet the quality requirements can be sold on a secondary market at a reduced price. Due to simplification, these papers ignore the demand of the secondary customers and assume instead that all imperfect quality items are salable. In this paper, we show that this ignorance can lead to unrealistic results as an unrestricted secondary market may reduce the manufacturer’s incentive for an error-free production process. By introducing a price- and quality-dependent demand for both the primary and secondary markets, we obtain the following main insights: (i) a restricted secondary market leads to more realistic results compared with the unrestricted case; the secondary demand is only ignorable under certain conditions; (ii) quality decisions are mainly influenced by the market parameters of both market segments as well as quality cost parameters; (iii) in some cases, it is advantageous not to serve the secondary market and to concentrate the sales process on primary customers.

This paper analyses the modal characteristics and dynamic stability of a whirling rotor with flexible blades. A rigid disk with flexible blades, which is called a rotor-blade system in this paper, is modelled as a Jeffcott rotor with Euler-Bernoulli beams. From the nonlinear equations of motion, we derived the equilibrium position and the linearized equations around the equilibrium position by the perturbation method. Based on the linearized equations, a modal analysis was performed for the variations of the supporting stiffness and rotating speed. Furthermore, we investigated the effects of the whirling motion on the natural frequencies and mode shapes and found the threshold speed of instability that cause instability in the system. In addition, we also studied the characteristics of stable and unstable regions in terms of the rotating speed and the supporting stiffness.

The classical formulation of reverse logistics inventory systems assumes an equal returned amount in all cycles. This assumption necessitates that manufacturing and remanufacturing processes are static in all cycles, i.e. it ignores the impact of the first cycle. This implies that the result of such modelling may not be an optimal policy. Another implicit assumption embedded in the classical formulation is that input parameters remain static indefinitely. In practice, however, it is often desirable that input parameters are adjusted to be responsive to a new policy due to acquired new knowledge. This paper develops a general joint model for production of new items together with remanufacturing of returned items “as good as new” condition. The proposed model focuses on a single production cycle and a single remanufacturing cycle and, as part of its formulation, considers the first time interval. This implies that the decision variables should vary from cycle to cycle until the system plateaus. Adopting this approach ensures a more realistic tractability of the impact of previous cycles, including the first time cycle. The mathematical formulation considers arbitrary functions of time, which allows the decision maker to assess and compare the consequences of a diverse range of time-varying forms by employing a single inventory model. All functions and input parameters may change their values for subsequent cycles and/or periodic review or remain static. A rigorous method is utilised to show that the solution, if it exists, is unique and global optimal. The paper provides illustrative examples that demonstrate the application of the theoretical model in different settings.

Elastothermodynamic damping is one of the most important energy loss mechanisms in micro/nano mechanical resonators, which is related to the high sensitivity and high resonance frequency characteristics of resonators. With the wide application of resonators, multilayer microplate resonators have been applied. In this paper, the elastothermodynamic damping models of three-layer of Kirchhoff–Love microplate under three typical boundary support conditions of Clamped-Clamped-Clamped-Clamped (C-C-C-C), Clamped-Free-Clamped-Free (C-F-C-F) and Clamped-Free-Free-Free (C-F-F-F) are established. Firstly, the thermoelastic mechanical equations of the microplate are given, and then the three-dimensional temperature distribution functions of the microplate are obtained based on the generalized orthogonal function method. Finally, the elastothermodynamic damping models are established based on the composite multilayer plate theory. The model in this paper is compared with the previous model and the finite element results.

The bipedal inverted pendulum with damping has been adopted to simulate human–structure interaction recently. However, the lack of analysis and verification has provided motivation for further investigation. Leg damping and energy compensation strategy are required for the bipedal inverted pendulum to regulate gait patterns on vibrating structures. In this paper, the Hunt–Crossley model is adopted to get zeros contact force at touch down, while energy compensation is achieved by adjusting the stiffness and rest length of the legs. The damped bipedal inverted pendulum can achieve stable periodic gait with a lower energy input and flatter attack angle so that more gaits are available, compared to the template, referred to as spring-load inverted pendulum. The measured and simulated vertical ground reaction force-time histories are in good agreement. In addition, the dynamic load factors are also within a reasonable range. Parametric analysis shows that the damped bipedal inverted pendulum can achieve stable gaits of 1.6 to 2.4 Hz with a reasonable first harmonic dynamic load factor, which covers the normal walking step frequency. The proposed model in this paper can be applied to human–structure interaction analysis.

In this paper a new method for online parameter identification and damage detection in smart building structures that are subjected to arbitrary seismic excitation is proposed. It uses real-time measurements of a structure's motion to identify its unknown constant or piecewise constant parameters such as stiffness, damping and mass over the time. The method is based on elements of system synchronization and adaptive control theories. First, a computational system, called the virtual system, is defined. Next, by using properly designed controller and estimations for the unknown parameters, the state of the virtual system is forced to follow the measured motion of the real structure. The mentioned estimations are computed from a proposed update law which depends on the measured motion of the real structure and the virtual system’s state. A major theoretical novelty of this paper is a proposed convergence condition which is applicable in case of arbitrary external forces or ground acceleration. It is shown that upon the satisfaction of that condition, as the synchronization completes, the computed estimation function converges to the true value of the vector of unknown parameters. In addition, an important practical contribution presented in this study is the introduction of a technique called scale factors. It helps to use available initial guesses of the unknown parameters to improve the speed of online identification. Numerical examples show that the proposed method is promising and has a good performance in both online identification and online damage detection problems.

Considering the importance of damage for the structural performance and for decreasing the identification error, this paper proposes an optimal sensor placement method based on a weighted standard deviation norm (WSDN) index. The standard deviation of the identified damage parameters is solved using the series expansion theory and probabilistic method to quantify the effect of a measurement error on damage identification. The damage estimation weight (DEW) index, which can reflect the importance of each element in the structural capabilities, is established based on a performance-damage curve. A significant DEW for a specified element indicates that the element is important for the structure and that its identification error should be small. The WSDN index is obtained from the Hadamard product of the standard deviations (SDs) and DEWs. Thus, the identification error of the entire structure is measured using the weighting coefficient. The optimal sensor placement (OSP) procedure is performed by minimizing the WSDN index. The proposed method can clearly decrease the uncertainties of the identification results for the important elements. Other OSP criteria, including the condition number, information entropy, and standard deviation norm, which aim to decrease the identification error, are discussed in this paper for comparison with the proposed method. Two numerical examples and an experiment, which pertain to the deformation performance, buckling features, and dynamic characteristics, are discussed to verify the advantages of the proposed method.

In this paper, we classify the asymptotic behavior for a class of stochastic SIR epidemic models represented by stochastic differential systems where the Brownian motions and Levy jumps perturb to the linear terms of each equation. We construct a threshold value between permanence and extinction and develop the ergodicity of the underlying system. It is shown that the transition probabilities converge in total variation norm to the invariant measure. Our results can be considered as a significant contribution in studying the long term behavior of stochastic differential models because there are no restrictions imposed to the parameters of models. Techniques used in proving results of this paper are new and suitable to deal with other stochastic models in biology where the noises may perturb to nonlinear terms of equations or with delay equations.

The paper is concerned with the nonlinear primary resonance of nano beams with axial load under the velocity time delay control. In order to have a deep insight into the system, the amplitude frequency response curve of the system is firstly obtained using the multiple scales method. The effects of the control gains and time delays on the system stability are then investigated. The analyses illustrated that both delay feedback gain coefficient and velocity time delay control can mitigate the system vibrations properties (e.g. hardening nonlinearity, resonance amplitude and the corresponding width) to an excellent level. The nonlinear primary resonance of nano beam is also discussed with the influences of small scale effect, axial initial load, wave number, Winkler foundation modulus and the ratio of the length to the diameter. This paper establishes the relationship between the time delay controller and vibration properties of a nano beam system which provides the guidance for applying time delayed active control for different types of nano devices in engineering practices.

This paper proposes a stochastic model for the evolutionary adaptive dynamics of species subject to trait-dependent intrinsic growth rates and the influence of environmental noise. The aim of this paper is twofold: (a) mathematically we make an attempt to investigate the evolutionary adaptive dynamics for models with noises; (b) biologically we investigate how the noises in environment affect the evolutionary stability. We first investigate the extinction and permanence of the population in the presence of environmental noises. Combining evolutionary adaptive dynamics with stochastic dynamics, we then establish a fitness function with stochastic disturbance and obtain the evolutionary conditions for continuously stable strategy and evolutionary branching. Our study finds that under intense competition among species, increasing stochastic disturbance can lead to rapidly stable evolution towards smaller trait values, but there is an opposite effect under weak competition among species. This yields an interesting evolutionary threshold, beyond which any increasing stochastic disturbance can go against evolutionary branching and promote evolutionary stability. We then carry out the evolutionary analysis and numerical simulations to illustrate our theoretical results. Finally, for demonstrating the emergence of high-level polymorphism we perform long-term simulation of evolutionary dynamics.

Concurrent multiscale method is a spatial and temporal combination of two different scale models for describing the micro/meso and macro mixed behaviors observed in strain localization, failure and phase transformation processes, etc. Most of the existing coupling schemes use the displacement compatibility conditions to glue different scale models, which leads to displacement continuity and stress discontinuity for the obtained multiscale model. To overcome stress discontinuity, this paper presented a multiscale method based on the generalized bridging domain method for coupling the discrete element (DE) and finite element (FE) models. This coupling scheme adopted displacement and stress mixed compatibility conditions. Displacements that were interpolated from FE nodes were prescribed on the artificial boundary of DE model, while stresses at numerical integration points that were extracted from DE contact forces were applied on the material transition zone of FE model (the coupling domain and the artificial boundary of FE model). In addition, this paper proposed an explicit multiple time-steps integration algorithm and adopted Cundall nonviscous damping for quasi-static problems. DE and FE parameters were calibrated by DE simulations of a biaxial compression test and a deposition process. Numerical examples for a 2D cone penetration test (CPT) show that the proposed multiscale method captures both mesoscopic and macroscopic behaviors such as sand soil particle rearrangement, stress concentration near the cone tip, shear dilation, penetration resistance vibration and particle rotation, etc, during the cone penetration process. The proposed multiscale method is versatile for maintaining stress continuity in coupling different scale models.

In this paper, new concepts of balanced systems are proposed based on real engineering problems. The system under study consists of l groups and each group has n functional sectors. The conception of balance difference is proposed for the first time. It is assumed that unbalanced systems are rebalanced by either forcing down some working units into standby or resuming some standby units into operation. In addition, a case that the forced-down units are subject to failure during standby is studied in this paper. Based on different balanced cases and system failure criteria, two reliability models for balanced systems are developed. The proposed systems have widespread applications in aerospace and military industries, such as wing systems of airplane and unmanned aerial vehicles with balanced engine systems. Markov process imbedding method is used to analyze the number of working units in each sector for each model. Finite Markov chain imbedding approach and universal generating function technique are used to obtain the system reliability for different models. Several case studies are finally presented to demonstrate the new models.

This paper considers the approach proposed by Zou et al. for determining the plastic zone around a shallow circular tunnel in an elasto-plastic semi-infinite space that incorporates the gravitational effect based on the bipolar coordinate system. The paper aims to analyze the correctness of the analytic expressions for elastic and plastic stress, as well as the reasonableness of the procedure building for solving the elasto-plastic interface. Finally, a promising way of solving this problem more exactly is presented.

The paper presents an elaboration of a well-known two-fluid model for the description of turbulence. The idea that a turbulent flow can be modeled by considering a heterogeneous mixture of two fluids that make a relative motion is proven mathematically in the paper. It is shown that the fluctuating nature of the turbulent flow is due to relative motion of these fluids. Therefore, the proposed mathematical model is based on the theory of the dynamics of two fluids. For each fluid its own equation of motion is written, that leads to a closed system of equations. To calibrate the unknown constants of a new model, a problem of turbulent flow around a flat plate is considered. It is shown that the model is a low-Reynolds-number model. To verify the calibrated model, numerical studies of round and plane turbulent jets for various flow sections are conducted. It is shown that the model is not affected by the well-known problem of “round jet anomaly”. Another problem is investigated - an abrupt expansion of a strongly swirling flow - to study the possibility of the proposed model to describe anisotropic turbulent flows. The derived model is validated by comparing the results of numerical calculations with existing experimental data.

The twin-web disk holds big promise for increasing efficiency of the aircraft engine. Its reliability-based multidisciplinary design optimization involves several disciplines including fluid mechanics, heat transfer, structural strength, and vibration. The solution to this optimization problem requires three-loop calculations including loops for optimization, reliability, and interdisciplinary consistence often making its computational cost unacceptably high. The lack of sufficient amount of probabilistic data, especially for this brand-new turbine disk, makes matters worse. In this paper, the non-probabilistic uncertain variables are described by an evidence theory-based fuzzy set method, which we extend to general structure of uncertain data. We also propose two modifications of the active learning kriging model: one of them for the purpose of optimization with respect to the distance from the optimum point and another one for the purpose of assessing reliability by introducing the importance concept. Applications of these two modifications are demonstrated in this paper. Finally, a multi-adaptive learning kriging strategy for non-probabilistic reliability-based multidisciplinary design optimization of twin-web disk is proposed to improve its power efficiency and reliability in a computationally effective way.

This paper presents a novel SPH scheme for modelling incompressible and divergence-free flow with a free surface (IDFSPH) associated with semi-analytical wall boundary conditions. In line with the projection method, the velocity field is decoupled from the pressure field in the momentum equation. A Poisson equation, serving as the pressure solver, is obtained by which pressure field is decoupled completely from the velocity field. In particular, an exact projection scheme is deployed to fulfil the requirement of the divergence-free velocity field. The condition of incompressibility is satisfied by iteratively updating the density field till the convergence. The two-equation k–e model is employed to describe the turbulence effects in Newtonian flows. It is shown that the discretised SPH schemes have the feature of both linear and angular momentum conservations. The semi-analytical wall method implements the appropriate integrals to evaluate the boundary contributions to the mass and momentum equations. In comparison to the boundary particle methods, it can greatly enhance the feasibility and efficiency with the complex geometries. The algorithm presented within this paper is applied to several academic test cases for which either analytical results or simulations with other methods are available. The comparisons verify that this scheme is provided with convincing efficiency and extensive applicability.

In this paper, we propose a new whale army optimization algorithm with a view to solving multifarious optimization problems. The key novelty of our approach is to modify the original whale optimization algorithm to make it effective to solve the complicated, large-scale and constrained optimization problems. Our modifications mainly embody two aspects: the beneficial strategic adjustment to set key parameters and to help establish base principles in the original optimizer and the introduction of armed force program which classifies the search whales into different categories to achieve efficient cooperation. We evaluate the performance of the proposed algorithm, using three simple benchmark test functions over thirty CEC-2014 real-parameter numerical optimization problems and three constraint engineering design problems. The test results indicate that this algorithm can provide a faster local convergence rate, a higher convergence accuracy, and a lower computational complexity in comparison to traditional whale optimization algorithms and other sophisticated state of the art whale optimizers. Performance wise, it also surpasses many advanced methods for large-scaled complex functions. Furthermore, in this paper we propose a variant of whale army optimization algorithm to specifically address and solve optimizing constrained problems with a high degree of precision.

In this paper fractional Hindmarsh Rose (HR) neuron, which mimics several behaviors of a real biological neuron is implemented on field programmable gate array (FPGA). The results show several differences in the dynamic characteristics of integer and fractional order Hindmarsh Rose neuron models. The integer order model shows only one type of firing characteristics when the parameters of model remains same. The fractional order model depicts several dynamical behaviors even for the same parameters as the order of the fractional operator is varied. The firing frequency increases when the order of the fractional operator decreases. The fractional order is therefore key in determining the firing characteristics of biological neurons. To implement this neuron model first the digital realization of different fractional operator approximations are obtained, then the fractional integrator is used to obtain the low power and low cost hardware realization of fractional HR neuron. The fractional neuron model has been implemented on a low voltage and low power circuit and then compared with its integer counter part. The hardware is used to demonstrate the different dynamical behaviors of fractional HR neuron for different type of approximations obtained for fractional operator in this paper. A coupled network of fractional order HR neurons is also implemented. The results also show that synchronization between neurons increases as long as coupling factor keeps on increasing.

The double-beam system with a viscoelastic layer is a classical mechanical model for many beam-type composite structures. However, few studies have been able to optimize the structure from the perspective of structural damping characteristics. To fully understand the damping characteristics of the viscoelastic double-beam system, an analysis method based on dynamic stiffness method and Wittrick-Williams algorithm is presented in this paper. Through numerical case studies, five typical parameters of the viscoelastic double-beam system are discussed to investigate their influence on the damping characteristic of the system. Finally, the conclusions are used to parametric analysis for a kind of double-sheathing cable systems. Results show that the damping coefficient of the connection layer have a significant effect on the damping characteristic of the double-sheathing cable system compared with other design parameters. The proposed methods and conclusions obtained in this paper are helpful to design and optimize the structural parameters of engineering structures, thus having certain application and promotional value.

A new methodology for modeling large-scale scheduling problems in low-volume low-variety production systems is proposed through this paper. Such scheduling problems are constrained by limited time and resources, where each work center is assigned a unique statement of work, to be completed on-time with the budgeted number of resources. Products assembled in low-volume low-variety production systems are processed through a series of stations referred to as work centers, where varying levels and classifications of resources are deployed onto the product. Aircraft, heavy aero-structures, and heavy military equipment are examples of products assembled in low-volume low-variety production systems. To ensure products are delivered on-time and on-budget, it is crucial to execute to a detailed schedule, such that all precedence, resource, zonal, and other constraints and characteristics inherent in such production systems are successfully satisfied. Despite the criticality of detailed schedules in delivering products on-time and on-budget, limited research is reported on mixed-integer programming approaches for scheduling optimization of activities in low-volume low-variety production systems. The discrete-time linear mixed-integer mathematical programming model developed in this paper fills the gap in the current literature with a direct impact on the organizations’ service levels and bottom line. The proposed mathematical programming models are validated through a real-world case-study of the assembly process of a narrow body aircraft to ensure compatibility in the modeling of large-scale industrial problems.

The paper is aimed to examine dynamic behaviors of a dual-disc bearing-rotor system in multi-fault state, and the crack detection based on the orbit morphological characteristics and vibration responses is proposed. Dynamic response and vibration signal analysis are two significant studies in rotor system. Most researchers have simulated the nonlinear dynamics and analyzed the fault signal using various methods separately. However, the fault feature from vibration signal is tightly connected with the dynamic mechanism in the rotor system, especially in rotor system with coupling multi-fault. In the paper, the dynamic model of the dual-disc bearing-rotor system is established, which takes into account the effects of crack, rub-impact and nonlinear oil-film forces. The vibration responses and the effect of crack on dual-disc rotor system with multi faults are investigated. The existence of crack and the coupling effect of multi faults enrich dynamic behavior of the dual-disc bearing-rotor system, and the response near the 1/2 subcritical speed provides a criterion for crack detection. Experiment investigation is attempted for the first time, which is based on the changes of crack depth and rotation speed for multi-fault dual-disc rotor system. The analysis of the dynamic response and the orbit morphological characteristics from experiment can effectively detect the crack information.

Image segmentation methods usually suffer from intensity inhomogeneity problem caused by many factors such as spatial variations in illumination (or bias fields of imaging devices). In order to address this problem, this paper proposes a Retinex-based variational model for image segmentation and bias correction. According to Retinex theory, the input inhomogeneous image can be decoupled into illumination bias and reflectance parts. The main contribution of this paper is to consider piecewise constant of the reflectance, and thereby introduce the total variation term in the proposed model for correcting and segmenting the input image. This is different from the existing model in which the spatial smoothness of the illumination bias is employed only. The existence of the minimizers to the variational model is established. Furthermore, we develop an efficient algorithm to solve the model numerically by using the alternating minimization method. Our experimental results are reported to demonstrate the effectiveness of the proposed method, and its performance is competitive with that of the other testing methods.

High-accuracy numerical method is of significance in nano-electromechanical systems. This paper focuses on the influence of nano-system uncertainties on the critical flow velocity of fluid-conveying carbon nanotubes (CNTs) under multi-physical fields. A hull iteration method is proposed to estimate the bounds of the critical flow velocity of fluid-conveying CNT. The nonlinear governing equations for a fluid-conveying CNT embedded in an elastic medium and subjected to thermal and magnetic fields are derived by using the nonlocal Euler–Bernoulli beam theory. Considering the limited available information on the uncertainties, the material and geometrical properties of the CNT are described by using unknown-but-bounded parameters. The iterative scheme is presented in the framework of interval mathematics. After the assessment of the proposed method by comparing Monte Carlo simulation and stochastic analysis method, the detailed parametric investigation is carried out to deeply understand the coupled effects of uncertainties and many parameters such as small scale, Knudsen number, elastic medium and magnetic field parameters, as well as temperature changes. Additionally, the effect of geometric nonlinearity on the critical flow velocity is also examined in detail. The results derived in this paper can provide useful insights on the further application of CNTs as nano-channels to convey fluids.

Magnetic energy release conversion, plasma heating and charged particle energization-acceleration in the magnetic fluctuation-induced self-generating-organization region and the plasma turbulence-induced self-feeding-sustaining region are key issues for large temporal-spatial scale turbulent magnetic reconnection (LTSTMR; observed current sheet thickness to characteristic electron length, electron Larmor radius for low-β and electron inertial length for high-β, ratios on the order of 1010∼1011; observed evolution time to electron cyclotron time ratios on the order of 1010∼1011) that ranges from Earth’s magnetosphere to solar eruptions and other astrophysical phenomena. As the first part of a two-paper series, this paper introduces a relativistic hybrid particle-in-cell and lattice Boltzmann (RHPIC-LBM) model that describes the continuous kinetic-dynamic-hydro fully coupled LTSTMR evolution process. First, based on the governing equations of resistive relativistic magnetohydrodynamics (MHD), the relativistic discrete distribution functions for a magnetic field (D3Q7), electric field (D3Q13), electromagnetic field (D3Q13), charged particle (D3Q19) and neutral particle (D3Q27) of different plasma species are obtained for the RHPIC-LBM lattice grid. Then, the numerical process, algorithm, pseudocode, flowchart and GPU-CPU heterogenous framework of the RHPIC-LBM are described. Finally, this model is tested and validated on Tianhe-2 from the National Supercomputer Center in Guang Zhou (NSCC-GZ) with 10,000 ∼ 100,000 CPU cores and 50∼120 hours per case. We investigate the solar atmosphere LTSTMR activities, including the picoscale ( 10 − 2 m ∼ 10 5 m), nanoscale ( 10 5 m ∼ 10 6 m), microscale ( 10 6 m ∼ 10 7 m), macroscale ( 10 7 m ∼ 10 8 m) and large hydroscale ( 10 8 m ∼ 10 9 m). All the simulation results are consistent with observations and theories. The validated model is applied to explore the turbulence evolution of the interactions of the magnetic helical structures in the 3D LTSTMR self-generating-organization magnetic field region and self-feeding-sustaining plasma region in Part II.

Pricing and inventory management make up together revenue management, which is a significant effort to boost revenues out of available resources. Firms use various forms of dynamic pricing, including personalized pricing, markdowns, promotions, coupons, discounts, and clearance sales, to respond to market fluctuations and demand uncertainty. In this paper, we study a temporary price increase policy, a form of dynamic pricing, for a non-perishable product, a practice used by several giant retailers such as Amazon, Walmart, and Apple. We develop a continuous review inventory model that allows for joint replenishment and pricing decisions, where the lead time is not zero. A replenishment decision controls supply, while a pricing decision controls demand. A manager exercises a temporary price increase to slow demand and avoid a stock-out situation while waiting for a shipment, which may not necessarily increase revenues, but decrease stock-out costs. The problem is to solve for the optimal replenishment and the pricing policy parameters that maximize the long-run expected profit. That is, when and how much to order and when to raise the price. In this paper, the inventory level and time trigger a price increase. We solve many numerical examples and perform extensive sensitivity analyses. Our results show that compared to a model that focuses on fixed pricing, our model brings an additional increase in profit of about 13%.

This paper deals with the modeling of a hybrid energy multisource network composed by a non-renewable energy source and a renewable energy source. The mathematical model is derived within the framework of the thermostatted kinetic theory where the external force field coupled to the thermostat term mimics the construction of the energy storage. The parameters of the mathematical model are set in order to promote the use of the renewable energy source thus improving the quality of the provided energy. A computational analysis is performed to show the emerging phenomena that the model is able to capture. Specifically the computational analysis is mainly addressed to a sensitivity analysis on the switching-source parameters and the transition-energy parameters. Moreover the construction of the energy storage is analyzed by performing a sensitivity analysis on the magnitude of the external force field. Discussions and future research perspectives are postponed to the last section of the paper.

Complex underground constructions in urban areas require strict predictions for ground movements and liner deformation induced by shield-driven tunneling, in which the complex interaction mechanics between ground and liner play a substantial role. Previous studies, however, provided little information on the ground-liner interaction and less attention to the effects of groundwater and compressed air during the shield operation. This paper presents a closed-form analytical solution for predicting long- and short-term ground deformation and liner internal forces induced by tunneling in saturated soils in which shield excavation effects with and without air pressure are both considered. The oval-shaped convergence deformation pattern is incorporated as the boundary condition of displacements around the tunnel section. This paper also investigates the difference between uniform radial and oval-shaped convergence deformation patterns on the ground and tunnel responses. Generally, the predicted ground movements by the oval-shaped deformation pattern aligns well with measured data of actual tunnels with and without considering the shield air pressure. It is comparatively observed that the shield excavation under air pressure obtains larger ground deformation than the non-pressure condition, and the long-term ground settlements induced by tunneling in saturated soils are confidently larger than the short-term. Moreover, the effects of sensitive parameters, including the shield air pressure, the long- and short-term effects on the tunneling-induced ground movements are assessed based on the oval-shaped deformation pattern. Furthermore, parametric analyses are conducted to measure the influences of concerned tunneling coefficients on the liner displacements and internal forces, namely, soil Young's modulus, soil unit weight, coefficient of lateral soil pressure, tunnel radius, tunnel buried depth and gap parameter. In summary, the analytical approach proposed in this research provides an effective insight into the ground-liner interaction mechanics related with the shield air pressure, which can serve as an alternative approach in the preliminary design for conservatively estimating the excavation influences caused by tunneling in saturated soils.

Quantitatively analyzing the role of turbulence in the magnetic fluctuation-induced self-generating-organization region and the plasma turbulence-induced self-feeding-sustaining region, which is closely related to the evolution of null points and the magnetic helical structure, represents the key issues in understanding the magnetic energy release-conversion, plasma heating, and charged particles energization and acceleration in three-dimensional large temporal-spatial scale turbulent magnetic reconnection (3D LTSTMR). The first part of this two-paper series developed and validated the continuous kinetic-dynamic-hydro fully coupled temporal-spatial scale relativistic hybrid particle-in-cell and lattice Boltzmann (RHPIC-LBM) model and code for investigating the fine structure evolution of the 2.5D solar atmosphere LTSTMR activities. Based on the model and code developed in Part I, in this paper, we investigate the turbulence of the magnetic helical structure, current density vector fields, self-generating-organization magnetic potential vector fields, self-feeding-sustaining plasma motion, and the ion and electron acceleration of 3D LTSTMR with 100,000 CPU cores on the Tianhe-2 from National Supercomputer Center in Guang Zhou (NSCC-GZ). According to the simulation evidence, we discovered and confirmed the following results: (i) Slipping magnetic reconnection (MR) exists in the adjacent magnetic field lines (MFLs) during the compress-stretch-slip process on the quasi-separatrix layers (QSLs and MFLs drastically change and form a linkage span) and the adjacent MFLs’ break-rejoin MR exists on the separatrix surfaces (SLs). Both are consistent to observations; (ii) The slipping MR (defined as 1st type MR) and the MFLs’ break-rejoin MR (defined as 2nd type MR) are closely linked with the oblique and resistive tearing instabilities, respectively. In the 3D model, the 1st type MR forms O -type null points, while the 2nd type MR forms X -type null points. The magnetic energy conversion is dominated by turbulence-induced oblique instabilities in the 3D model instead of the resistive tearing instabilities in the 2D/2.5D model, which is consistent with the 3D observations; (iii) Magnetic energy conversion occurs in the interaction of the plasmoid-to-flux rope, plasmoid-to-plasmoid, and flux rope-to-flux rope. The turbulent acceleration is an independent acceleration mechanism in LTSTMR, induced by the interaction of waves-to-waves and waves-to-particles, which is different from the original hybrid acceleration mechanism (composed of parallel electric fields, betatron, shock and Fermi) ; (iv) Particles can be energized and accelerated at a longer time scale (−30 s) and can be accelerated to relativistic energies after being pre-accelerated by a Fermi-Betatron-shock wave acceleration process. These results are in agreement with the observations.

This paper employs a discrete damped dynamic system to investigate the emerging concept of a digital twin. Dynamic systems are well understood across engineering and science domains, and represent a familiar and convenient platform for exploring the various aspects of a digital twin design. The aim is to create a framework accessible to engineering sciences related to the aerospace, electrical, mechanical and computational area. The virtual model of the physical system is expressed as a differential equation in two-time scales, with the concept of a slow time being used to separate the evolution of the system properties from the instantaneous time. Cases involving stiffness variation and mass variation are considered, individually and together. It is assumed that the damped natural frequency and the time response are measured through sensors placed on the physical system. Issues of errors and reduced sampling rate in sensor measurements on the digital twin are investigated. The digital twin is expressed as an analytical solution through closed-form expressions and the effect of sensor errors is brought out through the simulations. Several key concepts introduced in this paper are summarized and ideas for urgent future research needs are proposed. The current work breaks free from the qualitative description of digital twins pervading the literature and can be used as benchmark solutions to validate digital twin of experimental dynamic systems and their implementation using sensors, the internet of things and deployment on the cloud computing systems.

This paper presents investigations into classification and analysis of constraint singularities for parallel mechanisms. Parallel mechanisms (also called parallel manipulators or parallel robots) have wide applications in industry. The singularities tremendously affect their applications. Existing research works show that constraint singularity causes a mechanism to have instantaneous degree-of-freedoms (DoFs) or bifurcated finite motions. However, the intrinsic differences among the conditions under which the specific constraint singularities happen have not been discussed. This paper is focused on these topics by using differential manifolds as mathematical tools. Firstly, the general mathematical models of parallel mechanisms are formulated by respectively describing their finite motions and instantaneous motions in forms of differential manifolds and their tangent spaces. Then, parallel mechanisms having bifurcated finite motions and instantaneous DoFs are modelled accordingly, and the constraint singularities are thus classified into two kinds by considering their influences on motions of mechanisms in both finite and instantaneous motion levels. Finally, two examples are given to further illustrate the theoretical analysis. This paper lays foundations for mathematical modelling and applications of parallel mechanisms with constraint singularities.

In this paper, a chaos-enhanced bat algorithm is proposed to tackle the global optimization problems. Bat algorithm is a relatively new stochastic optimizer inspired by the echolocation behavior of bats in nature. Due to its effectiveness, it has been applied to many fields such as engineering design, feature selection, and machine learning. However, the classical approach is often prone to falling into local optima. This paper proposes an enhanced bat algorithm to alleviate this problem observed in the original algorithm. The proposed method controls the steps of chaotic mapping by a threshold and synchronizes the velocity of agents using a velocity inertia weight. These mechanisms are designed to boost the stability and convergence speed of the bat algorithm, instantly. Eighteen well-established and the state-of-the-art meta-heuristic approaches are considered to validate the effectiveness of the developed algorithm. Experimental results reveal that the proposed chaos-enhanced bat algorithm is not only superior to the well-established algorithms such as the original method but also the latest improved approaches. Also, the proposed method is successfully applied to I-beam design problems, welded beam design, and pressure vessel design. The results show that chaos-enhanced bat algorithm can deal with unconstrained and constrained feature spaces, effectively.

A fin serves as an extended surface to enhance the heat transfer from a larger heated mass to which it is attached. We consider a fin in steady-state and find its shape that maximizes the efficiency. Our approach is based on a piecewise linear approximation of design. We develop the corresponding algorithm and conduct numerical experiments. Our optimization problem is similar but not identical to the problem studied by L. Hanin. We discuss the applicability of Schmidt’s hypothesis for our model. This paper is an extension of the previous authors’ paper where we considered the shape of the fin that minimizes the cooling time of a heated mass.

The present paper introduces an approximation method for finding open geodesics on triangular surfaces. The algorithm is specifically designed to be able to solve real world problems where geodesic paths are needed. We use the model of geodesic curvature flow for open curves in the Lagrangian formulation. The model is enriched with a tangential term in order to have a control over the quality of the discretization grid during the computation. The governing equation of the flow is solved by a numerical method based on a semi-implicit time discretization and a finite difference space discretization. The paper presents the numerical scheme and various implementation details as well as numerous experiments to demonstrate the performance of the method and to provide comparison with several other well known methods. We also present a Grasshopper component for Rhinoceros for finding optimal paths on surface meshes that we developed and that includes our algorithm.

Accurate wind speed forecasting is important in power grid security, power system management, operation and market economics. However, most research has focused only on improving either accuracy or stability, with few studies addressing the two issues, simultaneously. Therefore, we proposed a novel combined model based on multi-objective particle swarm optimization, which is applied to optimize the key parameters of the echo state network. Most combined wind speed forecasting methods just use the combination theory to combine individual methods, this paper uses echo state network to combine the intermediate wind speed forecasting results of three artificial neural networks. Moreover, a new dataset division mechanism based on the moving window is applied in this paper. Firstly, the length of the input data is changed from 5 to 15 for 1-step, 2-step and 3-step wind speed forecasting, after that, the optimal length of the input vector can be got. And then we apply this optimal length of the input vector to another dataset for further verifying the proposed method. In order to verify the forecasting effectiveness of the proposed forecasting model, the 80/min wind speed data of M2 tower of the National Wind Power Technology Center of the United States were taken as an example. The experimental results indicate that the proposed algorithm is superior to the other ten comparative models in prediction accuracy and stability, and it also performs better than the combined model that we have proposed before.