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18 July 2018, Volume 50 Issue 4
Orginal Article
UNIFIED GAS-KINETIC SCHEME FOR TWO PHASE INTERFACE CAPTURING
Wang Zhao, Yan Hong
2018, 50(4):  711-721.  DOI: 10.6052/0459-1879-17-364
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The study of interfacial motion of gas-liquid phase is very important in science and engineering. Considering the non-equilibrium flow calculation, a unified gas-kinetic scheme for gas-liquid two phase interface capturing is presented in this paper. Since the free transport and particle collision are coupled to update the macroscopic variables and microscopic distribution functions, the unified gas-kinetic scheme can solve the non-equilibrium flow. The van der Waals (vdW) equation of state (EOS) is included to describe the coexistence of gas and liquid and the phase transition between them. Because of the characteristics of vdW EOS, the interface between gas and liquid can be captured naturally through condensation and evaporation processes. As a result, the new scheme can solve the gas-liquid two phase problems. Finally, the proposed method is used to obtain the numerical solution of Maxwell construction, which agrees well with the corresponding theoretical solution. Then, the Laplace’s theorem is verified by numerical calculation of the surface tension of the droplet corresponding to the van der Waals state equation. In addition, the collision of the two droplets is simulated, which proves the validity of the scheme further. However, due to the characteristics of the van der Waals equation of state, the constructed scheme is only applicable to the case where the liquid/gas two-phase density ratio is less than 5.

ANALYSIS OF DSMC REACTION MODELS FOR HIGH TEMPERATURE GAS SIMULATION 1)
Yang Chao, Sun Quanhua
2018, 50(4):  722-733.  DOI: 10.6052/0459-1879-18-056
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The non-equilibrium phenomenon of thermochemical coupling has been a difficult problem in high temperature aerothermal dynamics, and hinders to analyze phenomena such as cell structure of detonation wave and ignition speed of low temperature combustion. In this paper, typical chemical reaction models (TCE, VFD, QK models) employed in the direct simulation Monte Carlo (DSMC) simulation are analyzed using two examples (namely, N$2$ dissociation at high temperature, and chain displacement reaction in H$2$? O$2$ combustion) from microscopic reaction probability, vibrational state specific reaction rates, total reaction rate under thermal nonequilibrium condition, and post-collision redistribution of internal energy. It is found that the probability distribution of vibrational energy of reacted molecules deviates from the equilibrium Boltzmann distribution for both the high temperature dissociation reaction having high activation energy and the chain displacement reaction having low activation energy. The VFD model with strong vibrational favored contribution can predict well the high temperature dissociation reaction, whereas the TCE model (a special case of VFD model) and QK model are better for the chain displacement reaction. Besides, the post-collision redistribution of internal energy should follow the principle of detailed balance, as small deviations may cause inequality between the translational and vibrational energy under final equilibrium state. The DSMC simulation results also show that the vibrational favor of chemical reactions has an obvious effect on the thermochemical coupling process. Particularly, because molecules having high vibrational energy are more easily to have chemical reactions, the decrease of the average vibrational energy of the gas will affect the subsequent chemical reactions.

THE SINGLE STRIP-INDUCED CHANGE OF 2P-MODE VORTEX SHEDDING IN THE WAKE OF A TRANSVERSELY OSCILLATING CYLINDER
Cao Mengyuan, Jin Huabin, Shao Chuanping
2018, 50(4):  734-750.  DOI: 10.6052/0459-1879-18-005
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The control of vortex shedding from a forced oscillating cylinder is the basis of control of vortex-induced cylinder vibration, and of significance in civil and ocean engineering. A lock-on mode of vortex shedding called 2P mode exists in the wake of an oscillating cylinder, i.e., in a period of oscillation, a pair of vortices shed from each side. A strip element of width b/D=0.32 is used and set in the wake to influence the 2P mode vortex shedding at oscillating amplitude A/D=1.25, frequency f_e D/V_∞=0.22, and Reynolds number Re=V_∞ D/ν=1 200. Using the method of 2D large eddy simulation with experimental confirmation, we find that with the change of strip position in the area 0.8≤X/D≤3.2, 0.4≤Y/≤3.2, vortex shedding is changed between 2P, 2S, P+S and other 6 new modes. The detailed generation process of every mode of vortex shedding is described in the paper. The zones of the modes and the contours of vortex strength are figured out on the plane of strip position. It is shown that vortex strength can be reduced by 50% or more and the wake is narrowed if the strip is set at places in a considerably large region. The transverse shear flow induced by large amplitude cylinder oscillation plays a key role in the generation of vortex shedding. The influence of the strip on the transverse shear flow is discussed and the mechanism of the strip function to the formation of every mode of vortex shedding is analyzed.

RESEARCH ON DYNAMIC TEST TECHNOLOGY FOR WIND TURBINE BLADE AIRFOIL
Li Guoqiang, Zhang Weiguo, Chen Li, Nie Bowen, Zhang Peng, Yue Tingrui
2018, 50(4):  751-765.  DOI: 10.6052/0459-1879-18-108
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The dynamic oscillation process of wind turbine blades is usually accompanied by pitching and yaw. Due to the unclear understanding of many dynamic problems previously, a safer design is adopted at the expense of increasing the weight of the blade structure in engineering, usually neglecting the influence of the yaw oscillation. The design of large wind turbines has put forward higher requirements for obtaining more comprehensive and accurate dynamic loads of airfoils. It is of great significance to study the influence of yaw oscillation on the dynamic aerodynamic characteristics of airfoil. In view of this, the dynamic wind tunnel test on yaw oscillation of airfoil is carried out in this paper for the first time. The “electronic cam” technology is used instead of the mechanical cam to realize the stepless adjustment of oscillation frequency and oscillation angle. Based on the designed electronic external trigger device, the real-time measurement of the dynamic flow field is realized. Meanwhile, the synchronous acquisition of wind tunnel flow, model angular displacement and dynamic pressure data is realized. Furthermore, the static pressure measurement, pitching / yaw dynamic pressure measurement, PIV and fluorescent wire test are carried out respectively. The accuracy of the test results is high, and the regular pattern is reasonable. Besides, the influence mechanism of wall interference in dynamic test is analyzed. Research shows that: there is also obvious hysteresis effect on the dynamic aerodynamic parameters of yaw oscillation airfoil with the changing of the angle of attack. And with the increase of oscillation frequency, the aerodynamic hysteresis characteristics of the airfoil under pitching and yaw oscillation are all enhanced. The dynamic stall vortex at the positive stroke is delayed due to the pitching oscillation. The pressure distribution of the airfoil is greatly influenced by the strong three-dimensional effect at the intersection of the wind tunnel wall and the model tip. Overall, the dynamic test technique of yaw oscillation established in this paper can provide technical support for the study of the dynamic swept effect of the wind turbines.

GALLOPING IN VORTEX-INDUCED VIBRATION OF THREE TANDEM CYLINDERS AT LOW REYNOLDS NUMBERS AND ITS INFLUENCING FACTORS
Chen Weilin, Ji Chunning, Xu Dong
2018, 50(4):  766-775.  DOI: 10.6052/0459-1879-18-057
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Vortex-induced vibrations of three tandem cylinders at the spacing ratio of 1.2 and the Reynolds number of 100 shows that when the reduced velocity is larger than a critical value, the galloping is observed where the amplitudes of three cylinders increase with the reduced velocity. Three factors, including shift of equilibrium position, low vibration frequency and timing between vortex shedding and motion of cylinders, determine the appearance of galloping. Further investigations show that the galloping occurs at a range with lower mass ratio no more than 2.0 and Reynolds number no more than 100. When the mass ratio is large, the equilibrium position is unchanged and the displacements are irregular, which leads to the variation of timing between vortex shedding and motion of cylinders. When Re is large, the shift of the most upstream cylinder is zero and no more accommodate cylinders’ vibration, which makes the vibration irregular and change of timing between vortex shedding and motion of cylinders.

INVESTIGATION ON THE TURBULENT CHARACTERISTICS OF THE JET INDUCED BY A PLASMA ACTUATOR
Zhang Xin, Huang Yong, Yang Pengyu, Tang Kun, Li Huaxing
2018, 50(4):  776-786.  DOI: 10.6052/0459-1879-17-392
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In order to understand the controlling mechanism of plasma actuator and develop the mathematical model of plasma actuator, the turbulent characteristics of the jet induced by a dielectric barrier discharge (DBD) plasma actuator in quiescent air was studied in a closed chamber using particle image velocimetry (PIV). Here, an asymmetrical DBD plasma actuator was mounted on the plate model. First, measured time-averaged velocity field induced by the DBD plasma actuator indicated that voltage amplitude is an important parameter and could affect the flow characteristics of the induced jet. When the plasma actuator was driven at low voltage, the induced jet was a laminar wall jet. On the other hand, the Reynolds number of induced jet was improved and the shear layer was instability when the plasma actuator was actuated at high voltage. Then the induced jet became a turbulent wall jet. Secondly, the results of transient flow field structure suggested that the induced turbulent wall jet had some coherent structures, such as rolling up vortex and secondary vortex in the near-wall region. And these structures were linked to a dominant frequency of $f0=109$ Hz. The rolling up vortices had the process of formation, movement, merging and breakdown. Thirdly, the rolling vortex was stretched and collapsed due to the instability of flow field when the plasma actuator was actuated at high voltage. Then turbulence kinetic energy of induced flow filed was increased and the breakdown of rolling vortex was accelerated. The turbulent characteristics of the induced jet could enhance the entrainment effect of plasma actuator between the outside airflow and boundary layer flow, which is very important for flow control applications.

ANALYSIS FOR DYNAMIC RESPONSE OF FUNCTIONALLY GRADED MATERIALS USING POD BASED REDUCED ORDER MODEL $N$
Zheng Baojing, Liang Yu, Gao Xiaowei, Zhu Qianghua, Wu Zeyan
2018, 50(4):  787-797.  DOI: 10.6052/0459-1879-18-069
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In order to quickly analysis the response of heterogeneous materials under dynamic loads, a reduced order method was presented in this paper which only needed to compute dynamic characteristics of homogeneous material under sudden load and got the results for analysis complex non-homogeneous material. Firstly, we used the finite element method to compute the displacement field of homogeneous materials under sudden load, and then discretized data samples was obtained to establish a database which including every moment displacement information of all degrees of freedom (order of $L$) during a period of time. Secondly, dealing with database by specific way of time discretization, a snapshot matrix was formed. The matrix was decomposed into $H$ orthogonal basis by proper orthogonal decomposition method and we picked up the major $H<L?N$ basis from that. Till now we achieved the goal that reducing the model ($H$). Finally, the $[4-5]$ basis were used to obtain order-reduced governing dynamic equation. Different dynamic loads of time dependent were applied to the model, and the dynamic response of non-homogeneous material would be achieved by solving order-reduced governing dynamic equations. The displacement fields of traditional FEM and proposed ROM were compared. 2D and 3D examples showed that the computing scales reduced one or two orders of magnitude.
NUMBER-SHAPED TENSEGRITY STRUCTURES: CONFIGURATION DESIGN AND MECHANICAL PROPERTIES ANALYSIS
Zhu Shixin, Zhang Liyuan, Li Songxue, Zhang Boyang, Zhang Qingdong
2018, 50(4):  798-809.  DOI: 10.6052/0459-1879-18-043
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Due to the novel mechanical properties, tensegrity structures have found various applications in science and engineering, and the design of large-scale tensegrities becomes a vital issue. In this paper, a series of number-shaped tensegrity structures are proposed by assembling the cylindrical and spherical tensegrity elementary cells. Specifically, the quadruplex prismatic tensegrities and the truncated regular octahedral tensegrities are selected as the elementary cells and then connected by using the node-on-node assembly scheme. Furthermore, structural stiffness matrix-based numerical method is employed to simulate the mechanical responses of the assembled tensegrities. Our results show that the obtained number-shaped tensegrities are self-equilibrated and stable when the elementary cells satisfy their self-equilibrium and stability conditions, respectively. A physical sculpture is also constructed using the aluminium alloy bars and nylon strings. Taking the eight-shaped tensegrity structure as an example, the static mechanical responses of the structure subjected to self-weight loading and uniaxial tension/compression are simulated, as well as the structural natural frequencies and modes of its free vibration. The simulations show that the tensegrity could have enough rigidity to bear the self-weight when the structural pre-stress level, the mass density of the compressed bars, and the stiffness of the tensioned strings match well. The load-displacement curves of the tensegrity under uniaxial loading are nonlinear, that is, the tensile stiffness increases with the tensile deformation, while the compressive stiffness decreases with the compressive deformation. The structural natural frequencies are dependent on the pre-stress level, while the vibration modes change little. The present work enriches the shapes of large-scale tensegrities and would promote their applications in civil and material engineering.

NON-ORDINARY STATE-BASED PERIDYNAMIC THERMAL-VISCOPLASTIC MODEL AND ITS APPLICATION
Wang Han, Huang Dan, Xu Yepeng, Liu Yiming
2018, 50(4):  810-819.  DOI: 10.6052/0459-1879-18-113
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A three-dimensional non-local thermo-visco-plastic solid model considering the strain rate effect, plastic hardening, thermal softening and fracture characteristics of materials, together with corresponding non-local spatial integral-type numerical method, have been proposed under the configuration of the recently developed non-ordinary state-based peridynamic (NOSB-PD) theory, and the model and numerical method have been employed to analyze the high-rate thermal-viscoplastic deformation and failure behavior of metallic materials and components under impact loads. The validity of the proposed model and algorithms was established through simulating the three-dimensional classical Kalthoff-Winkler impact experiment and comparing the numerical results including the cracking initiation time and orientation, crack propagation path and propagation speed, and the distribution of temperature and equivalent stresses in the target with experimental observations and available numerical results in literature. The proposed model and method were further applied to simulate the deformation and failure mechanism of double-notched metallic plates subjected to impact loads with different impacting velocities. Numerical results show that the present model inherits the advantages of both peridynamics and classical thermo-visco-plastic models, and is able to describe the whole elastic and plastic deformation and crack propagation processes qualitatively as well as quantitatively. Moreover, the effect of impact velocity on crack initiation time, crack propagation path and crack propagation speed were investigated. When subjected to impact load with lower impacting velocity, the crack initiation time of the target plate will be later (until no crack propagation appears when impacting velocity is lower than some value), and both the crack propagation speed and the peak temperature in the target plate will decrease.

EXPERIMENTAL STUDY OF THE HIGH VELOCITY EXPANSION AND FRAGMENTATION OF PMMA RINGS
Li Tianmi, Zhang Jia, Fang Jisong, Liu Lizhi, Zheng Yuxuan, Zhou Fenghua
2018, 50(4):  820-827.  DOI: 10.6052/0459-1879-18-016
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The dynamic fracture and fragmentation of brittle solids under impact loading are important research subjects. However, the experimental study on the tensile fracture and fragmentation of brittle solids is relatively limited. A technique using liquid-driving expansion ring setup was developed for the dynamic tensile fracture and fragmentation testing of brittle materials. This technique was used to study the fragmentation properties of PMMA rings at different expansion velocities. From the observations of the fracture morphology and the residual internal cracks of the recovered fragments, it is concluded that the fracture of the rings is caused by the circumferential tensile stress. The unloading stress waves from the fracture points of the fragments inhibit the further development of other cracks close to the fracture points by unloading the tensile stress in the tension regions. The PMMA ring expansion process was captured using ultrahigh speed camera. The specimen surface expansion velocity was measured using laser interference device DISAR (displacement interferometer system for any reflector). The strain history and fracture strain of ring were captured using the strain gauge on the specimen. Preliminary experimental results conducted on PMMA rings show that: (1) In the range of tensile strain rate $150~500s-1$, the dynamic failure strain of PMMA is lower than that under the quasi-static tensile loading, which means that PMMA became brittle under higher strain rate loading; (2) Higher loading rates resulted in the more fragments and the smaller size of the PMMA rings; (3) The “non-dimensional fragment size vs. strain rate” data fall between the theoretical fragmentation predictions for ductile material and brittle material.

THE EVOLUTION OF INTRAGRANULAR VOIDS UNDER INTERFACE MIGRATION INDUCED BY STRESS MIGRATION
Yu Wentao, Huang Peizhen
2018, 50(4):  828-836.  DOI: 10.6052/0459-1879-18-015
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With the rapid development of microelectronics technology, the failure of interconnects in the integrated circuit raises wide attention. The interconnects inevitably exist some drawbacks, such as voids and cracks. If the drawbacks nucleate, grow and change their shape to form crack-like slits oriented perpendicular to an interconnect line, an open circuit could result. This is a common form of interconnects failure. And interface migration is one of the main mechanisms leading to the evolution of microstructure. Based on the classic theory and weak statement of interface migration, a finite-element method is developed to simulate the evolution of intragranular voids in copper interconnects caused by interface migration induced by stress migration. The validity of the method is confirmed by the agreement of the numerically simulated the undulating surface with that predicted theoretically. Through a large number of numerical simulations, we find that the evolution of the intragranular voids has two trends, namely, void growth and void shrinkage. And the shape of the void is governed by the stress, $β$ , the linewidth, $σ?c$ , and the initial aspect ratio of the intragranular void, $h?c$, and there exist critical values for these parameters ( $βc$, $h??h?c$ and $σ?$ ). When $h?$, $β$ or $σ?$, the intragranular void will grow along the major axis; otherwise, the intragranular void will shrink into a cylinder. The increase of the stress, or the aspect ratio, or the decrease of the linewidth is beneficial to void growth. And the area of void growth will increase faster with bigger $h?$ , smaller $β$ or bigger $~$ . But, the decrease of the stress or the aspect ratio, or increase the linewidth accelerates void shrinkage and the shrinkage area will decrease faster with smaller $~$ , bigger $β$ or smaller $σ$ .

STUDY ON NONLINEAR ELASTIC HOMOGENIZATION WITH ITERATIVE METHOD
Hou Shujuan, Liang Huiyan, Wang Quanzhong, Han Xu
2018, 50(4):  837-846.  DOI: 10.6052/0459-1879-18-039
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Microstructure is critical to affect or change the macroscopic mechanical properties of composites, and the desired material properties can be obtained by rationally designing the composite microstructure. As an effective design method, homogenization method is used to obtain and design the macro-mechanical properties on the basis of microstructure. However, once considering the nonlinear factors, the realization of homogenization can be very difficult. Therefore, this paper focuses on the nonlinear elastic homogenization of composite materials by theoretical deduction, and solves the problem by direct iteration method. In this study, the equation of nonlinear elastic homogenization is deduced by the asymptotic expansion homogenization method. The iterative steps of direct iteration method are given to solve the nonlinear elastic homogenization equation. According to the iterative steps and the nonlinear elastic homogenization equation, the program in MATLAB language is obtained. The porous materials with three typical constitutive relations are chosen to be the study object. The program and iterative method is verified by comparing the strain energy, maximum displacement and equivalent Poisson’s ratio with the results of detailed model. Then, the application of nonlinear elastic homogenization method is extended to three-dimensional composite materials with multi-scale periodic microstructure, a three-element rubber-based composite material. It is divided into core-scale and layer-scale and homogenized with multi-scale homogenization method. The equivalent elasticity modulus of the core-scale are obtained by linear elastic homogenization method and used as a parameter of a component in layer-scale. Then, the nonlinear elastic homogenization method is used for layer-scale. The macroscopic equivalent performance of the material is obtained and compared with experimental results. The nonlinear elastic homogenization method has certain guiding significance and reference value for the nonlinear homogenization and microstructure design of the composite material.

A SPECTRAL-DIFFERENTIAL QUADRATURE METHOD FOR 3-D VIBRATION ANALYSIS OF MULTILAYERED SHELLS
Ye Tiangui, Jin Guoyong, Liu Zhigang
2018, 50(4):  847-852.  DOI: 10.6052/0459-1879-18-083
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The equivalent single layer (ESL) theories can be grossly in error for predicting vibration characteristics of thick multilayered shells because the vibration displacement and stress field of such shells under vibration are in full 3-D coupling condition. It is necessary to develop more accurate and efficient methods which are capable of dealing with multilayered structures with different boundary conditions, general laminations as well as arbitrary thickness universally. In order to overcome the drawback of the existing three-dimensional methods that are only confined for very limited cases such as cross-ply laminated rectangular plates under simply-supported boundary conditions, a general spectral-differential quadrature method is proposed. This method is undertaken by the exact 3-D elasticity theory so that it’s able to study very well the dynamic behavior of thick multilayered structures which cannot be provided by the 2-D ESL theories. In each individual layer, the transverse domain is discretized by the differential quadrature technique. The displacement fields of the discretized surfaces are selected as fundamental unknowns. Then, each fundamental unknown is invariantly expanded by the general spectral method as a series of complete, orthogonal polynomials. The problems are stated in a variational form by the aid of penalty parameters which provides complete flexibilities to describe any prescribed boundary conditions. The current method can successfully avoid solving a highly nonlinear transcendental equation that is rely on roots-locating numerical method and all the modal information can be obtained just by solving linear algebraic equation systems. Numerical verification shows that the proposed method has high calculation precision. The method can be directly extend to the static and dynamic analysis of multilayered shells as well.

THREE-DIMENSIONAL GEOMETRIC NONLINEARITY ELEMENT-FREE METHOD BASED ON S-R DECOMPOSITION THEOREM
Song Yanqi, Zhou Tao
2018, 50(4):  853-862.  DOI: 10.6052/0459-1879-18-050
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Due to its overcoming the deficiencies of classic finite deformation theories, Strain-Rotation (S-R) decomposition theorem can provide a reliable theoretical support for the geometrically nonlinear simulation. In addition, due to it’s independent of the elements and meshes, the element-free method has more advantages to solve large deformation problems compared to finite element method (FEM), so that the accuracy is guaranteed as a result of avoiding the element distortions. Therefore, a more reasonable and reliable geometric nonlinearity numerical method certainly will be established by combining the S-R decomposition theorem and element-free method. But the studies of element-free methods based on S-R decomposition theorem in current literature are limited to two-dimensional problems. In most cases, three-dimensional mathematical-physical models must be established for the practical problems. Therefore it is very necessary to establish a three-dimensional element-free method based on the S-R decomposition theorem. Present study extends the previously work by authors into three-dimensional case: The incremental variation equation is derived from updated co-moving coordinate formulation and principle of potential energy rate in this paper, and three-dimensional discretization equations are obtained by element-free Galerkin method (EFG). By using the MATLAB programs based on the proposed 3D S-R element-free method in present study, the nonlinear bending problems for three-dimensional cantilever beam and simply supported plates subjected to uniform load are numerical discussed. The reasonability, availability and accuracy of 3D S-R element-free method proposed by present paper are verified through comparison studies, and the numerical method in present work can provide a reliable way to analysis 3D geometric nonlinearity problems.

MODEL SMOOTHING METHODS IN NUMERICAL ANALYSIS OF FLEXIBLE MULTIBODY SYSTEMS
Qi Zhaohui, Cao Yan, Wang Gang
2018, 50(4):  863-870.  DOI: 10.6052/0459-1879-18-111
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Dynamic equations of flexible multibody systems are usually a set of stiff differential equations. At present, the common numerical method for solving the stiff differential equations filters out the high frequency by using the numerical damping. The computational efficiency of this method is still unsatisfactory. In order to reduce the stiffness of dynamic equations of flexible multibody systems so greatly that the equations can be solved by regular ordinary differential equation (ODE) solvers such as MATLAB ODE45 solver, methods of filtering high frequency vibrations during the process of modeling are studied. Stresses of flexible bodies are homogenized by their mean value over a time interval from now to a short time later. The homogenized stress is then employed to replace its origin when computing the virtual deformation power. In this way, the obtained model of the flexible multibody system will not contain harmful high frequency elastic vibrations. The range of frequencies can be controlled by the length of the time interval used to homogenize stresses. As validated by the numerical examples in this paper, the precision and efficiency of the proposed method are comparable to some stiff ODE solvers. Moreover, it works well when the stiff ODE solver fails to give correct solutions in a reasonable time. Comparisons of numerical examples show that the proposed method can be a new available approach to numerical analysis of flexible multibody systems.

CONTROL METHOD OF AN UNDERACTUATED BIPED ROBOT BASED ON GAIT TRANSITION
Ge Yimin, Yuan Haihui, Gan Chunbiao
2018, 50(4):  871-879.  DOI: 10.6052/0459-1879-18-049
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The stability control of underactuated 3-D biped robot is still a hard nut to crack, as a result of locomotion characteristics which mix high dimension, strong nonlinearity and underactuation. Some traditional control methods, such as event-based feedback control and PD control, are poor in robustness and weak in resistance to external disturbances. Through observation, it is certain that humans adjust gaits tactically to regain stability when they are affected by external disturbances, by contrast with trying to keep the stability sustained by only one gait which is quite limited. Inspired by this, a control method based on gait transition is proposed for the underactuated 3-D biped robot. First of all, taking the minimum energy consumption as the optimization goal, a multi group of gait and step gait is designed as the reference gait to build a gait library by nonlinear optimization method. Secondly, to obtain an optimal performance in terms of the balance between the stability and input torques, a multi-objective gait transition function is established. Finally, a reference gait that minimizes the gait transition function is obtained by solving a quadratic optimization problem, and it is then used as the walking gait for the next step with the purpose of using gait library (multiple trajectories) method to reach the goal of improving robustness. In the simulation experiment, using the proposed gait transition control method, the underactuated 3-D biped robot can walk through the rough ground with the relative height varying within the range [$-$20,20] mm without falling down, in contrast to the failure of previous one-gait control method. The results show the effectiveness of the method.

STOCHASTIC RESONANCE OF A MEMORIAL-DAMPED SYSTEM WITH TIME DELAY FEEDBACK AND FLUCTUATING MASS
Gong Xulu, Xu Pengfei
2018, 50(4):  880-889.  DOI: 10.6052/0459-1879-18-051
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The stochastic resonance (SR) in the memorial under-damped system with time delay feedback and fluctuating mass is investigated in this paper. The non-Markovian original system is reformulated into two-dimensional Markovian linear system through introducing variable transformations and using the small time delay approximation. Further, the analytic expressions for the first moment of the response and the steady response amplitude are derived by using the Shapiro-Loginov formula and the Laplace transformation technique. All the research results show that when the Routh-Hurwitz stability is satisfied, the phenomenon of SR is shown with the variations of mass fluctuation noise intensity, driving frequency and time delay, respectively. The stochastic multi-resonance phenomenon is also observed. Moreover, the SR is enhanced with an increase in time delay by introducing the time delay feedback and instead, the SR is suppressed for large memory time and damping parameter. By adjusting the time delay feedback and the memory effects, the response of the system to a harmonic signal can be further improved. Finally, the theoretical results are well verified through numerical simulations

EFFECT OF PARALLEL MICRO-FRACTURES ON FLOODING BASED ON PORE-FRACTURE NETWORK MODEL
Liu Haijiao, Zhang Xuhui, Lu Xiaobing
2018, 50(4):  890-898.  DOI: 10.6052/0459-1879-18-025
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It is of great importance to understand the micro-seepage mechanism of water flooding in dual pore-fracture media, for the improvement of the recovery efficiency. The existence of micro-fractures can on the one hand increase the absolute permeability, on the other hand change the local fluid pressure distribution and flow in the porous media. The fracture flow prevails, the surrounding oil cannot be displaced, reducing the displacement efficiency. In this paper, a pore-fracture network model is used in the analysis, two parallel micro-fractures with equal length are set at the inlet. The effects of the relative interval (micro-fracture gap length/throat length) and the relative length (micro-fracture length/throat length) of the micro-fractures on micro-seepage are investigated. The results show that with the increase in relative length of micro-fractures, the displacement efficiency decreases, while the water saturation at the co-permeable zone and at the intersection of the relative permeability curve increase. With the increase of the relative interval between the two fractures, the pressure of the surrounding pores is approximately equivalent, and the oil displacement will not occur due to the capillary pressure, leading to water channeling and the decrease of the oil recovery.

DYNAMIC CHARACTERISTICS OF HYDROSTATIC THRUST BEARING WITH DOUBLE RECTANGULAR CAVITY UNDER EXTREME WORKING CONDITION
Yu Xiaodong, Yuan Tengfei, Li Daige, Qu Hang, Zheng Xuhang
2018, 50(4):  899-907.  DOI: 10.6052/0459-1879-18-041
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The dynamic characteristics of hydrostatic thrust bearings are influenced by the lubricating oil viscosity, oil film thickness, oil cavity area and other factors, and hydrostatic thrust bearing often withstands step load or sinusoidal load under extreme working conditions, and the sudden load will lead to change of dynamic characteristics, that is, the impact resistance of the bearing and the time required to restore the balance. In order to characterize the dynamics of the hydrostatic thrust bearing with double rectangular cavity under extreme working condition of high-speed heavy micro-gap, its dynamic characteristics were theoretically analyzed under different oil film thickness, lubricating oil viscosity, and oil cavity sizes. The influences of lubricating oil viscosity, oil film thickness and oil cavity area on the dynamic performance of the bearing under step load were also investigated. In further exploring the stability of the bearing under sinusoidal load, details on the dynamic changes of the oil film were revealed. Using the lubrication theory, mechanical control principles, and a theoretical research methodology, the influences of major oil film and design parameters on the dynamic performance of the bearing were derived, i.e. the influences of the viscosity of the lubricating oil, oil film thickness, and oil cavity size. The results show that the changes of the oil viscosity, oil film thickness, and oil cavity size have significant effects on the dynamic performance of the hydrostatic thrust bearing. The viscosity of the lubricating oil, the thickness of the oil film, and the area of the oil cavity are all positively correlated with the capability of the lubricating film to resist impact loading and the capability of the rotating table to maintain its equilibrium or recover its balance in a shorter time under high dynamic external loads. It is found that the oil film of the hydrostatic thrust bearing with double rectangular cavity has a larger damping coefficient with a high capability to resist sinusoidal loading.

ONE-DIMENSIONAL QUASI-STATIC CONSOLIDATION MODEL CONSIDERING INERTIA OF FLUID PHASE
Ding Zhouxiang
2018, 50(4):  908-928.  DOI: 10.6052/0459-1879-18-053
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Terzaghi’s one-dimensional classic consolidation theory ignores inertia of pore water in saturated soils, and has outstanding differences in its derivation and formulation during various publication periods. This leads to a strange phenomenon that considerable misunderstandings and confusion of it still prevail in the current literature. Within the previous framework of large strain dynamic consolidation theory, a one-dimensional infinitesimal strain consolidation wave model is obtained to consider the effect of inertia of pore water with necessary simplification and hypotheses. The present consolidation wave model is characterized by velocity dispersion and dissipative attenuation. The method of separation of variables is used to obtain an analytical solution for a consolidation wave model under the condition of one-way drainage and instantaneous loading. A numerical case study shows that behaviors of consolidation wave are actually controlled by the value of a dimensionless number $Dc$. For cases of higher values of $Dc$, jump and fluctuation of dimensionless excess pore water pressure between positive and negative values are prone to occur, in contrast with the cases of lower values of $Dc$ that result into special phenomena including the Mandel-Cryer effect observed in laboratory tests. The inherent ambiguities about Terzaghi’s classic theory models proposed in the early stage and the later stage respectively are investigated to draw a conclusion that the early Terzaghi’s (1923,1925) consolidation model can be interpreted as a large strain model with respect to the general soil coordinates contrasted with an infinitesimal strain model with respect to the solid-phase volume coordinates. Accordingly, the present consolidation wave model can be extended to various formulations based on the corresponding coordinate properties. Consolidation wave theory is of significance to probe into an innovative uncertainty principle which shows that for scale model testing, the observed consolidation wave response of undisturbed soil samples cannot equal the response of the same soil in practical conditions. Therefore, it is advisable to pay attention to the size effect of soil samples in consolidation study from the perspective of microscopic soil mechanics. The theoretical precision of evaluated excess pore water pressure by classic Terzaghi’s consolidation model varies with the value of dimensionless quantity $Dc$, which is a vital parameter in the proposed consolidation wave theory.

A CONSTITUTIVE MODEL FOR SAND UNDER COMPLEX LOADING CONDITIONS
Wan Zheng, Meng Da
2018, 50(4):  929-948.  DOI: 10.6052/0459-1879-18-047
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Abstract The test shows that stress-strain relationship of saturated sand has significant dependence on density and confining pressure. The above two factors can not be ignored to describe the deformation behavior of sand under static load conditions. In addition, saturated sand also exhibits obvious stress-induced anisotropy and phase transformation behaviors under complex loading, such as cyclic loading conditions. The distance $R$ between the current stress state and its corresponding point in critical state line (CSL) can be treated as a state parameter is introduced into the proposed model to reflect the density and confining pressure dependent behaviors based on the assumption that there is a unique CSL in $e$--$p$ space. The influence principle to stress-strain relationship under monotonic loading condition due to density and confining pressure is accurately described by using unified hardening parameter introduced by phase changing stress ratio and peak stress ratio expressed by exponential functions of state parameter. The shear volume compression, dilatancy, strain softening and hardening are all described for sand. By using non-associated flow rule, a water drop shape yield surface and an ellipse shape plastic potential surface are adopted in $p$--$q$ space. The liquefaction phenomenon under monotonic loading condition are also be described. To reflect the accumulation of plastic volume strain and hysteresis loops of deviatoric plastic strain under cyclic loading condition, the state parameter $R$ can be expressed as stress ratio parameter and the rotational hardening part can be adopted to describe the stress-induced anisotropy are introduced into the hardening parameter. The attenuation of shear modulus, stiffness weaken and strength decreasing behaviors are described effectively by using the proposed model. The cyclic mobility phenomenon is predicted under undrained cyclic loading conditions. The effectiveness and applicability of the proposed constitutive model is verified by the comparison of a series of simulation and test results.

MODIFIED SEMI-ANALYTICAL SENSITIVITY ANALYSIS AND ITS ERROR CORRECTION TECHNIQUES
Zhang Lidan, Zhang Sheng, Chen Biaosong, Li Yunpeng
2018, 50(4):  949-960.  DOI: 10.6052/0459-1879-18-058
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Modified semi-analytical sensitivity analysis algorithm and its error correction term method are presented, where the sensitivity analysis terms and the error correction term can be separated. The method can facilitates program implementation and the accuracy of the method won’t be influenced by perturbation step length and number of elements. Firstly, a modified semi-analytical sensitivity analysis technique with its error correction term is presented for static displacement, which is based on global structure equations of the sensitivity analysis, and its program implementations are provided. Then, the modified method is implemented on other analysis tasks including natural frequency and linear buckling analysis. Consequently, the error correction terms of both beam elements and shell elements are derived. Then, the specific deducing process of error correction terms concerning beam and shell elements is described. Next, the modified method is verified by typical finite element models with beam and shell elements. The results highlight the applicability of the modified method to various analysis types mentioned above, and the accuracy is not influenced by the number of elements and perturbation step length. Since sensitivity analysis parts and error correction term can be computed respectively, the error correction term can becomputed independently and added directly to the results of sensitivity analysis, which can make full use of existing sensitivity analysis programming. This modified method can help complex engineering structural design. Especially, compared to the original semi-analytical sensitivity analysis and error correction methods, the computational efficiency of the modified method is enhanced with respect to shape optimization design variables or shape combined with size optimization, which can provide new ideas for sensitivity analysis and its program implementation.

A DUAL-LEVEL SINGULAR BOUNDARY METHOD FOR LARGE-SCALE HIGH FREQUENCY SOUND FIELD ANALYSIS
Li Junpu, Chen Wen
2018, 50(4):  961-969.  DOI: 10.6052/0459-1879-18-100
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Numerical simulation of the large-scale high frequency sound field is a computational challenging task. To solve the difficulty that the traditional boundary collocation methods are not easy to be applied to large-scale problems thanks to the resulting large-scale fully-populated matrix, a dual-level singular boundary method is proposed in this study. By introducing a dual level structure, the fully-populated matrix is transformed to a large-scale locally supported sparse matrix on fine mesh. The bottleneck of excessive storage requirements and a large number of operations encountered by the traditional singular boundary method is hereby avoided. Secondly, the method uses only coarse mesh nodes to evaluate far-field contributions, and it is a kernel-independent algorithm. In comparison with the fast multipole method, the dual-level singular boundary method performs higher adaptability and flexibility. In addition, the dual level structure plays a role of preconditioner, which makes the method is very efficient for solving matrix with large scale, high rank and high condition number. In scattering sphere example, the dual-level singular boundary method simulates well the acoustic scattering problem with up to dimensionless wavenumber of 160 when the number of degrees of freedom is taken as 100 000. In the benchmark human head sound scattering, the dual-level singular boundary method using 80 000 degrees of freedom performs 78.13% faster than the COMSOL, and it is noted that the computational frequency is up to 25 kHz, which is far beyond the limit of hearing of human ear.

REVIEW OF THE FIRST NATIONAL SYMPOSIUM ON PHYSICAL MECHANICS FOR YOUNG SCHOLARS
Zhu Yunfei, Han Zengyao, Jiang Lixiang, Zhan Shige
2018, 50(4):  970-976.  DOI: 10.6052/0459-1879-18-244
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In this paper we gave a brief introduction to the First National Symposium on Physical Mechanics for Young Scholars and reviewed all the scientific reports presented at this symposium.