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- SIZE EFFECT OF THE INTERFACE ENERGY DENSITY IN BI-NANO-SCALED-METALLIC PLATES
- Wang Shuai, Yao Yin, Yang Yazheng, Chen Shaohua
- 2017, 49(5): 978-984. DOI: 10.6052/0459-1879-17-142
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- STUDY ON THE INFLUENCE OF THE NOSE SLENDERNESS RATIO OF HIGH-SPEED TRAIN ON THE AERODYNAMIC NOISE
- An Yi, Mo Huangrui, Liu Qingquan
- 2017, 49(5): 985-996. DOI: 10.6052/0459-1879-17-126
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- INSTABILITY STUDY OF AN ELECTRIFIED COAXIAL JET IN A COFLOWING GAS STREAM
- Li Shuaibing, Yang Rui, Luo Xisheng, Si Ting
- 2017, 49(5): 997-1007. DOI: 10.6052/0459-1879-17-082
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- SUPER-HARMONIC AND SUB-HARMONIC SIMULTANEOUS RESONANCES OF FRACTIONAL-ORDER DUFFING OSCILLATOR
- Jiang Yuan, Shen Yongjun, Wen Shaofang, Yang Shaopu
- 2017, 49(5): 1008-1019. DOI: 10.6052/0459-1879-17-105
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- NUMERICAL STUDY FOR LAMINAR FLOW OF NON-NEWTONIAN FLUID BASED ON FRACTAL DERIVATIVE
- Su Xianglong, Xu Wenxiang, Chen Wen
- 2017, 49(5): 1020-1028. DOI: 10.6052/0459-1879-16-318
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- AUTOMATIC VISCOUS BOUNDARY LAYER MESH GENERATION
- Gan Yangke, Liu Jianfei
- 2017, 49(5): 1029-1041. DOI: 10.6052/0459-1879-17-154
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- NONLINEAR NUMERICAL STUDY OF NON-PERIODIC WAVES ACTING ON A VERTICAL CLIFF
- Li Xiang, Zhang Chongwei, Ning Dezhi, Su Peng
- 2017, 49(5): 1042-1049. DOI: 10.6052/0459-1879-16-337
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- CONFINEMENT EFFECT ON THE RISING DYNAMICS OF A SKIRTED BUBBLE
- Zhang Yang, Chen Ke, You Yunxiang, Ren Wei
- 2017, 49(5): 1050-1058. DOI: 10.6052/0459-1879-17-212
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- EQUIVALENT FRACTIONAL ORDER MICRO-STRUCTURE STANDARD LINEAR SOLID MODEL FOR VISCOELASTIC MATERIALS
- Xu Yeshou, Xu Zhaodong, Ge Teng, Xu Chao
- 2017, 49(5): 1059-1069. DOI: 10.6052/0459-1879-17-134
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- AN EFFICIENT NUMERICAL METHOD FOR LARGE-SCALE MODAL ANALYSIS USING BOUNDARY ELEMENT METHOD
- Wang Junpeng, Xiao Jinyou, Wen Lihua
- 2017, 49(5): 1070-1080. DOI: 10.6052/0459-1879-17-040
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- STUDY OF DAMAGE IDENFICATION METHOD BASED ON THE COVARIANCE OF STRAIN IMPULSE RESPONSE FUNCTION
- Li Xueyan, Zhang Huimin
- 2017, 49(5): 1081-1090. DOI: 10.6052/0459-1879-17-039
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- TRANSITION LAW OF ADJACENT FUNDAMENTAL MOTIONS IN VIBRO-IMPACT SYSTEM WITH PROGRESSION
- Lü Xiaohong, Luo Guanwei
- 2017, 49(5): 1091-1102. DOI: 10.6052/0459-1879-17-037
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- RIGID BODY SYSTEM DYNAMIC WITH THE ACCURATE JACOBIAN MATRIX OF SPRING-DAMPER-ACTUATOR
- Kan Ziyun, Peng Haijun, Chen Biaoshong
- 2017, 49(5): 1103-1114. DOI: 10.6052/0459-1879-17-030
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- CONTACT-IMPACT ANALYSIS IN MULTI-BODY SYSTEMS BASED ON NEWTON-EULER LCP APPROACH
- Fu Li, Hu Hongkui, Fu Teng
- 2017, 49(5): 1115-1125. DOI: 10.6052/0459-1879-17-023
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- COMPUTATION OF INVARIANT MANIFOLD BASED ON SYMPLECTIC ALGORITHM OF MIXED LIE OPERATOR
- Zheng Dandan, Luo Jianjun, Zhang Renyong, Liu Lei
- 2017, 49(5): 1126-1134. DOI: 10.6052/0459-1879-17-079
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- A DUAL EXPLICIT MODEL BASED DP-EM METHOD FOR SOLVING A CLASS OF SEPARABLE CONVEX PROGRAMMING
- Sui Yunkang, Peng Xirong
- 2017, 49(5): 1135-1144. DOI: 10.6052/0459-1879-17-176
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- EFFECTS OF DIFFERENT EQUATIONS OF STATE ON THE OBLIQUE SHOCK WAVE REFLECTION IN SOLIDS
- Huang Xiao, Yu Xin
- 2017, 49(5): 1145-1153. DOI: 10.6052/0459-1879-17-015
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- ONE-DIMENSIONAL SEEPAGE FORCE AND BUOYANCY
- Ding Zhouxiang
- 2017, 49(5): 1154-1162. DOI: 10.6052/0459-1879-17-001
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- THE SUPPORTED PROJECTS ON MECHANICS OF NSFC IN 2017
- Zhan Shige, Bai Kunchao, Zhang Panfeng, Wang Jianshan, Cao Dongxing
- 2017, 49(5): 1163-1184. DOI: 10.6052/0459-1879-17-293
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15 September 2017, Volume 49 Issue 5

The interface free energy density is an important quantity characterizing the mechanical property of interface in nanocomposite systems. In this paper, molecular dynamics simulation method is adopted to investigate the interface energy density of different FCC metallic bi-nano-scaled plates. The morphology of the interface crystal structure and the interface effect on the atomic potential are analyzed. It is found that interface atoms have periodically wrinkled rarefied or serried configurations, and the potential energy of interface atoms is also periodically distributed. The potential energy of atoms near the interface is obviously different from that of atoms inside the nano-plates. Both the Lagrange interface energy and the Eulerian one increase with the increase of the thickness of the bi-material, which approach the interface energy of a bulk bi-material finally.

In the high-speed train design, the nose shape is a crucial control factor influencing not only aerodynamic performance but also the aerodynamic noise. In the engineering practice, the nose shape becomes more and more slender along with the increasing of the design speed, e.g. the Japanese high-speed maglev train L0 series even has a 15 m long slender nose (the slenderness ratio reach to 8.8). This study aims to discuss the influence of the slenderness ratio of the nose shape on the aerodynamic noise. The hybrid numerical method of nonlinear acoustics solver (NLAS) and Ffowcs Williams-Hawkings (FW-H) acoustic analogy method is employed to study the aerodynamics noise characteristics. The numerical method is validated with a standard wind mirror test case and a set of acoustics wind tunnel experiments of the CRH380A train. The shape of the CRH380A train is chosen as a bench mark, and four different nose shapes of different slenderness ratio under different running speed situation are studied with numerical simulation. The flow field, aerodynamic drag, and the aerodynamic noise are obtained and discussed. The result shows that the total drag decrease with the increase of the slenderness ratio, and this effect enhances when the train speed increases. However, the influence of the slenderness ratio on the aerodynamic noise is much complex as no simple trend is observed. Considering both the aerodynamic and aeroacoustics characteristics, the train with the most slender nose shape is the best while this advantage is not notable compared with the second-best. Thus, simply increase the slenderness does not necessarily result in better aerodynamic noise performance if the effect of tunnel boom is not considered.

Instability study of electrified coaxial jet coupling the electric and inertial forces is performed based on the simplified experimental model of gas-driven coaxial electro-flow focusing. Under the assumption that the fluids are inviscid, incompressible and irrotational, a triple-layer electrified fluid jet model is established and an analytical dispersion relation in the temporal regime is obtained. The dispersion equation is solved by the normal mode method, the unstable modes of the flow are calculated and the effects of mainly controllable parameters on the unstable modes are analyzed. The results indicate that the axisymmetric mode dominates the complete flow as the maximum growth rate of the axisymmetric mode is the largest among all unstable modes. As the velocity of outer gas stream increases, the inertial force can definitely promote the jet instability. The jet will become more unstable as the velocity difference between the inner and outer liquid jets increases. The surface tension also promotes the jet instability. The axial electric field has two-fold influence on the axisymmetric jet instabilities. There is a critical value for the axial electric voltage which is related to the free electric charge density at the interface and the perturbation propagations on the jet surfaces. The applied axial electric field can suppress the jet instability when its intensity is smaller than the critical value; otherwise, the applied axial electric field can promote jet instability. These results are in good agreement with the existing experimental results and can provide guidance on the process control of experiments.

The super-harmonic and sub-harmonic simultaneous resonance of Duffing oscillator with fractional-order derivative is studied in this paper. The first-order approximate analytical solution is obtained by averaging method. The definitions of equivalent linear damping coefficient and equivalent linear stiffness efficient for super-harmonic and subharmonic simultaneous resonance are presented. The analytical amplitude-frequency equation for steady-state solution of simultaneous resonance is established. A comparison of the analytical solution with the numerical results is made, and their satisfactory agreement verifies the correctness and higher-order precision of the approximately analytical results. Then, a further comparison between the fractional-order and traditional integer-order Duffing oscillator is fulfilled through the definitions of equivalent linear damping coefficient and equivalent linear stiffness coefficient, and the results prove that the fractional-order parameters has the effects of both damping and stiffness, which is similar in other fractionalorder system. At last the numerical simulation is used to analyze the effects of different fractional-order parameters on multi-value characteristics and jumping phenomena of the amplitude-frequency curve under simultaneous resonance, and the differences between the super-harmonic and sub-harmonic resonances under single-frequency excitation are analyzed in detail. It could be found that the fractional-order parameters not only affect the response amplitude and resonant frequency of the system, but also has significant influence on the number, existing area and the occurrence order of periodic solutions. Moreover, single super-harmonic resonance, single sub-harmonic resonance and both existing of these two resonances could be respectively found under different basic parameters, which is important to study the dynamic characteristics of the similar system.

Non-Newtonian fluid has complex rheological characteristics.It is very helpful to reveal these characteristics for the applications of non-Newtonian fluid in industry and agriculture.The classical rheological models of nonNewtonian fluid usually have sophisticated forms and the limitations of specific materials or rheological situations.Fractional models have been successfully applied to describe the motion of non-Newtonian fluid due to their simplicity and few parameters.As an alternative method,the Hausdorff fractal derivative possesses simpler form and higher computational efficiency compared with the fractional derivative.This paper proposes a fractal dashpot model that improves the current Newton's Law by using the Hausdorff fractal derivative.By investigating the apparent viscosity,the creep and recovery characteristics of the fractal dashpot,it shows that the proposed fractal dashpot model is suitable to describe the non-Newtonian fluid with viscoelasticity (the so-called fractal fluid).Combined the fractal dashpot model with the continuity and motion equations,the basic equation for the fractal fluid for the laminar flow between two parallel plates is derived.Moreover,the velocity distributions between two plates are numerically calculated in three cases,which can be obtained through whether there is horizontal pressure gradient or the initial velocity of upper plate.It is found that the horizontal pressure gradient can change the shape of velocity over time and delay the arrival of stable velocity.The fractal fluid with different orders has the same velocity distribution and evolution when the horizontal pressure gradient doesn't exist.In addition,the velocity of upper plate doesn't influence the difference of stable velocity between different orders of fractal fluid when the horizontal pressure gradient exists.

High Reynolds number fluid flows have boundary layers at the wall. To automatically generate robust and valid boundary layer mesh for the simulations is still the bottleneck problem of computational fluid dynamics. Prisms/Tetrahedra hybrid mesh leads to significant savings in mesh size and solution costs. However, it's still difficult to generate prismatic elements of high quality within boundary layers of complex models. Previous advancing layers techniques sometimes lead to invalid meshes and poor quality elements at concave/convex ridges and sharp corners. To improve these situations, we present a strategy for automatically generating viscous boundary layer mesh. In this method, multiple growth directions at ridges and corners are well defined by the dihedral angles around the vertices and the growth heights are adjusted appropriately. Therefore, boundary layer mesh grows well at sharp corners, convex and concave ridges of the domain. We also decrease the number of global intersection checks by predefining the total growth heights before generating elements through one global check, which improves the efficiency of mesh generation. At the same time, we develop a 3D strategy of mesh size control to get a size uniform triangular mesh of the outer boundary of the boundary layer mesh, which is beneficial to generate far-field isotropic mesh of high quality. Finally, mesh examples and the viscous flow simulations including 2D and 3D are presented. In 3D, the hybrid mesh over the standard ONERA M6 and DLR-F6 configurations are generated with the present method. The numerical results agree very well with experiment data which indicates that the hybrid viscous meshes generated by the proposed method are effective and efficient.

In this study, a 2D fully-nonlinear numerical wave tank is developed based on the time-domain higher-order boundary element method. Non-periodic waves acting on a vertical cliff are investigated. The fully nonlinear kinematic and dynamic boundary conditions are satisfied on the instantaneous free surface. The mixed Eulerian-Lagrangian method is adopted to track the transient water particle on the free surface and the fourth order Runge-Kutta method is used to predict the velocity potential and wave elevation on the free surface. Then the acceleration potential technique is adopted to calculate the temporal derivative of the potential on the vertical wall surface, and transient wave loads are obtained by integrating the Bernoulli equation along the wetted wall surface. The obtained nonlinear results are firstly compared with solutions of the Serre-Green-Naghdi (SGN) theory. It is observed that, for the highly nonlinear case of double-incidentwaves, the SGN model which only satisfies the weak dispersion relationship greatly underestimates the maximum wave run-up (MWR). Then, the nonlinear interaction between double-incident-waves and a vertical cliff is further studied. It is found that the linear prediction also underestimates the MWR. The nonlinearity not only leads to an evident increase of the MWR, but also results in a high-frequency oscillation of the free surface. During this process, nonlinear properties of wave loads are similar to those of the wave run-up. Finally, spectral analysis is performed on histories of wave run-up and wave loads. The dominant frequency wave component is found to transfer its energy to higher frequency components, as a typical nonlinear wave-wave interaction phenomenon.

In this work, the confinement effect on the buoyancy-driven, axisymmetric motion of a skirted bubble in a liquid-filled, circular cylinder is numerically studied. The gas and liquid phases are assumed to be isothermal, incompressible and immiscible. The volume of fluid (VOF) method is adopted to simulate the deforming interface between gas and liquid. A confinement ratio range of (1:1 ≤ *Cr* ≤ 10) is considered. The results reveal that the motion of a skirted bubble under *Cr* ≥ 8 resembles that in an infinite medium in terms of both shape and *Re*ynolds number in terminal state. With decreasing *Cr*, the wall plays a more significant role in determining the motion of the skirted bubble. For the range of *Cr* < 8, the drag on the bubble increases as *Cr* decreases, giving rise to the reduction of bubble rising velocity. As for the terminal shape, the skirted bubble is elongated in the axial direction and may evolve to an ellipsoidal cap or a bullet as a result of increasing wall proximity. The sensitivities of the thickness and length of trailing bubble skirts to the confinement ratio are examined. The skirt length reduces with the decrease of *Cr*, while the skirt thickness increases with decreasing *Cr*. The details of fluid field are analyzed both in the global reference frame and in a local reference frame moving with the bubble centroid. The wake effect of the skirted bubble is weaken by the increasing wall effect, suppressing the formations of vortex ring and skirt. Bubble break-up is captured under approximate conditions and can be enhanced by decreasing *Cr*, confirming the deduction in the literature. The present predictions on terminal velocities agree well with results by the correlation in the literature.

From the micro-molecular chain structures, based on the Gauss statistical model in hyperelasticity theory of the rubber matrix and the viscous rheological theory of viscoelastic materials, the effects of the micro molecular structures and fillers on the viscoelastic properties of the viscoelastic materials are studied. The temperature-frequency equivalent theory is introduced to investigate the effect of temperature on the mechanical properties of the viscoelastic materials, and an equivalent fractional order micro-structure standard linear solid model of the viscoelastic materials is established. The mechanical properties and energy dissipation capacity of the viscoelastic materials are tested by dynamic thermomechanical analyzer (DMA) device. The experimental results show that the storage modulus is large in low temperature region, and decreases significantly with increasing temperature; the loss factor is small in high and low temperature regions, but has peak values near the glass transition temperature. Then the validity of the model is verified based on the test results, and the equivalent fractional order micro-structure standard linear solid model can effectively describe the energy dissipation capacity of the viscoelastic materials. What's more, the validity of the model is further verified by the combination of 9050A and ZN22 viscoelastic materials. The results show that the viscoelastic materials have good energy dissipation capacity. The proposed equivalent fractional order micro-structure standard linear solid model can accurately describe the influence of microstructures and fillers on macro-properties of viscoelastic materials. And the dynamic mechanical properties of the viscoelastic materials at different temperatures and frequencies also can be accurately described.

Thanks to the great advances in fast boundary element method (BEM) achieved in the last two decades, the BEM has been increasingly used in the dynamic design of engineering structures, the analysis of noise and vibration. Consequently, solving large-scale eigenvalue problems and performing modal analysis for complicated structures and acoustic fields using the BEM becomes an very important but challenging task; so far there are no effective numerical methods for this purpose. This paper aims to extend the application of the newly-developed resolvent sampling based Rayleigh-Ritz projection method (RSRR) to the solution of the general nonlinear eigenvalue problems (NEP) in BEM. First, in order to generate reliable eigenvector search spaces, a series of BEM linear systerms in frequency domain are solved. Then the original NEP can be transformed to a reduced NEP based the classical Rayleigh-Ritz procedure, and the reduced NEP could be solved by those exiting NEP solvers easily. Second, to reduce the prohibitive computational burden involved in solving the projected NEP by the Rayleigh-Ritz procedure, a BEM matrix interpolation technique and a fast computation method for reduced NEP systerm matrix are proposed based on the discretized Cauchy integral formula of analytic functions. Then a simple rule for estimating the number of terms in the interpolation is proposed as well. Finally, the RSRR method is used to solve large-scale practical acoustic modal analysis problems using fast BEM with complicated sound absorbing boundary conditions. Numerical results indicate that the method can robustly dig out all the interested eigenvalues and the corresponding eigenvectors with good accuracy and high computational efficiency.

Structural damage identification based on vibration characteristics is the research topic in civil engineering in recent years. When the structure is damaged, the stress of the surrounding damage part of the structure will be redistributed obviously and the strain will have distinct change. So the damage detection can be performed by the comparison of the strain or the parameter from the strain responses between the damaged and intact states of the structure. In this paper the covariance of strain impulse response function (CoS) is proposed and it is proved that CoS is the function of structural modal parameters. It is the energy integral of the strain impulse response on the time interval. Compared to the traditional modal parameters, more high modes of modal parameters are preserved in the CoS and the errors produced in the modal parameter identification procedure are avoided. So the CoS can be used for structural damage identification. A simplysupported steel beam is studied to demonstrate the performance of CoS in the damage identification. From the results of numerical studies, it can be found that CoS can identify damage occurrence and location successfully. Moreover, any analytical structural model is not necessary for the damage identification procedure based on CoS. Only the computation or measurement of the strain response, strain impulse response function and CoS from the intact and the damaged states of the structure is required. It means that CoS is very suitable for health monitoring of real engineering structures.

The vibro-impact phenomena which widely exists in power mechanical system will make the system exhibit complex dynamic response. So far the research on stability and bifurcation of *p*/1 fundamental motions of vibro-impact system is still rare, and most studies of vibro-impact dynamics are based on single-parameter bifurcation analysis. In this paper, taking a small vibro-impact driver as engineering background, a mechanical model of a vibro-impact system with progressive motions is established. Types of impact between the vibration exciter and the cushion, and conditions of progressive motions of the slider are analyzed. Judgment conditions and motion equations of four probable motion states presented by the system are put forward. Based on bifurcation analysis of two-dimensional parameters, existence regions and distribution laws of different types of periodic motions of the system are obtained in the (*ω, l*) parameter plane. Transition laws of adjacent *p*/1 fundamental motions are analyzed in detail. In the right region of the existence region of 5/1 fundamental motion, there exists a singular point *X*_{p} on the boundary between adjacent regions of *p*/1 fundamental motions, which is the critical point of bifurcation characteristics of adjacent *p*/1 fundamental motions. In the region with *l* less than *l*_{Xp}, adjacent *p*/1 fundamental motions are transited mutually by real-grazing bifurcation and saddle-node bifurcation. Two periodic attractors can coexist in the hysteresis region, which exists between real-grazing bifurcation boundary and saddle-node bifurcation boundary. In the region with *l* more than *l*_{Xp}, there exists a transition region between adjacent regions of *p*/1 fundamental motions. The system exhibits (2*p* + 2)/2 and (2*p* + 1)/2 motions in the transition region. In the left region of the existence region of 5/1 fundamental motion, *p*/1 fundamental motion transits to (*p* + 1)/1 fundamental motion via multi-sliding bifurcation.

The spring-damper-actuator (SDA) is a common force element in multibody system and widely used in the field of engineering. The governing equations of multibody dynamic system established by absolute coordinate methods are differential-algebraic equations which are usually nonlinear and complex. To ensure the stability and accuracy of the numerical solutions, the implicit algorithms are commonly used to solve the dynamic equations. While the calculations of Jacobian matrices are the crucial process in implicit algorithms. For a multibody system containing the SDA, the additional Jacobian matrices induced by the SDA are highly nonlinear functions of the generalized coordinates and generalized velocities. A lot of current research works focus on the calculation of generalized force vector, however the calculations of additional Jacobian matrices are less concerned. This paper focuses on dynamic analysis of multi-rigid-body systems containing the SDA. Firstly, the construction of the accurate Jacobian matrices in solving the dynamic equations is investigated based on the Newmark algorithm. Then, the additional Jacobian matrices relating to the generalized force vector of the SDA are analytically derived. These matrices consist of the partial derivative of generalized force vector with respect to the generalized coordinates and the generalized velocities. Finally, the influence of additional Jacobian matrices on the convergence of dynamic analysis is investigated via two numerical examples. The numerical results indicate that when the values of stiffness, damping and active force are large, the additional Jacobian matrices induced by the SDA have a significant influence on the convergence of dynamic analysis. When the additional Jacobian matrices induced by the SDA are taken into account, the dynamic analysis can achieve convergence with less iteration steps and the computational time thus can be reduced.

The contact-impact analysis in multibody systems based on the nonsmooth dynamics approach is a hot topic in the research of multibody system dynamics. Newton-Euler approach is adopted to develop dynamics model of contactimpact analysis in non-smooth multi-body systems, and a new LCP formula is presented in this work. Different from Lagrange methods, Newton-Euler modeling method incorporate equality constraints into dynamic models with noninterpenetration constraints and frictional constraints together. In Newton-Euler modeling method, the basic system is derived by removing the non-interpenetration constraints and frictional constraints from the original multi-body system. Newton-Euler eqution of basic system is established by using the maximum coordinates method. Because the coordinates of the basic system are not independent of each other, equality constraints are involved in modeling, the basic system dynamic equations is a set of DAE (differential algebra equation). With the aid of constraint Jacobian matrix, Lagrangian multipliers corresponding to the non-interpenetration constraint forces and Coulomb friction forces are added to the basic system DAE to obtain the dynamic equations of global motion of the multi-body system with characteristics of variable topological structure. The complete dynamic model is composed of basic system DAE, equality and inequality constraints. In order to simplify the derivation process of LCP, a decomposed matrix form is built. The LCP -based Time-stepping method is adopted for numerical simulation. Time-stepping algorithm is a popular non-smooth numerical algorithm, Its prominent feature is that it can avoid the tedious event-detection process in numerical integration. In the process of numerical integration, the contact-detachment state of the system can be determined by solving the LCP. Our method is carried out in slider-crank mechanism with a translational clearance joint, the simulation results indicate that this method is effective.

Invariant manifolds of periodic orbit near the libration points attract a lot of attentions due to their importance in the low-energy orbits transfer problem. In the process of low-energy orbit design, the energy of the invariant manifolds must be matched, but the energy is dissipated when integrating with traditional numerical integration method. The explicit symplectic algorithm with energy conservation is more efficient than the implicit symplectic algorithm, but it requires the Hamiltonian system to be divided into two integral parts, while the circular restricted three-body problem in the rotating coordinate system being inseparable. It is difficult to solve the circular restricted three-body problem in the rotating coordinate system by explicit symplectic algorithm. In this paper, the mixed Lie derivative operator of kinetic energy is used to solve the circular restricted three-body problem in the rotating coordinate system, and the effectiveness of this explicit symplectic algorithm with the third derivation in dealing with this problem has been showed. Compared with the Runge-Kutta45 method and Runge-Kutta78 method, the symplectic algorithm with the third-order derivative term not only has high precision but also the smallest energy error and the highest efficiency. Finally, the invariant manifolds are calculated by the symplectic algorithm with the third derivative term, the patched point can match accurately with this method.

An explicit exact formula is derived for the objective function of the dual model of a class of separable convex programming problems. It makes more mature and efficient methods can be chose to solve the dual model. Therefore, the advantage of applying the duality theory of nonlinear programming to efficiently solve structural topology optimization problems is fully exploited. The research work is rooted in that the gap of a nonlinear convex programming with its dual programming is zero. Solving original programming can be equivalently transformed into solving its dual programming. The scale of the solved programming can usually be reduced greatly. But an explicit relationship is not existed between the original programming and dual programming has affected the application of the dual solution algorithm. Fortunately, the programming models of a large class of structural optimization problems, including the continuum topology optimization, are convex and separable. And an explicit relationship between the original variables and their dual variables is existed; therefore, the dual solution algorithm has become one of the effective methods for 38 years. However, the objective function of the dual problem is not explicit for a long time. It is because the dual problem is a parametric minimization problem which leads to the objective function is expressed as an implicit expression. The common explicit expression for the dual objective function is a two-order approximation. The regular thinking tendency that the dual problem is too difficult to be expressed explicitly and can only be expressed approximately is breakthrough. A dual programming explicit model (DP-EM) method is put forward for the topology optimization of continuum structures. Comparison of computational efficiency among the DP-EM method, the dual sequential quadratic program (DSQP) method and the method of moving asymptotes (MMA) is presented. The results showed that:(1) more external iterations are needed for the MMA algorithm than the DP-EM algorithm and DSQP algorithm; (2) same external iterations are needed for the DP-EM algorithm and DSQP algorithm, but internal iterations is less for the DP-EM method. It shows the advantage of the DP-EM algorithm due to its explicit dual function.

The equations of state of solids under high pressure are more complicated than that of gases in a variety of forms. While the existing investigations on the oblique shock wave reflection usually take one of the equations of state, lacking of the comparisons among them. Therefore, this paper aims at the oblique shock wave reflection in solids through shock polar methodology under four different forms of equations of state (principal shock taking with linear shock-particle velocity relationship and second shock taking with Grüneisen equation of state, principal and second shock both taking with shock-particle velocity relationship, principal shock taking with linear shock-particle velocity relationship and second shock taking with stiffened gas equation of state, and principal and second shock both taking with stiffened gas equation of state). The effects of different equations of state on the pressure behind the reflected shock wave are discussed. By conducting the dimensional analysis, we provide an applicable condition for employing a simplified equation of state to achieve high accuracy. Moreover, numerical simulations performed by ANSYS/LS-DYNA software are conducted to validate the results through shock polar methodology. The results of this paper could be helpful for the decision of the equation of state on the oblique shock wave reflection in solids.

For the sake of simplicity, two methods of deriving seepage force and buoyancy associated with one-dimensional fluid flow in saturated soil were developed to give a uselful supplement to conventional macro-scale method of isolated pore fluid. To study the seepage force and buoyancy, three fundamental governing equations, namely, equilibrium differential equation in elasticity, Terzaghi's effective stress equation and simplified Bernoulli's equation in fluid mechanics, are utilized for the derivation of seepage force and buoyancy. Based on the fundamental governing equations, it is readily to derive the differential equations of equilibrium corresponding to the isolated soil skeleton and the isolated pore fluid, and hence, the connotation and nature of seepage force and buoyancy can be disclosed. The seepage force per unit volume of saturated soil arises from the gradient of fluid pressure in terms of total head. The buoyancy originates in the vertical component of gradient of fluid pressure in terms of position head. Both the seepage force and the buoyancy can be embodied in equilibirium differential equation with respect to skeleton or pore fluid of satured soil. In geotechnical practice, the three governing equations can be directly applied to the calculation of effective stress. Only in some simplified cases, the distribution of effective stress can be obtained using the seepage force and buoyancy. Additionally, some hot topics are also discussed in this study, and emphasis shall be put on the discussion on the rationality and the potential application risk of the new definition of seepage force, i.e., *j*=*nγ*_{w}i. The strictness of one classic formulation of seepage force in history was thoroughly validated when considering the derivative of seepage velocity with respect to time. It is noteworthy that in the study of seepage force and buoyance from the perspective of soil mechanics, the Terzaghi's effective stress equation shall be reasonably implemented with practical significance.

The paper briefly introduced the supported NSFC projects for General Programs, Young Scientists Fund, Fund for Less Developed Regions on mechanics in 2017. The projects list is also given.