力学学报

ADVANCES IN DYNAMIC MODELING AND OPTIMIZATION OF FLEXIBLE MULTIBODY SYSTEMS

Sun Jialiang, Tian Qiang, Hu Haiyan

Flexible multibody system is a kind of mechanical system composed of many flexible components and kinematic pairs, such as flexible robot arms, helicopter rotors, deployable antennas of a satellite, and solar sail spacecraft. Flexible multibody systems serve as useful models in aerospace engineering, vehicle engineering, mechanical engineering, weapon engineering and so on. Recently, with the development of the engineering technology, new challenges have arisen to establish an accurate dynamic model of a flexible multibody system, as well as for the dynamic optimization design of such a flexible multibody system, especially of a flexible multibody system with variable-length components. As a matter of fact, when the component gets more and more flexible, the interactions between the component and the flexible multibody system cannot be disregarded when performing optimization design. The component-based structural optimization, hence, should be extended to the flexible multibody system-based structural optimization. In this review, the research background and significance of the dynamic optimization of flexible multibody systems are firstly surveyed. Three methods for investigating flexible multibody dynamics including flexible multibody systems with variable-length components are briefly outlined, i.e., floating frame of reference formulation (FFRF), geometrically exact formulation (GEF), and absolute nodal coordinate formulation (ANCF). Afterwards, the recent advances are systematically reviewed in the dynamic response optimization, the dynamic characteristics optimization, and the dynamic sensitivity analysis of flexible multibody systems, as well as the structural optimization, i.e., size optimization, shape optimization, and topology optimization of the flexible components in a flexible multibody system. Finally, several open problems are addressed for future studies.

2019, 51(6): 1565-1586. doi: 10.6052/0459-1879-19-212

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2019, 51(6): 1587-1588. doi: 10.6052/0459-1879-19-355

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THEORY, MODELLING AND COMPUTATION OF NONLINEAR OCEAN INTERNAL WAVES

Wang Zhan, Zhu Yuke

Salinity and temperature variations in the vertical direction lead to density stratification in oceans, and the fluctuation of isopycnal surfaces resulting from internal perturbations (such as stratified shear flow over a bottom topography) or external disturbances (such as the dead water phenomenon) is called the internal wave. Internal waves are ubiquitous in the ocean and usually arise in the situation when the density stratification is obvious and stable such as at the mouth of strait. Oceans are usually characterized by a sandwich-like structure: a mixing layer and a deep-water layer featuring an almost uniform density, and a transition layer in the middle with continuous density variation. Fluctuations of the transition layer have great impact on ocean engineering and ocean ecology, while waves inside the transition layer has potential applications in the non-acoustic detection of submarines (conversely, in the stealth operation of submarines). The main reason for these important influences lies in the ability of internal waves to propagate in both horizontal and vertical directions, which is the essential difference from that of ocean surface waves. In the current paper, two types of ocean density models, continuously stratified models and discontinuous layered models, are thoroughly discussed. Various nonlinear models used to study ocean internal waves (including celebrated weakly nonlinear models, such as the Korteweg-de Vries equation, the Benjamin-Ono equation, and the Kadomtsev-Petviashvili equation, and strongly nonlinear models, such as the Miyata-Choi-Camassa equation, the fully nonlinear potential theory, and the incompressible Navier-Stokes equation with density variations), as well as their respective scope of application, are reviewed from the aspects of theoretical analyses, numerical simulations, and laboratory experiments. Particular attention is paid to the important role of internal waves in transferring mass, momentum and energy in oceans.

2019, 51(6): 1589-1604. doi: 10.6052/0459-1879-19-326

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AN EXPERIMENTAL STUDY OF 3-D WAVE SURFACE AND HYDRODYNAMIC LOADS FOR INTERACTION BETWEEN SOLITARY WAVE AND SUBMERGED HORIZONTAL PLATE

Wang Qian, Liu Hua, Fang Yongliu, Shao Qi

An experiment of interaction between a solitary wave and a submerged plate with finite length and finite width is conducted in a wave basin. A new system of the multi-lens stereo reconstruction is proposed to measure the local deformation of the free surface in the horizontal area of 1.7 m$\times $1.6 m in the experiment. A set of underwater force measuring system consisted of four force balances is designed to obtain the wave loads on the submerged plate, without the interference on the surface elevation measurement. A solitary wave is generated in a wave basin of uniform water depth. The wave amplitude is 0.16 m and the still water depth is 0.4 m. The submerged depth of the horizontal plate is 0.1 m. No wave breaking occurs during the wave propagation. The three-dimensional deformations of the free surface elevation lead to the spatial and temporal variation of the solitary wave amplitude. It is found that, for the case of the present wave condition and the submerged plate, the surface elevation reaches the highest at the centerline of the plate near its trailing edge, and decreases along the span direction. The time series of the surface elevation measured by the multi-lens stereo reconstruction method agree well with the wave elevations measured by the wave gauges, which validates the new wave surface measurement system. The loading processes of the horizontal force, vertical force and the pitch moment are proposed as 6 typical phases in which the characteristics of the measured wave surface elevation are discussed. The wave surfaces measured by the multi-lens stereo reconstruction method are given at the corresponding time as the peaks of the vertical force and the pitch moment occurs. The multi-lens stereo reconstruction method could be used to measure the wave field for physical model experiments in a wave basin as a new tool.

2019, 51(6): 1605-1613. doi: 10.6052/0459-1879-19-244

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HYDROELASTIC RESPONSE OF A SEMI-INFINITE PLATE DUE TO OBLIQUELY INCIDENT WAVES IN A THREE-LAYER FLUID

Pu Jun, Lu Dongqiang

Wave scattering and hydroelastic response of semi-infinite elastic plates due to obliquely incident waves in a three-layer fluid are studied analytically. The densities of the three layers of fluid are different with sharp interfaces, but are constant at each layer. The fluid is assumed to be inviscid and incompressible and the motion to be irrotational. Within the frame of the linear potential flow, semi-analytical solutions for wave-plate interaction are derived with the aid of the methods of matched eigenfunction expansions and the inner product of eigenfunction. The critical angles for the incident waves of the surface mode and the interfacial wave mode are deduced in terms of the dispersion relation. As the physical parameters change, the critical angle varies accordingly. The critical angle are closely related with the existence of the surface or interfacial wave modes that propagate from the open water region to the plate-covered one, by which two problems can be answered: (1) Is there a transmitted wave on the incident interface in the plate-covered region; (2) Is there, on the interfaces above the incident surface, a transmitted wave on the plate-covered region and a reflected wave in the open water region. When the low interfacial wave is incident and the incident angle is large enough, the low interfacial wave mode in the open-water region becomes the solo wave mode in the entire fluid domain.

2019, 51(6): 1614-1629. doi: 10.6052/0459-1879-19-081

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HYSTERESIS IN VORTEX-INDUCED VIBRATIONS OF A NEAR-WALL CYLINDER

Liu Jun, Gao Fuping

The vortex-induced vibration (VIV) of a cylinder is a typical fluid-solid coupling problem. Previous investigations on VIV responses were mainly made under increasing-velocity flow and wall-free conditions. Nevertheless, the natural flow always features with alternately increasing or decreasing velocities, so that the VIV response of a near-wall cylinder holds different characteristics from that of a wall-free cylinder. In this study, a VIV device for a cylinder with low structural damping was designed and constructed in conjunction with a flume. Based on dimensional analyses, a series of flume model tests were carried out to investigate the critical velocities for the initiation and the cease of VIV (i.e., the upper critical and lower critical reduced velocities) of a near-wall cylinder under the action of increasing-velocity and decreasing-velocity flows, respectively. To examine wall-proximity effects on the VIV hysteresis, synchronous measurements were made for the time-variation of vibration displacement and the corresponding flow fields around the cylinder. Meanwhile, a specially designed PIV system with bottom-up laser scanning was employed to capture the flow field characteristics. Experimental observations indicate that the critical velocity for the initiation of VIV of a near-wall cylinder decreases with the decrease of gap-to-diameter ratio. The lower-critical reduced velocity for the cease of VIV under decreasing-velocity conditions is however much smaller than the upper-critical value for the initiation of VIV under increasing-velocity conditions. The deviation of the upper-critical reduced velocity from the lower-critical one is used for quantitative characterization of the hysteresis in VIVs, which increases approximately linearly with the decrease of gap-to-diameter ratio. Moreover, it was found that such VIV hysteresis is always accompanied with the jump of vibration amplitude, whose value decreases nonlinearly with the decrease of gap-to-diameter ratio.

2019, 51(6): 1630-1640. doi: 10.6052/0459-1879-19-293

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RESEARCH PROGRESS IN HYDRODYNAMICS OF WIND-WAVE COMBINED POWER GENERATION SYSTEM

Zhou Binzhen, Hu Jianjian, Xie Bin, Ding Boyin, Xia Yingkai, Zheng Xiaobo, Lin Zhiliang, Li Ye

With the increasing environmental problems such as depletion of fossil energy and global warming, marine renewable energy (offshore wind energy, tidal energy and wave energy) has become a research hotspot. In order to effectively develop marine renewable energy and reduce costs, comprehensive development of multiple energy sources has become a trend at this stage. The combination of offshore wind energy and wave energy has broad application prospects, and the combined power generation system continues to innovate. Hydrodynamic performance is an important basis for the interaction of combined system with waves. This paper introduces briefly a variety of hydrodynamic numerical simulation methods for combined power generation systems, including linear frequency domain, linear time domain, potential flow nonlinear method, and viscous method based on Navier-Stokes equation. The numerical simulation method is reviewed, and its advantages and disadvantages are analyzed from the aspects of computational efficiency and precision. The technical principle and main research difficulties of hydrodynamic control optimization and experiments are further elaborated, which provides a basis for the hydrodynamic design of the combined power generation system. The main conclusions are as follows. Firstly, from the perspective of computational efficiency, the linear frequency domain method is optimal, followed by linear time domain, potential flow nonlinearity, and viscous method. From the perspective of computational accuracy, it is the opposite of the former. Secondly, considering the computational efficiency and precision, it is a feasible solution to study the potential flow theory considering viscosity correction. Thirdly, at present, model experiment method and optimal control technology are not mature and still in the exploratory stage.

2019, 51(6): 1641-1649. doi: 10.6052/0459-1879-19-202

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MOTION SIMULATION OF FLOATING STRUCTURE WITH MULTIPLE SLOSHING TANKS BASED ON TIME-DOMAIN DECOUPLING ALGORITHM

Zhang Chongwei, Ning Dezhi

For a floating structure with multiple sloshing tanks, the structure motion, external hydrodynamics and sloshing dynamics of liquid tanks are mutually determined with complex coupling mechanism. This study introduces an effective time-domain decoupling algorithm for an accurate motion simulation of floating structure with multiple sloshing tanks. The algorithm is derived based on the modal decomposition approach. By decomposing the external hydrodynamic force and nonlinear sloshing forces in each liquid tank of the floating structure, this study gives a time-domain decoupling motion equation. With this algorithm, the instantaneous acceleration of a floating structure at any instant is calculated explicitly without iterations. Limitations of the conventional iterative method in terms of the iteration number, truncation errors and numerical convergences can be avoided. The CPU time consumption on dealing with the coupling effects can be greatly reduced. Combined with the boundary element method, the algorithm is applied to time-domain simulations of a floating structure with either a single liquid tank or multiple tanks. For single-tank cases, the time-domain decoupling algorithm is validated by comparing with the experimental measurements. This study first analyzes effects of the sloshing dynamics on a single-tank floating structure. A specific frequency range is found, outside which the floating structure shows a steady motion in the time domain. For lower wave frequency cases around this range, the sloshing force and external wave force can be in anti-phase or even cancelled, so that the motion of the structure is weakened. For higher wave frequency cases, the sloshing force can be in the same phase with the external wave force, and the liquid sloshing can eventually amplify the structure motion. Further, a floating structure with four liquid tanks is further investigated. It shows that the nonlinear sloshing forces can affect the surge and pitch motion of the structure, but with little effect on the heave motion.

2019, 51(6): 1650-1665. doi: 10.6052/0459-1879-19-210

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THE MECHANISM OF JETTING BEHAVIORS OF AN OSCILLATING BUBBLE

Li Shuai, Zhang Aman, Han Rui

The dynamic behaviors of an oscillating bubble (e.g., underwater explosion bubble, cavitation bubble and air-gun bubble) are well known to be strongly dependent on the nature of boundary conditions. Many experiments demonstrated that a high-speed liquid jet is formed away from a free surface or towards a nearby rigid wall. The violent jet impact is believed to be one of the most important mechanisms of cavitation erosion and damages by an underwater explosion. In the previously published literature, the Kelvin impulse based on spherical bubble theory is adopted to determine the gross migration and jet direction of bubbles. However, the underlying mechanisms of jet inception and development are not fully understood and the characteristics of the jet impact still lack exploration. In the present work, both experimental and numerical methods are adopted to do some fundamental studies on bubble dynamics beneath a free surface and near a rigid wall. The electric discharge method is used to generate a bubble and the bubble motion is captured by a high-speed camera. Meanwhile, the boundary integral method is adopted to conduct numerical simulation. The presence of a nearby boundary alters the pressure gradient surrounding the bubble, which has a significant influence on the jet inception. Additionally, a local high-pressure region is generated near the bubble bottom, and it results in a positive feedback mechanism that further accelerates the jet. This mechanism reveals the fact that the jet can speed up to a hundred meters per second within a relatively short time. A localized high-pressure region is caused by the jet impact around the jet tip and the maximum pressure decreases gradually as the rebound of the toroidal bubble. At last, the effect of the dimensionless standoff parameter (defined as $\gamma = d / R_{m}$, where $d$ is the distance between the initial bubble center and the rigid wall and $R_{m}$ is the maximum bubble radius) on the jet impact pressure is discussed.

2019, 51(6): 1666-1681. doi: 10.6052/0459-1879-19-219

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RESERCH PROGRESS ON INTERACTION BETWEEN A SINGLE CAVITATION AND FREE SURFACE

Guo Wenlu, Li Hongchen, Wang Jingzhu, Wang Yiwei, Huang Chenguang

The phenomena of cavitation have been observed in various natural and industrial processes like biomedical engineering and marine engineering. The interaction between a single bubble and free surface, involving the non-spherical collapse of cavitation bubble, nonlinear deformation and instability of free surface, etc., is frontier scientific issues in fluid mechanics and bubble dynamics. In this paper, we summarize research progress and achievements of this field in recent years and discuss typical phenomenon from the perspectives of cavitation dynamics and splash behaviors. For the non-spherical evolution of cavitation bubble, we focus on the key processes such as volume oscillation, jet generation, water hammer effect, collapsing shock wave, and centroid migration based on the dimensionless parameter of kelvin impulse. The mechanism of energy distribution during the cavitation collapse is obtained. For the splash dynamics of free surface, four typical phenomena are summarized based on the generation and development of the thin jet and thick jet: formation of transparent water layer, generation of water column, stable and unstable structure of water crown. Furthermore, we review physical mechanisms on the collapse of cavitation, splash and instability of the free surface by using the theories of Kelvin impulse, Singularity at the curved surface, and Rayleigh-Taylor instability. In addition, we present the effect of curved surfaces on the splash dynamics by investigating the behaviors of free surface of spherical and cylindrical shapes. Finally, remaining challenges and development tendency for future research are given.

2019, 51(6): 1682-1698. doi: 10.6052/0459-1879-19-328

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DYNAMIC DISTRIBUTION OF POWER-LAW DENSITY CURRENT IN ESTUARY UNDER WAVE

Xu Haijue, Wu Jinsen, Bai Yuchuan

The kinetic characteristics of floating mud in the estuarine bottom play an important role on the deposition process in harbors or channels and the maintenance of estuaries. And the study on them has become one of the focuses of coastal researchers. In this paper, the significance of the study on sediment density flow in estuaries is summarized firstly, the differences and the conditions of the functions applied in the rheological relation of the mud by many investigators are analyzed. Then, according to the actual needs of the research problem in this paper, a two-layer fluid model with the upper-layer fluid setting as a Newtonian one and the lower-layer fluid being simplified as Non-Newtonian, was established under the wave motion. Additionally, as one of the important functions, the power law is adopted as the constitutive relation of the fluid. The flow field of the density current is then studied, including the velocities, pressures of the two layers fluids, the wave amplitude ratio of the interface to the free surface and so on. Additionally, the influences of the mud density, the circular frequency and the power-law exponent on the flow field are discussed in details. It is found that the velocities of the both layers are consistent with each other at the interface, while the pressure has a sudden change due to the existence of the velocity gradient along z-axis. In the lower layer, there is an extreme point in the curve of horizontal velocity amplitude. The differences of the pressures between the two layers become greater, with the increase of the circular frequency or the decrease of the mud density as well as the power-law exponent. Finally, the comparisons between the calculated results and the experimental data prove the validity of the model.

2019, 51(6): 1699-1711. doi: 10.6052/0459-1879-19-073

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A HYBRID TURBULENCE MODEL FOR WAVE BREAKING SIMULATION

Zhan Jiemin, Li Yihua

On the one hand, the deformation and breaking of near-shore waves affect the transportation of water and sediment, and on the other hand, it is of guiding significance for wave elimination and revetment. In this paper, a 3-dimensional hybrid turbulence simulation model is proposed. Laminar flow mode is adopted in the wave-making region, and velocity wave-making is carried out at the boundary through the User Defined Function(UDF) developed based on fluent. This method can control the volume fraction of water by precise interpolation according to the known wave height under the condition of given inlet velocity. In the wave propagation region, large eddy simulation (LES) is used for simulation research, and in the wave elimination region, RANS model with the porous media wave absorber is used for wave elimination. The model uses the VOF method to capture the free surface changes in the process of wave breaking. This paper carried out simulation studies on the regular wave (M1) with a wave height of 5.5cm, the regular wave (M3) with a wave height of 13.5cm, the unidirectional irregular wave (U1) with a TMA spectrum with an effective wave height of 7.75cm and the multidirectional irregular wave (B5) with an effective wave height of 19cm. Simulation results show that the proposed model can accurately simulate the wave propagation in the process of refraction and diffraction phenomenon, and be able to catch the waves of the free surface in the process of change, broken for 3D wave propagation and broken numerical simulation provides a method of simulation.

2019, 51(6): 1712-1719. doi: 10.6052/0459-1879-19-321

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HIGH-PERFORMANCE DILATED POLYHEDRAL BASED DEM FOR ICE LOADS ON SHIP AND OFFSHORE PLATFORM STRUCTURES

Liu Lu, Yin Zhenyu, Ji Shunying

The ice loads on ship and offshore platform structures is the key factor in structure design for cold regions. The discrete element method (DEM) is an important approach to determine the ice load on structures. According to the Minkowski sum theory, the dilated polyhedra based DEM is employed to simulate the interaction between sea ice and ship and offshore platform structures in this paper. In the dilated polyhedra based DEM, the enveloped function of the dilated polyhedron is generated to establish the fast contact detection algorithm based on the optimization model. Meanwhile, the bond-break model between elements is established by considering the stiffness softening process between bonded elements. Accordingly, the high-performance algorithm based on CPU-GPU cooperative-heterogeneous environment is developed. The ISO standard is employed to validate the ice load determined by the dilated polyhedra based DEM for better engineering applications of the interaction between sea ice and marine structures. The ice load on ship hull is calculated by the proposed method while the line load distribution on ship hull is studied. The ice resistance of ship hull is compared with the result by Lindqvist empirical formula to validate the accuracy of DEM simulations. The interaction between level ice and multi-leg platform is simulated while the ice load on each leg is analyzed. For the ice management in broken ice regions, the ice load on ship and offshore structures is simulated when the ship navigates around the offshore platform in circle. The proposed method can be effectively applied in the analysis of ice load on marine structures, and can provide a scientific approach for the design and safety operation of ship and offshore structures.

2019, 51(6): 1720-1739. doi: 10.6052/0459-1879-19-250

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NUMERICAL INVESTIGATION OF MASS RATIO EFFECT ON FLOW-INDUCED VIBRATION OF TWO TANDEM SQUARE CYLINDERS AT LOW REYNOLDS NUMBER

Du Xiaoqing, Qiu Tao, Zhao Yan

To clarify the mass ratio effect on the flow-induced vibration of two tandem square cylinders, numerical simulation is employed to investigate the effect of mass ratio ($m^{\ast }=3$, 10, 20) on the vibration response characteristic of the downstream square cylinder at $Re=150$. The evolution of the wake modes are discussed, and the fluid-structure interaction (FSI) mechanism of the downstream square cylinder is analyzed as well. The results show that the mass ratio plays an important role in the flow-induced vibration of the downstream cylinder. When the mass ratio is small ($m^{\ast }=3$), the vibration response is very complicated for the downstream cylinder. With the increase of reduced flow speed, the downstream cylinder does not have the traditional lock-in phenomenon (with the lock-in frequency ratio around 1) but has the soft-lock-in phenomenon (with the lock-in frequency ratio less than 1). When the mass ratio is large ($m^{\ast }=10$ and 20), the soft-lock-in phenomenon disappears, while the traditional lock-in phenomenon occurs instead. The maximum transverse amplitude of the downstream cylinder decreases gradually with the increase of the mass ratio. Furthermore, the mass ratio has an obvious effect on the distance between two square cylinders. The distance between two cylinders severely decreases in the lock-in region for the small mass ratio but keeps almost constant for the larger mass ratio. In addition, the mass ratio also has a significant effect on the wake modes and FSI mechanisms of two tandem square cylinders. For the small mass ratio ($m^{\ast }=3$), the wake modes and the FSI mechanism are very diverse.

2019, 51(6): 1740-1751. doi: 10.6052/0459-1879-19-187

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NUMERICAL SIMULATION AND ANALYSIS OF ACTIVE JET CONTROL OF HYDROFOIL CAVITATION

Wang Wei, Tang Tao, Lu Shengpeng, Zhang Qingdian, Wang Xiaofang

In order to improve the cavitation characteristics of the flflow fifield on the suction side of the hydrofoil under high-speed flflow conditions, a method of active water jet arranged on the suction side is proposed to control the flflow around the hydrofoil. Based on a fifilter-based density correction turbulence model combined with Zwart-Gerber-Belamri cavitation model, the inflfluence of the water jet on the cavitation and hydrodynamic characteristics of the hydrofoil is analyzed when the cavitation number is 0.83, the angle of attack is 8◦ and the water jet is 0.19c from the foil leading edge. The intensity of the re-entrant jet is analyzed quantitatively to explore the relationship between the re-entrant jet and the cavitation characteristics of the flflow fifield. The numerical results show that the time-average cavity volume on the suction side of the hydrofoil with jet is 14/15 smaller than that of the original hydrofoil, which indicate that the water jet can signifificantly weaken the development of cavitation, and thus the cavitation pattern in the flflow fifield transforms from cloud cavitation to sheet cavitation. Moreover, the water injection greatly improves the hydrodynamic performance of the hydrofoil. The lift to drag ratio of the hydrofoil increases by 22.9% compared with that of the original hydrofoil, meanwhile, and the shedding frequency of the cavitation decreases by 26.2%, and the amplitude caused by the shedding of the cavitation decreases by 9.1%. The water jet shrinks low pressure area on the suction side sharply and reduces the reverse pressure difffference of flflow in the vicinity of the hydrofoil, as a result, intensity of the re-entrant jet declined. The water injection also thins the boundary layer which enhances the anti-reverse pressure gradient capability of the flflow and then blocks the re-entrant jet. Those explain the mechanism of cavitation flflow control by active water injection.

2019, 51(6): 1752-1760. doi: 10.6052/0459-1879-19-222

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DETAILED ANALYSIS OF VIBRATIONAL STATES OF OXYGEN IN HIGH TEMPERATURE NON-EQUILIBRIUM FLOWS

Hong Qizhen, Wang Xiaoyong, Sun Quanhua

Hypersonic flow is usually in a thermochemical nonequilibrium state due to high temperature after the bow shock. In this paper, the state-to-state method and two-temperature models are employed to study the thermochemical nonequilibrium processes of oxygen for a post-shock flow and a flow over a blunt body along the stagnation line. The state-to-state method treats each vibrational energy level of molecular oxygen as an independent species, and predicts the number density of each vibrational level by coupling the Euler equations or reduced Navier-Stokes equations along the stagnation line. The two-temperature models assume that all vibrational levels follow the Boltzmann distribution at a vibrational temperature, and a vibrational energy equation is solved to obtain the vibrational temperature. Simulation results show that the distributions of the temperature and species concentration predicted by the state-to-state method are in good agreement with the available experimental results in the literature, while the classical two-temperature models show large errors and the results of different two-temperature models are scattered. The state-to-state method gives detailed information of all vibrational levels along the streamline. After the normal shock or bow shock, the high vibrational levels are first excited but low levels with large number density will reach thermal equilibrium first, whereas high level molecules reach thermal equilibrium only after a long distance. Near the stagnation point, the recombination reaction produces oxygen molecules that are at high vibrational levels, thus the number density of a high vibration level is significantly higher than that of the equilibrium distribution. It is also found that the dissociation rate of classical two-temperature models deviates from the state-to-state result, which cannot accurately account for the coupling effects of vibration dissociation on the dissociation rate. However, it is reasonable for Park’s two-temperature model to take the vibration energy lost by dissociation to be 0.3$\sim$0.5 times of the molecular dissociation energy.

2019, 51(6): 1761-1774. doi: 10.6052/0459-1879-19-145

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STUDY ON INSTABILITY OF THE WIDE VANELESS DIFFUSER WITH DIFFERENT TREATMENTS OF VISCOSITY

Hu Chenxing, Yang Ce

The global stability of vaneless diffuser in the centrifugal compressor is often influenced by the main flow stability, backflow at exit and the boundary layer separation. For the vaneless diffuser with large axial width ratio, the effect of the main flow and boundary layer on the instability perturbations is the main subject. In this paper, the mean flow of the wide vaneless diffuser is firstly obtained with numerical simulations. The Eulers' and Navier-Stokes equations are linearized respectively based on the small perturbations assumption. Then the inviscid stability approach considering the inviscid main flow and the mixed stability approach considering the effect of both eddy viscosity and molecular viscosity are established. The prediction results are validated against the experimental results. At last, the structural sensitivity based on the adjoint method is adopted. And the wave-maker region is revealed under different treatments of viscosity. The instability perturbations are located at the middle of the flow field when inviscid main flow is only considered. And a centrifugal instability maybe the main cause of the vaneless diffuser stall. When the inviscid main flow and boundary layer are both considered, the wave-maker region not only lies at the main flow near the middle of the vaneless diffuser, but also lies at the reverse flow region of the boundary layer.

2019, 51(6): 1775-1784. doi: 10.6052/0459-1879-19-207

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NUMERICAL SIMULATION ON THE VORTEX-INDUCED COLLISION OF TWO SIDE-BY-SIDE CYLINDERS

Yang Ming, Liu Jubao, Yue Qianbei, Ding Yuqi, Wang Ming

Vortex-induced vibration of cylindrical structures is a common phenomenon in engineering. If the distance between cylindrical structures is small, vortex-induced collision will occur. Vortex-induced collision is more serious than vortex-induced vibration on the fatigue damage of the structures. The immersed boundary method was used to simulate the dynamic boundary problem in the fluid which avoided the numerical instability problem when the traditional boundary-fitting method was used to solve the collision problem between solids. The finite element method was used to simulate the motion and collision of the cylinders. The lubrication model under fluid flow condition was established by data regression method. The vortex-induced vibration and collision of two side-by-side cylinders at different initial gap ratios were simulated numerically. The numerical results show that if the collision occurs, there will be a continuous collision. Multiple frequencies occur in collisions and the main frequency of vibration is higher than that without collision. When the two cylinders collide, the relative velocity is smaller than that of free flow. When two cylinders are close to each other, the transverse fluid force decreases with the gradual inclination of vortex ring separation angle. When the vortex rings between two cylinders start to influence each other and squeeze, the transverse fluid force starts to increase gradually. When the two cylinders start to rebound, a low pressure area is formed between the two cylinders, which changes the direction of the transverse fluid force and makes the two cylinders move close to each other again. This repetition results in the oscillation of transverse fluid force and cylinder velocity after collision.

2019, 51(6): 1785-1796. doi: 10.6052/0459-1879-19-224

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DAMAGE CONSTITUTIVE MODEL FOR THERMAL SHOCKED-CERAMIC MATRIX COMPOSITE

Yang Zhengmao, Liu Hui, Yang Junjie

The ceramic-matrix composite (CMC) structure is inevitably subjected to cyclic thermal shocks in service, which induces the thermo-mechanical damage. Investigation of the damage constitutive model of the thermal shocked-CMC is of significance in the design and performance evaluation of such composites in aeronautical industry. In the present work, monotonic tensile damage tests were conducted for the thermal shocked-CMC. It is found that the degradation of elastic modulus for the thermal shocked-composites are directly related to the applied strain. Based on the framework of continuum damage mechanic, a nonlinear damage evolution model for the thermal shocked-CMC was proposed under plane stress assumption. The identification of the damage parameters involved in this model requires one off-axis (45$^\circ$), and two on-axis (parallel to tow directions) uniaxial tensile tests. Finally, the inelastic strain caused by matrix damage is described by classical plastic theory. The proposed strain damage macroscopic model can describe the damage evolution of CMC under thermo-mechanical loading, and also compensate the deficiency of the theoretical and experimental research for the damage evolution model of damaged-CMC under mechanical loading.

2019, 51(6): 1797-1809. doi: 10.6052/0459-1879-19-229

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RESEARCH ON MODELING OF COMPRESSIVE YIELD BEHAVIOR FOR HTPB COMPOSITE BASE BLEED GRAIN

Wu Zhihui, Niu Gongjie, Hao Yufeng, Qian Jianping, Liu Rongzhong

HTPB composite base bleed grain (CBBG), which has been widely applied to the base bleed extended-range technology, is a typical particle-filled energetic material and bears both impact and temperature loads in battlefield environments. In order to investigate impact compressive mechanical properties of HTBP CBBG, split Hopkinson pressure bar experiments were conducted at various temperatures and strain rates, ranging from 233 to 323 K and from 1100 to 7900 s$^{-1}$. True stress-true strain curves shows that HTPB CBBG yields and then deforms plastically with strain hardening effect and maintains high toughness under each experimental condition. The stress value at a certain strain increases with the increase of strain rate and the decrease of temperature, but temperature has a more significant influence on impact compressive mechanical behaviors of HTPB CBBG than strain rate. On the one hand, the time-temperature superposition principle was introduced into the cooperative model by taking the correlations between horizontal/vertical shift factor and temperature as WLF function-type equations based on the fact that the temperature range discussed here was higher than the glassy transition temperature of HTPB CBBG. One the other hand, the enhancement effect of strain rate of internal stress was also taken into consideration, and then a new stress model was proposed. The smooth horizontal shift factor-temperature curve, vertical shift factor-temperature curve and master curve of yield stress were built at a reference temperature according to experimental results to obtain the parameters in the proposed model. The comparison between the model prediction and experimental data indicates that the developed model can precisely describe the bilinear dependence of yield stress on strain rate at temperatures of 233$\sim $323 K. The proposed model points out that the strain rate effect is derived from internal stress at low strain rates while it is derived from drive stress at high strain rates.

2019, 51(6): 1810-1819. doi: 10.6052/0459-1879-19-200

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THE PROPAGATION PROCESS AND THE GEOMETRIC DISPERSION OF A TRAPEZOIDAL STRESS PULSE IN AN ELASTIC ROD

Yang Hongsheng, Li Yulong, Zhou Fenghua

Geometric dispersion effects are often difficult to avoid during stress wave propagation. Analytical analysis of the geometric dispersion of stress wave propagation in elastic rods is crucial for the study of fundamental wave problems and the dynamic mechanical behavioral characterization of materials. This paper briefly describes the one-dimensional Rayleigh-Love stress wave theory considering the lateral inertia correction in the elastic rod, and summarizes the derivation process of the control equation by the variation method. Aiming at the trapezoidal stress loading pulse commonly used in Hopkinson rod experiments, the corresponding model of the initial boundary value problem (IBVP) of the partial differential equations is established. The geometric dispersion phenomenon of pulse propagation in the rod is studied by using the Laplace transform method. The inverse Laplace transform is carried out according to the residue theorem. The analytic solutions of the stress waves at different positions and times are given in the form of series representation. The influence of the number of calculation terms on the convergence of the results is analyzed. These analytical calculation results are in good agreement with the results using three-dimensional finite element numerical simulation, which proves that the Rayleigh-Love lateral inertia correction theory can effectively characterize the geometric dispersions in typical Hopkinson bar experiments. Based on the analytic solutions, the parametric study of the trapezoidal loading pulse is conducted. The influences of propagation distance, Poisson's ratio, and the pulse slope on the geometric dispersions are quantitatively described. The analytical solution of the Rayleigh-Love rod under trapezoidal pulse loading reveals the essential law of geometric dispersion effect and can be used for the dispersion correction process in the real experiments.

2019, 51(6): 1820-1829. doi: 10.6052/0459-1879-19-183

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INVESTIGATION ON CRACK GROWTH IN CONCRETE BY MOMENT TENSOR ANALYSIS OF ACOUSTIC EMISSION

Ren Huilan, Ning Jianguo, Song Shuizhou, Wang Zonglian

On meso-scale, concrete can be considered as a kind of non-homogeneous composite mainly composed of aggregates, cement paste and cracks. Under uniaxial quasi-static loading condition, quasi-brittle characteristics can be observed in the stress-strain curves. Failure process of concrete is essentially a process of nucleation, propagation and convergence of internal micro-cracks.Therefore, investigation on the crack growth process on meso-scale is beneficial to understand the deformation and failure process of concrete. Acoustic emission is a physical detection method which can be used to obtain the physical information of the mesoscopic damage evolution inside many kinds of materials. Acoustic emission technique, modified time of arrival approach and moment tensor theory were applied to analyze the AE sources obtained in the Brazilian test and the locations, types and orientations of cracks in the specimens were investigated. Relationship between the failure on macro-scale and the mechanisms of cracking on meso-scale of concrete was revealed. The results show that the micro-cracks generate near two contact surfaces between specimen and loading plates firstly. Then the nucleation of the micro-cracks occur at the local zone. Finally, cracks propagate from top and bottom to the center of specimen along the loading direction in the elevation view. In the side view, cracks propagated from the internal zone to the surface. Volume of micro-cracks could be regarded as a measure of elastic energy released from the nucleation of micro-cracks. The volume of micro-cracks is small at the early stage of nucleation. When the load reached its peak value, the maximum volume of the micro-crack was $5.93\times 10^{ - 4}$ mm$^{3}$. Tensile cracks, shear cracks and mixed-mode cracks on meso-scale could be observed in the tensile failure process of concrete on macro-scale. The failure process of the concrete was dominated by tensile cracks (nearly 60%) and the shear cracks had the minimum proportion (nearly 10%). The motion of tensile cracks dominate the macro-scale failure of specimen.

2019, 51(6): 1830-1840. doi: 10.6052/0459-1879-19-170

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DAMAGE INDEX METHOD FOR DETECTION OF FLAWS BASED ON DYNAMIC RESPONSE

Du Chengbin, Jin Licheng, Wu Zhiqin

：The vibration response of the structure is of great importance for detecting the damage inside the structure. Based on the difference between the measured response of sensors at different positions in the real structure and the nondestructive structure, the different weights of each sensor are defined, and the damage indexes at different positions in the structure are constructed. In this paper, dynamic extended finite element method combined with level set method is used to describe the internal defects of the structure to avoid grid redivision in iterative computation. After obtaining the weights of sensors at different positions, the traditional linear weights are replaced by the nonlinear weights depending on the comparing the traditional linear weight with threshold value, in order to reduce and amplify the weight value at different locations. The double-cubic interpolation is introduced to obtain the damage index and its distribution of the structure, the number of defects and their potential areas are identified with interpolation imaging technology. In the process of inversion using intelligent algorithms, the unnecessary sensors are removed at first and the weighting coefficients are introduced to improve the objective function for accurate inversion. The analysis of several examples indicates that the damage index method can obtain the specific number and preliminary position information of defects through forward modeling quickly when the number of defects is unknown, and the objective function inverse analysis method with weighting coefficient can reach the convergence faster and obtain the precise location of defects more efficiently than those in the previous intelligent algorithms. In order to further prove the feasibility and application prospect of the model, we use the model presented in this paper to detect the defects of reinforced concrete slab with two circular holes, and the good inversion results are shown that the proposed model is feasible and has certain engineering significance.

2019, 51(6): 1841-1855. doi: 10.6052/0459-1879-19-221

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ROPE ELEMENTS WITH MOVING NODES IN ROPE-PULLEY SYSTEMS

Qi Zhaohui, Guo Shudong, Zhuo Yingpeng

In traditional finite element methods, a common hypothesis is the nodes of an element being fixed with material points, which place restrictions on the description of a system. In this paper, this hypothesis is released and a new kind of rope element with nodes being no longer fixed with either spatial coordinates or material coordinates is presented. The theory of obtaining material velocity and acceleration of a point from the corresponding spatial velocity and acceleration of the point is established. The velocities and accelerations in the principle of virtual power should being material velocities and accelerations respectively is emphasized and applied. According to the feature of a rope-pulley system, its movement can be described by a new group of variables, such as arc-lengths, azimuthal angles and strains of ropes at entrance and exit contacting points. Instead of traditional methods, conditions of contact between rope and pulley are modeled as the material velocity of a point on the contacting rope being equal to the corresponding point on the pulley. By means of the presented methods, some obstacles, such as frequently divergence and high time consuming that resulting from traditional finite element methods can be removed. Motion equation of the rope-pulley system is derived by the principle of virtual power. Arc-length coordinates, positions of contact boundary points, the movement of pulley centers and their body-fixed reference coordinate systems, shape and positions changing of ropes as well as the tension forces in every point of a rope, can be obtained high precisely. The presented method can provide a new theoretical basis for analysis of mechanical systems with ropes and pulleys. The presented theory of using spatial points as describing variables to replace traditional material points is of applicable widely. It can be a reference for theory and application of finite element methods.

2019, 51(6): 1856-1871. doi: 10.6052/0459-1879-19-168

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STABILITY ANALYSIS ON PARAMETRIC VIBRATION OF PIEZOELECTRIC ROTATING CANTILEVER BEAM WITH ACTIVE CONTROL

Tang Ye, Wang Tao, Ding Qian

In engineering application, rotating machines tend to be pulsating operation due to the errors of manufacturing and processing as well as the non-uniformity of assembly, which may cause parametric vibration of the system. Furthermore, if the pulsation parameters satisfy a certain relationship, the parametric vibration will cause the system to lose stability, which furtherly affects the normal operation of mechanical structures. In view of this problem, the piezoelectric material is introduced to suppress vibration of rotating cantilever beam subjected to parametric exciting. The problem about the parametric optimization and design of rotating cantilever beam with active control is studied in this paper. The first order approximate linear equation governing the piezoelectric rotating cantilever beam controlled by velocity feedback sensor is established based on the Hamilton' principle combining with the first-order Galerkin discretization method. Then, the multi-scale method is applied to obtain the governing equation of stability boundary of the piezoelectric rotating cantilever system with the 1/2 sub-harmonic parametric resonance. The correctness of the perturbation solution is verified by the direct analysis method. The critical damping ratio and the dimensionless parameter of pulsating amplitude of hub angular velocity in the perturbation solution are regarded as the indicators to evaluate the system stability. Numerical examples are presented to illustrate the effects of the hub radius, the average value and pulsating amplitude of hub angular velocity, the beam length and the feedback gain coefficient of velocity sensor on the dynamic stability. The results show that the stable region can be increased with the decrease of the beam length, the hub radius and pulsating amplitude of hub angular velocity, but the raise of the feedback gain coefficient, moreover, the relation between the average value of hub angular velocity and the stability is not monotonous. It provides a reference for the further design of piezoelectric rotating machinery structure.

2019, 51(6): 1872-1881. doi: 10.6052/0459-1879-19-211

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DYNAMIC ANALYSIS OF SANDWICH TAPERED POROUS MICRO-BEAMS CONSIDERING SIZE EFFECT

Pu Gang, Zhang Dingguo, Li Liang

Based on the modified couple stress theory, a size-dependent dynamic model of large overall rotating hub-flexible sandwich tapered Euler-Bernoulli micro-beams made from functionally graded materials with porosities is developed to study their dynamic characteristics. The functionally graded sandwich tapered micro-beams are composed of a core with inperfect functionally graded materials sandwiched between two homogeneous face sheets, which can decline the influence of the traditional sandwich structure debonding damage caused by the mismatch of stiffness properties between their core and face sheets. The high-order rigid-flexible coupled dynamic equations of the system applied to large deformation are derived by considering the von Kármán geometric nonlinear strain and adopting Lagrange's equation of the second kind. The method of assumed modes is used to describe the chordwise and axial deformations of micro-beams. The dynamic stiffening effect is captured the nonlinear coupling term obtained by longitudinal shortening caused by the transverse bending deformation of the micro-beams. Then, under the effects of different parameters, such as the width of face sheets of the tapered micro-beams, rotating angular velocity, material gradient index, size-dependency, porosities and volume fractions of each layer structure, the dynamic characteristics of functionally graded sandwich micro-beams in the plane are investigated. The functionally graded sandwich tapered micro-beams combine the characteristics of the functionally graded rectangular micro-beams and the tapered micro-beams. Compared with the functionally graded rectangular micro-beams, these characteristics make the natural frequency of the functionally graded sandwich tapered micro-beams increase and lead to different influences of porosities on the natural frequencies of the structures. Moreover, since the coupling potential energy of the chordwise and axial motions is involved in the strain energy of the flexible micro-beams, the equilibrium position of the system migrates in the steady state. It is observed that interesting frequency veering and mode shift phenomena of the system are observed when the size-dependency changes.

2019, 51(6): 1882-1896. doi: 10.6052/0459-1879-19-164

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VIBRATION CHARACTERISTICS OF AXIALLY MOVING TIMOSHENKO BEAM UNDER VISCOELASTIC DAMPING

Zhou Yuan, Tang Youqi, Liu Xingguang

Viscoelastic damping has always been one of the research hotspots of axial motion system. The influence of viscoelastic damping has not been considered in most previous researches on axial motion systems. In the present paper, the vibration characteristics of the axially moving Timoshenko beam with viscoelastic damping are studied. The dynamic equations of Timoshenko beams with axial viscoelastic motion and the corresponding boundary conditions of simply supported beams are obtained using the generalized Hamilton principle. The method of direct multiple scales is used to show the corresponding relationship between axial speed and parameters. The approximate analytical solutions of the first two natural frequency and attenuation coefficient are obtained. The differential quadrature method is applied to analyze the variation of the first two natural frequencies and attenuation coefficients with the axial speed under the presence or absence of viscoelasticity. The approximate numerical solutions of the first two natural frequencies and attenuation coefficients under viscoelastic action are given and the validity of approximate analytic solution is verified. It is shown that the natural frequency of the beam decreases gradually with the increasing axial speed. The natural frequency and attenuation coefficient of the beam decrease with the increasing viscoelastic coefficient. The attenuation coefficient is proportional to the viscoelastic coefficient. The viscoelastic coefficient has little effect on the first order attenuation coefficient and natural frequency. But it has a greater influence on the second-order attenuation coefficient and natural frequency.

2019, 51(6): 1897-1904. doi: 10.6052/0459-1879-19-205

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RESEARCH ON DYNAMICS OF A RIGID-FLEXIBLE-THERMAL COUPLING ROTATING HUB-FGM BEAM

Fan Jihua, Chen Liwei, Wang Mingqiang, Zhang Dingguo, Du Chaofan

The rigid-flexible-thermal coupling dynamics of the hub-FGM beam system under large overall rotating motion is studied. The FGM beam is an Euler-Bernoulli beam, and its physical properties follow certain kinds of power law gradient distribution and vary in the thickness direction. The longitudinal deformation and the transverse deformation of the flexible beam are considered and the coupling term of the deformation which is caused by transverse deformation of the flexible beam is included in the expression of longitudinal deformation. Considering the influence of the thermal coupling of the tapered hollow beam which is under the condition of external high temperature and internal cooling passage cooling on the dynamic characteristics of the system, the temperature field distributed along the thickness direction of the FGM beam is obtained, and the thermal strain is included in the constitutive relationship of the beam. By using the assumed mode method to describe the deformation of the beam, the rigid-flexible-thermal coupling dynamic equations of the system are derived via employing Lagrange's equations of the second kind, as well as to compile the dynamics simulation software. Then the dynamics of the system are studied through simulation examples. The results show that the dynamic response of beams with different sections is quite different, so it is necessary to model the actual system reasonably. When the large overall rotating motion is known, the FGM beam considering the thermal shock load will effectively suppress the transverse bending deformation, and the high-frequency oscillation will occur with the superposition of the thermal shock when the large overall rotating motion is constant; When the large overall rotating motion is unknown, the external torque and the thermal shock load interact to produce a thermal coupling effect, which causes the system to exhibit high-frequency oscillation, at the same time , the rigid-flexible thermal coupling effect of the hub-FGM beam system is appeared.

2019, 51(6): 1905-1917. doi: 10.6052/0459-1879-19-241

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EXPERIMENTAL STUDY ON HEMODYNAMICS OF AORTIC VALVE UNDER VARIED CARDIAC OUTPUT USING PIV

Liu Zhaomiao, Yang Gang, Pang Yan, Zhong Xixiang, Li Mengqi, Xue Hebo, Qi Yipeng, Shi Yi

Reduced cardiac output (CO) always occurs in aortic valve diseases. The hemodynamics of aortic valve is affected by reduced CO, causing secondary valvular diseases. In this paper, a three-dimensional reconstruction of aortic root geometric model with left coronary artery is complished based on medical imaging data, a highly smooth and transparent aortic root experimental model is casted, and an in vitro pulsating circulation system is constructed. Particle image velocimetry (PIV) is used to investigate the effect of CO on hemodynamics of aortic valve with or without the left coronary artery, such as the velocity, viscous shear stress (VSS), Reynolds shear stress (RSS), and so on. The results show that aortic sinus hemodynamics are influenced by left coronary artery that the presence of left coronary artery changes the vortex and vorticity in the sinus. In the case of the presence of left coronary artery, fluids in the aortic sinus flows out through the left coronary artery which leads to vortex gradually disappears and vorticity early decreases. At the peak systolic, regions of positive and negative VSS are exist in both sides of the centrosymmetric systolic jet and RSS is especially elevated in the ascending aorta on the side of left coronary artery. In addition, the hemodynamics of aortic valve, such as the velocity, VSS and RSS, are significantly affected by the CO. The maximum velocity, VSS and RSS increase with the increasing of CO, namely, the maximum velocity is 0.98, 1.13, 1.21 and 1.37 m/s, the maximum VSS is 0.87, 0.95, 0.96 and 1.02 N/m$^{2}$, and the maximum RSS is 103.76, 116.25, 138.68 and 146.55 N/m$^{2}$ when $CO=2.1$, 2.8, 3.5 and 4.2 l/min, respectively. At low CO, the values of transvalvular flow velocity and VSS of aortic valve are small, which may easily lead to thrombosis. The research findings can provide theoretical references for the aortic valve implantation.

2019, 51(6): 1918-1926. doi: 10.6052/0459-1879-19-231

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AN IMPROVED FIFTH-ORDER WENO-Z+ SCHEME

Luo Xin, Wu Songping

The performance improvement of the WENO-Z+ scheme depends on the role of the additional term, which is added to the WENO-Z weights to increase the weights of less-smooth substencils further. Since the additional term may lead to negative dissipation by over increasing the weights of less-smooth substencils in smooth regions, the coefficient $\lambda $ is set to control the role of this term and needs to be carefully determined. In this paper, the defects of the method the WENO-Z+ scheme adopts to determine the value of $\lambda $ are pointed out: It can neither fully utilize the potential of the scheme nor effectively avoid negative dissipation. We propose that to take the full role of the additional term in reducing numerical dissipation and improving resolution ability, the value of $\lambda $ should change with the local data of the flow field. Based on this idea, we design a new calculation formula for $\lambda $, which can adjust the role of the additional term adaptively: Weaken the role of the additional term only where the weights of less-smooth substencils are easy to be excessively increased. The new scheme employing the new $\lambda $ formula is named WENO-Z++, and its numerical performance is systematically analyzed. Theoretical analysis indicates that the new scheme maintains essentially non-oscillatory (ENO) property and has lower numerical dissipation at discontinuities. The investigation of approximate dispersion relation (ADR) shows that the new scheme effectively avoids the negative dissipation caused by excessive increase of the weights of less-smoothed substencils, and its spectral properties are significantly improved. The parameters set that allow the new scheme keeping the optimal order of accuracy at extreme points is deduced. A series of numerical experiments for solving the Euler equations show that both the shock-capturing ability and resolution for complex flow structure of the new scheme are significantly better than those of the original WENO-Z+ scheme.

2019, 51(6): 1927-1939. doi: 10.6052/0459-1879-19-249

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RECIPROCAL PROGRAMMING THEORY AND ITS APPLICATION TO ESTABLISH A REASONABLE MODEL OF STRUCTURAL TOPOLOGY OPTIMIZATION

Sui Yunkang, Peng Xirong, Ye Hongling, Tie Jun

Based on mathematical programming theory, the reciprocal programming is defined as a pair of programming which objective and constraint functions are changed each other. After that, it is pointed out that reciprocal programming and dual programming seem similar but there are five differences between them: (1) the difference in whether they are the same problem or not; (2) the difference in whether there exists a dual gap or not; (3) the difference in the number of design variables; (4) the difference between single-objective and multi-objective problems; (5) the difference between reasonable and unreasonable problems. Finally, based on the definition of the reciprocal programming, the structural topology optimization model is examined; and following results are obtained: (1) from this perspective, it is clear that there exists indeed unreasonable models that have been used in structural optimization; (2) a way to correct the unreasonable model and make it reasonable is put forward; (3) the reasons that the minimizing compliance model with volume constraint (MCVC for short) is unreasonable are presented; (4) according to the theory presented in this paper, the MCVC model is actually the m-aspect of reciprocal programming, so its corresponding s-aspect is established, that is, the minimizing volume model with multiple compliance constraints (MVCC for short); (5) the physical interpretation and algorithm of structural compliance constraints in the MVCC model are presented; and the concepts and methods of global stress constraint in the ICM (Independent continuous and mapping ) method are demonstrated; (6) numerical examples show the differences between the MCVC and MVCC model as a pair of reciprocal programming and verify the rationality of the MVCC model. Different optimized topologies are obtained for the MCVC model with different volume constraints and different weighting coefficients for multiple load cases. But a unique optimized topology can be achieved by the MVCC model with compliance constraints under multiple load case.

2019, 51(6): 1940-1948. doi: 10.6052/0459-1879-19-259

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