力学学报

TOPOLOGY OPTIMIZATION METHOD FOR INTEGRATED THERMAL PROTECTION STRUCTURE CONSIDERING TRANSIENT TEMPERATURE AND STRESS CONSTRAINTS

Li Shuai, Zhang Yongcun, Liu Shutian

The integrated thermal protection structure is usually in a severe unstable thermal environment, and the time effect of thermal load, namely transient thermal effect, is obvious. In order to avoid huge calculation consumption of transient thermal analysis, previous optimization design studies of integrated thermal protection structures usually equivalent transient heat transfer to steady-state heat transfer under the same thermal boundary conditions, and take the temperature field of steady-state heat transfer analysis as the design thermal load. However, previous studies have shown that the steady-state heat transfer cannot accurately equivalent the effect of transient heat transfer, and the transient thermal effect has an important influence on the structural design results. In this paper, the optimization design problem of integrated thermal protection structure considering transient thermal effect is studied, and a topology optimization method of integrated thermal protection structure considering transient temperature and stress constraints is established. Based on the Solid Isotropic Material with Penalization (SIMP) method, two kinds of topology optimization models for integrated thermal protection structures are constructed: 1. The stiffness design model taking minimizing the structural strain energy as objective function, considering material volume fraction, maximum stress and maximum bottom-face temperature constraints. 2. The strength design model taking minimizing material volume fraction as objective function, considering maximum stress and maximum bottom-face temperature constraints. By solving the transient thermodynamic coupling equation, the thermodynamic coupling static analysis results of the structure are obtained. The maximum value of structural response in time domain is represented by the condensed integral function in space and time domains, which was taken as constraint and objective functions. The sensitivity expressions of objective function and constraint functions are derived by adjoint method. The effectiveness of the proposed method is verified by three numerical results. Numerical examples showed that the proposed method could accurately reflect the influence of transient thermal effects on the design results of integrated thermal protection structures under the condition of transient heat transfer. Compared with the design results based on steady-state thermal analysis, the design results considering transient thermal effects were significantly improved.

, Available online , doi: 10.6052/0459-1879-22-598

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ANALYSIS OF CARANGIFORM UNDULATION BASED ON VIRTUAL POWER PRINCIPLE

Huang Kaijun, Yu Yongliang

Most fish in nature achieve propulsion through undulatory movements, which are the result of the interaction between the deforming fish body and the surrounding fluid. To study the response of the fluid can enhance our understanding of undulatory propulsion and flow control. A two-dimensional deforming airfoil is used to model the carangiform fish. The flow field generated by fish body and the fluid forces acting on the fish body were obtained by using computational fluid dynamics. Using the principle of virtual power, the thrust on the fish body was decomposed into four parts, which are the instantaneous contribution of the boundary acceleration, the contribution of the relative magnitude of fluid rotation and strain rate in the flow field, the wall friction-like component and the wall friction component. The results show that the instantaneous contribution of the boundary acceleration is the main source of positive thrust. The rear 80% of the thrust contribution of this term comes from the instantaneous boundary acceleration movement of the rear 20% of the fish body. The fluid rotation and strain rate in the boundary layer on both sides of the fish tail and the friction contribute to resistance. For low Reynolds number, the negative contribution of the relative magnitude of fluid rotation and strain rate is lower than that of wall friction, while for high Reynolds number, the negative contribution of the relative magnitude of fluid rotation and strain rate is stronger than that of wall friction. However, the wall friction-like component is always smaller compared to the other three terms. In the analysis of the scaling law of undulatory propulsion, it was found that there is a component independent of the Reynolds number which is provided by the first two parts, while the component that is dependent on the Reynolds number is provided by the last three parts. Furthermore, the frictional force and the friction-like force provide constant resistance.

, Available online , doi: 10.6052/0459-1879-23-076

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EXPERIMENTAL STUDY ON PERMEABILITY ANISOTROPY OF DEEP RESERVOIR SANDSTONE UNDER TRUE TRIAXIAL STRESS

Wang Zhechao, Yi Yunjia, Min Zhongshun, Feng Hao

Permeability anisotropy is a very typical phenomenon in sedimentary bedding structure. On one hand, it is determined by the primary sedimentary structure (i.e. primary anisotropy) and on the other hand, it is affected by stress and pore pressure (i.e. induced anisotropy). In order to study the primary and induced anisotropy of reservoir sandstone permeability under true triaxial stress, the reservoir sandstone of S6 gas storage in northeast China was taken as the research object. The true triaxial stress-seepage coupling device of hard rock independently developed by Northeastern University was adopted to carry out seepage experiment on reservoir sandstone, and the permeability test of the same sandstone in three mutually vertical directions was completed by steady-state method. The test results show that: in the range of applied stress and pore pressure, permeability of sandstone in parallel bedding direction${k_x}$ is 100.94 mD ~ 113.98 mD, ${k_y}$ is 98.34 mD ~ 111.41 mD, and permeability in vertical bedding direction ${k_z}$ is 54.98 mD ~ 63.29 mD. The permeability of sandstone in three orthogonal directions decreases with the increase of principal stress and increases with the load of pore pressure. The effect of stress perpendicular to the gas seepage direction on permeability is greater than that of stress parallel to the gas seepage direction. When the direction of external stress is perpendicular to the direction of gas flow, the effect of stress perpendicular to bedding on permeability is greater than that of stress parallel to bedding. The linear elastic response of pore pressure to reservoir sandstone permeability is not isotropic. The linear permeability increment generated by pore pressure to horizontal bedding direction is higher than that in vertical bedding direction. The research results provide a reference basis for accurate prediction of sandstone permeability of underground gas storage and a new petrophysical property data for operation and management of underground gas storage.

, Available online , doi: 10.6052/0459-1879-23-051

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NUMERICAL SIMULATION OF MULTI-SCALE FRACTURED RESERVOIR BASED ON CONNECTION ELEMENT METHOD

Zhan Wentao, Zhao Hui, Rao Xiang, Liu Wei, Xu Yunfeng

In order to solve the complex geometric characteristics description and dynamic connectivity identification problems of reservoir at different scales, a new method of reservoir numerical simulation, connection element method (CEM), based on non-European physical connectivity network with meshless characteristics has been developed in recent years. In this paper, CEM is extended to fractured reservoirs. From the perspective of fluid flow, the reservoir is discretized into physical connected network by the connection element. The generalized difference approximation of the pressure diffusion term is given according to the physical parameters of the node, the radius of the influence domain and the weighted least square method. Meanwhile, the control volume of nodes, the transmissibility between matrix nodes, the transmissibility between fracture nodes, and the transmissibility between matrix nodes and fracture nodes were calculated based on the material conservation equation. Thus, a fully implicit discrete scheme of seepage control equations is constructed to solve dynamic production parameters such as pressure, saturation and water cut. Based on the pressure gradient between nodes solved by each time step, the allocation factors of injection wells at each time step were calculated by the depth-first search algorithm of graph theory to quantitatively characterize the flow relationship and connectivity between well nodes. The algorithm validation shows that the method can freely and flexibly portray complex reservoir geometry including distribution of complex fractures networks and irregular reservoir boundaries. Compared with the traditional grid-based method, this method can retain more abundant flow topologies under the condition of coarser model, so as to achieve a better balance between computational accuracy and computational efficiency. As a result, CEM can better meet the demand of production dynamic simulation and prediction of actual large-scale fractured reservoirs, and provides a new idea for numerical simulation of fractured reservoirs with multi-scale geometric characteristics and complex boundary reservoirs.

, Available online , doi: 10.6052/0459-1879-23-069

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EXPLICIT SOLUTION OF STRESS COMPLEX POTENTIAL FUNCTION FOR SURROUNDING ROCK OF SHALLOW SUBSEA TUNNEL

Jin Bo, Tian Juntong, Fang Qihong

The elastic complex function theory is used to simplify the shallow-buried subsea tunnel into a semi-infinite plane problem. The stress distribution of surrounding rock after tunnel excavation is explored considering the effects of self-weight of surrounding rock and sea water pressure. The fractal mapping function is used to map the surrounding rock domain to a circular domain like a plane, and the complex potential single-value analytical function is expanded to a Laurent series in the circular domain. The power term of Laurent series is determined by using the stress boundedness at infinite distances. The iteration expression of Laurent series coefficient is obtained according to the surface boundary and the non-uniform stress boundary condition at the orifice. The determined Laurent series condition is substituted into the iteration expression to obtain the explicit solution of complex potential function, thus realizing the iteration of complex potential function coefficient from low power to high power. According to the complex function expression of the stress component, the stress component of all points around the tunnel can be obtained. The influence of two single-value analytical functions with different powers on the results is studied, and the influence of buried depth of shallow tunnel on the toroidal compressive stress is analyzed. The results show that the power series solution has high reliability, and it agrees well with the finite element solution in the first half of the tunnel. The final calculation results of power series solution in the second half of the tunnel are relatively conservative compared with the finite element results. Sufficient numbers of complex potential functions are required to ensure the accuracy of calculation results. As the buried depth of the tunnel increases, the circumferential compressive stress at the bottom of the tunnel and at the waist of the holes on both sides increases continuously. The difference in circumferential stress between the lumbar and the bottom increases as well.

, Available online , doi: 10.6052/0459-1879-23-077

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THE IMPACT OF ASPECT RATIO ON THE TRANSITIONS OF LID-DRIVEN CAVITY FLOW

An Bo, Meng Xinyu, Guo Shipeng, Sang Weimin

The critical characteristics of flow transitions refer to the changes of flow state and physical characteristics caused by flow bifurcations. It fundamentally determines the physical laws of flow evolution mode and flow characteristics and is of great significance to reveal the formation mechanism of flow phenomena. In the present paper, the numerical simulations and stability analysis of the classic lid-driven cavity flow with multiple aspect ratios ($ R \in [0.1,2.0] $) were performed. We predicted the critical Reynolds numbers for Hopf, Neimark-Sacker and period-doubling bifurcations and the initiation of turbulence. We found that some flows followed the classical Ruelle-Takens model as a routine, while others jumped from periodic flow to turbulent flow due to the period-doubling bifurcation. The mechanism of various flow phenomena was revealed and discussed, such as the loss of stability of flow field, energy cascade and flow topology changing along with aspect ratio etc.. The results are of great significance to reveal the influence of the aspect ratio R on the critical characteristics of the transitions in the cavities. It further improves the study of the internal flow. In the present study, some physical characteristics are found, for example, it is found that the Moffatt effect not only exists with sharp corners, but also in the elongated domain; Regardless of the value of R, the initial instability always starts with the appearance of Hopf bifurcation. For the shallow cavities ($ R < {\text{1}}{\text{.0}} $), as R increases, the critical Reynolds number of Hopf bifurcation decreases, indicating that the stability becomes more and more easily destroyed. For deep cavity ($ R > {\text{1}}{\text{.0}} $), compared with classical lid-driven square cavity flow ($ R = {\text{1}}{\text{.0}} $), the stability is more likely to be lost. Stretching along the longitudinal geometry is not a mandatory constraint to improve the stability of the flow field.

, Available online , doi: 10.6052/0459-1879-23-041

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PARALLEL SPECTRAL ELEMENT METHOD FOR 3D LOCAL-SITE GROUND MOTION SIMULATIONS OF WAVE SCATTERING PROBLEM

Yu Yanyan, Rui Zhiliang, Ding Haiping

For the wave scattering problem of 3D regional sites under plane wave incidences, the free field considering non-uniform distributed nodes of the spectral element method (SEM) is derived by using analytical method, which is used as the input wave filed for wave motion simulations based on the SEM. The motions of interior nodes are computed using the high-order SEM, and the motions of the boundary surface nodes are obtained by applying the multi-transmitting formula (MTF). In addition, a parallel computation across nodes is realized by using the message passing interface (MPI) technique. Then, the parallel simulations of 3D wave scattering problems base on SEM are achieved. Finally, the accuracy and stability performance of the proposed method are validated by some typical numerical examples. The results show that the presented method can achieve high simulation accuracy for the 3D local site scattering problem under plane wave incidence in different polarization directions. Up to the third order MTF, under the condition that the stability of the internal domain computation is ensured, a long-time stable calculation results can be obtained when the artificial wave velocity of the MTF is taken to be close to the shear wave velocity of the corresponding boundary nodes, and no other additional treatments are needed to eliminate the high-frequency oscillations. The low-frequency drift instability of the MTF can be easily eliminated by applying quite small modification parameters, and the simulation accuracy is generally unaffected by it. The method of this paper has promising application prospect in the numerical simulations of regional 3D ground motion under plane wave incidences.

, Available online , doi: 10.6052/0459-1879-23-052

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e^N-NEURAL NETWORK MODEL FOR PREDICTING TRANSITION OF 3-D SUPERSONIC SWEPT WING

Yu Shenghao, Yuan Jisen, Gao Liangjie, Qian Zhansen, Li Chunxuan

In order to improve the computational efficiency of 3-D supersonic boundary layer transition prediction, a neural network model for 3-D compressible boundary layer transition prediction using neural network models instead of linear stability analysis is developed. By the research on the linear stability analysis method and flowfield characteristics of supersonic swept wing, neural network model parameters of supersonic swept wing transition prediction are proposed. Using a series of supersonic swept blunt plate models as the sample set, the e^N-neural network model is established. The sensitivity of each input parameter is analyzed by taking the standard model of three-dimensional supersonic large swept back straight wing as the test set, and the calculation results and efficiency of e^N-neural network model and traditional stability analysis method are compared to verify the accuracy and efficiency of this method.

, Available online , doi: 10.6052/0459-1879-23-029

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EFFECT OF ELECTROSTATIC FORCE ON SPATIAL DISTRIBUTION AND INTERPHASE ENERGY TRANSPORT IN RADIANT HEATED PARTICLE-LADEN TURBULENT CHANNEL FLOW

Tang Xiaofeng, Feng Huanhuan, Pan Ming, Dong Yuhong

Interphase energy transfer in turbulence laden with particles is one of the focuses of scholars, and the effect of electrostatic force is an important factor affecting the particles propensity distribution and the efficiency of energy exchange between particles and turbulence in the turbulent channel flow laden with particles. In this paper, the spatial distribution of charged particles in vertical turbulent channel flow with radiation heating and the effect of spatial distribution on the energy transport between particles and turbulent flow were investigated. Direct numerical simulation was used for fluid, and Lagrange-point tracking model was used for particles. The momentum exchange and the heat exchange between particles and turbulent flow were considered. Based on the analysis of particle local aggregation characteristics, velocity correlation between particles and turbulent flow and interphase energy transport, the effect of electrostatic force on particle spatial distribution, kinetic energy exchange and heat exchange between particles and fluid were investigated. The results show that the electrostatic force of the same positive charged particles leads to the weak aggregation of particles in the low speed band area near the two wall, and the spatial distribution of particles is more uniform, which is positively correlated with the amount of charge carried by particles. At the same time, it is found that the electrostatic force attenuates the followability of particles to the fluid in the near wall region, and the electrostatic force is superior to the Stokes drag. Meanwhile, the uniform distribution of particles in the vertical channel improves the mean temperature of fluid and the mean streamwise velocity of the fluid. And it strengthens the kinetic energy exchange and the heat exchange between particles and fluid in the middle area of the vertical channel while weakens the kinetic energy exchange and the heat exchange between particles and turbulent flow near the wall.

, Available online , doi: 10.6052/0459-1879-23-163

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INFLUENCES OF KEY CONFIGURATION PARAMETERS ON FLOW-FOCUSING MICROFLUIDIC DROPLET GENERATION

Zhang Shuai, Wang Bo, Ma Zeyao, Chen Xiaodong

The flow-focusing droplet microfluidics achieves continuous generation of monodisperse microdroplets by means of flow-focusing effects and interfacial destabilization phenomenon of discrete-phase liquid filament. The multiphase interfacial flow in this technique exhibits dependence on configuration parameters and shows rich microfluidic device developed in our previous study, numerical simulations are used to investigate the influences of key configuration parameters on droplet generation modes and droplet dimensions. After reasonable simplifications, the study establishes an axisymmetric model of the actual device and combines the adaptive mesh refinement technique to improve the efficiency of the numerical simulation. The accuracy of the numerical simulation is verified through the comparison of several experimental operating conditions. It is found that within the selected fluid combination, geometry, and flow parameters, the droplet generation process exists in four modes: dripping, streaming, jetting, and unstable. Under the fixed discrete phase and continuous phase flow rate combinations, the variation of the distance between the upstream and downstream capillary ends changes the droplet length in the dripping and streaming modes, while it has little effect on the droplet size in the jetting mode. Under the fixed geometry parameters, when the flow rates vary, the change of droplet length is nearly continuous at the transition between dripping and streaming modes, but produces a sudden drop at the onset of the jetting mode. The internal diameter of the downstream capillary has a significant effect on the phase diagram, the dripping mode dominates for the large diameter internal diameter and the jet length changes more significantly in the jetting mode, while the jetting mode dominates for the small internal diameter and unstable modes are found at large continuous phase flows. The results of this paper show that the key configuration parameters have important effects on the flow-focusing microfluidic droplet generation, and the applicable alteration of these parameters can control the droplet size and improve the droplet monodispersity, which provides a basis for the design and optimization of flow-focusing microdroplet generation devices.

, Available online , doi: 10.6052/0459-1879-23-094

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THE LINEAR STABILITY ANALYSIS OF FLOW PAST A SPHERICAL BUBBLE UNDER THE EFFECT OF STREAMWISE MAGNETIC FIELD

Zheng Xiaolin, Pan Junhua, Ni Mingjiu

In electromagnetic metallurgy, argon is usually used as a power and carrier to blow desulfurizer and deoxidizer into liquid metal, so there is a problem of free movement of bubbles in liquid metal under a magnetic field environment. Flow past a fixed bubble as a special form of free movement, is the first step to study the problem of free movement. In this paper, the global linear stability analysis of the flow past a spherical bubble under the effect of a streamwise magnetic field is simulated by the finite element method. The response of the steady axisymmetric basic flow to the small perturbation of the independent time-azimuthal mode in the range of $\mathit{Re}\leqslant 1000,N\leqslant 60$ is discussed. Eight unstable stationary modes are found, and their neutral curves in the $ \mathit{Re}-N $ parameter plane or $ \mathit{Re}-Ha $ parameter plane are displayed. The results show that the stationary mode with azimuthal wave number m = 1 leads to the first regular bifurcation, this mode has been widely confirmed as the most unstable mode in the flow past axisymmetric objects, which transforms the axisymmetric wake into a plane symmetric wake composed of a pair of opposite vortices. In addition, the results of the neutral curve show the effect of the magnetic field on the instability of the flow past the spherical bubble. The subsequent bifurcations are successively caused by the unstable modes of m = 2, 3,..., 8, these bifurcations provide an important reference value for understanding the wake structure of the flow past a bubble in the magnetic field environment.

, Available online , doi: 10.6052/0459-1879-23-101

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ZONAL FINITE LINE METHOD AND ITS APPLICATIONS IN ANALYZING THERMAL STRESS OF COMPOSITE STRUCTURES

Liu HuaYu, Gao XiaoWei, Fan WeiLong

In this paper, we proposed a novel numerical method, Zonal Free Element Method (ZFLM), and used the proposed method to compute thermal stress in composite structures. ZFLM is a strong-form numerical method which solves the governing equations in differential form. For each node, we use two (two-dimension problems) or three (three-dimensions problems) lines to form a cross-line system. Then, we use Lagrange interpolating method to interpolate nodal coordinates and approximate the variables on each line. The gradients in the curvature direction are computed by the gradients of interpolating functions along the line. By a recursive procedure, the second or higher order of derivatives can be obtained by the expressions of the first order derivatives. Substituting the expressions of derivatives into the governing partial difference equations, we obtain the discretized linear system of equations. To solve the problem involving multiple kinds of composite structures efficiently, we use a zonal method. In the zonal method, we divide the computational domain into several regular zones by material types and geometric characteristics. We insert nodes in each zone by interpolating functions and use the finite line method to assemble the discretized governing equations at these nodes. For the nodes at the interfaces which are shared by two or more zones, the traction-equilibrium equations and the compatibility conditions of variables are used to construct the linear algebraic equations. For the irregular geometries and the nodes where the loads jump, we add up the traction-equilibrium equations of each neighbor faces in different directions to improve the robustness of the proposed method. We use the proposed method to solve several thermal stress problems in two- and three-dimension. The results of test cases indicate that the proposed method has good accuracy and a significant priority in problems involving stress concentration. Because the collocation method is used, the stress on the boundary is more accurate.

, Available online , doi: 10.6052/0459-1879-23-003

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ON THE MAGNETICALLY INDUCED ELECTROMECHANICAL COUPLING OF CENTROSYMMETRIC FLEXOELECTRIC SANDWICH PLATE

Guo Ziwen, Zhang Gongye, Mi Changwen

The development of modern industry inspires higher requirements for material properties and structural dimensions. The design of electromechanical devices is increasingly biased towards miniaturization, high frequency and intelligence. The most recent studies demonstrate that composite materials with magnetoelectric coupling can not only achieve mutual conversion of magnetic, mechanical, and electrical energy with high magnetoelectric conversion efficiencies, but can also avoid direct contact between the structure and the mechanical driving source to achieve non-contact control, which is crucial for the creation of multifunctional micro and nanoscale devices. Based on the multi-physics structural analysis framework developed by Mindlin, this paper studies the dynamic electromechanical coupling response of a sandwich plate composed of a flexoelectric dielectric layer and two symmetric piezomagnetic layers induced by external transverse magnetic fields. The macroscopic piezomagnetic and curvature-induced flexoelectric theories are employed and the classical electromechanical coupling theory is extended to centrosymmetric materials. The dynamic numerical examples of the sandwich plate driven by a sinusoidal global magnetic field and a uniformly distributed local magnetic field show that the magnitudes of displacement and potential are frequency dependent. When the excitation frequency reaches the natural frequency, the amplitude reaches the maximum. In addition, the distribution of symmetrical piezomagnetic layer tends to improve the electromechanical coupling performance of multilayer composite plates. Both the theoretical model and numerical results provide new ideas for the optimization design of magnetic-controlled electromechanical devices.

, Available online , doi: 10.6052/0459-1879-23-103

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EXPERIMENTAL STUDY ON DRAG REDUCTION CHARACTERISTICS OF POLYMER-SELF-POLISHING PAINT COMPOSITE COATING

Liu Hao, Xie Luo, Yao Boren, Sun Mengge, Hu Haibao

Based on the advantages of the drag reduction coating(easy engineering application)and the polymer(excellent turbulent drag reduction performance), the development of composite polymer drag reduction coating would have a broad application prospect in the field of ships and underwater vehicles, but there is still a relative lack of research in this field. Thus, the present work prepared domestic self-polishing paint composite coatings based on PAM and PEO drag reducing agents, and investigated the influence of coating types, flow velocity and PAM/PEO concentration on the drag reduction performance through the underwater plate resistance test, and surface roughness. The results showed that polymer-self-polishing paint composite coating has drag reduction effect, which can produce 9.4% drag reduction by using single-sided drag reduction coating; Flow velocity variations have a little influence on the drag reduction rate, and the induced variation of the drag reduction rate was smaller than 3.5%; When increasing the polymer concentration, the drag reduction rate of self-polishing paint-PAM coating first increased and then decreased, while the drag reduction rate of self-polishing paint-PEO coating first kept stable and then decreased rapidly; When the polymer concentration exceeded a certain critical value, the drag reduction rate decreased, and the coating even increased the flow drag. This is because the surface roughness of the coating increased and the viscosity of the solution changed when adding too many polymers. This work would provide supports for the engineering application of polymer drag reduction coating.

, Available online , doi: 10.6052/0459-1879-23-049

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PERIDYNAMIC OPERATOR METHOD

Li Zhiyuan, Huang Dan, Timon Rabczuk

A new method based on non-local theories, named as peridynamic operator method (PDOM), for solving ordinary and partial differential equations in physics, is proposed in the present work. By using the peridynamic operator method, the local differentiations of any orders as well as their products can be converted into corresponding nonlocal integral forms without any extra remedies or special treatment in the presence of discontinuities or singularities. It can be proved that both the so-called peridynamic differential operator (PDDO) and the nonlocal operator method (NOM), two nonlocal operators which have gained much concern of researchers in the field of computational mechanics in recent years, can be seen as special cases of this proposed PDOM. As a typical application example, linear elastic PDOM model for static and dynamic elasticity problems is developed by employing the variational principles and Lagrange's equations. Theoretical analysis shows that when the nonlocal interaction domain is defined by position-independent or dependent circles, PDOM elasticity model can be simplified to the classical peridynamic (PD) model or the dual-horizon peridynamic (DH-PD) model in literature correspondingly. The accuracy, convergence, as well as the numerical stability of the presented method are validated by analyzing three typical examples, including tension and wave motion in a bar, Helmholtz equation, and tensile deformation of plates with hole. It is shown that the proposed method can be effectively used with both uniform and non-uniform discretization, and it can naturally avoid the zero-energy modes and numerical oscillations which will occur when the original non-ordinary state-based peridynamic simulations, the peridynamic differential operator or the nonlocal operator method is employed without extra remedies. PDOM can provide a potential alternative to develop the nonlocal models for various physical problems especially involving discontinuities.

, Available online , doi: 10.6052/0459-1879-23-107

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STUDY ON THE INFLUENCE MECHANISM OF SEEPAGE FORCE ON THE STRESS FIELD AROUND OPEN HOLE WELLBORE AND FORMATION BREAKDOWN PRESSURE

Wang Haiyang, Zhou Desheng, Huang Hai, Li Ming

When fluid seepage enters the pore throat of reservoir rocks, it exerts a seepage force on the rock matrix, breaking the original stress balance state of the reservoir rock and affecting its deformation and failure. Although a large number of experiments and numerical simulation studies have confirmed the significant impact of seepage forces on rock failure, research on seepage forces in the field of petroleum engineering is scarce. The mechanism of seepage force on the initiation and expansion of hydraulic fracturing cracks remains unclear. Based on this, this paper first studied the mechanism of seepage force when fracturing fluid seeps into the rock pores, using the definition of geomechanical seepage force and Biot consolidation theory. Then, taking an open hole as an example, we analyzed the stress field formed by the seepage force, derived a formation breakdown pressure analytical solution formula considering the effect of seepage force, and revealed the influence of seepage force on the formation breakdown pressure of the open hole. The results show that when the pore pressure difference is equal to one atmospheric pressure, the volumetric seepage force per cubic centimeter of rock sample greatly exceeds the gravitational force on rock samples of the same size. Therefore, the effect of seepage force on reservoir rocks is significant and cannot be ignored. When fracturing fluid seeps into the pore throats of reservoir rocks, it exerts a seepage force on the rock matrix, which can significantly reduce the effective circumferential stress around the open hole and increase the likelihood of tensile failure on the well wall. The greater the Biot effective stress coefficient, the stronger the effect of seepage force, and the wider the range of the stress field around the wellbore affected by seepage force. Compared to impermeable reservoirs, seepage force substantially reduces the formation breakdown pressure of open hole. The impact of seepage force on the formation breakdown pressure of unconventional reservoirs with deeper depths and smaller bi-directional stress differences is more significant.

, Available online , doi: 10.6052/0459-1879-23-111

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MESHLESS ANALYSIS OF LINEAR BENDING AND FREE VIBRATION OF FUNCTIONALLY GRADED CARBON NANOTUBE-REINFORCED COMPOSITE PLATE ON ELASTIC FOUNDATION BASED ON IMPROVED REDDY TYPE THIRD-ORDER SHEAR DEFORMATION THEORY

Chen Wei, Fang Yaochu, Sun Bing, Peng LinXin

Based on the improved Reddy type third-order shear deformation theory (TSDT), and considering the orientation of carbon nanotubes (CNTs) and the inhomogeneity of functionally gradient materials, a meshless analysis model for the linear bending and free vibration of functionally graded carbon nanotube reinforced composite (FG-CNTRC) plates on elastic foundation is established. The potential energy and kinetic energy of FG-CNTRC plate are derived by the improved Reddy type TSDT, and then the expression of the potential energy of elastic foundation is given, and then they are respectively superposed. The linear bending and free vibration control equations of FG-CNTRC plate on elastic foundation are derived by the principle of minimum potential energy and Hamilton principle. Stable moving Kriging interpolation (SMKI) is used to discretize the nodes in the problem domain. The construction method of the approximate shape function satisfies the Kronecker condition and can directly apply the boundary conditions. In this paper, a meshless discrete model of linear bending and free vibration of FG-CNTRC plate on elastic foundation based on the third-order shear deformation theory is presented. Then, the effectiveness and accuracy of the proposed method are studied by a benchmark example. Finally, the effects of CNTs distribution, orientation angle, volume fraction, foundation coefficient, width thickness ratio and boundary conditions on the linear bending and natural frequency of FG-CNTRC plate are numerically analyzed. The results show that the proposed method has a good accuracy in calculating the linear bending and natural frequencies of FG-CNTRC thin, medium-thick, and even thick plates. As the volume fraction of CNTs and the foundation coefficient increase, the stiffness of the FG-CNTRC plate structure gradually increases. The stiffness of the FG-CTRC plate structure is positively correlated with the width-thickness ratio, and the shear effect of increasing thickness gradually reduces the influence of the CNTs orientation angle on the plate stiffness.

, Available online , doi: 10.6052/0459-1879-23-040

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BANDGAP OPTIMIZATION DESIGN OF PHONONIC CRYSTALS BASED ON SHAPE MEMORY ALLOY

Qiu Kepeng, Chen Zhimou, Zhang Jiangang, Zhang Weihong, Yan Qun, Sun Xiangyang, Peng Tao

Phononic crystal is a kind of periodic structures with the phononic band gap. The dynamically controllable design of its band gap could improve the vibration and noise reduction performance of major equipment in the aerospace field. In the work, smart materials are introduced for band gap design of phononic crystals. And the topological optimization method is used to design the multifunctional phononic crystal with the dynamically controllable band gaps. Firstly, the band gap of phononic crystals are computed by finite element analysis. Simultaneously, the temperature constitutive model of shape memory alloy is established. Secondly, based on variable density method, topology optimization model is established with maximizing the relative band gap under the specific volume ratio and strength constraints. At the same time, the connectivity constraints among phononic crystal unit cells must be ensured. Lastly, the band gaps of multifunctional phononic crystals are optimized by using the moving asymptotic method. During the optimization process, the design sensitivities are calculated with the improved material interpolation model. The optimization results show that the band gap is widened by 103.9% in XY mode with the transformation of shape memory alloy from martensite to austenite. And the bandwidth is increased by 3.75 times in Z mode. This research provides an effective design way for more actively control the phononic crystals band gaps in the complex application environments. And the novel phononic crystals have a wider application prospect.

, Available online , doi: 10.6052/0459-1879-23-024

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NUMERICAL SIMULATION OF PROPELLER TIP VORTEX CAVITATION INCEPTION CONSIDERING THE EFFECT OF NUCLEI GROWTH AND COLLAPSE

Wang Yongshuai, Wang Xincheng, Cheng Huaiyu, Ji Bin

Tip vortex cavitation (TVC) is the earliest type of cavitation that occurs on propellers, and it significantly enhances the underwater radiated noise level of ships once it happens. Therefore, TVC inception prediction is vital for cavitation inception speed determination and has attracted much attention from experts and scholars in the ship field. The explosive growth of microscopic nuclei under the action of low pressure of vortex core is an important mechanism for tip vortex cavitation inception, while the conventional macroscopic cavitation models in Eulerian framework use empirical parameters to model the influence of microscopic nuclei and cannot accurately simulate this process, which affects the accurate prediction of propeller cavitation inception. To overcome the limitations of traditional simulation methods, this paper develops and applies an Eulerian-Lagrange cavitation inception numerical method based on bubble dynamics and water phase compressibility to simulate TVC inception. Comparison with experimental results shows that this model can accurately predict propeller TVC inception. This paper not only investigates the effects of different incoming nuclei sizes on cavitation inception from a microbubble perspective, but also studies the significant influences of tip vortex flow characteristics on nuclei evolution, and thus reveals the sound generation mechanism of cavitation inception in propeller tip vortex flow field. Under optical criterion for cavitation inception, larger-sized gas nuclei are more likely to be captured by tip vortices and grow explosively. Nuclei gradually approach the vortex core low-pressure region under tip vortex suction. Nuclei grow explosively under continuous low-pressure action at the vortex core, and rapidly contract and collapse rapidly after reaching maximum size, producing strong acoustic pressure pulse.

, Available online , doi: 10.6052/0459-1879-23-080

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STUDY ON FRACTURE CHARACTERISTICS DIFFERENCE BETWEEN FRACTURING FLUID FLOWBACK AND GAS PRODUCTION STAGES OF SHALE GAS WELLS

Liu Wenchao, Qiao Chengcheng, Wang Ping, Huang Wensong, Liu Yuewu, Ding Wei, Sun Yuping

Horizontal well staged fracturing is the key technology to realize the economic development of shale gas. The closure of fracturing fractures in the production process will have adverse effects on the exploitation. Due to the large errors and serious oscillations of dynamic production data, it does not match the internal boundary conditions of the seepage flow mathematical model. Therefore, there are few quantitative methods to evaluate the difference of fracture characteristics between fracturing fluid flowback stage and shale gas production stage based on dynamic production data analysis currently. Based on this concern, a new method of production dynamic data analysis based on deconvolution is proposed to quantitatively evaluate the difference of fracture characteristics between flowback stage and production stage in this paper. Firstly, the seepage flow models and their Laplace solutions corresponding to flowback stage and production stage are given. Secondly, the pressure deconvolution algorithm is used to normalize the dynamic production data of the two stages. Then, the normalization parameter adjustment of deconvolution calculation and the parameter adjustment of theoretical seepage flow model calculation are mutually restricted in the process of typical curve fitting, and the fracture half-length and fracture conductivity of the two stages are interpreted respectively. Finally, the conductivity modulus is introduced to quantitatively evaluate the difference of fracture characteristics between flowback stage and production stage. The established method is used to analyze 10 wells in the field. The results show that this method can effectively quantify the difference of fracturing fracture characteristics between flowback stage and production stage; compared with the flowback stage, the fracture conductivity decreased by about two orders of magnitude in the production stage, and the fracture closed significantly. The analysis method established in this paper has important reference value for the optimization of stimulation measures in the later stage of shale gas reservoir.

, Available online , doi: 10.6052/0459-1879-23-031

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RESIDUAL STRESS AND HIGH-TEMPERATURE MECHANICAL BEHAVIOR OF THERMAL BARRIER COATED TURBINE BLADES

Hao Yiyi, Liang Lihong, Qiu Tian

Thermal barrier coated turbine blades can effectively improve the thermal efficiency and performance of aero-engines. They exert significant importance on security and stability of aero-engines. In the process of thermal shock service, the thermal barrier coating system is prone to various forms of damage such as surface cracks and interface cracks, which seriously affects the service stability of turbine blades. Considering that the residual stress generated during the preparation of turbine blades with thermal barrier coating will have a great impact on the quality of thermal barrier coating, this work firstly studied the residual deformation and stress during the natural convection cooling process after the thermal barrier coating was deposited into turbine blades with certain shape by using the finite element method. Furthermore, the temperature and stress state of turbine blades with thermal barrier coating under high temperature thermal shock were simulated and analyzed, and the stress mechanism of mechanical behavior difference between blade with thermal barrier coating and alloy blade without thermal barrier coating under high temperature was revealed. The results show that the distribution of deformation and residual stress after the preparation of thermal barrier coating blade is complex due to the geometrical structure of the curvature blade, and the maximum local compressive stress at the blade root is close to 200 MPa. The thermal barrier coating can provide obvious thermal protection for the blade under high temperature service, and the maximum Mises stress can be reduced 600 MPa, but the thermal protection effect in the trailing edge area is limited. The maximum principal stress in the suction surface near the trailing edge of the ceramic coated blade root reaches 159.5 MPa. Therefore, the thermal barrier coating turbine blade in high temperature service will preferentially show higher stress in the blade root and trailing edge of the ceramic layer, which becomes the starting position of crack initiation, propagation and spalling.

, Available online , doi: 10.6052/0459-1879-23-021

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CONSTITUTIVE MODELLING OF SILICONE ADHESIVE CONSIDERING MULLINS EFFECT

Guo Xin, Chen Suwen

Silicone adhesive has been widely used in assembly glass curtain walls. To achieve a reliable bonding system, an effective description of material behavior is required. However, commonly used phenomenological hyperelastic models have not considered the microstructure properties of materials and cannot describe the mechanisms of their mechanical behaviors, while the classical entropic hyperelastic models often do not consider the non-affine deformation, entanglement effect or other features of polymer network. The above deficiencies make it difficult for the existing models to effectively simulate the mechanical behavior of silicone adhesive, especially the significant Mullins effect under cyclic loading. For these reasons, in this paper, based on the non-affine network model and microsphere model of polymer chain distribution, we modified the macro-micro deformation transformation and the evolution of chain conformation to consider spatial distribution of polymer chains in finite directions. Based on the modified model, network alteration functions are proposed for crosslinked and entangled network respectively using network alteration theory. These functions describe the evolution of polymer network under cyclic loading to model Mullins effect. Considering the microstructure properties and deformation mechanisms of silicone adhesive, the modified non-affine network model can describe the characteristics of polymer network, including non-affine deformation, entanglement effect, finite chain extensibility and spatial chain distribution. The comparisons with the experimental data and other model results demonstrate the capability of the modified model to accurately predict the mechanical behavior of silicone adhesive under various loading conditions, as well as the permanent set and modulus degradation of Mullins effect, which shows good potential in engineering applications.

, Available online , doi: 10.6052/0459-1879-23-035

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AN EFFICIENT FE COMPRESSION METHOD FOR GENERATING THE RVES OF COMPOSITES

Tian Wenlong, Qi Lehua, Chao Xujiang

This work proposes an finite element (FE) compression method to establish periodic representative volume elements (RVEs) of composites with high inclusion volume fractions efficiently and simply. The main procedures of the proposed FE compression method are given as follows: (1) Generation of the RVEs of composites with periodic and sparse inclusions using the random sequential absorption (RSA) algorithm, (2) FE compression of the generated periodic and sparse RVEs in step-1 to obtain the RVEs of composites with periodic and packed inclusions (in the FE mesh format) under the constrain of a periodic boundary condition, and (3) postprocessing to obtain the centroids (and orientation) of all the inclusions in the compressed RVEs with periodic and packed inclusions and generate the periodic RVEs of composites in the CAD format. Based on the proposed FE compression method, the periodic RVEs of spherical inclusions composites with the inclusion volume fraction up to 50.0% are generated. The distribution of the spherical inclusions in the generated periodic RVEs of composites is analyzed using the probability distribution function of nearest neighbor distance, the cumulative probability distribution function of nearest neighbor orientation angle, the Ripleys-K function and the pair correlation function, and the results show that the distribution of the inclusions in the generated periodic RVEs of composites is completely spatial and random. The elastic properties of different types of composites are homogenized using the FE homogenization method based on the generated periodic RVEs, and are then compared with those of the double-inclusion model and available experimental tests. It is observed that the elastic properties of the studied composites obtained using the FE homogenization method based on the generated periodic RVEs, the experimental tests and the double-inclusion model agree well, and it thus concludes that the proposed FE compression method is capable of generating the RVEs of composites with high inclusion volume fractions.

, Available online , doi: 10.6052/0459-1879-23-061

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THE MIXING AND SEGREGATION OF BINARY PARTICLES TRANSPORTATION IN VERTICAL PIPE

Zhang Yan, Ren Wanlong, Zhang Xuhui, Lu Xiaobing

The development of deep-sea resources has attracted the attention of various countries in recent years where the mineral resources is an important part. This paper considers the internal flow in hydraulic conveying during the deep-sea mining, which is characterized by wide particle gradation and high particle volume concentration. The wide particle gradation will lead to the particle mixing and segregation, which may result in high local particle concentration. The mixing and segregation of binary particles transportation in vertical pipe is investigated based on the computational fluid dynamics-discrete element method (CFD-DEM). A virtual mass distribution function method is proposed for calculating the coarse particle volume fraction field. In addition, a weighted function method relating the particle size is given for the interpolation between the Eulerian and Lagrangian field. The two models are implanted in the open source code CFDEM based on the based on the C++ programming language. Then, the numerical method is verified by comparing the pressure drop and the minimum fluidization velocity of a fluidized bed case between the simulation results and analytical results. The study found that the mixing and segregation of binary particles will cause a gap between the front mixing area and the no-mixing area at the rear. The local particle concentration and the particle collision frequency increase also increased significantly. The particle collision stress and fluid-particle interaction stress are given, which are the ratios of unit particle collision force and unit fluid drag force to unit particle buoyant force, respectively, to explain the particle segregation mechanism. The particle mixing stage makes the particle collision stress increasing. Therefore, the moment from initial mixing to complete separation can be determined by the particle collision stress curve. In addition, the difference of fluid-particle interaction stress between the binary particle results the particle segregation because the fluid-solid interaction stress of fine particles is always greater than that of the coarse particles.

, Available online , doi: 10.6052/0459-1879-23-020

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THREE-DIMENSIONAL ANALYTICAL METHOD FOR CALCULATING VIBRATIONS OF A COUPLED TUNNEL-SOIL-FLUID SYSTEM

He Chao, Jia Yuanping, Zhou Shunhua

Train-induced vibrations from underground railway tunnels transmit to the ground surface or water, which may disturb adjacent residents or rare fish. This study presents a three-dimensional analytical method for calculating the dynamic response of the coupled tunnel-soil-fluid system. The twin tunnels and soils are simulated as the elastic solid, while the air or water is simulated as the ideal fluid medium. The three-dimensional dynamic problem in the time-space domain is transformed into the frequency-wavenumber domain by using the double Fourier transforms of time and longitudinal coordinates. The boundary conditions on the fluid-soil interface and the tunnel-soil interface are satisfied by introducing the transformation between the cylindrical waves and the plane waves and the transform between different cylindrical waves. The solution for dynamic response of the coupled tunnel-soil-fluid system under point loads is therefore derived. The accuracy of the proposed method is verified by comparing with the Green’s function and FE-BE model for a coupled soil-fluid system. The dynamic responses of soil, water, and air induced by point loads in a single tunnel and twin tunnels are compared and analyzed via two numerical cases. The results demonstrate that the existence of the adjacent tunnel can change the energy distribution of train-induced waves in the soil, thus affecting the propagation characteristics of waves in the water or in the air. The influence of the adjacent tunnel is highly dependent of the loading frequency, the observation position and the relative position between twin tunnels. When the distance between the two adjacent tunnels is less than four times of the diameter of the tunnel, the dynamic interaction between the twin tunnels play a relevant role in the response of the water or the air. This study can provide benefit for the evaluation of vibrations and radiated noise in air or water from underground railway tunnels.

, Available online , doi: 10.6052/0459-1879-22-543

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STOCHASTIC DYNAMIC RESPONSE ANALYSIS OF PIPE CONVEYING FLUID BASED ON THE PATH INTEGRAL METHOD

Sun Yibo, Wei Sha, Ding Hu, Chen Liqun

The study of pipes conveying fluid under stochastic excitation is of great importance as they are widely used in engineering. To predict the stochastic dynamic response of pipe conveying fluid system under Gaussian white noise excitation, a dynamic model of the nonlinear pipe conveying fluid under Gaussian white noise excitation is established based on the Hamilton’s principle. The Galerkin truncation method is employed to discretize the governing equation of pipes conveying fluid. The probability density function of the displacement and the probability density function of the velocity of the pipe conveying fluid are calculated by the path integral method based on the Gauss-Legendre formula. The results of the Monte Carlo method are compared with the results obtained by the path integral method to verify the accuracy of the path integral method in the calculation of the vibration response of the pipe conveying fluid. The effects of system parameters such as fluid speed, excitation strength and damping coefficient on the probability density function of the displacement and the probability density function of the velocity of the pipe conveying fluid are investigated. The critical fluid speed when the probability density function of displacement for the pipe conveying fluid has a double peak is determined. The results show that the path integral method is effective in calculating the response of the pipe conveying fluid system. The maximum possible displacement of the system will increase and the maximum possible speed will remain unchanged with the increase of fluid speed. The maximum possible displacement and the maximum possible speed of the system will increase with the increase of excitation strength. Increasing the damping coefficient results in that the maximum possible displacement and the maximum possible speed of the system decreases. In addition, it is found that the increase of fluid speed is one of the factors inducing stochastic bifurcation of the pipe conveying fluid.

, Available online , doi: 10.6052/0459-1879-23-032

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OPPOSING PLASMA SYNTHETIC JET FOR LOW-SPEED FLOW SEPARATION INHIBITION

Sun Zhikun, Shi Zhiwei, Li Zheng, Geng Xi, Zhang Weilin

The plasma synthetic jet is a comprehensive high-energy excitation with a strong ability to restrain flow separation. This paper uses experimental and numerical simulation methods to investigate the inhibition of flow separation by plasma synthetic jets on a low-speed airfoil. The actuator's electrodes are built into the wing, and the injection holes are located at the leading-edge point. The smoke particle concentration distribution and the numerical simulation results show that the opposing plasma synthetic jet can move the flow separation point of the low-speed airfoil and improve the lift characteristics of the airfoil. The ability of the opposing plasma synthetic jet to drive the separation point distance and enhance the airfoil's lift characteristics will increase with the angle of attack. At an angle of attack of 16°, plasma actuation pushes the distance of the airfoil flow separation point to about 16.5% of the chord length, increasing the airfoil's lift coefficient by about 17.3%. The results show that in the low-velocity flow, the thermal jet generated by the opposing plasma synthetic jet interacts with the mainstream to form a strip-like thermal structure. The strip-shaped thermal structure has a leading mixing effect, which can enhance the mixing of the mainstream and the fluid in the separated shear layer. The jet body has a mixing-inducing impact, which can induce the dynamic re-attachment of the separated shear layer. The interaction between the strip-like thermal structure and the mainstream and between the jet body and the mainstream are the primary mechanisms for the opposing plasma synthetic jet to inhibit the flow separation of the low-speed airfoil and improve the lift characteristics of the airfoil. The strip-like thermal structure acts differently from the jet body at different stages. The difference leads to the change in their coupling and makes the lift characteristics of the airfoil appear in five typical phases. In addition, the experimental results also show that when the discharge parameters are constant, the flow control effect of the serial array actuator is more potent than that of a single actuator.

, Available online , doi: 10.6052/0459-1879-23-005

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DIRECT COLLOCATION METHOD AND STABILIZED COLLOCATION METHOD BASED ON LAGRANGE INTERPOLATION FUNCTION

Hu Minghao, Wang Lihua

Since most of the approximation functions in the meshfree method are rational and do not have the Kronecker delta property, it is difficult to accurately impose the essential boundary conditions. Large errors on the boundary can easily lead to low accuracy of the solution in the whole solution domain and may even introduce the numerical instability in solution process. In this paper, the Lagrange interpolation function is introduced as the shape function in the meshfree direct collocation method and the stabilized collocation method, and the Lagrange interpolation collocation method (LICM) and the stabilized Lagrange interpolation collocation method (SLICM) are constructed. Since Lagrange interpolation has the Kronecker delta property, the essential boundary conditions can be imposed as simply and precisely as the finite element method, which promotes the numerical solution accuracy of the two methods. The stabilized collocation method is based on the subdomain integration, which can satisfy the high order integration constraints. That is, it can ensure that the shape function also meets the high-order consistency conditions in the integral form and achieve accurate integration. At the same time, the subdomain integration can also reduce the condition number of the discrete matrix, which improves the stability of the algorithm. By combining the Lagrange interpolation function and the stabilized collocation method, the accuracy and stability of the stabilized Lagrange interpolation collocation method is further improved. Numerical examples validate the accuracy, convergence and stability of the proposed Lagrange interpolation collocation method (LICM) and the stabilized Lagrange interpolation collocation method (SLICM). The results show that the accuracy of the collocation methods based on the Lagrange interpolation function is higher than that of the collocation method based on the reproducing kernel function, and the accuracy and stability of the stabilized Lagrange interpolation collocation method are superior to those of the Lagrange interpolation collocation method.

, Available online , doi: 10.6052/0459-1879-23-001

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DYNAMIC STABILITY AND INFLUENCE FACTORS OF THE WHOLE TRAJECTORY OF FIXED CANARD DUAL-SPIN PROJECTILES

Zhao Xinxin, Shi Jinguang, Wang Zhongyuan, Zhang Ning

To research the dynamic stability of fixed canard dual-spin projectiles in full trajectory flight, the state space model of the complex attack angle motion is established under the condition of small attack angle, and the general condition that the real parts of the characteristic roots are all negative is derived by using the Hurwitz method. Based on the motion characteristics of the front body’s rolling angle before/after starting control, the dynamic stability criterion of fixed canard dual-spin projectiles under different flight conditions is proposed by using the stability analysis method of the conventional rotating projectiles, whose form is similar to that of the conventional rotating projectiles. When flying without control, the control force and moment terms of control canard are added correspondingly in the lift force and static moment terms. Controlled flight further increases the relative increment effect of relevant terms. Accordingly, the constraint on canard parameters of control surface is deduced under the condition that the parameters of projectile body are determined, the effects of the control force coefficient’s derivative, the installation position and the deflection angle of control canard on the dynamic stability are discussed, and the reasons for the formation of the dynamic instability of this kind of projectile are revealed. The simulation analysis results of the complex attack angle motion under different conditions show that when the relative increment caused by control surface is within the boundary of both uncontrolled and controlled flight, the full trajectory flight of the fixed canard dual-spin projectile is dynamically stable, which verifies that the dynamic stability criterion and the constraint on canard parameters deduced in this paper are reasonable and feasible, and provides a theoretical basis and design reference for the design and development of this type of projectile.

, Available online , doi: 10.6052/0459-1879-21-636

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