应用数学和力学

2024 Vol. 45, No. 9

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Special Topic of China Congress on Computational Mechanics
Fluid Mechanics
Solid Mechanics

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Special Topic of China Congress on Computational Mechanics

Analytical Forced Vibration Solutions of Orthotropic Cantilever Rectangular Thin Plates With the Symplectic Superposition Method

WANG Senlin, LI Jinbao, MA Hongyan, LI Rui

2024, 45(9): 1117-1132. doi: 10.21656/1000-0887.440277

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Abstract:
The forced vibrations of orthotropic cantilever rectangular thin plates under harmonic loadings were investigated with the symplectic superposition method. The basic equations for the forced vibration of thin plates were introduced into the Hamiltonian system. The original problem was divided into some fundamental subproblems, and the analytical solutions of the subproblems were derived with the method of separation of variables and through eigenvector expansion in the symplectic space. The solution of the original problem was finally obtained by superposition. The main advantage of the symplectic superposition method is that the analytical solution can be obtained by step-by-step rigorous derivation, without any assumptions on the form of the solution, which breaks through the limitations of traditional semi-inverse methods. The numerical results calculated corresponding to different harmonic loads were compared with those obtained via the finite element method to verify the reliability and accuracy of the proposed method.

Research on the Dynamic Contact Angle Model for the Droplet Impact Process

WANG Xiangyu, KE Peng, DU Feng

2024, 45(9): 1133-1146. doi: 10.21656/1000-0887.440282

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Abstract:
The simulation of droplet-wall impact process based on computational fluid dynamics (CFD) is of great significance for understanding the dynamic behavior of droplets spreading on the solid wall, and can provide technical support for the design of superhydrophobic structures and the development of anti-icing coating. The difficulty lies in how to accurately describe the evolution process of the contact line and the dynamic contact angle in the model. Herein, 4 typical dynamic contact angle models were summarized, and their application ranges were analyzed theoretically. With the UDF function in FLUENT the dynamic contact angle model was applied to the wall boundary conditions, and the dynamic process of droplet impact on smooth wall was numerically simulated. The quantitative analysis of the changes of droplet shape parameters and the comparison with the experimental results show that, the Seebergh dynamic contact angle model is more suitable for simulating the motion of droplets with lower capillary numbers. The Kistler model and the Jiang model are more widely used and can accurately describe the motions of droplets with higher capillary numbers. Then, based on the Kistler dynamic contact angle model, the impact and spreading processes of droplets on the microstructure surface were simulated. It is found that, the application of the dynamic contact angle model will lead to the change of the internal flow fields of droplets with the surface tension playing a dominant role, and the simulated droplet contact angle value in equilibrium is close to the theoretical value.

Shell Structure Analysis Based on the Convected Particle Domain Interpolation

WANG Changsheng, YU Chuanze, ZHANG Xiangkui

2024, 45(9): 1147-1156. doi: 10.21656/1000-0887.440286

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Abstract:
The material point method (MPM) adopts the dual description of Lagrangian particles and Euler grids, so it can deal with large deformation and contact problems conveniently. The large deformation problem of thin shell structures was analyzed based on the framework of the convected particle domain interpolation material point method (CPDIMPM). The quadrilateral mesh was used to discretize the shell structure. The basis function was calculated by the double mapping from the material point to the shell element node and then to the background grid node. The momentum equation was solved on the background grid, and the internal force of the material point was updated based on the Belytschko-Tsay (BT) shell element theory. In the numerical example, the comparison of large deformations of the shell structure with reference solutions verifies the accuracy of the proposed method.

Hamiltonian System-Based Analytical Solutions to Free Vibration Problems of Functionally Graded Rectangular Plates

ZHANG Jichao, ZHONG Xinyu, CHEN Yiming, SHI Yueqing, GUO Chengjie, LI Rui

2024, 45(9): 1157-1171. doi: 10.21656/1000-0887.440279

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Abstract:
Finding vibration mode functions satisfying both high-order partial differential governing equations and various non-Lévy-type boundary conditions is extremely challenging for the free vibration problems of functionally graded rectangular plates, making it difficult to analytically solve the problem with traditional methods. Herein, the newly developed Hamiltonian system-based symplectic superposition method was extended and successfully applied to analytical solutions to the free vibration problems of functionally graded rectangular plates. For the solution methodology, the original vibration problem was divided into sub-problems and the physical neutral plane was introduced to eliminate the stretching-bending coupling effect caused by the transversely non-uniform materials. The sub-problems were analytically solved with some mathematical techniques, i.e., the variable separation and the symplectic eigenvector expansion, which are not applicable in the traditional Lagrangian system. The final solution to an original vibration problem was obtained through the superposition of sub-problems. The symplectic superposition method has the advantage of not requiring the pre-defined solution forms, which overcomes the limitations of traditional semi-inverse methods and allows for obtaining analytical solutions to more complex problems. Comparison of the obtained solutions with the numerical solutions proves the accuracy of the presented method. On this basis, quantitative parameter analyses on the natural frequencies were conducted to reveal the effects of boundary conditions, material distributions and aspect ratios.

Study on Mechanical Properties of Barbed Contact Metamaterials

TIAN Gengxin, CAO Shenghu, ZHANG Jian

2024, 45(9): 1172-1181. doi: 10.21656/1000-0887.440285

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Abstract:
Inspired by the differences in the barbed structures of some plant stems and cat tongues in different directions, a reusable and easy-to-recover barbed metamaterial was designed, with its mechanical properties analyzed theoretically and numerically. The results show that, in the reciprocating motion of the barbs, when the barb rectangular section sizes are 1 mm × 1 mm, the length is 20 mm, and the angle with the vertical direction is 60°, the maximum reaction force in the forward contact process with the blocking rib will be about 20 times of the maximum reaction force in the reverse contact process, and the former energy consumption is about 200 times of the latter. With the decrease of the angle between the barb and the vertical direction, the barb structure will exhibit a higher energy absorption capacity and a reduced energy requirement for recovery. Similarly, with the increase of the barb length, the structure energy absorption will be lower and the energy required for recovery will reduce. These characteristics indicate that, the structure possesses excellent impact resistance and energy absorption capacity. Structures with significant energy disparities between forward and backward motions are more easily recoverable, and the energy absorption efficiency can be enhanced by carefully designing the angles and lengths of the barbs.

A Buckling Analysis Method for Composite Panels in Multiweb Box Structures Based on Elastic Boundaries

ZHAO Bei, XIONG Sijun, CHEN Liang, WANG Chengbo, LI Rui

2024, 45(9): 1182-1199. doi: 10.21656/1000-0887.440283

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Abstract:
The multiweb box structures in the wings are paid special attention in aircrafts' structural design. The multiweb box is mainly composed of skins and stiffeners. The skins are approximately divided into many rectangular panels by stiffeners. During the service of an aircraft, the wing majorly bears bending, torsion, and bending-torsion coupling loads, etc., so the panels in box structures are susceptible to instability. In traditional buckling analysis of composite panels, the boundary conditions were typically simplified as either clamped or simply supported boundaries, with significant deviations from experimental results. On the other hand, comprehensive simulations with the finite element method are generally inefficient. Aimed at the above issues, a rapid buckling analysis method combining the unit cell model with the differential quadrature method for composite panels was proposed. Firstly, the unit cell model was established to calculate the stiffness coefficients of elastic boundaries of rectangular panels. The governing equations were then solved with the differential quadrature method to obtain the buckling loads on the panels. Finally, the buckling loads on composite panels in different types of box structures were calculated and compared to the results obtained with the finite element method to verify the accuracy of the presented buckling analysis method.

Fluid Mechanics

Schur Forms and Normal-Nilpotent Decompositions

LI Zhen

2024, 45(9): 1200-1211. doi: 10.21656/1000-0887.450129

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Abstract:
Real and complex Schur forms have been receiving increasing attention from the fluid mechanics community recently, especially related to vortices and turbulence. Several decompositions of the velocity gradient tensor, such as the triple decomposition of motion (TDM) and normal-nilpotent decomposition (NND), have been proposed to analyze the local motions of fluid elements. However, due to the existence of different types and non-uniqueness of Schur forms, as well as various possible definitions of NNDs, confusion has spread widely and is harming the research. This work aims to clean up this confusion. To this end, the complex and real Schur forms are derived constructively from the very basics, with special consideration for their non-uniqueness. Conditions of uniqueness are proposed. After a general discussion of normality and nilpotency, a complex NND and several real NNDs as well as normal-nonnormal decompositions are constructed, with a brief comparison of complex and real decompositions. Based on that, several confusing points are clarified, such as the distinction between NND and TDM, and the intrinsic gap between complex and real NNDs. Besides, the author proposes to extend the real block Schur form and its corresponding NNDs for the complex eigenvalue case to the real eigenvalue case. But their justification is left to further investigations.

Curriculum-Transfer-Learning-Based Physics-Informed Neural Networks for Simulating Long-Term-Evolution Convection-Diffusion Behaviors on Curved Surfaces

MIN Jian, FU Zhoujia, GUO Yuan

2024, 45(9): 1212-1223. doi: 10.21656/1000-0887.440320

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Abstract:
Physics-informed neural networks (PINNs) encode prior physical knowledge into neural networks, alleviating the need for extensive data volume within the network. However, for long-term problems involving time-dependent partial differential equations, the traditional PINN exhibits poor stability and struggles to obtain effective solutions. To address this challenge, a novel physics-informed neural network based on curriculum learning and transfer learning (CTL-PINN) was introduced. The main idea of this method is to transform the problem of long-term course simulation into multiple short-term course simulation problems within this time domain. Under the concept of curriculum learning, and step by step from simpleness to difficulty, the scope of the time domain to be solved was gradually expanded by training the PINN within small time quanta. Furthermore, the transfer learning method was adopted to transfer across the time domain based on the curriculum learning, and the PINN was gradually employed for solution, thus to achieve long-term simulation of convection-diffusion behaviors on curved surfaces. The CTL-PINN was combined with the extrinsic surface operator processing technology to simulate long-term convection-diffusion behaviors on complex surfaces, and the effectiveness and robustness of the improved physics-informed neural network were verified through multiple numerical examples.

Solid Mechanics

Residual Axial Deformation and Buckling Analysis of Thin-Walled Carbon Fiber Fully Wound Composite Gas Cylinders

ZHAO Liang, LI Yawei, WANG Jianbao, LI Dongdong, DING Shuai, BI Qingjie

2024, 45(9): 1224-1234. doi: 10.21656/1000-0887.450086

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Abstract:
Experimental research and finite element analysis were carried out to study the phenomenon of axial shortening and buckling instability of carbon fiber fully wound composite gas cylinders with titanium alloy thin-wall liners after hydraulic test. The results show that, after self-tightening and unloading, the area near the polar hole of the head will be concave axially, the area near the equator of the head will expand radially, and the whole head will become shorter axially. The axial shortening of the head will be 6.15 mm and 6.363 mm, respectively, and the error of the finite element calculation will be 3.46%. The finite element simulation results are in good agreement with the experimental results. Finally, the multi-pole hole method was used to optimize the thickness distribution of the fiber layer of the head, and the extreme thickness of the head was reduced by 32.6%, and the transition was made smoother. After the optimization, the gas cylinder will be slightly extended along the axis, with an average elongation of 0.6 mm. By CT and endoscope detections, no buckling instability would appear in the liner, which means an effective solution of the problems of axial shortening and liner buckling after hydraulic tests.

Stochastic Responses and Stability Analysis of Vibro-Impact Systems With Friction Under Wideband Noise Excitation

MA Lan, TIAN Lili, LIU Li

2024, 45(9): 1235-1242. doi: 10.21656/1000-0887.440313

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Abstract:
The stochastic responses and the asymptotic stability with probability 1 of vibro-impact systems with friction under wideband noise excitation were investigated. The Zhuravlev non-smooth transformation and the stochastic averaging method were extended to obtain the steady-state probability density functions of the system. The accuracy of the method was verified through comparison of the theoretical results with those from the Monte Carlo simulations. The effects of the friction force and the vibro-impact restitution coefficient on the system responses were studied. Furthermore, The Lyapunov exponent of the linearized averaged Itô equation was derived and the stability of the trivial solution was determined with the Lyapunov exponent. The results show that, changing the frictional coefficient and the vibro-impact restitution coefficient could adjust the system stochastic stability.