应用数学和力学

2023 Vol. 44, No. 4

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Solid Mechanics
Applied Mathematics

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Solid Mechanics

Numerical Study of Nonlinear Scattering Characteristics of SH₀ Waves Encountering Cracks in Prestressed Plates

CHEN Huijian, ZHU Qingfeng, MIAO Hongchen, FENG Zhiqiang

2023, 44(4): 367-380. doi: 10.21656/1000-0887.440029

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Abstract:
Ultrasonic guided waves are widely used in structural health monitoring (SHM) for their long propagation distances and small energy attenuation. Understanding the scattering characteristics of guided waves encountering defects is essential for the design of transducer arrays and wave signal interpretation in SHM. The contact nonlinear scattering characteristics of the SH₀ wave (zero-order shear horizontal wave) encountering cracks in prestressed plates were investigated. Based on the previously developed bi-potential spectral method, the spectral finite elements (SFEs) and the finite elements (FEs) were further coupled with the mortar method to make full use of the high efficiency of the spectral element method in calculating guided wave propagation and the strong ability of the finite element method in discretizing complex structures. The nonlinear scattering fields of SH₀ waves interacting with microcracks at different angles in plates under free and loaded conditions were calculated with the developed numerical method. The results show that, the induced 2nd harmonic scattering field is approximately symmetrical with respect to the crack direction. Moreover, the existence of uniaxial prestress will not change the symmetry of the 2nd harmonic scattering field, so the orientation of the microcrack can still be determined by the distribution of the scattering field.

Surface Acoustic Wave Characterization of Equivalent Young's Moduli for Patterned Films

CHEN Long, XIAO Xia, ZHANG Li, QI Haiyang

2023, 44(4): 381-393. doi: 10.21656/1000-0887.430050

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Abstract:
Based on the equivalent elasticity theory for layered materials, the micro-mechanics equivalent models for single and dual damascene structures were established. The equivalent elastic constant of the patterned structure was introduced, to establish the propagation model for the surface acoustic waves propagating in the layered structure of the patterned film/substrate, and the theoretical dispersion curves of the surface acoustic waves were calculated with Green's function and the matrix method. The finite element method was used to calculate 24 numerical examples of damascene structures with different volume ratios, and the results were compared with those of the strain energy method. The results show that, the average relative errors of the equivalent Young's moduli of the 300 nm-thick dual damascene film and the 100 nm-thick single damascene film are 2.06% and 2.27%, respectively. The research verifies the correctness of the equivalent patterned structure model and the feasibility of the surface acoustic wave method to characterize the mechanical properties of patterned films, and provides a reference for the development of suitable chemico-mechanical polishing technologies for patterned films under low pressure.

Blast Damage Simulation With the Discontinuous Galerkin Finite Element Method of Bond-Based Peridynamics

CHENG Jiahe, GU Xin, ZHANG Qing

2023, 44(4): 394-405. doi: 10.21656/1000-0887.430338

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Abstract:
The peridynamics (PD) as an integral non-local continuum mechanics theory, is widely used in the discontinuous deformation and failure analyses of solid materials and structures, mainly with the explicit dynamic solution method for meshless particles. In recent years, the discontinuous Galerlin finite element method for weak-form peridynamic equations has been developed. This method can not only describe the non-local action effects and discontinuous deformation characteristics of the investigated body, but also make full use of the finite element method. It has the advantages of efficient solution, direct application of local boundary conditions, and effective avoidance of the surface effects in peridynamics. The basic principle of the discontinuous Galerkin finite element method of bond-based peridynamics was expounded, the calculation formula was derived, the specific algorithm flowchart and details were given, and the dynamic cracking and bifurcation problems of brittle glass plates were simulated. The damage processes of concrete slabs under blast impact loads were calculated and analyzed. The research results show that, the proposed method can predict the complex rupture mode and the damage process of the structure under blast impact loads, with high computational efficiency, and makes an effective way to the simulation of the structural blast damage effects.

Identification of Pipeline Inner Wall Geometry Based on the POD-RBF Method

YU Bo, TAO Yingying

2023, 44(4): 406-418. doi: 10.21656/1000-0887.430168

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Abstract:
Based on the proper orthogonal decomposition-radial basis function (POD-RBF), a geometric identification method for pipeline inner wall was proposed to solve the internal corrosion detection problem of natural gas and oil pipelines. In view of the static magnetic field, the simplified finite element model for the pipelines was established, and the variable-geometry sample library was constructed, to realize the response prediction of arbitrary geometry by the POD-RBF. The proposed method achieves reduced-order analysis and avoids repeated solution of the stiffness matrix due to the geometrical change during the identification process. Hence, it can significantly improve the computation efficiency. Finally, the grey wolf optimization (GWO) algorithm was used to optimize the objective function and avoid the calculation of the sensitivity in the process of geometry change. The numerical examples show that, the proposed method has high efficiency and accuracy in the geometric identification of the pipeline inner wall, with good stability even under introduced noises.

Thermal Buckling Analysis of FGM Sandwich Circular Plates Under Transverse Nonuniform Temperature Field Actions

GONG Xuebei, ZHAO Weidong, GUO Dongmei

2023, 44(4): 419-430. doi: 10.21656/1000-0887.430094

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Abstract:
Based on the von Kármán geometric nonlinear plate theory, the displacement-type geometric nonlinear governing equations for FGM sandwich circular plates under transverse nonlinear temperature field actions were derived. With the immovable clamped boundary condition, the analytical formula for dimensional critical buckling temperature differences of the system was obtained from the solution of the linear eigenvalue problem. Moreover, the 2-point boundary value problem of ordinary differential equations was solved with the shooting method. The effects of geometric parameters, constituent material properties, gradient indexes, temperature field parameters and layer-thickness ratios on the critical buckling temperature differences, the thermal postbuckling equilibrium paths, and the buckling equilibrium configurations of FGM sandwich circular plates, were investigated. The results show that, with the increases of the thickness-radius ratio, the relative thickness of the FGM layer and the gradient index, the FGM sandwich circular plate's critical buckling temperature difference will increase monotonically. Given a fixed radius and a fixed total thickness, the postbuckling deformation of the FGM sandwich circular plate will decrease significantly with the relative thickness of the FGM layer.

Research on the Fictitious Source Points of the Hybrid Fundamental Solution-Based Finite Element Method for Heat Conduction Problems

ZHANG Kai, WANG Keyong, QI Dongping

2023, 44(4): 431-440. doi: 10.21656/1000-0887.430077

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Abstract:
The hybrid fundamental solution-based finite element method was proposed for heat conduction problems. Firstly, 2 independent fields were assumed: the intra-element temperature field approximated through the linear combination of fundamental solutions, and the auxiliary frame temperature field in the same form as that in the conventional finite element method. Then, a modified variational functional was employed to link the 2 independent fields and derive the finite element formulation. However, the accuracy of the method is strongly dependent on the distribution and the number of source points. The source points were usually placed on 2 fictitious boundaries outside the element: one is similar to the element shape, the other is a circular one. Furthermore, the dual fictitious boundary scheme was proposed for comparison with the above fictitious boundaries. With different configurations of source points, 2 typical numerical examples were given to demonstrate the validity and the insensitivity to mesh distortion of the proposed method.

A Prediction Model for Skin Wound Suture Forces With Uncertain Material Parameters

WEN Guangquan, JI Xiaogang, DUAN Yushun, DENG Lin

2023, 44(4): 441-449. doi: 10.21656/1000-0887.430067

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Abstract:
To evaluate the forces required for the suture of skin wounds quickly and effectively, the nonlinear finite element method was used to calculate the suture forces for skin wounds with different sizes and material parameters. With the calculated results as samples, the prediction model for skin wound suture forces was constructed by means of the EBF neural network model. Given the uncertain skin material parameters influencing the reliability of numerical results, the Monte-Carlo method was used to analyze the uncertainty propagation of skin material parameters. Finally, the prediction analysis and measuring experiment of wound suture forces were carried out with pig skin specimens to verify the reliability of the method. The results showed that, the suture force increases first and then decreases according to the suture point sequence, and the peak force occurs before the center of the wound. For a 40 mm×10 mm wound, the peak suture force is about 1.7 N, and that for a 40 mm×14 mm wound is about 2.5 N. Influenced by the uncertainty of material parameters, the prediction results of suture forces fluctuate by as much as ±0.6 N. The proposed theoretical prediction model provides an effective solution to the problem of parameter uncertainty propagation for biological soft tissue materials such as skins, and makes an important mechanical reference for robotic surgical suture.

Applied Mathematics

UGV Path Programming Based on the DQN With Noise in the Output Layer

LI Yang, YAN Dongmei, LIU Lei

2023, 44(4): 450-460. doi: 10.21656/1000-0887.430070

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Abstract:
The path programming of the unmanned ground vehicle (UGV) was studied under the framework of the deep Q-network (DQN) algorithm. To improve the exploration efficiency, the DQN algorithm was applied through discretization of the continuous state into the discrete state. To balance between exploration and exploitation, the Gaussian noise was added only in the output layer of the network, and a progressive reward function was designed. Finally, experiments were carried out in the Gazebo simulation environment. The simulation results show that, first, this strategy can quickly program a collision-free route from the initial point to the target point, and the convergence speed is significantly higher than those of the Q-learning algorithm, the DQN algorithm and the noisynet_DQN algorithm; second, this strategy has the generalization ability about the initial point, the target point and the obstacles, as well as verified effectiveness and robustness.

Nonlinear Stability of Traveling Wavefronts for a Discrete Cooperative Lotka-Volterra System With Delays

YAN Rui, LIU Guirong, LI Xiaocui

2023, 44(4): 461-470. doi: 10.21656/1000-0887.430172

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Abstract:
The stability of traveling wave solutions of the reaction diffusion model is a very important research topic. The globally nonlinear stability of traveling wavefronts for a discrete cooperative Lotka-Volterra system with delays was studied. More precisely, for the initial perturbation decaying exponentially to the traveling wavefronts with a relatively large speed at infinity, but arbitrarily large speeds in other positions, by means of the L²-weighted energy method, the comparison principle and the squeezing technique, such traveling wavefronts were obtained and proved to be of exponentially asymptotic stability. Moreover, the problem of establishing the energy estimates was solved under the actions of the discrete dispersal operator and the time delays. In short, the extension of the weighted energy method to discrete systems with delays, enriches the relative research.

Bistable Periodic Traveling Wave Solutions to Lattice Competitive Systems

LI Jian

2023, 44(4): 471-479. doi: 10.21656/1000-0887.430071

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Abstract:
The existence of bistable periodic traveling wave solutions to lattice competitive systems was studied. Firstly, the lattice competitive system of 2 species was transformed into a cooperative system. Then, the principle of comparison was established and a pair of upper and lower solutions were given to obtain the convergence of the solution at infinity, with the initial function satisfying certain conditions. By means of the vanishing viscosity method and the principle of comparison, the existence of the traveling wave solution connecting 2 stable periodic equilibrium points of the system, was proved.

Finite-Time Synchronization of Coupled Neutral-Type Neural Networks With Stochastic Disturbances and Uncertainties

WANG Kejie, CHEN Qiaoyu, TONG Dongbing, MAO Qi

2023, 44(4): 480-488. doi: 10.21656/1000-0887.420411

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Abstract:
The finite-time synchronization problems were solved for coupled neutral-type neural networks with stochastic disturbances and uncertainties. Based on the Lyapunov stability theory and the inequality techniques, the finite-time synchronization criterion was proposed for this system. Then the finite-time synchronization was realized for the master-slave system through the construction of an appropriate state feedback controller. At last, a numerical simulation was given to verify the effectiveness of the proposed theory.