应用数学和力学

Current Issue
2026 Vol. 47, No. 9

column

Cover And Contents
Chief Editor’s Note
Solid Mechanics
Fluid Mechanics
Applied Mathematics
Chief Editor's Note

Previous Issue

Select articles

Display Method:

Cover And Contents

2026, 47(9): .

Abstract(84) PDF(10)

Abstract:

Chief Editor’s Note

Solid Mechanics

A Small Sample Data Driven and Constraint Fusion Multi-Objective Topology Optimization Method

Ying Jiahuan, Zhou Huanlin, Meng Zeng

2026, 47(9): 1107-1116. doi: 10.21656/1000-0887.460112

Abstract(126) HTML (49) PDF(14)

Abstract:
The traditional volume-compliance multi-objective topology optimization methods rely on intensive finite element calculations, often suffering from low computational efficiency and sparse coverage of the Pareto solution set. To address these issues, an efficient multi-objective topology optimization method based on small sample data driving and constraint fusion was proposed. The core of this method is the establishment of a small sample conditional generative adversarial network embedded with physical constraints, which learns from a limited number of finite element calculations to generate a high-precision and comprehensive Pareto optimal solution set for multi-objective topology optimization. By means of the small sample data driven techniques, the relationships between volume fractions and multi-objective topological structures were established. The results show that, the proposed method ensures computational accuracy while minimizing reliance on large-scale finite element calculations. Furthermore, the fusion of multiple physical constraints guarantees high precision mapping between physical fields and topologies, providing an efficient and reliable solution for multi-objective optimization design.

Numerical Simulation of Stand-Off Distance Effects on Explosive Welding With an Intermediate Layer

Miao Guanghong, Ni Yadong, Wei Zhengmei, Wu Weida, Tian Guangkuo

2026, 47(9): 1117-1131. doi: 10.21656/1000-0887.460147

Abstract(110) HTML (52) PDF(12)

Abstract:
the stand-off distance effects on explosive welding with an interlayer were investigated, with the ANSYS/LS-DYNA finite element software. The 3D numerical simulations of the explosive welding process were conducted for the commercially pure titanium (TP270C), the duplex stainless steel (SUS821L1), and the interlayer (SUS821L1), with 2 distinct algorithms: the arbitrary Lagrangian-Eulerian (ALE) method and the smoothed particle hydrodynamics-finite element method (SPH-FEM) coupling algorithm. The behaviors of the flyer plate under varying stand-off distances (5 mm, 10 mm, 15 mm/2 mm), including the vertical displacement, the collision velocity, the collision pressure, and the plastic deformation of the interlayer, were systematically analyzed across both algorithms. The simulation results show that, the vertical displacement of the flyer plate exceeds the stand-off distance between the base plate and the flyer plate, and the collision velocity falls within the explosive welding window. The collision velocity, the collision pressure of the flyer plate, the plastic deformation of the interlayer, the partitioned kinetic energy of the flyer plate and the collision pressure by the interlayer, all increase with the stand-off distance. All 3 stand-off distances achieve successful welding under the action of the interlayer. The highest explosive welding quality occurs at a stand-off distance of 15 mm, which matches the experimental results. Furthermore, numerical simulations based on 2 control groups' explosive thicknesses: 40 mm with stand-off distances of 30 mm/2 mm, and 38 mm with stand-off distances of 55 mm/2 mm, were performed. The results demonstrate that, reducing the explosive thickness and increasing the stand-off distance between the interlayer and the flyer plate during the acceleration phase achieve welding effects identical to those of the original experiments. This work ensures welding quality, reduces the cost of explosive welding experiments, and provides theoretical guidance for engineering applications.

Research on Transverse Crack Problems in Fiber-Reinforced Composites With the Distributed Dislocation Method

Wang Hao, Jiang Xiaoyu

2026, 47(9): 1132-1143. doi: 10.21656/1000-0887.460153

Abstract(132) HTML (52) PDF(9)

Abstract:
Under uniaxial tensile loading, the interaction between matrix cracks and interfacial cracks in fiber-reinforced composites within an infinite plane medium was investigated with the distributed dislocation method (DDM). With crack surfaces represented as continuously distributed dislocations, the composite fracture problem was transformed into a 2D boundary-value problem containing circular elastic inhomogeneities. Based on the superposition principle, the problem was decomposed into 2 sub-problems. The stress intensity factors (SIFs), the stress fields, the maximum tensile stress at the interface, and the energy release rates at the tips of both the matrix crack and the interfacial crack were derived. The crack propagation direction was determined through comparison of the ratio of maximum energy release rates at the crack tips. The results demonstrate that, when inclusions are located in the region ahead of the matrix crack tip, they will exert a shielding effect on crack propagation. Conversely, inclusions in other regions will amplify the crack growth. The initiation of interfacial cracks is primarily attributed to the maximum tensile stress at the interface. When the matrix crack tip approaches the interfacial crack tip, mutual promotion of propagation will occur between both cracks, and the closer their proximity is, the more pronounced the interfacial crack propagation will be due to reduced resistance to debonding.

Quantitative Assessment of Soil Landslide Impact Risk Based on the GA-BP Improved Random Material Point Method

Wang Zongyao, Zhou Xiaomin, Wu Rui, Sun Zheng

2026, 47(9): 1144-1158. doi: 10.21656/1000-0887.460137

Abstract(108) HTML (47) PDF(8)

Abstract:
The quantitative assessment of landslide risk is an important means to achieve landslide risk management. However, the existing assessment methods still have certain limitations in accuracy and applicability, and it is difficult to effectively characterize the uncertainty characteristics of the landslide process. The accuracy of the Monte Carlo simulation method in slope stability analysis has been widely recognized by the industry, but its application in engineering practice is greatly limited due to its high computational cost and long computational time. In addition, soil parameters show significant spatial variability under the long-term action of the geological environment, and traditional methods are difficult to describe this characteristic. To more realistically characterize the variation characteristics of soil in space, the spatial variability was considered and the stochastic material point method (SMPM) was developed based on the random field theory. Based on the 4 quantitative evaluation indicators of the maximum sliding kinetic energy, the maximum sliding pressure, the half bandwidth and the starting point, the risk of the entire soil landslide process was quantitatively assessed. At the same time, to further improve the computational efficiency and prediction accuracy, the genetic algorithm was combined with the BP neural network to construct a GA-BP neural network model to be integrated with the MC-SMPM method. The model uses the BP neural network to estimate the nonlinear relationship between soil parameters and quantitative risk assessment indicators, and optimizes the network structure through genetic algorithms, including key parameters such as the number of hidden layer neurons and the number of training times, to adapt to the characteristics of soil parameters, thereby improving the computational efficiency of the quantitative landslide risk assessment. The research results show that, the BP neural network model based on the GA optimization has significant improvements in overall prediction accuracy and stability, and can effectively promote the efficiency of quantitative landslide risk assessment and provide an efficient and reliable numerical calculation method for engineering applications.

Fluid Mechanics

A Multi-Fidelity Surrogate Model for Aerodynamic Coefficients of Iced Bundle Conductors

Yang Hui, Yan Bo, Gao Yingbo, Zhang Bo, Lü Zhongbin, Liu Guanghui, Wu Xingwang, Deng Huachao

2026, 47(9): 1159-1178. doi: 10.21656/1000-0887.460131

Abstract(166) HTML (63) PDF(8)

Abstract:
For quad-, six-, and eight-bundle conductors with crescent-shaped ice, the air flow fields around the iced conductors under different icing and wind conditions were numerically simulated with the computational fluid dynamics, and the variations of drag, lift, and torque coefficients of each iced sub-conductor vs. the wind attack angle were obtained. With the high-fidelity data of aerodynamic coefficients measured by wind tunnel tests and the low-fidelity data by numerical simulations, a multi-fidelity surrogate model was built by means of the BP neural networks, the random forest algorithm and the feature enhancement technique, with the sub-conductor position, the ice thickness, the ice accretion angle, the wind speed, the turbulence intensity and the wind attack angle as the input, and the aerodynamic coefficients of each sub-conductor as the output. The efficiency of the surrogate model was verified through the analysis of aerodynamic stability regions based on the Den Hartog and Nigol coefficients, and through the galloping simulations of typical iced bundle conductors. By this multi-fidelity surrogate model, a software with the bundle number of conductor lines, the ice thickness, the ice accretion angle, the wind speed and the turbulence intensity as the input, and the aerodynamic coefficient vs. the wind attack angle for each iced sub-conductor as the output, was developed. This work lays a foundation for the investigation of galloping of iced transmission lines and its prevention technologies.

The Research Progress of Dynamic Behaviors of Nematodes in Fluids

Shi Si, Gu Zheng, Luo Yige, Duan Ying, Chen Yeyu, Ouyang Zhenyu

2026, 47(9): 1179-1196. doi: 10.21656/1000-0887.460128

Abstract(204) HTML (52) PDF(5)

Abstract:
The advancements in understanding the dynamic behaviors of nematodes in fluid environments were reviewed, with foci on their individual locomotion and mechanical responses in various environments, collective aggregation behaviors, theoretical modeling, numerical simulations, and external control strategies. The Caenorhabditis elegans, owing to its advantages such as ease of observation, high culturing survival rates, and short life cycles, has become a widely used model organism in fluid dynamics research. It serves as a valuable platform for uncovering the physical principles of life motions under non-equilibrium conditions and for advancing the development of bioinspired micro/nano systems. Current research faces major challenges, including deciphering the multi-scale coupling mechanisms among neural, muscular, and fluidic systems, in situ capturing of the microscopic emergence processes of collective behaviors, establishing a unified multi-scale computational framework, and achieving synergistic and precise control via multi-physical fields. To date, studies have mechanistically elucidated the adaptive principles of individual locomotion and the emergence mechanisms of collective behaviors, developed cross-scale theoretical models and high-fidelity numerical simulation methods, and innovated external control strategies utilizing multi-physical fields. Future research will focus on advancing high-resolution multimodal measurement technologies, validating multi-scale integrated models, innovating multi-physics field control strategies, developing AI-driven methodologies, and improving open-source scientific ecosystems. These efforts aim to deepen the understanding of fundamental principles of biological motions and provide theoretical supports for breakthroughs in advanced biomimetic systems and biomedical technologies.

Applied Mathematics

A 3rd-Order Numerical Method for 2D Burgers Equations Based on the Generalized Trapezoid Formula

Liu Tao, Liu Libin

2026, 47(9): 1197-1210. doi: 10.21656/1000-0887.470066

Abstract(154) HTML (64) PDF(18)

Abstract:
The Burgers equation has been widely applied in many fields such as fluid mechanics due to its accurate description of nonlinear transport and dissipation mechanisms. A numerical method for a class of 2D nonlinear Burgers equations was studied. Firstly, the Burgers equation was transformed into a class of heat conduction equations via the Hopf-Cole transformation. Then, the high-order barycentric rational spectral approximation was used in the spatial direction to obtain the semi-discrete scheme, and the generalized trapezoidal formula with parameter δ was adopted in the temporal direction, thus to obtain the fully discrete scheme for the transformed equation. For parameter δ=1/3, the truncation error in the temporal direction has 3rd-order accuracy. Finally, numerical examples verify the correctness of the truncation error analysis.

Homotopy Analysis Solutions to Bubble Equations Through the Laplace Transform and Machine Learning

Qin Yupeng, Wang Shuo, Jin Hao, Wang Zhen, Zou Li

2026, 47(9): 1211-1221. doi: 10.21656/1000-0887.460144

Abstract(148) HTML (57) PDF(15)

Abstract:
The homotopy analysis method (HAM) is an important approach for solving nonlinear problems. The HAM was applied to construct high-precision approximate analytical solutions to the Rayleigh-Plesset equations for describing the periodic pulsation of bubbles in an ideal fluid. In the HAM framework, the Laplace transform was used to solve high-order deformation equations and obtain HAM solutions to the bubble equations. By means of the machine learning to optimize the convergence-control parameter in the HAM approximations, high-precision HAM approximations of the bubble equations were obtained. The results show that, the Laplace transform improves the efficiency of solving complex high-order deformation equations, and the machine learning technique provides an effective way to optimize the convergence control parameter in HAM approximations. Finally, as applications of the HAM approximations, dynamic characteristics and laws of bubble pulsation were discussed.

Adaptive Isogeometric Topology Optimization of Shell Structures Based on RMB-Splines

Li Jiawei, Li Hangchuan, Wang Xuan, Pan Maodong

2026, 47(9): 1222-1234. doi: 10.21656/1000-0887.460154

Abstract(142) HTML (61) PDF(10)

Abstract:
An adaptive isogeometric topology optimization algorithm based on RMB-splines (reachable minimally supported b-splines) was proposed to optimize the material distribution problem of shell structures. This algorithm combines the SIMP model to interpolate material density, and uses a new type of local refinement spline RMB-splines to represent the displacement field and density field. By taking advantages of its local linear independence, minimum support, and efficient local refinement ability, it achieves the unification of the analysis domain and the design domain. An adaptive mesh refinement strategy was constructed to identify refinement regions from the density distribution and to refine the mesh locally along material boundaries. This effectively reduces the overall degree of freedom while ensuring the optimization accuracy. In the numerical implementation, the RMB-splines provides stable and efficient numerical support for the optimization process. The projection function was introduced for density filtering to smooth the material distribution, and the MMA algorithm was used to solve the optimization problem. The numerical examples of multiple shell structures show that, compared with the B-spline method, this method can significantly reduce the number of basis functions, improve computational efficiency, and obtain a clearer and more engineering-feasible topological layout, promising great application potential in the topological optimization of complex structures.

Chief Editor's Note

Innovation is King: Only What Truly Advances a Problem Deserves to be Called Innovation

Lu Tianjian, Xiong Ke

2026, 47(9): 1093-1106. doi: 10.21656/1000-0887.472087

Abstract(190) HTML (64) PDF(24)

Abstract:
The fifth editorial, After Tools, Problems Remain, argued that human judgment must not be surrendered to powerful tools. The sixth, Problem as Foundation, brought research back to real objects, contradictions, and boundaries. This editorial takes the next step: once a real problem has been established, what does it mean for research to genuinely move forward? A problem governs the starting point; innovation governs the advance. No problem means empty novelty; no advance means that the problem remains where it began.Placed in the historical arc from Science: the endless frontier (1945) to science: a new golden age (2026), this editorial defines innovation as a substantive increment on a real problem. Such an increment may enlarge knowledge, capability, or the accessible boundary of inquiry: we may know what was previously unknown, do what was previously impossible, or bring a problem previously beyond observation, computation, design, or verification into reach. Zero-to-one opens an entry point, whereas one-to-N develops that entry point into reliable, usable, and transferable knowledge or capability. Pseudo-innovation is not a third mode of advance, but an appearance of novelty without a substantive increment.Innovation may occur in problem formulation, method, mechanism, system capability, or paradigm. Real problems may arise from national missions, engineering practice, and clinical needs, but also from fundamental questions whose use cannot yet be foreseen. Artificial intelligence is expanding the accessible domain of research by entering hypothesis generation, model discovery, experimental design, and validation; at the same time, it lowers the threshold for work to look innovative. A defensible claim must therefore identify the real problem, state the baseline and increment, present evidence commensurate with the claim, and define its boundary. For Applied Mathematics and Mechanics, the final question remains whether a study has produced a real advance on a real problem.