This review presents a comprehensive and critical examination of the integration of fractal analysis and symmetric analysis within tsunami modeling, proposing a new paradigm for predicting tsunami behavior. Tsunamis are among the most devastating natural hazards, yet traditional models, often predicated on smooth-boundary assumptions, struggle to capture the complex dynamics influenced by irregular seafloor topography and coastlines. This work systematically demonstrates how incorporating the inherent fractal and symmetric characteristics of tsunami-related phenomena significantly enhances predictive accuracy, particularly concerning wave propagation direction and energy distribution. We provide a rigorous exposition of the mathematical foundations, including fractal dimension, two-scale fractal theory, and Lie point symmetry, and detail their application to fractal solitary wave theory and governing equations. The review further explores the emerging synergy with artificial intelligence, specifically deep learning architectures that embed fractal and symmetry constraints, creating a robust framework for data-driven and physics-informed prediction. Key validation studies against historical tsunamis and recent events, such as the 2025 Kamchatka tsunami, are presented to substantiate the model's efficacy. Finally, we discuss the profound implications of this integrated approach for advancing tsunami early warning systems and outline strategic directions for future research, underscoring the need for more efficient algorithms, comprehensive global seafloor databases, and the development of fully coupled physical-AI models.

Taking the parameters of the flat-swing dual operation mode linear vibrating screen, namely amplitude, vibration frequency, vibration direction Angle and swing Angle, as the research object, the influence of the interaction of different parameters on the stratification effect is explored. [Objective] In order to more intuitively analyze the complex influence relationship between the screening parameters of the vibrating screen and the delamination effect and the dynamic characteristics of the material group in the screening process, it is convenient to carry out the optimization research of the parameters of the vibrating screen to improve the screening and layering effect and screening efficiency. [Method] Firstly, solid works are used to establish a vibrating screen to simplify the three-dimensional model, the orthogonal test method is used to design the test, and the corresponding factors in the test scheme are imported into EDEM, and the neural network is used as the carrier to explore the change law of the four parameters of the amplitude, vibration frequency, vibration direction angle and swing angle of the vibrating screen and the layering effect, and the data results are obtained for deep learning. [Result] The influence weight analysis of screening parameters and different layering effects showed that the vibration frequency, vibration amplitude, vibration direction angle and swing angle had the most significant effects, so these four parameters were combined to characterize the operating state of the vibrating screen to clarify the change law of different parameters on the layering effect of the material body. When the amplitude is 3.4 mm, the vibration frequency is 14.8 Hz, the vibration direction angle is 44.1°, and the swing angle is 0.6°, the layering effect of the material group is obvious, and the overall distribution of fine-grained materials and the screen surface end. [Conclusion] This article takes the vibration parameters of the sieve machine as the variable and the material body layering effect as the optimization goal, in order to provide inspiration for the optimal design of the vibrating screen.

This study addresses the complex loading conditions experienced by moving printed films during drying and curing processes. The external excitation caused by hot air blowing in drying ovens and the time-varying tension caused by the rewinding and unwinding process are comprehensively considered. The primary focus is on investigating the nonlinear vibration of viscoelastic PET films undergoing axial motion under coupled timevarying tension and velocity. Based on the Kelvin model and incorporating Hamilton's principle, the differential equations governing the motion of a viscoelastic PET film undergoing coupled time-varying tension and velocity under uniform temperature conditions are established. The multiscale method is employed to solve the amplitude-frequency response equations of the film system under parametrically coupled resonance. Stability analysis of the steady-state solution is conducted using the Routh-Hurwitz criterion. The amplitude-frequency characteristic curve and the relationship between various parameters and resonance amplitude under parametrically coupled resonance are obtained by MATLAB programming. The study systematically investigated the influence of initial velocity, tension fluctuation coefficient, and drying temperature on the nonlinear vibration response of PET film.

The chaotic vibration analysis of fractional-order viscoelastic polyethylene terephthalate (PET) films under air resistance and non-uniform temperature fields is investigated. During the printing production process, moving viscoelastic PET films in the oven hot-air system are not only subjected to non-uniform temperature distributions and hot air, but they are also influenced by the friction forces generated by the airflow field on the film surface, causing the printed PET film to vibrate and affecting print quality. Based on the von Karman theory of large-deflection thin plates and Hamilton's principle, a fractional-order nonlinear vibration differential equation is established for the motion of PET films in a non-uniform temperature distribution field during roll-to-roll printing processes. The finite difference method and discrete Caputo-type fractional derivatives are applied to obtain the nonlinear vibration response of the fractional-order viscoelastic PET film under non-uniform temperature distribution. Through programming and calculation using MATLAB, bifurcation diagrams, phase trajectory diagrams, time history diagrams, phase diagrams, and Poincaré section diagrams are obtained to analyze the effects of air resistance, fractional-order degree, and non-uniform temperature on the nonlinear vibration of the film.

First, a Kurchatov-type derivative-free single-parameter method without memory for solving nonlinear systems is constructed, and the convergence order of the method is proved to be third order by using the higher-order Fr & eacute;chet derivative. By using the Kurchatov-type difference operator to represent the variable parameter, the method without memory develops into two Kurchatov-type methods with memory, and the convergence order of the two methods with memory can reach 3.303 and 3.414, respectively. The new Kurchatov-type methods with and without memory use LU decomposition only once per iteration and require less computational cost when applying them to solve nonlinear systems. Second, in order to enlarge the application range of the novel Kurchatov-type method, the local convergence of the new Kurchatov-type method with and without memory is proved by using the omega-continuity condition of the first Fr & eacute;chet derivative. In this way, the influence of higher-order derivatives on the proof of convergence is avoided, and the application range of the method is expanded. In addition, the radius of convergence around the solution is obtained by theoretical proof, and the uniqueness of the solution is proved. Finally, convergence radius is calculated by solving nonlinear systems with Kurchatov-type methods in numerical experiments, and the error and calculation time of different iteration methods are compared.

To further enhance the performance of integrated quasi-zero stiffness (IQZS) isolator, this work presents a novel IQZS isolator with segmented variable cross-section thickness (IQZS-SVCST) based on a non-uniform thickness distribution (NUTD) strategy. The proposed design features an inclined-wall with tunable local thickness and width, yielding two configurations: Type A, which is the variable end cross-section thickness isolator, and Type B, which is the variable central cross-section thickness isolator. Subsequently, analytical derivation of contact stiffness and restoring force was performed using energy methods, while the amplitude-frequency response and force transmissibility were derived through harmonic balance analysis. Based on the developed models, the influence of various factors on the system's vibration attenuation capability was investigated. Compared with conventional configurations and combined with optimization results from genetic algorithms, the proposed NUTD strategy demonstrates enhanced parameter adjustability and significant structural lightweight effects while maintaining excellent isolation performance. Experimental results verify the accuracy of the developed theoretical model and related calculations for IQZS-SVCST isolator, showing that the introduced NUTD parameters achieves 46.3

In the field of nonlinear equation research, especially for solitary wave equations, obtaining exact solutions is extremely challenging. However, variational theory offers crucial insights. This paper focuses on the fractal-fractional Korteweg-de Vries-Zakharov-Kuznetsov (KdV-ZK) equation with viscoelasticity and conducts a comprehensive and in-depth study. By using the semi-inverse method to carefully construct the variational formulation, setting the trial functional in the energy integral ingeniously, and determining the unknown function according to the Euler-Lagrange equation, the variational principle of this equation is successfully obtained. In this process, a new modified equation form is derived with the help of the potential function. Further in-depth exploration reveals that under specific conditions, the equation can be transformed into an ordinary differential equation. Through detailed analysis, the conditions for the existence of periodic solutions are obtained, and effective methods such as He's frequency formulation or the homotopy perturbation method are used to determine the frequency-amplitude relationship. Meanwhile, the variational formulation obtained in this paper has important application value in numerical simulation and analytical analysis. It not only opens up new directions for the energy method research of nonlinear wave equations but also provides valuable references and guidance for theoretical and applied research in related fields, effectively promoting the development of this field.

Ultrafine tailing sand was a solid waste generated by the ore dressing process, which seriously affects the efficient large-scale production of mining enterprises due to its fine particle size and slow settling speed. In response to the difficulty of settling and concentration of ultrafine tailings, two kinds of technology for strengthening sedimentation and concentration of tailings were proposed in this paper, namely process strengthening and structure strengthening. In the process strengthening approach, a novel flocculant CF, exhibits optimal sedimentation performance at a concentration of 45% and a dosage of 120 mg/L, achieving a maximum sedimentation velocity of 2.93 mm/s. In the structural enhancement approach, a new deep cone concentrator was designed, which, at a feed rate of 400 mL/min and a scraper rotation speed of 5 rpm, results in a bottom flow concentration of 70.53% after 30 min of sedimentation, demonstrating superior sedimentation concentration performance. The type and content of water in the tailing sand were investigated using Nuclear Magnetic Resonance (NMR), and adsorbed water, pore water and free water were mainly present in the tailing sand, and the active water content in the tailing sand decreased significantly and migrated to more stable water as the settling proceeded.

This study investigates the interaction between highly nonlinear solitary waves (HNSWs) in particle chain and hyperelastic materials (e.g., silicone and fluorine rubber) through simulations and theoretical modeling. A discrete element/finite element (DE/FE) coupled model was developed based on Hertz contact law and Newton’s second law, analyzing two contact methods: direct particle-material contact and the addition of a face sheet. Results demonstrate that hyperelastic material properties (Young’s modulus, compressive strength) and incident particle velocity significantly influence the amplitude and delay of reflected solitary waves. The inclusion of a face sheet enhances sensitivity, enabling precise differentiation between material types. This work advances HNSW-based health diagnosis theory for hyperelastic materials, offering practical applications in non-destructive testing and material characterization.

The fracture failure of dual-axis micromirrors under the AEC-Q100 qualification test could not be mitigated by structural designs alone due to the need for compatibility in bending and torsional stiffness. To address this, a passive MEMS vibration isolator was proposed to protect the micromirrors within a broad frequency range of 20 Hz to 1200 Hz, unlike conventional designs limited to a fixed frequency. The proposed method was based on a two-degree-of-freedom (DOF) micromirror-isolator system, in contrast to the single-DOF systems employed in existing methods. The isolator's stiffness was matched to the micromirror's stiffness to maximize the mirror plane's movement, and an air damping mechanism was incorporated using a 20 mu m gap to control the dynamic response time. The designed isolator was fabricated using a novel SOI-on-glass process and tested on a high-precision vibration shaker equipped with a laser Doppler vibrometer. Results showed that the proposed isolator attenuated vibration amplitude by 25.55 dB, closely aligning with the design value of 24.89 dB. Additionally, an automotive-grade vibration test demonstrated successful isolation under a 50g vibration within the 20 Hz to 1200 Hz frequency range, without introducing parasitic modes that could disrupt the micromirror's operational modes.

Solitary waves possess extensive potential for application in non-destructive testing due to their role as efficient information carriers. This study investigates the coupling effect between highly nonlinear solitary waves and functionally graded porous plates reinforced with graphene platelets (FGP-GPLs). An improved Halpin-Tsai micromechanics model and an improved two-variable precision plate theory are employed to derive a differential equation system for the coupling of particle chains and FGP-GPLs. The system is solved using the fourth-order Runge-Kutta method to obtain velocity and displacement solutions of the particles. The time and amplitude of the rebound waves are analysed, and it is found that the pore distribution, graphene distribution, porosity coefficient, thickness ratio, and graphene weight fraction impact the solitary wave. The results of this study provide a theoretical basis for the non-destructive detection of FGP-GPLs by solitary waves, which enables rapid inspection and controllability studies of structures. Moreover, this technology expands the application fields of nonlinear solitary waves based on one-dimensional spherical particle chains.

This study analyzes the vibration and stability characteristics of a submarine fluid-conveying pipeline with local variation in stiffness under the influence of external flow fields. Utilizing the absolute nodal coordinate formulation (ANCF), the dynamic equation for the submarine pipeline with local stiffness variation under arbitrary boundary conditions is derived. Through numerical solutions, curves depicting complex frequency variations as a function of the internal flow velocity of the pipeline system are obtained. Additionally, the effects of external flow velocity, geometrical parameters, and local stiffness variation on the vibration and stability characteristics of the submarine pipeline under three different boundary conditions are investigated. Results indicate that the stability of the pipeline system decreases across the three boundary conditions under the influence of the external flow field. Increased external flow velocities lead to further reductions in system stability. Furthermore, as the length-to-diameter ratio increases and the thickness-to-diameter ratio decreases, the first three orders of frequencies of the pipeline gradually diminish. Enhancing local stiffness contributes to improving the stability of the system, whereas reducing local stiffness has the opposite effect.

In this research, a new passive-type integrated quasi-zero stiffness isolator with variable cross-section (IQZS-VCS) characteristic is proposed in order to extend the application fields. Compared with the integrated quasi-zero stiffness isolator with uniform cross-section (IQZS-UCS), the IQZS-VCS isolator has more adjustable structural parameters and can enhance vibration attenuation performance in lower-frequency region. Based on the developed static model, the nonlinear force and nonlinear stiffness characteristics of the IQZS-VCS isolator are evaluated using different design parameters. Furthermore, the effects of different factors on vibration reduction ability of the isolating system are addressed in terms of amplitude-frequency response and force transmissibility. Compared with a traditional linear, IQZS-UCS and conventional three-spring-type quasi-zero stiffness isolators, the proposed isolator exhibits better low-frequency and wide-band isolation performance. Quasi-static and electromagnetic exciting tests reveal that the developed theoretical models and related calculation results of the IQZS-VCS isolator are correct, providing an innovative solution and insight for broadband isolator.

Every fluid problem is greatly affected by its boundary conditions, especially the near-shore seabed could produce an irrevocable harm when a tsunami wave is approaching, and a real-life mathematical model could stave off the worst effect. This paper assumes that the unsmooth seabed is a fractal surface, and fractal-fractional governing equations are established according to physical laws in the fractal space. The geometrical potential theory is used to explain the force produced by the wave surface, and Kong-He friction law is applied to further figuring out the local and memory properties of the friction along the fractal boundary. This paper aims at studying tsunami waves in a fractal space, rendering a reliable mathematical model for both prediction of the tsunami motion and the coastal protection.

The variational theory has triggered skyrocketing interest in the solitary theory, and the semi-inverse method has laid the foundation for the search for a variational formulation for a nonlinear system. This paper gives a brief review of the last development of the fractal soliton theory and discusses the variational principle for fractal Boussinesq-like [Formula: see text] equation in the literature. The paper establishes a variational formulation for [Formula: see text] equation to show the effectiveness of the semi-inverse method, and a general trial-Lagrange function with two free parameters is established for [Formula: see text] equation, the identification of the unknown parameters and the unknown function involved in the trial-Lagrange function is shown step by step. This paper opens a new path for the fractal variational theory.

A flexible printing electronical membrane is an electron equipment made by precisely spraying conductive metal ink such as silver on a soft membrane substrate. With its advantages of light weight and flexibility, it can adapt to changing working environments and is widely used in aerospace, wearable electronics and other fields. Nevertheless, during the manufacturing preparation of roll-to-roll printing membranes, the high-speed movement of printing electronical membranes under tension is affected by the impact of hot air from the drying oven and the electrostatic interference generated by friction in transmission, which restricts the overprint accuracy and preparation velocity of flexible electronical membranes. To address this issue, the nonlinear forced vibrational characteristics of a traveling flexible printing electronical membrane on temperature coupling subjected to nonlinear electrostatic force were investigated. The roll-to-roll printed intelligent RFID electron membrane is the research target. On the basis of the energy approach and the heat conduction equation considering the effect of deformation, the nonlinear vibrational equations of an axially traveling flexible printing electronical membrane coupled with temperature under the function of nonlinear electrostatical excitation force were derived. The Bubnov–Galerkin algorithm was applied to discretize the vibration partial differential equations; by making full use of the quartic Runge–Kutta numerical algorithm to calculate the approximate solution of equations, the phase portraits, Poincaré maps, time history diagrams, power spectra, and bifurcation plots of the nonlinear vibrations of the traveling printing electronical membrane were used to explore the effects of movement velocities, electrostatical field, and thermal coupling coefficients. The findings obtained the stable working domain and the divergence instability domain of the traveling flexible printing electronic membrane, which provided a theory fundamental for enhancing the stable craft of a printing electronical membrane.

The coupling interaction between highly nonlinear isolated waves and honeycomb sandwich panels is examined based on the characteristics of isolated waves. The set of differential equations for the connection between the particle chain and the honeycomb sandwich panel is derived in accordance with Hertz's law. To obtain the displacement and velocity curves of each particle in the chain of particles, a set of differential equations is solved using the fourth-order Longe-Kutta technique, and the influence of the structure's skin thickness and core layer thickness on the rebounding isolated waves is examined. It was discovered that rebound isolated waves are sensitive to the skin thickness and core layer thickness of honeycomb sandwich panels by analyzing the time of appearance of rebound waves, the energy carried by rebound waves, and the effects of skin thickness and core layer thickness on isolated waves. This finding provides a theoretical foundation for further research on core-skin peeling and damage detection in honeycomb sandwich panels.

TPU (thermoplastic polyurethane) films are subjected to heating in production. As the temperature rises, thermal stress will be generated inside the TPU film. It causes defects, such as wrinkles and cracks, in the moving TPU film; therefore, it is of great significance to study the thermoelastic coupling effect of the moving TPU film in the printing process. To study the thermoelastic coupling vibration characteristics of the moving TPU film under the opposite edge tension, it includes combining the thermal conduction equation with temperature coupling term and the vibration differential equation with variable temperature effect to get the coupling equation. The differential quadrature method is used to discretize the vibration differential equation. The influence of the thermoelastic coupling coefficient, aspect ratio, and tension ratio on the vibration characteristics of the TPU film is studied.

The main purpose of this paper is to detect the damage of interior delamination in composite laminate using a new non-destructive testing method of highly nonlinear solitary waves. We studied the coupling mechanism between highly nonlinear solitary waves and interior delamination composite plate. The discrete element method is used for the one-dimensional particle chain, based on the residual stiffness model, Hertz’s contact law and the composite plate vibration differential equation are used, we obtained the coupled differential equations for one-dimensional particle chain and interior delamination composite plate. We used the fourth-order Runge–Kutta method to solve the differential equations, and obtained the velocity curve and displacement curve of each particle in the particle chain. The influence of the thickness of the damaged composite plate, the average diameter of the interlaminar cracks, and the density of the interlaminar cracks on the reflected solitary waves are analyzed. We used one-dimensional homogeneous particle chain and one-dimensional composite particle chain to detect damaged composite plate with different material properties. The results shows that the reflected waves are sensitive to the thickness of the plate, the average diameter of the interlaminar cracks and the density of the interlaminar cracks. We find that the propagation time of solitary waves in the composite particle chain is shorter and faster, and the detection efficiency can be increased by 23.5% at most. The characteristic information carried by reflected waves can be used for non-destructive testing of damaged composite plate. The research results of this paper enrich and improve the non-destructive testing of composite materials, and accelerate the practical application of one-dimensional particle chain.