Weight Function Method Research Articles

The construction conditions of a Hilbert-type local fractional integral operator and the norm of the operator

<p>The parameterized local fractional singular integral operator $ \mathit{ T}^{(\tau)} $ is defined on the space $ L_p^{\tau\mu}(\mathbb{R}_+^\tau) $ as $ \mathit{T}^{(\tau)}: L_p^{\tau \mu}(\mathbb{R}_+^\tau)\rightarrow L_p^{\tau\nu(1-p)}(\mathbb{R}_+^\tau) $, $ \mathit{T} ^{(\tau)}(f_\tau)(y) = \, _0\mathscr{Y}_{+\infty}^{(\tau)}\Big[\frac{ |x-y|^{\tau\alpha}}{(x+y)^{\tau\beta}}f_\tau(x)\Big], y\in\mathbb{R}_+ $. By employing the weight function method and analysis techniques on the fractal real line number set $ \mathbb{R}_+^\tau $, a general Hilbert-type local fractional integral inequality has been established, thereby demonstrating the boundedness of the defined integral operator. Through optimization of parameters, it was determined that the necessary and sufficient condition for the constant factor in this general Hilbert-type local fractional inequality to be the best possible is that the power parameters $ \sigma $ and $ \sigma_1 $ satisfy $ \sigma+\sigma_1 = \beta-\alpha $. Consequently, the formula for calculating the operator norm has been derived.</p>

Two-Dimensional Numerical Method for Predicting the Resistance of Ships in Pack Ice: Development and Validation

This study presents a 2D numerical simulation method for predicting the resistance of ships navigating in pack ice. The key contribution of this study lies in the derivation of analytical closed-form solutions for calculating the flexural deformation and stress distribution in an elastic plate using Symplectic Mechanics and Hooke’s laws. These solutions are used to determine the failure mode of ice floes. Linear Elastic Fracture Mechanics (LEFMs) and the weight function method are utilized to analyze crack initiation, propagation, and fracture. Ice is broken when a crack propagates to 14.5% of the ice length. The compressive strength of ice and the contact area are used to calculate the ice load. A collision method was developed based on the Sweep and Prune (SAP) and Gilbert–Johnson–Keerthi (GJK) algorithms. A program for predicting the resistance of ships navigating in pack ice was developed based on MATLAB and the aforementioned theories. The navigation resistance of RV Xuelong at different ice concentrations and speeds was simulated and compared with the model test results from an ice tank. The comparison shows that the simulation results are consistent with the test results, with an average error of 9.05%, indicating the effectiveness and reliability of this numerical method. This study lays a solid foundation for future research on autonomous ship navigation in pack ice.

The proposal and calculation method of the annular domain weight function for the electromagnetic flowmeter under annular conductivity distribution

Researching on the characteristic of weight function distribution under non-uniform conductivity distribution is of significant importance for expanding the engineering applications of electromagnetic flowmeter. In this paper, the characteristic of weight function distribution is firstly explored for annular conductivity distribution where the center and outer ring are both conductive. The research results indicate an important finding that the average value of the weight function within any small annular domain of the center or outer ring remains constant as the radial position changes. Therefore, the concept of ‘annular domain weight function’ is then innovatively proposed. Furthermore, the relationship is studied between the ratio of the annular domain weight function of the center and outer ring and the annular feature parameters. Ultimately, the correlations are obtained for calculating the annular domain weight functions of the center and outer ring.

Necessary and Sufficient Conditions for the Boundedness of Multiple Integral Operators with Super-Homogeneous Kernels in Weighted Lebesgue Space

Super-homogeneous functions including homogeneous functions, quasi-homogeneous functions, and several non-homogeneous functions are considered. Using the weight function method, the construction conditions of Hilbert-type multiple integral inequalities with super-homogeneous kernels are first discussed. Then, using the obtained results, the construction problem of bounded multiple integral operators with super-homogeneous kernels in weighted Lebesgue space is discussed, and the necessary and sufficient conditions for operator boundedness and the operator norm formula are obtained.

The equivalent conditions for norm of a Hilbert-type integral operator with a combination kernel and its applications

Introducing adaptation parameters σ,σ1, formal parameters λi(i=1,2,3,4),κ,τ, and type parameters μ,ν, the integration operator is defined as T:Lpp(1−μσˆ)−1(R+)→Lppνσˆ−1(R+), Tf(y)=∫R+eλ1xμyν+κe−λ2xμyνeλ3xμyν+τe−λ4xμyνf(x)dx,y∈R+. Using the weight function method, a general Hilbert-type integral inequality is obtained, thereby proving the boundedness of the operator. The constant factor of the general Hilbert-type inequality is the best possible if and only if the adaptation parameters satisfy σ=σ1. From this, the formula for calculating the operator norm is obtained. In terms of application, some results from the references have been consolidated by discussing the combination of formal parameters and type parameters, and many new operator inequalities of different types and forms have been derived.

Inverse method of eigenstrain determination of laser shock peening

The determination of eigenstrain from residual stress is a crucial aspect in process predicting and optimizing of laser peen forming. However, direct measurement of eigenstrain is not possible, and therefore an indirect method based on the Fredholm integral equation is proposed for determining eigenstrain from residual stress. The residual stress of full-coverage laser shock peening (LSP) is analyzed, and the governing equation of eigenstrain is derived from the elastic deformation of a linear elastic plate under full-coverage LSP. The governing equation is a second-kind Fredholm integral equation, which can be solved analytically using the weight function method. Since residual stress is difficult to measure accurately in LSP, an inverse method based on the integral governing equation is proposed for determining eigenstrain from an incomplete set of measurements. The validity of this method is demonstrated through simulation and experimental results, highlighting its high efficiency and engineering application value.

Effect of local laser heating on fatigue crack propagation rate of AA2524 thin plate

To extend the operational lifespan of the AA2524 aircraft panel, the surface of the AA2524 specimen was locally heated by laser. The specimen’s microhardness distribution and surface residual stress (RS) were tested, as was its fatigue crack growth rate (FCGR). The fatigue fractography of specimen was also observed. The FCGR of specimen after laser heating was predicted by the weight function method (WFM). The consequences indicate that when the laser power is lower than 900 W, the hardness of the heating zone on the surface of the specimen decreases by no more than 10 HV0.5. Compared to the base material (BM) specimen, transverse heating specimen creates a maximum compressive residual stress (CRS) of 32 MPa along the crack extension line. This can effectively stop the crack propagation. When laser power with the transverse heating is 900 W, the crack propagation suppression effect is optimal. The fatigue life of heated specimen is increased by 108.5 % compared to the BM specimen under a constant amplitude load with the maximum load of 2600 N and R of 0.1, and the fatigue life of heated specimen is increased by 46.4 % under a constant amplitude load with the maximum load of 3300 N and R of 0.1. The spacing among 6 adjacent striations of specimen after transverse heating is shortened from 2.79 μm to 0.88 μm compared with the BM specimen. The Walker model, which is based on the WFM, has high accuracy in predicting FCGR of specimen after laser heating under various load conditions. It is evident that local laser heating technology can extend the service life of aviation aluminum alloy panels.

A fast adaptive spatio-temporal fusion method to enhanced Fit-FC.

Space-time fusion is an economical and efficient way to solve "space-time contradiction". Among all kinds of space-time fusion methods, Fit-FC space-time fusion method based on weight Function is widely used. However, this method is based on the linear model to depict the phase change, but the phase change in the real scene is complicated, and the linear model is difficult to accurately capture the phase change, resulting in the spectral distortion of the fusion image. In addition, pixel-by-pixel scanning with moving Windows leads to inefficiency issues, limiting its use in large-scale and long-term tasks. To overcome these limitations, this paper developed a simple and fast adaptive remote sensing image Spatio-Temporal fusion method based on Fit-FC, called Adapt Lasso-Fit-FC (AL-FF). Firstly, the sparse characteristics of time phase change between images are explored, and a time phase change estimation model based on sparse regression is constructed, which overcomes the fuzzy problem of fusion image caused by the failure of linear regression to capture complex nonlinear time phase transition in the weighted Function method, making the algorithm better at capturing details. Secondly, an adaptive window selection Function is established to overcome the problem of manually setting parameters on different data sets, improve the convenience of the algorithm and robustness of the application on different data sets, and make the algorithm simpler and more efficient. Finally, the improved AL-FF algorithm is compared with other algorithms to verify the performance improvement. Compared with the current advanced Spatio-Temporal fusion methods, AL-FF algorithm has stronger detail capture ability and can generate more accurate fusion results. In addition, the computational efficiency is significantly improved, and the efficiency is increased by more than 20 times compared with the current mainstream method.

P-wave propagation in attenuative elliptical VTI media

Elliptical anisotropy is convenient to use as the reference medium in perturbation methods designed to study P-wave propagation for transverse isotropy (TI). We make the elliptically anisotropic TI model attenuative and discuss the corresponding P-wave dispersion relation and the wave equation. Our analysis leads to two conditions in terms of the Thomsen-type parameters, which guarantee that the P-wave slowness surface and the dispersion relation satisfy elliptical equations. We also obtain the viscoacoustic wave equation for such elliptically anisotropic media and solve it for point-source radiation using the correspondence principle. For the constant- Q TI model, we use the weighting function method to derive the viscoacoustic wave equation in differential form. Numerical examples validate the proposed elliptical conditions and illustrate the behavior of the P wavefield in attenuative elliptical TI models.

Effect of crack surface friction on stress intensity factor for double-edge cracked Brazilian disk specimens under parabolic distribution load

The weight function method was employed to derive the closed-form solutions of stress intensity factors (SIFs) for double-edge cracked Brazilian disk (DECBD) specimens subjected to parabolic distribution loads, taking into account the influence of crack surface friction. Additionally, the accuracy and validity of these solutions were discussed in comparison with previous numerical results. Furthermore, the impact of friction coefficient and load distribution angle on SIFs was further investigated. The results indicate that the effect degree of load distribution angle on SIFs is closely correlated with both the loading angle θ and crack length α. Both the mode I and mode II SIFs increase with increasing load distribution angle γ for θ = 15° and α < 0.4. However, when α ≥0.60, the impact of load distribution angle γ on SIFs becomes negligible. Furthermore, the friction between crack surfaces of DECBD specimens significantly affects the dimensionless mode II SIF YII, leading to a decrease in dimensionless mode II SIF YII as the friction coefficient μ increases. Moreover, it further demonstrates that both the relative crack length α and friction coefficient μ play important roles in determining the critical loading angle θIIc for crack sticking. The critical loading angle θIIc for crack sticking increases with an increasing relative crack length; however, an increase in the friction coefficient leads to a reduction of this critical loading angle θIIc.

Optimal matching parameters of the inverse Hilbert-type integral inequality with quasihomogeneous kernels and their applications

UDC 517.9 By using the inverse Hölder inequality and the weight function method, we establish the inverse Hilbert-type integral inequality. In the case of a quasihomogeneous kernel, we obtain the necessary and sufficient conditions for the optimal matching parameters. Finally, their applications in the operator theory are discussed.

Multi-stage voltage sag frequency evaluation based on process immunity in the distribution network

Abstract The reclosing action of the distribution network may cause multi-stage voltage sag in the grid nodes, which leads to more serious losses for sensitive users. In order to more reasonably characterize the disturbance degree of multi-stage voltage sag on sensitive equipment, a frequency evaluation method based on process immune time (PIT) and improved analytic hierarchy process is proposed. Firstly, multi-stage voltage sag waveforms caused by the reclosing corresponding action under transient fault and permanent fault are analyzed, and the PIT characteristic curve of sensitive equipment under multi-stage voltage sag is characterized. Secondly, the weight function method is used to calculate the influence degree of each stage sag. The analytic hierarchy process is improved by utilizing the correlation and influence degree of each stage sag. The weight of each stage sag is determined, and then the amplitude of the multi-stage voltage sag is calculated equivalently. The multi-stage voltage sag frequency evaluation method is proposed in the distribution network. Finally, the IEEE 30-bus system is used for simulation verification. The results show that the proposed method is more accurate and reasonable for the evaluation of multi-stage voltage sag frequency. This method provides a basis for the frequency evaluation and governance of multi-stage voltage sag.

Zone‐based failure risk assessment of fatigue crack growth caused by initial defects in powder turbine disc

AbstractIn this research, A zone‐based failure risk assessment (FRA) method of fatigue crack growth (FCG) caused by initial defects in the FGH96 alloy turbine disc is developed. Firstly, the initial defects distribution in the FGH96 alloy turbine disc is calculated based on the defect data. Subsequently, a probabilistic short FCG life model is established, taking into account the dispersion in grain size. Meanwhile, a probabilistic long FCG life model is established, incorporating the life dispersion factor. To calculate the Stress Intensity Factor (SIF) at any position of the disc, the general weight function method and the rectangular plate model are established. Finally, the zoning process is established, enabling a FRA that considers the FCG due to initial defects. The results indicate that the number of cycles corresponding to a 0.13% failure probability of the turbine disc is 7150, and the percentage of failures in each zone is analyzed.

Digital twin-driven framework for fatigue life prediction of welded structures considering residual stress

The welding of steel generates substantial welding residual stress (WRS), which exerts a significant impact on the fatigue life of steel bridges. In this study, a physical model for calculating the fatigue crack growth (FCG) life of welded specimens in the WRS field is established based on the weight function method. Experimental data validates the reliability and precision of the proposed physical model. The impact of WRS on the fatigue life of structural components is scrutinized and analyzed. On this basis, a digital twin (DT) framework driven by a physical-data model is proposed to consider the inherent parameter uncertainty in FCG behavior within the WRS domain. A dynamic Bayesian network (DBN) is used to characterize the evolution characteristics of fatigue crack states over time in the digital space. Particle filter algorithm is used as DBN inference method. The results show that the calculation of the physical model is in good agreement with the experimental values. Neglecting the influence of WRS distribution may lead to an overestimation of fatigue life in welded structures. The DT framework can update uncertain parameters online and realize the accurate prediction of the FCG life in the WRS field.

A HILBERT-TYPE LOCAL FRACTIONAL INTEGRAL INEQUALITY WITH THE KERNEL OF A HYPERBOLIC COSECANT FUNCTION

By using Yang’s local fractional calculus theory, the method of weight function, and real-analysis techniques in the fractal set, a general Hilbert-type local fractional integral inequality with the kernel of a hyperbolic cosecant function is established. The necessary and sufficient condition for the constant factor of the general Hilbert-type local fractional integral inequality to be the best possible is discovered. Furthermore, two equivalent inequalities with the best constant factors were obtained.

Strategic Analysis of Profit Structure and Quality Improvement in Digital Transformation of Advanced Manufacturing Industries

Abstract Digital transformation is the engine to promote the high-quality development of the advanced manufacturing industry. This paper constructs a measurement model for the development level of digital transformation in the advanced manufacturing industry and proposes a measurement method for the development level of digital economy and digital transformation in the advanced manufacturing industry. The data are preprocessed through text mining, the vertical and horizontal pull-out grade method is used to determine the weights and calculate the final measurement value, and the time weights are obtained with the help of the nonlinear programming method to realize the measurement of the development level of the digital economy. Use the maximum-minimum value method to process the data related to digital transformation, utilize the entropy value method to calculate the index weights, and calculate the digital transformation development level through the linear weighting function method. The representative provinces of each region are selected to analyze the digital transformation of the advanced manufacturing industry in the east, middle, and west regions of China. The composite index of the digital economy development level in the East, Central, and West is 0.62, 0.45, and 0.37, respectively. The value of the digital transformation development level of Guangdong, Shandong, and Sichuan in 2023 reaches 0.9857, 0.3949, and 0.3747, respectively, ranking in the top three, whereas Jilin and Guizhou are only 0.1479 and 0.1128, and the level of digitalization of China’s advanced manufacturing industry is marked by obvious geographical imbalance.

Weight function analyses and stress intensity factors for corner/surface crack in straight and tapered lugs

AbstractLug is one of the most important lap joints in many engineering structures, especially airframes. Through‐thickness and part‐through cracks emanating from pin‐loaded lug hole are frequently encountered due to stress concentration. Stress intensity factors for various crack geometries are the prerequisite for damage tolerance design of attachment lug. In this paper, analytical 2D weight functions for through‐thickness crack in lug are derived considering the contact condition between the pin and lug. With the derived 2D weight function, a highly efficient 3D slice synthesis weight function method (SSWFM) is then developed to determine 3D SIFs of hole‐edge corner/surface crack in straight and tapered lugs under various pin loadings. The resulting 3D SIFs are verified by comparisons with the numerical FEM/Franc3D solutions and literature data. The SSWFM is about 1800 times faster than the numerical FEM/Franc3D and therefore will significantly enhance damage tolerance analysis capability for attachment lugs.

A review of three‐dimensional weight function methods for the analysis of various surface/corner crack problems

AbstractThe stress intensity factor (SIF) is the foundation of fracture mechanics analysis. Accurate determination of SIFs is at the very heart of damage tolerance design and fatigue crack growth life prediction. The weight function method (WFM) is a powerful method for SIF‐determination involving complex load conditions. This article presents a review of the historical development over the past five decades and the current state‐of‐the‐art in three‐dimensional (3D) WFMs. The discussions are focused on the slice synthesis weight function method (SSWFM) and the point weight function method (PWFM). The powerfulness of the 3D WFMs is shown by a variety of examples with complex part‐through crack configurations under uni‐ and bi‐variant loadings. Solution accuracy is verified by comparisons of SIFs with various numerical methods. Use of the substitute geometry concept to expand the capability of 3D WFMs for solving real‐world engineering 3D crack problems is demonstrated. Some remaining challenges are briefly discussed.

Multimodal discourse analysis of English reading instruction in colleges and universities based on weighted function algorithm

Abstract This paper proposes a kernel clustering method that uses the weighted function method to address the problem of excessive similarity calculation in the kernel function clustering process. Based on function-based clustering with orthogonal basis expansion, function-based principal component analysis is used to reduce the dimensionality of function-based data and extract the first few principal components that can contain the most information of the original data. The factors of each component are assigned with characteristic weights so that the scores of the principal components replace the original functional data for clustering analysis of English reading teaching in colleges and universities. A systematic functional language framework was used to conduct multimodal discourse analysis based on clustering results. The symbolic annotation of pure language reached 14.5, and the symbolic resources of visual imagery reached 85.7 during the CB stage of English reading instruction. The number of semiotic resources was relatively small, so the complexity of the modality was lower than in other stages.

Development types and design guidelines for the conservation and utilization of spatial environment in traditional villages in Southern China

ABSTRACT In China, despite the continuous efforts to preserve traditional villages in recent years, a significant amount of cultural heritages continue to vanish. Commonly observed issues such as the lack of coordination between preservation and development, the imbalance in spatial environment conservation and utilization, are pervasive. Regrettably, there is a conspicuous absence of theoretical underpinning in addressing these challenges. In this study, The evaluation factor of spatial environmental protection was selected, and these factors were weighted through expert surveys and the Analytic Hierarchy Process, and applied the multi-objective linear weighting function method to obtain a comprehensive evaluation calculation formula. Subsequently, we conducted an assessment of 28 representative traditional villages in the southern Chinese regions of Guangdong, Fujian, and Jiangxi. Their current issues were systematically analyzed and cluster analysis was applied to discern 11 distinct typologies of traditional village development. Besides, the temporal evolution patterns of these villages were delved into, and their development were categorized into five stages: initiation, enhancement, prosperity, over-development, and decline. Ultimately, based on the spatial environment characteristics of each development type and evolution stage, corresponding guidelines for spatial environment conservation and utilization are proposed, which provides necessary theoretical frameworks for future work this region.