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This paper aims to employ a multivariate nonlinear regression analysis to establish a predictive model for the final fracture area, while accounting for the impact of individual parameters.

This analysis is based on the numerical simulation data obtained, using the hybrid finite element–discrete element (FE–DE) method. The forecasting model was compared with the numerical results and the accuracy of the model was evaluated by the root mean square (RMS) and the RMS error, the mean absolute error and the mean absolute percentage error.

The multivariate nonlinear regression model can accurately predict the nonlinear relationships between injection rate, leakoff coefficient, elastic modulus, permeability, Poisson’s ratio, pore pressure and final fracture area. The regression equations obtained from the Newton iteration of the least squares method are strong in terms of the fit to the six sensitive parameters, and the model follow essentially the same trend with the numerical simulation data, with no systematic divergence detected. Least absolutely deviation has a significantly weaker performance than the least squares method. The percentage contribution of sensitive parameters to the final fracture area is available from the simulation results and forecast model. Injection rate, leakoff coefficient, permeability, elastic modulus, pore pressure and Poisson’s ratio contribute 43.4%, −19.4%, 24.8%, −19.2%, −21.3% and 10.1% to the final fracture area, respectively, as they increased gradually. In summary, (1) the fluid injection rate has the greatest influence on the final fracture area. (2)The multivariate nonlinear regression equation was optimally obtained after 59 iterations of the least squares-based Newton method and 27 derivative evaluations, with a decidability coefficient R2 = 0.711 representing the model reliability and the regression equations fit the four parameters of leakoff coefficient, permeability, elastic modulus and pore pressure very satisfactorily. The models follow essentially the identical trend with the numerical simulation data and there is no systematic divergence. The least absolute deviation has a significantly weaker fit than the least squares method. (3)The nonlinear forecasting model of physical parameters of hydraulic fracturing established in this paper can be applied as a standard for optimizing the fracturing strategy and predicting the fracturing efficiency in situ field and numerical simulation. Its effectiveness can be trained and optimized by experimental and simulation data, and taking into account more basic data and establishing regression equations, containing more fracturing parameters will be the further research interests.

The nonlinear forecasting model of physical parameters of hydraulic fracturing established in this paper can be applied as a standard for optimizing the fracturing strategy and predicting the fracturing efficiency in situ field and numerical simulation. Its effectiveness can be trained and optimized by experimental and simulation data, and taking into account more basic data and establishing regression equations, containing more fracturing parameters will be the further research interests.

The main objective of this paper is to investigate the response of human skin to an intense temperature drop at the surface. In addition, this paper aims to evaluate the efficiency of finite difference and finite volume methods in solving the highly nonlinear form of Pennes’ bioheat equation.

One-dimensional linear and nonlinear forms of Pennes’ bioheat equation with uniform grids were used to study the behavior of human skin. The specific heat capacity, thermal conductivity and blood perfusion rate were assumed to be linear functions of temperature. The nonlinear form of the bioheat equation was solved using the Newton linearization method for the finite difference method and the Picard linearization method for the finite volume method. The algorithms were validated by comparing the results from both methods.

The study demonstrated the capacity of both finite difference and finite volume methods to solve the one-dimensional and highly nonlinear form of the bioheat equation. The investigation of human skin’s thermal behavior indicated that thermal conductivity and blood perfusion rate are the most effective properties in mitigating a surface temperature drop, while specific heat capacity has a lesser impact and can be considered constant.

This paper modeled the transient heat distribution within human skin in a one-dimensional manner, using temperate-dependent physical properties. The nonlinear equation was solved with two numerical methods to ensure the validity of the results, despite the complexity of the formulation. The findings of this study can help in understanding the behavior of human skin under extreme temperature conditions, which can be beneficial in various fields, including medical and engineering.

Robotic arms’ interactions with the external environment are growing more intricate, demanding higher control precision. This study aims to enhance control precision by establishing a dynamic model through the identification of the dynamic parameters of a self-designed robotic arm.

This study proposes an improved particle swarm optimization (IPSO) method for parameter identification, which comprehensively improves particle initialization diversity, dynamic adjustment of inertia weight, dynamic adjustment of local and global learning factors and global search capabilities. To reduce the number of particles and improve identification accuracy, a step-by-step dynamic parameter identification method was also proposed. Simultaneously, to fully unleash the dynamic characteristics of a robotic arm, and satisfy boundary conditions, a combination of high-order differentiable natural exponential functions and traditional Fourier series is used to develop an excitation trajectory. Finally, an arbitrary verification trajectory was planned using the IPSO to verify the accuracy of the dynamical parameter identification.

Experiments conducted on a self-designed robotic arm validate the proposed parameter identification method. By comparing it with IPSO1, IPSO2, IPSOd and least-square algorithms using the criteria of torque error and root mean square for each joint, the superiority of the IPSO algorithm in parameter identification becomes evident. In this case, the dynamic parameter results of each link are significantly improved.

A new parameter identification model was proposed and validated. Based on the experimental results, the stability of the identification results was improved, providing more accurate parameter identification for further applications.

The purpose of this study is to obtain the optimal frequency for low-frequency transmission lines while minimizing losses and maintaining the voltage stability of low-frequency systems. This study also emphasizes a reduction in calculations based on mathematical approaches.

Telegrapher’s method has been used to reduce large calculations in low-frequency high-voltage alternating current (LF-HVac) lines. The static compensator (STATCOM) has been used to maintain voltage stability. For optimal frequency selection, a modified Jaya algorithm (MJAYA) for optimal load flow analysis was implemented.

The MJAYA algorithm performed better than other conventional algorithms and determined the optimum frequency selection while minimizing losses. Voltage stability was also achieved with the proposed optimal load flow (OLF), and statistical analysis showed that the proposed OLF reduces the frequency deviation and standard error of the LF-HVac lines.

The optimal frequency for LF-HVac lines has been achieved, Telegrapher’s method has been used in OLF, and STATCOM has been used in LF-HVac transmission lines.

In this paper, the authors study the nonlinear matrix equation Xp=Q±A(X-1+B)-1AT, that occurs in many applications such as in filtering, network systems, optimal control and control theory.

The authors present some theoretical results for the existence of the solution of this nonlinear matrix equation. Then the authors propose two iterative schemes without inversion to find the solution to the nonlinear matrix equation based on Newton's method and fixed-point iteration. Also the authors show that the proposed iterative schemes converge to the solution of the nonlinear matrix equation, under situations.

The efficiency indices of the proposed schemes are presented, and since the initial guesses of the proposed iterative schemes have a high cost, the authors reduce their cost by changing them. Therefore, compared to the previous scheme, the proposed schemes have superior efficiency indices.

Finally, the accuracy and effectiveness of the proposed schemes in comparison to an existing scheme are demonstrated by various numerical examples. Moreover, as an application, by using the proposed schemes, the authors can get the optimal controller state feedback of $x(t+1) = A x(t) + C v(t)$.

This paper aims to derive a reduced-order model for the heat transfer across the interface between a millimetric thermocapillary liquid bridge from silicone oil and the surrounding ambient gas.

Numerical solutions for the two-fluid model are computed covering a wide parametric space, making a total of 2,800 numerical flow simulations. Based on the computed data, a reduced single-fluid model for the liquid phase is devised, in which the heat transfer between the liquid and the gas is modeled by Newton’s heat transfer law, albeit with a space-dependent Biot function Bi(z), instead of a constant Biot number Bi.

An explicit robust fit of Bi(z) is obtained covering the whole range of parameters considered. The single-fluid model together with the Biot function derived yields very accurate results at much lesser computational cost than the corresponding two-phase fully-coupled simulation required for the two-fluid model.

Using this novel Biot function approach instead of a constant Biot number, the critical Reynolds number can be predicted much more accurately within single-phase linear stability solvers.

The Biot function for thermocapillary liquid bridges is derived from the full multiphase problem by a robust multi-stage fit procedure. The derived Biot function reproduces very well the theoretical boundary layer scalings.

The purpose of this paper is to provide a comprehensive review of a non-contact full-field optical measurement technique known as digital image correlation (DIC).

The approach of this review paper is to introduce the research pertaining to DIC. It comprehensively covers crucial facets including its principles, historical development, core challenges, current research status and practical applications. Additionally, it delves into unresolved issues and outlines future research objectives.

The findings of this review encompass essential aspects of DIC, including core issues like the subpixel registration algorithm, camera calibration, measurement of surface deformation in 3D complex structures and applications in ultra-high-temperature settings. Additionally, the review presents the prevailing strategies for addressing these challenges, the most recent advancements in DIC applications across quasi-static, dynamic, ultra-high-temperature, large-scale and micro-scale engineering domains, along with key directions for future research endeavors.

This review holds a substantial value as it furnishes a comprehensive and in-depth introduction to DIC, while also spotlighting its prospective applications.

Bayesian cubature (BC) has emerged to be one of most competitive approach for estimating the multi-dimensional integral especially when the integrand is expensive to evaluate, and alternative acquisition functions, such as the Posterior Variance Contribution (PVC) function, have been developed for adaptive experiment design of the integration points. However, those sequential design strategies also prevent BC from being implemented in a parallel scheme. Therefore, this paper aims at developing a parallelized adaptive BC method to further improve the computational efficiency.

By theoretically examining the multimodal behavior of the PVC function, it is concluded that the multiple local maxima all have important contribution to the integration accuracy as can be selected as design points, providing a practical way for parallelization of the adaptive BC. Inspired by the above finding, four multimodal optimization algorithms, including one newly developed in this work, are then introduced for finding multiple local maxima of the PVC function in one run, and further for parallel implementation of the adaptive BC.

The superiority of the parallel schemes and the performance of the four multimodal optimization algorithms are then demonstrated and compared with the k-means clustering method by using two numerical benchmarks and two engineering examples.

Multimodal behavior of acquisition function for BC is comprehensively investigated. All the local maxima of the acquisition function contribute to adaptive BC accuracy. Parallelization of adaptive BC is realized with four multimodal optimization methods.

The purpose of this study is to find the suitable trajectory path of the Numerical model of the Quadcopter. Quadcopters are widely used in various applications due to their compact size and ease of assembly. Because they are quite unstable, autonomous control systems would be used to overcome this problem. Modelling autonomous control is predominant as the research scope faces challenges because of its highly non-linear, multivariable system with 6 degree of freedom.

Quadcopters with antonym systems can operate in an unknown environment by overcoming unexpected disturbances. The first objective when designing such a system is to design an accurate mathematical model to describe the dynamics of the system. Newton’s law of motion was used to build the mathematical model of the system.

Establishment of the mathematical model and the physics behind a four propeller drone for the frame TAROT 650 carbon was done. Simulink model was developed based on the mathematical model for simulating the complete dynamics of the drone as well as location and gusts were included to check the stability.

The control response of the system was simulated numerically results are discussed. The trajectory path was found. The phases with their own parameters can be used to implement the mathematical model for another type of quadcopter model and achieve quick development.

Ensuring the early detection of structural issues in aircraft is crucial for preserving human lives. One effective approach involves identifying cracks in composite structures. This paper employs experimental modal analysis and a multi-variable Gaussian process regression method to detect and locate cracks in glass fiber composite beams.

The present study proposes Gaussian process regression model trained by the first three natural frequencies determined experimentally using a roving impact hammer method with crystal four-channel analyzer, uniaxial accelerometer and experimental modal analysis software. The first three natural frequencies of the cracked composite beams obtained from experimental modal analysis are used to train a multi-variable Gaussian process regression model for crack localization. Radial basis function is used as a kernel function, and hyperparameters are optimized using the negative log marginal likelihood function. Bayesian conditional probability likelihood function is used to estimate the mean and variance for crack localization in composite structures.

The efficiency of Gaussian process regression is improved in the present work with the normalization of input data. The fitted Gaussian process regression model validates with experimental modal analysis for crack localization in composite structures. The discrepancy between predicted and measured values is 1.8%, indicating strong agreement between the experimental modal analysis and Gaussian process regression methods. Compared to other recent methods in the literature, this approach significantly improves efficiency and reduces error from 18.4% to 1.8%. Gaussian process regression is an efficient machine learning algorithm for crack localization in composite structures.

The experimental modal analysis results are first utilized for crack localization in cracked composite structures. Additionally, the input data are normalized and employed in a machine learning algorithm, such as the multi-variable Gaussian process regression method, to efficiently determine the crack location in these structures.