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This study aims to investigate two newly developed (3 + 1)-dimensional Kairat-II and Kairat-X equations that illustrate relations with the differential geometry of curves and equivalence aspects.

The Painlevé analysis confirms the complete integrability of both Kairat-II and Kairat-X equations.

This study explores multiple soliton solutions for the two examined models. Moreover, the author showed that only Kairat-X give lump solutions and breather wave solutions.

The Hirota’s bilinear algorithm is used to furnish a variety of solitonic solutions with useful physical structures.

This study also furnishes a variety of numerous periodic solutions, kink solutions and singular solutions for Kairat-II equation. In addition, lump solutions and breather wave solutions were achieved from Kairat-X model.

The work formally furnishes algorithms for studying newly constructed systems that examine plasma physics, optical communications, oceans and seas and the differential geometry of curves, among others.

This paper presents an original work that presents two newly developed Painlev\'{e} integrable models with insightful findings.

The primary objective of this study is to tackle the enduring challenge of preserving feature integrity during the manipulation of geometric data in computer graphics. Our work aims to introduce and validate a variational sparse diffusion model that enhances the capability to maintain the definition of sharp features within meshes throughout complex processing tasks such as segmentation and repair.

We developed a variational sparse diffusion model that integrates a high-order L₁ regularization framework with Dirichlet boundary constraints, specifically designed to preserve edge definition. This model employs an innovative vertex updating strategy that optimizes the quality of mesh repairs. We leverage the augmented Lagrangian method to address the computational challenges inherent in this approach, enabling effective management of the trade-off between diffusion strength and feature preservation. Our methodology involves a detailed analysis of segmentation and repair processes, focusing on maintaining the acuity of features on triangulated surfaces.

Our findings indicate that the proposed variational sparse diffusion model significantly outperforms traditional smooth diffusion methods in preserving sharp features during mesh processing. The model ensures the delineation of clear boundaries in mesh segmentation and achieves high-fidelity restoration of deteriorated meshes in repair tasks. The innovative vertex updating strategy within the model contributes to enhanced mesh quality post-repair. Empirical evaluations demonstrate that our approach maintains the integrity of original, sharp features more effectively, especially in complex geometries with intricate detail.

The originality of this research lies in the novel application of a high-order L₁ regularization framework to the field of mesh processing, a method not conventionally applied in this context. The value of our work is in providing a robust solution to the problem of feature degradation during the mesh manipulation process. Our model’s unique vertex updating strategy and the use of the augmented Lagrangian method for optimization are distinctive contributions that enhance the state-of-the-art in geometry processing. The empirical success of our model in preserving features during mesh segmentation and repair presents an advancement in computer graphics, offering practical benefits to both academic research and industry applications.

Additive manufacturing (AM), a rapidly evolving paradigm, has shown significant advantages over traditional subtractive processing routines by allowing for the custom creation of structural components with enhanced performance. Numerous studies have shown that the technical qualities of AM components are profoundly affected by the discovery of novel metastable substructures in diverse alloys. Therefore, the purpose of this study is to determine the effect of cell structure parameters on its mechanical response.

Initially, a methodology was suggested for testing porous materials, focusing on static tensile testing. For a qualitative evaluation of the cellular structures produced, computed tomography (CT) was used. Then, the CT scanner was used to analyze a sample and determine its actual relative density, as well as perform a detailed geometric analysis.

The experimental research demonstrates that the mechanical properties of a cell’s structure are significantly influenced by its shape during formation. It was also determined that using selective laser melting to produce cell structures with a minimum single-cell size of approximately 2 mm would be the most appropriate method.

Further studies of cellular structures for testing their static tensile strength are planned for the future. The study will be carried out for a larger number of samples, taking into account a wider range of cellular structure parameters. An important step will also be the verification of the results of the static tensile test using numerical analysis for the model obtained by CT scanning.

The fabrication of metallic parts with different cellular structures is very important with a selective laser melted machine. However, the determination of cell size and structure with mechanical properties is quiet novel in this current investigation.

Feature extraction from 3D datasets is a current problem. Machine learning is an important tool for classification of complex 3D datasets. Machine learning classification techniques are widely used in various fields, such as text classification, pattern recognition, medical disease analysis, etc. The aim of this study is to apply the most popular classification and regression methods to determine the best classification and regression method based on the geodesics.

The feature vector is determined by the unit normal vector and the unit principal vector at each point of the 3D surface along with the point coordinates themselves. Moreover, different examples are compared according to the classification methods in terms of accuracy and the regression algorithms in terms of R-squared value.

Several surface examples are analyzed for the feature vector using classification (31 methods) and regression (23 methods) machine learning algorithms. In addition, two ensemble methods XGBoost and LightGBM are used for classification and regression. Also, the scores for each surface example are compared.

To the best of the author’s knowledge, this is the first study to analyze datasets based on geodesics using machine learning algorithms for classification and regression.

Surface curvature is needed to analyze the range data of real objects and is widely applied in object recognition and segmentation, robotics, and computer vision. Therefore, it is not easy to estimate the curvature of the scanned data. In recent years, machine learning classification methods have gained importance in various fields such as finance, health, engineering, etc. The purpose of this study is to classify surface points based on principal curvatures to find the best method for determining surface point types.

A feature selection method is presented to find the best feature vector that achieves the highest accuracy. For this reason, ten different feature selections are used and six sample datasets of different sizes are classified using these feature vectors.

The author examined the surface examples based on the feature vector using the machine learning classification methods. Also, the author compared the results for each experiment.

To the best of the author's knowledge, this is the first study to examine surface points according to principal curvatures using machine learning classification methods.

A multi-laser sensors-based measurement instrument is proposed for the measurement of geometry errors of a differential body and quality evaluation. This paper aims to discuss the aforementioned idea.

First, the differential body is set on a rotation platform before measuring. Then one laser sensor called as “primary sensor”, is installed on the intern of the differential body. The spherical surface and four holes on the differential body are sampled by the primary sensor when the rotation platform rotates one revolution. Another sensor called as “secondary sensor”, is installed above to sample the external cylinder surface and the planar surface on the top of the differential body, and the external cylinder surface and the planar surface are high in manufacturing precision, which are used as datum surfaces to compute the errors caused by the motion of the rotation platform. Finally, the sampled points from the primary sensor are compensated to improve the measurement accuracy.

A multi-laser sensors-based measurement instrument is proposed for the measurement of geometry errors of a differential body. Based on the characteristics of the measurement data, a gradient image-based method is proposed to distinguish different objects from laser measurement data. A case study is presented to validate the measurement principle and data processing approach.

The study investigates the possibility of correction of sensor data by the measurement results of multiple sensors to improving measurement accuracy. The proposed technique enables the error analysis and compensation by the geometric correlation relationship of various features on the measurand.

The proposed error compensation principle by using multiple sensors proved to be useful for the design of new measurement device for special part inspection. The proposed approach to describe the measuring data by image also is proved to be useful to simplify the measurement data processing.

Bi-slant submanifolds of S-manifolds are introduced, and some examples of these submanifolds are presented.

Some properties of D_i-geodesic and D_i-umbilical bi-slant submanifolds are examined.

The Riemannian curvature invariants of these submanifolds are computed, and some results are discussed with the help of these invariants.

The topic is original, and the manuscript has not been submitted to any other journal.

When the temperature of an air conditioning unit’s fin surface goes below its dew point temperature, condensation forms on the unit’s surface. As a result, the cooling coil’s performance is compromised. By altering the cross-section and heat conductivity of the fins, the performance of such systems can be improved. This study aims to analyze the thermal performance of longitudinal fins made up of a variable thickness (assuming constant weight) and functionally graded material.

Different grading parameters are considered for an exponential variation of thermal conductivity. The humidity ratio and the corresponding fin temperatures are assumed to follow a cubic relationship. The Bvp4c solver in MATLAB^® is used to solve the differential heat transfer equation resulting from balancing heat transfer in a small segment.

Validation of the methodology is provided by previous research presented in this area. For different combinations of grading parameters, geometry parameters and relative humidity, the normalized temperature distribution along the fin length and fin efficiency contours are plotted, and the results are very promising.

When compared to the efficiency of an isotropic homogenous rectangular longitudinal fin with optimal geometry and grading parameters, a 17% increase in efficiency under fully wet conditions is measured. When it comes to fin design, these efficiency contour plots are extremely useful.

The mixing of fuel and air plays a pivotal role in enhancing combustion in supersonic regime. Proper mixing stabilizes the flame and prevents blow-off. Blow-off is due to the shorter residence time of fuel and air in the combustor, as the flow is in supersonic regime. The flame is initiated in the local subsonic region created using a flameholder within the supersonic combustor. This study aims to design an effective flameholder which increases the residence time of fuel in the combustor allowing proper combustion preventing blow-off and other instabilities.

The geometry of the strut-based flameholder is altered in the present study to induce a streamwise motion of the fluid downstream of the strut. The streamwise motion of the fluid is initiated by the ramps and grooves of the strut geometry. The numerical simulations were carried out using ANSYS Fluent and are validated against the available experimental and numerical results of cold flow with hydrogen injection using plain strut as the flameholder. In the present study, numerical investigations are performed to analyse the effect on hydrogen injection in strut-based flameholders with ramps and converging grooves using Reynolds-averaged Navier–Stokes equation coupled with Menter’s shear stress transport k-ω turbulence model. The analysis is done to determine the effect of geometrical parameters and flow parameter on the flow structures near the base of the strut where thorough mixing takes place. The geometrical parameters under consideration include the ramp length, groove convergence angle, depth of the groove, groove compression angle and the Mach number. Two different strut configurations, namely, symmetric and asymmetric struts were also studied.

Higher turbulence and complex flow structures are visible in asymmetric strut configuration which develops better mixing of hydrogen and air compared to symmetric strut configuration. The variation in the geometric parameters develop changes in the fluid motion downstream of the strut. The fluid passing through the converging grooves gets decelerated thereby reducing the Mach number by 20% near the base of the strut compared to the straight grooved strut. The shorter ramps are found to be more effective, as the pressure variation in lateral direction is carried along the strut walls downstream of the strut increasing the streamwise motion of the fluid. The decrease in the depth of the groove increases the recirculation zone downstream of the strut. Moreover, the increase in the groove compression angle also increases the turbulence near the base of the strut where the fuel is injected. Variation in the injection port location increases the mixing performance of the combustor by 25%. The turbulence of the fuel jet stream is considerably changed by the increase in the injection velocity. However, the change in the flow field properties within the flow domain is marginal. The increase in fuel mass flow rate brings about considerable change in the flow field inducing stronger shock structures.

The present study identifies the optimum geometry of the strut-based flameholder with ramps and converging grooves. The reaction flow modelling may be performed on the strut geometry incorporating the design features obtained in the present study.

Present study aims to extend the lattice Boltzmann method (LBM) to simulate radiation in geometries with curved boundaries, as the first step to simulate radiation in complex porous media. In recent years, researchers have increasingly explored the use of porous media to improve the heat transfer processes. The lattice Boltzmann method (LBM) is one of the most effective techniques for simulating heat transfer in such media. However, the application of the LBM to study radiation in complex geometries that contain curved boundaries, as found in many porous media, has been limited.

The numerical evaluation of the effect of the radiation-conduction parameter and extinction coefficient on temperature and incident radiation distributions demonstrates that the proposed LBM algorithm provides highly accurate results across all cases, compared to those found in the literature or those obtained using the finite volume method (FVM) with the discrete ordinates method (DOM) for radiative information.

For the case with a conduction-radiation parameter equal to 0.01, the maximum relative error is 1.9% in predicting temperature along vertical central line. The accuracy improves with an increase in the conduction-radiation parameter. Furthermore, the comparison between computational performances of two approaches reveals that the LBM-LBM approach performs significantly faster than the FVM-DOM solver.

The difficulty of radiative modeling in combined problems involving irregular boundaries has led to alternative approaches that generally increase the computational expense to obtain necessary radiative details. To address the limitations of existing methods, this study presents a new approach involving a coupled lattice Boltzmann and first-order blocked-off technique to efficiently model conductive-radiative heat transfer in complex geometries with participating media. This algorithm has been developed using the parallel lattice Boltzmann solver.