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A new mesh generation procedure is suggested for the generation of 2D adaptive finite element meshes with strong element gradation and stretching effects. Metric tensors are employed to define and control the element characteristics during the mesh generation process. By using the metric tensor specification and a new, robust and refined advancing front triangulation kernel, triangles with nearly unit edge length with respect to the normalized space are generated. Highly graded and stretched elements can be generated without much difficulty and the operation complexity of the mesh generation process is exactly the same as the usual 2D advancing front mesh generator. A set of mesh quality enhancement procedures has also been suggested for the further improvement of the quality of the finite element meshes. A simple and effective mesh conversion scheme is used to convert the output triangular mesh to a pure quadrilateral mesh while all the essential element characteristics are preserved. Mesh generation examples show that high quality finite element meshes with element characteristics compatible with the specified metric tensors are generated within a reasonable time limit in a common small computing environment.

With the increasing demand for surveillance and smart transportation, drone technology has become the center of attraction for robotics researchers. This study aims to introduce a new path planning approach to drone navigation based on topology in an uncertain environment. The main objective of this study is to use the Ricci flow evolution equation of metric and curvature tensor over angular Riemannian metric, and manifold for achieving navigational goals such as path length optimization at the minimum required time, collision-free obstacle avoidance in static and dynamic environments and reaching to the static and dynamic goals. The proposed navigational controller performs linearly and nonlinearly both with reduced error-based objective function by Riemannian metric and scalar curvature, respectively.

Topology and manifolds application-based methodology establishes the resultant drone. The trajectory planning and its optimization are controlled by the system of evolution equation over Ricci flow entropy. The navigation follows the Riemannian metric-based optimal path with an angular trajectory in the range from 0° to 360°. The obstacle avoidance in static and dynamic environments is controlled by the metric tensor and curvature tensor, respectively. The in-house drone is developed and coded using C++. For comparison of the real-time results and simulation results in static and dynamic environments, the simulation study has been conducted using MATLAB software. The proposed controller follows the topological programming constituted with manifold-based objective function and Riemannian metric, and scalar curvature-based constraints for linear and nonlinear navigation, respectively.

This proposed study demonstrates the possibility to develop the new topology-based efficient path planning approach for navigation of drone and provides a unique way to develop an innovative system having characteristics of static and dynamic obstacle avoidance and moving goal chasing in an uncertain environment. From the results obtained in the simulation and real-time environments, satisfactory agreements have been seen in terms of navigational parameters with the minimum error that justifies the significant working of the proposed controller. Additionally, the comparison of the proposed navigational controller with the other artificial intelligent controllers reveals performance improvement.

In this study, a new topological controller has been proposed for drone navigation. The topological drone navigation comprises the effective speed control and collision-free decisions corresponding to the Ricci flow equation and Ricci curvature over the Riemannian metric, respectively.

In any machining process, the surface profile of the workpiece is continuously changing with respect to time and input parameters. In a conventional machining process, input parameters are feed and depth of cut whilst other parameters are considered to be constant throughout the process.

The direct and indirect participation of this instantaneous curvature can be used to optimize the strategy of cutting operation in terms of different parameters like heat generation-induced stresses, etc. The concepts of the metric tensor and Riemannian curvature tensor are made use in this study as a representation of curvature itself. The objective of this study is to create a mathematical methodology that can be implemented on a highly flexible machining process to find an optimum cutting strategy for a particular output parameter.

The study also includes different case studies for the validation of this newly introduced mathematical methodology.

The study will also find its position in other mechanical processes like forging and casting where instantaneous curvature affects various mechanical properties.

The purpose of this paper is to study the Bertotti–Kasner space-time and its geometric properties.

This paper is based on the features of λ-tensor and the technique of six-dimensional formalism introduced by Pirani and followed by W. Borgiel, Z. Ahsan et al. and H.M. Manjunatha et al. This technique helps to describe both the geometric properties and the nature of the gravitational field of the space-times in the Segre characteristic.

The Gaussian curvature quantities specify the curvature of Bertotti–Kasner space-time. They are expressed in terms of invariants of the curvature tensor. The Petrov canonical form and the Weyl invariants have also been obtained.

The findings are revealed to be both physically and geometrically interesting for the description of the gravitational field of the cylindrical universe of Bertotti–Kasner type as far as the literature is concerned. Given the technique of six-dimensional formalism, the authors have defined the Weyl conformal λW-tensor and discussed the canonical form of the Weyl tensor and the Petrov scalars. To the best of the literature survey, this idea is found to be modern. The results deliver new insight into the geometry of the nonstatic cylindrical vacuum solution of Einstein's field equations.

Most of the cloth simulation and modelling techniques rely on the energy function of the system. The geometric deformation is related to the energy function by the fabric material characteristics, which are usually difficult to measure directly. This paper discusses how the fabric material properties are related to the measurable mechanical properties of the fabric such as tensile modulus, Poisson's ratio etc. These properties are incorporated into a cloth simulator to produce draping results. The simulated image and real object are then compared to show the realism.

This paper is concerned with a numerical study on evolution of a deformed bubble and propagation of the fields around the bubble. The model uses a numerical procedure with the direct‐predictor method and the alternating dependent variables (ADV) skill developed by Li and Yan. The aim of this paper is to study the characteristics of interfacial transport for a deformed inviscid bubble rising in a quiescent hot or bi‐solution liquid. The effects of the bubble deformation on temperature and concentration fields are calculated and simulated. The results demonstrate that the current numerical procedure is effective for solving such unsteady deformation problems of bubble accompanied with heat and mass transfer.

Presents a finite element formulation of the layout optimization and design sensitivity applied to doubly‐curved shells of revolution. The objectives of the optimization are to maximize buckling pressures and first‐ply‐failure pressures. The problem is formulated and solved with the use of geometrically non‐linear transverse shear shell theory. However, the optimization method proposed limits the sensitivity analysis to a geometrically linear problem. Focuses special attention on the formulation of the optimization problem taking into account various factors, such as the form of geometrical and physical relations, types of design variables and the finite element discretization. Demonstrates several numerical examples to illustrate the capability of the proposed optimization procedures.

A 3D surface mesh generation scheme is suggested for the triangulation of general bi‐variate surfaces. The target surface to be meshed is represented as a union of bi‐variate sub‐surfaces and hence a wide range of surfaces can be modelled. Different useful features such as repeated curves, crack lines and surface branches are included in the geometrical and topological models to increase the flexibility of the mesh generation scheme. The surface metric tensor specification is employed to define and control the element characteristics in the mesh generation procedure. A robust metric triangulation kernel is used for parametric space mesh generation. The shape qualities of the sub‐surface meshes generated are then improved by using some ad hoc mesh quality enhancement schemes before they are combined together to form the final mesh. Numerical examples indicate that high quality surface meshes with rapid varying element size and stretching characteristics can be generated within a reasonable time limit in a few mesh adaptive iterations.

The authors set the goal to find the solution of the Eisenhart problem within the framework of three-dimensional trans-Sasakian manifolds. Also, they prove some results of the Ricci solitons, η-Ricci solitons and three-dimensional weakly  symmetric trans-Sasakian manifolds. Finally, they give a nontrivial example of three-dimensional proper trans-Sasakian manifold.

The authors have used the tensorial approach to achieve the goal.

A second-order parallel symmetric tensor on a three-dimensional trans-Sasakian manifold is a constant multiple of the associated Riemannian metric g.

The authors declare that the manuscript is original and it has not been submitted to any other journal for possible publication.

Cotton soliton is a newly introduced notion in the field of Riemannian manifolds. The object of this article is to study the properties of this soliton on certain contact metric manifolds.

The authors consider the notion of Cotton soliton on almost Kenmotsu 3-manifolds. The authors use a local basis of the manifold that helps to study this notion in terms of partial differential equations.

First the authors consider that the potential vector field is pointwise collinear with the Reeb vector field and prove a non-existence of such Cotton soliton. Next the authors assume that the potential vector field is orthogonal to the Reeb vector field. It is proved that such a Cotton soliton on a non-Kenmotsu almost Kenmotsu 3-h-manifold such that the Reeb vector field is an eigen vector of the Ricci operator is steady and the manifold is locally isometric to.

The results of this paper are new and interesting. Also, the Proposition 3.2 will be helpful in further study of this space.