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A fitting algorithm is presented that can be applied to any surface described numerically or analytically. The algorithm that is based on differential geometry is more robust and faster than the traditional kinetic‐model based algorithms and it also allows for more flexible initial conditions.

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.

When establishing a mathematical model to simulate solid mechanics, considering realistic geometries, special tools are needed to translate measured data, possibly with noise, into idealized geometrical entities. As an engineering application, wheel-rail contact interactions are fundamental in the dynamic modeling of railway vehicles. Many approaches used to solve the contact problem require a continuous parametric description of the geometries involved. However, measured wheel and rail profiles are often available as sets of discrete points. A reconstruction method is needed to transform discrete data into a continuous geometry.

The authors present an approximation method based on optimization to solve the problem of fitting a set of points with an arc spline. It consists of an initial guess based on a curvature function estimated from the data, followed by a least-squares optimization to improve the approximation. The authors also present a segmentation scheme that allows the method to increment the number of segments of the spline, trying to keep it at a minimal value, to satisfy a given error tolerance.

The paper provides a better understanding of arc splines and how they can be deformed. Examples with parametric curves and slightly noisy data from realistic wheel and rail profiles show that the approach is successful.

The developed methods have theoretical value. Furthermore, they have practical value since the approximation approach is better suited to deal with the reconstruction of wheel/rail profiles than interpolation, which most methods use to some degree.

The purpose of this paper is to review the mutual coupling of electromagnetic fields in the magnetic vector potential formulation with electric circuits in terms of (modified) nodal and loop analyses. It aims for an unified and generic notation.

The coupled formulation is derived rigorously using the concept of winding functions. Strong and weak coupling approaches are proposed and examples are given. Discretization methods of the partial differential equations and in particular the winding functions are discussed. Reasons for instabilities in the numerical time domain simulation of the coupled formulation are presented using results from differential-algebraic-index analysis.

This paper establishes a unified notation for different conductor models, e.g. solid, stranded and foil conductors and shows their structural equivalence. The structural information explains numerical instabilities in the case of current excitation.

The presentation of winding functions allows to generically describe the coupling, embed the circuit equations into the de Rham complex and visualize them by Tonti diagrams. This is of value for scientists interested in differential geometry and engineers that work in the field of numerical simulation of field-circuit coupled problems.

This paper seeks to give an outline about the geometric concept of electronic circuits, where the jump behavior of nonlinear circuits is emphasized.

A sketch of circuit theory in a differential geometric setting is given.

It is shown that the structure of circuit theory can be given in a much better way than by means of a description of circuits using concrete coordinates. Furthermore, the formulation of a concrete jump condition is given.

In this paper, an outline is given about the state of the art of nonlinear circuits from a differential geometric point of view. Moreover, differential geometric methods were applied to two example circuits (flip flop and multivibrator) and numerical results were achieved.

The purpose of this paper is to propose a new way of prototyping surfaces, taking the mathematical background into account, without involving drawing environments.

The authors thicken surfaces from a mathematical point of view to obtain solids. Next they look for an operative procedure to build virtual models and interchange files. The authors build a sample of Enneper thickened surface by fused deposition modelling and verify the prototype by reverse engineering techniques.

The authors provide a formulation able to thicken surfaces in mathematical terms. An operative procedure generates virtual solids and interchange files in the same environment. The approximations necessary for additive fabrication, such as triangulations and mesh geometry, can be chosen at this stage.

The approach is useful at the product/process development stage, in which surfaces are delivered by theoretical analysis. At this stage a prototype can give useful advice permitting functional tests. The limitation is that, when the mathematical formulation is not available, it is difficult to translate a concept without fundamentals of differential geometry.

Approximations of drawing environments typically lead to fault models, not ready for fabrication by additive manufacturing (AM) technologies, needing empiric, not at all obvious and not rapid repair interventions. The authors' approach eliminates this stage, permitting a faster and simple managing of modifications due to functional and technological requirements, that are frequent at concept stage. This leads to a time‐to‐market reduction in the course of product/process development.

This paper extends the capability of a mathematical approach to solve surface prototyping problems. By reducing the required stages, the proposed methodology finds a theoretical and practical shorter route to direct fabrication.

Cables are widely used, and they play a key role in complex electromechanical products such as vehicles, ships, aircraft and satellites. Cable design and assembly significantly impact the development cycle and assembly quality, which is be-coming a key element affecting the function of a product. However, there are various kinds of cables, with complex geo-metric configurations and a narrow assembly space, which can easily result in improper or missed assembly, an unreasonable layout or interference. Traditional serial design methods are inefficient and costly, and they cannot predict problems in installation and use. Based on physical modeling, computer-aided cable design and assembly can effectively solve these problems. This paper aims to address virtual assembly (VA) of flexible cables based on physical modeling.

Much research has focused recently on virtual design and assembly-process planning for cables. This paper systematically reviews the research progress and the current state of mechanical models, virtual design, assembly-process planning, collision detection and geometric configuration and proposes areas for further research.

In the first instance, the main research groups and typical systems are investigated, followed by extensive exploration of the major research issues. The latter can be reviewed from five perspectives: the current state of mechanical models, virtual design, assembly-process planning, collision detection and geometric configuration. Finally, the barriers that prevent successful application of VA are also discussed, and the future research directions are summarized.

This paper presents a comprehensive survey of the topics of VA of flexible cables based on physical modeling and investigates some new ideas and recent advances in the area.

The waste recycling industry increasingly relies on magnetic density separators. These devices generate an upward magnetic force in ferro-fluids allowing to separate the immersed particles according to their mass density. Recently, a new separator design has been proposed that significantly reduces the required amount of permanent magnet material. The purpose of this paper is to alleviate the undesired end-effects in this design by altering the shape of the ferromagnetic covers of the individual poles.

The paper represents the shape of the ferromagnetic pole covers with B-splines and defines a cost functional that measures the non-uniformity of the magnetic field in an area above the poles. The authors apply an iso-geometric shape optimization procedure, which allows us to accurately represent, analyze and optimize the geometry using only a few design variables. The design problem is regularized by imposing constraints that enforce the convexity of the pole cover shapes and is solved by a non-linear optimization procedure. The paper validates the implementation of the algorithm using a simplified variant of the design problem with a known analytical solution. The algorithm is subsequently applied to the problem posed.

The shape optimization attains its target and yields pole cover shapes that give rise to a magnetic field that is uniform over a larger domain.

This increased magnetic field uniformity is obtained at the cost of a pole cover shape that differs per pole. This limitation has negligible impact on the manufacturing of the separator. The new pole cover shapes therefore lead to improved performance of the density separation.

Due to the larger uniformity the generated field, these shapes should enable larger amounts of waste to be processed than the previous design.

This paper treats the shapes optimization of magnetic density separators systematically and presents new shapes for the ferromagnetic poles covers.

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.