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The purpose of this study is to present the effectiveness and robustness of a numerical algorithm based on the block Newton method for the nonlinear kinematic hardening rules adopted in modeling ductile materials.

Elastoplastic problems can be defined as a coupled problem of the equilibrium equation for the overall structure and the yield equations for the stress state at every material point. When applying the Newton method to the coupled residual equations, the displacement field and the internal variables, which represent the plastic deformation, are updated simultaneously.

The presented numerical scheme leads to an explicit form of the hardening behavior, which includes the evolution of the equivalent plastic strain and the back stress, with the internal variables. The features of the present approach allow the displacement field and the hardening behavior to be updated straightforwardly. Thus, the scheme does not have any local iterative calculations and enables us to simultaneously decrease the residuals in the coupled boundary value problems.

A pseudo-stress for the local residual and an algebraically derived consistent tangent are applied to elastic-plastic boundary value problems with nonlinear kinematic hardening. The numerical procedure incorporating the block Newton method ensures a quadratic rate of asymptotic convergence of a computationally efficient solution scheme. The proposed algorithm provides an efficient and robust computation in the elastoplastic analysis of ductile materials. Numerical examples under elaborate loading conditions demonstrate the effectiveness and robustness of the numerical scheme implemented in the finite element analysis.

This study aims to propose and numerically assess different ways of discretising a very weak formulation of the Poisson problem.

We use integration by parts twice to shift smoothness requirements to the test functions, thereby allowing low-regularity data and solutions.

Various conforming discretisations are presented and tested, with numerical results indicating good accuracy and stability in different types of problems.

This is one of the first articles to propose and test concrete discretisations for very weak variational formulations in primal form. The numerical results, which include a problem based on real MRI data, indicate the potential of very weak finite element methods for tackling problems with low regularity.

This study aims to ensure safe operation of buildings in the mining area.

The strain energy value was taken as one of the parameters characterizing the deformation process at critical stages in these problems and providing a link between them. Based on the data obtained for the structural element of loading diagrams and assessment of the stress–strain state of the structure as a whole, the maximum permissible horizontal deformations of the soil around the foundation are determined, at which the building elements reach the stress–strain state preceding the loss of bearing capacity. For this purpose, a parameter is used that characterizes the deformation process at the stages of critical deformation in these problems and provides a link between them. This parameter is the value of strain energy.

Based on the obtained force behavior diagrams of structural elements and assessment of the stress–strain state of the structure as a whole, the maximum permissible horizontal ground deformations in the vicinity of the foundation are determined, at which the building elements reach the stress–strain state preceding the loss of bearing capacity.

The research provides new data in the form of regularities of deformation behavior of building structures in the zones of mine workings. These data formed the basis for the normative documentation being developed. The research results were used for the development of internal instructions of a large mining enterprise.

The main focus is to provide a non-similar solution for the magnetohydrodynamic (MHD) flow of Casson fluid over a curved stretching surface through the novel technique of the artificial intelligence (AI)-based Lavenberg–Marquardt scheme of an artificial neural network (ANN). The effects of joule heating, viscous dissipation and non-linear thermal radiation are discussed in relation to the thermal behavior of Casson fluid.

The non-linear coupled boundary layer equations are transformed into a non-linear dimensionless Partial Differential Equation (PDE) by using a non-similar transformation. The local non-similar technique is utilized to truncate the non-similar dimensionless system up to 2nd order, which is treated as coupled ordinary differential equations (ODEs). The coupled system of ODEs is solved numerically via bvp4c. The data sets are constructed numerically and then implemented by the ANN.

The results indicate that the non-linear radiation parameter increases the fluid temperature. The Casson parameter reduces the fluid velocity as well as the temperature. The mean squared error (MSE), regression plot, error histogram, error analysis of skin friction, and local Nusselt number are presented. Furthermore, the regression values of skin friction and local Nusselt number are obtained as 0.99993 and 0.99997, respectively. The ANN predicted values of skin friction and the local Nusselt number show stability and convergence with high accuracy.

AI-based ANNs have not been applied to non-similar solutions of curved stretching surfaces with Casson fluid model, with viscous dissipation. Moreover, the authors of this study employed Levenberg–Marquardt supervised learning to investigate the non-similar solution of the MHD Casson fluid model over a curved stretching surface with non-linear thermal radiation and joule heating. The governing boundary layer equations are transformed into a non-linear, dimensionless PDE by using a non-similar transformation. The local non-similar technique is utilized to truncate the non-similar dimensionless system up to 2nd order, which is treated as coupled ODEs. The coupled system of ODEs is solved numerically via bvp4c. The data sets are constructed numerically and then implemented by the ANN.

A finite element computational methodology on a curved boundary using an efficient subparametric point transformation is presented. The proposed collocation method uses one-side curved and two-side straight triangular elements to derive exact subparametric shape functions.

Our proposed method builds upon the domain discretization into linear, quadratic and cubic-order elements using subparametric spaces and such a discretization greatly reduces the computational complexity. A unique subparametric transformation for each triangle is derived from the unique parabolic arcs via a one-of-a-kind relationship between the nodal points.

The novel transformation derived in this paper is shown to increase the accuracy of the finite element approximation of the boundary value problem (BVP). Our overall strategy is shown to perform well for the BVP considered in this work. The accuracy of the finite element approximate solution increases with higher-order parabolic arcs.

The proposed collocation method uses one-side curved and two-side straight triangular elements to derive exact subparametric shape functions.

A novel method for modelling permanent magnets is investigated based on numerical approximations with rational functions. This study aims to introduce the AAA algorithm and other recently developed, cutting-edge mathematical tools, which provide outstandingly fast and accurate numerical computation of potentials and vector fields.

First, the AAA algorithm is briefly introduced along with its main variants and other advanced mathematical tools involved in the modelling. Then, the analysis of a circular Halbach array with a one-pole pair is carried out by means of the AAA-least squares method, focusing on vector potential and flux density in the bore and validating results by means of classic finite element software. Finally, the investigation is completed by a finite difference analysis.

AAA methods for field analysis prove to be strikingly fast and accurate. Results are in excellent agreement with those provided by the finite element model, and the very good agreement with those from finite differences suggests future improvements. They are also easy programming; the MATLAB code is less than 200 lines. This indicates they can provide an effective tool for rapid analysis.

AAA methods in magnetostatics are novel, but their extension to analogous physical problems seems straightforward. Being a meshless method, it is unlikely that local non-linearities can be considered. An aspect of particular interest, left for future research, is the capability of handling inhomogeneous domains, i.e. solving general interface problems.

The authors use cutting-edge mathematical tools for the modelling of complex physical objects in magnetostatics.

This paper aims to model and analyze Jeffery Hamel’s channel flow with the magnetohydrodynamics second-grade hybrid nanofluid. Considering the importance of studying the velocity slip and temperature jump in the boundary conditions of the flow, which leads to results close to reality, this paper intends to analyze the mentioned topic in the convergent and divergent channels that have significant applications.

The examination is conducted on a EG-H_2 O <30%–70%> base fluid that contains hybrid nanoparticles (i.e. SWCNT-MWCNT). To ensure comprehensive results, this study also considers the effects of thermal radiation, thermal sink/source, rotating convergent-divergent channels and magnetic fields. Initially, the governing equations are formulated in cylindrical coordinates and then simplified to ordinary differential equations through appropriate transformations. These equations are solved using the Explicit Runge–Kutta numerical method, and the results are compared with previous studies for validation.

After the validation, the effect of the governing parameters on the temperature and velocity of the second-grade hybrid nanofluid has been investigated by means of various and comprehensive contours. In the following, the issue of entropy generation and its related graphical results for this problem is presented. The mentioned contours and graphs accurately display the influence of problem parameters, including velocity slip and temperature jump. Besides, when thermal radiation is introduced (Rd = +0.1 and Rd = +0.2), entropy generation in convergent-divergent channels decreases by 7% and 14%, respectively, compared to conditions without thermal radiation (Rd = 0). Conversely, increasing the thermal sink/source from 0 to 4 leads to an 8% increase in entropy generation at Q = 2 and a 17% increase at Q = 4 in both types of channels. The details of the analysis of contours and the entropy generation results are fully mentioned in the body of the paper.

There are many studies on convergent and divergent channels, but this study comprehensively investigates the effects of velocity slip and temperature jump and certainly, this geometry with the specifications presented in this paper has not been explored before. Among the other distinctive features of this paper compared to previous works, the authors can mention the presentation of velocity and temperature results in the form of contours, which makes the physical analysis of the problem simpler.

This paper aims to study qualitative properties and approximate solutions of a thermostat dynamics system with three-point boundary value conditions involving a nonsingular kernel operator which is called Atangana-Baleanu-Caputo (ABC) derivative for the first time. The results of the existence and uniqueness of the solution for such a system are investigated with minimum hypotheses by employing Banach and Schauder's fixed point theorems. Furthermore, Ulam-Hyers (UH) stability, Ulam-Hyers-Rassias UHR stability and their generalizations are discussed by using some topics concerning the nonlinear functional analysis. An efficiency of Adomian decomposition method (ADM) is established in order to estimate approximate solutions of our problem and convergence theorem is proved. Finally, four examples are exhibited to illustrate the validity of the theoretical and numerical results.

This paper considered theoretical and numerical methodologies.

This paper contains the following findings: (1) Thermostat fractional dynamics system is studied under ABC operator. (2) Qualitative properties such as existence, uniqueness and Ulam–Hyers–Rassias stability are established by fixed point theorems and nonlinear analysis topics. (3) Approximate solution of the problem is investigated by Adomain decomposition method. (4) Convergence analysis of ADM is proved. (5) Examples are provided to illustrate theoretical and numerical results. (6) Numerical results are compared with exact solution in tables and figures.

The novelty and contributions of this paper is to use a nonsingular kernel operator for the first time in order to study the qualitative properties and approximate solution of a thermostat dynamics system.

The purpose of this study is to develop a new numerical algorithm to simulate the phase-field model.

First, the derivative of the temporal direction is discretized by a second-order linearized finite difference scheme where it conserves the energy stability of the mathematical model. Then, the isogeometric collocation (IGC) method is used to approximate the derivative of spacial direction. The IGC procedure can be applied on irregular physical domains. The IGC method is constructed based upon the nonuniform rational B-splines (NURBS). Each curve and surface can be approximated by the NURBS. Also, a map will be defined to project the physical domain to a simple computational domain. In this procedure, the partial derivatives will be transformed to the new domain by the Jacobian and Hessian matrices. According to the mentioned procedure, the first- and second-order differential matrices are built. Furthermore, the pseudo-spectral algorithm is used to derive the first- and second-order nodal differential matrices. In the end, the Greville Abscissae points are used to the collocation method.

In the numerical experiments, the efficiency and accuracy of the proposed method are assessed through two examples, demonstrating its performance on both rectangular and nonrectangular domains.

This research work introduces the IGC method as a simulation technique for the phase-field crystal model.

Loitering aerial vehicle (LAV) swarm safety flight control is an unmanned system control problem under multiple constraints, which are derived to prevent the LAVs from suffering risks inside and outside the swarms. The computational complexity of the safety flight control problem grows as the number of LAVs and of the constraints increases. Besides some important constraints, the swarms will encounter with sudden appearing risks in a hostile environment. The purpose of this study is to design a safety flight control algorithm for LAV swarm, which can timely respond to sudden appearing risks and reduce the computational burden.

To address the problem, this paper proposes a distributed safety flight control algorithm that includes a trajectory planning stage using kinodynamic rapidly exploring random trees (KRRT*) and a tracking stage based on distributed model predictive control (DMPC).

The proposed algorithm reduces the computational burden of the safety flight control problem and can fast find optimal flight trajectories for the LAVs in a swarm even there are multi-constraints and sudden appearing risks.

The proposed algorithm did not handle the constraints synchronously, but first uses the KRRT* to handle some constraints, and then uses the DMPC to deal with the rest constraints. In addition, the proposed algorithm can effectively respond to sudden appearing risks by online re-plan the trajectories of LAVs within the swarm.