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The purpose of this study is to investigate the post-buckling analysis of functionally graded columns by using three analytical, approximate and numerical methods. A pre-defined function as an initial assumption for the post-buckling path is introduced to solve the differential equation. The finite difference method is used to approximate the lateral deflection of the column based on the differential equation. Moreover, the finite element method is used to derive the tangent stiffness matrix of the column.

The non-linear buckling analysis of functionally graded materials is carried out by using three analytical, finite difference and finite element methods. The elastic deformation and Euler-Bernoulli beam theory are considered to establish the constitutive and kinematics relations, respectively. The governing differential equation of the post-buckling problem is derived through the energy method and the calculus variation.

An incremental iterative solution and the perturbation of the displacement vector at the critical buckling point are performed to determine the post-buckling path. The convergence of the finite element results and the effects of geometric and material characteristics on the post-buckling path are investigated.

The key point of the research is to compare three methods and to detect error sources by considering the derivation process of relations. This comparison shows that a non-incremental solution in the analytical and finite difference methods and an initial assumption in the analytical method lead to an error in results. However, the post-buckling path in the finite element method is traced by the updated tangent stiffness matrix in each load step without any initial limitation.

The purpose of this paper is to address unsteady heat conduction in two subsets of ordinary bodies. One subset consists of a large plane wall, a long cylinder and a sphere in one dimension. The other subset consists of a short cylinder and a large rectangular bar in two dimensions. The prevalent assumptions in the two subsets are: constant initial temperature, uniform surface heat flux and thermo-physical properties invariant with temperature. The engineering applications of the unsteady heat conduction deal with the determination of temperature–time histories in the two subsets using electric resistance heating, radiative heating and fire pool heating.

To this end, a novel numerical procedure named the enhanced method of discretization in time (EMDT) transforms the linear one-dimensional unsteady, heat conduction equations with non-homogeneous boundary conditions into equivalent nonlinear “quasi–steady” heat conduction equations having the time variable embedded as a time parameter. The equivalent nonlinear “quasi–steady” heat conduction equations are solved with a finite difference method.

Based on the numerical computations, it is demonstrated that the approximate temperature–time histories in the simple subset of ordinary bodies (large plane wall, long cylinder and sphere) exhibit a perfect matching over the entire time domain 0 < t < ∞ when compared against the rigorous exact temperature–time histories expressed by classical infinite series. Furthermore, using the method of superposition of solutions in the convoluted subset (short cylinder and large rectangular crossbar), the same level of agreement in the approximate temperature–time histories in the simple subset of ordinary bodies is evident.

The performance of the proposed EMDT coupled with a finite difference method is exhaustively assessed in the solution of the unsteady, one-dimensional heat conduction equations with prescribed surface heat flux for: a subset of one-dimensional bodies (plane wall, long cylinder and spheres) and a subset of two-dimensional bodies (short cylinder and large rectangular bar).

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.

This paper aims to construct a sixth-order weighted essentially nonoscillatory scheme for simulating the nonlinear degenerate parabolic equations in a finite difference framework.

To design this scheme, we approximate the second derivative in these equations in a different way, which of course is still in a conservative form. In this way, unlike the common practice of reconstruction, the approximation of the derivatives of odd order is needed to develop the numerical flux.

The results obtained by the new scheme produce less error compared to the results of other schemes in the literature that are recently developed for the nonlinear degenerate parabolic equations while requiring less computational times.

This research develops a new weighted essentially nonoscillatory scheme for solving the nonlinear degenerate parabolic equations in multidimensional space. Besides, any selection of the constants (sum equals one is the only requirement for them), named the linear weights, will obtain the desired accuracy.

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.

The space fractional PDEs (SFPDEs) play an important role in the fractional calculus field. Proposing a high-order, stable and flexible numerical procedure for solving SFPDEs is the main aim of most researchers. This paper devotes to developing a novel spectral algorithm to solve the FitzHugh–Nagumo models with space fractional derivatives.

The fractional derivative is defined based upon the Riesz derivative. First, a second-order finite difference formulation is used to approximate the time derivative. Then, the Jacobi spectral collocation method is employed to discrete the spatial variables. On the other hand, authors assume that the approximate solution is a linear combination of special polynomials which are obtained from the Jacobi polynomials, and also there exists Riesz fractional derivative based on the Jacobi polynomials. Also, a reduced order plan, such as proper orthogonal decomposition (POD) method, has been utilized.

A fast high-order numerical method to decrease the elapsed CPU time has been constructed for solving systems of space fractional PDEs.

The spectral collocation method is combined with the POD idea to solve the system of space-fractional PDEs. The numerical results are acceptable and efficient for the main mathematical model.

In recent times, there has been a growing interest in buoyancy-induced heat transfer within confined enclosures due to its frequent occurrence in heat transfer processes across diverse engineering disciplines, including electronic cooling, solar technologies, nuclear reactor systems, heat exchangers and energy storage systems. Moreover, the reduction of entropy generation holds significant importance in engineering applications, as it contributes to enhancing thermal system performance. This study, a numerical investigation, aims to analyze entropy generation and natural convection flow in an inclined square enclosure filled with Ag–MgO/water and Ag–TiO2/water hybrid nanofluids under the influence of a magnetic field. The enclosure features heated slits along its bottom and left walls. Following the Boussinesq approximation, the convective flow arises from a horizontal temperature difference between the partially heated walls and the cold right wall.

The governing equations for laminar unsteady natural convection flow in a Newtonian, incompressible mixture is solved using a Marker-and-Cell-based finite difference method within a customized MATLAB code. The hybrid nanofluid’s effective thermal conductivity and viscosity are determined using spherical nanoparticle correlations.

The numerical investigations cover various parameters, including nanoparticle volume concentration, Hartmann number, Rayleigh number, heat source/sink effects and inclination angle. As the Hartmann and Rayleigh numbers increase, there is a significant enhancement in entropy generation. The average Nusselt number experiences a substantial increase at extremely high values of the Rayleigh number and inclination.

This numerical investigation explores advanced applications involving various combinations of influential parameters, different nanoparticles, enclosure inclinations and improved designs. The goal is to control fluid flow and enhance heat transfer rates to meet the demands of the Fourth Industrial Revolution.

In a 90° tilted enclosure, the addition of 5% hybrid nanoparticles to the base fluid resulted in a 17.139% increase in the heat transfer rate for Ag–MgO nanoparticles and a 16.4185% increase for Ag–TiO2 nanoparticles compared to the base fluid. It is observed that a 5% nanoparticle volume fraction results in an increased heat transfer rate, influenced by variations in both the Darcy and Rayleigh numbers. The study demonstrates that the Ag–MgO hybrid nanofluid exhibits superior heat transfer and fluid transport performance compared to the Ag–TiO2 hybrid nanofluid. The simulations pertain to the use of hybrid magnetic nanofluids in fuel cells, solar cavity receivers and the processing of electromagnetic nanomaterials in enclosed environments.

Magneto-quasi-static fields emanated by inductive charging systems can be potentially harmful to the human body. Recent projects, such as TALAKO and MILAS, use the technique of wireless power transfer (WPT) to charge batteries of electrically powered vehicles. To ensure the safety of passengers, the exposing magnetic flux density needs to be measured in situ and compared to reference limit values. However, in the design phase of these systems, numerical simulations of the emanated magnetic flux density are inevitable. This study aims to present a tool along with a workflow, based on the Scaled-Frequency Finite Difference Time-Domain and Co-Simulation Scalar Potential Finite Difference schemes, to determine body-internal magnetic flux densities, electric field strengths and induced voltages into cardiac pacemakers. The simulations should be time efficient, with lower computational costs and minimal human workload.

The numerical assessment of the human exposure to magneto-quasi-static fields is computationally expensive, especially when considering high-resolution discretization models of vehicles and WPT systems. Incorporating human body models into the simulation further enhances the number of mesh cells by multiple millions. Hence, the number of simulations including all components and human models needs to be limited while efficient numerical schemes need to be applied.

This work presents and compares four exposure scenarios using the presented numerical methods. By efficiently combining numerical methods, the simulation time can be reduced by a factor of 3.5 and the required storage space by almost a factor of 4.

This work presents and discusses an efficient way to determine the exposure of human beings in the vicinity of wireless power transfer systems that saves computer simulation resources and human workload.

The purpose of this study is to investigate the sealing performance of S-CO₂ dry gas seals (DGSs) by considering the effects of pressure-induced deformation, thermal deformation and coupling deformation.

A hydrodynamic lubrication flow model of S-CO₂ DGS was established, and the model was solved using the finite difference and finite element methods. The pressure-induced deformation and thermal deformation of the sealing ring, as well as the sealing performance under the effects of pressure-induced deformation, thermal deformation and coupling deformation, were obtained.

The deformation of the sealing ring is mainly thermal deformation. The influence of pressure-induced deformation on leakage and gas film stiffness is greater than that of thermal deformation and coupling deformation. However, thermal deformation has a greater impact on friction torque and minimum film thickness than pressure-induced deformation and coupling deformation. The influence of deformations on sealing performance is important.

The sealing performance of S-CO₂ DGSs was analyzed considering the effect of pressure-induced deformation, thermal deformation and coupling deformation, which can provide a theoretical basis for S-CO₂ DGS optimization design.

The peer review history for this article is available at: https://publons.com/publon/10.1108/ILT-04-2023-0120/

Intumescent coatings are nowadays a dominant passive system used to protect structural materials in case of fire. Due to their reactive swelling behaviour, intumescent coatings are particularly complex materials to be modelled and predicted, which can be extremely useful especially for performance-based fire safety designs. In addition, many parameters influence their performance, and this challenges the definition and quantification of their material properties. Several approaches and models of various complexities are proposed in the literature, and they are reviewed and analysed in a critical literature review.

Analytical, finite-difference and finite-element methods for modelling intumescent coatings are compared, followed by the definition and quantification of the main physical, thermal, and optical properties of intumescent coatings: swelled thickness, thermal conductivity and resistance, density, specific heat capacity, and emissivity/absorptivity.

The study highlights the scarce consideration of key influencing factors on the material properties, and the tendency to simplify the problem into effective thermo-physical properties, such as effective thermal conductivity. As a conclusion, the literature review underlines the lack of homogenisation of modelling approaches and material properties, as well as the need for a universal modelling method that can generally simulate the performance of intumescent coatings, combine the large amount of published experimental data, and reliably produce fire-safe performance-based designs.

Due to their limited applicability, high complexity and little comparability, the presented literature review does not focus on analysing and comparing different multi-component models, constituted of many model-specific input parameters. On the contrary, the presented literature review compares various approaches, models and thermo-physical properties which primarily focusses on solving the heat transfer problem through swelling intumescent systems.

The presented literature review analyses and discusses the various modelling approaches to describe and predict the behaviour of swelling intumescent coatings as fire protection for structural materials. Due to the vast variety of available commercial products and potential testing conditions, these data are rarely compared and combined to achieve an overall understanding on the response of intumescent coatings as fire protection measure. The study highlights the lack of information and homogenisation of various modelling approaches, and it underlines the research needs about several aspects related to the intumescent coating behaviour modelling, also providing some useful suggestions for future studies.