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Numerical three-dimensional formulations using vector potential A have been examined for magnetic fields, with emphasis on the finite difference (FDM) and edge element (EEM) methods, with the view to establish common features. The paper aims to discuss these issues.

It has been shown that for hexahedral elements the FDM equations may be presented in the form similar to the EEM equations, providing the products of the nodal potentials and distances between the nodes are used as unknowns in FDM, instead of the usual nodal potentials.

The analogy between the FDM and the EEM approach has been established.

It has been demonstrated, following from this and previous publications, that analogy exists between all fundamental methods of field solutions relying on space discretisation. This is helpful in terms of classification of the methods and aids the understanding of physical processes involved.

The purpose of this paper is to provide a brief introduction of the finite difference based parabolic equation (PE) modeling to the advanced engineering students and academic researchers.

A three-dimensional parabolic equation (3DPE) model is developed from the ground up for modeling wave propagation in the tunnel via a rectangular waveguide structure. A discussion of vector wave equations from Maxwell’s equations followed by the paraxial approximations and finite difference implementation is presented for the beginners. The obtained simulation results are compared with the analytical solution.

It is shown that the alternating direction implicit finite difference method (FDM) is more efficient in terms of accuracy, computational time and memory than the explicit FDM. The reader interested in maximum details of individual contributions such as the latest achievements in PE modeling until 2021, basic PE derivation, PE formulation’s approximations, finite difference discretization and implementation of 3DPE, can learn from this paper.

For the purpose of this paper, a simple 3DPE formulation is presented. For simplicity, a rectangular waveguide structure is discretized with the finite difference approach as a design problem. Future work could use the PE based FDM to study the possibility of utilization of meteorological techniques, including the effects of backward traveling waves as well as making comparisons with the experimental data.

The proposed work is directly applicable to typical problems in the field of tunnel propagation modeling for both national commercial and military applications.

Finite difference method (FDM) is a very useful and simple tool in determining electrical potential field of two‐dimensional geometries, such as integrated circuit (IC) planar resistors. It is very accurate and its accuracy can be easily controlled by changing the grid size. One limitation of the FDM, however, is that it computes potentials at predetermined grid points only. Unlike the finite element method (FEM), it does not compute potential functions that can be used to interpolate potentials at the points that are not located at the grid, or to use these functions in determining other quantities based upon the computed potential such as electric field intensity. This paper describes a method that is a combination of the FDM and FEM. It retains the simplicity and accuracy of the FDM. Yet, like the FEM, it provides potential functions that can be used for interpolation and post‐processing of potential. The combined FDM‐FEM method is used to determine the potential functions of an IC planar resistor. The results are in agreement with analytically derived results. The approach we have developed is simple yet accurate and thus of use in professional engineering work.

This paper employs a hybrid numerical method combining the differential transformation method (DTM) and the finite difference method (FDM) to study the bifurcation and nonlinear behavior of a rigid rotor supported by a relative short gas lubricated journal bearing system with herringbone grooves. The analysis reveals a complex dynamic behavior comprising periodic, subharmonic and quasi‐periodic responses of the rotor center. The dynamic behavior of the bearing system varies with changes in the rotor mass and bearing number. The current analytical results are found to be in good agreement with those of other numerical methods. This paper discusses these issues.

In this paper, DT is used to deal Reynolds equation and is also one of the most widely used techniques for solving differential equations due to its rapid convergence rate and minimal calculation error. A further advantage of this method over the integral transformation approach is its ability to solve nonlinear differential equations. In solving the Reynolds equation for the current gas bearing system, DTM is used for taking transformation with respect to the time domain τ, and then the FDM is adopted to discretize with respect to the directions of coordinates.

From the Poincaré maps of the rotor center as calculated by the DTM&FDM method with different values of the time step, it can be seen that the rotor center orbits are in agreement to approximately four decimal places for the different time steps. The numerical studies also compare the results obtained by the SOR&FDM and DTM&FDM methods for the orbits of the rotor center. It is observed that the results calculated by DTM&FDM are more accurately than SOR&FDM. Therefore, the DTM&FDM method suits this gas bearing system and provides better convergence than SOR&FDM method.

This study utilizes a hybrid numerical scheme comprising the DTM and the FDM to analyze nonlinear dynamic behavior of a relative short gas lubricated journal bearing system with herringbone grooves. The system state trajectory, phase portraits, the Poincaré maps, the power spectra, and the bifurcation diagrams reveal the presence of a complex dynamic behavior comprising periodic, subharmonic and quasi‐periodic responses of the rotor center. Therefore, the proposed method provides an effective means of gaining insights into the nonlinear dynamics of relative short gas lubricated journal bearing systems with herringbone grooves.

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 deals with structural shape and thickness optimization of axisymmetric shell structures loaded symmetrically. In the finite element stress analysis use is made of newly developed linear, quadratic, and cubic, variable thickness, C(0) elements based on axisymmetric Mindlin‐Reissner shell theory. An integrated approach is used to carry out the whole shape optimization process in a fully automatic manner. A robust, versatile and flexible mesh generator is incorporated with facilities for generating either uniform or graded meshes, with constant, linear, or cubic variation of thickness, pressure etc. The midsurface geometry and thickness variations of the axisymmetric shell structure are defined using cubic splines passing through certain key points. The design variables are chosen as the coordinates and/or the thickness at the key points. Variable linking procedures are also included. Sensitivity analysis is carried out using either a semi‐analytical method or a global finite difference method. The objective of the optimization is the weight minimization of the structure. Several examples are presented illustrating optimal shapes and thickness distributions for various shells. The changes in the bending, membrane and shear strain energies during the optimization process are also monitored.

Finite difference and finite element methods have serious limitations when applied to unbounded regions. This paper describes a hybrid method which uses a conformal transformation to map the original boundaries, including those at infinity, to a bounded region and only then applies a numerical method based on finite differences or finite elements when no direct solution is obvious. Testing this approach by means of examples for which exact solutions are obtainable, the hybrid method is applied to determine the electrical potential at specific points in the field of a capacitor with long plates that in their cross‐sectional view are parallel to each other, and in the field of a microstrip line at some distance from it. In both the cases, the results are in agreement with analytically derived results. The method is simple, readily applied by undergraduate students, yet accurate and thus of use in professional engineering work as well.

The purpose of this paper is the application of the finite difference method (FDM) for numerical modeling of the microscale heat transfer processes occurring in the domain of thin metal film subjected to a laser pulse. The problem discussed is described by the different variants of the second-order dual-phase-lag equation (DPLE). The laser action is taken into account by the introduction of internal volumetric heat source to the governing equation. The capacity of the source is dependent on the geometrical co-ordinates and duration of the laser beam. The modified forms of DPLE presented in the paper, resulting from the certain substitutions introduced to the basic equation.

At the stage of numerical computations, the different variants of the FDM are applied. Both the explicit and implicit FDM schemes are used. The formula determining the capacity of the internal heat source suggests the formulation of the task discussed using the cylindrical co-ordinate system. The in-house programs realizing the numerical computations concern the axially-symmetrical tasks. In this paper, the metal films made of the nickel and gold are considered.

The algorithms presented make possible to analyze the heating/cooling processes occurring in the domain of metal film having a thickness Z for the different laser parameters (laser intensity, characteristic time of laser pulse and laser beam radius) and the different materials (optical penetration depth, reflectivity of irradiated surface, lag times, thermal conductivity and volumetric specific heat).

Not for all metals, one can find information on lag times. In the literature, analytical formulas can be found to calculate these values, but they are strongly approximated. It should be pointed out that there are some limitations concerning the delay times of material considered, which assure the physical correctness of the second-order DPLE.

The FDM algorithm concerns the three-dimensional cylindrical domain while a large majority of the second-order DPLE numerical solutions have been obtained for the one-dimensional tasks. Both the implicit and explicit numerical schemes are proposed and the testing computations confirm the correctness and effectiveness of the algorithms presented.

This paper aims to compare different dynamical models, cavitation procedures and numerical methods to simulate hydrodynamic lubricated bearings of internal combustion engines.

Two dynamical models are considered for the main bearing of combustion engines. The first is a fluid-structure interaction multi-body dynamics coupled with lubricated bearings, where the equilibrium and Reynolds equations are solved together. The second model finds the equilibrium position of the bearing subjected to previously calculated dynamical loads. The Traditional p-? procedure and Giacopini’s model described in Giacopini et al. (2010) are adopted for cavitation purposes. The influence of the finite difference and finite element numerical methods is investigated.

Simulations were carried out considering small-, mid- and large-sized engines and the dynamical models differed mainly in predicting the journal orbits. Finite element method with Giacopini’s cavitation model had improved numeric stability for the three engines.

The dynamic models do not consider the flexibility of the components of the main mechanism of combustion engines which may overestimate the oil pressure and journal orbits.

It can help researchers and engineers to decide which combination of methods is best suited for their needs and the implications associated with each one.

The used methods may help engineers to design better and more efficient combustion engines.

This paper helps practitioners to understand the effects of different methods on the results. Additionally, depending on the engine, one approach can be more effective than the other.

This article aims to develop a fast numerical method for solving the one-dimensional heat and mass transfer problem within a desiccant rotor.

The collocation method is used for discretizing the axial dimension and reducing the number of dependent variables. The resulting system of equation is then solved through backward differentiation formulas.

The numerical results obtained here focus on verifying the accuracy and the computation time of the proposed method with respect to the finite difference method. The proposed numerical solution method resulted faster than, and as much accurate as, the finite difference method, over a large range of operating conditions that are of interest in desiccant cooling applications.

For heat and mass transfer analysis, constant average transfer coefficients are used. The results are calculated for NTU between 2 and 15 and for Le number between 0.5 and 2.

The results can be used in designing desiccant heat exchangers and desiccant cooling systems including complex rotor arrangements.

Different from other simplified solution techniques, the proposed method relies on few parameters that retain physical meaning and applies also to complex rotor configurations.