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Gives introductory remarks about chapter 1 of this group of 31 papers, from ISEF 1999 Proceedings, in the methodologies for field analysis, in the electromagnetic community. Observes that computer package implementation theory contributes to clarification. Discusses the areas covered by some of the papers ‐ such as artificial intelligence using fuzzy logic. Includes applications such as permanent magnets and looks at eddy current problems. States the finite element method is currently the most popular method used for field computation. Closes by pointing out the amalgam of topics.

This paper aims to develop a hybrid finite difference‐finite element method and apply it to solve the three‐dimensional energy equation in non‐isothermal fluid flow past over a tube.

To implement the hybrid scheme, the tube length is partitioned into uniform segments by choosing grid points along its length, and a plane perpendicular to the tube axis is drawn at each of the points. Subsequently, the Taylor‐Galerkin finite element technique is employed to discretize the energy equation in the planes; while the derivatives along the tube are discretized using the finite difference method.

To demonstrate the validity of the proposed numerical scheme, three‐dimensional test cases have been solved using the method. The variation of L²‐norm of the error with mesh refinement shows that the numerical solution converges to the exact solution with mesh refinement. Moreover, comparison of the computational time duration shows that the proposed method is approximately three times faster than the 3D finite element method. In the non‐isothermal fluid flow around a tube for Re=250 and Pr=0.7, the results show that the Nusselt number decreases with the increase in the tube length and, for the tube length greater than six times the tube diameter, the average Nusselt number converges to the value for the two‐dimensional case.

A hybrid finite difference‐finite element method has been developed and applied to solve the 3D transient energy equation for different test cases. The proposed method is faster, and computationally more efficient, compared with the 3D finite element method.

This study aimed to solve the engineering problem of free vibration disturbance and local mesh refinement induced by microcrack damage in circularly curved beams. The accurate identification of the crack damage depth, number and location depends on high-precision frequency and vibration mode solutions; therefore, it is critical to obtain these reliable solutions. The high-precision finite element method for the free vibration of cracked beams needs to be developed to grasp and control error information in the conventional solutions and the non-uniform mesh generation near the cracks. Moreover, the influence of multi-crack damage on the natural frequency and vibration mode of a circularly curved beam needs to be detected.

A scheme for cross-sectional damage defects in a circularly curved beam was established to simulate the depth, location and the number of multiple cracks by implementing cross-section reduction induced by microcrack damage. In addition, the h-version finite element mesh adaptive analysis method of the Timoshenko beam was developed. The superconvergent solution of the vibration mode of the cracked curved beam was obtained using the superconvergent patch recovery displacement method to determine the finite element solution. The superconvergent solution of the frequency was obtained by computing the Rayleigh quotient. The superconvergent solution of the eigenfunction was used to estimate the error of the finite element solution in the energy norm. The mesh was then subdivided to generate an improved mesh based on the error. Accordingly, the final optimised meshes and high-precision solution of natural frequency and mode shape satisfying the preset error tolerance can be obtained. Lastly, the disturbance behaviour of multi-crack damage on the vibration mode of a circularly curved beam was also studied.

Numerical results of the free vibration and damage disturbance of cracked curved beams with cracks were obtained. The influences of crack damage depth, crack damage number and crack damage distribution on the natural frequency and mode of vibration of a circularly curved beam were quantitatively analysed. Numerical examples indicate that the vibration mode and frequency of the beam would be disturbed in the region close to the crack damage, and a greater crack depth translates to a larger frequency change. For multi-crack beams, the number and distribution of cracks also affect the vibration mode and natural frequency. The adaptive method can use a relatively dense mesh near the crack to adapt to the change in the vibration mode near the crack, thus verifying the efficacy, accuracy and reliability of the method.

The proposed combination of methodologies provides an extremely robust approach for free vibration of beams with cracks. The non-uniform mesh refinement in the adaptive method can adapt to changes in the vibration mode caused by crack damage. Moreover, the proposed method can adaptively divide a relatively fine mesh at the crack, which is applied to investigating free vibration under various curved beam angles and crack damage distribution conditions. The proposed method can be extended to crack damage detection of 2D plate and shell structures and three-dimensional structures with cracks.

This study aimed to overcome the challenging issues involved in providing high-precision eigensolutions. The accurate prediction of the buckling load bearing capacity under different crack damage locations, sizes and numbers, and analysing the influence mechanism of crack damage on buckling instability have become the needs of theoretical research and engineering practice. Accordingly, a finite element method was developed and applied to solve the elastic buckling load and buckling mode of curved beams with crack damage. However, the accuracy of the solution depends on the quality of mesh, and the solution inevitably introduces errors due to mesh. Therefore, the adaptive mesh refinement method can effectively optimise the mesh distribution and obtain high-precision solutions.

For the elastic buckling of circular curved beams with cracks, the section damage defect analogy scheme of a circular arc curved beam crack was established to simulate the crack size (depth), position and number. The h-version finite element mesh adaptive analysis method of the variable section Euler–Bernoulli beam was introduced to solve the elastic buckling problem of circular arc curved beams with crack damage. The optimised mesh and high-precision buckling load and buckling mode solutions satisfying the preset error tolerance were obtained.

The results of testing typical examples show that (1) the established section damage defect analogy scheme of circular arc curved beam crack can effectively realise the simulation of crack size (depth), position and number. The solution strictly satisfies the preset error tolerance; (2) the non-uniform mesh refinement in the algorithm can be adapted to solve the arbitrary order frequencies and modes of cracked cylindrical shells under the conditions of different ring wave numbers, crack positions and crack depths; and (3) the change in the buckling mode caused by crack damage is applicable to the study of elastic buckling under various curved beam angles and crack damage distribution conditions.

This study can provide a novel strategy for the adaptive mesh refinement for finite element analysis of elastic buckling of circular arc curved beams with crack damage. The adaptive mesh refinement method established in this study is fundamentally different from the conventional finite element method which employs the user experience to densify the meshes near the crack. It can automatically and flexibly generate a set of optimised local meshes by iteratively dividing the fine mesh near the crack, which can ensure the high accuracy of the buckling loads and modes. The micro-crack in curved beams is also characterised by weakening the cross-sectional stiffness to realise the characterisation of locations, depths and distributions of multiple crack damage, which can effectively analyse the disturbance behaviour of different forms of micro-cracks on the dynamic behaviour of beams.

The purpose of this paper is to provide an efficient and robust second-order monotone hybrid scheme for singularly perturbed delay parabolic convection-diffusion initial boundary value problem.

The delay parabolic problem is solved numerically by a finite difference scheme consists of implicit Euler scheme for the time derivative and a monotone hybrid scheme with variable weights for the spatial derivative. The domain is discretized in the temporal direction using uniform mesh while the spatial direction is discretized using three types of non-uniform meshes mainly the standard Shishkin mesh, the Bakhvalov–Shishkin mesh and the Gartland Shishkin mesh.

The proposed scheme is shown to be a parameter-uniform convergent scheme, which is second-order convergent and optimal for the case. Also, the authors used the Thomas algorithm approach for the computational purposes, which took less time for the computation, and hence, more efficient than the other methods used in literature.

A singularly perturbed delay parabolic convection-diffusion initial boundary value problem is considered. The solution of the problem possesses a regular boundary layer. The authors solve this problem numerically using a monotone hybrid scheme. The error analysis is carried out. It is shown to be parameter-uniform convergent and is of second-order accurate. Numerical results are shown to verify the theoretical estimates.

The real challenge in the solution of contact problems is the lack of an optimal adaptive scheme. As the contact zone is a priori unknown, successive refinement and iterative method are necessary to obtain a high-accuracy solution. The purpose of this paper is to provide an optimal adaptive scheme based on second-generation finite element wavelets for the solution of non-linear variational inequality of the contact problem.

To generate an elementary multi-resolution mesh, the authors used hierarchical bases (HB) composed of Lagrange finite element interpolation functions. These HB functions are customized using second-generation wavelet techniques for a fast convergence rate. At each step of the algorithm, the active set method along with mesh adaptation is used for solving the constrained minimization problem of contact case. Wavelet coefficients-based error indicators are used, and computation is focused on mesh zones with a high error indication. The authors take advantage of the wavelet transform to develop a parameter-free adaptive scheme to generate an appropriate and optimal mesh.

Adaptive wavelet Galerkin scheme (AWGS), a newly developed method for multi-scale mesh adaptivity in this work, is a combination of the second-generation wavelet transform and finite element method and significantly improves the accuracy of the results without approximating an additional problem of error estimation equations. A comparative study is performed taking a solution on a highly refined mesh and results are generated using AWGS.

The proposed adaptive technique can be utilized in the simulation of mechanical and biomechanical structures where multiple bodies come into contact with each other. The algorithm of the method is easy to implement and found to be successful in producing a sufficiently accurate solution with relatively less number of mesh nodes.

Although many error estimation techniques have been developed over the past several years to solve contact problems adaptively, because of boundary non-linearity development, a reliable error estimator needs further investigation. The present study attempts to resolve this problem without having to recompute the entire solution on a new mesh.

Discusses the 27 papers in ISEF 1999 Proceedings on the subject of electromagnetisms. States the groups of papers cover such subjects within the discipline as: induction machines; reluctance motors; PM motors; transformers and reactors; and special problems and applications. Debates all of these in great detail and itemizes each with greater in‐depth discussion of the various technical applications and areas. Concludes that the recommendations made should be adhered to.

The purpose of this paper is to provide a robust second-order numerical scheme for singularly perturbed delay parabolic convection–diffusion initial boundary value problem.

For the parabolic convection-diffusion initial boundary value problem, the authors solve the problem numerically by discretizing the domain in the spatial direction using the Shishkin-type meshes (standard Shishkin mesh, Bakhvalov–Shishkin mesh) and in temporal direction using the uniform mesh. The time derivative is discretized by the implicit-trapezoidal scheme, and the spatial derivatives are discretized by the hybrid scheme, which is a combination of the midpoint upwind scheme and central difference scheme.

The authors find a parameter-uniform convergent scheme which is of second-order accurate globally with respect to space and time for the singularly perturbed delay parabolic convection–diffusion initial boundary value problem. Also, the Thomas algorithm is used which takes much less computational time.

A singularly perturbed delay parabolic convection–diffusion initial boundary value problem is considered. The solution of the problem possesses a regular boundary layer. The authors solve this problem numerically using a hybrid scheme. The method is parameter-uniform convergent and is of second order accurate globally with respect to space and time. Numerical results are carried out to verify the theoretical estimates.

The purpose of this paper is to develop a time implicit discontinuous Galerkin method for the simulation of two‐dimensional time‐domain electromagnetic wave propagation on non‐uniform triangular meshes.

The proposed method combines an arbitrary high‐order discontinuous Galerkin method for the discretization in space designed on triangular meshes, with a second‐order Cranck‐Nicolson scheme for time integration. At each time step, a multifrontal sparse LU method is used for solving the linear system resulting from the discretization of the TE Maxwell equations.

Despite the computational overhead of the solution of a linear system at each time step, the resulting implicit discontinuous Galerkin time‐domain method allows for a noticeable reduction of the computing time as compared to its explicit counterpart based on a leap‐frog time integration scheme.

The proposed method is useful if the underlying mesh is non‐uniform or locally refined such as when dealing with complex geometric features or with heterogeneous propagation media.

The paper is a first step towards the development of an efficient discontinuous Galerkin method for the simulation of three‐dimensional time‐domain electromagnetic wave propagation on non‐uniform tetrahedral meshes. It yields first insights of the capabilities of implicit time stepping through a detailed numerical assessment of accuracy properties and computational performances.

In the field of high‐frequency computational electromagnetism, the use of implicit time stepping has so far been limited to Cartesian meshes in conjunction with the finite difference time‐domain (FDTD) method (e.g. the alternating direction implicit FDTD method). The paper is the first attempt to combine implicit time stepping with a discontinuous Galerkin discretization method designed on simplex meshes.

The purpose of this work is to introduce an efficient, global second-order accurate and parameter-uniform numerical approximation for singularly perturbed parabolic differential-difference equations having a large lag in time.

The small delay and advance terms in spatial direction are handled with Taylor's series approximation. The Crank–Nicholson scheme on a uniform mesh is applied in the temporal direction. The derivative terms in space are treated with a hybrid scheme comprising the midpoint upwind and the central difference scheme at appropriate domains, on two layer-resolving meshes namely, the Shishkin mesh and the Bakhvalov–Shishkin mesh. The computational effectiveness of the scheme is enhanced by the use of the Thomas algorithm which takes less computational time compared to the usual Gauss elimination.

The proposed scheme is proved to be second-order accurate in time and to be almost second-order (up to a logarithmic factor) uniformly convergent in space, using the Shishkin mesh. Again, by the use of the Bakhvalov–Shishkin mesh, the presence of a logarithmic effect in the spatial-order accuracy is prevented. The detailed analysis of the convergence of the fully discrete scheme is thoroughly discussed.

The use of second-order approximations in both space and time directions makes the complete finite difference scheme a robust approximation for the considered class of model problems.

To validate the theoretical findings, numerical simulations on two different examples are provided. The advantage of using the proposed scheme over some existing schemes in the literature is proved by the comparison of the corresponding maximum absolute errors and rates of convergence.