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The purpose of this paper is to emphasise the analogies between variational and network formulations using geometrical forms, with the purpose of developing alternative but otherwise equivalent derivations of the finite element (FE) method.

FE equations for electromagnetic fields are examined, in particular nodal elements using scalar potential formulation and edge elements for vector potential formulation.

It is shown how the equations usually obtained via variational approach may be more conveniently derived using integral methods, employing a geometrical description of the interpolating functions of edge and facet elements. Moreover, the resultant equations describe the equivalent multi‐branch circuit models.

The approach proposed in the paper explores the analogy of the FE formulation to loop or nodal magnetic or electric networks and has been shown to be very beneficial in teaching, especially to students well familiar with circuit methods. The presented methods are also helpful when formulating classical network models. Finally, for the first time, the geometrical forms of edge and facet element functions have been demonstrated.

This paper seeks to develop 3D finite element methods for the electromagnetic field calculation in electrical machines and to present the discrete methods of winding description.

The 3D finite element models of electrical machine windings are considered. Attention is paid to the windings with stranded conductors. The finite element equations are considered as the equations of magnetic networks. The formulation of matrix that transforms winding currents into the field sources is discussed. This matrix is also used in the calculations of flux linkages. In the proposed method, the winding loops are replaced by a set of plane loops. The field sources are defined by the numbers of these loops around the element edges and loops associated with element facets.

The presented description is the 3D finite element representation of MMF description used in the classical models of electrical machines. The advantage of the proposed approach is that the source description can be successfully applied in the FE method using single scalar potential. In addition, the presented approach guarantees a good convergence of ICCG procedure of solving edge element equations for ungauged formulation using magnetic vector potential.

The applied analogies between the finite element formulation and the equivalent magnetic network models help to formulate an efficient method of field source description. The developed method allows one to apply single magnetic scalar potential in the 3D finite element analysis of electrical machines.

The paper presents a method for the 3D edge element analysis of saturation effects in the classical rotating electrical machines of cylindrical structure. The edge element (EE) method using vector magnetic potential has been applied. Special attention is paid to the saturation effects in permanent magnet motors. In order to solve the non‐linear EE equations the authors propose to apply the modified Newton algorithm with block relaxation solver and Cholesky decomposition procedure for block matrices. The convergence of the algorithm is analysed. The influence of core non‐linearity on the values of electromagnetic torque and armature inductances is considered. The results for 3D and 2D models are compared.

Resolving eddy currents in three dimensions with finite elements, especially in geometrically complex structures, is very time consuming. Notable additional efforts will be required, if these eddy currents are influenced by magnetic fields arising from larger parts or range over widespread regions. The purpose of this article is to present a new sub-modelling simulation technique, based on the finite-element approach. This method offers remarkable advantages for solving this type of problems.

A novel sub-modeling technique is developed for the finite-element method addressing this problem by dividing the process into two steps: firstly, a simulation of a “source”-model is carried out providing magnetic field distributions within the entire domain neglecting local eddy current effects and without modeling it in full detailed geometry. A subsequent “sub”-model comprises only the region of interest in higher resolution and is solved while being constrained with boundary conditions derived from the previous source-model. An implementation in ANSYS Mechanical is carried out with the objective to validate finite-element simulation against measurement results.

The proposed simulation technique performs robustly and time efficiently. Applying this method to an end-region of a turbogenerator allows comparisons with test data of this region for validation purposes. The comparison between measured and simulated radial flux densities shows good correspondence.

This work is novel in many aspects: a new technique for three-dimensional (3D) finite-element method using edge-elements is introduced. To the best of the authors’ knowledge, for the first time, these 3D sub-models are compared against measurement results of an electric machine with net currents. Leveraged from this work, detailed analyses of eddy current phenomena under influences of external magnetic fields can be investigated in higher detail within shorter calculation times.

The aim of this paper is to develop network models of an electromagnetic field containing both eddy and displacement currents. The proposed network models provide good physical insight, help understanding of complicated electromagnetic phenomena and aid explanation of methods of analysis of electromagnetic systems.

The models consist of magnetic and electric networks coupled via sources. The analogy between the finite element method and the loop and nodal formulations of electric circuits is emphasised. The models include networks containing branches associated with element edges (edge networks) or facets (facet networks).

Methods of determining mmf sources of magnetic networks from loop and branch currents in electric circuits, as well as emf sources in electric networks on the basis of the rate of change of loop and branch fluxes in electric networks, have been carefully considered. The models are general and allow creation of networks of electromagnetic systems containing non‐homogenous materials and multiply‐connected conducting regions.

The presented analogies between the finite element formulation and the equivalent network models not only facilitate understanding of the methods of field analysis but also help to formulate efficient computational algorithms.

This paper aims to investigate the performance of two novel numerical methods, the face-based smoothed finite element method (FS-FEM) and the edge-based smoothed finite element method (ES-FEM), which employ linear tetrahedral elements, for the purpose of strength assessment of a high-speed train hollow axle.

The calculation of stress for the wheelset, comprising an axle and two wheels, is facilitated through the application of the European axle strength design standard. This standard assists in the implementation of loading and boundary conditions and is exemplified by the typical CRH2 high-speed train wheelset. To evaluate the performance of these two methods, a hollow cylinder cantilever beam is first used as a benchmark to compare the present methods with other existing methods. Then, the strength analysis of a real wheelset model with a hollow axle is performed using different numerical methods.

The results of deflection and stress show that FS-FEM and ES-FEM offer higher accuracy and better convergence than FEM using linear tetrahedral elements. ES-FEM exhibits a superior performance to that of FS-FEM using linear tetrahedral elements, showing accuracy and convergence close to FEM using hexahedral elements.

This study channels the novel methods (FS-FEM and ES-FEM) in the static stress analysis of a railway wheelset. Based on the careful testing of FS-FEM and ES-FEM, both methods hold promise as more efficient tools for the strength analysis of complex railway structures.

This paper gives a bibliographical review of the finite element methods (FEMs) applied to the analysis of ceramics and glass materials. The bibliography at the end of the paper contains references to papers, conference proceedings and theses/dissertations on the subject that were published between 1977‐1998. The following topics are included: ceramics – material and mechanical properties in general, ceramic coatings and joining problems, ceramic composites, ferrites, piezoceramics, ceramic tools and machining, material processing simulations, fracture mechanics and damage, applications of ceramic/composites in engineering; glass – material and mechanical properties in general, glass fiber composites, material processing simulations, fracture mechanics and damage, and applications of glasses in engineering.

The focus of this paper is on the efficient numerical computation of 3D electromagnetic field problems by using the finite element (FE) and multigrid (MG) methods. The magnetic vector potential is used as the field variable and the discretization is performed by Lagrange (nodal) as well as Ne´de´lec (edge) finite elements. The resulting system of equations is solved by applying a preconditioned conjugate gradient (PCG) method with an adapted algebraic multigrid (AMG) as well as an appropriate geometric MG preconditioner.

Propose post processing methods for the edge finite element (FE) method on a tetrahedral mesh. They make it possible to deduce vector values on the vertices from scalar values defined on the edges of the tetrahedra.

The new proposed techniques are based on a least squares formulation leading to a sparse matrix system to be solved. They are compared in terms of accuracy and CPU time on a FEs formulation for open boundary – frequency domain problems.

A significant improvement of vector values accuracy on the vertices of the tetrahedra is obtained compared to a classical approach with a very small additional computation time.

This work presents techniques: to obtain the values at the initial nodes of the mesh and not inside the tetrahedra; and to take into account the discontinuity to the interface between two media of different electromagnetic properties.

This work is concerned with the development and the numerical investigation of a hybridizable discontinuous Galerkin (HDG) method for the simulation of two‐dimensional time‐harmonic electromagnetic wave propagation problems.

The proposed HDG method for the discretization of the two‐dimensional transverse magnetic Maxwell equations relies on an arbitrary high order nodal interpolation of the electromagnetic field components and is formulated on triangular meshes. In the HDG method, an additional hybrid variable is introduced on the faces of the elements, with which the element‐wise (local) solutions can be defined. A so‐called conservativity condition is imposed on the numerical flux, which can be defined in terms of the hybrid variable, at the interface between neighbouring elements. The linear system of equations for the unknowns associated with the hybrid variable is solved here using a multifrontal sparse LU method. The formulation is given, and the relationship between the considered HDG method and a standard upwind flux‐based DG method is also examined.

The approximate solutions for both electric and magnetic fields converge with the optimal order of p+1 in L₂ norm, when the interpolation order on every element and every interface is p and the sought solution is sufficiently regular. The presented numerical results show the effectiveness of the proposed HDG method, especially when compared with a classical upwind flux‐based DG method.

The work described here is a demonstration of the viability of a HDG formulation for solving the time‐harmonic Maxwell equations through a detailed numerical assessment of accuracy properties and computational performances.