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Purpose The optimization of the cross section of an axisymmetric induction heating device is performed by means of genetic algorithms (GAs). Design/methodology/approach The hybrid finite element method–Dirichlet boundary condition iteration method is used to deal with the unbounded nature of the field. The formulation of the electromagnetic problems takes into account skin and proximity effects in the source currents. Findings The convergence of GAs towards the optimum is very fast, since less than a thousand analyses have been necessary. Originality/value A special derivation of the finite element global system is presented which allows us to save computing time.

In this paper, the hybrid finite-element method (FEM) singular Dirichlet boundary condition iteration method, already developed for 2-D skin-effect problems, is extended to the 3-D solution of time-harmonic eddy current problems. The method is similar to the well-known FEM-boundary-element method (BEM), but it assumes a Dirichlet boundary condition on the truncation boundary instead of a Neumann one. Shorter solving times are obtained with respect to FEM-BEM.

In this paper an improved placement of the integration surface for the integral equation in the hybrid finite element method-Dirichlet boundary condition iteration (FEM-DBCI) method is achieved by selecting it as lying in the middle of the tetrahedral or hexahedral finite elements. A better accuracy is obtained with respect to the case in which the integration surface lies along the element faces. Some numerical examples are provided.

PurposeThis paper aims to discuss various numerical implementations of the integral equation in the hybrid finite element method‐Dirichlet boundary condition iteration (FEM‐DBCI) method for the numerical solution of unbounded static and quasi‐static electromagnetic field problems.Design/methodology/approachThree numerical implementations are described and compared from the point of view of accuracy and complexity, by means of two examples regarding simple electrostatic problems.FindingsThe implementation by means of a pair of integration surfaces made of element sides leads to accuracy levels which are much better than that of a single surface (made of element sides) and only a little worse than that of a single surface connecting point in the middle of finite element sides.Practical implicationsThe former implementations, however, are simpler since they are practically the same as that of a standard boundary element method integral equation.Originality/valueThe paper constitutes a useful guide to the implementation of the FEM‐DBCI method.

In multi‐ and many‐objective optimization problems, the optimization target is to obtain a set of non‐dominated solutions close to the Pareto‐optimal front, well‐distributed, maximally extended and fully filled. Comparing solution sets is crucial in evaluating the performance of different optimization algorithms. The use of performance indicators is common in comparing those sets and, subsequently, optimization algorithms. Therefore, an effective performance indicator must encompass these features as a whole and, above all, it must be Pareto dominance compliant. Unfortunately, some of the known indicators often fail to properly reflect the quality of a solution set or cost a lot to compute. This paper demonstrates that the Degree of Approximation (DOA) quality indicator is a weakly Pareto compliant unary indicator that gives a good estimation of the match between the approximated front and the Pareto‐optimal front.

Purpose – The purpose of this paper is to present a modified version of the hybrid Finite Element Method-Dirichlet Boundary Condition Iteration method for the solution of open-boundary skin effect problems. Design/methodology/approach – The modification consists of overlapping the truncation and the integration boundaries of the standard method, so that the integral equation becomes singular as in the well-known Finite Element Method-Boundary Element Method (FEM-BEM) method. The new method is called FEM-SDBCI. Assuming an unknown Dirichlet condition on the truncation boundary, the global algebraic system is constituted by the sparse FEM equations and by the dense integral equations, in which singularities arise. Analytical formulas are provided to compute these singular integrals. The global system is solved by means of a Generalized Minimal Residual iterative procedure. Findings – The proposed method leads to slightly less accurate numerical results than FEM-BEM, but the latter requires much more computing time. Practical implications – Then FEM-SDBCI appears more appropriate than FEM-BEM for applications which require a shorter computing time, for example in the stochastic optimization of electromagnetic devices. Originality/value – Note that FEM-SDBCI assumes a Dirichlet condition on the truncation boundary, whereas FEM-BEM assumes a Neumann one.

FEM/DBCI (Dirichlet Boundary Condition Iteration) is a hybrid method which has been successfully applied to the finite element solution of static and quasistatic unbounded field problems. It is based on the iterative improvement of a Dirichlet condition on a fictitious boundary enclosing all the conductors, making use of the free-space Green's function. In this paper it is shown that for two-dimensional skin effect problems a more robust solving strategy can be pursued by means of GMRES, obtaining a reduction in both CPU and storage requirements.

The Robin iteration procedure is a technique for the FEM computation of electromagnetic scattering fields in unbounded domains. It is based on the iterative improvement of the known term of a non‐homogeneous Robin condition on a fictitious boundary enclosing the scatterer. In this paper it is shown that the procedure is equivalent to the application of the Richardson method to a reduced system and that the use of GMRES significantly reduces the computational effort.

Charge iteration is an FEM procedure, devised by the authors for the computation of electrical fields in open boundary domains. In the procedure, a closed fictitious boundary is introduced enclosing all the conductors. On this boundary the electrical potential is first guessed and then iteratively improved according to the current solution by means of the free-space Green's function until convergence takes place. In this paper the procedure is generalized by allowing not only Dirichlet conditions, but also Neumann and Robin ones to he imposed on the fictitious boundary. Some reduction in computational effort comes from this generalization as pointed out by theoretical and numerical examples.

Current iteration is a purely FEM iterative procedure devised by the authors to deal with boundless skin-effect problems in terms of the magnetic vector potential. If material non-linearities occur, a Newton-Raphson solver, nested inside the current iteration, has to be used. A computing strategy is described which allows us to achieve overall computational efficiency in carrying out the two nested iterations.

This paper presents an iterative FEM procedure for the solution of unbounded non-linear transient magnetic field problems. The boundlessness of the domain is dealt with by introducing a fictitious boundary defining a bounded region. At each discrete time, by assuming a first guess for the Dirichlet condition on such a boundary a standard FEM solution for the bounded non-linear transient problem is carried out; the results of the FEM analysis are used to update the Dirichlet condition on the fictitious boundary; the above steps are then iterated until a solution to the unbounded problem is reached. The procedure is rapidly convergent, accurate and easy to implement.

This paper extends an iterative method to efficiently solve the global algebraic system of equations obtained by applying the hybrid FEM-BEM method to cases in which the field is due to charge distributions only, that is, when conductors with assigned potential (Dirichlet conditions) are not present. In the proposed approach, a conjugate gradient solver is used to solve the singular FEM equations, whereas the BEM equations are modified by means of the Gauss law.

Purpose – The purpose of this paper is to compare the hybrid FEM-BEM and FEM-DBCI methods for the solution of open-boundary static and quasi-static electromagnetic field problems. Design/methodology/approach – After a brief review of the two methods (both coupling a differential equation for the interior problem with an integral equation for the exterior one), they are compared in terms of accuracy, memory and computing time requirements by means of a set of simple examples. Findings – The comparison suggests that FEM-BEM is more accurate than FEM-DBCI but requires more computing time. Practical implications – Then FEM-DBCI appears more appropriate for applications which require a shorter computing time, for example in the stochastic optimization of electromagnetic devices. Conversely, FEM-BEM is more appropriate in cases in which a high level of precision is required in a single computation. Originality/value – Note that the FEM-BEM considered in this paper is a non standard one in which the nodes of the normal derivative on the truncation boundary are placed in positions different from those of the potential.

Charge iteration is an iterative procedure for the finite element computation of unbounded electrical fields, created by voltaged conductors. It makes use of a fictitious boundary, enclosing all the conductors, on which the electrical potential is first guessed and then iteratively improved according to the charge lying on the conductor surfaces. Highlights the theoretical foundations of the procedure outside any numerical context. From this useful insight, obtains a model which can aid the user in utilization of the numerical version of the procedure.

Purpose – The purpose of this paper is to improve the accuracy of the integral equation of the hybrid FEM‐RBCI (Finite Element Method‐Robin Boundary Condition Iteration) method for the numerical solution of two‐dimensional electromagnetic (or acoustic) scattering problems.Design/methodology/approach – This accuracy improvement is achieved by selecting the integration curve as straight segments lying in the middle of the triangular finite elements. An accuracy improvement is obtained as compared with selecting the integration curve as constituted by element sides.Findings – The improved FEM‐RBCI method described in this paper leads to accuracies of the numerical results which are better than those obtained by selecting the integration curve by element sides.Originality/value – The paper presents results for a simple two‐dimensional structure: a dielectric circular cylinder.

The numerical treatment of unbounded electrostatic field problems can be conveniently integrated in standard FE codes for bounded problems, by using charge iteration, an efficient, N-dimensional FE procedure recently presented by the authors. In this paper its convergence characteristics (accuracy and number of iteration steps) are investigated. Comparisons with known analytical solutions are used in order to assess the precision achievable. >

In this paper, stochastic optimization of the shielding of an axisymmetric induction heating system is performed by means of the self-adaptive low-high evaluations evolutionary algorithm (SALHE-EA). The finite element formulation of the electromagnetic problem is based on the magnetic vector potential. The hybrid finite element method Dirichlet boundary condition iteration (FEM-DBCI) method is used to deal with the unbounded nature of the field problem and to reduce the computing effort required for optimization.

Purpose – This paper aims to extend an efficient method to solve the global system of linear algebraic equations in the hybrid finite element method – boundary element method (FEM‐BEM) solution of open‐boundary skin effect problems. The extension covers the cases in which the skin effect problem is set in a truncated domain in which no homogeneous Dirichlet conditions are imposed.Design/methodology/approach – The extended method is based on use of the generalized minimal residual (GMRES) solver, which is applied virtually to the reduced system of equations in which the unknowns are the nodal values of the normal derivative of the magnetic vector potential on the fictitious truncation boundary. In each step of the GMRES algorithm the FEM equations are solved by means of the standard complex conjugate gradient solver, whereas the BEM equations are not solved but used to perform fast matrix‐by‐vector multiplications. The BEM equations are written in a non‐conventional way, by making the nodes for the potenti...

PurposeThis paper aims to propose a hybrid method, called finite element method‐Dirichlet boundary condition iteration (FEM‐DBCI), for the computation of time‐harmonic eddy current problems inside a conductor heated by coils in 3D open‐boundary geometry.Design/methodology/approachThe method assumes the electrical field as unknown on a mesh of tetrahedral edge elements. The heating power density inside the conductor is then computed and a steady‐state thermal analysis is performed on the same mesh of nodal tetrahedra to calculate the temperature distribution inside the heated piece, taking radiation and convection into account. A numerical example is also provided.FindingsThe method couples a differential equation for the interior problem in terms of the electric fields with an integral equation for the exterior one. The global algebraic system is efficiently solved in an iterative way.Originality/valueThe paper illustrates the computation of time‐harmonic eddy current problems inside a conductor heated by coils.

This paper shows that the global system of linear equations in the hybrid finite element method/boundary element method (FEM-BEM) solution of open-boundary skin effect problems can be efficiently solved by means of the generalized minimal residual (GMRES) solver. This solver is applied virtually to the reduced system of equations in which the unknowns are the nodal values of the normal derivative of the magnetic vector potential on the fictitious truncation boundary. In each step of the GMRES algorithm, the solution of the FEM equations is performed by means of the standard complex conjugate gradient solver, whereas the BEM equations are not solved but used to perform fast matrix-by-vector multiplications. The BEM equations are written in a nonconventional way, by making the nodes for the potential noncoinciding with the nodes for its normal derivative.

This paper describes an iterative method to solve efficiently the global algebraic system of equations obtained by applying the hybrid finite element method-boundary element method (FEM-BEM) to the solution of open-boundary electrostatic problems. In the proposed approach, a conjugate gradient (CG) solver is used to solve the FEM equations, whereas the BEM equations are solved by making use of the direct LU solver. To do so, it is convenient to implement the BEM equations in a nonconventional way, by choosing the nodes of the potential as not coinciding with those of its normal derivative

The authors show that the global matrix in the finite element integro-differential formulation of two-dimensional skin effect problems can be partitioned into the sum of a sparse matrix and a product of two equal arrays for each conductor of the multiconductor system. This partition allows both a great memory saving and a drastic reduction in the conjugate gradient solution time.

In this paper optimization of the cross section of the coils of an axisymmetric electromagnetic actuator is performed by means of two stochastic methods: genetic algorithms and simulated annealing. The finite element formulation of the electromagnetic problems is based on the magnetic vector potential. The hybrid FEM/DBCI (Dirichlet boundary condition iteration) method is used to deal with the unbounded nature of the field problem and to reduce the computing effort of the whole optimization.

This paper discusses the perfectly matched layer method recently proposed for the computation of static or quasistatic fields in open boundaries. In particular it is shown how the method can be derived by means of a particular co‐ordinate transformation applied to a finite‐size isotropic domain surrounding the system of interest. The method is therefore equivalent to a trivial truncation from the point of view of both accuracy and computing time.

In this paper, the optimization of the shape of the magnetic channel inside a superconducting cyclotron is performed. The objective is to ensure a proper shaping and reduction of the intensity of the cyclotron magnetic field so as to facilitate the extraction of the charged particles. The optimization problem is solved by following two different approaches: a genetic algorithm and a simulated annealing approach. The relative advantages of each of the methods employed for the optimization of the magnetic channel are discussed.

An iterative procedure is described for the finite-element solution of scalar scattering problems in unbounded domains. The scattering objects may have multiple connectivity, may be of different materials or with different boundary conditions. A fictitious boundary enclosing all the objects involved is introduced. An appropriate Robin (mixed) condition is initially guessed on this boundary and is iteratively improved making use of Green's formula. It will be seen that the best choice for the Robin boundary condition is an absorbing-like one. A theorem about the theoretical convergence of the procedure is demonstrated. An analytical study of the special case of a circular cylindrical scatterer is made. Comparisons are made with other methods. Some numerical examples are provided in order to illustrate and validate the procedure and to show its applicability whatever the frequency of the incident wave. Although particular emphasis is laid in the paper on electromagnetic problems, the procedure is fully applicable to other kinds of physical phenomena such as acoustic ones. © 1998 John Wiley & Sons, Ltd.

In this paper an iterative finite-element procedure is described for the solution of nonlinear time-varying eddy current problems in open boundaries, A fictitious boundary delimiting a bounded domain is introduced. At each discrete time, two algebraic systems are set up and solved for the predictor and corrector solutions. Both these systems are solved iteratively by assuming a first guess for the Dirichlet condition on the fictitious boundary and improving it by making use of the free-space Green's function until convergence takes place. A great reduction of computing time is obtained with respect to the solution with a Newton-Raphson solver.

The charge iteration procedure, recently presented by the authors for the FEM solution of unbounded electrical fields, makes use of a fictitious boundary whose position strongly affects convergence properties and performance in terms of computational effort. In this paper the problem of the optimal placement of this boundary is first addressed analytically in the case of simple systems and then it is illustrated how the obtained conditions can be used for more complex systems so as to offer simple criteria for the application of the procedure to practical cases. >