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In this paper it is shown how the Robin iteration procedure, already proposed by the authors for the solution of electromagnetic scattering problems, can be easily adapted to scattering from cavities embedded in a perfectly-conducting wedge. The procedure couples a differential equation for the interior problem with an integral equation for the exterior one. A suitable choice of the Robin (mixed) boundary condition on the fictitious boundary dividing the interior and exterior domains avoids resonances.

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.

In this paper the charge iteration procedure is extended to cover the computation of spatially-periodic electrostatic fields. A fictitious boundary which encloses all the conductors is introduced inside the basic space period. The unknown electric potential on the fictitious boundary is initially guessed and then iteratively improved making use of a suitable periodic Green's function.

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.

A method is proposed for the computation of the transient heating of a conductor in which eddy currents flow, induced by time-harmonic source currents. The electrical field is assumed as unknown on a mesh of edge elements and is computed by a time-harmonic analysis. The heating power density inside the conductor is computed and a transient thermal analysis is started on the same mesh of nodal elements. This analysis is continued until the temperature-dependent electrical conductivity changes enough to require another time-harmonic eddy-current analysis. The combined procedure is iterated.

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.

Charge iteration is an N-dimensional (N=1,2,3) FEM procedure for the solution of unbounded electrostatic fields created by voltaged conductors. In the procedure a fictitious boundary enclosing all the conductors defines a bounded region where the unbounded problem is solved iteratively using the charge lying on conductors to improve the Dirichlet condition on the fictitious boundary. The convergence properties of charge iteration in the presence of overrelaxation are investigated. A remarkable reduction in computational effort is obtained due to the reduction in the number of iterations and the restriction of the bounded domain to the vicinity of the conductors where the unrelaxed procedure exhibits a divergent behaviour. The above advantages are illustrated by an analytical verification and by a numerical example.

An iterative finite-element procedure is described for the solution of skin-effect problems, under the assumption of quasi-static TEM conditions. The systems considered consist of parallel straight conductors of arbitrary cross section, with inhomogeneous conductivity and permeability, embedded in an unbounded homogeneous dielectric medium. A fictitious boundary enclosing all the conductors is introduced; an initial guess for the magnetic vector potential is assumed on this boundary and is iteratively improved according to the Laplacian behaviour of the potential outside the conductors. An example of application to a simple transmission line is given. >

FEM-RBCI method is applied to the analysis of the scattering of time-harmonic electromagnetic waves at optical frequencies from systems of several metallic nanoparticles, exhibiting a ring shape. Plasmons are taken into account by modelling the metallic material by means of appropriate complex permittivities. System of circular nanorings are studied.

This paper presents a modified version of the hybrid FEM-DBCI method for the solution of time-harmonic skin effect problems in open boundaries. In this new version of the method the fictitious truncation boundary coincides with the integration one, as in the FEM-BEM method. The global algebraic system obtained is solved iteratively by means of the GMRES solver. Some validation examples are provided.

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. >

An improved placement of the integration surface for the integral equation in the hybrid FEM-RBCI method is achieved by selecting such surface as lying in the middle of tetrahedral finite elements. A better accuracy is obtained with respect to the case in which the integration surface lies along the finite element faces.

In this paper the authors consider a real-world set of cables on which Power Line Communication (PLC) signals are transmitted in the power distribution network of a ship. The electrical signals which convey PLC data are carried by unshielded cables and may therefore interact with the external environment and affect the functioning of other devices. On the basis of a set of working assumptions, several electromagnetic field analyses are performed by means of the Finite Element Method in order to identify the distribution close to the conductors of the electromagnetic field generated by PLC signals and common mode currents.

An iterative procedure is presented for the finite-element computation of axisymmetric electrostatic fields in unbounded domains. In the procedure the original unbounded problem is solved by using successive evaluations of the potential on a fictitious boundary enclosing all the conductors of the problem, according to the charge lying on their surface. For simplex Lagrangian elements universal matrices are used for a fast and efficient computation of this charge. The main advantage of this procedure lies in its ease of implementation in a standard bounded problem finite-element code and in the accuracy of the computed results. >

This paper compares the hybrid FEM-BEM and FEM-DBCI methods for the solution of open-boundary electrostatic field problems. Both methods couple a differential equation for the interior problem with an integral equation for the exterior one. The comparison shows that FEM-BEM is more accurate than FEM-DBCI but requires more computing time.

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.

This paper describes a new family of axisymmetric nodal boundary elements to be used in the hybrid FE-BE method to deal with unbounded static and quasi-static electromagnetic field problems. The scalar field is developed by means of classical polynomial shape functions of a given order, whereas its normal derivative is developed with lower-order shape functions.

Power Line Communications (PLC) are a very popular technology having numerous applications ranging in a wide area of industrial and information engineering. In this paper the authors focus on the recently stated promising field of application of PLC to naval transportation systems. Specifically, they take into consideration a real-world set of cables in the power distribution network of a ship, on which the PLC signals are transmitted. On the basis of a set of working assumptions, several electromagnetic field analyses are performed by means of the Finite Element Method. The analyses are intended to investigate a typical cable configuration in order to identify undesirable couplings between the cables caused by power apparatuses connected to the network during their operation.

This paper reviews some solving procedures for unbounded electromagnetic field problems, based on the integral representation of the unknown boundary condition on the truncation fictitious boundary. Three problems are considered: electrostatic, time-harmonic skin effect, and time-harmonic scattering in unbounded domains. These problems are formulated, respectively, by means of the hybrid FEM-BEM, FEM-DBCI and FEM-RBCI methods, which in all cases lead to global algebraic systems which are partly sparse and partly dense. The paper shows that a very good solving strategy is based on the use of the GMRES solver, virtually applied to a suitably reduced algebraic system, in which the unknowns are only the truncation boundary ones.

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.

This paper presents a novel family of nodal boundary elements to be used in the context of the hybrid FEM-BEM method to deal with unbounded static and quasi-static electromagnetic field problems. In this new type of boundary elements the scalar field is developed by means of classical polynomial shape functions of a given order, whereas its normal derivative is developed with lower-order shape functions.

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...

This paper presents a new evolutionary algorithm (EA) for the optimization of electromagnetic devices called SALHE-EA (self-adaptive low-high evaluations – evolutionary algorithm). Its main aspects are identification of the optima of the objective function and evaluation of their sensitivity. Moreover, SALHE-EA works well if combined with the deterministic pattern search (PS) method, forming a good hybrid method. It performs well in the design of electromagnetic devices when the optimization of a multimodal function is required.

Non-Destructive Evaluation techniques are of great relevance nowa-days and are extensively employed in many areas of engineering to detect defects in products and structures, or to point out deteriorations in industrial plants. This paper focuses on techniques which make use of transient ultrasonic waves, the aim being numerical modelling and simulation of their propagation in concrete structures affected by defects, possibly in the presence of material inhomogeneities and/or voids. For this purpose a fully three-dimensional finite element code has been developed by the authors, who have validated and used it to perform numerous simulations, the results of which are presented and discussed.

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 compares the hybrid FEM-BEM and FEM-DBCI methods for the solution of open-boundary electrostatic field problems. Both methods couple a differential equation for the interior problem with an integral equation for the exterior one. The comparison shows that FEM-BEM is more accurate than FEM-DBCI but requires more computing time.

This paper presents a novel family of nodal boundary elements to be used in the context of the hybrid FEM-BEM method to deal with unbounded static and quasi-static electromagnetic field problems. In this new type of boundary elements the field variable is developed by means of classical polynomial shape functions of a given order, whereas its normal derivative is developed with lower-order shape functions. A numerical example is provided regarding a simple electrostatic problem.

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

A new niching genetic algorithm (NGA) called self-adaptive NGA (SANGA) is proposed. The main innovation is that the niche radius is estimated as an additional variable of the optimization problem and is not assigned a priori, as is usually done in other standard NGAs. SANGA, coupled with the deterministic pattern search method, forms a hybrid optimization method which works well in the optimization of multimodal functions and in the design of electromagnetic devices

The paper describes an efficient technique for optimizing the shape of the patch of a multi-band antenna by means of genetic algorithms and a hybrid finite element - robin boundary condition iteration method, for the analysis of open-boundary scattering and radiation electromagnetic field problems. The admissible rectangular patch area is logically and regularly subdivided into rectangular sub-areas, coinciding with the trace of the tetrahedral edge element mesh on the patch surface. In this way the relevant matrices of the finite element algebraic system, computed at the beginning of the optimization, remain unchanged, even if the patch is changed by inserting some metallic sub-areas. Moreover, in order to reduce the computing time of the iterative solver, the solution to a similar patch configuration is used as the initial guess for the solver

An automatic mesh generator of triangular finite elements, based on an artificial neural network (ANN), is optimized by means of genetic algorithms. The optimization concerns the various parameters which are defined in the generator, whereas the objective function is related to the mesh quality. By means of this optimization the generator obtains very high-quality meshes

The simulated annealing (SA) algorithm is often used in the optimization of electromagnetic devices, but in multi-modal continuous problems it may be affected by premature convergence. This paper proposes an improvement to prevent the algorithm from becoming trapped in local minima, called SA with restarts. This algorithm allows an enhancement of performance and a reduction in computing effort, appreciated especially in the optimization of electromagnetic devices, where the single evaluation of an objective function calls for the numerical solution of an electromagnetic field problem

In this paper, the accuracy of the integral equation of the hybrid finite-element-method-Dirichlet boundary condition iteration method is improved for the computation of open boundary electrostatic fields. This is achieved by selecting the integration curve (or surface) as lying in the middle of the triangular (or tetrahedral) elements. A notable improvement in accuracy is obtained compared with selecting the integration curve (or surface) as constituted by element sides

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 an induction heating system is optimized using the hybrid FEM-DBCI (Finite Element Method — Dirichlet Boundary Condition Iteration) to evaluate the objective function. In order to reduce the computing time for the whole procedure, Design of Experiments (DOE) techniques are used to screen significant geometric parameters. Once the number of problem variables is decreased, the optimization is performed by means of the Hooke-Jeeves algorithm. The results are compared with those obtained by stochastic optimization of the device.

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.

A new accurate procedure for the FEM solution of unbounded electromagnetic scattering problems is described. The scatterers are enclosed in a fictitious boundary on which a Robin condition is initially guessed, and then iteratively improved using the Green formula. The method does not present interior resonances and is easy to implement.

Charge iteration is an iterative finite-element procedure to deal with unbounded electrical field problems 1,2,3. In the procedure a fictitious boundary BF enclosing all the conductors, defines a bounded domain. By assuming an initial guess for the potential on BF a standard FEM solution can be carried out. Starting from this first-step solution one can compute the charge density lying on the conductor surfaces and from it a new estimate (closer to the true one) of the potential on BF, by means of an appropriate free-space Green’s function. The procedure is iterated until a convergence test is satisfied.

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. >