An overview of the ELFIN code for finite element research in electrical engineering G. Aiello, W S. Alfonzetti, ^ G. Borzi, an

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An overview of the ELFIN code for finite

element research in electrical engineering

G. Aiello, W S. Alfonzetti, ^ G. Borzi, and N. Salerno ^

^ Dipartimento Elettrico, Elettronico e Sistemistico,

Universita di Catania, Viale A. Doria 6, 1-95125 Catania, Italy

^ Dipartimento di Fisica della Materia e Tecnologie Fisiche Avanzate,

Universita di Messina, Salita Sperone 31, 1-98166 Messina, Italy

Abstract

This paper gives a general overview of the basic features and latest enhancements

of the finite element code ELFIN, developed by the authors for research in the

electrical engineering field. The basic features are N-dimensional (ID, 2D and

3D) geometrical discretization, wide application area, easy treatment of coupled

problems, and generalised iterative procedures. The latest enhancements of the

code refer to the computation of non-linear transient eddy currents in open

boundary domains, to the solution of electromagnetic scattering problems from

inhomogeneous bodies (in 2D and 3D), to the automatic generation of meshes of

tetrahedra based on an artificial neural network and to the stochastic optimization

of electromagnetic devices.

1 Introduction

Starting in 1984, the authors have developed a finite element code in order to

perform research in the field of computational electromagnetics for electrical

engineering [1]. The code is called ELFIN (from the Italian words ELementi

FINiti) [2,3]. The reasons for such an effort are mainly due to the difficulties

which arise when using commercial codes in a research environment. In fact,

such codes seldom exhibit the following characteristics: i) openness of the source

code (in order to operate modifications and add new capabilities easily); ii)

modularity (new problems should be implementable mainly by performing simple

operations of recombining existing functions); N-dimensionality (ID, 2D and 3D

problems should be dealt with by means of the same algorithms); iii) mesh

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144 Software for Electrical Engineering Analysis and Design

freedom (higher-order and curved-side elements should be individually defined);

iv) FEM/BEM integration. Many commercial finite element codes, based on

well-known and efficient procedures, are developed to solve a specific class of

problems and therefore they generally have limitations such as a fixed

dimensionality, a fixed kind and order of elements, and a fixed number and kind

of unknowns. These limitations are aimed at optimising performance, but they

practically prevent the use of such codes in a research environment.

This paper briefly describes the basic characteristics of ELFIN and outlines the

research conducted by means of it.

2 The ELFIN code

The ELFIN code is written in Fortran-77 and is now running on several

computers under the OpenVMS, Unix, Linux and DOS/Windows operating

systems. The code is structured according to the classical scheme of FE codes, in

which three distinct main programs are devoted respectively to pre-processing,

processing and post-processing. These programs recall a set of more than 600

routines, which can be grouped into three main categories: a) specific

subroutines, which refer to the code data structure, including: I/O subroutines;

geometrical discretization and material catalogue subroutines; geometrical

window and list subroutines; shape function subroutines; universal matrix

subroutines; b) non-specific subroutines, which do not refer to the code data

structure, including: matricial and polynomial ordering; combinatorial calculus;

polynomial algebra and calculus; polynomial matrix algebra; N-dimensional

integration of polynomial rational functions; geometrical computations; the

solution of linear algebraic equation systems; c) graphic subroutines, which

interface the external graphic library used (Regis, GKS, Xlib, Display Postscript).

2.1 Pre-processing

In ELFIN four geometrical entities of the space discretization are defined: nodes,

elements, boundary sides and, optionally, edges. Nodes are the primary entities of

the discretization. A node is identified by its absolute Cartesian coordinates X%

(k=l,...,N), in an N-dimensional space. The other entities of the discretization

(elements, boundary sides and edges) are derived from nodes in terms of ordered

sets of them. An element is an ordered set of nodes, the ordering having a

geometrical meaning. The following data are individually specified for each finite

element: the element family (e. g. isoparametric Lagrangian simplex and N-

rectangular); the element dimensionality N' (with N'<N); the element order m; the

element curved-side indicator (for orders m > 1). The local sides of an element

are implicitly defined as ordered subsets of its nodes. Boundary sides are globally

defined local sides, which generally lie on the domain boundary, even if internal

boundary sides are allowed. Optionally edge elements can be used, based on the

definition of edge as an ordered pair of nodes. Region "colours" can be

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Software for Electrical Engineering Analysis and Design 145

independently attributed to nodes, elements, boundary sides and edges in order to

treat particular subsets differently during the pre-processing and/or processing

phases. These definitions of the geometrical entities imply hierarchy: nodes are

more important than elements and edges, elements more important than boundary

sides. This structure is clearly independent of the space dimensionality N (which

only affects the cardinality of the co-ordinates of nodes) but has a cost in terms of

larger memory occupation and computing time with respect to codes employing

only elements of a fixed kind and order.

The ELFIN code, although developed in an electrical engineering department, has

been designed as general-purpose environment to be used in any other

engineering area (mechanical, civil, etc.). For this reason great freedom has been

reserved in the definition of the field variables and sources. A set of MS scalar

variables Sj (i=l,...,Mg) and a set of My vector variables Vj (j=l,...,My) are

associated to each node and to each edge, respectively, of the discretization.

These variables can be all real or all complex, according to the real/complex

indicator IRC, which is set to 1 for real problems and to 2 for complex ones. For

the field variables, boundary conditions (Dirichlet, Neumann or mixed) are

imposed on the nodes or edges of the boundary sides. Four kinds of sources are

defined: node sources, element sources, boundary-side sources and edge sources.

The real/complex indicator also applies to sources. Note that parameterization

with respect to the number of field variables and sources is the basis to allow

unified treatment of coupled problems. Moreover, no semantics need to be

preliminarily attributed to each variable in the pre-processing: a great variety of

problems (scalar, vector, coupled, real or complex) with different field variables

and sources can be implemented with no modifications of data structures.

Due to the fact that the code has been implemented by a small research group, the

pre-processor has medium capabilities for solid modeling, but quite sophisticated

features for defining boundary conditions and sources and for managing high-

order and/or curved-side elements; however, functions are expressly foreseen to

interface more powerful external solid modeling programs that may be available.

The pre-processor exhibits a mixed interactive-batch user interface: before a pre-

processing session the user can edit a text file, containing the sequence of pre-

processing commands (a command is constituted by a 7-character identifier

followed by some mumerical parameters), which can be recalled (once or more)

at any time in the interactive session. The pre-processing commands refer to the

following items: a) the relative frame (Cartesian, polar or cylindrical), optionally

defined for more simple reference to the absolute spatial co-ordinates; b) the

geometrical window, defined in the relative frame to restrict the application of

some pre-processing commands to this domain; c) the geometrical list, defined as

a set of nodes, elements, boundary sides or edges to restrict the application of

some commands to this set; d) the geometrical functions, which are real functions

of the spatial co-ordinates in the relative frame, defined to impose boundary

conditions and sources in an automatic way; e) the mesh generation, which is

relative to the current geometrical window and allows the generation of nodes,

elements and boundary sides in an N-dimensional way. These features allow a

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146 Software for Electrical Engineering Analysis and Design

very efficient imposition of sources and boundary conditions. This operation is

divided in two phases: 1) selection of the boundary sides on which the boundary

conditions are to be imposed; geometrical (volume, surface, line), indexed and

mixed selections are performed by using the geometrical window and list

concepts; 2) assignment of the pertaining nodal numerical values, either by using

the internal geometrical functions or by reading them from external files.

2.2 Processing

The ELFIN processing program works in a batch way, starting from the data

prepared in the pre-processing session and stored in a suitable file. In the

following some aspects of the program will be highlighted.

- Generalised iteration structures: a general iteration structure was implemented in

order to treat the various iterative procedures independently; four kinds of

iteration are implemented: non-linearity, adaptive mesh refinement, boundary

conditions (see DBCI and RBCI in Sect. 3), and time discretization [4];

- Numerical techniques: classical variational and weighted residual (Galerkin)

approaches are used. For each equation type a routine exists which builds the

global algebraic system, by resorting to a set of lower-level routines dealing with

the most common differential, integral and algebraic operators. This structure

allows a simple treatment of equations by splitting them into elementary

differential, integral and algebraic terms. Great ease in composing new problems

is given: the addition of new problem-solving capabilities only involves the

construction of a new limited-size routine, addressing the existing basic ones.

- Universal matrices: in addition to numerical (Gauss) integration, great care has

been devoted to the universal matrix technique for simplex Lagrangian elements

with straight sides. Since universal matrices, initially introduced by Silvester, are

continuously growing in number and kind, a set of basic routines was developed

to allow on-line computation of universal values in integer form. At present about

twenty kinds of universal matrices are available [5,6].

- Coupled problems: the treatment of coupled problems is based on

parameterization with respect to MS and My and is dealt with in a generalised

way: the ordered set of field variables (whose semantics is unknown to the

preprocessor) is correctly interpreted only by the routine which builds the global

algebraic system for the specific coupled problem. In this way great flexibility

exists to implement new coupled problems into the code, because only one

limited-size routine has to be developed and added to the processor.

2.3 Post-processing

The post-processing program exhibits the same interactive-batch user interface as

the pre-processing one, with the provision of interfacing to more powerful

commercial post-processing programs. The user can define up to three (real or

complex, scalar or vector) variables VR, related to the computed field variables by

means of algebraic or differential operations. After one of them has been selected

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Software for Electrical Engineering Analysis and Design 147

(that is, the token has been assigned to it) all the subsequent restitutions will refer

to the variable currently selected. With respect to discretization four types of VR

are possible: node, element, boundary-side and edge variables. Restitutions are

available such as: displaying/printing of nodal values, 2D contour line plotting,

axonometric 3D contour line plotting [7], 2D and 3D vector plotting. Global

quantities (energy, flux, current, etc.) can also be evaluated by means of integrals

on elements, boundary (or local) sides or edges.

3 Research conducted by means of ELFIN

In this section an overview is given of the research conducted by means of the

ELFIN code. For the sake of conciseness, only a brief outline is given; more

details can be found in the papers referenced.

3.1 Static and quasi-static problems in open boundaries

These problems have been dealt with by means of a hybrid method, now called

Dirichlet Boundary Condition Iteration (DBCI) and formerly referred to as charge

iteration or current iteration. Referring to a simple electrostatic problem, in which

a system of voltaged conductors is embedded in an unbounded homogeneous

dielectric medium, DBCI is applied by introducing a fictitious boundary BF which

includes all the conductors and truncates the unbounded domain to a bounded one

D [8-18]. The method is based on the following equations:

V-(eVv) = 0

in D

(1)

^da'

reBp

(2)

Sn'

where BC is the conductor surface and G is the free-space Green's function. By

discretising D by means of nodal finite elements, the following linear system of

equations is obtained:

(3)

This system can be efficiently solved iteratively: initially guessing the Dirichlet

condition Vp on BF, equation (3) is solved for V, which is used in (4) to improve

Vp. The procedure is iterated until convergence takes place. A better convergence

can be obtained by means of a relaxation coefficient in (4) or by means of

GMRES. Similar procedures have been successfully applied to the solution of the

integro-differential equation [19] of the two-dimensional time-harmonic skin

effect [20-24] and of transient non-linear eddy current problems [25-26].

Further research in this context was reported on in [27] in which it was shown

that the Pefectly Matched Layer (PML), recently proposed for static fields, is

equivalent to the utilization of a trivial truncation followed by simple co-ordinate

transformations. For this reason no advantage with respect to truncation derives

from the use of such a method.

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148 Software for Electrical Engineering Analysis and Design

3.2 Electromagnetic scattering problems

Let us consider a system of conductors and/or dielectrics infinitely extended in

the z-direction, embedded in a homogeneous dielectric medium, lit up by an

incident monochromatic wave <|>o, E- or H-polarised along the z-axis. An

unbounded scattering problem is set up in terms of the total field ((>. To deal with

this problem, the hybrid method illustrated in the previous Section has been

modified into the RBCI (Robin Boundary Condition Iteration) method [28-33], in

which the Dirichlet condition on the fictitious boundary Bp has been replaced by

a Robin (mixed) one of the type:

on Bp

(5)

where ko is the free-space wavenumber and \\i is an unknown function on Fp.

Then the working equations are:

in D

(6)

(7)

where a and TJ are relative constitutive parameters, BS is the scatterer surface and

G is the two-dimensional free-space Green's function

-r'|)

(8)

Equations (6) and (7), taking into account the boundary condition (5), are

discretised as:

AO =OF

(9)

T-TO + MO

(10)

and solved iteratively as before. Initially guessing *F (a good guess is %),

equation (9) is solved for O, which is used in (10) to improve *F. This procedure

is iterated until convergence is obtained. Note that if one tries to use Dirichlet

conditions on the fictitious boundary Bp, as in standard FE-BI methods, possible

singularities may arise in (6), due to the resonance of the domain D and also in

(7), due to the existence of non-vanishing incident fields §Q which may vanish on

the boundary Bp. On the contrary, by using the Robin boundary condition (5)

resonances are completely avoided whatever the frequency of the incident wave.

This very good property is due to the fact that the boundary condition (5) works

as an absorbing-like one. Another important property is that the air gap in

between the scatterer surface BS and the fictitious boundary Bp can be very thin

(two or three layers of elements are enough) still obtaining accurate results with

an acceptable computing time (normally from 4 to 8 iterations are sufficient to

obtain convergence). By means of a simple indicator, the accuracy of the

solution can be tested in post-processing when the bistatic radar cross section is

computed.

The RBCI method has been successfully adapted to the computation of scattering

from cavity-backed apertures on a perfectly-conducting plane or wedge, and also

to the solution of three-dimensional scattering problems, employing tetrahedral

edge elements.

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Software for Electrical Engineering Analysis and Design 149

3.3 Mesh generation by neural networks

Starting from a rough initial mesh of triangles or tetrahedra with a small number

of nodes the neural network algorithm increases the number of nodes until a user-

selected value is reached [34-37]. Both internal and boundary nodes are

separately increased according to two distinct probability density functions,

specified by the user using the initial mesh as support. New nodes are inserted in

the middle of triangle or tetrahedra edges. After an example point is generated,

the nearest nodes are moved to it, taking into account the mesh quality; moreover

the classical Delaunay triangulation algorithm is inserted in the growing process

to further increase the quality of the final mesh. The main advantages of this

neural approach are the lack of a need for classical deterministic programming,

the simplicity of use (the user only needs to specify an initial rough mesh and the

probability density functions) and the very good qualities of the simplex elements

obtained.

3.4 Optimization by means of stochastic methods

Recently the ELFIN code has been adapted to deal with stochastic optimization of

electromagnetic devices. This goal has been reached by allowing the use of

symbolic variables in the pre- and post-processing command parameters instead

of fixed numeric values. In this way both the pre- and post-processing batch

sessions (and then the whole code) can be made parametric with respect to a set

of variables, typically representing geometrical or constitutive data of the device

to be optimised. Two stochastic optimisation methods have been implemented:

genetic algorithms and simulated annealing. They have been applied to the

optimization of some electromagnetic devices, ranging from magnetostatic to

wave-propagation ones [38-40].

4 Conclusions

In this paper the finite element code ELFIN, developed for research purposes in

the field of computational electromagnetics in electrical engineering, has been

presented. A general overview of the code has been given, briefly describing

some non-standard implementation aspects as well as the main research

conducted by means of it.

The main merit of ELFIN is the facility it offers in building, testing and verifying

new numerical procedures devised for the analysis and design of electromagnetic

devices.

At present the code is composed of about 600 subroutines (for 60,000

statements). It will be expanded according to research subjects rather than

widening the range of standard problem-solving capabilities.

Further information on ELFIN can be found on the Internet at the following Web

address: http://wwwelfin.dees.unict.it/elfin/.

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150 Software for Electrical Engineering Analysis and Design

Acknowledgements

This work was partially supported by the MURST (the Italian Ministry for

University and Scientific and Technological Research).

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