code for electrical engineering research:

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1 An FEM ELFIN code for electrical engineering research: G. Aiello, S.Alfonzetti, S. Coco, N. Salerno Dipartimento Elettrico, Elettronico e Sistemistico, Universita di Catania, VialeA. Doria 6, Catania, Italy Abstract This paper discusses the basic philosophy and design guidelines of the ELFIN code for electromagnetic CAD research. ELFIN is basically a finite-element code, conceived as a sort of numerical analysis laboratory, expressly tuned to research purposes, where innovative procedures can be easily implemented and tested. The main features of the code are N-dimensional (ID, 2D and 3D) parameterization of procedures, wide application to PDE, IE and IDE problems, and easy treatment of coupled problems, mixed approaches and generalized iterative procedures. Examples of application to 2D and 3D electromagnetic CAD research are given. 1 Introduction Basic research in the area of electromagnetic CAD is concerned with general investigation of new strategies for problem solving and the study of their properties and performance [1], Typical subjects in this context are: the development of new numerical approaches, comparative evaluations of various methods and formulations, validation studies, modelling for special applications, sensitivity and error analyses, inverse problems, optimal design. Among the various numerical methods for electromagnetic analysis the Finite Element Method (FEM) is usually preferred because of its flexibility and generality [2]. The main characteristics that an FE code for electromagnetic CAD research should have are: - openness: the source code should be accessible to the researcher in order to operate modifications easily and to add new capabilities; - modularity: new problems should be implementable by performing simple operations of recombining existing functions and/or by adding suitable modules not interfering with the pre-existing ones [3],[4];

2 358 Software for Electrical Engineering - N-dimensionality: the code should allow the treatment of ID, 2D and 3D problems by following the same algorithms; note that this capability is more than just having three separate codes according to a fixed dimensional index; - mesh freedom: the code should allow great freedom in the discretization of the domain of interest, in the sense that it should be possible to specify elements individually, including higher-order and curved-side elements; - FEM/BEM integration: BEM (Boundary Element Method) should be conveniently integrated in an FE context, to take full advantage of mixed FEM/BEM approaches [5] [6]. Commercial FE codes seldom offer all the above capabilities. In fact, although they are based on well-known, efficient and robust procedures, many of them are meant 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 characteristics are intended to optimize code performance, but they practically prevent the exclusive use of such codes in a research environment. On the other hand, the development of special programs for each individual research requirement leads to an uncontrollable proliferation of software, with difficulties in integrating different data structures and procedures. To overcome such difficulties, the authors have designed a sort of unitary environment, where research items can be conveniently treated by resorting to a set of highly integrated routines. A basic requirement of this design was its implementability in a small research centre, where limited resources are normally available. The implementation of this environment led to the development of the ELFIN code [7], which is still in progress. This paper highlights the basic philosophy of the design and gives examples of application to CAD research. 2 The basic philosophy of ELFIN The ELFIN code is written in Fortran-77 and was developed on DEC VAX and ALPHA computers under the VMS operating system. Although the code architecture follows the classical scheme of FE codes, in which three distinct main programs are devoted respectively to preprocessing, processing and postprocessing phases, some aspects of the ELFIN structure have been expressly conceived in order to offer the capabilities usually required in a research environment. In the following attention is mainly focused on such aspects. More precisely, the organization of the data structure adopted for the purpose of dimensional and problem parameterization, which are a peculiarity of the ELFIN code, is firstly addressed; a brief overview of the code functions is then offered and the reasons leading to the implemented options are also discussed. 2.1 Geometrical discretization for N-dimensional parameterization The complete N-dimensional parameterization of the geometrical algorithms of

3 Software for Electrical Engineering 359 ELFIN is based on the choice of nodes as fundamental entities of the geometrical discretization. A node is identified by its absolute Cartesian coordinates Xfc (k=l,...,n), in an N-dimensional space. The other geometrical entities of the discretization (elements and boundary sides) are derived from nodes, in the sense that they are defined in terms of particular sets of nodes exhibiting certain geometrical properties. An element is an ordered set of nodes and is identified by an integer The geometrical characteristics of the element are defined every time a geometrical meaning is attributed to the ordering. In this way no limitations are imposed a priori on the kind of element (dimensionality, shape, order) and the following items can be 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). Local sides of an element are implicitly defined as an ordered subset of its nodes. Boundary sides are explicitly defined local sides; they generally lie on the boundary of the domain, even if internal boundary sides are allowed Region colours can be independently attributed to nodes, elements and boundary "hse" * ***"W"^ specified collections of them during the processing The above organization of geometrical information is clearly structured in a hierarchical way; nodes have dominance over elements and elements over boundary sides. The operations performed on dominated geometrical entities are affected only by hierarchical constraints posed by higher-level entities However, this structure has the great advantage of independence of space dimensionality (which affects only the cardinality of the coordinates of nodes but does not cause any structural modification) and the capacity to cover a wide range of problems. This agility and flexibility has a cost in terms of larger memory occupation and computing time if compared, for example, with codes employing only elements of a fixed kind and order. 2.2 Parameterization of problem variables and sources The conception of a suitable organization of problem variables when the problem itself is not known a priori requires careful balancing between the capacity of treating even more new problems and the rapidly increasing complexity of the structures and relationships involved. In the ELFIN code the objective of maximum freedom of these structures from the kind of problem has been reached by parameterizing problem variables and sources. An ordered set ofm, scalar variables V, (i=l,...,m,) is associated to each node of the discretization. These variables can be all real or all complex according to the real/complex indicator V, which is set to 1 for real problems and to 2 for complex ones (so the total number of real variables associated to n rir?m* *^' ** ^ P^lem Cables V, boundary conditions Dmchlet, Neumann or mixed) are imposed on the nodes of each boundary side (see Section 2.3). Parameterization with respect to the number of problem

4 360 Software for Electrical Engineering variables is the basis to allow unified treatment of coupled problems (see Section2.4). Like problem variables, three kinds of sources are defined: node sources, element sources, boundary-side sources. The real/complex indicator also applies to sources. This organization of the problem-dependent data allows no semantics to be preliminarily attributed to each variable and offers the valuable advantage of the capacity to implement a great variety of problems (scalar, vectorial, coupled, real or complex) exhibiting different kinds of variables and sources with no modifications of structures. 2.3 Preprocessing Since the main objective of the preprocessing functions is to facilitate experimental work for research purposes, the preprocessor has medium capabilities for solid modeling, but quite sophisticated features for defining boundary conditions and sources and for managing high-order and/or curvedside elements; moreover functions are expressly foreseen to interface more powerful external solid modeling programs, if available. A very usercomfortable mixed interactive/batch modality is implemented: before a preprocessing session the user can edit a SPICE-like command file, which can be recalled (once or more) at any time in the interactive session. In the following a brief outline of some preprocessing features is given. - Relative frame (Cartesian, polar or cylindrical) is optionally defined by the user for more simple reference to the absolute spatial coordinates. - Geometrical window is defined by the user (in the relative frame) to restrict the application of some pre- and post-processing commands to this domain. - Geometrical list is a set of nodes, elements or boundary sides, defined by the user to restrict the application of some commands to this set. - Geometrical functions are real functions of the spatial coordinates in the relative frame, defined by the user to impose boundary conditions and sources in an automatic way. - Mesh generation 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 very efficient treatment of the crucial issue of imposing sources and boundary conditions when new formulations are experienced; this operation is articulated 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 geometrical function (internally generated values) or by reading them from an external file. 2.4 Processing The ELFIN processing program works in a batch way, starting from the data specified in the preprocessing session and stored in a suitable file. In the

5 Software for Electrical Engineering 361 following only some aspects of the program will be highlighted, neglecting details which reflect standard implementations. - Generalized iteration structures: in the code the problem definition is accomplished by means of definition of the equation and, optionally, of the iteration. The equation definition specifies the kind of PDE, IE or IDE equation (e.g. Laplace, Poisson, Helmholtz, 2D skin effect, motional magnetic diffusion, etc.) from a specific menu. The iteration definition specifies the iteration type (non-linearity, mesh refinement, boundary conditions, time discretization), the convergence norm and the maximum number of iteration steps. A general iteration structure was designed for the main program, in order to treat these independent aspects of the problem definition separately. The equation definition affects the renumbering of the unknowns, the building of the row and column pointers of the non-null entries in the coefficient matrix and the computing of the matrices of the global algebraic system. The iteration definition affects the computation of the convergence norm and the enditeration adjustments, which depend on the kind of iteration selected. - Numerical techniques: from the standpoint of numerical treatment of problems, classical variational and weighted residual (Galerkin) approaches are used. For each equation type a routine exists to build the global system, by resorting to a set of lower-level routines, which deal 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. Moreover, great ease in composing new problems is given since the addition of new problem-solving capabilities only involves the construction of a new limited-size routine, opportunely addressing the existing basic ones. - Universal matrices: as far as element matrix computing is concerned, 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 online computation of universal values in integer form. - Coupled problems: the treatment of coupled problems is based on parameterization with respect to My and is faced in a generalized way: the unstructured set of problem 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. 2.5 Postprocessing For postprocessing most of the considerations made for preprocessing apply, with the provision of interfacing to more powerful commercial postprocessing programs. The postprocessing program performs restitutions of a user-defined (real or complex, scalar or vectorial) variable V%, related to the computed

6 362 Software for Electrical Engineering problem variables by means of algebraic or differential operations. With respect to discretization three types ofv% are possible: nodal, elemental and boundarial. Restitutions are available such as: displaying/printing of nodal values, 2D contour line plotting, axonometric 3D contour line plotting, 2D and 3D vectorial plotting [8]. Global quantities (energy, flux, current, etc.) can also be evaluated by means of N-dimensional integrals on elements or boundary (or local) sides. 2.6 Libraries The ELFIN libraries are the core of the code because they constitute the base on which new problems and research can be dealt with by a prevalent work of assembling. More than 500 subroutines have been developed, 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; solution of linear algebraic equation systems. c) graphical subroutines, which, for modularity reasons, interface the external graphical library adopted (Regis, GKS, Xlib, Display Postscript) 3 Examples of research In this section examples are presented of electromagnetic CAD 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 relative papers. 3.1 Charge iteration procedure Charge iteration [9] is a new N-dimensional purely FEM iterative procedure for the computation of electrical fields in unbounded domains [10]. Consider a system of voltaged conductors embedded in an unbounded homogeneous dielectric medium. A suitable bounded region D is obtained by introducing a fictitious boundary Bp including all the conductors. By discretizing this domain by means of Lagrangian finite elements of a certain order and assuming an initial guess for the DiricWet condition on Bp (as for example a homogeneous one), the FEM for the Laplace equation leads to the global algebraic system: [A][V] = [B] (1) where [V] is the vector of the unknown potential values; [A] is a symmetrical square matrix of geometrical coefficients and [B] is the vector of the known

7 Software for Electrical Engineering 363 terms, whose generic entry depends on the geometry and on the various Dirichlet conditions. The solution of (1) is affected by a systematic error, due to the untrue values of the potential on Bp; a better approximation to the true solution is obtained by modifying the Dirichlet condition on Bp with the potential values induced by the charge lying on the conductors. These values are computed by means of an appropriate Green function as: V; = ZGimQm (2) mes<. where: G^ is the Green function relative to the i-th node and the barycentre of the m-th element side S^ lying on the conductor surfaces and Q^ is the charge lying on it which can be computed as a linear combination of the potentials in the nodes of the element. The procedure is iterated and will continue until a suitable convergence test is satisfied. In this way the original unbounded problem is converted into a set of iteratively correlated bounded electrostatic problems. The procedure exhibits aspects of computational efficiency if the fictitious boundary is placed in the vicinity of the conductor surfaces [11] and is numerically stable and rapidly convergent (typically after few iterations) [12]; it has also been validated to assess its accuracy by solving classical problems for which analytical solutions are available in literature; the results show an excellent agreement with the known solutions [9]. As an example in Fig. 1 the contours of the potential are drawn in the vicinity of one quarter of a conductor cube on a ground plane. For this analysis a mesh of 1035 nodes and 500 2ndorder tetrahedra was used. The results were obtained with 7 iterations starting with a homogeneous Dirichlet condition on the fictitious boundary and assuming an end-iteration tolerance of 0.1% for the mean absolute difference of the potential on thefictitiousboundary nodes. At this point some aspects of the above procedure can be highlighted, showing how the ELFIN code represents a suitable environment to deal with such studies. The charge iteration procedure fits perfectly into the generalized iteration structure of the processing program; only a limited-size routine has been added to the code to perform the end-iteration adjustments (at thefirstcall the routine computes and stores the relevant coefficients of the procedure). Naturally the code characteristics have allowed an N-dimensional implementation of the procedure, which is also parameterized with respect to the order m of the element; this capability allows useful comparisons between solutions employing elements with different orders. 3.2 Mutual matrices for 2D skin effect problems The application of FEM to the solution of integrodifferential equations, such as those modeling skin effect problems [13], leads to a discretization scheme in which different treatments are used for the various terms of the equation. The differential part, which establishes local links between the unknowns of the problem, is treated in the usual way by assembling the element contributions

8 364 Software for Electrical Engineering Fig. 1 -Contours of the electrical potential in the cube proximity. Fig. 2 -Contours of the vector potential in a two-wire transmission line. into the global algebraic system, whereas the integral part, which establishes direct links between all the unknowns of the problem, is generally treated in a global way by means of quadrature coefficients without following any perelement assembly procedure. In [14] the authors proposed a unitary approach, utilizing a per-element procedure for assembly of the global algebraic system: this was obtained by expressing the global node coupling in terms of element coupling, by means of a new class of universal numerical matrices, named mutual matrices, which are similar to those introduced by Silvester. This approach allows easy inclusion of integrodifferential formulations into generalpurpose N-dimensional finite-element codes. Moreover the resulting assembly algorithm exhibits better performance than conventional techniques. In Fig. 2 the contours of the magnetic vector potential magnitude are reported for one conductor of a two-wire transmission line. For the solution of the relative unbounded problem a procedure similar to that of the previous section was employed [15],[16],[17]. 3.3 Automatic Mesh Generation by the Let-It-Grow Neural Network Let-It-Grow is a self-organizing neural network in which the number of nodes (neurons), connected in a triangular mesh, grows according to examples generated randomly by means of a given probability density function [18]. Starting from the basic idea of the network several changes were made in order to obtain an automatic mesh generator which was implemented in the ELFIN code [19].

9 Software for Electrical Engineering " * Fig. 3 -LIG initial mesh. Fig. 4-LIGfinalmesh. Starting from a rough initial mesh of triangles with a small number of nodes the neural algorithm increases the number of nodes until a user-selected value is reached. 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 barycentre of the triangles or in the middle of triangle sides on the boundary. 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 and simplicity of use since the user only needs to specify an initial rough mesh and the probability density functions. As an example in Fig. 3 the initial mesh (with the associated probability values for elements and boundary sides) is shown for a domain arising in the electrostatic analysis of a conductor disk on a ground plane (r-axis), taking into account the axisymmetry around the z-axis. The final mesh, reported in Fig 4 is constituted of 1149 nodes and 2108 triangles. 4 Conclusions In this paper the FEM code ELFIN for electromagnetic CAD research has been presented, discussing its basic philosophy and design guidelines. The major contribution of ELFIN is the facility it offers in studying new approaches and building, testing and verifying procedures for the analysis and design of electromagnetic devices in a unitary context At present the code is composed of about 500 subroutines (for 50,000 statements). It will be expanded according to research subjects rather than widening the range of standard problem solving capabilities.

10 366 Software for Electrical Engineering Acknowledgements This work was partially supported by the MURST (the Italian Ministry for University and Scientific and Technological Research) through the research project "Integrated CAD Methods for Electromagnetic Fields and Circuits". References 1. Trowbridge, C.W. Electromagnetic computing: the way ahead?, IEEE Trans. Magn., 1988, Silvester, P.P., Ferrari, R.L. Finite Elements for Electrical Engineers, Cambridge University Press, Cambridge, Girdinio, P., Mofino, P., Molinari, G. The versatility of a multiformulational approach in the solution of electromagneticfieldproblems, IEEE Trans. Ma#%.,1987, Molinari, G. et alii, A modular finite-element package for research in electromagnetic analysis developed in a group of Italian universities", Proceedings of BISEF, Peking, China, Oct , 1988, pp Me Donald, B., Wexler, A. Mutually constrained partial differential and integral equation field formulations, Chapter 4, Finite Elements in Electrical and Magnetic Field Problems, ed. M. Chari and P. Silvester (Ed.), J. Wiley & Sons, Salon, S.J., D'angelo, J. Applications of the hybrid finite element - boundary element method in electromagnetics". IEEE Trans. Magn., 1988, Alfonzetti, S., Coco, S. ELFIN: an N-dimensional finite-element code for the computation of electromagneticfields,ieee Trans. Magn., 1988, Alfonzetti, S. An N-dimensional algorithm to draw contour lines over triangular elements", Compumag'95, Berlin, July 10-13,1995. To appear on IEEE Trans. Magn. 9. Aiello, G., Alfonzetti, S., Coco, S. Charge iteration: a procedure for the finite element computation of unbounded electricalfields, Int. J. Numer. Methods Eng., 1994, Emson, C.R.I. Methods for the solution of open-boundary electromagnetic field problems, Proceedings IEE, Pt. A, 1988, Aiello, G., Alfonzetti, S., Coco, S., Salerno, N. Placement of thefictitiousboundary in the charge iteration procedure for unbounded electrical field problems," IEEE Trans. Magn., 1995, Aiello, G., Alfonzetti, S., Coco, S., Salerno, N. Convergence analysis of the charge iteration procedure for unbounded electricalfields,ieee Trans. Magn., 1994, Konrad, A. Integrodifferential finite element formulation of two-dimensional steadystate skin effect problems, IEEE Trans. Magn., 1981, Aiello, G., Alfonzetti, S., Coco, S. Mutual finite-element universal matrices in skin effect problems, IEEE Trans. Magn., 1990, Aiello, G., Alfonzetti, S., Coco, S., Salerno, N. Current iteration for unbounded skin.effect problems, COMPEL, 1994, Aiello, G., Alfonzetti, S., Coco, S., Salerno, N. Treatment of unbounded skin-effect problems in the presence of material inhomogeneities, IEEE Trans. Magn, 1995, Aiello, G., Alfonzetti, S., Coco, S., Salerno, N. Finite element iterative solution of skin effect problems in open boundaries, to appear on Int. J. Numer. Modelling, special issue on Computational Magnetics. 18. Fritzke, B. Let-it-grow self-organizing feature maps with problem dependent cell structure, Artificial Neural Network, 1991,7, Alfonzetti, S., Cavalieri, S., Coco, S., Malgeri, M, Automatic Mesh Generation by the Let-It-Grow Neural Network, Compumag'95, Berlin, July 10-13, To appear on IEEE Trans. Magn.