[PDF] 96 - Frequency-Domain Finite Element Methods for Electromagnetic

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Frequency-Domain Finite Element Methods

for Electromagnetic Field Simulation:

Fundamentals, State of the Art,

and Applications to EMI/ Analysis

Andreas C. Cangellaris

Center for Electronic Packaging Research, ECE Department

University of

Arizona, Tucson, AZ 85721, U.S.A.

Abstract-This paper provides a critical review of

frequecy-domain finite element methods and their ap- plications to the modeling of electromagnetic interac- tions in complex electronic components and systems. Emphasis is placed on latest advances in finite element grid generation practices, element interpolation func- tion selection, and robust, highly absorbing numerical grid truncation techniques for modeling electromag- netic interactions in unbounded domains. These ad- vances have helped enhance the robustness and accu- racy of the method. Finally, the advantages of domain decomposition techniques for the modeling of com- plex geometries are examined. Such domain decom- position techniques are expected to play an important role in the continuing effort to extend the applica- tions of frequency-domain finite methods beyond the subcomponent-level to component and system model- ing for electromagnetic interference and electromag- netic compatibility analysis and design. I.

INTRODUCTION

There are two attributes of the method of finite ele- ments that have prompted the rapid growth of its appli- cation to the modeling of electromagnetic interactions in electronic systems. One of them is its superior modeling versatility where structures of arbitrary shape and compo- sition can be modelled as precisely as the desirable model complexity and available computer resources dictate. The second, is common to all differential equation-based nu- merical methods, and has to do with the fact that the matrix resulting from the discretization of the governing equations is very sparse, which implies savings in com- puter memory for its storage as well as in CPU time for its inversion. Clearly, these two attributes come at the expense of an increase in the degrees of freedom used in the numerical approximation of the problem since now, contrary to integral equation methods, the entire space surrounding all sources of electromagnetic fields needs be incorporated in the numerical model. Nevertheless, be- cause of the sparsity of the resulting matrix and the sim- plicity with which complex geometries can be modeled, this increase in the degrees of freedom of the approxima- tion is an acceptable penalty.

Over the past ten years, a significant volume of liter- ature has been generated on the application of the finite element method to a variety of electromagnetic scattering

and radiation problems. The book by J.M. Jin [l] serves both as a tutorial on the procedures for the application of the finite element method to the approximation and solu- tion of electromagnetic boundary value problems, and as a rather thorough survey of the classes of problems that have been tackled. Considering the power of the afore- mentioned attributes, one would have expected that the method of finite elements would have gained in popularity among EMC/EMI engineers and would have established itself as the method of choice in the analysis and predic- tion of EM1 and the design of electromagnetically compat- ible systems. Nevertheless, a literature review indicates that this is not the case. As an example we mention that in a special issue of the IEEE Transactions on Electromag- netic Compatibility, dedicated to computational methods for EMI/ analysis, very few papers on finite elements appeared, and the applications presented where limited to rather simple problems of low complexity [2]-[3]. Before one attempts to search for drawbacks in the method of finite elements that have prevented its prolifer- ation as an EMI/ prediction tool, one has to keep in mind that application of electromagnetic CAD for compo- nent and system EMI/ is actually still at its infancy. The reason for this is that the complexity of an integrated electronic component, subsystem or system is such that accurate modeling of source, coupling mechanism, and re- ceiver of electroma.gnetic noise is almost prohibitive using a single numerical method for solving Maxwell's equa- tions, irrespective of the type of the method used. More specifically, considering the tremendous variation in fea- ture size from chip, to package, to board, to cables, to shields, it becomes clear that the number of elements re- quired for the discretization of such a system for finite element analysis of electromagnetic interactions is out of the reach of today's most powerful supercomputers. In view of the above and recognizing that an elec- tromagnetic analysis tool will be useful as a CAD tool only if simulation times are in the order of minutes or at most a few hours, this paper examines the latest ad- vances in the method of finite elements that are expected to help the method establish itself as a reliable candidate for EMI/ problem solving either at the component level or in conjunction with reduced-order electromagnetic

O-7803-3207-5/96/$5,00 0 1996lEEE 107

models of subsystems. As a matter of fact, it is this area where the method of finite elements can have an impor- tant impact. Indeed, current practices of EMI/ anal- ysis concentrate on rather simplistic, individual source-to- victim models, which often suffer from their inability to capture the impact of surrounding conducting, dielectric, and magnetic material topology on the electromagnetic interaction. The finite element method allows for the de- velopment of a more precise model that will lead to higher accuracy in noise prediction and thus facilitate the design of electromagnetically compatible electronic modules. Finally, the potential of domain decomposition meth- ods for reducing the complexity of the original problem will be examined. The basic idea behind such methods is the partitioning of the domain of interest into smaller ones and the development of the solution in a piecewise manner, one subdomain at a time, using different types of both numerical and analytic techniques. The inherent parallelism of such approaches combined with the smaller size of the subdomains makes them extremely well-suited for massively-parallel computation.

II. MATHEMATICAL FRAMEWORK FOR FINITE

ELEMENT ANALYSIS

The focus of this paper is on the numerical approxi- mations of Maxwell's equations with time-harmonic field variation. Therefore, the following discussion pertains to linear sources and materials. However, time-domain finite methods that can handle transient electromagnetic inter- actions in the presence of nonlinear sources and nonlinear media are possible and are currently the topic of vigorous research within the computational electromagnetics com- munity. As a matter of fact, the finite-element formula- tion in [2] is such that both transient and time-harmonic electromagnetic simulations can be effected within a sin- gle mathematical framework. In order to review the basic steps involved in the fi- nite element approximation of electromagnetic boundary- value problems, let us consider the double-curl equation for the electric field, E, which, in a source-free, isotropic and linear medium with position-dependent magnetic and electric properties has the form

Vx (&VxE)+jwiE=O,

The time dependence exp(jwt) is assumed (j = G),

and the complex permittivity, E = c - ja/w, is used to account for any conduction and/or dielectric losses in the medium. For the purposes of finite element solutions, a weak form of (1) is required. For node baaed finite element expan- sions the unknown vector field is approximated in terms of scalar basis functions, &,

E=CEidi, (2)

where Ei denotes the unknown vector field value at node i. The relevant weak form, in the spirit of Galerkin's approximation, is (( &V X E) x V&) + (j&E&) = f

1 - -ii x (V x E)gS& jwp (3)

where ( ) and $ indicate integration over the domain of interest and its boundary, respectively, while ii is the out- ward unit normal on the boundary. For edge element expansions, vector basis functions, Ni, are used for the expansion of the field,

E = C EiNiy

where Ei are the unknown coefficients in the expansion.

The relevant weak form is

(( LO X E) * (V X Ni)) + (j&E * Ni) = 3WP -f

1 -G X (V X E) . Nids jwp (5)

For two-dimensional problems, a scalar version of (3) is readily obtained. More specifically, for a transverse magnetic to z (TM,) polarization, the fields, E = bE, H = kH, + 9Hy, are independent of z and (3) reduces to (C &Vx,,> . Vxyh) + (jwtEq$) = f '(6. VE)(bjdZ jwp (6) where VW = S/&Z + ?a/ay. For transverse electric to z (TE,) polarization of the two-dimensional fields, where H = i?H, E = S?, + QE,, the weak form is easily found from (6) by duality. For static problems (w = 0), a scalar potential, @, is often introduced, and the electric or magnetic fields are obtained in terms of the gradient of the potential. For such cases, it is a weak statement of Laplace's equation for the scalar potential that is being derived. The most well-known attribute of Galerkin's method, where the solution is sought in a finite-dimensional sub- space of the class of admissible functions for the problem of interest using the same set of functions as trial and test functions, is the symmetry of the resulting stiffness matrix given a symmetric weak formulation. However, another important merit of Galerkin's method is that, if a sym- metric weak formulation is used, Galerkin's approximate 108
solution exactly conserves energy in the electromagnetic field despite the fact that it satisfies the vector Helmholtz equation only approximately over the domain of inter- est. This is easily shown starting from (5) and using the complex conjugate of the field, E*, as the test function and Faraday's law to introduce the magnetic field in the boundary integral (( $V x E) . (V x E')) + (jw?E . E*) = f (fi x H) . E"ds (7) The complex conjugation of (7), use of Faraday's law to simplify the integrand in the first term on the left, and use of the constitutive relations B = pH, D = 1E, result in the following equation (-jwB . H') + (jwD* . E) = f (E x H*) . iids (8) Clearly, the resulting expression is Poynting's theorem for time-harmonic fields. Thus energy conservation in the electromagnetic field is satisfied exactly by the approxi- mate solution. The importance of the aforementioned result is rather significant. Considering the various potential sources of error in the development of a numerical solution to a boundary value problem, it is definitely advantageous to be able to work with a weak statement that is consistent with the correct physics of the field we are attempting to calculate. For the electromagnetic field problems of in- terest, the calculated field quantities will be acceptable only if they satisfy both energy conservation and elec- tric charge conservation. As a matter of fact, the latter has been found to be extremely important in the finite element solution of three-dimensional vector electromag- netic problems. In later sections, it is pointed out that modifications to the weak statement in (3) and careful se- lection of the vector basis functions in (5) are needed to prevent the contamination of the numerical solution from spurious fields caused by the lack of enforcement of charge conservation in the original weak statement. In order to illustrate the development of the numeri- cal approximation of the electromagnetic boundary value problem, let us consider the weak statement in (5). Sub- stitution of (4) into (5) and testing with each and every one of the vector basis functions N. results in a linear J system of simultaneous equations M c AijEi=fj, j=l,Z&...,M id where the elements, Aij, of the so-called stiffness matrix are given by -VxNi).(VxNj))+(jv Ni.Nj)(lO) while the elements of the forcing vector are .fj=-f &G X (V X E) * Njds

M is the number of degrees of freedom in

01) the approxi- mation. One of the important attributes of the method of finite elements is that that the basis functions used have local support, i.e. they are non-zero only over a set of adjacent elements. This is what leads to the spar- sity of the resulting system since most of the elements Aij are zero. The forcing vector is formed by contribu- tions of the surface integrals over the domain boundaries. From the uniqueness theorem, the tangential component of the magnetic field, -(l/jwp)G x (V x E), on the do- main boundaries is all that is needed for a unique solution of Maxwell's equations inside the domain. This tangen- tial magnetic field on the domain boundary is used to account for all sources exterior to the domain of inter- est. The surface term in (5) is used also for enforcing tangential magnetic field continuity conditions at mate- rial interfaces. As far as tangential electric field boundary conditions are concerned, they are taken into account in the construction of the basis functions. This is discussed in more detail in Section 4.

III. GRID GENERATION

Numerical grid generation is probably the most critical step in a finite element analysis of electromagnetic wave interactions. During the early stages of the application of the finite element method to modeling of electromagnetic interactions, the emphasis was on mathematical model and weak statement formulations and their subsequent use in the analysis of propagation, radiation and scat- tering problems in conjunction with rather simple geome- tries. Consequently, the important issue of automatic gen- eration of finite element grids appropriate for electromag-quotesdbs_dbs6.pdfusesText_12

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