A surface integral formulation of Maxwell equations for ... - IEEE Xplore

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 53, NO. 12, DECEMBER 2005

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A Surface Integral Formulation of Maxwell Equations for Topologically Complex Conducting Domains Giovanni Miano and Fabio Villone

Abstract—A general and effective method is presented to numerically solve the electric field integral equation (EFIE) for topologically complex conducting domains by the finite element method. A new technique is proposed to decompose the surface current density into a solenoidal part and a nonsolenoidal remainder to avoid the low frequency breakdown. The surface current density field is approximated through div-conforming (facet) elements. The solenoidal part is represented through the of the surface divernull space of the discrete approximation gence operator in the subspace spanned by the facet elements, whereas the nonsolenoidal remainder is represented through its complement. The basis functions of the null space and its complement are evaluated, respectively, by the null and pseudo-inverse of the matrix . The completeness of the null-pinv basis functions is studied. Unlike the loop-star and loop-tree basis functions, the null-pinv basis functions allow to deal with topologically complex conducting domains in a general and readily applicable way. A topological interpretation of the “null-pinv” decomposition is given and a general and simple method to evaluate the null and pseudo-inverse of is proposed. The computational complexity of the proposed method is discussed.

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Index Terms—Div-conforming (facet) elements, low frequency breakdown, multiply connected domains, null space, null-pinv decomposition, surface integral equations.

I. INTRODUCTION

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NE OF THE MOST widely used approaches in the electromagnetic modeling of electrically large and/or geometrically complex conducting bodies is based on the electric field integral equation (EFIE) formulation of the Maxwell equations (e.g., [1]–[4]). In many applications (e.g., design of interconnects and packaging structures in high density and high speed electronic circuits, EMI-immune devices [5]) a broadband electromagnetic modeling, from zero frequency to multigigahertz frequencies, is needed. The main advantage of the integral approach is in the fact that only the conducting part of the domain must be discretized. Nevertheless, the EFIE approach exhibits two shortcomings: first of all, both the numerical solution time and memory requirements grow at least as the square of the number of unknowns involved in the finite element approximation of the integral equations; second, the so-called low frequency breakdown of the solution, which occurs when the wavelength becomes larger than the characteristic size of the finite element grid [6]–[8].

Manuscript received May 6, 2005; revised July 28, 2005. The work was supported by the Italian Ministry of University (MIUR), by the Program for the Development of Research of National Interest under Grant 2004093025. G. Miano is with the Università di Napoli Federico II, Napoli 80125, Italy (e-mail: [email protected]). F. Villone is with the Università di Cassino, Cassino 03043, Italy. Digital Object Identifier 10.1109/TAP.2005.859898

The computational complexity and large memory requirements of finite element approximations to integral equations may be effectively reduced by fast multipole methods and its multilevel modifications [1], or by the use of fast Fourier transform (FFT) algorithms (see [3] and references therein). To avoid the low frequency breakdown of the solution it is required that: the current density field should be expanded in terms of div-conforming basis functions, that is, basis functions that have continuous normal components across adjacent elements (e.g., [9], [10]); the solenoidal component of the current density field should be separated from the nonsolenoidal one [7], [8], [11]. In the literature two schemes have been proposed to decompose asurfacecurrentdensityinto asolenoidalpart andanonsolenoidal remainder [7], [8], [11]. The loop-star decomposition uses loop-type basis functions to represent the solenoidal part and the so-called “star” basis functions to represent the nonsolenoidal part, whereas in the loop-tree decomposition the nonsolenoidal part is represented by “tree” basis functions. Indeed, both the loop-star and loop-tree basis functions are suitable linear combinations of the Rao–Wilton–Glisson basis functions [9]. The loop-type basis functions have a drawback. In a multiconnected domain or in presence of electrodes, as it will be shown, a generic solenoidal field may be not expressed through loop basis functions, unless some tricks are used: making some suitable cuts in the domain or introducing some additional, in general nonlocal, basis functions depending on the electrodes and the connection degree of the domain [8]. In this paper, we propose a new decomposition of the current density field that does not show this disadvantage. We will call it “null-pinv” decomposition. It is based on the null and pseudo-inverse of the matrix representing the discrete approximation of the divergence operator in the subspace spanned by the facet elements [12]. The facet elements are div-conforming functions, which are in the stream of the Rao-Wilton-Glisson basis functions. The solenoidal part of the current field density is represented through the null of and the nonsolenoidal part is represented through its pseudo-inverse. The null-pinv basis functions so obtained may be considered as a generalization of the loop-star and loop-tree basis functions. The null basis functions, as it will be shown, can represent solenoidal fields both in multiconnected domain and in presence of electrodes. The only condition is that the normal to the conducting surfaces should be continuous. On the other hand, contrary to the loop-star and loop-tree basis functions, the null-pinv basis functions are in general nonlocal, in the sense that their support is in general spread on all over

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the domain. This is not a problem in the EFIE because the matrices arising from the discretization are fully populated, regardless the support of the basis functions. Some preliminary results, obtained by applying the null-pinv decomposition, have already been described in [13]. The core of the null-pinv decomposition is the computation of the null and the pseudo-inverse of the matrix . This can be done with standard algorithms, e.g., based on singular value decomposition. However, the interpretation of the matrix as the incidence matrix of the graph obtained by connecting the centroids of adjacent elements of the mesh, allows us to devise an alternative “topological” way of making these calculations. As it will be shown, the transpose of the fundamental loop matrix, corresponding to a given tree of , is a null of , whereas the transpose of the fundamental path matrix along the tree is a pseudo-inverse of . To assembly the impedance matrix to be inverted by using the null-pinv decomposition a pre- and post-multiplication of the standard facet elements-based impedance matrix by the null and pseudo-inverse of is required. This apparently could increase the computational cost of the method. Indeed, the elements of the null and pseudo-inverse of computed by the topological method are equal to 0 and 1, and the aforementioned multiplications simply are algebraic summations of rows and columns of the facet element-based impedance matrix. Thanks to this consideration, all the “fast” techniques (e.g., multipolar, FFT-based, zooming, etc.), which are applied to facet-element based formulations to improve the scaling of the computational cost, are equally applicable to the proposed formulation. The paper is organized as follows. In Section II, the EFIE formulation for different kinds of excitations is briefly reviewed. In Section III, the Galerkin equations relevant to EFIE previously described are derived, by representing separately the solenoidal and nonsolenoidal parts of the current density field to avoid the low frequency breakdown. In Section IV the null-pinv decomposition is described and it is applied to generate the solenoidal and nonsolenoidal basis functions of the current density field. Furthermore, a topological way to evaluate the null-pinv basis functions, based on the graph theory, is discussed. In Section V the differences and similarities between the null-pinv decomposition and the conventional loop-star and loop-tree basis decompositions are highlighted; particular care is devoted to the topologically complex conducting domains. The numerical studies of Section VI are used to demonstrate the validity of the null-pinv decomposition methodology. Finally, Section VII concludes the paper with a brief summary of the method, a few remarks about its features and an outline of future developments. II. MATHEMATICAL FORMULATION In this Section the derivation of the integral equation for the surface current distribution induced on a 3-D perfect conducting body is reviewed. The extension to conducting sheets of arbitrary shape (at the end of this section), and to imperfect conducting bodies (at the end of next section) will be briefly outlined. The current in the conducting body may be generated in two different manners: an electromagnetic field illuminates the body

(a typical example is the electromagnetic scattering); an electric circuit feeds the conducting body through physical contacts (typical examples are the antennas and the interconnects in electronic circuits). We will deal with both of them. Let denote the conducting body surface on which the surface current density and the surface charge density are defined and denote the unit normal to , outgoing from the body. We will assume that is continuous over (e.g., the surface is not twisted). On the external page of , the electric field has to satisfy the following: on

(1)

Let and denote, respectively, the incident electric field on the body and the scattered electric field from the body. The electric field may be expressed as (2) In terms of the electromagnetic potentials electric field can be expressed as

and , the scattered

(3) The electromagnetic potentials are expressed in terms of and by means of the integral relations (according to the Lorentz gauge condition) (4) 1

(5)

where and are, respectively, the vacuum dielectric constant and vacuum magnetic permeability, indicates the distance between the “field” point and the “source” point (6)

4

is the vacuum Green function, and 1 . To the (1)–(5) we have to add the current continuity equation 0 on

(7)

where “ ” is the surface divergence operator defined over . Div-conforming vector fields are square integrable and have a square integrable divergence (e.g., [10]). As a consequence, they denote the have a continuous normal component over . Let linear vector space of complex-valued div-conforming vector fields defined on and tangent to it. Since by definition , the boundary equation (1) may be enforced by requiring that 0

(8)

MIANO AND VILLONE: A SURFACE INTEGRAL FORMULATION OF MAXWELL EQUATIONS

Substituting in (8) the expression of (2) and (3), we have

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obtained by combining

(9)

Equation (9) is the weak form of the boundary equation (1). Substituting (4) and (5) into (9) we get a set of integral equations in the unknowns and . The set of equations so obtained has to be solved together with the current continuity equation (7). If the surface is closed (i.e., there are not contacts), the first of (9) is equal to zero and the only forcing term on the term is due to the incident field. The excitation by a delta gap across a line [see Fig. 1(a)] may be also described trough an equivalent incident electric field given by

Fig. 1.

(a) Delta-gap voltage excitation and (b) conducting body with contacts.

(10) is the voltage across the gap, is a generic point where is the Dirac function and is the belonging to the curve , normal to the curve lying over , oriented according to the reference direction for the voltage [as indicated in Fig. 1(a)]. Let us now consider a conducting body fed only through 0), as schematically shown in Fig. 1(b). contacts (i.e., The areas of the body, where the circuit connections are made, is are defined as contact regions. The entire body surface divided into subsurfaces: the subsurfaces of the contacts and the remaining part of the body surface . The contact curves 1 that delimit, respectively, the contact surfaces 1 are the boundaries of the open surface . In this case, (9) becomes

(11) is the outward normal to the curve lying over . where Equation (11) may be solved by assigning either the value of the potential , or the value of the normal component of the , along the contact curves with current density, 1 . If the electric potential is assigned along the contact curves along

for

(12)

the right-hand side of (11) is known. Instead, if the normal component of the current density is assigned along

1 with

it is sufficient to impose (11) for any 0 along

for

hence, the right-hand side of (11) vanishes.

(13)

1

(14)

Fig. 2. Conducting sheet.

Remark: In the impressed current or electric potential excitation the contributions of the currents and charges that are found inside the devices that feed the conducting body are disregarded. In fact, the effects of these contributions may be considered negligible only if the lengths of the devices are small in relation to the smallest characteristic wavelength and to the smallest characteristic length of the conducting body. As a consequence of this approximation, the potential expressions (4) and (5) do not wholly satisfy the Lorentz gauge condition. However, as already stressed in [14], the Lorentz gauge condition would be satisfied if, to the scalar potential expression, given by (5), we were to add the contribution of a suitable charge distribution located on the contacts. Remark: Conducting Sheets: Let us consider a conducting thin layer, that is, a conductor with a depth much smaller than its width and the smallest characteristic wavelength , 0 (with we mean Fig. 2(a). In the limit for which ) we have , , that and , and the thin layer may be repand resented as a sheet, Fig. 2(b). Let us indicate with , , the surface current and charge distributions on the faces “ ” and “ ” of the sheet , respectively. The tangential component of the vector potential and the electric scalar potential 0 . are assumed to be continuous across in the limit This assumption does not allow us to treat situations in which and in the the excitation is such that limit 0 . Since, in the limit 0 the contribution of the electrical charges and currents located on the lateral surface may be disregarded, the tangential component of and the potential are completely determined by the equivalent current and charge surface distributions (15) (16)

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defined on . The quantities and satisfy the same equaon the boundary of tions reported above with (except along the contacts), where is the outward normal to the curve lying over .

and any . According to the current conany tinuity equation (7), the surface charge density is approximated as (22)

III. GALERKIN EQUATIONS In this Section we briefly review the derivation of the Galerkin equations for the problems described in the previous Section. It , and is helpful to introduce the products

where (23) and (24)

(17)

where is the domain on which the fields and are defined, is its boundary. In the definition (17), if both and and are vector fields the symbol “ ” stands for the scalar product, otherwise it stands for the ordinary arithmetic product. By substituting the expressions of the potentials (4) and (5) in the (9) we obtain the weak form equations for the surface current density and the charge density

Let us introduce the vectors

(25) representing the unknown coefficients in the expansions (21) and (22), and the vectors (whose components are vector or scalar fields)

1 (18) To avoid the low frequency breakdown we have to separate the solenoidal and nonsolenoidal parts of in the numerical soludenote a subspace of tion of (18) (e.g., [7], [8], [11]). Let such that for any vector field

(26) representing the corresponding basis functions. Immediately, it follows that: (27)

0 and field

the complement of

on

(19) , such that for any vector

This is the discrete form representation of the current continuity equation (7). By substituting expressions (21) and (22) in (18), and any by imposing that it is satisfied both for any , and by using (27) we obtain the Galerkin equations

(20) (28) at least in a subset of of area different from zero. The suband are, respectively, the space of solenoidal spaces and nonsolenoidal surface vector fields defined on . An approximate solution of the weak form equation (18) is found by representing the solenoidal part of in a suitable finite dimensional subspace of , and the nonsolenoidal part of . The in a suitable finite dimensional subspace surface current density is approximated as

where (29) (30) (31) (32)

(21)

where

1

are the basis functions of , are the basis functions of and 1 , 1 are unknown coefficients to be determined by imposing that (18) were satisfied both for 1

for , , . Matrices (30) and (31) are symmetrical and fully populated regardless of basis functions used. In the case of external incident field and contact potential excitations, both terms in (32) are known. For impressed current 0, but some coefficients in the expansion (21) excitation are known.


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The kernel of the double surface integrals in (30) and (31) is evidently singular. Hence, particular attention must be paid and . We when performing the calculation of the matrices extracted the singularity with the standard position 1

1

(33)

where the first term is regular when 0, and hence it can be integrated numerically using Gauss formulae over triangles, 0, is integrated analytwhile the second term, singular for ically using the results presented in [15]. One could think of solving (28) by blocks (34)

(35) 0 (for The important thing to notice is that, in the limit which the Green’s function approaches the static case) the matrix is exactly the same as for an integral formulation of the is the same as for electrostatic problem, while the matrix an integral formulation of the magnetostatic and magneto-quasi static problems [16], [17]. This means that if the frequency is low enough, so that

Fig. 3. Triangularization of the body surface.

The boundary condition (38) is a low order surface impedance boundary condition. To improve its accuracy high order surface impedance boundary conditions may be used, [19], [20]. Fur(low thermore, for thin layers with depth such that frequency limit) the current density is practically uniformly distributed in the depth of the layer, and the boundary condition . Instead, in the high frequency (38) still holds with limit, , it is easy to verify that the boundary condiis replaced with 2. By imposing tion (38) still holds if boundary condition (38) [instead of that given by (1)], (28) still , in the expression of the matrix holds if we add to given by (29), the matrix given by

(36) (40)

(37) the results will be consistent with solutions of the static proband alone. lems, obtained inverting matrices Remark: Imperfectly Conducting Bodies: Through (28) we may also describe the situations in which the effects due to the finite value of the electrical conductivity are not negligible. the frequency value for which the penetration depth Let be is equal to the smallest characteristic dimension of the body, and the frequency value for which the wavelength is equal to the smallest characteristic radius of curvature of the body surface . If the frequency is very large compared with and we have, with good approximation, that the induced currents behave as surface currents and the electric field on the external page of the surface is related to the magnetic through the Shchukin-Leontovich boundary condition field (e.g., [18]). Under these conditions the boundary equation for the surface current density is

IV. NULL-PINV DECOMPOSITION In this section we describe a numerical procedure to build up the solenoidal and nonsolenoidal basis functions in terms of facet elements. It is based on the null and pseudo-inverse of the matrix approximating the divergence operator. The domain of definition of the current density field may be either an open surface or a closed surface. The only difference between these two cases is in the total electric charge lying on the surface: as a consequence of the continuity equation (7) the total electric charge must be equal to zero for closed surfaces, whereas it may be different from zero for open surfaces. The facet elements, as we will see, allow us to enforce naturally these properties. A. Facet Elements

on

(38)

is the intrinsic impedance of the metal, which is given where ) by (under the assumption that 1

(39)

Let us refer to the case in which the solution domain is made of a single surface. The results can be easily generalized to the case of multiple surfaces. We suppose that a triangular finite elements discretization of (Fig. 3) has been given, with elements, edges and nodes. In presence of contacts, every is approximated by a number of edges of the contact curve mesh.

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The facet element basis function [12] associated with the th edge is defined as (41) where the functions and are the standard nodal functions related to the initial and final nodes of the edge, see Fig. 3, and “ ” is the surface gradient defined on . It is eass to see that the vector field is div-conforming, in particular its normal component along the th edge is continuous. This means that no linear charge density is present along denote the normal to the th edge the edges of the mesh. Let along the th edge is lying over . The integral of and to 0 if equal to 1 if (42)

Let us now consider a surface current density linear combination of facet elements

expressed as

(43)

Since (42), the th degree of freedom represents the current flowing across the th edge. To represent the divergence of (for 1 2 ) we (for introduce the piecewise constant functions 1 2 ) vanishing on all the elements but the th, where it is equal to the inverse of its area in element elsewhere.

(44)

Let us introduce the vectors

(45)

Fig. 4. Graph associated to the finite element mesh.

Hence, the matrix is the discrete representation of the surface divergence operator in the linear space spanned by the facet elements. The matrix is the incidence matrix of the graph obtained by connecting the centroids of adjacent elements of the mesh, see Fig. 4: the nodes and the branches of correspond, respectively, to the elements and to the edges of the mesh. Hence, the th component of the vector represents the algebraic sum of the currents incident in the th element through its edges. This analogy allows us to easily discuss the rank of the matrix, in all cases of i.e., the number of independent rows (since interest), analogously to what is done in standard circuit theory, e.g., [21]. If the solution domain is a closed surface, the rank of is not maximum. Indeed, in this case each edge is shared by exactly two elements, with opposite signs with respect to their orientation; hence, in each column of the matrix there is exactly one 1 and one 1. Consequently, when summing all the rows of the matrix, an identically vanishing row is obtained, indicating a rank deficiency. Eliminating one row of the matrix, this reasoning is not valid any longer, since there is at least one edge appearing in only one row of . On this basis, it is possible . to demonstrate [21] that the rank of is exactly If a certain number of disjoint closed conducting surfaces is . Instead, if the solution present, the rank of will be domain is open, there is at least one edge belonging to one ele. ment alone and, hence, is a full rank matrix of rank Let us consider the surface integral of over the surface . From (48) it follows that:

(49)

Then, the surface divergence of the facet elements is given by (46) where

is the

sparse matrix defined as if the edge does not belong to the element if the edge belongs to the element

(47) the sign depending on the relative orientation of the element and the edge. By using (46) we obtain for the surface divergence of

(48)

is the sum of all the rows of the matrix where the vector . Therefore, if the surface is closed we have 0 and, hence, the total electric charge lying on is always equal to zero independently of . Instead, if the surface is open we 0 and, hence, the total electric charge lying on may have be different from zero. B. Generation of the Solenoidal and Non-Solenoidal Basis Functions onto . The rank The matrix is a linear operator from is the dimension of the range of . The null space of is the mapped onto the null vector of . The nullity subspace of


is the dimension of the null space, and it satisfies the relation [22] (50) As now we will show the solenoidal basis functions can be obtained using the vectors belonging to the null space of . matrix whose columns constitute a be a Let complete basis of the null space of , then, we have 0 From the definition of

(51)

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(56) does not exist because the rank of is not maximum . Let denote the reduced matrix obtained by simply removing one row of , for instance, the last one; the matrix , so obtained, is a full rank matrix. As a consequence there exist a matrix such that (59) where now is the sum of the rows of

1 1 identity matrix. Since the is equal to zero, it follows that:

, it immediately follows that: (60) (52)

forms a basis of the solenoidal part of the space spanned by the facet elements. Indeed, by using (48) we obtain 0

where 1 is a column vector of dimension whose elements are all equal to 1. Substituting (60) into (55) we finally obtain

(53) 1

for any vector . The dimension of the solenoidal so generated is equal to the nullity of the matrix subspace , . We must now find a basis for the nonsolenoidal part of the space spanned by the facet elements. Such basis must evidently functions; hence, we need a matrix consist of such that

•

has a nonzero divergence. By using (48) we obtain

Now we have to distinguish between the two cases of open and closed solution domain . If the solution domain is open, is a full-rank matrix and a matrix exists such that

• • •

(56) where I is the identity matrix. By substituting (56) in the relation (53) we obtain 0

(57)

To accomplish relation (56), we could choose

• •

we have generated solenoidal and nonsolenoidal basis functions (52) and (54) for the current density, which we call “null” and “pinv” basis functions, respectively, introducing the null space and the pseudo-inverse of the discrete divergence matrix ; the charge density is expanded over piecewise constant functions; , the number of solenoidal basis functions is the nullity of ; the number of nonsolenoidal basis functions and , the rank of : charge basis functions is if the solution domain is an open surface, since 1 if the a net charge can enter the domain; solution domain is a closed surface, since the total charge is constrained to be zero; the total number of current density basis functions is after (50); the edge-based degrees of freedom in (43) are related to the solenoidal and nonsolenoidal degrees of freedom and in (21) by: (62)

(58) where “ ” indicates the Moore-Penrose pseudo-inverse, e.g., [22]. With this particular choice for the nonsolenoidal basis functions, the charge density (22) is expanded through piecewise constant functions. Let us consider the case in which the solution domain is a closed surface. In this case a matrix verifying the condition

(61)

of the nonsolenoidal subspace In conclusion, the dimension so generated is always equal to the rank of the matrix , . To sum up, with reference to the notation introduced in the previous section we can conclude the following:

(54)

(55)

0

C. Evaluation of the Null and Pseudo-Inverse of the Graph

Through

The core of the proposed approach is the calculation of the null and the pseudo-inverse of the matrix . This can be done

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with standard algorithms, e.g., based on singular value decomposition [22]. However, since the matrix is the incidence matrix of the graph there is an alternative “topological” way of making these calculations. To start with, let us suppose that the solution domain is made of a closed surface. In this case, the rank of the matrix is , as discussed above. This means that we must arbitrarily define one node of (i.e., one element of the mesh) as reference, and eliminate the corresponding row of to get a maximum . rank matrix Let us compute a tree of the graph , and the corresponding . Since has nodes and twigs, is made of co-tree 1 twigs and is made of 1 twigs. We can 1 fundamental loops of , made uniquely define of one twig of the co-tree and a number of twigs of the tree that uniquely close a loop. The quantity coincides with the nullity , . Standard circuit theory considerations [21] allows us fundamental loop matrix (which to conclude that the is of full rank ) if the edge does not belong to the fundamental loop if the edge belongs to the fundamental loop .

(a) Closed surface and (b) open surface.

a nonzero divergence. Nevertheless, the null-pinv basis functions, proposed in this paper, show a superior ability to cope with topologically complex domains. Since we have assumed that the conductor surface is not twisted, Euler’s formula states that [23]

(63)

is such that its transpose can be chosen as null matrix . Conversely, for each of the 1 (nonreference) nodes, we can (uniquely) define a fundamental path over the tree (hence closing no loops) linking the node to the reference one. We can 1 fundamental path matrix , which is introduce a 1 as of full rank if the edge does not belong to the fundamental path if the edge belongs to the fundamental path .

Fig. 5.

(64)

It is easy to be convinced that the transpose of (although ) can be not equal to the Moore-Penrose pseudo-inverse of chosen as matrix , since it satisfies condition (59). If the surface is open, then the matrix is of full rank, because there are some edges that belong to one element only. With reference to the graph introduced above, we notice that there are a number of twigs that apparently terminate with no node, since the related edges are not shared by two elements. We solve this contradiction by introducing one additional node (per separate piece), which is a fictitious additional “super-element” in the original mesh that acts as the outside connections. Choosing this additional node as reference, we immediately fall in the previous case, and we can apply all the considerations made above. V. TOPOLOGICAL ISSUES With respect to the popular loop-star decomposition, e.g., defined in [11], we observe that the “null” basis functions (52) can be regarded as generalized loop basis functions, since detheir divergence is zero, while the “pinv” basis functions fined in (54) are generalized star basis functions, since they have

21

(65)

where is the number of “handles” (also called the genus of the polyhedron) and is the number of apertures of the domain (i.e. boundary contours). In [11] the solenoidal part of the surface current density is generated by introducing a scalar function such that (66) The loop basis functions are obtained simply assuming that is represented in terms of standard nodal functions . In addi1 loops are added around the apertures of the dotion, 1 main. Conversely, the star basis functions are related to elements, assuming global charge neutrality. Evidently, as already mentioned explicitly in [23], also a number of additional degrees of freedom should be added, related to the handles of the domain. Indeed, let us consider the case in which the solution domain is a torus [see Fig. 5(a)]. It is easily seen that in this case 0 (no apertures) and 1 (one handle). This means that 0

(67)

1 elementary Following [11] and [23], we can introduce loop basis functions (the scalar function can be arbitrarily set star basis functions (the total to 0 in one node), and charge is zero). Using (68), the total number of basis functions then becomes 1

1

2

(68)

Using the standard rooftop basis functions [9], or the facet elements the number of degrees of freedom would be . This means that in the loop-star basis 2 degrees of freedom are missing. In our example, the two missing degrees of freedom are the total toroidal and poloidal solenoidal currents flowing in the


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torus around the handle. Indeed, by using (66) we have [refer to Fig. 5(a)]

0

(69)

and similarly (70) In other words, solving the problem using the loop-star basis functions would lead to an unphysical solution with a vanishing net poloidal and toroidal solenoidal currents unless some ad hoc tricks are used (e.g., cuts or nonautomatic addition of some nonlocal basis functions) [7], [8]. This problem does not affect the null-pinv basis functions, because, thanks to (50), the number of “null-pinv” basis functions is exactly , as already pointed out in the previous section. Indeed, in the present case the rank of 1 . Consequently, the the discrete divergence matrix is 1 columns, while the pinv manull matrix has 1 columns, so that the total null-pinv basis trix has functions are , as expected. There are also other cases in which the standard loop-star approach fails. Let us now consider the case in which is a may be cylinder [Fig. 5(b)], whose boundary crossed by a current. In this case, we have 2 and 0, so that again

Fig. 6.

Reference geometry for the Hallén problem.

1, since it is not The number of loop basis functions is is constant over possible to impose that the scalar function separate pieces of , as it is done in [11], due to the possible . Similarly, the number of star basis current flow through , since the total charge is not necessarily zero. functions is Using (71), the total number of loop-star basis functions then becomes

through a flux function [as it is done by (66)], unless some special care is used (e.g., making some cuts in the domain). This point has been deeply studied in the low frequency magnetoquasistatic limit, when the current density is solenoidal. Indeed, the position (66) is the surface counterpart of the introduction of the electric vector potential, as for instance reported in [16]. With this choice, the degrees of freedom used to expand the electric vector potential may be not sufficient. Indeed, when complicated geometries and electrodes are considered, a number of additional (in general nonlocal) basis functions must be introduced [17], as suitable combinations of electric vector potential basis functions. Their number and definition depends in a complicated way on the electrodes and the topological properties of the domain. Hence, as reported in [17], finding such additional degrees of freedom for general topologically complex domains in the presence of electrodes is a very hard task. This is also acknowledged by [8], “It might not be possible to design software that reliably and efficiently finds the required sets of topologically equivalent loops and then picks only one of them”. On the contrary, the null-pinv basis functions automatically provide always the right number of degrees of freedom.

1

VI. EXAMPLES OF APPLICATIONS

0

(71)

1

(72)

This means that one degree of freedom is missing; in our exto . ample, this is the net solenoidal current flowing from In fact, by using (66) we have 0

(73)

as in the previous case. Conversely, again the null-pinv decomposition is not affected by such problem, as it can trivially be proven as before. In general, we can say that, as well known, in multiconnected domains a generic divergence-free field may not be expressed

The formulation described above has been implemented in a code called SURFCODE. In this section, some examples are presented, which illustrate the potentiality and generality of the proposed approach. A. Antennas First of all, we present some results for the Hallén problem: a tubular antenna of radius and height with a (theoretically infinitesimal) gap (Fig. 6) fed by a voltage source. This problem can be solved analytically (e.g., [24]); many numerical solutions are also available (e.g., [13]). We have applied SURFCODE to two sets of data, described in Table I, where also some infor-

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TABLE I DATA FOR THE HALLÉN PROBLEM

Fig. 7.

Spatial behavior of the current in the tubular antenna of Fig. 6 (case 1).

mation about the discretization are reported. Fig. 7 reports the spatial behavior of the current for Case 1, at a frequency such . The agreement with the results presented in [24] that is clearly satisfactory. In addition, Fig. 8 reports the frequency dependence of the input admittance for case 2, for two values of the gap. Clearly, the real part of the input admittance is not influenced by the choice of the gap, while the input susceptance is quite different, due to the different gap capacitance in the two cases. This figure can be positively compared with Fig. 6 of [13]. We also studied a wire antenna over a small ground plane, as described in [1]. Fig. 9 reports the frequency behavior of the input admittance, while Fig. 10 depicts a zoom of the mesh used (consisting of 1994 elements, 2944 edges and 2391 nodes) and the imaginary part of the current density at 2 GHz (the scale factors are different in the plate and in the wire). The input conductance is in good agreement with the results presented in [1], while the input susceptance shows a disagreement which is probably due to the different treatment of the feeding, and hence of the feeding gap capacitance. B. Interconnects First of all, we have solved the problem for a pair of parallel conductors (length 3 cm) with a rectangular section (2 mm 1 mm); Fig. 11 shows the mesh used, consisting of 1696 elements, 2560 edges, and 876 nodes. The impedance matrix at the two ports is computed, and compared to the value estimated by the standard transmission line (STL) model, and a recently proposed enhanced transmission line model (ETL) [14], [25]. As the STL model, the ETL model is a one-dimensional description of the interconnects made of very long and quasiparallel straight conductors, based on the approximation that the transverse distributions of the electric and magnetic fields are

Fig. 8. Input impedance of antenna of Fig. 6 (Case 2): (a) real part and (b) imaginary part.

those of the quasistationary limit. Instead, the ETL model, unlike the STL one, is able to take into account the effects of radiation when the distance between the wires is not higher than the smallest characteristic wavelength of the signals [14]. The comparison shows a good agreement between SURFCODE and ETL, and a significant difference with respect to STL, in terms of position and amplitude of the resonance peaks, Figs. 12 and 13. The agreement becomes less satisfactory at higher frequencies, probably because of the fact that some assumptions at the basis of ETL model start to fail [14], [25]. For what concerns the computational cost we notice that each frequency point takes around 173 seconds, once all the pre-calculations have been carried out, on Pentium based PC computer. The computational cost of the ETL model is significantly lower because it is essentially a one-dimensional model. However, resorting to a code like SURFCODE is unavoidable whenever the interconnect configuration is such that the one-dimensional approximated model is no longer valid (e.g., very short interconnects, nonuniform interconnects, interconnects with holes, vias, ).


Fig. 11.

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Pair of conductors: mesh used.

Fig. 9. Input impedance of the wire antenna considered in [1]: (a) real part and (b) imaginary part.

Fig. 10. Zoom of the mesh used for the wire antenna considered in [1]. The imaginary part of the current density at 2 GHz is reported (the scale factor is different in the plate and in the wire).

We also considered the case of a thin interconnect with a rectangular cross section of dimensions 2 mm (horizontal)

Fig. 12. (a) Amplitude and (b) phase of the self-impedance of the interconnect of Fig. 11.

0.2 mm (vertical), with a resistivity of 1.7857e-8 m. We solved this problem both with the right rectangular cross section and with the sheet approximation discussed above, assuming

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Fig. 13. (a) Amplitude and (b) phase of the mutual impedance of the interconnect of Fig. 11.

that the sheet is concentrated in the middle of the cross section. Retaining the same discretization level, the number of edges decreases from 1549 to 651. Fig. 14 shows the results, for a frequency range in which the penetration depth is small as compared to the geometrical dimensions, and capacitive effects are negligible. First of all, for the computations with the rectangular cross section, we find a good agreement with the data presented in [26]. Secondly, we notice that the sheet approximation slightly overestimates the per-unit-length reactance, since it neglects the magnetic field shielding in the conductor volume. Conversely, the per-unit-length resistance is underestimated, since we neglect the contribution to the overall resistance of the vertical sides of the cross section. This contribution is 10% higher than what expected (what should come out from geometrical considerations) due to the fact that the current density is not uniformly distributed along the conductor, but is denser near the shorter edges. The last example realistically reproduces the geometry of vertical via-holes connecting two coplanar interconnects. The mesh (consisting of 1750 nodes, 4818 edges, and 3068 elements) is

Fig. 14. (a) Per-unit-length resistance and (b) reactance of an interconnect with a flat rectangular cross-section.

Fig. 15. Mesh for the three via-holes.

shown in Fig. 15. The structure is made of two coplanar differential signal lines (ground-signal-ground, dimensions 120 64 120 m, distance 24 m) interconnected through ver90 m, external radius tical cylinders (internal radius 160 m, height 250 m) representing three via-holes; a total length of 2 mm is considered. The thickness of the conductors


Fig. 16.

Typical low frequency current density distribution.

has been neglected and the sheet model has been used. Fig. 16 reports the zoom of a typical current density distribution, for a low frequency case in which the wavelength is much longer than the typical dimensions of the device. Evidently, this pattern is qualitatively satisfactory, showing that SURFCODE is able to deal with such complex geometries with no additional effort. On the contrary, the treatment of such structures with standard loop-tree or loop-star basis functions may be difficult. In this case, it would be indeed necessary to specify some ad hoc basis functions able to represent the solenoidal net current flowing through the electrodes and along the cylinders.

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aforementioned multiplications simply account for algebraic summations of rows and columns of the facet elements-based impedance matrix, which do not alter the overall computational cost. In addition, thanks to this consideration, all the “fast” techniques (e.g. multipolar, FFT-based, zooming, etc.) that can in principle be applied to facet-elements based formulations to improve the scaling of the computational cost, are equally applicable also to the present formulation. This will indeed be one of the issues that will be addressed in future work. Another point that deserves particular attention is the introduction of nonhomogeneous dielectrics. This is evidently fundamental to treat several important real-world applications (e.g. PCBs, patch antennas, etc.), and hence will be addressed in our future activity. In particular, two approaches can be pursued: the first one is to fully discretize also the volume of the dielectric bodies, in order to represent numerically the displacement current flowing through them. This obviously allows the treatment of dielectric bodies of generic shape, but dramatically increases the computational cost. Another approach, that seems more viable in the context of the present surface integral formulation, is the introduction of a dyadic Green’s function able to represent dielectric bodies of canonical shape (e.g., layered, which is often the case in applications). However, the numerical examples presented demonstrate that, also in its present form, SURFCODE is able to efficiently and effectively solve technically significant problems, both in the context of interconnects and antennas. We are currently planning to enhance the code to allow the study also of waveguides and resonant cavities.

VII. CONCLUSIONS AND PERSPECTIVES In this paper, we have presented a surface integral formulation for the solution of Maxwell equations in a homogeneous medium in the presence of good electric conductors, and the related code called SURFCODE. The main feature of the proposed approach is the so-called null-pinv decomposition. We introduce new basis functions, as linear combinations of surface facet elements using (as weights) the basis of the null space of matrix (the discrete divergence in the subspace spanned by the facet elements) and its pseudo-inverse. This allows us to separate the solenoidal and nonsolenoidal part of the current density, thus retaining all the advantages of the standard loop-star and loop-tree decomposition, as the regular behavior at low frequencies. In addition, the present approach allows us to treat topologically complex domains (e.g., with holes, handles, etc.) completely automatically, without introducing any ad hoc “super-loop” basis functions or cuts in the solution domain. The price to be paid to get the matrix to be inverted is a preand post-multiplication of the standard facet elements-based impedance matrix by matrices and (null and pseudo-inverse). This apparently could increase the computational cost of the method, and cause its unfavorable scaling with the number of discrete unknowns. In fact, the computation of such matrices with graph theory considerations practically removes this problem. Since we end up with matrices made of 0 and 1, the

ACKNOWLEDGMENT The authors thank Prof. A. Maffucci and Dr. W. Zamboni for useful support in performing via-holes simulations.

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Giovanni Miano received the Laurea (summa cum laude) and Ph.D. degrees in electrical engineering from the University of Napoli “Federico II,” Naples, Italy, in 1983 and 1989, respectively. From 1984 to 1985, he was engaged in research on magnetic plasma lenses at the CERN, Geneva, Switzerland, in the PS Division. From 1989 to 1992, he was a Researcher in electrical engineering and, from 1992 to 2000, he was an Associate Professor of Circuit Theory in the Engineering Faculty of the University of Napoli “Federico II.” Currently, he is a Full Professor of Electrical Engineering at the same Faculty. In 1996, he was a Visiting Scientist with the laboratories of the GSI, Darmstadt, Germany, and in 1999, a Visiting Professor with the Department of Electrical Engineering, University of Maryland, College Park. Since 2001, he has been Coordinator of the Ph.D. courses in electrical engineering and since November 1, 2005, the Director of the Department of Electrical Engineering at the University of Napoli “Federico II.” of the He has undertaken research in electromagnetism applied to ferromagnetic materials, nonlinear dielectrics, plasmas, experimental machines for thermonuclear controlled fusion, and particle accelerators. Currently, he is mainly involved in research on the electrodynamics of continuum media, the electromagnetic modeling of high-speed interconnects, and the modeling of lumped and distributed circuits. He is also the author of more than 70 papers published in international journals, 60 papers published in international conference proceedings, two items in the Wiley Encyclopedia of Electrical and Electronic Engineering (New York: Wiley, 1999) and the monograph Transmission Lines And Lumped Circuits (New York: Academic, 2001). Dr. Miano was a Member of the Conference on Electromagnetic Field Computation (CEFC) and COMPUMAG Editorial Boards. He is a reviewer for various journals of the IEEE.

Fabio Villone received the Laurea degree (summa cum laude) in electronic engineering from the University of Napoli “Federico II,” Naples, Italy, in 1994, and the Ph.D. degree in industrial engineering from the University of Cassino, Cassino, Italy, in 1998. In May 1997, he became a Research Scientist with the Faculty of Engineering, University of Cassino, where, in October 2001, he became an Associate Professor and currently teaches courses in basic electrical engineering, numerical models for electromagnetic fields and circuits, and plasma engineering. In 1996, he was a Visiting Scientist at the CRPP-Lausanne, Switzerland, and in recent years, he has been a Visiting Scientist at the Joint European Torus (JET), located near Oxford, U.K. He is Scientific Coordinator of several experiments carried out at JET, and Principal Investigator of a national research project on fusion plasma modeling and control at the University of Cassino Research Unit. His scientific interests are in the field of computational electromagnetics, with particular reference to fusion plasma modeling and engineering, eddy-current nondestructive testing, and electromagnetic compatibility. He is coauthor of more than 50 papers published in international journals and essays in books, and more than 40 contributions to international conferences, among which several were invited papers. Dr. Villone is a Member of the Editorial Board of the COMPUMAG Conference and is a reviewer for several international journals, including the IEEE TRANSACTIONS ON MAGNETICS, Nuclear Fusion, Cryogenics, and Fusion Engineering and Design.