ACES JOURNAL, VOL. 15, NO. 2, JULY 2000
103
A Novel Preconditioning Technique and Comparison of Three Formulations forHybrid FEM/MoM Methods
Yun Ji, Hao Wang, and Todd H. HubingUniversity of MissouriRolla
Abstract
– Hybrid FEM/MoM methods combine thefinite element method (FEM) and the method of moments (MoM) to model inhomogeneous unboundedproblems. These two methods are coupled by enforcingfield continuity on the boundary that separates the FEMand MoM regions. There are three ways of formulatinghybrid FEM/MoM methods: outwardlookingformulations, inwardlooking formulations andcombined formulations. In this paper, the threeformulations are compared in terms of computerresource requirements and stability for four sampleproblem geometries. A novel preconditioning techniqueis developed for the outwardlooking formulation. Thistechnique greatly improves the convergence rate of iterative solvers for the types of problems investigated inthis study.
Index Terms
: FEM, MoM, EMC, sparse matrix,permutation, preconditioning, iterative solvers.
I. INTRODUCTIONHybrid FEM/MoM methods, which are also referred toas FEBI, FEMM, or FEM/BEM in the literature, combinethe finite element method (FEM) and the method of moments (MoM) to model inhomogeneous unboundedproblems. FEM is used to analyze the details of the structureand MoM is employed to terminate the FEM meshes and toprovide an exact radiation boundary condition (RBC). Thesetwo methods are coupled by enforcing tangentialfieldcontinuity on the boundary separating the FEM and MoMregions. Hybrid FEM/MoM techniques were introduced inthe early seventies by Silvester and Hsieh [1], andMcDonald and Wexler [2] as attempts to apply FEM tomodel unbounded radiation problems. FEM/MoM was notwidely used until the late eighties due to its largecomputational requirements. Yuan [3], and Jin and Volakis[4], [5] were among the first to apply FEM/MoM to 3Delectromagnetic problems using vector basis functions.Ang
é
lini
et al.
[6], and Antilla and Alexopoulos [7] laterapplied FEM/MoM to 3D scattering in anisotropic media.FEM/MoM has been used to analyze electromagneticcompatibility (EMC) problems since the midnineties.Ali
et al
. [8] employed FEM/MoM to analyze scattering andradiation from structures with attached wires. Shen and Kost[9] used FEM/MoM to analyze EMC problems in powercable systems. FEM/MoM has also been utilized to modelthin shielding sheets and microstrip lines [10], [11].Electronic devices with printed circuit boards (PCBs) areusually composed of many detailed structures: dielectrics,traces, cables, holes and vias. MoM is not well suited tomodel this kind of complex geometry efficiently. With ahybrid FEM/MoM technique, the details of a printed circuitboard can be modeled using FEM and an exact radiationboundary can be provided using MoM to terminate the FEMmeshes. When the structure has long cables, a FEM/MoMmethod is particularly efficient because the cables can bemodeled by MoM without meshing the empty space aroundthe cable.There are three formulations for hybrid FEM/MoMmethods [12][14]. The first formulation constructs an RBCusing MoM and incorporates the RBC into the FEMequations. The second formulation derives an RBC fromFEM and incorporates the RBC into the MoM equations.The third formulation combines the FEM and MoM matrixequations to form a large matrix equation and solves for allunknowns simultaneously.
The first and secondformulations are referred as
outwardlooking
and
inwardlooking
, respectively, in [13], [14]. The last formulation isreferred to as the
combined formulation
in this paper.This paper compares the three formulations for hybridFEM/MoM methods and presents a novel preconditioningtechnique that can be applied to outwardlookingformulations. Section II describes the matrix equationsgenerated by FEM/MoM. Section III introduces four sampleproblems used to compare the three formulations. In SectionIV, preconditioning and permutation techniques arepresented. Section V presents the outwardlookingformulation and the new preconditioning technique. Theinwardlooking formulation is described in Section VI.Section VII presents the combined formulation. Section VIIIcompares the three formulations in terms of computerresource requirements. Finally, conclusions are drawn inSection IX.II. MATRIX EQUATIONS GENERATED BY FEM/MoMFullwave hybrid FEM/MoM methods are well suitedfor solving problems that combine small complex structuresand large radiating conductors. The srcinal problem mustbe divided into an exterior equivalent problem and aninterior equivalent problem. MoM is used to model theexterior equivalent problem and FEM is employed toanalyze the interior equivalent problem. The two equivalentproblems are related by enforcing the continuity of tangential fields on the boundary separating the FEM andMoM regions [14][16].The electricfield integralequation (EFIE) is generallyused to describe the exterior equivalent problem [17],
10544887 (c) 2000 ACES
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ACES JOURNAL, VOL. 15, NO. 2, JULY 2000
Sdk jk j
S
′′∇′′•∇′−′′′∇′×′∫ +
),(G)( ),(G)(+),(
G)( )(21=)(
0000000inc
rrr J
rrr JrrrM
rErE
ηη
(1)where
k
0
and
η
0
are the wavenumber and the intrinsic waveimpedance in freespace, and
S
is the surface enclosing theexterior equivalent problem. The integral term with a bar inEquation (1) denotes a principalvalue integral. Thesingularity at
r
=
r
′
is excluded. The threedimensionalhomogeneous Green’s function is given by,
rrrr,
rr
′−=′
′−−
4)(G
0
0
π
e
j k
. (2)If
S
is a closed surface, the EFIE is not immune to falseinterior resonances [15], [17], [18]. If the interior resonancescause serious problems, the combined field formulation maybe employed [12], [18].Triangular basis functions (RWG functions) [19] maybe employed to approximate surface fields. A Galerkinprocedure can be used to test Equation (1). The resultingMoM matrix equation follows [8],
−=
chdcdhdchccchhchh
FF0E0D0D J JCCCC
(3)where {
J
h
} and {
J
c
} are sets of unknowns for the electriccurrent densities on the dielectric surface and perfectelectricconductor (PEC) surface, respectively; {
E
d
} is a setof unknowns for the electric field on the dielectric surface;
C
hh
,
C
hc
,
C
ch
,
C
cc
,
D
hd
and
D
cd
are dense coefficient matrices;
F
h
and
F
c
are source terms. The matrix formed by
C
hh
,
C
hc
,
C
ch
and
C
cc
in Equation (3) is called the MoM matrix ormatrix
C
in this paper.The interior equivalent problem is modeled using FEM.The goal is to solve the weak form of the vector waveequation as follows [14], [20]. (This equation can also bederived using a variational approach [16], [21].)
dV)()(+))((
)(
r0r0V
1
•∈∈×∇• ×∇∫
rwrErw
rE
ωµµω
j j
dV)()(dS)())(
ˆ(=
intVS
11
rwr JrwrHn
•∫ •×∫
(4)where S
1
is the surface enclosing the interior equivalentproblem,
w
(
r
) is the weighting function, and
J
int
is animpressed source. Vector tetrahedral elements [22] can beused to approximate the E field. A Galerkin procedure canbe used to test Equation (4). The resulting FEM matrixequation follows [8],
dihdhdidddiidii
gg + J0 B000 =EE AAAA
(5)where {
E
i
} and {
E
d
} are sets of unknowns for the electricfield within the FEM volume and on the dielectric surface,respectively; {
J
h
} is a set of unknowns for the electriccurrent density on the dielectric surface;
A
ii
,
A
id
,
A
di
,
A
dd
and
B
dh
are sparse coefficient matrices;
g
i
and
g
d
are source terms.The matrix formed by
A
ii
,
A
id
,
A
di
, and
A
dd
in Equation (5) iscalled the FEM matrix or matrix
A
in this paper. Both theFEM and the MoM matrices are symmetric. Note thatneither the FEM matrix equation nor the MoM matrixequation can be solved independently. They are coupledthrough the
J
h
and
E
d
terms.One objective of this study is to determine whichformulation works best for various problems. A couplingindex,
ρ
,
is defined in this paper as follows,
ρ
=
unknownsMoMof Number
unknownsFEMof Number
. (6)The value of
ρ
is determined by the problem geometry andhow it is meshed. As shown in later sections, the couplingindex
ρ
can be used as a rough
measure to determine whichformulation is preferred for a given problem.III. SAMPLE PROBLEMSFour sample problems are presented to compare theoutwardlooking, inwardlooking and combinedformulations and to validate the preconditioning techniquesdiscussed in later sections. Three of the problems includePCB structures, which are key elements of devices that arefrequently modeled by EMC and signal integrity (SI)engineers. Each of these three problems has a thinrectangular shape and presents a unique challenge. Theremaining problem has a spherical shape and provides acontrast to the PCBlike structures.
A. Problem 1: A PCB Power Bus Structure
The first problem is to model the input impedance of aPCB power bus structure. As shown in Figure 1, the boarddimensions are 5 cm
×
5 cm
×
1.1 mm. The top and bottomplanes are PECs. The dielectric between the PEC layers hasa relative dielectric constant of 4.5. A source is placed in themiddle of the board between the planes. The MoMboundary is chosen to coincide with the physical boundaryof the board. The E fields tangential to the top and bottomplanes are zero, thus no Efield unknowns are assigned onthe two planes and the number of FEM unknowns is small.Table 1 summarizes the discretization of this problem andthe other problems presented in this section.
B. Problem 2: Scattering from a Dielectric Sphere
The second problem is to model the scattering fieldsfrom a dielectric sphere. As illustrated in Figure 2, theradius of the sphere is 0.15
λ
. The relative dielectric constantof the sphere material is 4.5. The incident wave travelsalong the zaxis. The polarization of the E field is along thexaxis. The goal is to model the far fields. The discretizationof this problem is summarized in Table 1.
JI, WANG, HUBING: A NOVEL PRECONDITIONING TECHNIQUE & COMPARISON OF THREE FORMULATIONS
105
Figure 1. A PCB power bus structure.Figure 2. Scattering from a dielectric sphere.
C. Problem 3: A Gapped Power Bus Structure
The third problem is to model a gapped power busstructure. As shown in Figure 3, the board dimensions are152.4 mm
×
101.6 mm
×
2.39 mm. The board has a solidPEC plane on the bottom and a gapped PEC plane on thetop. The dielectric between the top and bottom planes has arelative permittivity of 4.5. The gap is 5.1 mm wide andlocated in the center of the top plane. The discretization of this problem is summarized in Table 1. This board is muchlarger than the board in Problem 1. A fine mesh is used inthe vicinity of the gap. To reduce the number of MoMelements, the MoM boundary is placed 9.56 mm above thegap, resulting in a large number of FEM unknowns.
D. Problem 4: A Microstrip Line
The fourth problem is to model the behavior of amicrostrip line. The board dimensions are 5 cm
×
5 cm
×
1.1mm as shown in Figure 4. The bottom is a solid PEC plane.The trace placed on the top plane is 3 cm long and 0.5 mmwide. The dielectric has a relative permittivity of 4.5. Thegoal of this problem is to determine the input impedance of the microstrip line at one end when the other end isterminated by a resistor. The discretization of this problemis summarized in Table 1. To reduce the number of boundary elements, the MoM boundary is placed 3.3 mmabove the microstrip line. A fine FEM mesh is required nearthe vicinity of the microstrip line as shown in Figure 5. As aresult, this problem has a large coupling index.Figure 3. Configuration of a gapped power bus structure.Figure 4. A microstrip line configuration.Figure 5. The FEM mesh in the plane of the trace.Table 1. Summary of the discretization of the four sample problems# of FEM unknowns # of MoM unknownsE
i
E
d
J
h
J
c
Total # of unknownsCoupling index
ρ
Problem 1 402 80 80 575 1,137 0.74Problem 2 699 612 612 0 1,923 2.14Problem 3 4,521 1,223 1,223 454 7,421 3.43Problem 4 2,277 360 360 136 3,133 5.32
ACES JOURNAL, VOL. 15, NO. 2, JULY 2000
106IV. TECHNIQUES FOR SOLVING SPARSE MATRIXEQUATIONS
A. Preconditioning
Iterative solvers are widely used to solve large sparsematrix equations of the form,
bMx
=
(7)where
M
is a square matrix and
b
and
x
are column vectors.
b
is the source vector and
x
is the unknown vector.Equation (7) is also called a
linear system
.To have a nontrivial solution, the matrix
M
must benonsingular (det(
M
)
≠
0). The convergence rate of iterativesolvers depends mainly on the condition number of thematrix
M
, which is defined as [14],
minmax
)(
λλ
=
MK
(8)where
min
λ
and
max
λ
are the smallest and largesteigenvalues of the matrix
MM
H
, where
H
M
is thetranspose conjugate of
M
. The condition number provides ameasure of the spectral properties of a matrix. The identitymatrix has a condition number of 1.0. A singular matrix hasa condition number of infinity. A matrix with a largecondition number is nearly singular, and is called
illconditioned
. An illconditioned linear system is verysensitive to small changes in the matrix. Iterative solversmay not converge smoothly, or may even diverge whenapplied to illconditioned systems.The coefficient matrices generated by FEM and MoMusually have very large condition numbers. It may bedifficult to apply iterative solvers to the srcinal FEM andMoM matrix equations. However, a linear system can betransformed into another linear system so that the newsystem has the same solution as the srcinal one, but hasbetter spectral properties. For instance, both sides of Equation (7) can be multiplied by a square matrix
1
−
P
,
bPMxP
11
−−
=
(9)where
P
has the following properties,
)( )A(
1
K(M)MPK
<<
−
0)det( )B(
1
≠
−
MP
(C) It is inexpensive to solve
Px
=
b.
Such a matrix
P
is called a
preconditioner
. This technique iscalled
preconditioning
. Condition (A) assures favorablespectral properties for the new linear system. Condition (B)guarantees that the new system, Equation (9), has the samenontrivial solution as Equation (7). Condition (C) isessential to ensure the efficiency of preconditioned iterativesolvers. In preconditioned iterative algorithms, it is notnecessary to solve
1
−
P
explicitly. Instead, a linear systemof the form
Px
=
b
is solved at each step.If the preconditioner
P
is chosen to be
M
,
MP
1
−
becomes an identity matrix. However, finding
1
−
M
isgenerally more difficult than solving Equation (7). It is morepractical to find a preconditioner
P
that is an approximationof
M
, and satisfies all three conditions. There is a tradeoff between the cost of constructing and applying thepreconditioner, and the gain in the convergence rate [23].LU factorization and incomplete LU (ILU) factorizationare commonly used to construct preconditioners. LUfactorization decomposes a matrix
M
into a lower triangularmatrix
L
and an upper triangular matrix
U
, which satisfy,
M
=
LU
.
(10)ILU factorization ([23], [24]), decomposes matrix
M
into alower triangular matrix
L
and an upper triangular matrix
U
so that the residue matrix
R
=
M

LU
is subject to certainconstraints, such as levels of fillin, or drop tolerance.
B. Permutation
Because the FEM matrix,
A
, is sparse, LU factorizationmay generate a lot of
fillin
elements
, which refer to matrixentries that are zero in the matrix
A
but are nonzero in the
L
and
U
matrices [24].
Permutation
is a technique that can beused to reduce the number of fillins in LU factorization byreordering the matrix. Generally, a symmetric permutationon matrix
M
is defined as follows [24],
P
M
=
P
M
T
P
(11)where
P
M
is the new matrix after permutation and
P
is thepermutation matrix.
P
is a unitary matrix [24], whichsatisfies,
1
−
P
=
T
P
. (12)Figure 6 illustrates the sparsity pattern of the srcinalFEM matrix in Problem 1. The number of unknowns in theFEM matrix is 482. A fully populated matrix has 482
×
482 =232,324 entries. Figure 6 shows only 3,772 nonzero entries.The percentage of nonzero elements is 1.6%, indicatingthat the FEM matrix is highly sparse. Figure 7 illustrates thesparsity patterns of the
L
and
U
matrices after applying LUfactorization to the FEM matrix in Problem 1. The data inFigure 7 was generated using MATLAB
®
[25]. The
L
matrixobtained by MATLAB is a "psychologically lowertriangular matrix” (i.e. a product of lower triangular andpermutation matrices) [26]. This explains why the
L
matrixis not a strictly lower triangular matrix. The total number of nonzero entries in
L
and
U
is 34,640 + 35,379 = 70,019.The total number of fillins is 70,0193772 = 66,247.The
reverse CuthillMcKee algorithm
can be used tominimize the bandwidth of a matrix [16], [27]. Bandwidthreduction techniques are useful because they save bothstorage and operation counts in LU factorization. Figure 8shows the sparsity pattern of the FEM matrix in Figure 6after performing a symmetric reverse CuthillMcKeepermutation. Figure 9 illustrates the sparsity patterns of the
JI, WANG, HUBING: A NOVEL PRECONDITIONING TECHNIQUE & COMPARISON OF THREE FORMULATIONS
107
L
and
U
matrices after the permutation. The number of fillins is 10,457+12,457 – 3,772 = 19,142. Compared withFigure 7, the number of fillins has been reduced by 71%.The
minimum degree permutation
is a complicated andpowerful technique that has many advantages over otherpermutation techniques [16], [26]. One widely usedimplementation was proposed by George and Liu [28]. Thistechnique reduces fillins during Gaussian elimination basedon graph theory [16], [29]. In this study, the authors usedFigure 6. Sparsity pattern of Problem 1 FEM matrix (“nz”is # of nonzero entries).Figure 7. Sparsity pattern of the Problem 1
L
and
U
matrices after LU factorizationFigure 8. Sparsity pattern of the Problem 1 FEM matrixafter symmetric reverse CuthillMcKee permutation.the symmetric minimum degree permutation provided byMATLAB
®
. Figure 10 shows the sparsity pattern of theFEM
matrix in Figure 6 after performing the symmetricminimum degree permutation. Figure 11 illustrates thesparsity patterns of the
L
and
U
matrices after performingthe symmetric minimum degree permutation. The number of fillins is 7,901+9,628 – 3,772 = 13,757. Compared withFigure 7, the number of fillins has been reduced by 79%.Figure 9. Sparsity pattern of the Problem 1
L
and
U
matrices after symmetric reverse CuthillMcKeepermutation.Figure 10. Sparsity pattern of the Problem 1 FEM matrixafter symmetric minimum degree permutation.Figure 11. Sparsity pattern of the Problem 1
L
and
U
matrices after symmetric minimum degree permutation.