1

REFORMULATION OF THE NOVEL SCALED BOUNDARY FINITE E LEMENT METHOD FOR COMPUTATIONAL ELECTROMAGNETICS

V. S. Prasanna Rajan

Dept. of Electrical & Electronics Engg., BITS, Pilani, Rajasthan 333031, INDIA Email: vsprajan@yahoo.com

Abstract: The scaled boundary finite-element method is a novel semi-analytical method jointly developed by Chongmin Song and John P. Wolf to solve problems in elastodynamics and allied problems in civil engineering. This novel method is reformulated recently for problems in electromagnetics. In this paper, the reformulation of the novel method for the case of metallic cavity structures is explained. The closeness of the values obtained by the novel method with those obtained from theoretical analysis is depicted numerically in the form of a table.

Key words: Computational Electromagnetics, Scaled boundary finite element method, Vector finite element method, Cavity structures. 1. Introduction: The vector finite element method is one of the most successful and popular

methods for electromagnetic analysis [1-8]. In spite of the success of the method, some key aspects

with regard to its implementation are worth noting. When the method is employed to solve a

problem, the finite element discretization is to be performed throughout the structure under

consideration, for those geometries which lack symmetry. If the geometry under consideration is

uniform along a particular axis, the finite element discretization need not be performed along that

particular axis, resulting in the reduction in the dimension of the discretization. However, if the

geometry under consideration does not posses any uniformity in any of the axes, finite element

discretization is to be performed along all the axes. For the case of non-standard two and three-

dimensional structures, this amounts to considerable requirement on computer memory, processor

and computational time. The requirement on memory and becomes more pronounced when dealing

with lossy, in-homogenous and anisotropic [9,10,11] materials. A reduction in the dimension of

discretization has significant impact in the computing time and resources [12].

As an alternative, the boundary element method has the advantage of only surface discretization

for three-dimensional geometries with analogous reduction in the dimension for discretization in the

case of 2-D geometries.

2

But the method requires the fundamental solution (Greens function) to be known in advance.

Also, singularity appears in the integrals involving the fundamental solution and hence provisions

are to be made to avoid the singularity during the evaluation of such integrals. Also, the knowledge

of the fundamental solution is extremely difficult if not impossible except for some simple

structures. In the case of a general anisotropic medium, it is extremely difficult to find the greens

function. Moreover, the matrix resulting from the boundary element method is full and non-

symmetric unlike the finite element matrix, which is sparse and symmetric. In spite of these

disadvantages, the boundary element method has the advantage that the radiation condition at

infinity is exactly satisfied, whereas in the case of finite element method, it is only approximately

satisfied due to the application of the absorbing boundary condition [5] in the unbounded medium.

Hence this makes the boundary element method favorable while dealing with unbounded domains

usually encountered in open boundary problems.

In this connection, it is mentioned that, a novel semi-analytical finite element method called

The Scaled Boundary Finite element method was developed jointly by Chongmin Song and

John. P. Wolf in 1997 [13,14] to successfully solve Elastodynamic and allied problems of Civil

Engineering and Soil structure interaction. The novel method is based entirely on finite elements,

but with a discretization only on the boundary [13,14]. The method combines the advantages of

both the finite and boundary element methods [13,14]. This method doesnt require any

fundamental solution to be known in advance, unlike the boundary element method [13,14]. This

novel method is analytical in its approach in the radial direction with respect to an origin called as

scaling center, and implements the finite element method in the circumferential direction [13,14].

Hence the method is semi-analytical.

The key advantages of this method are as follows: [13,14]

a) Reduction of the spatial dimension by one, reducing the discretization effort.

3

b) No fundamental solution required which permits general anisotropic material to be addressed

and eliminates singular integrals.

c) The method being analytical in the radial direction, permits the radiation condition at infinity, to

be satisfied exactly for unbounded media .

d) No discretization on that part of the boundary and interfaces between different materials passing

through the scaling center.

e) Converges to the exact solution in the finite-element sense in the circumferential directions.

f) Tangential continuity conditions at the interfaces of different elements are automatically

satisfied.

In light of the above discussion, it is imperative to develop a vector finite element method based

on the scaled boundary transformation [13,14] which address the problem of discretization and at

the same time retains the advantages of both the tangentially continuous vector finite element

method, as well as the scaled boundary finite element method. The development of such a novel

vector finite element method for solving problems in electromagnetics is the main aim of this paper.

To the best knowledge of the author, till date, the reformulation of the scaled boundary finite

element method for electromagnetics has not been done before. In this paper, the reformulation is

done specifically for determining the resonant frequencies of metallic cavity structures.

2. Development of the scaled boundary finite element method: The initial development of the

novel method was based on an approach, using the concept of assemblage and similarity familiar to

engineers. The method was then called as the Consistent Infinitesimal Finite- Element cell method

[15], reflecting its derivation. Successive developments of the method led to its reformulation based

on the scaled boundary transformation [13,14]. In this approach, the governing differential equation

is transformed using a Galerkin weighted residual technique. This results in the scaled boundary

finite-element equation of the problem [13,14]. This method is called the Scaled Boundary Finite

Element method. The scaled boundary finite element method, is based entirely on finite elements.

4

As a prelude to the reformulation of the scaled boundary finite-element method for

electromagnetics, the concept of the scaled boundary transformation is explained in the forthcoming

section.

3. Concept of the scaled boundary transformation: In order to apply this novel method, a scaling

center is first chosen in such a way that the total boundary under consideration is visible from it

[13,14]. In case of geometries where it is impossible to find such a scaling center, the entire

geometry is sub-structured [12]. In each sub-structure, the scaling center is independently chosen,

and the method is applied in each sub-structure. The sub-structures are combined together, which

corresponds to the analysis of the whole geometry.

The concept of the scaled boundary transformation is that, by scaling the boundary in the radial

direction with respect to a scaling center O, with a dimensionless numerical factor varying in the

range from 0 to 1, the whole domain is covered [13,14]. For bounded domains, the upper and lower

bounds of the scaling factor are 1 and 0 respectively. For unbounded domains, the corresponding

lower and upper bounds of the scaling factor are 1 and . The figures (1a) and (1b), illustrate the

concept of the scaled boundary transformation for unbounded and bounded domains respectively.

Fig. (1a) Unbounded medium Fig.(1b) Scaled boundary (section) with scaling center inside the medium (section)

5

The scaling applies to each surface finite element. Its discretized surface on the boundary is

denoted as Se (superscript e for element). Continuous scaling of the element yields a pyramid with

volume Ve. The scaling center O is at its apex.

The base of the pyramid is the surface finite element. The sides of the pyramid forming the

boundary Ae follow from connecting the curved edge of the surface finite element to the scaling

center by straight lines. No discretization of Ae occurs. Assembling all the pyramids by connecting

their sides, corresponds to enforcing compatibility and equilibrium conditions. This results in the

total medium with volume V and the closed boundary S. No boundaries Ae passing through the

scaling center remain. Mathematically, the scaling corresponds to a transformation of the

coordinates for each finite element resulting in two curvilinear local coordinates along the

tangential directions and a single dimensionless radial coordinate representing the scaling factor.

This transformation becomes unique due to the choice of the scaling center from which the total

boundary of the geometry is visible [13,14].

The scaled boundary transformation is basically a relation between the derivatives in the

cartesian coordinates and the derivatives expressed in the scaled boundary variables [13,14].

4. The scaled boundary finite element method in electromagnetics: The scaled boundary

transformation equations [13,14] are quite general. It can be applied to differential equations

governing the phenomena in any discipline. This feature of the scaled boundary transformations is

used in the reformulation of the novel method for electromagnetics [17]. However, the actual

formulation of the scaled boundary finite-element equation depends upon the additional constraints

that are specific to the discipline, which are to be satisfied. This approach ensures that, the scaled

boundary finite-element equation takes into account, the specific features of that discipline. Hence,

a closest possible representation of the system, represented by the original differential equations

along with the constraints in the form of boundary conditions, is achieved.

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In this context, when the scaled boundary finite-element method is reformulated in

electromagnetics in H formulation, it is necessary that apart from satisfying the essential boundary

conditions, the fields should satisfy the solenoidality property of the magnetic field [17]. This

condition should be necessarily incorporated while formulating the scaled boundary finite-element

equation in electromagnetics [17]. This is necessary so that no spurious solutions occur as eigen

solutions of the boundary value problem [17].

Considering the implementation of the essential boundary condition that occurs in the problems

in electromagnetics that, the tangential Electric/Magnetic fields vanish at those points lying on the

Electric/Magnetic walls respectively is accomplished by setting to zero, all the unknown numerical

coefficients of the tangential variables h and z in the corresponding tangential field components,

when the scaled boundary passes through those points [17]. This is achieved numerically by the

quadrature method of integration. The upper and lower limits for the integrals involving the

integration by quadrature method, is chosen in such a way that they contain the points lying on the

electric/magnetic walls and form a part of the quadrature nodes for integration. While performing

the integration, when the nodes of the quadrature lie on the electric/magnetic walls, all the unknown

numerical coefficients of the tangential variables h and z in the corresponding tangential field

components are set to zero, for that point. By this process, the essential boundary conditions

pertaining to the problem gets satisfied [17].

The scaled boundary finite element method posses a distinct feature, when formulated in

electromagnetics. The accuracy of the eigen solutions depend upon the number of terms included in

the radial expansion of the field [17]. This dependence is apart from the conventional factors like

the choice of the interpolation functions, size of the element, accuracy of the Jacobian

transformation, etc. This fact is demonstrated numerically in section 5.7 for the case of a spherical

metallic cavity.

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5. The Scaled Boundary Finite Element formulation for Cavity Structures: An ideal metallic

cavity structures represent the total confinement of the electromagnetic field in a volume of space

bounded externally by metallic surfaces. The eigen-modes of the cavity structures are in fact, the

standing wave field patterns corresponding to various eigen values. The tangential component of

the electric field and the normal component of the magnetic field vanish on the boundary

comprising the metallic surfaces. It is assumed that, the metallic surface is a perfect conductor.

In this section, the scaled boundary finite-element equation is developed for a general metallic

cavity structure, and a numerical implementation is illustrated for the case of a spherical metallic

cavity.

5.1. Theory

The first step in the development of the appropriate scaled boundary finite element formulation

is the formulation of the variational functional. This is considered in the following discussion.

The vector helmholtz equation in H formulation [5] is given by

0201 = - H-H rr k me , 00

220 emw=k (5.1.1)

where er and mr are the local material properties corresponding to the relative permittivity and

relative permeability of the medium respectively, w is the angular frequency.

The associated essential boundary condition [5] is given by

tn H1H = (5.1.2)

The essential boundary condition implies that the magnetic field is purely tangential to the

bounding metallic surface. tH denotes the tangential magnetic field component, and n1 denotes unit

normal vector to the surface.

The associated functional for Eq.(5.1.1) can be developed by taking the dot product of Eq.(5.1.1) by

an arbitrary test vector W, and integrating over the whole domain W,

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( ) 0d 201 = -W k rr H-HW me (5.1.3)

Using the following identity [5]

( ) ( ) ( ) ( ) dSd d n1QPQPHP -= WW S

(5.1.4)

Eq.(5.1.3) can be written as,

( ) ( ) ( ) W

--

W

=W- 0d -dS d 2011 HW1HWHW n r

Srr k mee (5.1.5)

The boundary integral can be re-arranged as

( )dS 1 -S

r n1HW e (5.1.6)

Also,

EH wej= (5.1.7)

for the time harmonic variation of the magnetic field.

Hence, the rearranged boundary integral vanishes over a perfect metallic surface since the

tangential electric field over the metallic surface is zero and the cross product of the normal to the

surface 1n with the normal component of the electric field will be zero.

Hence, Eq.(5.1.5) can then be written as,

( ) ( ) 0d d 201 =W- W

-

W

HWHW rr k me (5.1.8)

The Eq. (5.1.8) represents the weighted residual form of the vector helmholtz equation in H

formulation.

The associated variational form of Eq.(5.1.1) is obtained by substituting W = H in Eq.(5.1.8) and

by using the identity given in [18] for all the three Cartesian components of the vectors occurring

in the integral form given in (5.1.8). This gives the variational functional F given by,

( ) ( )( ) kF rr d 21 2

01

W

- -= HHHH me (5.1.9)

9

The expression for F given in (5.1.9) is the variational functional associated with the vector

helmholtz equation (5.1.1) subject to the essential boundary condition given in (5.1.2).

The variational functional given in (5.1.9) is made stationary, with respect to the unknown

coefficients in the expansion of H, which corresponds to the solution of the vector helmholtz

equation.

5.2. Expression of the field components in terms of the scaled boundary coordinates

The first step in the development of the scaled boundary finite element formulation, is

expressing the vector variable H in terms of the scaled boundary variables zhx ,, . The vector

variable H is expanded in terms of the ortho-normal vectors in the scaled boundary co-ordinate

system [13,14] as

( ) ( ) ( ) ( ) ( ) ( ) ( ) zzzhhhxxx VhxVhxVhxVhx nnnH ,H H ,H H ,H H,, nnnnnn ++= (5.2.1)

From (5.2.1), the corresponding representation of ( )zhx ,, H in Cartesian coordinates obtained by

using the transformation equations given in [17] is given by,

( ) ( ) ( ) ( ) ( ) ( ) ( )[ ]

=

k

j

i

H ,H H,H H,H H ,, zhxzhxzhxzhx zzyyxx (5.2.2)

In Eq.(5.2.2) i ,j, k represent the ortho-normal vectors in the Cartesian coordinate system.

The individual terms in the R.H.S of (5.2.2) are given as follows.

( ) ( ) ( ) ( ) ( )[ ]( )( )( )

=z

z

hh

xx

zhx

Vh

Vh

Vh

xxxzhx

x

x

x

xx

n ,H

n ,H

n ,H

H H H,H H

n

n

n

nnn (5.2.3)

10

( ) ( ) ( ) ( ) ( )[ ]( )( )( )

=z

z

hh

xx

zhx

Vh

Vh

Vh

xxxzhx

y

y

y

yy

n ,H

n ,H

n ,H

H H H,H H

n

n

n

nnn (5.2.4)

( ) ( ) ( ) ( ) ( )[ ]( )( )( )

=z

z

hh

xx

zhx

Vh

Vh

Vh

xxxzhx

z

z

z

zz

n ,H

n ,H

n ,H

H H H,H H

n

n

n

nnn (5.2.5)

The terms of the form ( ) ( )zhx ,H H occurring in the matrix multiplication have the form,

( ) ( ) ( ) ( )zhzhxzhx , )( )( ,H H0

nhhhf jim

oi

n

jij

= =

= (5.2.6)

In the above expression, ( )xf denotes the unknown radial function expressed in terms of the radial

variable x and ijh denotes the unknown numerical coefficients occurring in the expansions of Hx,

Hy and Hz respectively. )(hih denotes the single variable functions of x, y and z components of H

expressed in terms of the tangential variable h . )(z jh denotes the single variable functions of x, y

and z components of H expressed in terms of the tangential variable z . The form of the functions

expressed as )(hh and )(zh are given as [5]

rrh -=1)(0 (5.2.7)

rrh +=1)(1 (5.2.8)

2for )1()( 22 -= - irrrh ii (5.2.9)

It is shown in detail in [5] that, while employing the vector finite element method, only the

tangential continuity between adjacent elements are sufficient to impose inter-element field

continuity. In (5.2.6), ( )zh,n represent the scalar components of the orthogonal vectors in the

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scaled boundary coordinate system. The function f(x) occurring in the field expansions is expanded

in the form of the power series expansion in x as

( ) =

=N

k

kkaf

01 xx (5.2.10)

Eqs.(5.2.1-5.2.10) enables the field components to be expressed in terms of the scaled boundary

variables. The next step is the derivation of the solenoidality (divergence) condition of the magnetic

field, in terms of the scaled boundary coordinates. This is detailed in the following section.

5.3. Derivation of the solenoidality condition in terms of the scaled boundary coordinates

The derivation of the solenoidality condition plays a crucial role in the development of the

scaled boundary finite element formulation. As mentioned earlier, this is essential for the

elimination of spurious modes in the eigen spectrum of the boundary value problem.

Since the problem is formulated in terms of the scaled boundary variables( )zhx ,, [13,14] it is

necessary that the divergence condition should be written in terms of ( )zhx ,, . This is achieved as

follows.

The divergence condition for the magnetic field is given by,

( ) 0= Hm (5.3.1)

When m is a non-zero scalar constant, Eq.(5.3.1) can be written as

0= Hm (5.3.2)

Expanding Eq.(5.3.2),

0HHH

=

+

+

zyxzyxm (5.3.3)

Since 0m , Eq.(5.3.3) gives,

12

+

+

zyxzyx HHH (5.3.4)

Rewriting the Eq.(5.3.4) in terms of scaled boundary coordinates by using the three dimensional

scaled boundary transformation [13,14] and using the convention that xH , yH and zH respectively

denote the scalar part of the i , j , and k components given in (5.2.2), we get

0H

|J|H

|J|1H

|J|

H

|J|

H

|J|1H

|J|H

|J|H

|J|1H

|J|

=

+

+

+

+

+

+

+

+

zhxx

zhxxzhxx

zz

hh

xx

zz

hh

xx

zz

hh

xx

zz

zz

zz

yy

yy

yy

xx

xx

xx

ng

ng

ng

ng

ng

ng

ng

ng

ng

(5.3.5)

Multiplying both sides of Eq.(5.3.5) by x,

0H

|J|H

|J|H

|J|

H

|J|

H

|J|

H

|J| H

|J|H

|J|H

|J|

=

+

+

+

+

+

+

+

+

zhxx

zhxx

zhxx

zz

hh

xx

zz

hh

xx

zz

hh

xx

zz

zz

zz

yy

yy

yy

xx

xx

xx

ng

ng

ng

ng

ng

ng

ng

ng

ng

(5.3.6)

Rewriting Eq.(5.3.5) using Eq.(5.2.10) and grouping the terms of x k,

[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ]

[ ] ( )[ ] [ ] ( )[ ] [ ] ( )[ ][ ] 0 ,N ,N ,N |J|

dNd

dNd

dNd

|J|dNd

dNd

dNd

|J|

3210

0

321321

=++

+

++

+

++

=

=

kzkykxk

m

k

km

kkzkykxkzkykx

ncnbnakg

cnbnang

cnbnang

xzhzhzh

xhhhhhh

xxxx

zzzh

hhhh

(5.3.7)

The Eq.(5.3.7) holds for all values of x . This is possible if the coefficient terms of kx are zero.

Equating the coefficient of kx to zero, gives,

[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ]

[ ] [ ] [ ] [ ] (5.3.8) 0dNd

|J|dNd

|J|N

|J|

dNd

|J|dNd

|J|N

|J|

dNd

|J|dNd

|J|N

|J|

333

222

311

=

+++

+++

++

zh

zhzh

zz

hh

xx

zz

hh

xx

zz

hh

xx

zzzk

yyykxxxk

ng

ng

kng

c

ng

ng

kng

bng

ng

kng

a

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Eq.(5.3.8) is valid for 0k and [N1] ,[N2] and [N3] are matrices containing the functions in terms of

the variables h and z of the double summation series given in (5.2.6). [ak], [bk] and [ck] are the

matrices containing the unknown constants of the radial power series expansion given in (5.11) and

J occurring in Eqs.(5.3.5-5.3.8) is the surface discretization factor [17].

The relationship expressed in Eq.(5.3.8) is the solenoidality condition expressed in terms of the

scaled boundary coordinates. It is a point wise constraint between the unknown coefficients of

the field vector variable H valid for every point in the domain.

The next section deals with the expression of the variational functional in the scaled boundary

coordinates.

5.4. The variational form of the functional in terms of the scaled boundary coordinates

To express the functional given in (5.1.9) in terms of the scaled boundary variables, the three

dimensional scaled boundary transformation given in [17] is used. The expression H occurring

in the functional is written in terms of the scaled boundary variables using the scaled boundary

transformation. The expressions for H given in Eqs.(5.2.3-5.2.5) are used to evaluate the term

HH in the functional.

Following this procedure, the variational form of the functional given in Eq.(5.1.9) written in terms

of the variables of the scaled boundary coordinates is given by,

( )( ) ( )( ) ( )( )[ ] ( )( ) ( )( ) ( )( )[ ]6541

3

2

2

2

1

21

0

1

TTT|J|

2TTT|J|2

1 zxxhzhzhx

h z

ex

egggggggggF rr +++++

=

--

( ) ( )( ) ( ) ( )( ) ( ) ( )( )[ ] 0d d d ,N,N,N 23322221120 =++- zhxzhxzhxzhxm fffk r (5.4.1)

14

The terms T1 to T6 are given by,

222

1

HHHHHHT

-

+

-

+

-

=xxxxxx

xxxxxx xy

yx

zx

xz

xz

xy nnnnnn (5.4.2)

222

2

HHHHHHT

-

+

-

+

-

=hhhhhh

hhhhhh zy

yx

zx

xz

yz

zy nnnnnn (5.4.3)

222

3

HHHHHHT

-

+

-

+

-

=zzzzzz

zzzzzz zy

yx

zx

xz

yz

zy nnnnnn (5.4.4)

-

-

+

-

-

+

-

-

=

zzhh

zzhhzzhh

zzhh

zzhhzzhh

xy

yx

xy

yx

xx

xz

zx

xz

yz

zy

yz

zy

nnnn

nnnnnnnn

HH

HH

HH

HHHH

HHT4

(5.4.5)

-

-

+

-

-

+

-

-

=

hhxx

hhxxhhxx

hhxx

hhxxhhxx

xy

yx

xy

yx

zx

xz

zx

xz

yz

zy

yz

zy

nnnn

nnnnnnnn

HH

HH

HH

HHHH

HHT5

(5.4.6)

-

-

+

-

-

+

-

-

=

zzxx

xxzzhxzz

zzxx

xxzzxxzz

xy

yx

xy

yx

zx

xz

zx

xz

yz

zy

yz

zy

nnnn

nnnnnnnn

HH

HH

HH

HHHH

HHT6

(5.4.7)

5.5. Implementation of the solenoidality condition in the functional

Having expressed the functional in terms of the scaled boundary variables, and the next step is to

implement the solenoidality condition given in Eq.(5.3.8), in the variational functional given in

Eq.(5.4.1). The Lagrange multiplier technique [19] is made use of which results in a modified

functional containing )1( +k Lagrange multiplier terms for every value k. These Lagrange

multiplier terms account for the implementation of the solenoidality condition of the magnetic field.

The radial coordinate x in the modified functional is independent of the two circumferential

15

coordinates h and V. The functional is integrated with respect to x with its lower and upper limits

being 0 and 1. This renders the functinal entirely in terms of the circumferential variables h and V.

The resulting modified functional is given by,

[ ] zhz

d d TTTTTTTTT21

9872065432

,1 ++-+++++= kF

+ )1( +k Lagrange multiplier terms

(5.4.8)

In (5.4.8), the letters with a prime denote the terms after the integration with the radial variable

x. An important observation that is to be noted is that, the terms in Eq.(5.4.8) contain only

the surface finite element discretization factor denoted by | J | even for the general 3-D

structures.

This unique feature of the scaled boundary finite element method, in contrast with the

conventional finite element method, where the finite element discretization is to be necessarily

performed in all the three dimensions, for arbitrary three dimensional structures devoid of

uniformity along a particular axis. The forth-coming section deals with the generation of the

finite element matrices.

5.6. Generation of the scaled boundary finite element matrices and the formation of matrix

equations

After the expression of the functional purely in terms of the tangential variables, as given in

Eq.(5.4.8), the next step is the generation of the scaled boundary finite element matrices.

The expression given in (5.4.8) is evaluated for every surface element characterized by the

circumferential variables (h,V). Then the variation with respect to each undetermined coefficient is

set to zero. This process leads to a set of linear equations. Imposing only tangential continuity of the

field component between the adjacent elements, the final system equation is of the form

0 B A 20 =+ hkh (5.4.9)

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The Eq.(5.4.9) is a standard form of the matrix eigen value equation which can be solved

numerically. The solution of Eq.(5.4.9) gives the solution of the vector helmholtz equation given in

Eq.(5.1.1) along with the essential boundary condition given in (5.1.2).

An important feature that is to be noted in the theoretical formulation developed above is

that, there is no specific assumption on the shape of the geometry under consideration. Hence,

the formulation thus developed, holds independent of geometry.

5.7. Numerical Implementation

The theoretical formulation thus developed, is implemented numerically for the case of a

spherical metallic cavity, with air being the dielectric medium. The schematic diagram of the

spherical metallic cavity considered for the numerical implementation is shown in Figure 5.1 in the

following page..

Figure 5.1 Spherical metallic Cavity

The units of the radius of the spherical metallic cavity denoted as a is taken to be in cms. For

the finite element discretization of the surface, an eight node curvilinear quadrilateral elements were

used with the mesh of one octant consisting of three finite elements. The discretized boundary of

the solid sphere of one octant is shown below in Figure 5.2 in the following page.

17

Figure 5.2 Finite Element mesh of one octant of boundary of solid sphere.

After the discretization of the boundary of the spherical metallic cavity by surface finite

elements, the functional given in Eq.(5.4.8) is evaluated for every finite element followed by the

numerical integration along the radial and tangential coordinates. During this process, only the

inter-element tangential continuity is imposed. The integration which is performed involving the

circumferential variables are done numerically using 5 point gaussian quadrature. Then, the

variation with respect to every undetermined coefficient is taken, and set to zero.

The eigen value equation resulting from the process of assembling the element matrices were

solved by using the standard LAPACK [20] collection of Fortran subroutines. The number of terms

were used in the radial expansion of the fields were five. The resonant frequencies of TM 011 mode

for the spherical cavity were computed by varying the radius of the spherical cavity, and the

numerically obtained results show close agreement to the values obtained by a full theoretical

analysis [21], and is shown below in Table 5.1.

Table 5.1 The close agreement between the theoretical and the numerical values of the resonant frequency of

TM (011) mode.

S.No Radius of the spherical metallic cavity.

a (in cms)

Resonant frequency computed for the TM011 mode by theory.

f r (10 9 Hz)

Resonant frequency computed by scaled boundary

Finite element method. Fr (10

9 Hz) 1 3 4.367 4.3668

2 4 3.277 3.278

3 5 2.622 2.6215

4 6 2.185 2.1848

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By theoretical modal field analysis [21], the resonant frequency fr for a spherical cavity of radius

3 cms for TM mode satisfies the condition TM (even) (011) = TM (even) (111) = TM(odd) (111).

Accordingly, the resonant frequency obtained through the scaled boundary formulation for the

spherical cavity of radius 3cms for the above mentioned modes were also constant with a value of

4.3668 x 109 Hz, satisfying the above property, confirming the validity of the new method.

Another important feature of the scaled boundary finite element formulation, is the effect of the

number of terms in the radial expansion of the field variable on the accuracy of the eigen values.

Table 5.2, shows the effect of increase in the number of terms in the radial expansion of the field

variable, on the accuracy of the resonant frequency.

Table 5.2 The effect of increase in the number of terms in the radial expansion of the field variable, on the resonant frequency of TM (011) mode.

No. of terms in the radial

expansion of the field variable

Radius of the spherical metallic cavity.

a (in cms)

Resonant frequency computed for the TM011 mode

by theory. f r (10

9 Hz)

Resonant frequency computed by scaled

boundary Finite element method.

Fr (10 9 Hz)

5 3 4.367 4.3654

6 3 4.367 4.3660

7 3 4.367 4.3664

8 3 4.367 4.3670

From Table 5.2, it can be inferred that, the number of terms in the radial expansion of the field

variable, has a marked influence on the accuracy of the resonant frequency. It is observed from the

above table that, the increase in the number of terms in the radial expansion of the field variable is

accompanied by the corresponding increase in the accuracy of the eigen values.

5.8. Conclusion: The scaled boundary finite element method is a novel semi analytical method

based on finite elements, originally developed in the field of civil engineering to study problems

pertaining to elastodynamics and Soil structure interaction. The crucial aspects of the reformulation

19

of the novel method for the full-wave analysis of cavity structures, is reported in this paper. The

closeness of the numerical results obtained from the scaled boundary finite element method to those

obtained from analytical approach validates the methodology developed in this paper, for analyzing

cavity structures.

An important consequence of the method being semi-analytical is that, the accuracy of the

numerical values depend not only on the element size as in the conventional finite element method,

but also on the number of terms that are used in the radial series expansion of the fields. This

enables to get accurate numerical results even by increasing the number of terms in the radial

expansion of the fields without tampering with the element size. Also, the novel formulation

reported in this paper, is independent of the geometry under consideration. The scaled boundary

finite element formulation in electromagnetics is further developed to analyze multi-layered and

multi-conductor micro-strip transmission lines, and VLSI interconnects.

Acknowledgements: The author thanks Prof. John. P. Wolf, Institute of Hydraulics and Civil

Engineering, Swiss Federal Institute of Technology, Lausanne, Switzerland, and Dr. Chongmin

Song, Department of Environmental and Civil Engineering, University of New South Wales,

Australia, for their valuable suggestions and providing their research articles on Scaled Boundary

Finite Element method.

References:

[1]. Monk, P. Finite Element Methods for Maxwells Equations Oxford University Press, Oxford, 2003. [2]. J. Jin, The Finite Element Method in Electromagnetics J. Wiley & Sons, New York, 2002 [3]. Salazar-Palma, M., Sarkar, T. K., Garcia-Castillo, L. E., Roy, T. and Djordjevic, A. R. Iterative and Self-Adaptive Finite-Elements in Electromagnetic Modeling Artech House Publishers, Inc., 1998. [4]. Volakis, J. L., Chatterjee, A. and Kempel, L. C., Finite Element Method for Electromagnetics: Antennas, Microwave Circuits, and Scattering Applications IEEE Press and Oxford University Press, New York, 1998.

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[5]. P.P. Silvester, and R.L. Ferrari, Finite Elements for Electrical Engineers,3 rd Ed., Cambridge University Press., 1996. [6]. P.P. Silvester and G.Pelosi, Finite Elements for Wave Electromagnetics, IEEE Press, 1994. [7]. Wang, X. H., Finite Element Methods for Nonlinear Optical Waveguides Gordon & Breach, New York, 1996 [8]. Fernandez, F. A. and Lu, Y., Microwave and Optical Waveguide Analysis by the Finite Element Method, J. Wiley & Sons, New York, 1996. [9] Luis valor and Juan Zapata, Efficient Finite Element Analysis of Waveguides with Lossy Inhomogeneous Anisotropic materials Characterized by Arbitrary Permittivity and Permeability Tensors, IEEE Trans. Microwave Theory Tech., vol. MTT-43, no.10, pp.2452 2459, Oct.1995. [10] Luis valor and Juan Zapata, An Efficient Finite Element Formulation to Analyze Waveguides with Lossy Inhomogeneous Bi-Anisotropic materials, IEEE Trans. Microwave Theory Tech.,vol. MTT-44, no.2, pp.291 -296, Feb.1996. [11] Javier Arroyo and Juan Zapata, Subspace Iteration Search Method for Generalized Eigenvalue problems with Sparse Complex Unsymmetric Matrices in Finite-Element Analysis of Waveguides, IEEE Trans. Microwave Theory Tech., vol. MTT-46, no.8, pp.1115 -1123, Feb.1996. [12] Andrew J. Deeks and John P. Wolf, An h-hierarchical adaptive procedure for the scaled boundary finite-element method, Int. J. Numer. Meth. Engng, 54, pp.585-605,2002. [13] Chongmin Song and John. P. Wolf, The Scaled boundary finite-element method- alias Consistent infinitesimal finite-element cell method for elastodynamics, Computer Methods in applied mechanics and engineering, (1997), No.147, pp.329-355. [14] John. P.Wolf and Chongmin Song, The Scaled boundary finite-element method a primer : derivations, Computers and Structures, (2000), 78, pp.191-210. [15] Chongmin Song and John P. Wolf, Consistent Infinitesimal Finite-Element Cell Method : Three-Dimensional Vector Wave Equation, International Journal for Numerical Methods in Engg, (1996), Vol.39, pp. 2189-2208. [16] Andrew J. Deeks, John P. Wolf, An h-hierarchical adaptive procedure for the scaled boundary finite-element for elastodynamics, Int. J. Numer. Meth. Engng, (2002), 54, pp.585-605. [17] V.S. Prasanna Rajan, The Theory and application of a Novel Scaled Boundary Finite Element method in computational electromagnetics, Ph. D thesis, University of Hyderabad, India, Dec.2002. [18] J.N.Reddy, An Introduction to the Finite Element Method, McGraw Hill, p.11-138, 1984.

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[19]. Irving H. Shames, Clive L. Dym, Energy and Finite Element Methods in Structural Mechanics, New Age International Publishers Ltd., Wiley Eastern Ltd, pp.671- 674,1995. [20] LAPACK users guide, 3rd Ed., SIAM, Philadelphia. [21] C.A. Balanis, Advanced Engineering Electromagnetics, John Wiley & Sons, New York, pp.560-562.