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MODELLING OF INDUSTRIAL CONVEYORIZED
APPLICATORS USING HIGHER ORDER
VECTOR FINITE ELEMENTS
A.Hallac1 and A. C. Metaxas2
Engineering Dept, University of Cambridge, UK
2
St Johnʼs College Cambridge, UK
1
This paper presents a finite element time domain method for the solution of Maxwellʼs equations
in microwave heating applicators using first and second order vector finite elements. Results are
compared with experimental data and it has been shown that second order vector finite elements
have many advantages over first order elements. Capitalising on the high accuracy and low
computational cost attainable by higher order elements, an industrial conveyor belt system is
numerically analyzed.
Submission Date: February 21, 2006
Acceptance Date: October 24, 2006
INTRODUCTION
Numerical techniques have been used
widely in the past to model microwave heating
applicators loaded with a variety of loads.
The methods include finite differences, finite
elements, transmission line methods and other
techniques [Hussain M.Al-Rizzo et al, 2005;
Dominguez-Tortajada et al, 2005; Yakovlev,
2006] while some earlier attempts of modelling
using these techniques can be found in Dibben
and Metaxas [1994].
This paper discusses the issue of using
higher order finite elements for modelling large
conveyorised microwave applicators as opposed
to the usual formulation using first order finite
elements. The most computationally intensive
part of a finite element (FE) approximation is
Keywords: industrial applications, FE approximation,
modeling
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the solution of a large system of linear equations
produced by the discretization of the domain
being analyzed. In the frequency domain a
single solution is required at the frequency
of interest whereas in the time domain the
system has to be solved at each time step over
a band of frequencies. However, for practical
microwave heating problems, the time domain
solution is considerably easier to obtain than
its frequency domain counterpart. This is due
to highly ill conditioned matrices arising from
the frequency domain method [Dibben and
Metaxas, 1994 and 1997]. It has been proven
that for first order tangentially continuous
vector finite elements (TVFE) the condition
number of matrices produced by the frequency
domain is strictly proportional to the size of the
solution domain [Dibben and Metaxas, 1996].
It is therefore difficult to achieve convergence
when using existing iterative methods on high
density meshes. For higher order elements this
problem is even more severe [Hallac, 2003;
101
Hallac and Metaxas, 2003 and 2003a]. In
the time domain, however, the second order
basis functions produce real, symmetric and
positive definite matrices allowing one to use a
simple iterative method, such as the conjugate
gradient method, while still achieving adequate
convergence [Hallac and Metaxas, 2001]. Also,
the time domain method is particularly well
suited to the calculation of field distribution
within the applicator and hence power density
which is a key design parameter.
In this paper we present the use of second
order 3D time domain TVFE for the study of
microwave heating problems. The formulation
is described in Hallac and Metaxas [2003a] and
uses interpolatory [Yioultsis and Tsiboukis,
1998] non-interpolatary [Savage and Peterson,
1996] second order vector finite elements for
the electric field interpolation. These elements
have degrees of freedom associated with their
edges and faces and are well suited to model
Maxwellʼs equations because they only enforce
the tangential continuity of vector fields and do
not generate spurious contributions or nonphysical resonances that are characteristic of
nodal-based expansions [Carpes and Razek,
2000].
The efficiency and accuracy of the higher order
bases are demonstrated through the analysis of
batch and continuous multimode cavities with
different dielectric loads including the use of
a conveyor belt. The results are compared
with those from a network analysis and other
numerical solutions where applicable.
THEORY
This paper uses the well documented
theoretical treatise starting from the time
dependent vector equation describing the
electric field and defining the weak form of that
equation which includes the surface integral
which surrounds the domain to be investigated
[Hallac and Metaxas, 2003a]. This equation is
102
solved numerically using tetrahedral elements.
The usual format of the vector basis functions
is used to expand the electric field associated
with the edges and facets of each element.
Such a formulation produces a set of ordinary
differential equations:
[ S ]e(t ) + [Tσ + TABC ]
d
d2
e(t ) + [Tε ] 2 e(t ) = b(t )
dt
dt
(1)
with the term b(t) representing the excitation
currents and the surface integral term. The
components of the FEntal matrices are:
[ S ]i , j = ∫ (∇
Ωe
Wi ) ⋅ (∇ W ) dΩe
[Tσ ]i , j = σ eµo ∫ (Wi ⋅ W j )dΩe
Ωe
[Tε ]i , j = ε oε ' µo ∫ (Wi ⋅ W j )dΩe
Ωe
(2)
(3)
(4)
and TABC is the elemental matrix resulting from
the application of absorbing boundary conditions
(ABC). A number of methods can be used to
descretise the time derivatives, however, as
we have reported previously, we have used the
Newmark method [Dibben and Metaxas, 1994;
Ehlers and Metaxas, 2003].
TVFE Basis
It is well known that for practical microwave
heating problems it is difficult to produce high
quality unstructured meshes. For a simulation
using first order basis elements, to increase the
quality of the mesh the number of degrees of
freedom must be substantially increased leading
to excessive memory requirement. It is therefore
necessary to increase the order of TVFE bases
so as to reduce errors caused by the distorted
meshes [Hallac, 2003], and to minimize
computational costs.
We consider a tetrahedral element with nodes
Journal of Microwave Power & Electromagnetic Energy
Vol. 40, No.2, 2006
numerical scheme but removes the necessity of
specialized mesh generation software [Hallac,
2003].
Boundary Conditions
Figure 1. Illustration of a tetrahedral element
and the numbering of nodes.
i, j, k and l as shown in Figure 1. The volume
of a tetrahedron is denoted by V. Simplex (or
volume) coordinates æi, æj, æk and æl at a
point Pi are defined by Savage and Peterson
[1996]. Using simplexes the second order
interpolatory Yioultsis and Tsiboukis [1998]
TVFEʼs constitute 12 linearly independent edgebased vector bases and 8 face-based functions.
For example, on edge {i,j} the two bases ( {i,j}
and {j,i} ) can be written as:
r
Wij = 8ζ i2 − 4ζ j ∇ζ j + −8ζ iζ j + 2ζ j ∇ζ i
(
)
(
)
(5)
and for face {i,j,k}, two basis functions ( {i,j,k}
and {i,k,j} ) can be written as:
r
Wijk1 = 16ζ iζ j ∇ζ k − 8ζ jζ k ∇ζ i − 8ζ kζ i ∇ζ j 

r
Wikj2 = 16ζ iζ k ∇ζ j − 8ζ jζ k ∇ζ i − 8ζ kζ i ∇ζ j  (6)
The conformity of degrees of freedom between
adjacent elements is guaranteed by using a
consistent global numbering. In contrast to the
well known classic finite element method where
all tetrahedrons have the same parent element,
second order TVFE bases are non symmetric
and each tetrahedron has its own parent element,
where element parents have dissimilar local
numbering. This adds a complication to the
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As it is usual the walls of the cavity and the feed
waveguide are assumed to be perfect conductors.
In the feed waveguide it is assumed that at the
operating frequency only the dominant mode
TE10 is supported while a first order ABC is
used to model open boundaries [Dibben and
Metaxas, 1994] [Hallac, 2003]. The boundary
integral terms for the ABC and the source can
be written as:
[ S ]i , j = µo ∫ ν TE
Γ
10
(Wi ⋅ W j ) dSo
)
bi , j = µ0 ∫ Wi ⋅ Einc × n dS1
Γ
( )(
)
(7)
(8)
where So and S1 are the ABC and excitation
planes respectively in the feed waveguide, n) is a
unit vector, W’s are the basis functions, _TE_10 is
the waveguide phase velocity for the TE10 mode,
and the incident field can be written as:
πx
)
Einc = ny sin( )sin(ωt − ν TE10 )
a
(9)
where ς is a unit vector in the y direction which
defines the direction of the incident field. In
practical microwave heating applications the
source modelling described in equation (8) is
adequate to represent the magnetron source
[Hallac, 2003].
RESULTS
Frequency domain matrices are known to
suffer from ill conditioning when multimode
applicators are analyzed [Dibben and Metaxas,
1996; Boffi et al, 1999]. The time domain solution
overcomes the problem of ill conditioning and
103
provides a symmetric, positive definite system of
equations to be solved at each time step. In this
section for the solution of matrices the Conjugate
Gradient method with zero fill-in incomplete LU
preconditioner is used [Hallac, 2003].
shows the consistent performance of the second
order edge elements with respect to the mesh
quality in that any choice of the mesh parameters
that increases the mesh quality would reduce
computational errors.
Mesh Quality Consideration
Experimental Verification
First order edge elements are very sensitive
to the quality of the unstructured mesh. Mesh
quality affects both the efficiency and accuracy
of the FE solutions, since meshes with distorted
elements make accurate results more difficult to
achieve. For complex geometries it is difficult
to mesh the domain using well-shaped elements
with neither very small nor very large angles
at corners. Therefore to obtain a high quality
mesh it is necessary to refine the mesh, which
increases substantially the number of degrees
of freedom. Higher order TVFEʼs, on the other
hand, are less sensitive to highly distorted
meshes [Hallac and Metaxas, 2001]. The most
common approach in estimating errors within
a mesh is to view the mesh quality as being
independent of the solution.
The mesh quality for each element in a 3D
mesh can be defined as [Berzins, 1998]:
A complete characterization of microwave
heating involves coupling the electromagnetic
and the thermal fields through the temperature
dependence of the material dielectric properties.
Such a coupled problem, particularly when solved
using finite elements, presents a severe CPU
requirement and is not considered in this paper.
Instead, by solving only Maxwell’s
equations, the electromagnetic field distribution
within a loaded applicator is obtained which,
given a knowledge of the loss factor, yields the
power density established in the material to
be heated. In the past such power density data
from simulations were compared with thermal
imaging from experimental tests during the
initial stages of heating but otherwise under
similar conditions of load, dielectric properties,
cavity size and so on and gave good agreement
in multimode heating applicators. This suggests
that such an approach gives sufficient insight
and adequate accuracy for assisting in the design
of microwave heating applicators [Ehlers and
Metaxas, 2003].
To validate the numerical method, a different
method is adopted whereby the reflection
coefficient is measured using a network
analyzer and compared with that derived
from the simulations. The cavity tested was of
rectangular shape of 40 cm x 30 cm, and height
35 cm. The position of the load measuring 12 cm
by 12 cm and height 4.1 cm, is offset as shown in
Figure 3, while the feed waveguide type WG9A
is placed 15.8 cm and 13.4 cm from one side of
the cavity box applicator. The load chosen for
verification purposes is rubber whose dielectric
properties were 8.3-j0.4. Reasonable correlation
was obtained between the return loss measured
M Q = 8.45528Ve
6
hi3
(10)
where Ve and hi are the tetrahedron volume
and average edge length respectively. For each
element, MQ ranges from 1 for an equilateral
tetrahedron down to very small values for
highly irregular tetrahedrons. Figure 2 shows the
average error in the first 10 resonant frequencies
versus the mesh quality for a 1 m x 0.5 m x 0.75 m
rectangular cavity. In all simulations the number
of unknowns were kept around 3500. To obtain
a different mesh quality whilst maintaining the
number of degrees of freedom the nodes in the
mesh are moved in arbitrary directions until
the desired mesh quality is obtained. Figure 2
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Journal of Microwave Power & Electromagnetic Energy
Vol. 40, No.2, 2006
memory requirement of 120 MB. It took 4 hours
and 9 hours to reach 4% and 0.5 % convergence
respectively.
A.
Figure 2. Comparison of error norms of the
first 10 resonant modes with respect to the
mesh quality. The number of degrees of freedom are kept at 3500.
using a network analyser and that computed
from the finite element method [Hallac, 2003;
Hallac and Metaxas, 2003a]. The discrepancies
were attributed to the modelling of the ABC at a
single frequency of 2.45 GHz and the fact that the
experimental setup has some additional losses
caused by geometrical imperfections that are
not accounted for numerically. Using a personal
computer with 1000 MHz Athlon processor the
model is descretised using 30000 tetrahedral
elements producing 186,118 unknowns with a
Conveyor Belt Systems
As conveyor belt systems are widely used in
industry, it is necessary to develop an ability
to model such a system including passive
components and a conveyor belt. The conveyor
belt carries the load through the input and output
ports of an industrial applicator [Metaxas and
Meredith, 1988]. First order ABCʼs are used
to truncate the ports. A continuous industrial
microwave applicator with a conveyor belt,
similar to a microwave tempering unit for
processing blocks of butter, is shown in Figure
4. Due to the complex nature of such a unit,
experimental verification has not been carried
out. The system is initially simulated in the
absence of the load and conveyor belt so as to
find out how the power propagates through the
input and output ports.
For comparison the results for a loaded
system are also presented. Figures 5 and 6
show respectively the electric field patterns
and the return loss characteristic for the
industrial conveyor applicator. It is evident
from Figure 5 that the energy dissipation within
Figure 3. Plan diagram showing test model for numerical verification and the
position of the load.
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105
b)
a)
Figure 4. Continuous microwave conveyor-belt system. (a) Side plan and 3D view
(b) Front plan and top plan. Dimensions are in centimetres.
a)
b)
Figure 5. Normalised electric field distribution at the centre of the conveyor belt (y=20 cm). The
frequency of operation is 2.45 GHz and the simulation is terminated at 1 percent convergence.
(a) Unloaded (b) Loaded
Figure 6. Return loss characteristic for
a loaded and unloaded conveyor belt
system. The frequency of operation is
2.45 GHz and the simulation is terminated at 1 percent convergence.
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Journal of Microwave Power & Electromagnetic Energy
Vol. 40, No.2, 2006
a)
b)
Figure 7. Computed field distributions at two
different cross sections in the workload. The
frequency of operation is 2.45 GHz and the
simulation is terminated at 1 percent convergence. (a) Slice at the top of the loads (b) Slice
at the conveyer belt
a)
the unloaded oven (that is, empty applicator)
is more pronounced than in the loaded cavity
when it contains a lossy material. Furthermore,
as expected and clearly depicted in Figure 6,
the bandwidth of an individual response in the
loaded case is wider as the effective Q-factor
of the system decreases due to the increased
loss. Sharper resonances are characteristic of
unloaded systems.
For the simulation with the load and
conveyor belt in situ, the conveyor belt needs
to be transparent to the microwave energy,
so polypropylene is used, having a relative
permittivity of 2.55-j0.0008 at 2.45 GHz. A
material with such a low loss factor allows the
energy to also enter the workload from below.
For the load, a butter-like material is used which
has a relative permittivity of 4.05-j0.39 at 2.45
GHz in the temperature range -20 to –2oC, a
value which remains nearly constant throughout
this temperature range. Figures 7 and 8 show the
field and power distributions at different cross
sections respectively. The numerical simulations
are carried out using 170,944 tetrahedral
elements, with an average edge length of 3.5
cm, resulting in 1,066180 degrees of freedom
and a memory requirement of 600 MB. If the
model were to be discretised using first order
tetrahedral elements with an average edge length
of 1 cm, then up to 2 million unknowns would
have to be solved for.
CONCLUSIONS
b)
Figure 8. Computed power distributions at two
different cross sections in the workload. The frequency of operation is 2.45 GHz and the simulation is terminated at 1 percent convergence. (a)
Slice on upper surface of butter loads (b) Slice
at y=12.5 cm
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In this paper, we have presented a three
dimensional time domain TVFE method using
second order vector finite element bases for
the numerical modelling of microwave heating
applicators. The comparison between first and
second order TVFE show that higher order
elements require less computer resources,
especially as the problem size increases. One
reason for the superiority of higher order
elements is that errors are kept low even for
107
highly distorted meshes that have to be used
for some realistic system geometries. The
method then presents another promising tool
for modelling large industrial systems.
ACKNOWLEDGEMENTS
The authors wish to acknowledge the many
helpful discussions they held with Drs Wai
Fu, Richard Ehlers and other members of the
Electricity Utilization Group, Engineering
Department at the University of Cambridge,
UK.
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