An electromagnetic solver comparison for various use cases of antennas mounted on vehicles is presented. For this purpose, several modeling approaches, called transient, frequency and integral solver, including the features fast resonant method and autoregressive filter, offered by CST MWS, are investigated. The solvers and methods are compared for a roof antenna itself, a simplified vehicle, a roof including a panorama window and a combination of antenna and vehicle. With these examples, the influence of different materials, data formats and parameters such as size and complexity are investigated. Also, the necessary configurations for the mesh and the solvers are described.

For solving electromagnetic problems in complex environments, the choice of
the most appropriate method does not only determine the time efficiency, but
has an influence on the accuracy of the gained results, as well. There is not
a single combination of a numerical method and algebraic solver, in the
following called solver, which can fulfill all requirements. Moreover, many
parameters, such as size in relation to wavelength, complexity and resonating
behavior must be considered. In the following, the transient (T), the
frequency (F) and the integral (I) solver offered in CST MWS

The T solver is based on the finite integration technique. The geometrical
model is here divided into hexahedra

The F solver uses the finite element method. A limit for this method is the
availability of random access memory (RAM) which is used mainly dependent on
the number of mesh cells. The resulting matrix is sparsely populated as
elements are only non-zero if nodes in the discretized geometry are
neighboring. The numerical system size can be reduced by a model order
reduction technique (MOR)

The I solver uses the Method of Moments. As only the surface must be meshed, the method is well suitable for large solution domains. Dielectrics are not meshed for this solver method in CST MWS.

In the following, the mentioned solvers and methods are investigated in order to simulate a complex roof antenna mounted on a vehicle accurately and efficiently. Therefore, several simulations with the roof antenna itself are conducted. In a second step, the solvers are compared for the purpose of simulating extended simulation domains as vehicles and roofs. In these models, monopoles are used as simplified antennas in order to isolate the problems from each other. Finally, a vehicle including the roof antenna is simulated. Also, the influence of data formats and materials is taken into consideration. The values of interest for the feasibility and efficiency of a simulation are majorly the RAM and time consumption. Especially the time consumption is only a rough value. The benchmark computer has 2 processors of the type Intel(R) Xeon(R) CPU with E5640@2.67 GHz and 24 GB RAM. Each of the processors consists of 4 cores and moreover the Intel(R) Hyper-Threading Technology is enabled. Some simulations could not be performed on this computer so the necessary time was estimated. Simulations for the purpose of time comparability were started in order to estimate the differences in computation speed. Finally, the given times can be seen as benchmarks.

The first part of the investigation is a roof antenna itself. The antenna
designed for the north American market consists of a SDARS patch, a GPS patch
and a telephone antenna, which are contained in one antenna assembly as shown
in Fig.

Photograph of the antenna structures.

Photograph of the antenna plastic cap.

Services joined within one antenna assembly.

All solvers explained above, except the I solver, are evaluated for the
antenna model and finally compared to measurement results. All connections
are modeled as coaxial structures. The antenna needs to be slightly modified
for each solver. For the transient solver all ports were implemented as
perfect conducting wires between two points realizing a source, called edge
ports, whereas with the frequency solver a face is used instead of a thin
wire, called discrete face ports, were used. This port modification does not
relevantly change the simulation behavior as the results for the Global
Positioning System (GPS) and Satellite Digital Audio Radio Services (SDARS)
antennas do correspond well to each other.
For the simulations with the T solver, the antenna is meshed using hexahedra
as shown in Fig.

Hexahedral mesh.

Tetrahedral mesh.

The simulated and measured reflection parameters of the SDARS antenna are
shown in Fig.

Comparison of different solver for the roof antenna.

For useful investigations with the T solver, the AR filter is necessary. Once some experience with the meshing of the structure could be achieved, the most efficient simulations still can be undertaken with the F solver. A further advantage of the F solver is the fact that single frequencies can be simulated at frequency points of interest after the solver run has finished without performing adaptive meshing.

Vehicles feature an extended and at the same time complex environment which
strongly influences the far field patterns of roof antennas. A common data
format for vehicles is the Computer Aided Three-Dimensional Interactive
Application (CATIA) format. In this format, every detail is included and the
total amount of data is by far too extensive for the import into
electromagnetic field solver programs. To reduce the amount of data and for
reasons of compatibility the data is simplified to Nasa Structural Analysis
System (NASTRAN) data, in which the surface is represented by triangles as
shown in Fig.

Nastran mesh of a vehicle.

Far field simulation results at 2 GHz for T, F and I solver with different accuracies.

The simulation with the T solver is carried out in a frequency range from 1
to 2.5 GHz and the impulse is propagated through the structure until the
energy level decreased to

Meshing of NASTRAN structure with triangles.

The time efficiency of the solvers is dependent on the number of frequencies of interest. In case only one frequency point is investigated, the I solver is faster than the T solver. As soon as scattering parameters should be simulated at the same time, a larger bandwidth is necessary for reasonable investigations and the time consumption with the I solver will increase. Additionally it must be considered that windows are important for the far field behavior which were not considered in the I solver as they are dielectrics.

Comparison of different solvers for the monopole on a metallic vehicle modeled in NASTRAN.

For the I and F solver a reduction of the overall model by deleting parts
which do not influence the far field patterns, brings advantages as there are
less triangles. With the T solver this effect is less distinctive because the
whole box including air is meshed. For this reason, in the following only the
roof is taken into consideration.
Another vehicle model had to be used for the roof comparisons. Usually
vehicle models are prepared in NASTRAN format at AUDI AG. The disadvantage of
the NASTRAN format in CST MWS is that the mesh gets unnecessary fine as the
triangles cannot be loaded as the mesh itself but are meshed a second time as
shown in Fig.

Roof with panorama glass window and monopole as simplification of an antenna.

Far field patterns at 2 GHz simulated with T and with F solver.

In Table

Simulated far field results at 2 GHz comparing the simulation using a roof model in CATIA and NASTRAN format.

The differences between the two plots can be explained by the deviations of
the models which result from the conversion to NASTRAN. The problem changes
when dielectrics as glass are introduced. For that a rectangular glass window
is introduced into the roof as shown in Fig.

Comparison of different solvers for the monopole on a roof and influence of a panorama glass window.

The results of the simulated far field patterns correspond to typically
observed results. A typical effect with panorama windows is the damping of
the far field in the horizontal direction

The far field pattern is, in contrast to the scattering parameters, not only
dependent on a small area around the antenna. This is why a simulation of the
model including the vehicle from Fig.

With these settings the same reflection parameters as with the T solver in
Figs.

In this paper, the theoretical background of the F, T and I solver and their accuracy and efficiency for a roof antenna mounted on a vehicle were discussed. The results show that the choice of the solver is not only dependent on the structure of the simulation domain, but also on the demanded results. The scattering parameters are more dependent on the structure itself, whilst the far field is strongly dependent on the environment. For the simulation of the roof antenna itself the T solver under usage of the AR filter and the F solver give good results whereas the vehicle is most efficiently simulated using the T solver, especially in case it contains dielectrics as glass. For this reason, the roof antenna including the vehicle was simulated with the T solver using the AR filter. The meshing of both the vehicle and the antenna works out the best when importing the data in CATIA format. The scattering parameters were validated with measurements and the far field patterns agreed with experiences from similar measurements. By comparing the different ways of simulations, an efficient way for investigating further antenna systems concerning scattering parameters as well as far field patterns could be described.

The authors wish to thank AUDI AG for providing CAD Data which are used in the simulation models and for the measurement data which serve to validate the simulation results. Also, a special thanks goes to the company CST AG for parts of the investigations and the simulation support.Edited by: R. Schuhmann Reviewed by: S. Lindenmeier and one anonymous referee