A Finite Element Mesh Truncation Technique for
Scattering and Radiation Problems in HPC
Environments
Adrian Amor-Martin1 , Daniel Garcia-Doñoro2 , and Luis Emilio Garcia-Castillo1
1
Department of Signal Theory and Communications, Leganes, 28911, Spain
2
Xidian University, Xi’an, China
[email protected], [email protected], [email protected]
Abstract—An in-house electromagnetic code based on the finite
element method (FEM) called HOFEM, a higher order finite
element method, has been developed by the authors in the last
years. HOFEM has been developed to work in high-performance
computing (HPC) environments. Among other features, it makes
use of a non-standard mesh truncation technique for open region
wave propagation problems as those of scattering and radiation.
This mesh truncation technique works in an iterative fashion
providing an asymptotically exact absorbing boundary condition
while minimizing the size of the FEM domain and preserving the
original sparse nature of the finite element matrices. Thus, it is
suitable for large scale computations. Numerical results obtained
on an HPC system corresponding to the scattering analysis of
the radar cross section of a military helicopter are shown.
II. M ESH T RUNCATION T ECHNIQUE
I. I NTRODUCTION
If an integral equation (IE) representation of the exterior
field is used, a non-standard matrix system of equations,
partially sparse and partially dense is obtained. Instead of this
straightforward approach, with FE-IIEE the electromagnetic
problem is rather reformulated as a two domain decomposition
problem by dividing the original infinite domain into two
overlapping domains limited by a finite FEM domain bounded
by a surface S and by an infinite domain exterior to an
auxiliary boundary S 0 . On the surface S, a Cauchy type
boundary condition is used as
−1
n̂ × f¯r ∇ × V + γ n̂ × n̂ × V = Ψ over S, (1)
An in-house electromagnetic code based on the finite element method (FEM) called HOFEM, a higher order finite
element method, has been developed by the authors in the
last years [1]. The electromagnetic kernel has already been
verified in a systematic way, [2]. The code has been written
from scratch to adopt high-performance computing (HPC)
paradigms such as message passing interface (MPI) and open
multi-processing (OpenMP) in order to use HPC environments in an efficient way. The language used for HOFEM
is FORTRAN 2003 making use of some features of object
oriented programming paradigm with a strong emphasis on
code maintenance. HOFEM is currently used for research purposes in problems related to the modeling of electromagnetic
wave propagation phenomena, e.g., analysis of waveguides,
microwave passive devices, antenna radiation, radar cross
section (RCS) prediction and so on.
For open region problems, a mesh truncation technique
called finite element iterative integral equation evaluation (FEIIEE) [3], is implemented. This mesh truncation technique
works in an iterative fashion providing an asymptotically exact
absorbing boundary condition while minimizing the size of
the FEM domain and preserving the original sparse nature of
the finite element matrices. Thus, it is suitable for large scale
computations. Details of FE-IIEE are given next in Section II.
Numerical results of a scattering problem are presented in
Section III. Finally, conclusions are given in Section IV.
where V stands for the electromagnetic field (either E or
−1
H depending on the formulation used), f¯r
is a tensor
representing the appropriate material property (electric or magnetic permeability) and γ is the complex propagation constant
(typically, the one for free space). Symbol Ψ represents a
vector right hand side function.
When S is placed very far from the sources (ideally at
infinite distance from the sources) Ψ is either null when the
sources are internal (radiation problem) or analytically known
when the sources are external (e.g., RCS prediction). When S
is getting closer to the sources Ψ is no longer known. FE-IIEE
provides an iterative methodology to estimate Ψ by using an
integral equation representation of the exterior field from the
equivalent currents given by FEM on the auxiliary boundary
S0.
Thus, this technique provides an asymptotically exact absorbing boundary condition while hiding the hybrid nature
(sparse and dense) of the standard matrix system of equations
provided by using an integral equation representation of the
exterior field as boundary condition to FEM. By using two
different boundary surfaces singularities in the computation
of the matrix IE coefficients are avoided. Furthermore, the
overlapping between the domains provides a right characterization of the scattered/radiated evanescent modes and a
nice monotonically convergence behavior is observed, [4]. The
rate of convergence of the method depends on the amount
of overlapping; the larger the overlapping the faster the convergence. However, the size of the computational domain is
also increased, and hence, the number of unknowns. Also, the
rate of convergence is changed by the specific physics of the
problem.
A nice feature of FE-IIEE is that it provides a numerical
boundary condition with a prescribed level of accuracy (set by
the user) independently of the distance between the exterior
boundary and the sources of the problem (antenna or RCS
target). That is, if the mesh is truncated very close to the
sources of the problem with a standard method the numerical
solution would be limited by the quality of the radiation
boundary condition provided. However, with FE-IIEE it would
simply take a few more iterations to reach the prescribed
level of accuracy. Obviously, for a given problem, there is an
optimum location/distance of the exterior truncation boundary
in terms of minimizing the computer resources (either CPU of
memory).
Fig. 2. Electromagnetic field on Apache helicopter at 100 MHz.
III. N UMERICAL R ESULTS
Numerical results of a bistatic RCS calculation at 100
MHz of an Apache helicopter are presented. The helicopter
model is detailed with 1,751,478 tetrahedra and 11,342,475
unknowns. Note that the blades of the helicopter are modeled
as dielectrics. The distance between the exterior boundary S
and auxiliary boundary S 0 is equal to 0.20λ with λ denoting
the free space wavelength. The number of FE-IIEE iterations
is 4 for a relative error threshold of 10−4 . The simulation
time is 22 minutes, using 46 compute nodes with 24 cores
each. The parallel configuration chosen is 2 MPI processes
on each node with 12 OpenMP threads for each process. The
total RAM used in this problem is 502 GB. Model and RCS
results can be found in Figs. 1–3.
Fig. 3. Bistatic RCS of Apache helicopter at 100 MHz.
IV. C ONCLUSIONS
The use and features of a non-standard mesh truncation
technique for FEM wave propagation open region problems
within an electromagnetics code operating in HPC environments has been illustrated.
ACKNOWLEDGMENT
This work has been financially supported by TEC201347753-C3, TEC2016-80386-P, CAM S2013/ICE-3004 projects
and “Ayudas para contratos predoctorales de Formación del
Profesorado Universitario FPU”.
R EFERENCES
Fig. 1. Mesh of Apache helicopter for 100 MHz.
[1] D. Garcia-Doñoro, I. Martinez-Fernandez, L. E. Garcia-Castillo, and
M. Salazar-Palma, “HOFEM: A higher order finite element method electromagnetic simulator,” 12th International Workshop on Finite Elements
for Microwave Engineering, Mount Qingcheng, Chendu, China, May
2014.
[2] D. Garcia-Doñoro, L. E. Garcı́a-Castillo, and S. W. Ting, “Verification
process of finite-element method code for electromagnetics: Using the
method of manufactured solutions,” IEEE Antennas Propag. Mag.,
vol. 7, no. 2, pp. 28–38, Apr. 2016.
[3] I. Gómez-Revuelto, L. E. Garcı́a-Castillo, M. Salazar-Palma, and
T. K. Sarkar, “Fully coupled hybrid-method FEM/high-frequency technique for the analysis of 3D scattering and radiation problems,” Microw. Opt. Tech. Lett., vol. 47, no. 2, pp. 104–107, Oct. 2005.
[4] R. Fernández-Recio, L. E. Garcia-Castillo, S. L. Romano, and I. GómezRevuelto, “Convergence study of a non-standard Schwarz domain decomposition method for finite element mesh truncation in electromagnetics,” Progress In Electromagnetics Research (PIER), vol. 120, pp. 439–
457, 2011.