Large Scale FD-TD, A PSCI project
Note:This project has now been concluded.
It has been succeeded by the
- Ulf Andersson, C2M2/Nada, firstname.lastname@example.org
- Gunnar Ledfelt, C2M2/Nada, email@example.com
- Ulf Thibblin, Ericsson Saab Avionics, Ulf.Thibblin@esavionics.se
The goal of this PSCI-project is to develop methods for numerical
simulation of electromagnetic phenomena in complex systems, for
instance an aircraft. Within the time frame of this project a 3D
Finite-Difference Time-Domain (FD-TD) method will be implemented on the
IBM SP-2, Strindberg, at the Center for Parallel Computers
(PDC) at KTH. In parallel, new alternative types of time domain
methods, such as finite volume or finite element approaches on
unstructured grids, will be considered for the future. Small
geometrical details and fast field variations, compared with the
exciting wavelength, can be resolved by an unstructured grid. The
memory size and speed of todays computers makes it necessary for the
traditional FD-TD method to introduce subcell models to model thin
structures (thin compared with the overall resolution). Examples of
such structures are wires, gaps, coatings, and semiconducting
walls. The project will also study such subcell models as well as
developing new ones. Particularly it will study the possibility to use
Wavelet homogenization as a general technique to produce subcell
Electromagnetic fields are governed by the Maxwell equations, a first
order hyperbolic system of PDE:s. These can not be solved analytically
except for a few simple geometries, such as spheres and cylinders. For
more complex structures one has to rely on numerical methods, i.e.
Computational Electromagnetics (CEM), and/or experiments.
There is a wide range of applications for CEM. Some of the more
The methods developed in this project are general and will have impact
on all the CEM areas mentioned above. They are well suited for a
modern industrial computational environment. There automatic
geometrical input and meshing from a CAD description of the object and
advanced data visualization are necessary both from a feasibility and
cost effectiveness point of view.
- Electromagnetic compatibility (EMC).
- Antenna analysis and synthesis.
- Radar cross section (RCS) calculations.
- Cellular phone-human body interaction.
- Microwave ovens.
- Target recognition.
- Hybrid/monolithic microwave integrated circuits.
The main interests of the industrial participants Ericsson Space
Avionics, FOA and SP are EMC, RCS and antenna analysis. Regarding EMC
there are several threats that an aircraft has to be verified against
where some examples are lightning, radio and radar fields,
electrostatic discharge, fields emitted by the electronic equipment
and directed high power microwave weapons. All these threats can, if
the electronic equipment of the aircraft is not sufficiently
protected, disturb or damage the electronic equipment causing EMC
problems which would jeopardize flight safety. A well known example is
the prohibition of using cellular phones during take off and landing.
The position of the antennas on an aircraft has to be optimized so
that the antenna-antenna coupling is minimized and the desired
coverage is achieved. Minimizing the RCS will make an aircraft harder
to detect by radar.
The Yee Method
In 1966 an FD-TD method for the Maxwell equations was introduced by
K. S. Yee. This method uses a Leapfrog scheme on staggered Cartesian
grids. Calculating the electromagnetic field outside an object usually
leads to an open problem, i.e. an infinite computational
domain. Therefore, some artificial boundary conditions (ABC) is needed
to truncate the computational domain. These boundary conditions must
be such that they minimize the reflection of waves trying to leave the
computational domain. One common way is to use the so called Mur
boundary condition. These are the electromagnetic version of the
Engquist-Majda boundary condition. A more modern approach is to use
the Perfectly Matched Layer (PML) introduced by Berenger in
1994. Incident plane waves can be generated by Huygens' surfaces.
The Yee method has several advantages. It is robust, fast and
simple to understand. Furthermore it is possible to achieve the
response in a chosen frequency band in one calculation by using a
pulse excitation. This can not be achieved with a frequency domain
method. On the other hand the Cartesian grid conforms badly to the
real geometry, thus introducing so called stair stepping
errors. Furthermore, the absence of a general subgridding scheme means
that structures smaller than the resolution have to be treated by
subcell models. These models are limited to relatively simple
geometrical structures such as wires, gaps, layers, etc. In spite of
these disadvantages we believe that the Yee method will remain in use
for many years. Therefore the subcell models have to be improved and
new subcell models must be developed. Some of the areas in need of
(better) subcell models are:
The project will also consider the possibility to use wavelet
homogenization as a general subgridding scheme.
- thin wires. Todays models make it necessary to put
the wires along edges of the Cartesian grid. This means that thin
wires close to surfaces must be placed one cell above the surface
which may not be good enough. Models for arbitrarily placed thin
wires are therefore desired.
- losses in thin layers. Good models for the frequency domain
exists, but how shall it be done in the time domain?
- frequency dispersive materials.
- thin slots. Lids for electromagnetic equipment is not
electromagneticly dense and they can not be resolved.
- bundled wires.
The project is also interested in many other topics related to
time-domain methods. Some of these are related to the Yee method, but
others are entirely new methods. Such topics are:
- Hybrid methods. Unstructured local body fitted grids inserted into the
global Cartesian grid. The main issue here is to construct the interface
so that stability and second order accuracy are maintained.
- Higher order methods. The Yee method is second order
accurate in both time and space. Higher order methods will reduce the
dispersion error but is computationally more expensive per point and
time step and they also demand higher order in the boundary conditions.
- Outer boundary conditions, i.e. absorbing boundary
condition. The recently introduced Perfectly Matched Layer (PML)
algorithm has been shown to be superior to classical absorbing boundary
conditions, such as the Mur boundary condition.
- Semi explicit methods. Using implicit methods in local fine
grained grids and explicit methods in the coarse grained grids will
preserve a relatively large time step.
- Signal identification. When has an FD-TD calculation reached its
time harmonic state? What is an appropriate measure of when an FD-TD
calculation reaches its time harmonic state?
- Semi empirical methods.
High Frequency Methods
When using FD-TD methods one has to resolve the waves with at least 10
points per wavelength and sometimes with as many as 30 points per
wavelength. Memory size and execution times increases with frequency
and with the ``electrical size'' of the object. With ``electrical
size'' we mean the quotient between the physical size and the
wavelength. FD-TD methods can not be used for arbitrarily large
objects. One must instead use high frequency methods such as
Geometrical Theory of Diffraction (GTD). The CEM program within PSCI
is also interested in looking at such methods.
The following persons was involved in the project during its later stages:
- Erik Abenius, Graduate student, C2M2/Nada, KTH.
- Ulf Andersson, Graduate student, C2M2/Nada, KTH.
- Mats Bäckström, Director of Research, Electromagnetic Effects, FOA, Linköping.
- Jan Carlsson, Researcher (EMC), Swedish National Testing and Research Institute. (SP)
- Björn Engquist, Professor in Numerical Analysis, C2M2/Nada, KTH
- Stefan Hagdahl, Graduate student, C2M2/Nada, KTH.
- Gunnar Ledfelt, Graduate student, C2M2/Nada, KTH.
- Thorleif Martin, Research Officer, FOA, Linköping.
- Olof Runborg, Graduate student, C2M2/Nada, KTH.
- Bo Strand, CEM Program director, C2M2/Nada, KTH.
- Ulf Thibblin, Ericsson Saab Avionics.
- Implementation of a Yee code on Strindberg.
- Deliveries: A parallel platform code, pscyee, using
the Yee method with Mur's first order boundary condition and
Huygens' surfaces for wave excitation has been implemented on
PDC:s IBM SP-2.
- Responsible: Ulf Andersson and Gunnar Ledfelt.
- Short description: The Yee method uses structured grids
and was therefore relatively easy to parallelize. MPI was
used since it is more efficient than PVM and more portable than
MPL. The code is written in Fortran 90.
- Studies of Wavelet homogenization.
- Deliveries: PSCI Report 3: Wavelet-Based Subgrid Modeling:
1. Principles and Scalar Equations
- Authors: Ulf Andersson, Björn Engquist, Gunnar Ledfelt and Olof Runborg.
A systematic technique for the derivation of subgrid scale models
in the numerical solution of partial differential equations, is described.
The technique is based on wavelet projections of the discrete
operator followed by a sparse approximation. The resulting numerical
method will accurately represent subgrid scale phenomena on a
coarse grid. Applications to numerical homogenization and wave
propagation in materials with subgrid inhomogeneities are presented.
- Studies of subcell models.
- Deliveries: PSCI Report: Subcell Models in the Finite
Difference Time Domain Method for the Maxwell Equations.
- Responsible: Erik Abenius and Gunnar Ledfelt
- Short description: This was carried out as a master thesis
by Erik Abenius with Gunnar Ledfelt as supervisor. Abstract
from the report:
Electromagnetic problems are described by the Maxwell equations
which have been successfully solved numerically since the early days
of computers. The report discusses algorithms for treating
geometrically small objects in the Finite-Difference Time-Domain
method, FD-TD. Existing models for thin wires, narrow slots and thin
sheets have been implemented and tested for a variety of geometries.
Particularly the thin wire model works very well.
- Implementation of PML.
- Deliveries: Master thesis report, TRITA-NA-E9755:
PML as Absorbing Boundary
Condition in FDTD-methods for the Maxwell Equations.
- Responsible: Sina Moghaddassi and Ulf Andersson
- Short description: This was carried out as a master thesis
by Sina Moghaddassi with Ulf Andersson as supervisor. Abstract
from the report:
Free space simulation of the Maxwell equations requires some kind of
Absorbing Boundary Conditions (ABC) to truncate the computational
grid. In this work, the Perfectly Matched Layer (PML) was implemented
as absorbing boundary condition for the traditional Yee
FDTD-scheme. The results were compared with the existing Mur ABC and a
numerical reference solution which was simulated by using a much
larger domain of calculation. The results computed with the PML
technique were much better than the results with the MUR ABC.
- An FD-TD calculation with one billion cells.
- Deliveries: The results are described on a separate
- Responsible: Ulf Andersson and Gunnar Ledfelt
- Short description: A calculation simulating an
lightning strike on the Saab 2000 aircraft with one billion cells was
performed using the IBM SP at PDC. The grid of the Saab 2000 was
supplied by Ericsson Saab Avionics. The results was first demonstrated at
SC98 in Orlando Florida.
This project have now been concluded. It has been succeeded by the
project. The GEMS project began 1998-01-01. These two projects ran in
parallel for a while before it was decided to conclude the Large Scale FD-TD
Updates of this document
- 1995/09/13 First version.
- 1996/03/04 Second version. Updates on Results and deliveries.
- 1996/05/29 Third version. General update.
- 1998/03/0X Fourth version. Mainly updates on Results.
- 1999/07/09 Draft for Final version.
Up to PSCI homepage.
Responsible for this page: <firstname.lastname@example.org>
Content last modified Fri Jul 09, 1999.
(Grammatical error corrected Fri May 19, 2000)
Technical support: <email@example.com>