Hindawi Publishing Corporation
Journal of Computational Engineering
Volume 2014, Article ID 634269, 8 pages
http://dx.doi.org/10.1155/2014/634269

Research Article
FDTD Acceleration for Cylindrical Resonator
Design Based on the Hybrid of Single and Double Precision
Floating-Point Computation
Hasitha Muthumala Waidyasooriya, Masanori Hariyama,
Yasuhiro Takei, and Michitaka Kameyama
Graduate School of Information Sciences, Tohoku University, Aoba 6-6-05, Aramaki, Aoba, Sendai, Miyagi 980-8579, Japan
Correspondence should be addressed to Hasitha Muthumala Waidyasooriya; hasitha@ecei.tohoku.ac.jp
Received 25 June 2014; Accepted 18 November 2014; Published 4 December 2014
Academic Editor: Fu-Yun Zhao
Copyright © 2014 Hasitha Muthumala Waidyasooriya et al. This is an open access article distributed under the Creative Commons
Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is
properly cited.
Acceleration of FDTD (finite-difference time-domain) is very important for the fields such as computational electromagnetic
simulation. We consider the FDTD simulation model of cylindrical resonator design that requires double precision floating-pointand cannot be done using single precision. Conventional FDTD acceleration methods have a common problem of memorybandwidth limitation due to the large amount of parallel data access. To overcome this problem, we propose a hybrid of single
and double precision floating-point computation method that reduces the data-transfer amount. We analyze the characteristicsof the FDTD simulation to find out when we can use single precision instead of double precision. According to the experimental
results, we achieved over 15 times of speed-up compared to the CPU single-core implementation and over 1.52 times of speed-up
compared to the conventional GPU-based implementation.

1. Introduction
Computational electromagnetic simulation shows a rapid
development recently due to the introduction of processorsthat have parallel processing capability such as multicore
CPUs and GPUs (graphic processing units). The FDTD
(finite-difference time-domain) algorithm [1, 2] is one of
the most popular methods of computational electromagnetic
simulations due to its simplicity and very high computational
efficiency. It has been widely used in many applications suchas coil modeling [3] and resonance characteristics analysis ofa cylindrical cavity [4, 5]. Many of these applications require
double precision floating-point computation to satisfy thestability condition [6].
The FDTD simulation requires a large amount of data.
When more processor cores are used in parallel, more datatransfers occur between the memory and the processor
cores. Therefore, the memory-bandwidth limitation is amajor problem in the FDTD simulation using computers. To
overcome this problem, we have to reduce the data transfer

amount, so that we can use more cores in parallel. To do this,
we propose a hybrid precision computation method that uses
both single and double precision. Single precision data use
4 bytes compared to 8 bytes used in double precision data.
Therefore, using single precision reduces the data amountand increases the processing speed. However, using single
precision could bring inaccurate results. In some cases, the
FDTD simulation does not converge when it is executed fora large number of iterations.
In this paper, we consider the FDTD simulation of acylindrical resonator [5]. It is one of the most fundamental
types of resonant cavities and has been used to construct,
for example, wavelength filters for microwaves [7, 8]. It also
has a number of important applications in the lightwave field,
such as couplers and laser cavities [9–11]. In such applications,the quality factor (Q) of a cylindrical cavity, which is a basic
characteristic, depends on the performance of the cavity
walls. The fundamental characteristics, such as resonancewavelength, Q factor, and modal fields, are calculated bynumerical simulation. The same simulation is executed many

2

Journal of Computational Engineering

times by changing the parameters such as the radius of thecavity and the thickness and the depth of the cavity wall.
Therefore, the processing time of the simulation is very large.
This paper proposes a hybrid of single and double precision computation method to reduce the processing time of
the FDTD simulation of a cylindrical resonator. The proposed
method can be used in both multicore CPUs and GPU
accelerators. We analyze the characteristics of the application
to find out where we can use single precision instead of
double precision. According to the experimental results, weachieved 1.41 and 5.19 times of speed-up, respectively, for CPU
and GPU implementations, compared to the conventional
double precision implementation using 12 threads and 6 CPU
cores. Compared to the conventional GPU implementation,
the proposed hybrid precision method using GPU has over
1.52 times of speed-up. Using the proposed method, we canextract more performance from the same hardware without
any additional overhead.

RBC𝜌

OutsideΛ

2ad0r
InsideCross section of
the cavity (ring)

PBC
RBC

Figure 1: Cross section of a cylindrical cavity.

RBCk

2. The FDTD Simulation of
Cylindrical Resonator
A cylindrical cavity surrounded by a curved resonant gratingwall is proposed in [5]. Its resonance characteristics aredescribed using the FDTD simulation. In this section, we
briefly explain how the FDTD simulation is performed
for this application. Figure 1 shows a cross section of thecylindrical cavity which is similar to a ring. The cavity wall
has a curved resonant grating. PBC and RBC stand for
periodic and radiation boundary conditions, respectively.
The depth and the base thickness of the grating are 2𝑎 and𝑑0 , respectively. The pitch of the grating is Λ. When the freespace wavelength is 𝜆 0 , 𝜆 0 ≈ 2Λ. The FDTD simulation is
performed by simplifying the structure as a two-dimensional(2-D) one. Figure 2 shows the 2-D grid for the simulation.
The polar coordinates of the computation area in Figure 1
are transformed to a 2-D grid of orthogonal coordinates.
Note that the boundary conditions are calculated separately
and a small portion of the area inside the ring is calculatedusing 1-D FDTD. The rest of the area is calculated using 2D FDTD simulation. According to the experimental results,
this simulation does not converge when using only single
precision floating-point computation.
The flow-chart of the FDTD simulation is shown in
Figure 3. The total number of iterations and the iteration
number are given by 𝐼tot and 𝑛, respectively. The electric andmagnetic fields are computed one after the other. In between
these computations, the boundary conditions are applied.
To design a cylindrical resonator for a particular resonancewavelength, the FTDT simulation is executed many times bychanging the parameters such as the radius of the cavity andthe thickness and the depth of the grated wall. This requires a
lot of processing time.
There are many recent works such as [12–16] that use
GPUs to accelerate FDTD. GPU acceleration of 2-D and 3D FDTD for electromagnetic field analysis is proposed in[12, 13], respectively. Multi-GPUs are used to accelerate the
FDTD in [14, 15, 17]. Although [15] gives good results forOn the ringi

PBC

InsideOutsideOn the ring

Figure 2: Computation grid for the FDTD simulation.

Start: n = 0

Electric field computation

Periodic boundary condition

Magnetic field computation

Periodic boundary conditionn > Itot

No
Yes
FinishFigure 3: Flow chart of the FDTD computation.

Journal of Computational Engineering
few GPUs, the communication overhead between GPUs via
host memory could be a serious bottleneck for a systemwith a large number of GPUs. Periodical data exchange
between GPU and CPU is proposed in [16] to overcome
the GPU memory limitation. However, it comes with a 10%
performance loss. Moreover, recent GPUs have GPU-to-GPU
data transfer capability so that multi-GPU method can bebetter than this approach. The speed-up of these methods
comes with an additional hardware cost.
In this paper, our approach is different from the previous
works. We focus on increasing the throughput of the GPUs
without adding extra hardware. We propose a method to
reduce the processing time of the FDTD simulation byusing a hybrid of single and double precision floating-pointcomputation. We evaluate using both multicore CPU andGPU to show the effectiveness of our method. Moreover,the proposed method can be used together with most of theprevious works to increase the speed-up further.

3. Hybrid of Single and Double Precision
Floating-Point Computation
3.1. Hybrid Precision Computation. Since the FDTD simulation usually requires a large amount of data, its processing
speed is decided by the memory-bandwidth. Our idea inthis paper is to reduce the data amount by using more single precision floating-point computations instead of double
precision. Double precision requires 64-bit data while thesingle precision requires only 32-bit data. Therefore, the dataamount can be reduced by half if only the single precision isused. However, some areas on the computation domain havebig fluctuations of the electromagnetic field so that the single
precision cannot be used. Therefore, we use single precisioncomputation for the area where the fluctuation of the fieldis very low. We use double precision on the rest of the area.
We set a partition boundary on the computation domainthat separates the area of single precision computation andthe double precision computation. The optimal position of
the partition boundary in hybrid precision computation is
the grid coordinates that separate the single and double
precision computation area in such a way that the processing
time is minimized, under which the condition of simulation
converges. Note that we assume the simulation converges
when using double precision computations. The amount offluctuations of the electromagnetic field depends on thesimulation model and it could only be found by doing thesimulation. Therefore, the optimal partition boundary is very
hard to determine theoretically.
Due to this problem, we propose a practical approach
to partition the computation domain to achieve a sufficient
processing time reduction. This practical approach depends
on two factors. The first one is that the electromagnetic field
reduces exponentially with the distance from its source. It
is proportional to 𝑒−𝛼𝜌 , where 𝜌 is the distance from thecenter of the ring (the source). The term 𝛼 depends on thegrating thickness Λ of the cylindrical cavity (refer to [5]). The
second one is that the partition boundary is placed outside
the ring (or cavity wall). Since the resonator is designed to3kDouble precisionr + 2𝜆0

Single precisionirPartitionboundary
Cross section of
the cavity (ring)(a) Partition boundaryon the ring

Inside On the ring

Outside(b) Partition boundary on the grid

Figure 4: Partitioning of the computation domain.

preserve the field of a certain wavelength, a strong field could
exist in the cavity and computing it using single precision
may not be possible. Considering these two factors, we
conduct experiments and observed the electromagnetic field
at different distances from the center. According to theseexperimental results, we found that the field measured 𝑟 +4Λ away from the center is very weak and has very small
fluctuations. Since 𝜆 0 ≈ 2Λ, the partition boundary is givenby (1), where 𝑟 is the radius of the ring as shown in Figure 4(a):Partition boundary = 𝑟 + 2𝜆 0 .

(1)

The partition boundary on the grid is shown by Figure 4(b).
It is calculated by converting the polar coordinates of thepartition boundary to the orthogonal coordinates of the grid.
We know that the grid-points away from the partition
boundary have small fluctuations so that the computations
can be done by single precision floating-point computation.
However, we do not know the degree of the fluctuations of theelectromagnetic field on or near the cavity wall. It depends
on the parameters of the simulation model and we have todo the simulation first to find it out. Therefore, similar to theconventional method, we use double precision floating-point
for the computation for the grid-points on or near the cavity
wall and the boundaries.
Figure 5 shows the flow-chart of the proposed hybrid
precision floating-point computation. The total number of
iterations and the iteration number are given by 𝐼tot and 𝑛,
respectively. A part of the area is computed using double
precision while the rest is computed using single precision
as shown in Figure 4. In the GPU implementation, the initialand output data transfers are done by the CPU and all theother computations are done by the GPU. The single anddouble precision areas are identified by the thread-ID [18].
Moreover, this hybrid precision computation can also be usedin multicore CPUs. In the CPU implementation, single anddouble precision computation areas are identified by the gridcoordinates.

4. Evaluation
For the evaluation, we use an “Intel Xeon ES-2620” CPU and“Nvidia Tesla C2075” GPU [19]. CPU has the hyperthreading

4

Journal of Computational Engineering
Start: n = 0

Initial data transfer

For the threads done insingle precision

For the threads done indouble precision

Check thread ID

Electric field computationin single precision

Electric field computationin double precision

Single to double data conversion on partition boundary
Periodic boundary condition
For the threads done insingle precision

For the threads done indouble precision

Check thread ID

Magnetic field computation
single precision

Magnetic field computation
double precision

Single to double data conversion on partition boundary
Periodic boundary condition

n=n+1

Done in GPU

No

n > Itot

Yes
Output data transfer
FinishFigure 5: Hybrid single/double precision computation on GPU.

technology with 6 physical cores. The detailed specifications
are given in Table 1. We use gcc compiler and CUDA 4.2 [18]for the compilation. GPU is programmed by CUDA which isbasically the “C language” with GPU extensions. A detailed
description of CUDA architecture and its programming
environment are given in [18, 20, 21]. The specifications of the
GPU and CPU used for the evaluation are given in Table 1.
Figure 6 shows the electromagnetic field fluctuations
against time (iteration) and area. Figure 6(a) shows theelectric field at the distance from the center of the ring after
100,000 iterations. The field on and near the cavity wall isstrong while the field outside the cavity is weak. We observed
similar characteristics from the magnetic field data analysis
as well. Figure 6(b) shows the fluctuations of the electricfield against the number of iterations. We plot the electricfield values of three points: one point inside the ring, oneon the ring, and the other outside the ring. The electricfield inside the ring remains quite strong while the fieldoutside the ring gets weaker with the time. According toTable 1: Specifications of the evaluation environment.

Type
Frequency
Number of cores
Maximum power
Global memory(ECC)
Memory bandwidth

CPU

GPU

Xeon ES-2620
2.0 GHz
695 W
8 GB(on motherboard)
42.6 GB/s

Tesla C2075
1.15 GHz
448
225 W
6 GB
(on card)
144 GB/s

the observations in Figure 6, the electromagnetic field values
on and near the cavity walls show large fluctuations, so thathigh-precision computation is required. Other areas show
small fluctuations, so that low-precision computation canbe applied. In the proposed hybrid precision computation
method, we use single precision for the most of the area

Journal of Computational Engineering

5
×10−4
2

On and near the cavity wall

0.3

1
Electric field

Electric field (×10−4 )

0.2
0.1
0

0

−1

−0.1

−0.2
−2

4

−0.3
0

400

800

1200

1600

600
500 i
400

5

6

7
Time step

8

9

10
×104

10

12

Inside the cavity
On the cavity
Outside the cavity

2000k(a) Variation of electric field with area(b) Variation of electric field with time

Figure 6: Variation of electric field.

4000

Estimated memory bandwidth (GB/s)
Number of threads
2
3
4
6
12
25.99
35.00
39.56
39.60
41.71
18.86
26.66
33.91
36.63
39.24

3500

Precision
DoubleHybrid

1
13.63
9.63

outside the cavity wall. The rest of the computation area andthe boundaries are done in double precision. Note that someof the computation area inside the cavity is done by 1-D
FDTD, so that the processing time required is very small.
Since the boundary area is very small compared to the total
grid size, the time required to process the boundary data isalso very small.
Figure 7 shows the processing time of multicore CPU
implementation using conventional double precision andthe proposed hybrid precision methods. The parallel thread
execution is done using OpenMP with C language. Compiler
is gcc. According to the results of the double precision computation, we can see a significant processing time reduction
from 1 to 3 threads. However, from 4 threads, the processing
time does not change much. The results for the hybrid
computation are similar to those of the double precisioncomputation. However, the gap between the curves of double
precision and hybrid precision widens after 3 threads. To
explain this, we estimated the memory bandwidth of each
implementation.
Table 2 shows the estimated memory bandwidth in each
implementation. It is estimated by calculating the data capacity and dividing it with the processing time. The estimateddata capacity that needs to be transferred in a single iteration
is about 490 MB. Since there are 100211 iterations, total ofover 49 GB of data are transferred. Since the data transfer
amount is so large and the cache memory of the CPU is only

Processing time (s)Table 2: Estimated memory bandwidth versus number of threads.

3000
2500
2000
1500
1000
500
0
0

2

4

6
8
Number of threads

Double precisionHybrid precision

Figure 7: Precision versus processing time of FDTD on CPU.

15 MB, we assume that the impact of the cache memory is
the same for all single core and multicore implementations.
Since we are only comparing the performance of different
implementations, we did not include the impact of cachememory for the memory bandwidth estimation. According
to the results in Table 2, memory bandwidths of double andhybrid precision computations near their peaks after 3 and 6
parallel threads. After 3 threads, the hybrid precision that has
a smaller data capacity than the double precision gives better
results. Therefore, the proposed method gives better results
for multicore processors with a lot of parallel processing.
Note that, in 1 to 2 thread implementations, the memory
bandwidth is not a bottleneck so that both double and hybrid
precision take similar amount of execution time (althoughhybrid precision is slightly better).

6

Journal of Computational Engineering

1000
1.41
times

800

5.19
times

3.41
times

600
400

1.52
times

CPU
double precision0GPU
double precision

3.67
times

GPU
hybrid precision

200
CPU
hybrid precision

Processing time (s)1200

Processing time (s)Figure 8: Processing time comparison.
360
340
320
300
280
260
240
220
200
180
160

Processing timeusing the proposed
partition boundary

Minimum processing time

010

20 30 40 50 60 70 80
Single precision computation area

90

100

Figure 9: Processing time versus single precision computation area.

Figure 8 shows the comparison between double andhybrid precision computation on CPU and GPU. The CPU
implementation is done with 12 threads on 6 cores, using
Intel hyperthreading technology. According to the results,
GPU implementation with hybrid precision is 5.19 times
and 1.52 times faster than the conventional double precision
implementations on the CPU and GPU, respectively. Hybrid
precision computation on the GPU is 3.67 times faster thanthat on the CPU. Moreover, hybrid precision computationon the CPU is 1.41 times faster than the conventional double
precision implementation on the CPU. The GPU speed-up
factors compared to the CPU in double and hybrid precisioncomputations are 3.41 and 3.67, respectively. Interestingly,
these figures are very close to the memory bandwidth ratio in
Table 1 between GPU and CPU which is 3.38. Therefore, we
can assume that the processing times of multicore-CPU and
GPU implementations are almost decided by the memory
bandwidth. Since single precision floating-point data used in
hybrid precision computation need only 4 bytes compared tothe 8 bytes in double precision, more data can be transferredwith the same bandwidth. As a result, the processing time is
reduced.
Figure 9 shows the processing time reduction against thesingle precision computation area for a simulation model.
According to these results, the processing time decreases
when the percentage of the single precision computation
increases. That is, if we can push the partition boundary

towards the center of the ring, we can reduce the processing
time further. In this example, we push the partition boundary
by increasing the single precision area to find the smallest
processing time. According to the results, the minimum
processing time is observed when 71% of the area is computed
using single precision, while the speed-up is 1.52 times. The
proposed partition boundary defined in (1) allows only 64%
of the computation to be done by single precision and the
speed-up is 1.41 times. Since the speed-up of the proposed
method is very similar to the best speed-up, we can say thatthe proposed partition boundary is very effective. Moreover,
if the same simulation is executed numerous times by slightly
changing the parameters, it would be useful to move the partition boundary towards the center for aggressive processing
time reduction, even if there is a risk of a divergence. If thesimulation does not converge, we can always move back thepartition boundary away from the wall.
As shown in Figure 9, more than 10% of the computation
area must be done in single precision to reduce the processing time. Otherwise, the processing time increases dueto the single-double precision conversion overhead on thepartition boundary. Since the electric (or the magnetic) fieldis calculated by its near-by magnetic (or electric) field data,
both single and double precision data are required to process
the data on the partition boundary. Therefore, if the single
precision area is too small, the proposed method cannot beapplied.
We compare the FDTD simulation results of double andhybrid precision computations in Figure 10 of the revised
paper. The electric fields calculated using double precisionand hybrid precision computations are shown in Figures 10(a)and 10(b), respectively. The two electric fields are almost
identical where a strong field can be seen on the ring area.
We observed that the magnetic fields obtained by double
and hybrid precision computations are also very similar.
Figure 10(c) shows the absolute difference between the double
and hybrid precision computations. The absolute differenceis smaller than 1.0 × 10−9 and is very small compared tothe electric field values shown in Figures 10(a) and 10(b).
Considering the almost identical electric field distribution
and very small computation difference, we conclude that the
hybrid precision computation is accurate enough to be usedin FDTD simulations of a cylindrical resonator.

5. Conclusion
In this paper, we proposed an FDTD computation acceleration method that uses hybrid of single/double precision
floating-point to extract the maximum performance fromGPU accelerators and multicore CPUs. In multicore parallel
processing, the speed of the FDTD computation depends onthe memory access speed and the memory bandwidth. Since
the amount of data required for double precision floatingpoint is two times larger than that required for the single
precision floating-point, we can increase the processing speed
by doing more computation in single precision. Otherwise,
we can reduce the cost by using cheaper medium-range GPUs
(or CPUs) to extract the same performance of the expensive

7

2000
1600
1200

2500

800
400

1500

0
−400
−800
−1200
−1600

Electric field (×10−8 )Electric field (×10−8 )Journal of Computational Engineering

2000
1000
500
0
−500

0
400
k

0
400 i

800
1200

800
400
0
−400
−800
−1200
−1600

2500

2000
1500
1000
500
0
0

−500
400

−1000
−1500

0

−1000

400 i

−1500

800

k

(a) Electric field obtained by double precision computation
Absolute difference (×10−8 )

2000
1600
1200

1200

(b) Electric field obtained by hybrid precision computation

z
0.10

0.12

0.08

0.10

0.06

0.08

0.04

0.06

0.02
0

0.04
0

400
k

800

1200

1600

400

0.02

0
i

0.00

(c) Absolute difference of the double and hybrid computations

Figure 10: Comparison of the simulation results using double and hybrid precision computations.

high-end GPUs (or CPUs). According to the results, weachieved over 15 times of speed-up compared to the singlecore CPU implementation and over 1.52 times of speed-up
compared to the conventional GPU acceleration.

[3] T. S. Ibrahim, R. Lee, B. A. Baertlein, Y. Yu, and P.-M. L.
Robitaille, “Computational analysis of the high pass birdcage
resonator: finite difference time domain simulations for highfield MRI,” Magnetic Resonance Imaging, vol. 18, no. 7, pp. 835–
843, 2000.

The authors declare that there is no conflict of interests
regarding the publication of this paper.

[4] E. Kuramochi, H. Taniyama, T. Tanabe, K. Kawasaki, Y.-G.
Roh, and M. Notomi, “Ultrahigh-Q one-dimensional photonic
crystal nanocavities with modulated mode-gap barriers on SiO2
claddings and on air claddings,” Optics Express, vol. 18, no. 15,
pp. 15859–15869, 2010.

Acknowledgment[5] Y. Ohtera, S. Iijima, and H. Yamada, “Cylindrical resonator utilizing a curved resonant grating as a cavity wall,” Micromachines,
vol. 3, no. 1, pp. 101–113, 2012.

Conflict of Interests

This work is supported by MEXT KAKENHI Grant no.
24300013.

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Antennas andPropagation

Hindawi Publishing Corporation
http://www.hindawi.com

Hindawi Publishing Corporation
http://www.hindawi.com

Volume 2014

Journal ofVolume 2014

Hindawi Publishing Corporation
http://www.hindawi.com

Volume 2014

Shock and Vibration

Electrical and Computer
Engineering

Hindawi Publishing Corporation
http://www.hindawi.com

Hindawi Publishing Corporation
http://www.hindawi.com

Volume 2014

Volume 2014