a conformal mesh generating technique for conformal finite difference time domain (cfdtd) method

7/28/2019 A Conformal Mesh Generating Technique for Conformal Finite Difference Time Domain (CFDTD) Method

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A Conformal Mesh Generating Technique

for Conformal Finite Difference Time Domain (CFDTD) Method

Tao Su, Yongjun Liu, Wenhua Yu and Raj Mittra

Electromagnetic Communication Laboratory, 319 EE East

The Pennsylvania State University, University Park, PA, 16802, USA

Abstract: The article presents an efficient conformal mesh generating technique suitable

for the Conformal Finite Difference Time Domain (CFDTD) technique. We describe the

mesh generation for different types of objects including planar structures (patch antennaand microwave circuits), as well as for arbitrary 3-D structures created by using

AutoCAD, GID or other commercial solid modelers. The versatility of the mesh

generation tool is illustrated through several examples.

Keywords: Mesh Generation, Computational Electrocmagnetics, Conformal FDTD

I. INTRODUCTION

Field solvers based on the Finite Difference Time Domain (FDTD) method [1], whichis one of the most powerful and versatile computational electromagnetics tools, have

been successfully applied to simulate electromagnetic phenomena in a variety of devices

widely used in engineering applications. The FDTD engine is simple to develop and hasthe added advantage that it operates on a Cartesian grid, which has a relatively simple

mesh structure. However, the original Yees scheme does not produce accurate results for

objects with curved edges or surface--owing to the staircasing approximation--unless afine mesh is employed to mitigate the error introduced by the staircasing approximation.A mesh generation technique for the conventional Yees scheme has been described in

the literature [2]. The conformal FDTD techniques [3-6] that offer an alternative to fine

discretization, require somewhat more complex mesh information to describe the curvededges or surfaces without resorting to staircasing, yet their demands on the mesh

generation schemes are still relatively simple compared to those required by the MoM [7]

and FEM [8]. In this article, we present an efficient mesh generation technique for theConformal FDTD technique described in [5]. It can interface with commonly used

commercial software packages such as I-DEAS, PATRAN, AutoCAD and GID, and

generates a group of output files based on the geometry of the object and the application

of the Yees scheme. It provides the intersections of the object being simulated with theCartesian coordinates. The mesh generator is not limited to the conformal algorithm in

[5], and can be applied to other conformal techniques such as the ones described in [3, 4]as well. We discuss the mesh generation procedure for two classes of objects, viz., planar

and arbitrary 3-D structures. Most patch antennas and microwave circuit configurations

typically have a complex shape only in one plane, and are often uniform in the directionnormal to the plane with the exception of a few simple geometries such as cylinders,

spheres, and waveguides. Such arbitrary 3-D objects are typically drawn by using


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commercial software packages, and the triangular patch modeling of the object is an

option often used in these softwares. The article presents a technique that can be appliedto calculate the conformal mesh information based on the objects with a triangular patch

model and Yees grid. Compared to the MoM and FEM methods, the conformal FDTD

technique only requires the intersection between the surface of the objects being

simulated and the Cartesian grid, while retaining the Yee format for the grid, be ituniform or non-uniform, both in the interior of the and exterior of the object. The mesh

generation technique is described in the following sections, where two illustrativeexamples are also included to validate the proposed mesh generation scheme.

II. CONFORMAL FDTD TECHNIQUE

2.1. The Conformal FDTD (CFDTD) technique

In the section, we briefly describe the conformal CFDTD technique presented in [5].

Figure 1 shows the geometry of the intersection between a curved PEC object and the

FDTD mesh.

Fig. 1. 2-D configuration of intersection between

a curved PEC and Cartesian mesh.

The conventional Yees FDTD method can introduce significant errors due to the

staircasing approximation, unless a relatively fine mesh is used. The conformal FDTD(CFDTD) technique, described in [5], can be used to obviate this problem and to improve

the accuracy of modeling the curved boundaries. In this technique, the electric field

update algorithm remains unchanged from that in Yees scheme, but the magnetic field isupdated differently. For instance, for the Hy component we use

( ) ( )( )

( ) ( ) ( ) ( )( ) ( )

( ) ( ) ( ) ( )( ) ( )

+

+=+

kzix

kjiEkjixkjiEkjix

kzix

kjiEkjizkjiEkjiz

kji

tkjiHkjiH

nx

nx

nz

nz

y

n

y

n

y

00

0021

21

,,,,1,,1,,

,,1,,1,,,,

,,

,,,,

(1)


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In contrast to Yees scheme, the magnetic field update equation above requires the

geometrical information pertaining to the intersection of the object surface with theCartesian grid.

2.2. Curved Dielectric Surfaces

In common with curved PEC surfaces and edges, the conventional FDTD can also

introduce significant staircasing errors when dealing with curved interfaces between twodissimilar dielectrics. The conformal FDTD technique has been generalized in [6] to

improve the simulation accuracy of such geometries. This method also requires the mesh

truncation information of dielectric objects to calculate the effective dielectric constantalong the deformed cell edge, whose geometry is shown in Fig.2.

Fig. 2. Mesh truncation of a dielectric object.

The article presents a conformal mesh generation technique that can be applied to

produce the conformal mesh information involved in both curved PEC and dielectric

objects.

III. MESH GENERATION FOR PLANANR STRUCTURE

In the conformal FDTD algorithm, the magnetic fields in the partially-filled cells are

updated by using (1), while retaining the location of the H field at the center of the cell,

just as in the conventional FDTD method. For each partial cell, we need to obtain theequivalent cell size for a PEC object or the equivalent material constants for dielectric

ones. These can be determined if we know the intersections of the grid lines with the

object surface and can uniquely distinguish between the interior and exterior of theobject. The intersection points are determined relatively easily when a geometrical

description of the object is available. Hence the outstanding step that remains to be

attended is the identification of the interior region of the object.


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To explain how we address this problem we begin with a two-dimensional one,

shown in Fig. 3. The 2-D object is described by a polygon, which is a piecewise linearapproximation of a curved edge. This polygon represents the interface between the

interior and the exterior of the object. To determine which grid points are inside a

polygon, we move from one end of a grid line to the other. Each time it intersects with an

edge of the polygon, the side changes. In our algorithm, we first find the intersectionsbetween a grid line and the edges of the polygon, sort the intersections by their

coordinates, pair them, and find the grid points located between each pair of intersections.

Fig. 3. Inside and outside region for a 2-D object represented by a polygon.

In the determination of the intersection of the grid lines and the polygonal edges,

there can be some special cases if one or both ends of an edge fall on the grid line. If oneend of an edge is on the grid line, as is the case for the lines ab and bc shown in Fig. 3,

we need both this edge and its neighboring edge to determine whether or not this point is

crossing into the object.

If both ends of the edge are on the grid line, for instance, the edges gh and de in Fig.3, we need to utilize the information derived from both of the neighbor edges to

determine whether or not the grid line crosses into the object. In Fig. 3, for example, a

grid line along y crosses into the object after intersection with the edge gh, while thecorresponding line that intersects the edge de stays outside the object after intersection.

To distinguish between these special cases, without modifying the algorithm extensively,

we introduce the virtual grid lines, shown as the dashed lines in Fig. 4. For the grid linedefined byx=xi, the virtual grid lines are at the locationsx=xiandx=xi+, where is a

very small distance. If a grid line crosses with an edge in the middle, both virtual lines are

declared as crossing. On the other hand, if one end of the edge is on the grid line, onlyone virtual line is found to cross. Also, if an edge overlaps the grid line, neither of the

virtual lines is interpreted as crossing. When all of the intersections have both been

identified and tagged, we determine the interior region for both the virtual lines,


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recognizing there is an OR relationship between the two set of interior points, i.e., the

grid points declared inside on either of the virtual line are inside. We then repeat theprocess for all of the grid lines so that the interior and exterior regions can be separated

and the exterior dimensions of the partial cells can be calculated.

Fig. 4. Virtual grid lines.

A planar structure created from AutoCAD is shown in Fig. 5. The thickness ismanually added. The mesh generation technique described above is used to generate the

conformal mesh for this structure (see Fig. 6). We display the conformal mesh by

connecting the partial cells to its neighborhood so that we can recover the boundary ofthe object. This is explained in more detail in the next section.

Fig. 5. A planar structure created from AutoCAD.


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Fig. 6. Conformal mesh of the planar structure.

IV. MESH GENERATION FOR ARBITRARY 3D OBJECTS

In the three-dimensional case, we begin by describing the surface of an object using

triangular patches, as illustrated in Fig. 7. We first intersect the object by using differentgrid planes. For an enclosed object, the trajectory on the grid plane will be one or more

polygons and we can use the 2-D technique to determine the interior and exterior regions.

We denote the edges of the polygon as segments in order to avoid confusion with theedges of the triangles. The segments fall in the following three categories: (1) the

common edge of two triangles; (2) the connections between two edges of a triangles or

(3) the connection between a point on one edge of a triangle and the opposite vertex. Forthe latter two cases, we are certain that the grid plane crosses into the object. For the first

case, however, the relationship may depend on the locations of many other triangles.Similar to the 2-D approach, we solve this problem by utilizing the concept virtual grid

planes, as illustrated in Fig. 8.

Fig. 7. Meshing of a 3-D object.


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Let us suppose that we are looking at a grid line (xi,yj) in the z-direction. We first

determine the relationship between all of the triangles and the grid plane defined by x=xi.If a triangle intersects with the grid plane, both the virtual grid planes at x=xi+andx=xi-

are declared as crossing. Alternatively, if only one edge of a triangle is inside the grid

plane but the opposite vertex is not, we conclude that only one virtual grid is crossing. If

the entire triangle resides in the plane x=xi, neither of the virtual plane is crossing. Next,we determine all of the intersecting points and segments and this leads to one or more

polygons in the y-zplane. The 2-D approach is then applied and the crossing points are

determined for the virtual grid lines (xi, yj). Once the interior regions have beenidentified for the four virtual grid lines, they are combined by using an OR operation todetermine the interior region for the actual grid line.

Fig. 8. Virtual grid lines in 3-D.

To examine the effectiveness of the mesh generation algorithm, we have developed

an interactive program to display the conformal mesh. We first output the cells inside theobject, as well as the cell size for the incomplete cells on the surface, and then connect

the four neighboring cells in the same grid plane, forming a polygon for the incomplete

cells, or a simple rectangle for normal cells. The display program, powered by theOpenGL graphics library, processes all the polygons or rectangles and generate a 3-D

view of the FDTD mesh, as shown in Fig. 9. We use different colors to represent the cells

in different grid planes, viz., red, green and blue to correspond to the y-z, x-z and x-y

planes, respectively. By examining the figure, we can easily identify the position andlength of each partial cells and the overall shape of the object. Figure 9(a) is the mesh for

the step structure shown in Fig. 7. The coordinates are carefully chosen so that the first

step does not overlap with a grid plane. The OpenGL enables the easy implementation of

shading of different polygons, so that we can observe the half-cells on two sides of thestructure. The user is also able to rotate the 3-D graphics to view the object from different

angles. Figure 9(b) shows a very coarse mesh of a sphere. Although such a mesh is notsufficiently fine for the conformal FDTD simulation, it nonetheless exhibits the meshstructure clearly, especially for the partial cells.


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(a) Rectangular step

(b) 3-D mesh examplesFig. 9. 3-D mesh examples.

To further demonstrate the capability of the technique, a triangular patch model ofhuman head, which is shown in Fig. 10(a), has been downloaded from the website

http://gts.sourceforge.net/samples.html. Since the mesh generation technique requires the

object to be closed, we manually added some triangular patches to enclose the headmodel at the eyes and neck positions. The conformal mesh, generated by our algorithm, is

shown in Fig. 10(b). Figure 10(c) is a helix antenna created from FEMAP and saved in

the triangular patch format. Figure 10(d) shows the conformal mesh of the helix antennamounted on a PEC circular plate. The PEC circular patch is created from the graphics

interface of our CFDTD package. Although the wire thickness is only 2~3 unit cell size,

the curved surface can still be correctly modeled.


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(a) Human head model (b) Conformal mesh of the human head

(c) Helix antenna model (d) Conformal mesh of a helical antennaon a circular PEC plate

Fig. 10. Conformal mesh examples.

A demo of our CFDTD software package, which has a built-in module for the

conformal mesh generation, is available on-line at our websites: http://www.rm-

associates.biz and http://www.personal.psu.edu/faculty/w/x/wxy6/. The triangular patchmodel files, obtained from different software packages may have slightly different

formats, and the input format required by this demo package needs to be converted intothe following:
http://www.rm-associates.biz/http://www.rm-associates.biz/http://www.rm-associates.biz/http://www.rm-associates.biz/


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Fig. 11. Triangularpatch format required by the CFDTD software package.

V. CONFRMAL MESH FOR THIN PEC STRUCTURE

Thin PEC structures are often encountered in electromagnetic simulations. Suchstructures cannot be handled in the conventional FDTD algorithm unless it coincides with

the grid plane, or is approximated by using a staircase approach. However, the conformalFDTD algorithm, given the effective cell size on both sides, is able to simulate thin PEC

structures, as shown in Fig. 12. The electric field components Ex(i,j) and Ey(i+1,j) are

separated by the thin PEC, dividing the space into regions I and II. The fields in the tworegions need to be updated separately. Thus we define the magnetic fields H

Iz(i,j) and

HII

z(i,j), and electric fields Ex(i,j), Ex(i,j+1), Ey(i,j) and Ey(i+1,j) are also redefined for the

two regions. The update equation then becomes:

( ) ( )

( )

( ) ( ) ( ) ( )

( ) ( )

( ) ( ) ( ) (( ) ( )

)

, ,

1 1, , 0 02 2

, ,

0 0

, 1, , 1, , , , ,

, , , ,

, , , , , , 1, , 1, ,

I I n I I n

x x

I n I n

z z I I n I I n

y y y

x i j k E i j k x i j k E i j k

x i y jtH i j k H i j k

i j k y i j k E i j k y i j k E i j kx i y j

+

+ +

= +

+ + +

(2)

and a similar companion equation may be written for Region II. Since the electric fieldcomponents Ex(i,j+1) and Ey(i,j) do not contribute to the magnetic field in Region II, their

corresponding values are set to be zero in this region. For the field components such asHz(i,j-1), that resides outside of both regions I and II but are still affected by the partial

cell, we use the effective value of the electric field:

0

( , , ) ( , , ) ( , , ) ( , , )( , , )

( , , )

I I II IIeff x x

x

E i j k x i j k E i j k x i j kE i j k

x i j k

+ =

(3)


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Fig. 12. Conformal FDTD mesh for thin PEC structures.

By updating the fields separately in both regions, we can perform the FDTD

simulation for thin PEC structures. However, there is a concern regarding to the

implementation of this algorithm because the conventional FDTD storage structure isspoiled and the field update equations become more complex. To overcome this problem,

we leave the original E- and H-field storage locations unchanged, but rename them as E0

and H0. In addition, we introduce a new block of memory to store the field in different

regions. We then perform the following operations in each iteration step to obtain the

field values:1. Update the global electric field E0

by using conventional FDTD updateequations in the entire domain.

2. Restore the H- field in Region I to the global H-field, let HI

=> H0, and then

update EI.

3. Repeat Step 2 for Region II.

4. Calculate the effective E-field for the partial cells using (3), and store them as

the global E field, by letting Eeff

=> E0.

5. Update the global magnetic field H0

using conventional FDTD update equation.

6. Restore the E-field in Region I to the global E-field, EI=> E

0, update H

I.

7. Repeat Step 6 for Region II.

In this approach, the original FDTD storage structure and update equations remain thesame. The field update in both regions is only performed for the partial cells, that are

located either on or close to the PEC surface, and thus its order of complexity is lower by

one dimension than that of the entire domain. Consequently, the penalty in CPU time toperform the updates in the two regions is negligible compared to that needed to solve the

entire problem. To store the field components in different regions, we also need to store


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their location and the effective cell size, but the increase in memory requirement is also

negligible.The mesh structure for thin PEC objects is also different from normal 3-D objects, as

shown in Fig. 13. In this figure, the cells connecting the black dots are in Region I, and

those connecting the white dots are in Region II. However, since the two regions are

topologically connected, we cannot use the approach in the previous section to separatethe two regions. The only property we can use to separate the regions is that the cells in

different regions do not connect to each other. Thus, we first find all of the intersectionsof the grid lines with the thin PEC surface, and then declare the two grid points

corresponding to the partial cell as in different regions. After all such cells have been

identified, we further examine the connectivity between those special grid points. If twosuch grid points can be connected by a complete cell, they are marked as in the same

region.

Fig. 13. Two sides of a thin PEC structure.

To illustrate the above procedure we show the mesh of a slanted PEC plate in Fig.14(a). The common boundary of the two colors shows the position of the plate, where the

conformal cell size in each region is represented by the colored polygons. This plot

provides a clear picture of all the field components that need to be split. Note that thismesh generation algorithm is not limited to the two-region scenario; in fact, it can be

applied to any number of regions separated by thin PEC structures. The FDTD algorithm

is also not limited to two regions, since the steps (2, 3) and (6, 7) in the above updatingscheme can be extended to any number of regions. Figure 14(b) shows the mesh for two

PEC plates positioned in a T shape. To demonstrate the mesh generation algorithm wehave chosen the positions and orientations of the plates such that they do not coincidewith the grid planes. The space is now divided into three regions where the fields need to

be updated separately. This is shown by three different colors in the plot.


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(a) Slanted PEC plate with zero thickness

(b) T-shape PEC plate with zero thickness

Fig.14. Thin PEC mesh examples.

VI. NUMERICAL EXAMPLES

In this section, we present two examples to validate the mesh generation code

described above. These examples are simulated by our Conformal FDTD software

package [4], and the mesh generator code is a built-in module of this package. The inputsto the conformal mesh generator are the uniform or non-uniform grids and the

information pertaining to object geometry. The procedure is as follows: (1) the user

draws a 2D or 3D object using the FDTD interface, AutoCAD or GID; (2) next, he

generates either a uniform or non-uniform grid from the FDTD interface; (3) followingthis he calls the conformal mesh generator to create the necessary mesh information for

the FDTD algorithm; (4) finally, the user specifies the boundary conditions and the

excitation source and executes the CFDTD solver. The examples in this section include acircular horn antenna and a zero thickness PEC plate.


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6.1. Circular horn antenna

The first example is the so-called Potter horn design [9], shown in Fig. 15, which has

a very low sidelobe level.

Fig.15. Circular horn antenna.

The mesh code is used to generate the conformal mesh required by the CFDTD.

Three colors in Fig. 16 show the deformed dx, dy and dzdistribution.


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Fig. 16. Conformal mesh distribution of the circular horn antenna.

The excellent result obtained via the Conformal FDTD package is shown in Fig.17.

Fig. 17. Far field pattern of the circular horn antenna on the E plane.

6.2. Thin PEC plate

We simulate the radiation problem from a finite line source above a square PEC plate.

At first, we let the plate overlap with a plane in the Yees grid, as shown in Fig. 18(a), sothat the problem can be solved by using the conventional FDTD. The side length of the


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plate is 1m and the length of the line source is 0.2m. We measure the time domain

electric field at points 1 and 2, and derive the far field pattern as well. Next, we rotate theplate as shown in Fig. 18(b), while keeping the relative positions of the source and

observation points unchanged. The second problem is then solved by using the conformal

algorithm with the thin PEC mesh. We compare the near fields at the observation points 1

and 2 in Figs 19(a) and 19(b), respectively. The solid curve corresponds to the firstconfiguration while the dashed one is computed by using the conformal FDTD, and the

agreement between the two sets of fields is seen to be very good. The comparisonbetween the far field patterns in the E plane at 200MHz is shown in Fig. 20, which shows

that the two results are in excellent agreement.

(a) configuration 1 (b) configuration 2

Fig. 18. Thin PEC test case: line source radiation above square plate.

0 50 100 150 200 250 300 350 400-1.5

-1

-0.5

0

0.5

1

Time step

Ey1

(a) E field observed at point 1


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0 50 100 150 200 250 300 350 400-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

Time step

Ey2

(b) E field observed at point 2

Fig. 19. Near field computed using the thin PEC mesh.

5

10

15

20

30

210

60

240

90

270

120

300

150

330

180 0

Fig. 20. E-plane far field pattern computed using the thin PEC mesh.

To test the stability of the conformal algorithm, we slightly vary the position of theslant plate and repeat the above simulation. The plate is shifted in x-direction by 0.1, 0.2,

0.4, 0.6, 0.8 and 1 cell, and the near fields at the two observation points are plotted in

Figs. 21(a) and (b). We can see that the variations due to the direct incidence and platereflection at point 1 are very small, while the diffracted fields received at point 2 are a

little more deviated. This is reasonable because the diffracted field is sensitive to the path

from the source to the edge then to the observation point. In addition, the level of thefield received at point 2 is approximately 50 times lower than that at point 1, and has very

limited effect on the consequent computations.


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0 50 100 150 200 250 300 350 400-1.5

-1

-0.5

0

0.5

1

Time step

Ey1

(a) E field observed at point 1

0 50 100 150 200 250 300 350 400-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

Time step

Ey2

(b) E field observed at point 2

Fig. 21. Received near field when the plate is shifted0, 0.1, 0.2, 0.4, 0.6, 0.8 and 1 cell in the x direction.

VII. CONCLUSIONS

In this article we have presented a mesh generation technique that can be used toproduce the conformal mesh required by the conformal FDTD method. Even though the

examples presented in this article are associated with the algorithm described in [4], the

meshing technique can be readily generalized to adapt to other conformal FDTDalgorithms as well.

AKNOWLEDGEMENT

The authors wish to thank Dr. Nader Farahat, Dr. Hany E. Abd El-Raouf, Dr. Kai Du,

and the Ph.D. students in the EMC lab including Junho Yeo, Lai-Ching (Kit) Ma,Yoonjae Lee, Bing Wang and Kyung Dae Ko for their assistantce in the testing of

software package. Two (Wenhua Yu and Raj Mittra) of the authors also thank Dr. Terry

J. Hilsabeck of SPAWAR System Center and Dr. Peter Chow of Fujitsu Laboratories ofEurope Limited for their helps in the testing of software package.


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REFERENCES:

[1] K. S. Yee, Numerical Solution of Initial Boundary Value Problems Involving

Maxwells Equations in Isotropic Media, IEEE Transactions on Antennas andPropagation, vol. 14, pp. 302-307, May 1966.

[2] Y.Srisukh, J. Nehrbass, F. L. Teixeira, F. F. Lee, and R. Lee, An approach forautomatic grid generation in three-dimensional FDTD simulations of complex

geometries, IEEE Antennas and Propagation Magazine, Volume: 44, Issue: 4,

August, 2002, Page(s): 75- 80.[3] T. G. Jurgens, A. Taflove, K. Umashankar, and T. G. Moore, Finite-Difference

Time-Domain Modeling of Curved Surfaces,IEEE Transactions on Antennas and

Propagation, vol. 40, no. 4, pp. 357-366, April 1992.[4] S. Dey and R. Mittra, A Locally Conformal Finite Difference Time Domain

(FDTD) Algorithm for Modeling Three-Dimensional Perfectly Conducting

Objects, IEEE Microwave and Guided Wave Letters, vol.7, no.9, pp. 273-275,September 1997.

[5] Wenhua Yu and Raj Mittra, A Conformal FDTD Software Package for Modeling

of Antennas and Microstrip Circuit Components,IEEE Antenna and Propagation

Magazine. Vol. 42, no. 5, pp. 28-39, October 2000.

[6] Wenhua Yu, and Raj Mittra, A Conformal Finite Difference Time Domain

Technique for Modeling Curved Dielectric Surfaces,IEEE Microwave and Guided

Wave Letters, vol. 11, no. 1, pp. 25-27, January 2001.[7] R. F. Harrington,Field Computations by Moment Methods, MacMillan, New York,

1968.[8] Jin-Fa Lee and R. Dyczij-Edlinger, Automatic mesh generation using a modified

Delaunaytessellation,IEEE Antennas and Propagation Magazine , Volume: 39Issue: 1 , Feb. 1997 Page(s): 34 45.

[9] P. D. Potter, A New Horn Antenna with Suppressed Sidelobes and Equal

Beamwidths,IEEE Trans. Antennas Propagat., vol. AP-15, pp. 307-308, 1961.


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Biographies

Tao Su received his Bachelor's degree from Tsinghua University, China in 1996,majoring in Electronics Engineering. He obtained his MSE and PhD degrees from theUniversity of Texas at Austin in 1997 and 2001, respectively.

Since Sept. 2001, he has been working as a postdoctoral research associate in theElectromagnetics Communications Lab at Pennsylvania State University. His researchinterests include physics-based signal processing in electromagnetics, antenna mutualcoupling and array signal processing, and advanced numerical modeling techniques.

Yongjun Liu received his Bachelor degree in Electrical Engineering from TsinghuaUniversity, China, in 1991, Master degree in Electrical Engineering from BeijingBroadcasting Institute, China, in 1994, respectively. From 1994 to 2001, he worked inBeijing Broadcasting Institute as an associate professor. He has jointed the EMC lab ofPennsylvania State University since September 2001. His research interests include thecomputational electromagnetics and its visualization techniques and visual languages.

Wenhua Yu is a Visiting Professor in the Pennsylvania State University Department ofElectrical Engineering. Dr. Yu received the BS degrees in Physics from Henan NormalUniversity, XingXiang, in 1984, MS in Electrical Engineering from Beijing BroadcastingInstitute, Beijing, in 1989, and PhD in Electrical Engineering from Southwest JiaotongUniversity, Chengdu, China, in 1994, respectively. Dr. Yu worked in Beijing Institute ofTechnology as a Postdoctoral Research Associate from February 1995 to August 1996.


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He worked in the Pennsylvania State University Department of Electrical Engineering asa Research Associate from May 1999 to August 2001 and a Postdoctoral ResearchAssociate from September 1996 to April 1999. He has published 43 technical papers, 27proceeding articles and one book chapter. Dr. Yu has also developed two softwarepackages for modeling MMIC, RF antennas and microstrip circuit components,

waveguide and cavity. He is a Senior Member of the Institute of Electrical andElectronics Engineers.

His research interests include the areas of Antenna, RF circuit design, RCS prediction,packaged software development, computational electromagnetics, electromagneticmodeling and simulation of electronic packages, EMC analysis, frequency selectivesurfaces, microwave and millimeter wave integrated circuits, and satellite antennas.

Raj Mittra is Professor in the Electrical Engineering department of the PennsylvaniaState University. He is also the Director of the Electromagnetic Communication

Laboratory, which is affiliated with the Communication and Space Sciences Laboratoryof the EE department. Prior to joining Penn State he was a Professor in Electrical andComputer Engineering at the University of Illinois in Urbana Champaign. He is a LifeFellow of the IEEE, a Past-President of AP-S, and he has served as the Editor of theTransactions of the Antennas and Propagation Society. He won the GuggenheimFellowship Award in 1965, the IEEE Centennial Medal in 1984, and the IEEEMillennium medal in 2000. He has been a Visiting Professor at Oxford University,Oxford, England and at the Technical University of Denmark, Lyngby, Denmark.Currently, he serves as the North American editor of the journal AE. He is the Presidentof RM Associates, which is a consulting organization that provides services to industrialand governmental organizations, both in the U. S. and abroad.

His professional interests include the areas of RF circuit design, computationalelectromagnetics, electromagnetic modeling and simulation of electronic packages,communication antenna design including GPS, broadband antennas, EMC analysis, radarscattering, frequency selective surfaces, microwave and millimeter wave integratedcircuits, and satellite antennas.

He has published over 500 journal papers and 30 books or book chapters on varioustopics related to electromagnetics, antennas, microwaves and electronic packaging. Healso has three patents on communication antennas to his credit. For the last 15 years hehas directed, as well as lectured in, numerous short courses on Electronic Packaging,


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Wireless antennas and Computational Electromagnetics, both nationally andinternationally.

a conformal mesh generating technique for conformal finite difference time domain (cfdtd) method

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