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Research Article

Creeping Ray Tracing Algorithm for Arbitrary NURBS

Surfaces Based on Adaptive Variable Step Euler Method

Song Fu,1Yun-Hua Zhang,1Si-Yuan He,1Xi Chen,2and Guo-Qiang Zhu1

1School of Electronic Information, Wuhan University, Wuhan 430072, China

2No. 29 Research Institute, China Electronics Technology Group Corporation, Chengdu 610000, China

Correspondence should be addressed to Yun-Hua Zhang; zhangyunhua@whu.edu.cn

Received April ; Revised July ; Accepted July

Academic Editor: Ana Alejos

Copyright © Song Fu et al. is 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.

Although the uniform theory of diraction (UTD) could be theoretically applied to arbitrarilyshaped convex objects modeled

by nonuniform rational B-splines (NURBS), one of the great challenges in calculation of the UTD surface diracted elds is the

diculty in determining the geodesic paths along which the creeping waves propagate on arbitrarilyshaped NURBS surfaces. In

dierential geometry, geodesic paths satisfy geodesic dierential equation (GDE). Hence, in this paper, a general and ecient

adaptive variable step Euler method is introduced for solving the GDE on arbitrarilyshaped NURBS surfaces. In contrast with

conventional Euler method, the proposed method employs a shape factor (SF) to eciently enhance the accuracy of tracing

and extends the application of UTD for practical engineering. e validity and usefulness of the algorithm can be veried by the

numerical results.

1. Introduction

In the high-frequency electromagnetic problems, the UTD

analysis of the diraction of electromagnetic (EM) waves

[]bysmoothconvexsurfacesisofgreatimportance,such

as the EM compatibility, the analysis of coupling between

two antennas [], and the estimation of scattering properties.

And, as known to all, ray tracing of creeping waves plays

an important role in the process of obtaining the solution

of UTD surface diracted ied. erefore it is essential to

investigate the creeping ray tracing (geodesic paths) on sur-

faces rstly. In fact, however, except some typically geometric

objects which have analytical solution of the geodesic path, it

is a great challenge in determining the geodesic paths along

which the creeping waves propagate on arbitrarily shaped

smooth convex surfaces.

Jha et al. [–]putforwardanoveltechniqueknown

as the geodesic constant method, an analytical method,

and it is applicable to the creeping ray tracing on general

paraboloids of revolution and hybrid surfaces of revolution.

Usually, for some slightly more complex models such as

simple aircra, satellite, they are usually approximated by

some typically geometric objects, such as boards, cylinders,

cones, and spheres, which already have analytical solution

in tracing the creeping rays, for engineering application.

But it is dicult to use such typically geometric objects

to approximate arbitrarily shaped objects, which apparently

limits the application of UTD method.

Hence it is very essential to introduce the numerical

ray tracing methods of creeping waves to handle arbitrarily

shaped convex objects. In [], the models are characterized

by many discrete triangular meshes. And creeping ray tracing

numerically on discrete triangular meshes is presented by

Surazhsky et al. However, such methods cannot be used

directly for the UTD solution. Except modeling arbitrarily

shaped convex object by discrete triangular meshes, it can

also be described in terms of free-form parametric surfaces

such as NURBS surfaces []. Moreover, due to the advantages

ofhighprecisioninmodelingwithlownumberofpatchesand

great exibility in simulation algorithms, NURBS has been

introduced into the area of high-frequency electromagnetic

analysis [,]. When object is described by NURBS surfaces,

geodesic path can be obtained by solving the geodesic

dierential equation []. Hence, some numerical schemes

wereemployedtohandletheGDE.In[], to obtain the

tracks of creeping rays on NURBS surfaces, the geodesic

Hindawi Publishing Corporation

International Journal of Antennas and Propagation

Volume 2015, Article ID 604861, 12 pages

http://dx.doi.org/10.1155/2015/604861

International Journal of Antennas and Propagation

u

v

x

y

z

Mapping

oo

S

(a) uv-parameter domain (b) x

y

z coordinates

0

u0

P:(u0,

0)

=

0u=u

0

P: (u0,

0

)

⃗

r

u

⃗

r

⃗

r

F : Parametric surface.

dierential equations are solved by Euler method which is

the rst-order scheme for dierential equations. And in [],

the Runge-Kutta method of order was applied to solving

the GDE for determining the tracing on NURBS surfaces.

Euler method is highly ecient but with low precision. On

the contrary, the Runge-Kutta method is of high precision but

time-consuming in the process of creeping ray tracing.

In this paper, aiming at improving the accuracy and

eciency of creeping ray tracing on arbitrarily shaped

NURBS surface, we present a creeping ray tracing algorithm

basedonanoveladaptivevariablestepEulermethod.Since

theadaptivevariablestepEulermethodisbasedonthe

conventional Euler method, the eciency is ensured. And in

the process of numerically iteratively solving the GDE, due

to the introduction of shape factor , the discrete step sizes

are adaptively corrected timely. Hence,compared with the

conventional Euler method, the proposed method can easily

ensure the accuracy of creeping ray tracing on arbitrarily

shaped NURBS surface. Namely, it is more suitable for

engineering applications. e eciency and accuracy of the

proposed method will be analyzed and veried by comparing

the results of dierent methods.

2. NURBS Modeling for Arbitrarily

Shaped Convex Objects

NURBS are generations of nonrational B-splines and rational

and nonrational Bezier curves and surfaces []. And NURBS

surface is the rational generalization of the nonrational B-

spline surface. It is dened as follows:

(,V)∈[0,1]2→

(,V)

=∑𝑚

𝑖=0∑𝑛

𝑗=0𝑖𝑗P𝑖𝑗𝑝

𝑖()𝑞

𝑗(V)

∑𝑚

𝑖=0∑𝑛

𝑗=0𝑖𝑗𝑝

𝑖()𝑞

𝑗(V),()

where 𝑖𝑗 are the weights and P𝑖𝑗 are the control points;

𝑝

𝑖() are the normalized B-spline basis functions of

degree dened recursively as

0

𝑖()=

1,

𝑖≤≤𝑖+1,

0,others,

𝑝

𝑖()=−𝑖

𝑖+𝑝 −𝑖𝑝−1

𝑖()+𝑖+𝑝+1−

𝑖+𝑝+1−𝑖+1𝑝−1

𝑖+1(),

()

where 𝑖are the so called knots forming a knot vector =

{0,1,...,𝑚}. e degree, number of knots, and number of

control points are related by the formula =++1.

In many applications of computational geometry, NURBS

surfaces are oen expanded in terms of rational Bezier

patches by applying the Cox-De Boor transform algorithm

[] since Bezier patches are numerically more stable than

NURBS surfaces, while NURBS are more ecient for the

storage and representation of a model. e surface points of

a rational Bezier surface are given by

(,V)∈[0,1]2→

(,V)

=∑𝑚

𝑖=0∑𝑛

𝑗=0𝑖𝑗P𝑖𝑗𝑚

𝑖()𝑛

𝑗(V)

∑𝑚

𝑖=0∑𝑛

𝑗=0𝑖𝑗𝑚

𝑖()𝑛

𝑗(V),()

where P𝑖𝑗 are the control points and 𝑖𝑗 are the weights. 𝑚

𝑖()

and 𝑛

𝑗(V)are Bernstein polynomials of degrees and ,

respectively, which are dened for all integers ( = 0),

( = 0)by

𝑚

𝑖()=!

!(−)!𝑖(1−)𝑚−𝑖 ,

𝑛

𝑗(V)=!

!−!V𝑗(1−V)𝑛−𝑗 .()

A NURBS or Bezier surface can be dened as a vector-

valued mapping from two-dimensional Vparameter domain

to a set of three-dimensional coordinates. And they can

be generally described as

:

(,V)=(,V),(,V),(,V)𝑇.()

e mapping of V-parameter domain to three-space can be

seen in Figure .

International Journal of Antennas and Propagation

(a) Cylinder (b) Cone (c) Sphere

F : NURBS models.

In Figure (b), when we x V=V0and let vary, then

(,V0)depends on one parameter and is, therefore, a curve

called a -parameter curve. Similarly, if we x =

0,then

the curve

(0,V)is a V-parameter curve. Note that both

curves pass through :

(0,V0)in R3.Tangentvectors

for the -parameter and V-parameter curves are given by

dierentiating the components of

(,V)with respect to and

V,respectively.Wewrite

𝑢(,V)=

(,V)

=

𝑢(,V),

𝑢(,V),

𝑢(,V)𝑇,

V(,V)=

(,V)

V=

V(,V),

V(,V),

V(,V)𝑇.()

In this paper, if |

𝑢(,V)|and |

V(,V)|are both constant

on parametric surfaces, we call the surface uniform grid sur-

face, otherwise nonuniform grid surface. Here some typical

NURBS models are given in Figure . e surfaces of cylinder

are uniform grid surfaces, and others are nonuniform grid

surfaces.

e surfaces of models above are digitized by networks of

V-isoparametric curves. And the cylinder, cone, and sphere

are, respectively, constructed by Bezier patches highlighted

in dierent colors. And in the following, for convenience,

dierent Bezier patches will be shown in the same color.

3. Algorithm of Creeping Ray Tracing on

Arbitrary NURBS Surfaces

According to the positions of the source and the observation

points, the diraction problems of convex surfaces can be

classied as three types. First, the source and the observation

points are both out of the surfaces and the observation points

are far away from the surfaces. is case belongs to the

scattering problem of the smooth convex surfaces. Second,

the source is directly on the surfaces, and the observation

points are far away from the surfaces. is can be called

theradiationproblemofantennas.ird,thesourceandthe

observationpointsarebothonthesurfaces.isisthemutual

coupling problem among antennas on the surface. Hence,

the ray tracing should also be divided into three cases, as

in Figures –. In all of these cases, the ray trajectories on

the surfaces are called creeping rays, which are restrained

to propagate along geodesic paths. And this paper mainly

focuses on the tracing of creeping rays since it is the most

dicult part in the process of tracing.

3.1. e Start Points and Exit Points of the Creeping Rays.

As shown in Figure ,atthestartingpoint

0, the grazing

incident condition satises

0,V0−

0

0,V0−

0⋅

0,V0=0,()

where

(0,V0)is the unknown

0on the surface

(,V)and

(0,V0)is the corresponding unit normal vector of at

0.

At the exiting point

𝑠:

(𝑖,V𝑖),

𝑖,V𝑖−

𝑠

𝑖,V𝑖−

𝑠⋅

𝑖,V𝑖=0.()

According to () and (),wecanobtainanumberof

the eective starting and exiting points of creeping rays on

NURBS surfaces.

3.2. Creeping Ray Tracing by Solving GDE with Adaptive Vari-

able Step Euler Method. As we know, the paths of creeping

rays on arbitrarily shaped surfaces satisfy GDE. en the

problem of obtaining creeping rays is changed into solving

GDE. Generally, GDE are solved by Euler method which is

a simple and fast method. However, according to study, in

most of the cases conventional Euler method does not work

because of its low accuracy and stability.

For the nonlinear problems, the step controlled correc-

tion procedure is essentially needed. Hence, in this paper, in

order to improve the accuracy of ray tracing and make sure

of the eciency, an adaptive variable step Euler method is

presented to solve the GDE.

International Journal of Antennas and Propagation

Incident rays

Creeping rays

Emergent rays

Plane waves

(a) Side view

Incident rays

Creeping rays

Emergent rays

Pn

P0

P1

P0,...,P

n: observation points

(b) Top view

F : Ray tracing of the plane wave incidence.

Q: source point

Q

Creeping rays

Emergent rays

P0,...,P

n: observation points

Pn

P0

(a) Side view

Creeping rays

Emergent rays

Q

Q: source point

P0,...,P

n: observation points

Pn

P0

(b) Top view

F : Ray tracing of source on surface.

Firstly, GDE are given as follows:

2

2+Γ1

11

2+2Γ1

12

V

+Γ1

22 V

2

=0, ()

2V

2+Γ2

11

2+2Γ2

12

V

+Γ2

22 V

2

=0, ()

where Γ𝑘

𝑚𝑛 are the Christofel symbols of the second kind, is

arc length along the geodesic path, and (,V)are parametric

coordinates of creeping rays.

en, the rst and second derivatives of in () can be

approximately expressed by

()=(+)−()

,()

()=(+)−()

=(++)−(+)−(+)+()

2,()

where is the step size between two adjacent discrete points,

the determination of which is of great importance. is shape

factor (SF), which is introduced to adaptively control every

discrete step size. And the value of is subject to the shape

International Journal of Antennas and Propagation

Creeping rays

Q: source point

P: observation point

P

Q

(a) Side view

Creeping rays

Q: source point

P: observation point

Q

P

(b) Top view

F : Ray tracing of source and observation point on surface.

Incident

Creeping ray

Exiting point

Starting point Q0

Qs

direction ̂

t

Source p0

Observation point ps

Surface S:r(u, )

F : Creeping ray on surface .

of the object; more details about will be presented in

Section ..

In numerical calculation, a number of discrete points

(𝑖,V𝑖)can be computed to express the creeping ray, where

=0,1,2,....So, according to discretization, () and ()

canberewrittenas

𝑖=𝑖+1−𝑖

𝑖,()

𝑖=𝑖+2−

𝑖+1−𝑖+1+𝑖

𝑖+1,𝑖𝑖2,()

where

𝑖+1 =(+𝑖+1,𝑖𝑖),whichisatestpoint.Since

𝑖=𝑖+1−𝑖

𝑖≈

𝑖+1−𝑖

𝑖+1,𝑖𝑖,()

then

𝑖+1=𝑖+𝑖+1,𝑖 ⋅𝑖+1−𝑖. ()

Next, substituting () into () we obtain

𝑖=𝑖+2−𝑖+1−𝑖+1,𝑖 ⋅𝑖+1−𝑖

𝑖+1,𝑖𝑖2.()

Similarly, the rst and second derivatives of Vcan be

V

𝑖=V𝑖+1−V𝑖

𝑖,

V

𝑖=V𝑖+2−V𝑖+1−𝑖+1,𝑖 ⋅V𝑖+1−V𝑖

𝑖+1,𝑖𝑖2.()

Finally, substituting (),(),and() into the GDE ()

and (),wehave

𝑖+2=𝑖+1+𝑖+1,𝑖 ⋅𝑢,

V𝑖+2=V𝑖+1+𝑖+1,𝑖 ⋅V,()

where

𝑢=𝑖+1−𝑖−Γ1

11 𝑖+1−𝑖2

+2Γ1

12 𝑖+1−𝑖V𝑖+1−V𝑖+Γ1

22 V𝑖+1−V𝑖2,

V=V𝑖+1−V𝑖−Γ2

11 𝑖+1−𝑖2

+2Γ2

12 𝑖+1−𝑖V𝑖+1−V𝑖+Γ2

22 V𝑖+1−V𝑖2.

()

According to (),tocalculate(𝑖+2,V𝑖+2)on the creeping

path, the previous two points (𝑖+1,V𝑖+1),(𝑖,V𝑖)and 𝑖+1,𝑖 are

needed. erefore the rst two points (0,V0),(1,V1)and 1,0

should be calculated at the beginning of creeping ray tracing.

Here, the starting point 0(0,V0)canbeobtainedby().

And according to the dierential geometry [], the second

International Journal of Antennas and Propagation

v-parameter curve

S

u-parameter curve Q0

0

u0

r(u0,

0)·(

1−

0)̂

t·h

0

ru(u0,

0)·(u

1−u

0)

Incident direction ̂

t

F : e determination of the second tracing point.

point is approximated by the sum of the two tangent vectors

on tangent plane at the starting point 0shown in Figure .

Hence, the second point can be expressed by

1=0+0⋅

𝑢0,V0⋅

𝑢0,V0,

V1=V0+0⋅

V0,V0⋅

V0,V0,()

where 0is initial step size and

𝑢(0,V0)and

V(0,V0)

are the unit tangent vectors for the -parameter and V-

parameter curves at 0(0,V0). And tangent vectors

𝑢(0,V0)

and

V(0,V0)for the -parameter and V-parameter curves

can be obtained by ().

is the unit vector along the incident

direction.

e general expression of shape factor is derived in

Section .. According to the expression, 1,0can be obtained.

Aer the calculation of (0,V0),(1,V1)and 1,0,() can

iterate step by step with the increase of .

3.3. e Derivation of Shape Factor .As we know, the smaller

the step size is, the more accurate that derivative approx-

imate expression is. However, in the process of iteratively

solving the discrete points on creeping ray path, the eciency

of algorithm will decrease with the increase of the number

of discrete points. More importantly, the more the discrete

points, the larger the accumulative error which may lead to

wrong results.

Clearly, for the approximation of

𝑖or V

𝑖,wecould

achieve greater eciency if we could provide more discrete

points where the variables are changing rapidly to achieve

accuracy, while less discrete points should be provided in

regions where variables are changing slowly. erefore it is

necessary to allocate discrete points during the process of

Euler approximation.

However the fact is that the parametric expressions

((),V())ofthecreepingraypathareunknown.Wecannot

directly obtain the variable change speed degree which the

policy of allocating discrete points is based on. So, the shape

factor is introduced to determine the discrete step size.

e discrete value of shape factor 𝑖+1,𝑖 is dened to reect

therelativechangerateofparametersattwoadjacentpoints

on the creeping path. Since the creeping rays include two

parametric expressions () and V(), we should consider

C

P

S

v-parameter curve

u-parameter curve

0

u0

r

̂

t

ru

F : Curve on the parametric surface.

the change of ,Vsimultaneously for well determining 𝑖+1,𝑖.

Hence,

𝑖+1,𝑖 =

𝑖

𝑖+1

,

𝑖

𝑖+1

≤

V

𝑖

V

𝑖+1

,

V

𝑖

V

𝑖+1

,else,()

where

𝑖+1,

𝑖,V

𝑖+1,andV

𝑖are unknown, which need to be

solved out. And the relationship between the SF and step

size is

𝑖+1=𝑖+1,𝑖 ⋅𝑖.()

Let be an arc length parametric curve (creeping ray) on

surface :

(,V)which pass through point as shown in

Figure and denote that

()=

((),V()).()

Let

be a unit tangent vector of at ,andletthenumberof

the discrete points be .

Tangent vector for the creeping ray at the point can

be obtained by dierentiating equation () with respect to

the arc length

𝑖=

((),V())

=

𝑢,𝑖 𝑖

+

V,𝑖 V𝑖

,()

where

𝑢,𝑖 and

V,𝑖 canbecalculatedby().Since|

𝑖|=1,

𝑢,𝑖 𝑖

+

V,𝑖 V𝑖

=1.()

However, 𝑖/and V𝑖/cannot be obtained from ().

Now, we approximate 𝑖/ and V𝑖/ at point on

by 𝑖/at point on -parameter curve (V=V0)and

V𝑖/at point on V-parameter curve, respectively. For

on -parameter curve,

V,𝑖 =0,() can be

𝑢,𝑖 𝑖

=1. ()

For on V-parameter curve,

𝑢,𝑖 =0,() can be

V,𝑖 V𝑖

=1.()

International Journal of Antennas and Propagation

T : Comparison of creeping rays on cylinder obtained from three dierent methods.

Source point (, , )

Creeping ray number

Last point

Analytical method (., −., .) (−., ., .) (., ., .)

Euler method (., −., .) (−., ., .) (., ., .)

Proposed method (., −., .) (−., ., .) (., ., .)

Length (m)

Analytical method . . .

Euler method . . .

Proposed method . . .

So, according to () and (),

𝑖

=1

𝑢,𝑖,

V𝑖

=1

V,𝑖.()

Similarly, at the +1point on ,

𝑖+1

=1

𝑢,𝑖+1,

V𝑖+1

=1

V,𝑖+1.()

Substituting () and () into (), nally the shape factor

expression is

𝑖+1,𝑖 =

𝑢,𝑖+1

𝑢,𝑖

,

𝑢,𝑖+1

𝑢,𝑖

≤

V,𝑖+1

V,𝑖

,

V,𝑖+1

V,𝑖

,else.

()

So, based on the 𝑖+1,𝑖, we can obtain discrete variable step

adaptively.

4. Results and Discussions

4.1. Creeping Ray Tracing Algorithm Validation. e creeping

rays can be calculated theoretically on some typical objects,

such as cylinder, cone, and sphere. erefore, the proposed

method can be veried from the analytical results of these

canonical objects.

(1) Creeping Ray Tracing on Cylinder (Uniform Grid Surfaces).

e cylinder which has a radius of m and height of m

is considered for creeping ray tracing. We consider three

dierent incident wave directions, all of which incident start

fromthesamepoint(1,0,0). In order to validate the

accuracy of the proposed creeping ray tracing method, it is

compared against the analytical method and the conventional

Euler method. Figure shows the creeping ray tracing results

obtained from the above mentioned methods, for three

dierent incident directions, respectively. It is shown that,

0

0

0.5

1

1.5

2

2.5

3

Analytical method

Euler method

Proposed method

1

3

2

−1

1

1

0

−1

Q

F : Creeping ray tracing on cylinder.

for each incident direction, the creeping ray obtained from

theproposedmethodoverlayswiththosefromtheother

two methods. To further validate the proposed method, the

creeping ray ending point and the length of the creeping rays

are compared, as shown in Tab l e .

From Table ,thenumericalresultsbytheproposed

method are in good agreement with the analytical method,

which can conrm its rationality. And we can nd that the

numerical results by the proposed method and Euler method,

respectively, are identical, and this is because cylinder is

so special that the () and V() of creeping ray on its

surfaces are rst-order linear. Namely, the discrete step sizes

canbechosenequally.Andinthisspecialcase(uniform

grid surfaces), 𝑖+1,𝑖 =1,theproposedmethodisreduced

to conventional Euler method. (Comparing to the Euler

method, the advantages of the proposed method can be

demonstrated in the following examples.)

(2) Creeping Ray Tracing on Cone (Nonuniform Grid Surfaces).

e cone which has a radius of m and height of m is consid-

ered for creeping ray tracing. We consider three dierent inci-

dent wave directions, all of which incident start from the same

International Journal of Antennas and Propagation

0

0.5

1

1

0

0

0.5

1

1.5

2

Analytical method

Euler method

Proposed method

−1

−0.5 −1

Q

(a) Side view

0

0.2

0.4

0.6

0.8

1

Analytical method

Euler method

Proposed method

Q

1

2

3

−0.2

−0.4

−0.6

−0.8

−1

−1 −0.5 0 0.5 1

(b) Top view

F : Creeping ray tracing on cone.

T : Comparison of creeping rays on cone obtained from three dierent methods.

Source point (, , )

Creeping ray number

Last point

Analytical method (−., −., .) (−., −., .) (., −., .)

Euler method (−., −., .) (., −., .) (., −., .)

Proposed method (−., −., .) (−., −., .) (., −., .)

Length (m)

Analytical method . . .

Euler method . . .

Proposed method . . .

Time (s) Euler method . . .

Proposed method . . .

point (1,0,0).InFigure , the creeping ray tracing results

on a cone are shown. e numbers , , and in Figure (b)

are three creeping rays of dierent incident directions.

Firstly, from Tab l e ,wecanndthenumericaltracing

results by proposed method are in good agreement with the

analytical method. As a comparison, the results obtained by

theEulermethodareinloweraccuracy.Moreover,thecreep-

ing ray results obtained by the Euler method deviate from

that of the analytical method when the creeping ray trajectory

is closer to cone-tip. is is clearly shown in Figure and

Table , the Euler method results deviate from the analytical

resultsfortheraynumber,anditevenfallsintofalseresults

when the ray runs too close to the cone-tip as for the ray

number . is indicates that the conventional Euler method

cannot be applied to nonuniform grids surfaces while the

proposed method is capable of handling the nonuniform

grids surfaces. Secondly, the proposed method is as ecient

as conventional Euler method.

(3) Creeping Ray Tracing on Sphere (Nonuniform Grid Sur-

faces). espherewhichhasaradiusofmisconsidered

for creeping ray tracing. According to dierential geometry,

the geodesic paths on sphere are orthodromes. Hence, the

numerical results of sphere can be validated by the theoretical

results.

From Figure (a), the result by proposed method is

in good agreement with the theoretical results. And from

Table and Figures (b) and (c), although, with the

decrease of discrete step size, the error by Euler method is

reduced, time consumption is increased a lot. More impor-

tantly, if the discrete step size continues to decrease, from

Figure (d), the result is totally wrong, which means Euler

method is unstable.

4.2. Electromagnetic Calculation with UTD on the Base of

Creeping Ray Tracing Algorithm. To assess the accuracy of

UTD diraction based on the proposed creeping ray tracing

International Journal of Antennas and Propagation

0

0.5

0

0.5

0

0.5

1

Q

−0.5

−0.5

−0.5

−1

(a) Proposed method

0

0.5

00.5

0

0.5

1

Q

−0.5

−1

−0.5

−0.5

(b) Euler method

0

0.5

0

0.5

0

0.5

1

Q

−0.5

−0.5

−0.5

−1

(c) Euler method

0

0.5

0

0.5

0

0.5

1

Q

−0.5

−0.5

−0.5

−1

(d) Euler method

F : Creeping ray tracing on sphere.

T : Results of three creeping rays tracing on sphere.

Source point (, , )

Methods (a) Proposed method (b) Euler method (c) Euler method (d) Euler method

Initial step size (m) . . . .

Last point (., ., .) (., ., .) (., ., .) (., ., −.)

Time (s) . . . .

algorithm, the analytical results of canonical targets are given.

Figures and show the bistatic scattering electric elds of

PEC sphere and cylinder in shadow region, respectively, and

the UTD solutions are compared with the analytical results.

Moreover, in reality, lots of targets are low detectable.

When analyzing the scattering properties of those targets

especially for some constructed by smooth convex curved

surfaces, the contributions of creeping waves cannot be

neglected. Here, taking an antiradar stealth screen compo-

nent of a radar stealth satellite as example, since the antiradar

stealth screen component is constructed by some smooth

convex curved surfaces (see Figure ), the contributions of

creeping waves will play important roles in this case. In

the following example, the monostatic RCS of the antiradar

stealth screen are studied by comparing the PO + UTD

solution with the numerical solution to demonstrate the

contribution of creeping waves.

() e bistatic diracted electric elds of a PEC cylinder

inshadowregionarecalculatedbyUTD.eradiusofthe

cylinder is m, and the distance between the observation

International Journal of Antennas and Propagation

x

y

R

Diracted ray

𝜌

30∘

Ps

−30∘

Incident rays

Creeping rays

Emergent rays

Field points

F : Diracted contribution in shadow region.

0 10−10 20−20 3

0

−30

0

0.2

−0.2

0.4

0.

6

Bistatic angle (deg.)

Magnitu

d

e o

f

e

l

ectric

e

ld

E (V/m)

Analytical result

UTD

F : Scattered electric eld of PEC cylinder.

point and the origin is m. e frequency of incident plane

waveisGHz.esketchmapofthediractedcontribution

is given in Figure .

From Figure , the UTD results of PEC cylinder in the

shadow region based on the proposed creeping ray tracing

algorithm are in good agreement with the analytical results,

and the good agreement conrms the validity of the creeping

ray tracing algorithm.

() e bistatic diracted electric elds of a PEC sphere

inshadowregionarecalculatedbyUTD.eradiusofthe

sphere is m, and the distance between the observation

point and the origin is m. e frequency of incident plane

waveisGHz.esketchmapofthediractedcontribution

is given in Figure .

As shown in Figure , the UTD results of PEC sphere

in the shadow region based on the proposed creeping ray

tracing algorithm are in good agreement with the analytical

results, which denotes the validity of the creeping ray tracing

algorithm.

() Monostatic RCS of the antiradar stealth screen com-

ponent of a radar stealth satellite are calculated by the PO

method combined with the UTD method based on the

proposed creeping ray tracing algorithm, and these results

are then compared with that of the numerical method to

validatetheaccuracyoftheproposedmethod.esideand

D view of the antiradar stealth screen component are shown

in Figure ,where=1.12m, =0.3m, and =3.0m.

Figure shows the trajectories of creeping rays on the

antiradar stealth screen component. In space, the conical tip

of the antiradar stealth screen is directed to the earth. Hence

we focus on the analysis of scattering property about the

antiradar stealth screen nearby region of the conical tip. Here,

the RCS will be calculated in the range of =90∘∼180∘.e

frequency of incident plane wave is GHz.

Next, we compare the monostatic RCS of the target in the

range of =90∘∼180∘by three dierent methods, namely,

the numerical method Multilevel Fast Multipole Algorithm

(MLFMA), the PO method, and the PO + UTD method. In

International Journal of Antennas and Propagation

x

y

Diracted ray

Ps

R

𝜌

150∘

210∘

Incident rays

Creeping rays

Emergent rays

Field points

1

0.5

0

−0.5

−1

2

1

0

−1

−2 1

0.5

0

−0.5

−1

F : Diracted contribution in shadow region.

150 160 170 180 190 200 21

0

0

0.2

0.4

0.6

0.8

1

Bistatic angle (deg.)

Magnitu

d

e o

f

e

l

ectric

e

ld

E (V/m)

Analytical result

U

TD

F : Scattered electric eld of PEC sphere.

Incident rays

Creeping rays

r

r

r

r

h

R

x

z

113∘

23∘

F : Creeping ray tracing on an antiradar stealth screen.

International Journal of Antennas and Propagation

90 100 110 120 130 140 150 160 170 18

0

0

10

20

30

RCS (

d

Bsm)

MLFMA

PO

PO + UTD

Monostatic angle 𝜃(deg.)

−10

−20

−30

−40

−50

0∘

90∘

180∘

270∘

F : Monostatic RCS of the antiradar stealth screen.

thePO+UTDscheme,theUTDsolutionbasedonthepro-

posed algorithm is used to obtain the diracted contribution

of creeping rays. In the following, the monostatic RCS of the

antiradar stealth screen are given in Figure .

From Figure , it is observed that the contribution of

creeping waves is very obvious when the incident direction

=145∘∼175∘.Inthisrange,thankstothecontributionsof

the creeping waves, the PO + UTD results are about dBsm

higher than the PO results. So, the contributions of creeping

waves cannot be neglected. And compared with PO, the PO

+ UTD solution has a better agreement with the MLFMA

solution.

Hence, the rationality and validity of the creeping ray

tracing algorithm presented in this paper are proved.

5. Conclusion

An accurate and ecient creeping ray tracing algorithm

basedonanadaptivevariablestepEulermethodispre-

sented in this paper. e proposed algorithm is applicable to

arbitrary NURBS surface. is adaptive variable step Euler

method, based on the conventional Euler method, employs

ashapefactor(SF)to solve the GDE. e SF can

timely reect the relative change rate of the creeping ray

parameters at two discretely adjacent points on surfaces.

And according to , every adaptive step size, which is very

important for the accuracy of creeping ray tracing, can be

obtained in turn. Finally based on the fast and accurate

creeping ray tracing, the contribution of UTD diraction can

be calculated. Bearing in mind the accuracy and eciency,

the algorithm of NURBS-UTD will be very useful in practical

engineering.

Conflict of Interests

e authors declare that there is no conict of interests

regarding the publication of this paper.

Acknowledgments

is work was supported by the National Natural Sci-

ence Foundation of China (Grant no. , Grant no.

, and Grant no. ), the China Postdoctoral

Science Foundation, and the Fundamental Research Funds

for the Central Universities.

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