Computing GTD Coefficients via Direct Numerical Simulation

HUIDONG YANG

Master’s Degree Project Stockholm, Sweden 2004

TRITA-NA-E04150

Numerisk analys och datalogi Department of Numerical Analysis KTH and Computer Science 100 44 Stockholm Royal Institute of Technology SE-100 44 Stockholm, Sweden

HUIDONG YANG

TRITA-NA-E04150

Master’s Thesis in Numerical Analysis (20 credits) at the Scientific Computing International Master Program,

Royal Institute of Technology year 2004 Supervisor at Nada was Olof Runborg

Examiner was Axel Ruhe

Computing GTD Coefficients via Direct Numerical Simulation

Computing of GTD-coefficient

via direct numerical simulation

Abstract : Geometrical optics (GO) and geometrical theory of diffraction (GTD),a generalization of GO, are two classical methods to study high frequency which isimportant in many applications. However, GO does not include diffraction effects andGTD, although including them, is difficult to derive mathematically for complicatedcases.

In this paper, we propose a method to compute diffraction coefficients forgeometrical theory of diffraction (GTD) via direct numerical simulation of the waveequation. It uses finite difference approximation to wave equation and fast FourierTransform (FFT) to compute GTD coefficients for different frequencies.

Beräkning av GTD-koefficienter

via direkt numerisk simulering

Sammanfattning : Geometrisk optik (GO) och dess generalisering geometriskdiffraktionsteori (GTD) är två klassiska metoder för studiet av högfrekventa vågor, vilkaär viktiga i många tillämpningar. GO inkluderar dock inte diffraktionseffekter som GTDgör, och GTD-modellen är svår att härleda analytiskt för komplicerade fall.

I den här artikeln presenterar vi en metod som beräknar diffraktionskoefficienterför GTD via direkt numerisk simulering av vågekvationen. Den använder en finitadifferensapproximation av vågekvation och den snabba Fourier-transformen för attberäkna GTD-koefficienterna för olika frekvenser.

Table of Contents1. Introduction .....................................................................................12. Mathematical Theory.......................................................................2

2.1 Wave equation..............................................................................................22.2 Geometrical optics ........................................................................................32.3 Geometrical theory of diffraction....................................................................4

3. Numerical Method............................................................................73.1 Numerical solver............................................................................................7

3.11 Numerical scheme..................................................................................73.12 Boundary conditions ...............................................................................83.13 Initial conditions......................................................................................9

3.2 Angles.........................................................................................................113.3 Numerical calculation of diffraction coefficients ..........................................123.4 Complexity and accuracy............................................................................14

3.4.1 Numerical error sources.......................................................................153.4.2 Computational cost..............................................................................19

4. Numerical Simulation Results........................................................214.1 Example 1...................................................................................................214.2 Example 2...................................................................................................314.3 Example 3...................................................................................................374.4 Example 4...................................................................................................424.5 Example 5...................................................................................................44

5. Acknowledgment............................................................................506. Reference.......................................................................................51

1. Introduction The wave equation occurs in many fields, for instance, acoustics, elasticity, and

electromagnetism. It is the simplest equation for modeling wave propagation. Thenumerical approximation of high frequency wave propagation is important in manyapplications such as seismic, acoustic and optical waves, microwaves and so on.

For relatively high frequency, the wavelengths are short compared to thecomputational domain and direct simulations using standard numerical methods require avery refined grid size, and therefore takes long time and needs lots of memory.

Instead, one can use geometrical optics (GO), or ray optics, which is frequentlyused to study such high frequencies. However, the standard geometrical optics does nottake into account the effects of geometry and boundary conditions, which often gives riseto diffracted waves. This happens when the pure geometrical optics solution isdiscontinuous.

In our case, the application of the geometrical theory of diffraction to the waveequation is considered. The geometrical theory of diffraction (GTD) can be seen as ageneralization of geometrical optics. It provides a systematic technique for addingdiffraction effects to the geometrical optics approximation.

In some simple cases the diffraction coefficients for the geometrical theory ofdiffraction can be calculated by mathematical analysis but for more complicated cases itis hard to do so and we want to be able to do it numerically instead. Examples includeproblems with complicated geometry, or non-homogeneous medium materials where thelocal speed of wave propagation may vary in different medium parts. These cases couldinstead be simulated via numerical methods.

The goal of this paper is to compute diffraction coefficients for geometrical theoryof diffraction(GTD) via direct numerical simulation of the wave equation. The task isdivided into two parts:

Part I deals with the numerical simulation of the wave equation in two space-dimensional cases by finite difference methods.

Part II uses Fast Fourier Transform (FFT) to compute the GTD coefficients fordifferent frequencies from the numerical results obtained in part I.

The programming tool is Matlab 6.5.

1

2. Mathematical Theory

2.1 Wave equation

The equation

utt−c x 2u=0, t ,x ∈ℝ× ,⊂ℝd , (2.1)

is known as the linear scalar wave equation. The coefficient c x is the local speed ofwave propagation in the medium. The wave equation is the simplest model for wavephenomena that occurs in many fields such as acoustics, elasticity, electromagnetism, andso on [1]. For example, in acoustics, it describes the evolution of pressure disturbances(sound waves). Another application is the transverse vibration in strings and membranesor the longitudinal and torsional waves in bars. The solution to Maxwell's equations for anon-conducting medium can also, under appropriate conditions, be reduced to thesolution of the scalar wave equation. It is clearly interesting to do numerical simulation of

2.1 .

To better understand typical solutions to (2.1), we will look at two types of exactsolutions. One type of exact solutions for (2.1) is the plane wave shown in figure 2.1below.

Figure 2.1 Plane waves in the coordinates

For two space dimension x=x , y , suppose c x , y ≡constant , and defineS=−cosinc ,sininc , where inc is the incident angle of the plane wave as shown

in figure 2.1 above. If characteristic coordinates =x⋅S−ct and =x⋅Sct areintroduced, a solution is then

2

X

Y

u((x , y) , 0)

f(α)

g(β)

Ŝ

θinc

u= f g = f x⋅S−ct g x⋅Sct , (2.2)

where f and g are arbitrary functions. This solution thus represents one wave pulsewith profile f moving in the forward S direction and a pulse with profile g moving in the opposite direction.

Another type of exact solutions is spherical wave in three dimensions. For wavessymmetric about the origin u=u R ,t , where R is the distance from the originR= x2 y2z2 , the wave equation reduces to utt=c

2uRR2uR /R . This may bewritten as Rutt=c

2RuRR , which is the one dimensional wave equation. Thus forR0 , the general solution takes the simple form

u R , t = f R−ct R

g Rct R

, (2.3)

where f and g are arbitrary functions.

Figure 2.2 Spherical waves

Again, we have two wave pulses with profiles f and g , this time moving inthe radial direction from the origin and decaying with rate 1/R .

2.2 Geometrical optics

We give some introduction to the geometrical optics (GO) model [2] . Whenu t ,x =eiktuk x , insert it into wave equation (2.1), and we obtain

eikt ukk 2/c2uk =0

3

u(R , 0)

f(α)

g(β)

Then uk x is the solution of the Helmholtz equation:

ukk22uk=0 , (2.4)

where the coefficient =x =1/c x is the index of refraction and k=2/0 is thewave number with 0 being the reference wavelength. Geometrical optics relies on theassumption that the complex valued solution uk x can be approximated,asymptotically in k , by the “ansatz”

uk x ≃A x eikx , (2.5)

where the amplitude Ax and the phase x are frequency independent real valuedsolutions of the Eikonal/Transport GO system of equations:

∣∇∣2=2 , (2.6)

2∇⋅∇ AA=0 . (2.7)

The Eikonal/Transport GO system of equations may be obtained from (2.4) and(2.5) by inserting (2.5) into (2.4). We get

eik A2∇⋅∇ AAik A²−A∣∇∣2k 2=0 , (2.8)

For different k , (2.8) should be satisfied and then the coefficients for k 1 andk 2 should be zero. Upon neglecting the k 0 term, this leads to the GO system of

equations (2.6) and (2.7).

2.3 Geometrical theory of diffraction

The geometrical theory of diffraction (GTD) is a generalization of geometricaloptics. More details about this background can be found in [2]. In a general case as infigure 2.3 below, when a plane wave hits the corner of a boundary, it will produceinfinitely many diffracted rays around the corner.

4

Reflected ray

Ŝ

θinc θd

Region B

Diffracted rays

Region A

Half plane

Incident ray

Region C

Figure 2.3 Generation of diffracted waves in a general case

Consider a typical case as shown in figure 2.4 below, when an incident ray hits thetip of the half plane, it will give rise to infinitely many diffracted rays coming out in alldirections from the tip of the half plane. We may see the whole domain as divided intothree region A, B and C by the incident ray and half plane, where the GO solution has 2,1and 0 crossing waves, respectively. In GTD, there is also the additional diffracted waveoriginating in the tip of the plane. The amplitude of the diffracted wave is proportional tothe incoming wave and a diffraction coefficient D . In this article, we mainly focus onpart of regions A and B near the tip to do the numerical simulation.

Figure 2.4 Generation of diffracted waves in a typical case

For the case in figure 2.4, in a two-dimensional homogeneous medium, letuincx , t be a plane wave with wave number k in direction S=−cos inc ,sininc ,

5

Plane wave direction

i.e., uinc x , t =Aeik x⋅S−ct . Then the diffracted wave is

udiff x , t =uinc x tip ,0

rD d ,inc , k eik r−ct = A

rD d ,inc , k eik x tip⋅Sr−ct , (2.9)

where r is the distance to the tip of the half plane, r=∣x− x tip∣ . The diffractioncoefficient D d ,inc , k for a half plane can be derived mathematically. It is :

D d ,inc , k =ei/4

22k 1

cosd−inc

2

± 1

cosdinc

2

, (2.10)

with the definition of the parameters

inc , angle from the half plane to the incident ray,

d , angle form the half pane to one of the diffracted rays,

k , wave number.

Formula (2.9) and (2.10) are accurate for high frequency waves.

Another type of diffraction is called “ creeping rays ” which is generated fromsmooth scatterers as shown in figure 2.5 below. The incident field U inc inducescreeping rays U c at the north (and south) pole of the cylinder. As the creeping raypropagates along the surface, it continuously emits surface-diffracted rays U d . Anumerical simulation for this type of diffraction is shown in example 4 in section 4.

Figure 2.5 Diffraction by a smooth cylinder.

6

Uinc

Uc

UdUc

3. Numerical Method

In our numerical simulations, we only consider the 2D problems. The typical testcase is shown in figure 3.1 below. We suppose most of the test box consists of the samematerial, but, a small part of it contains another type of material, which means there isdifferent local speed of wave propagation, c , in this part. The half plane is inserted inthe proper position and we assume it has infinite length. We sample the solution in thepoints on the half circle around the tip of the half plane when we compute the diffractioncoefficients.

Figure 3.1 The whole white box contains the same material except the gray

rectangle part and the black plate. The half circle contains the sample points.

3.1 Numerical solver

3.11 Numerical scheme

A second order numerical approximation method is applied in the numericalcomputations. The method is called leapfrog and uses central difference approximationformula both in time and space. Suppose the analytical solution to the wave equation

(2.1) u x i , y j , tn can be approximated by the numerical solution u nij

, i.e.

u nij≈u x i , y j , tn , where x i=i x , y j= j y and tn=n t .

Equation (2.1) can be approximated with the numerical method at x i , y j , tn .Under the assumption h= x= y , i.e. a uniform grid, we have

7

un1ij

−2u nijun−1

ij

t2 =cij2u ni1 j

−2u niju n

i−1 j

h2 cij2u nij1

−2u niju n

ij−1

h2

, (3.1)

where the local speed of wave propagation is cij=c x i , y j .

We verify that it is the second order of accuracy both in time and space bychecking a sequence of numerical values at a point when the grid size is refined. Wecalculate the difference between two successive numerical solutions at that point for grid

sizes h /2,h ,2h . Then Rh :=∣u2h−uh∣∣uh−uh/2∣

≈2−p,where p is the order of accuracy. The

resulting Rh is shown in figure 3.2 below. Clearly, p≈2 in our case.

Figure 3.2 (a) Order of accuracy in space, (b) order of accuracy in time

3.12 Boundary conditions

Periodic boundary conditions are used in the numerical simulation of the foursides and Dirichlet boundary condition is used on the plate in figure 3.3 below, i.e.,

8

u ni N

=u ni1, i=1, ... ,M , u n

Mj=u n

1 j, j=1,... , N for the sides of the box, where

M ,N are the number of grid points in X- and Y-direction respectively, and u nij=0

when x i , y j∈plate . The periodic conditions means we are actually solving aninfinite problem as in figure 3.3 and we extracted one from all the boxes. Also note thatadding the plate to the problem introduces a singularity at the tip of the plate whichreduces the formal numerical order of accuracy in the simulation.

Figure 3.3 The gray box is extracted from the whole testing boxes. Dirichlet boundary

condition is applied on the plate. In this special case the incident angle is 3/4 .

3.13 Initial conditions

We use a plane wave as initial data. The wave has an incident angle inc as infigure 2.4. The initial conditions are set in (3.2) and (3.3) such that it satisfies the periodicboundary conditions and the plane wave propagate in the forwarding S direction:

u0=u x ,0= f x⋅S , (3.2)

u1=u x , t = f x⋅S−c t , (3.3)

where f is the profile of the plane wave f =ecos2/d −1

w , d is a scaling factor

given by d=S⋅ x L , yL

2, and x L , yL are the size of the box in the X- and Y-direction

respectively, c is the local speed of wave propagation, and w is the 'width' of theprofile f as shown in figure 3.4 below. The exact solution of this problem without

9

Plate : Dirichlet BC

Box sides: Periodic BC

the half plane, is given by (2.2) with g ≡0 .

Figure 3.4 The definition for w

For simplicity, a special case, inc= as shown in figure (3.5) below, is takenas a starting point. In this special case, the plane wave moves in one direction along theX-axis and then the initial conditions (3.2) and (3.3) can be reduced to the ones in(3.4)and (3.5), in which we do not need to care about the Y-direction,

u0=u x ,0= f x−x p , (3.4)

u1=u x , t = f x−x p−c t . (3.5)

Here S=1 ,0 , d=xL , and x p is the location of the initial pulse as in figure 3.5.Together with the periodic boundary conditions and central difference formula, we cannow do the numerical simulation of the wave equation (2.1).

10

Figure 3.5 Special case inc=

3.2 Angles

For arbitrary angles, we cannot set our periodic boundary conditions soconveniently if we fix the box size. We can see the inconvenience in figure 3.6 below.

Figure 3.6 Arbitrary incident angles for a fixed box size .

Since the periodic boundary conditions are employed, the initial data also needs tobe periodic. Then if the initial wave pulse with an arbitrary incident angle is putsomewhere in the fixed box as the bold one in figure 3.6 above, because of the restriction

11

Plane wave direction

Diffracted rays

Xp

of the periodic conditions, more and more initial wave pulse should be added in the box.The number of pulses is not limited and will in general eventually fill up the box. Instead,we always put the initial wave pulse on the diagonal of the rectangle, but change the boxsize such that it can easily satisfy periodic boundary conditions for different incidentangles. See figure 3.7 below.

Figure 3.7 The initial wave pulse is put on the diagonals of the rectangles. The incident

angle only depends on the box size: inc=−=−arctan x L/ yL

The incident angle inc depends on and can be determined by the box size

xL and yL .

3.3 Numerical calculation of diffraction coefficients

The discrete Fourier transform is applied to transform the time domain intofrequency domain or the reverse. Suppose the function g is 2− periodic. Let ussample g by gn :=g tn , where tn=n t=2n /N for some integer N . Definethe discrete Fourier transform for the sampled data

gk :=1N ∑

n=0

N−1

gne−ikt n , k=−N /2,... , N /2−1 , (3.6)

The inversion formula is then given by

12

Y

X

Φ

Φ

Φ

θinc

Ŝ

gn= ∑k=−N /2

N /2−1

gk eikt n ,n=0,... , N−1 , (3.7)

The Fourier coefficients gk represents the amplitude and the phase of wave numberk in the function g .

To compute the discrete Fourier transform (DFT) of an N− point sequenceusing equation (3.6) would take O N 2 multiplications and additions if a naïve methodwas used. The fast Fourier transform (FFT) algorithm computes the DFT using onlyO N ln N multiplications and additions.

According to the GTD theory, a simple wave uinc x , t would generate a simplediffracted wave udiff x0, t given by the formula (2.9). Consider some fixed point

x= x0 for the problem in figure 2.4. Let r=∣xo− x tip∣ , where x tip is the tip positionof the half plane. Then,

uincx , t =Ae ik x⋅S−ct , (3.8)

udiff x0 , t =A

reik x tip⋅Sr−ct D d ,inc , k , (3.9)

A general solution is generated as a superposition of simple waves:

uinc x , t = ∑k=−∞

∞

Ak e−ik x⋅S−ct , (3.10)

udiff x0 , t = ∑k=−∞

∞ Akr

Dinc ,d , k e−ik x tip⋅Sr−ct , (3.11)

On the other hand, by Fourier series:

uinc x0 , tn= ∑k=−N /2

N /2−1

U inck

eikt n , (3.12)

udiff x0 , tn= ∑k=−N /2

N /2−1

U diffk

eikt n , (3.13)

Let us now assume that uinc and udiff are well approximated by its first NFourier modes. Formulas 3.10~3.13 are then all finite Fourier series, and by itsuniqueness property

13

Ak e−ik x0⋅S= U inck

, (3.14)

1

rAk D inc ,diff , k e−ik x tip⋅Sr = U diff

k. (3.15)

From this, we obtain the formula (3.16) for the diffraction coefficient, which is theessential point in the programming.

D inc ,diff , k =U diffk

U inck

r eik [ x tip− x0⋅Sr ], (3.16)

The total solution in x0 is u x0 , t =uinc x0 , t u

diff x0 , t ,and the diffractedwave at the point is then

udiff x0 , t =u x0 , t −uinc x0 , t , (3.17)

Since u x0 , t can be obtained directly from the numerical simulations anduinc x0 , t from the initial condition at the sample points we can can get a time-trace

for udiff x0 , t at any fixed point x0 . Therefore, both U diffn

and U incn

can be

calculated directly by the Matlab tool FFT for vectors of length N . In Matlab, thefrequency is scaled by 2 and in order to get real frequency, and it has to be re-scaled.

U diffk

=1/N ∑n=−N /2

N /2−1

udiff x0 , t n−n−1k−1 , (3.18)

U inck

=1/N ∑n=1

N

uinc x0 , t n−n−1k−1 , (3.19)

where =e−2 i/N .

3.4 Complexity and accuracy

In order to analyze the numerical error sources and computational cost, we nowdiscuss some of the “numerical parameters” used in the method as shown in table 3.1below.

14

Parameters Its meaning

h Grid size

t Time step size

w “Width”of the profile of the initial plane wave

T Simulation time

cfl CFL number

Table 3.1 Parameters VS its meanings

3.4.1 Numerical error sources

The “width” of the profile of the initial plane wave, w , determines how manyfrequencies we can compute. For different w the absolute values of the Fourier

transform of uinc x0 , tn , ∣

U inck

∣ is given by the Fourier transform of the function for

the initial plane wave profile in figure 3.4. That is because a Fourier transform ofuinc x , tn in time is the same as a Fourier transform in space of uinc x0 , t0 , i.e., ofu0 , the initial plane wave. A brief proof for a simple case as shown in figure 3.5 is

given below. In this special case S=1,0 and the local speed c≡1 , anduinc x0 , tn only depends on x0 and t . At the initial time t0 , no diffracted waves

happen at the tip of the plate, using formula (3.17), and we have

uinc x0 ,0=u x0 ,0=u0 . Let x=x0−t , and then ∣

U inck

∣ can be obtained as

follows.

∣ U inck

∣=∣∫ ∞−∞

uinc x0 , tne−ikt dt∣=∣∫ ∞

−∞Aeik x0−t e−ikt dt∣

=∣∫ ∞−∞

Aeikx e−ik x0−x d x0−x ∣=∣∫ ∞−∞

Ae2 ikx dx∣

=∣∫ ∞−∞

uinc x ,0eikx dx∣=∣∫ ∞−∞

u0eikx dx∣ . (3.20)

15

For a more general case:

∣ U inck

∣=∣∫ ∞−∞

uinc x0 , tne−ikt dt∣=∣∫ ∞

−∞f x0⋅S−t e−ikt dt∣

={x= x0⋅S−t }=∣∫ ∞−∞

f x e−ik x0⋅S−x d x0⋅S−x ∣

=∣∫ ∞−∞

f x eikx dx∣ . (3.21)

Figure 3.8 Absolute values for the Fourier transform of

uinc x0 , t0 with different w and fixed grid size h=1/1280 .

See figure 3.8 above for an example of U inck

for w varying from 1/200 to

1/100000 . The numerical errors in the coefficients will go to infinite values asU inck

0 . Therefore, for larger w , fewer frequencies can be used to compute the

diffraction coefficients as in formula (3.16). In order to check this point, we take

16

w=[1/2001/5001/1000] as examples to compute the coefficients. The results togetherwith the expected mathematical value from (2.10) are shown in figure 3.9 for a typicalcase. It would clearly be better to take smaller w in our test examples to capture thediffraction coefficient D for higher frequencies.

Figure 3.9 Larger w introduces larger error for high frequencies. Simulation

time T=1.414 , grid size h=1/1280 , w=[1/2001/5001/10001/5000]

However, on the other hand, in the direct numerical simulation of (2.1), theaccuracy of the solution is determined by the number of grid points per wavelength. Inorder to obtain a reasonable numerical solution for (2.1), the grid size h must be smallenough such that the solution in the pulse is resolved by sufficiently many points, i.e., weneed h≪w . In conclusion, small w means we can compute higher frequencies but ata higher cost. In figure 3.10, we show a pulse with w=1/500 resolved with differentgrid size h=1/50~1/500 .

17

Figure 3.10 Number of points inserted in the peak of the wave varies with

varying grid size h and fixed 'width' of the initial value w=1/500

When doing FFT, a sufficiently long time trace should be obtained to get anaccurate FFT solution. As shown in figure 3.11 below, the time tracing for u x0 , t atsome fixed point 0.4 ,0.5 . Because of the periodic boundary conditions the incidentplane wave “comes back” at t≈1.00 . It would be better to do FFT on the time interval

00.9 than the interval 00.7 , but we cannot use an interval 0t for t1 .We can imagine if let time evolve even longer before the incident plane wave comesback, it would become more and more accurate for the FFT because of the smaller “timeevolving error” (see figure 3.11). However, the computational domain must then beenlarged, which leads to higher computational cost. In addition, the diffracted waves will“come back” because of the periodic conditions, c.f. figure 4.2. It cannot be subtractedlike the incident waves as in formula (3.17).

18

Figure 3.11 Time evolving error

The effect of the time evolving error can be reduced by using “windowing”, i.e.,multiplying the time trace by a smooth cut-off function before Fourier transforming. Thecutoff function would be constant one inside the time interval and zero outside aneighborhood of the interval. Using formula (3.17), we get the time trace udiff x0 , t upto t≈t0 . Before applying FFT to it, we multiply it by a “windowing” function definedas follows:

=1

2

−atant−t0

. (3.22)

We tested windowing for Example 1 in Section 4.1.

3.4.2 Computational cost

Since an explicit method is applied here, the complexity for one time step is

19

O xL/h yL /h , in particular O 1/h2 for the box with size xL= yL=1 . Thetime step t is limited by the CFL (Courant-Friedrich-Levy) condition for numericalstability(two space dimensions) with x= y=h on the uniform grid,

t 1

c 1

x2 1

y2

= 1

c 1

h2 1

h2

= hc2 , (3.23)

and cfl=c th

1

2. The total number of time-steps up to time T is

T t

= Th⋅cfl .

Then the complexity for the whole problem is O T⋅xL⋅yLh3⋅cfl

. It is obvious that h ,

T and box size xL , yL are sources of the computational cost for this type ofnumerical simulation. In order to get reasonable solutions, proper values of theseparameters should be considered as discussed above. We can check the computationalcost with the relationship to one of the these parameters. For example, let the grid sizeh vary, but fix the others. The computational cost is obtained in figure 3.12 below. For

small h , the experimental cost results agree with the theoretical results.

Figure 3.12 Computational cost for varying grid size h

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4. Numerical Simulation Results

4.1 Example 1

In this example, the initial wave pulse is put in x p=0.4 , the dashed line infigure 4.1 below. The thin plate is the solid line inserted in y=0.5 and x=0.51 .The circle points are the sample points we used to compute the diffraction coefficients byFFT. In this case, the “width” of the profile: w=1/1000 , incident angle: inc= andgrid size h=1/1280 . The test model is shown in figure 3.5 in section 3.

Figure 4.1 Numerical simulation for the diffracted waves,

solid line: position of the half plane,

dash line: pulse position of the initial plane wave,

circle points: sample points to calculate the diffraction coefficient

From the simulation we get u x0 , t , the time-trace at the sample points. Thevalues in figure 4.3 below are the ones extracted from the central position of the totalsamples, u 0.4 ,0.5 , t . At the initial time, the pulse of the plane wave just goesthrough this point. Then the plane wave goes along the positive X-direction until it hitsthe tip of the half plane and then the diffracted rays happen at that moment and they willshoot out in all directions from the tip . After a while some diffracted ray will reach the

21

central point 0.4 ,0.5 at time t≈0.2 . Since periodic boundary conditions areemployed here, the pulse of the plane wave will arrive at the this point at time 1.00 andthe new diffracted waves also come to it later.

Figure 4.2 Diffracted waves at sequential time points. h=1/640 , w=1/500

22

Figure 4.3 Time-trace for point 0.4,0.5

Using formula (3.17), udiff x0 , t can be calculated and uinc x0 , t can be

obtained from the initial plane wave. Then U diffk

and U inck

can be obtained at certain

points by formula (3.18) and (3.19). They are functions of wave number k as defined inMatlab. The numerical results are shown in figure (4.4) below. Then by the numericaldiffraction coefficients formula (3.16), the coefficients at these sample points can easilybe calculated. We just consider the amplitude in the simulation:

∣D inc ,diff , k ∣=∣U diffk

U inck

r eik x tipx0r ∣=∣U diffk

U inck

r∣ , (4.1)

23

Figure 4.4 U diffk

and U inck

at 0.4,0.5

At first, we calculate the diffraction coefficients at a fixed sample pointx0=0.4 , y0=0.5 for all frequencies and compare the numerical results with the

mathematical ones, formula (2.10) stated in section 2. Just the amplitudes are considered.For the incident angle inc= in this special case,

∣D d ,inc , k ∣=∣ 1

22k 1

cosd−

2

± 1

cosd

2

∣

=∣ 1

42k sininc2

∣ . (4.2)

The comparison between the numerical and mathematical solutions are shown in figure(4.5) below. Good results are obtained up to k≈400 .

24

Figure 4.5 Absolute values of numerical and mathematical

diffraction coefficients at 0.4,0.5

In the sequence of plots in figure 4.6, the diffraction coefficients at all of thesample points, but, just for one special frequency in each of the plot is shown. Thismeans, in formula (2.10), the diffraction angle d is varied and wave number k isfixed in each of these plots. Comparing the numerical solutions with the mathematicalones, they agree well for not too low and not too high frequencies. For high ones, weneed to insert more grid points in order to resolve the solutions in the wave pulse asdiscussed in section 3. The formula (2.10) is accurate for high frequency and it may be aapproximation for the lower ones. That is the reason why the numerical solutionsconform better to the mathematical ones as the frequency increasing. See figure 4.6below. The diffraction coefficient has been scaled by k to simplify comparisons.

25

26

Figure 4.6 Scaled diffraction coefficients at sample points with different wave number sequentially

27

To test the “windowing” technique we note that the efficient time interval ist∈[0 ,0.90] . We therefore take t0=0.75 and =0.01 in formula (3.22) for our

numerical test. We get the modified trace by multiplying P x0 , t =t ⋅udiff x0 , t .

See figure 4.7 below.

Figure 4.7 Diffracted waves at a sample point ( 0.4 , 0.5 ) in the efficient time interval [0 ,0.9] .

(a), without multiplying by “windowing” function, (b), after multiplying by “windowing” function

Now, we do the calculation of the diffraction coefficients. First, U diffk

and

U inck

are calculated as in figure 4.8 below. Comparing the result with the one in figure

4.4, we see it has a little improvement for higher frequencies. From that, diffractioncoefficients for higher frequencies can be calculated as shown in figure 4.9, c.f. figure4.5.

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Figure 4.8 U diffk

and U inck

at 0.4,0.5 with “windowing”

Figure 4.9 Absolute values of diffraction coefficients at ( 0.4 , 0.5 ) with “windowing”

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Figure 4.10 Scaled diffraction coefficients as function

of diffraction angle, wave number k=481.4668

We also calculate the diffraction coefficients as function of diffraction angle for alarge wave number, k up to 481.4668 , see figure 4.10. By this “windowing”numerical scheme, we can thus to compute the diffraction coefficients for higherfrequency, c.f. figure 4.6.

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4.2 Example 2

This example is an arbitrary incident angle case. The test model is given in figure3.7 in section 3. We did the same simulation and calculation for the diffractioncoefficients as in the previous example. From these numerical results, we arrive at similarconclusions as the previous ones. In this case, the incident angle is

inc=−arctan 1.8/1.4=2.2318 , the “width” of the profile of the initial plane waveis w=1/500 , and the grid size is h=1/800 .

Figure 4.11 Numerical simulation for the diffracted waves with an arbitrary incident angle,

solid line: position of the half plane,

dashed line: pulse position of the initial plane wave,

circle points: sample points to calculate the diffraction coefficient,

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Figure 4.12 Diffracted waves at sequential time points

32

Figure 4.13 Time-trace at the center of the sample points

Figure 4.14 U diffk

and U inck

at the center of the sample points

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Figure 4.15 Diffraction coefficient at the center of the sample points

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35

Figure 4.16 Scaled diffraction coefficients at sample points with different wave number sequentially,k=7.9337,15.8674,31.7349 ,71.4035,111.0721,150.7407,190.4093,230.0779

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4.3 Example 3

This type of test model is show in figure 3.1 in section 3. The gray part contains adifferent type of material compared with the other part of the box, which leads todifferent local speeds of wave propagation: the white part, C1=1.0 , the gray part,C2=0.5 . The numerical size for the box, [0,1.0]×[0,1.0] , the gray parts,[0.453125,1.0]×[0.475,0.525] , and the plate [0.503125,1.0 ]×[0.4875,0.5125] . For

this case, the incident angle is inc= , the “width” of the profile is w=1/500 andthe grid size is h=1/640 .

Figure 4.17 Numerical simulation for the diffracted waves,

smallest rectangle: position of the plate,

circle points: sample points to calculate the diffraction coefficient,

larger rectangle: the position inserted with another type of material

For this more complicated simulation case, we have no mathematical solutions tocompare with because of the complicated geometry and different types of materials. Fromthe numerical calculation, it is interesting to note that D does not decay with k asbefore (Figure 4.21) and also we find diffraction coefficients D≈0 for some d andwave numbers k (Figure 4.22).

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Figure 4.18 Diffracted waves at sequential time points

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Figure 4.19 Time-trace at 04 ,0.5

Figure 4.20 U diffk

and U inck

at 0.4 ,0.5

39

Figure 4.21 Diffraction coefficient at 0.4,0.5

40

Figure 4.22 Diffraction coefficients at sample points with different wave number

sequentially, k=6.9702,13.9404,27.8809,62.7319,97.5830,132.4341

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4.4 Example 4

This example just shows the diffraction caused by a smooth scatterer. We did notcalculate the diffraction coefficients at some sample points. The diffraction phenomenaare recorded at a sequence of time points. They can be used as an experimentalexplanation to the type of diffraction called “creeping rays” in figure 2.5 in section 2.

Figure 4.23 Diffraction caused by a smooth scatterer at time=0.3536

As we expected from the point of view of GTD, we can see the diffracted wavesbehind the cylinder where GO would give zero solutions. Compared with GO, GTDremoves the discontinuity in the GO solution and gives a more reasonable description ofthis type of phenomena.

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Figure 4.24 Diffracted waves at sequential time points

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4.5 Example 5

This is another special case with an incident angle inc=/2 . The numericaldiffraction coefficients are calculated and compared with the mathematical formula(2.10). However, at some sample points at the most left or right positions of the samplecircle, the diffracted angles are d≈/2 or d≈3/2 . Applying them into theformula (2.10), these two terms will go to infinities as follows:

1

cosd−inc

2

≈ 1

cos3/2−/2

2

= 1

cos2

=∞, (4.3)

1

cosd−inc

2

≈ 1

cos/2/2

2

= 1

cos2

=∞. (4.4)

That is the reason why the mathematical solutions of the diffraction coefficients at themost left and right sample points lose their accuracy by formula (2.10). However, it stillworks for some sample points near the central part of the samples. For example, the timetracing values for u 0.6 ,0.7 , t at that point is recorded in figure 4.27. With the same

procedure as in example 1, U diffk

and U inck

can be calculated and the absolute value

of coefficients at that point ∣D∣ as well. See figure 4.28 and 4.29.

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Figure 4.25 Numerical simulation for the diffracted waves with an special incident angle,

solid line: position of the half plane,

dashed line: pulse position of the initial plane wave,

circle points: sample points to calculate the diffraction coefficient,

the incident angle inc=/2 , the “width” of the profile w=1/1000

45

Figure 4.26 Diffracted waves at sequential time points

46

Figure 4.27 Time-trace at the sample point 0.6 ,0.7

Figure 4.28 U diffk

and U inck

at 0.6 ,0.7

47

Figure 4.29 Absolute values of numerical and mathematical

diffraction coefficient at 0.6 ,0.7 , h=1/1000 , w=1/1000

The absolute values of the diffraction coefficients at the sample points for onespecial frequency are calculated as in example 1. The comparison between the numericaland mathematical solutions are given in figure 4.30. Skipping the two outermost samplepoints and comparing the results with the numerical ones as in figure 4.30, the numericalsolutions do not conform to the mathematical ones close to d=±/2 as expected.

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Figure 4.30 Diffraction coefficients at sample points excluded two special points, k=45.568

49

5. AcknowledgmentI would like to express the great appreciation to my supervisor, Professor Olof

Runborg who provide me such a good opportunity to study this interesting project andlots of good suggestions and instructions through my thesis work.

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6. Reference[1]. G.B.Whitham(1974) , Linear and nonlinear waves, Wiley-interscience.

[2]. Olof Runborg and Björn Engquist, 'Computational high frequency wave propagation',Acta Numerica (2003), 1-86.

[3]. Jean-David Benamou, Francis Collino and Olof Runborg, 'Numerical microlocalanalysis of harmonic wavefields', September 7, 2003.

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