**4. Verification of numerical results**

The numerical results discussed in the results section were obtained after implementation of the above schemes in a MATLAB code. A number of tests were applied to verify its correctness, including those applied in [5]. Analytical solutions were derived for a circular scatterer for the three boundary conditions and the Mie series method was used to compute an actual solution. This enabled comparison of the scattered field computed by the methods described in this section for a circular scatterer. For all three boundary conditions the error was in the order of 10−<sup>15</sup> which was considered a suitable tolerance. Also, the condition number of the systems was checked to ensure that uniqueness problems arising for wavenumbers *k* near an interior Dirichlet eigenvalue of the scatterer were avoided.

However, there is no analytical expression for the scattered field from a non-circular scatterer and as such, there is no true solution to which we can compare results. For this study, we use a significant digit measurement to determine the convergence of the solution.

We choose a point x in the domain external to the scatterer and compute the field. As the number of quadrature points increases, if the solution is convergent, the number of significant digits in agreement increases. Thus we measure the number of unchanging digits in the approximate solution as the number of quadrature points *N* increases, and terminate the calculation when the truncation of the computed value to a prespecified number of significant digits does not change as *N* increases.

Two measures were used determine the convergence of the solutions. Firstly, a near field measure of the real and imaginary parts of the scattered field *usc*. This measurement was taken at a radius *r* = 10 from the origin in the direction x = (−1, 1).

The second measure employs the far-field. It is measured in a specified direction ˆx. For the Dirichlet boundary condition, the far field pattern is calculated as

$$\mathfrak{u}\_{\infty}(\hat{\mathfrak{x}}) = \frac{e^{-i\frac{\pi}{4}}}{\sqrt{8\pi k}} \int\_{\partial D} \left\{ k n(y) \cdot \hat{\mathfrak{x}} + \eta \right\} e^{-ik\hat{\mathfrak{x}}\cdot\mathfrak{x}} \phi\left(y\right) ds\left(y\right), \qquad |\hat{\mathfrak{x}}| = 1,\tag{79}$$

and for Neumann and impedance boundary conditions the calculation is

$$\mu\_{\infty} \left( \hat{\mathfrak{x}} \right) = \frac{e^{-i \frac{\mathfrak{x}}{4}}}{\sqrt{8 \pi k}} \int\_{\partial D} e^{-ik \hat{\mathfrak{x}} \cdot \mathbf{y}} \phi \left( y \right) ds \left( y \right) \, \qquad \left| \hat{\mathfrak{x}} \right| = 1. \tag{80}$$

### **5. Results and discussion**

### **5.1. Effect of corner rounding on a domain with a single corner**

Consider the curve given by the parametric representation (it is half of the so-called lemniscate of Gerono):

$$x = x(t) = a \left( 2 \sin \frac{t}{2}, -\sin t \right), \qquad t \in [0, 2\pi], \tag{81}$$

where *a* is a parameter. It has the corner at *t* = 0 and with an interior right angle. Henceforth the parameter *a* is set equal to 1 length unit.

We will also consider a family of curves in which the corner has been rounded; the family is parameterized by the quantity *ε*(0 ≤ *ε* ≤ 1) :

$$x = x(t) = a \left( 2\sqrt{\varepsilon^2 + (1 - \varepsilon^2)\sin^2\frac{t}{2}}, -\sin t \right), \qquad t \in [0, 2\pi]. \tag{82}$$

Figure 1 illustrates the two shapes, with *a* = 1. The radius of curvature *ρ* at the corner point *t* = 0 is 2*ε*/ <sup>1</sup> <sup>−</sup> *<sup>ε</sup>*2 .

What Effect does Rounding the Corners have on Diffraction from Structures with Corners? http://dx.doi.org/10.5772/61152 17

in the approximate solution as the number of quadrature points *N* increases, and terminate the calculation when the truncation of the computed value to a prespecified number of

Two measures were used determine the convergence of the solutions. Firstly, a near field measure of the real and imaginary parts of the scattered field *usc*. This measurement was

The second measure employs the far-field. It is measured in a specified direction ˆx. For the

<sup>−</sup>*ikx*ˆ·*y<sup>φ</sup>* (y) *ds*(y), <sup>|</sup>xˆ<sup>|</sup> <sup>=</sup> 1, (79)

, *t* ∈ [0, 2*π*], (81)

, *t* ∈ [0, 2*π*]. (82)

<sup>−</sup>*ikx*ˆ·*y<sup>φ</sup>* (y) *ds*(y), <sup>|</sup>xˆ<sup>|</sup> <sup>=</sup> 1. (80)

{*k*n(*y*) · xˆ + *η*} *e*

and for Neumann and impedance boundary conditions the calculation is

4 √ 8*πk* 

**5.1. Effect of corner rounding on a domain with a single corner**

 2 sin *<sup>t</sup>* 2 , − sin *t*

Consider the curve given by the parametric representation (it is half of the so-called

where *a* is a parameter. It has the corner at *t* = 0 and with an interior right angle. Henceforth

We will also consider a family of curves in which the corner has been rounded; the family is

Figure 1 illustrates the two shapes, with *a* = 1. The radius of curvature *ρ* at the corner point

*<sup>ε</sup>*<sup>2</sup> <sup>+</sup> (<sup>1</sup> <sup>−</sup> *<sup>ε</sup>*2) sin<sup>2</sup> *<sup>t</sup>*

2 , − sin *t*

*∂D e*

significant digits does not change as *N* increases.

16 Advanced Electromagnetic Waves

*<sup>u</sup>*<sup>∞</sup> (xˆ) <sup>=</sup> *<sup>e</sup>*−*<sup>i</sup> <sup>π</sup>*

**5. Results and discussion**

lemniscate of Gerono):

*t* = 0 is 2*ε*/

 <sup>1</sup> <sup>−</sup> *<sup>ε</sup>*2 .

4 √ 8*πk* 

*∂D*

*<sup>u</sup>*<sup>∞</sup> (xˆ) <sup>=</sup> *<sup>e</sup>*−*<sup>i</sup> <sup>π</sup>*

x = x(*t*) = *a*

 2 

the parameter *a* is set equal to 1 length unit.

parameterized by the quantity *ε*(0 ≤ *ε* ≤ 1) :

x = x(*t*) = *a*

taken at a radius *r* = 10 from the origin in the direction x = (−1, 1).

Dirichlet boundary condition, the far field pattern is calculated as

**Figure 1.** Leminscate (blue). The interior (red) curve with rounded corner has parameter *ε* = 0.05 (*ρ* = 0.1).

The near- and far-fields were computed for each of the boundary conditions using the graded mesh (61) for the lemniscate (81). A variety of angles of incidence were tested and in the case of the impedance loaded lemniscate a number of impedance parameters were tried. All tests were performed for *ka* = 1, 5, 10 and 2*π*. Colton and Kress [4] have published results for the Dirichlet boundary condition. We were able to reproduce these results. In all cases an examination of the convergence rate as a function of *N* was observed to be exponentially fast (super-algebraic). Some typical results are as follows. Table 1 shows the scattered near- and far-field patterns for the lemniscate illuminated by a plane wave incident at angle *θ*<sup>0</sup> = 0 with *ka* = 2*π*. For the impedance boundary condition, the impedance parameter shown is *λ* = 1 + *i*.

We then examined the effect of rounding the corner of the lemniscate. The near- and far-fields were computed for each of the boundary conditions using a uniform mesh *tj* <sup>=</sup> *<sup>π</sup><sup>j</sup> <sup>n</sup>* , for *j* = 0, 1, ..., 2*n* − 1, in the parameterisation (36) of the rounded lemniscate (82) and the lemniscate (81).

A variety of angles of incidence were tested and in the case of the impedance loaded scatterers a number of impedance parameters were tried. All tests were performed for *ka* = 1, 5, 10 and 2*π*, and radii of curvature *ρ* = 0.1, 0.08, ..., 0.02, 0.01. The results were similar in all cases. As expected a decrease in the radius of curvature shows a decrease in the rate of convergence. For radius of convergence *ρ* = 0.1 use of a uniform mesh achieves 10 significant digits of agreement and eventually for small radii (*ρ* < 0.04) the solution fails to converge (agreement of less than 6 significant digits). The results for the lemniscate, as expected, exhibit non-convergence.

The same series of experiments were then re-run using the graded mesh (61). In all cases this discretization method exhibits superior results. Super-algebraic convergence was


**Table 1.** Direction of incident plane wave *<sup>θ</sup>*<sup>0</sup> = 0 with *ka* = <sup>2</sup>*π*, *<sup>d</sup>* = (1, 0) and *<sup>u</sup>sc* (*x*) for *<sup>x</sup>* = (−1, 1) with *kr* = <sup>20</sup>*π*.

observed in all cases when examining the convergence rate as a function of *N*, demonstrating the advantage of using the graded mesh. In all cases 15 significant digit convergence was achieved. Of interest is the observation that even though the rounded lemniscate (82) has a smooth boundary *∂D*, as the radius of curvature decreases use of the uniform mesh for discretization fails to produce a convergent solution for small radii of curvature. This suggests that the type of discretization method chosen should be decided on a more sophisticated approach rather than a simplistic smooth versus not smooth criterion.

A set of typical results is provided in Table 2 which shows the values for the near-field using uniform mesh and then using graded mesh for a scatterer with the impedance boundary condition with impedance parameter *λ* = 1 + *i*, illuminated by a plane wave incident at angle *θ*<sup>0</sup> = 0 with *ka* = 2*π*. Table 3 shows the results of the far-field for the same experiments.

Having established that the graded mesh gives superior results for the rounded lemniscate, we attempt to answer the concern that rounding the corner will produce significant deviation from the solution where corners are not rounded. The difference between the actual solution, *uL* <sup>∞</sup> (xˆ) for ˆ<sup>x</sup> <sup>∈</sup> [0, 2*π*] , and that produced by rounding, *<sup>u</sup><sup>R</sup>* <sup>∞</sup> (xˆ), is measured using the *L*<sup>2</sup> norm

$$\left\| u\_{\infty}^{L} - u\_{\infty}^{R} \right\|\_{2} = \left( \int\_{0}^{2\pi} \left| u\_{\infty}^{L} \left( \left. \hat{\mathfrak{a}} \right| - u\_{\infty}^{R} \left( \left. \hat{\mathfrak{a}} \right| \right)^{2} d\hat{\mathfrak{a}} \right)^{\frac{1}{2}} \right) \right. \tag{83}$$

and *L*<sup>∞</sup> norm

$$\left\| u\_{\infty}^{L} - u\_{\infty}^{R} \right\|\_{\infty} = \max\_{\hat{\mathbf{x}} \in [0, 2\pi]} \left| u\_{\infty}^{L} \left( \hat{\mathbf{x}} \right) - u\_{\infty}^{R} \left( \hat{\mathbf{x}} \right) \right|. \tag{84}$$


**Lemniscate using graded mesh** *N* Re *usc*(x) Im *usc*(x) Re *u*∞(d) Im *u*∞(d)

32 -0.07494830903628 -0.07116098685594 -1.87242780404153 1.24490326848555 64 -0.07494835562512 -0.07116093299795 -1.87243588474719 1.24489457829233 128 -0.07494835564211 -0.07116093293816 -1.87243588474320 1.24489457829267 256 -0.07494835564212 -0.07116093293813 -1.87243588474320 1.24489457829268

32 0.04164120404373 0.03521714231575 1.59458645194898 0.92713146438117 64 0.04164071915392 0.03521722967811 1.59457738453702 0.92713314620758 128 0.04164071916034 0.03521722965359 1.59457738456520 0.92713314620969 256 0.04164071916034 0.03521722965358 1.59457738456522 0.92713314620969

32 0.00222570467763 -0.04334130654021 1.26634214415129 1.65780088985777 64 0.00222588468293 -0.04334146584422 1.26633733538116 1.65780860014239 128 0.00222588466664 -0.04334146583637 1.26633733538197 1.65780860013947 256 0.00222588466664 -0.04334146583637 1.26633733538197 1.65780860013947

**Table 1.** Direction of incident plane wave *<sup>θ</sup>*<sup>0</sup> = 0 with *ka* = <sup>2</sup>*π*, *<sup>d</sup>* = (1, 0) and *<sup>u</sup>sc* (*x*) for *<sup>x</sup>* = (−1, 1) with *kr* = <sup>20</sup>*π*. observed in all cases when examining the convergence rate as a function of *N*, demonstrating the advantage of using the graded mesh. In all cases 15 significant digit convergence was achieved. Of interest is the observation that even though the rounded lemniscate (82) has a smooth boundary *∂D*, as the radius of curvature decreases use of the uniform mesh for discretization fails to produce a convergent solution for small radii of curvature. This suggests that the type of discretization method chosen should be decided on a more

sophisticated approach rather than a simplistic smooth versus not smooth criterion.

 2*π*

0

<sup>∞</sup> <sup>=</sup> max *x*ˆ∈[0,2*π*]

 *uL*

<sup>∞</sup> (xˆ) <sup>−</sup> *<sup>u</sup><sup>R</sup>*

 *uL* <sup>∞</sup> (xˆ) 2 *d*xˆ 

<sup>∞</sup> (xˆ) <sup>−</sup> *<sup>u</sup><sup>R</sup>*

<sup>∞</sup> (xˆ) 

<sup>∞</sup> (xˆ) for ˆ<sup>x</sup> <sup>∈</sup> [0, 2*π*] , and that produced by rounding, *<sup>u</sup><sup>R</sup>*

 *uL* <sup>∞</sup> <sup>−</sup> *<sup>u</sup><sup>R</sup>* ∞ 2 =

> *uL* <sup>∞</sup> <sup>−</sup> *<sup>u</sup><sup>R</sup>* ∞

A set of typical results is provided in Table 2 which shows the values for the near-field using uniform mesh and then using graded mesh for a scatterer with the impedance boundary condition with impedance parameter *λ* = 1 + *i*, illuminated by a plane wave incident at angle *θ*<sup>0</sup> = 0 with *ka* = 2*π*. Table 3 shows the results of the far-field for the same experiments. Having established that the graded mesh gives superior results for the rounded lemniscate, we attempt to answer the concern that rounding the corner will produce significant deviation from the solution where corners are not rounded. The difference between the actual solution,

<sup>∞</sup> (xˆ), is measured using the *L*<sup>2</sup>

, (83)

. (84)

1 2

Dirichlet

Neumann

*uL*

norm

and *L*<sup>∞</sup> norm

Impedance *λ* = 1 + *i*

18 Advanced Electromagnetic Waves

**Table 2.** Direction of incident plane wave *<sup>θ</sup>*<sup>0</sup> = 0 with *ka* = <sup>2</sup>*<sup>π</sup>* and *<sup>u</sup>sc* (*x*) for *<sup>x</sup>* = (−1, 1) with *kr* = <sup>20</sup>*π*. Impedance paramater *λ* = 1 + *i*.


**Table 3.** Direction of incident plane wave *θ*<sup>0</sup> = 0 with *ka* = 2*π* and *d* = (1, 0). Impedance parameter *λ* = 1 + *i*.

These tests were run for all three boundary conditions for *ka* = 1, 5, 10, and 2*π*, and radii of curvature *ρ* = 0.1, 0.08, ..., 0.02, 0.01 and, in the case of the impedance loaded scatterers, a number of impedance parameters. The results were similar for all wave numbers and Table 4 presents the results for *ka* = 2*π*. The smaller the radius of curvature used for the rounding, the smaller the measured error. Both the absolute and relative errors were measured. The relative error is expressed as a percentage of the same norm of the lemniscate far-field. Using a radius of curvature of *ρ* = 0.02, using the *L*<sup>2</sup> norm measures an error of 2.4% in the Dirichlet case, 0.9% in the Neumann case and 1.7% for the impedance boundary condition. Similarly, the *L*<sup>∞</sup> norm measures an error of 1.4% in the Dirichlet case, 0.4% in the Neumann case and 0.8% for the impedance boundary condition. Using a radius of curvature of *ρ* = 0.01, using the *L*<sup>2</sup> norm measures an error of 0.9% in the Dirichlet case, 0.03% in the Neumann case and 0.8% for the impedance boundary condition. Similarly, the *L*<sup>∞</sup> norm measures an error of 0.6% in the Dirichlet case, 0.1% in the Neumann case and 0.4% for the impedance boundary condition.

**Rounded Lemniscate with Impedance boundary condition: far field**

16 1.26738976138034 1.66105526777388 1.25530278020801 1.64198030138557 32 1.26778462660553 1.66045261136046 1.26781160112455 1.66041662811791 64 1.26780743084847 1.66042387986373 1.26780750283287 1.66042376203697 128 1.26780750283088 1.66042376204068 1.26780750283287 1.66042376203697 256 1.26780750283287 1.66042376203697 1.26780750283287 1.66042376203697

16 1.26679319682521 1.66100369319241 1.25514611227298 1.64136408633317 32 1.26723239948607 1.66019266418943 1.26728244518451 1.66011880476277 64 1.26727771729194 1.66012685149464 1.26727812591656 1.66012608299574 128 1.26727812584623 1.66012608315554 1.26727812591656 1.66012608299574 256 1.26727812591656 1.66012608299574 1.26727812591656 1.66012608299574

16 1.26628254876688 1.66094266668662 1.25496277020654 1.64065624538918 32 1.26675739510909 1.65984471852085 1.26684907150266 1.65968759452492 64 1.26684233823722 1.65969990319333 1.26684456821701 1.65969501037167 128 1.26684456565199 1.65969501752357 1.26684456821701 1.65969501037167 256 1.26684456821700 1.65969501037171 1.26684456821701 1.65969501037167

16 1.26581970905186 1.66101371994892 1.25479198100542 1.63987507228103 32 1.26636387623415 1.65947078536106 1.26652994925162 1.65913395755207 64 1.26651395959820 1.65917079145125 1.26652530266827 1.65914149765931 128 1.26652520551250 1.65914182632024 1.26652530266827 1.65914149765931 256 1.26652530265039 1.65914149772878 1.26652530266827 1.65914149765931

16 1.26516216215893 1.66171371564432 1.25467501349357 1.63904489275113 32 1.26602051309766 1.65923831994890 1.26634907227985 1.65848212898845 64 1.26629109761250 1.65864493833610 1.26634432336604 1.65848977294699 128 1.26634078340330 1.65850339846154 1.26634432336604 1.65848977294699 256 1.26634429273403 1.65848992335866 1.26634432336604 1.65848977294699

16 1.26439990092700 1.66296118057458 1.25465229232071 1.63863275250060 32 1.26579068367998 1.65938609457059 1.26632066601268 1.65813602895562 64 1.26619618222096 1.65849076674951 1.26631588055722 1.65814371438451 128 1.26629637930118 1.65821726963456 1.26631588055722 1.65814371438451 256 1.26631455164602 1.65815016944201 1.26631588055722 1.65814371438451

16 1.26753738579244 1.65531926109144 1.25466535533964 1.63824863702138 32 1.26675455678284 1.65668969319373 1.26634214415129 1.65780088985777 64 1.26649349543064 1.65733861326598 1.26633733538116 1.65780860014239 128 1.26639670753949 1.65761574700699 1.26633733538197 1.65780860013947 256 1.26636023315794 1.65773061188777 1.26633733538197 1.65780860013947

**Table 3.** Direction of incident plane wave *θ*<sup>0</sup> = 0 with *ka* = 2*π* and *d* = (1, 0). Impedance parameter *λ* = 1 + *i*.

**Uniform Mesh Graded Mesh** *N* Re *u*∞(d) Im *u*∞(d) Re *u*∞(d) Im *u*∞(d)

*ρ* = 0.1

20 Advanced Electromagnetic Waves

*ρ* = 0.08

*ρ* = 0.06

*ρ* = 0.04

*ρ* = 0.02

*ρ* = 0.01

Lemniscate


**Table 4.** Direction of incident plane wave *θ*<sup>0</sup> = 0 with *ka* = 2*π*.

### **5.2. Effect of Corner Rounding on a Domain with Two Corners**

Consider the curve given by the parametric representation:

$$x = x(t) = a \left( \frac{\cos t}{1 + |\sin t|}, \frac{\sin t}{1 + |\sin t|} \right), \qquad t \in [0, 2\pi], \tag{85}$$

where *a* is a parameter. It has the corners at *t* = 0 and *t* = *π* respectively, with interior right angles. Henceforth the parameter *a* is set equal to 1 length unit.

We will also consider a family of curves in which the corner has been rounded; the family is parameterized by the quantity *ε* (0 ≤ *ε* ≤ 1):

$$x = x(t) = a \left( \frac{\cos t}{1 + \sqrt{\varepsilon^2 + \sin^2 t}}, \frac{\sin t}{1 + \sqrt{\varepsilon^2 + \sin^2 t}} \right), \qquad t \in [0, 2\pi]. \tag{86}$$

Figure 2 illustrates the two shapes, with *a* = 1. The radius of curvature *ρ* at the corner points *t* = 0 and *π* is

$$\rho(x) = \left| \frac{\left(\mathbf{x}\_1'(t)^2 + \mathbf{x}\_2'(t)^2\right)^{3/2}}{\mathbf{x}\_1'(t)\mathbf{x}\_2''(t) - \mathbf{x}\_2'(t)\mathbf{x}\_1''(t)} \right|, \qquad t \in [0, 2\pi]. \tag{87}$$

**Figure 2.** Two-corner scatterer (blue). The interior (red) curve with rounded corners has parameter *ε* = 0.05 (*ρ* ≈ 0.05).

The near- and far-fields were computed for each of the boundary conditions using the graded mesh (67) for the two-corner scatterer (85). A variety of angles of incidence were tested and in the case of the impedance loaded two-corner scatterer a number of impedance parameters were tried. All tests were performed for *ka* = 1, 5, 10 and 2*π*. In all cases an examination of the convergence rate as a function of *N* was observed to be exponentially fast (super-algebraic). We note that unlike the case of the lemniscate (81) we obtained 12 significant digit convergence rather than 15. This is attributed to the choice of function (67) used for the graded mesh for the two-corner scatterer: the derivatives at the points *s* = 0, *π*, 2*π* vanish up to order 6 whereas the function (61) used for the lemniscate vanish up to order 8. Some typical results are as follows. Table 5 shows the scattered near- and far-field patterns for the two-corner scatterer illuminated by a plane wave incident at angle *θ*<sup>0</sup> = 0 with *ka* = 2*π*. For the impedance boundary condition, the impedance parameter shown is *λ* = 1 + *i*.

**5.2. Effect of Corner Rounding on a Domain with Two Corners**

 cos *t* 1 + |sin *t*|

cos *t*

*ε*<sup>2</sup> + sin2 *t*

<sup>1</sup>(*t*)<sup>2</sup> <sup>+</sup> *<sup>x</sup>*

<sup>2</sup> (*t*) − *x*

, sin *<sup>t</sup>* 1 + |sin *t*|

where *a* is a parameter. It has the corners at *t* = 0 and *t* = *π* respectively, with interior right

We will also consider a family of curves in which the corner has been rounded; the family is

Figure 2 illustrates the two shapes, with *a* = 1. The radius of curvature *ρ* at the corner points

2(*t*)23/2

2(*t*)*x* <sup>1</sup> (*t*)


**Figure 2.** Two-corner scatterer (blue). The interior (red) curve with rounded corners has parameter *ε* = 0.05 (*ρ* ≈ 0.05).

, sin *<sup>t</sup>* <sup>1</sup> <sup>+</sup>

*ε*<sup>2</sup> + sin<sup>2</sup> *t*

  , *t* ∈ [0, 2*π*], (85)

, *t* ∈ [0, 2*π*]. (86)

, *t* ∈ [0, 2*π*] . (87)

Consider the curve given by the parametric representation:

angles. Henceforth the parameter *a* is set equal to 1 length unit.

<sup>1</sup> <sup>+</sup>

 

 *x*

*x* 1(*t*)*x*

x = x(*t*) = *a*

parameterized by the quantity *ε* (0 ≤ *ε* ≤ 1):

*ρ*(x) =

x = x(*t*) = *a*





0

0.1

0.2

0.3

0.4

0.5

*t* = 0 and *π* is

22 Advanced Electromagnetic Waves


**Table 5.** Direction of incident plane wave *<sup>θ</sup>*<sup>0</sup> = 0 with *ka* = <sup>2</sup>*π*, *<sup>d</sup>* = (1, 0) and *<sup>u</sup>sc* (*x*) for *<sup>x</sup>* = (−1, 1) at *kr* = <sup>20</sup>*π*.

As in the case of the lemniscate (81), we then examined the effect of rounding the two corners of the scatterer. The near- and far-fields were computed for each of the boundary conditions using a uniform mesh *tj* <sup>=</sup> *<sup>π</sup><sup>j</sup> <sup>n</sup>* , for *j* = 0, 1, ..., 2*n* − 1, in the parameterisation (36) of the rounded scatterer (86) and the two-corner scatterer (85).

A variety of angles of incidence were tested and in the case of the impedance loaded scatterers a number of impedance parameters were tried. All tests were performed for *ka* = 1, 5, 10 and 2*π*, and radii of curvature *ρ* = 0.1, 0.05, 0.04, ..., 0.01. The results were similar in all cases. As expected a decrease in the radius of curvature shows a decrease in the rate of convergence and eventually for small radii (*ρ* ≤ 0.05) the solution fails to converge. The results for the two-corner scatterer, as expected, exhibit non-convergence.

The same series of experiments were then re-run using the graded mesh (67). In all cases this discretization method exhibits superior results. Super-algebraic convergence was observed in all cases when examining the convergence rate as a function of *N*, demonstrating the


**Table 6.** Direction of incident plane wave *<sup>θ</sup>*<sup>0</sup> = 0 with *ka* = <sup>2</sup>*<sup>π</sup>* and *<sup>u</sup>sc* (*x*) for *<sup>x</sup>* = (−1, 1) with *kr* = <sup>20</sup>*π*. Impedance parameter *λ* = 1 + *i*.

advantage of using the graded mesh. In all cases 15 significant digit convergence was achieved for the rounded scatterer. As in the case of the rounded lemniscate, we observe that even though the rounded two-corner scatterer (86) has a smooth boundary *∂D*, as the radius of curvature decreases use of the uniform mesh for discretization fails to produce a convergent solution for small radii of curvature. It demonstrates the need to consider an appropriate distribution of quadrature points.


**Rounded two-corner scatterer with impedance boundary condition: near field**

 0.03470677457659 0.03798379196971 0.03843618042815 0.03338842573544 0.03897149015734 0.03486730759441 0.03940928962261 0.03387147655152 0.03939900905631 0.03392160276388 0.03940929911450 0.03387163276154 0.03940928308095 0.03387177435745 0.03940929911442 0.03387163276137 0.03940929911413 0.03387163276369 0.03940929911442 0.03387163276137

 0.03481359170705 0.03603686016428 0.03973252285591 0.03005487424932 0.04004351467396 0.03216490387031 0.04069913302070 0.03060510102866 0.04067397485734 0.03074186288893 0.04069909427737 0.03060516627501 0.04069899786205 0.03060639554837 0.04069909427756 0.03060516627559 0.04069909426695 0.03060516643891 0.04069909427756 0.03060516627559

 0.03403888671802 0.03482751799820 0.04067458127216 0.02679979723427 0.04063667872250 0.02981632754250 0.04162851228396 0.02737654283524 0.04156807903825 0.02773425361138 0.04162851666947 0.02737622783093 0.04162795338930 0.02738656204370 0.04162851666923 0.02737622782875 0.04162851649254 0.02737623967346 0.04162851666923 0.02737622782875

 0.03217155484158 0.03507379516897 0.04121333829419 0.02405994803154 0.04067690299734 0.02828422652111 0.04216254273554 0.02454661770912 0.04202756957945 0.02536776882377 0.04216259142752 0.02454704861633 0.04215961390872 0.02461376091346 0.04216259142616 0.02454704861049 0.04216259955285 0.02454764108611 0.04216259142616 0.02454704861049

 0.02764745305810 0.03908271244059 0.04145153747908 0.02129232793218 0.03994042453183 0.02818804933763 0.04242372188117 0.02166516985259 0.04207481477062 0.02365888639225 0.04242374291713 0.02166462301523 0.04239986579142 0.02210111122351 0.04242374292179 0.02166462302828 0.04242431719588 0.02170326321857 0.04242374292179 0.02166462302828

 0.04613478369077 0.02381635331120 0.04141767031255 0.01902504141843 0.04354116342657 0.02025415654551 0.04240217311338 0.01943563484368 0.04274414603582 0.01951718789786 0.04240224749500 0.01943484871137 0.04250241810608 0.01939520893768 0.04240224762591 0.01943484562635 0.04243092698390 0.01939826345460 0.04240224762614 0.01943484561999

**Table 6.** Direction of incident plane wave *<sup>θ</sup>*<sup>0</sup> = 0 with *ka* = <sup>2</sup>*<sup>π</sup>* and *<sup>u</sup>sc* (*x*) for *<sup>x</sup>* = (−1, 1) with *kr* = <sup>20</sup>*π*. Impedance

advantage of using the graded mesh. In all cases 15 significant digit convergence was achieved for the rounded scatterer. As in the case of the rounded lemniscate, we observe that even though the rounded two-corner scatterer (86) has a smooth boundary *∂D*, as the radius of curvature decreases use of the uniform mesh for discretization fails to produce a convergent solution for small radii of curvature. It demonstrates the need to consider an

**Uniform Mesh Graded Mesh** *N* Re *usc* (*x*) Im *usc* (*x*) Re *usc* (*x*) Im *usc* (*x*)

*ρ* = 0.05

Advanced Electromagnetic Waves

*ρ* = 0.04

*ρ* = 0.03

*ρ* = 0.02

*ρ* = 0.01

2 Corners

parameter *λ* = 1 + *i*.

appropriate distribution of quadrature points.

**Table 7.** Direction of incident plane wave *θ*<sup>0</sup> = 0 with *ka* = 2*π* and *d* = (1, 0). Impedance parameter *λ* = 1 + *i*.

A set of typical results is provided in Table 6 which shows the values for the near-field using uniform mesh and then using graded mesh for a scatterer with the impedance boundary condition with impedance parameter *λ* = 1 + *i*, illuminated by a plane wave incident at angle *θ*<sup>0</sup> = 0 with *ka* = 2*π*. Table 3 shows the results of the far-field for the same experiments.

Having established that use of the graded mesh gives excellent results for the two-corner scatterer, we may now examine the effect of rounding the corners and determine the relationship between the radius of curvature of the rounding and the deviation from the solution produced by the two-corner scatterer. The difference between the actual solution and that produced by rounding is measured using the *L*<sup>2</sup> norm (83) and *L*<sup>∞</sup> norm (84).

The tests were run for all three boundary conditions for *ka* = 1, 5, 10, and 2*π*, and radii of curvature *ρ* = 0.05, 0.04, ..., 0.01 and, in the case of the impedance loaded scatterers, a number of impedance parameters. The results were similar for all wave numbers and Table 8 presents the results for *ka* = 2*π*. The smaller the radius of curvature used for the rounding, the smaller the measured error. Both the absolute and relative errors were measured. The relative error is expressed as percentage of the same norm of the far-field of the two-corner scatterer.

Using a radius of curvature of *ρ* = 0.02 , using the *L*<sup>2</sup> norm produces an error of 3.8% in the Dirichlet case, 1.6% in the Neumann case and 2.4% for the impedance boundary condition. Similarly, the *L*<sup>∞</sup> norm measures an error of 2.4% in the Dirichlet case, 1.5% in the Neumann case and 1.4% for the impedance boundary condition. Using a radius of curvature of *ρ* = 0.01, using the *L*<sup>2</sup> norm produces an error of 1.2% in the Dirichlet case, 0.4% in the Neumann case and 1% for the impedance boundary condition. Similarly, the *L*<sup>∞</sup> norm measures an error of 0.9% in the Dirichlet case, 0.4% in the Neumann case and 0.6% for the impedance boundary condition.


**Table 8.** Direction of incident plane wave *θ*<sup>0</sup> = 0 with *ka* = 2*π*.
