Illustrative simulations

A series of illustrtive simultions were creted using the inite element sowre COMSOL Mul- tiphysics®. Perfectly mtched lyers were used in the vicinity of the oundry of the comput- tionl domin in order to simulte n ininite domin. For the purposes of these computtions, the following non-dimensionl prmeter vlues were chosen: a = 0.5,w = 0.5, = ϱ = 1,

0=0.1,ϱ0=0, =1×10−6. Figures 7.4 nd 7.5 show the displcement mplitude ieldu(x)

for  cylindricl source oscillting t =5 nd =10 respectively. e igures clerly illustrte roughout this chpter, ll numericl prmeters re normlised such =ϱ=1 unless otherwise stted.

Chapter Seven A microstructured invisibility cloak

Source Scttering MesureE

Position Frequency Uncloked Cloked Q Scttering regionR₁ [−3,0]T 5 0.1529 4.351×10−4 0.9972 [−3,0]T 10 0.1455 4.514×10−4 0.9969 [−3,3]T/√2 5 0.2002 3.941×10−4 0.9980 [−3,3]T/√2 10 0.3286 4.068×10−4 0.9988 Scttering regionR₂ [−3,0]T 5 0.3224 3.664×10−4 0.9989 [−3,0]T 10 0.3093 1.167×10−3 0.9962 Scttering regionR₃ [−3,3]T/√2 5 0.2988 3.654×10−4 0.9988 [−3,3]T/√2 10 0.2988 7.803×10−4 0.9974

Table 7.1: e scttering mesures corresponding to the simultions shown in igures 7.4 nd 7.5.

(a) Uncloked,x0=[−3,0]T (b) _Cloked,x

0=[−3,0]T

(c) Uncloked,x0=[−3,3]T/√_{2 (d) Cloked,x}

0=[−3,3]T/√₂

Figure 7.4: Plots of the ieldufor the uncloked nd cloked squre inclusion, where the ngulr frequency of excittion is = 5. e positionx₀of the source is indicted under the

Chapter Seven A microstructured invisibility cloak

(a) Uncloked,x0=[−3,0]T _{(b) Cloked,x}

0=[−3,0]T

(c) Uncloked,x0=[−3,3]T/√₂ (d) _Cloked,x₀₌[_−3,3]T/√₂

Figure 7.5: Plots of the ieldufor the uncloked nd cloked squre inclusion where the ngulr frequency of excittion is =10. e positionx₀of the source is indicted under the

relevnt plot nd the inclusion is locted t the centre of the imge in ll cses. e colour scle is s indicted in igure 7.4.

(a) (b)

Figure 7.6: () e scttering mesure plotted ginst ngulr frequency. () e log of the sct- tering mesure plotted ginst ngulr frequency. e solid line corresponds to the continuum in the sence of oth n inclusion nd clok. e dshed line represents the cloked inclusion nd the dsh-dot line corresponds to the uncloked inclusion. e regionR₁(see igure 7.3 nd the ssocited text) ws used to compute the error mesure.

Chapter Seven A microstructured invisibility cloak Source Scttering MesureE

Boundry Condition Frequency Uncloked Cloked Q Scttering regionR₁ Neumnn 5 0.1624 4.351×10−4 0.9973 Neumnn 10 0.1558 4.540×10−4 0.9971 Dirichlet 5 0.2931 1.038×10−2 0.9646 Dirichlet 10 0.2553 7.875×10−3 0.9692 Scttering regionR₂ Neumnn 5 0.3436 3.664×10−4 0.9989 Neumnn 10 0.3258 1.163×10−3 0.9964 Dirichlet 5 0.4864 1.566×10−2 0.9678 Dirichlet 10 0.5030 1.673×10−2 0.9667 Table 7.2: e scttering mesures for  void with Neumnn nd Dirichlet oundry conditions.

Here the source is locted t[−3,0]T_.

the efficcy of the squre clok, even t reltively high frequencies. Tle 7.1 shows the corre- sponding scttering mesures s introduced in section 7.1.5. It is cler tht this squre “push out” clok is highly effective. Indeed, for the illustrtive simultions presented here, the clok reduces the scttering mesure y not less thn 99.62%compred with the uncloked inclusion.

Figure 7.6 shows the scttering mesure plotted ginst non-dimensionl ngulr frequency (with =ϱ=1). e solid curve in igure 7.6 corresponds to the continuum, in the sence of

oth clok nd inclusion. is curve gives n indiction of the numericl error in the simultion induced y, for exmple, the use of perfectly mtched lyers nd the numericl discretistion. e dshed curve corresponds to the cloked inclusion, whilst the dsh-dot curve corresponds to the uncloked inclusion. It is oserved tht the numericl mesure of the cloked inclusion remins close to tht of the intct continuum for  lrge rnge of frequencies. Moving to di- mensionl quntities, suppose the simultion corresponded to  prticulr polriztion of n electric wve trvelling through glss t  speed of pproximtely 2×108m/s. e line =10

on igure 7.6 then corresponds to  frequency of pproximtely 340 MHz.

Boundary considerations

Whilst cloking vi trnsformtion geometry hs een extensively treted in the literture, the sensitivity of the cloking effect to the oundry conditions is rrely discussed. e clok is formed y deforming  smll region ( point in the cse of the clssicl rdil trnsformtion 126), into  lrger inite region. If the region is n inclusion, then the nturl interfce conditions my e determined following the method outlined in section 7.1.2. If the cloked region is  void or rigid inclusion, however, there is some freedom in choosing the oundry condition, suject to the constrints of the physicl prolem. Figure 7.7 shows the ieldu(x)for  cloked void, with Neumnn (prts () nd ()) nd Dirichlet (prts (c) (d)) conditions pplied to the interior of the cloked region. e corresponding scttering mesures re shown in tle 7.2.

Although the squre clok is effective in oth cses, it is cler from oth the igures nd the tle of scttering mesures tht the type of oundry condition imposed on the cloked oject ffects the qulity of the cloking. Indeed, for  void (Neumnn) the cloking reduces the sct-

Chapter Seven A microstructured invisibility cloak

(a) Uncloked, Neumnn (b) Cloked, Neumnn

(c) Uncloked, Dirichlet (d) Cloked, Dirichlet

Figure 7.7: Plots of the ield ufor the uncloked nd cloked squre inclusion with Neumnn oundry conditions on the oundry of the inclusion in prts () nd (), nd Dirich- let oundry conditions on the oundry of the inclusion in prts (c) nd (d). Here the source is locted tx=[−3,0]T_{nd oscilltes t =10. e colour scle is s indicted}

Chapter Seven A microstructured invisibility cloak

tering mesure y etween 99.7%nd 99.9%for oth =5 nd =10. In contrst, cloking

reduces the scttering mesure of  rigid inclusion (Dirichlet) y etween 96.5%nd 96.8%for

= 5 nd etween 96.7%nd 96.9%for = 10. e effect of the oundry condition my

e interpreted in the following wy. As  result of the trnsformtion, the cloked oject nd clok together ehve s if the void is smll. In this sense, the cloked inclusion represents  singulr perturtion of the fundmentl solution of the Helmholtz eqution. In the cse of  free void with Neumnn conditions, the leding order term in the symptotic expnsion is the dipole term, which is of order 2_{nd decys like the irst derivtive of the fundmentl solu-} tion. On the other hnd, for  ixed void with Dirichlet conditions, the leding order term in the expnsion is the monopole term which is of order nd decys like the fundmentl solution. us, the perturtion from the free void is smller thn the perturtion from the ixed void, leding to improved cloking