3.4 Transfer Matrix Method Employed in Numerical Simulation Domain with
3.4.2 Effective parameter retrieval
dipolar case, the relative impedance curve exhibits a totally different aspect. The impedance matching condition is fulfilled above the resonance gap whereas it was achieved below the gap in the Meff-substructure case. The originally lowered impedance of the Keff-substructure is compensated “above” the resonance frequency.
In other words, the impedance matching condition for the Meff-substructure occurs at the upper end of the lower passband whereas for the Keff-substructure, it happens at the lower end of the upper passband. Also, the effect of the cos
qd 2
term isinsignificant in this case. Therefore, this cell is mainly dominated by the effective properties Meff and Keff, which will be thoroughly discussed in the next subsection.
Consequently, the transmission curve decreases near the resonance gap and then fulfills the total transmission condition as the frequency increases.
1 1
11 11
2 12
cos ( )sin(cos ( ))
eff
T T
kZ
S Sd T
, (3.58a)
1 11 1
12 11
sin(cos ( )) cos ( )
eff
T d E Z
kS T T
. (3.58b)
where S is the cross-sectional area. Here, numerically derived (or experimentally obtained) transfer matrix components are required to calculate the effective mass density and Young’s modulus properties. Based on the similar procedures, the effective properties of the lumped element models will be derived.
Effective parameter retrieval for the Meff-substructure (dipolar cell)
With Eq. (3.1) and Eq. (3.12), the effective parameters for the Meff-substructure (dipolar cell) can be retrieved as
21 eff 2
M T
, (3.59a)
21
2( 11 1)
eff
K T
T . (3.59b)
Because the components of the transfer matrix are frequency dependent, the corresponding effective parameters are exclusively obtained at every frequency.
The semi-analytical results from the actual cell in Fig. 3.7(a) are shown in Fig.
3.9(b). As intended and expected when designing the cell, only the Meff term undergoes resonant state whereas Keff remains almost constant. In particular, as frequency increases, Meff first drastically exceeds to the positive infinite value
and then after the resonance frequency, it starts from the negative infinite value and eventually recovers to the positive value. Therefore, for the impedance increasing phenomenon just below the resonance frequency, it can be concluded that both the amplified Meff value and the effect of lowered cos
qd 2
terms are mainly responsible. In other words, the Meff-substructure is capable of compensating the impedance for total transmission just below its resonance frequency.Effective parameter retrieval for the Keff-substructure (monopolar cell)
With Eq. (3.1) and Eq. (3.16) the effective parameters of the Keff-substructure (monopolar cell) can be deduced as
11
2 12
1 2
eff
M T
T
, (3.60a)
12
1 Keff
T . (3.60b)
The results from the actual monopolar cell (Fig. 3.8(a)) is shown in Fig. 3.10(b). In contrast to the Meff-substructure, only the Keff term experiences resonance state whereas Meff remains rather constant as intended. Also, as analytically calculated in Chapter 2, Keff first goes to negative infinity first and then starts from positive infinity after the resonance frequency. In other words, the resonance phenomena of Keff and Meff have somewhat opposite aspects. Most importantly, it can be inferred from the impedance term that the amplified Keff is the one that is mainly
responsible for impedance compensation. Therefore, total transmission occurs at the beginning of the higher branch where Keff is extremely amplified.
Remarks on effective parameter retrieval procedure with transfer matrix method Compared to the analytic method with static estimation by Oh et al. [38], our method is capable of computing the actual ‘dynamic’ values of the effective properties that absolutely correlate with the transmission, dispersion curve, and the impedance values. Even more, compared to the S-parameter retrieval method [67]
which is only feasible with homogenized continuum materials as in Eq. (3.58), our method is optimal to retrieve the proper values for metamaterials containing local resonators. In other words, the conventional method could not retrieve meaningful values for metamaterials especially near the resonance where dynamic fluctuations take place.
For example, the effective properties of the two cells have been retrieved with the S-parameter method (Eqs. (3.58a-b)) and the results are shown in Fig. 3.11. In other words, the metamaterial cells were assumed to behave as a homogenized material with anomalous properties as depicted inside the graphs. In contrast to the appropriate values that had been discussed above, both eff and Eeff are shown to experience resonance phenomena for both the dipolar case (Fig. 3.10(a)) and monopolar case (Fig. 3.10(b)).
It can be confirmed that wrong assumption of the cell provides meaningless values
of properties. In other words, the conventional S-parameter retrieval method is not suitable for estimating the properties of metamaterials with resonant characteristics.
The S-parameter retrieval method has been limited to estimate the effective property values by considering it as a fully homogenized material. Yet, the local dynamics within the metamaterials cannot be completely expressed with homogenized material assumptions. However, our method fully describes the geometrical features of the metamaterial cell by the lumped element model and thus enables retrieval of the correct values.
In a similar context, the lumped element model must also be chosen carefully regarding the eigen mode shape of the continuum cell. For example, the properties of the Meff-substructure (dipolar continuum cell) is retrieved by the Keff-substructure model (monopolar lumped element model) as shown in Fig. 3.12(a). Similar to the former inappropriate assumption, the properties are somewhat coupled near the resonance frequency. The main reason is that the monopolar model cannot characterize the dipolar mode shape into valid properties. The inertia term is managed by both the spring and mass term, thus inducing unwanted resonance state of effective stiffness values. Although one may question that this could indicate a meaningful value, the different aspect of the Keff -curve from the theoretical analysis (Fig. 2.8(b)) proves it wrong. Such invalid values have occurred to just satisfy the boundary responses of the Meff-substructure.
On the other hand, the retrieved values of the Keff-substructure (monopolar cell) by assuming it as a Meff-substructure model (dipolar lumped element model) are
shown in Fig. 3.12(b). Alike the former case, the properties seem correlated such that both of them undergo resonance states in a similar manner. Here, unwanted properties of Meff have occurred because the monopolar mode shape is forcefully expressed by the inertia term of the rigid mass outer mass. One should be aware that this is an invalid value because the aspect of the Meff-curve does not follow the theoretically evaluated form as derived in Fig. 2.8(a).
As such, choosing the proper basic unit cell does matter in characterizing a target material although they make up the same system (mass-spring lattice) in infinite series. As discussed in Chapter 2, each model has totally different physics behind them, which goes the same for the continuum cells. Consequently, it should be noted that a proper model must be considered to retrieve the meaningful effective properties from the target cell. In other words, contrary to the wave characteristics (dispersion relation, impedance, and transmission), effective parameter retrieval can be subjective and it depends on proper choice from various models of which the cell is assumed to be.