If better superdirectivity is required to be obtained in practice, the maximum order of the retained mode-beams is unnecessarily selected to be large.
Efficient and highly accurate solutions are obtained with a small number of retained modes.
The results in this table show that with few numbers of retained modes we obtained a maximum pointwise error of order 10−8.
In the third step, by properly selecting the number of retained modes, the original time series can be approximated to a numerical model.
In Table 2, we list the maximum pointwise error E and the time elapsed τ for different values of M. The results in this table show that with few numbers of retained modes we obtained a reasonable accuracy 10−13.
Also, various definitions of the residual are briefly discussed, along with the number and distribution of snapshots, the number of retained modes, and the effect of CFD errors, to conclude that the method is numerically robust.
The choice of the number of retained modes is made after considering both the image reconstruction errors as well as the convergence of the effective material constitutive behavior obtained by numerical homogenization.
In this approach, the response is expressed as a sum of two parts: (1) the dynamic part, representing the contribution of the retained modes; and (2) the pseudo-static part, representing the contribution of the truncated modes.
In other words, POD computations are performed using only a set of collocation points, whose number is comparable to the number of retained modes, in a similar fashion as in collocation spectral methods.
In the transition probability, the beta distribution of Q a at trial t retained the mode value of Q a at trial t−1.
A free interface method is presented in this paper and six test cases are used to evaluate the number of retained substructure modes necessary to ensure convergence.
