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Efficient Z-transform implementa-tion of the complex frequency-shifted perfectly matched layer (CFS-PML) based on the stretched coordinate PML (SC-PML) formulations and the D-B formulations is proposed for truncating meta-material finite-difference time-domain (FDTD) lattices. In the proposed PML formulations, the Z-transform method is incorporated into the CFS-PML FDTD implementation. The main advantage of the proposed formulations can allow direct FDTD implementation of the Maxwell’s equations in the PML regions. A numerical test has been carried out in a three dimensions (3-D) FDTD domain to validate the proposed formulations. It is shown that the proposed formulations with CFS scheme are efficient in holding good absorbing performances.

We developed analogous parallel algorithms to implement CostRank for distributed memory parallel computers using multi processors. Our intent is to make CostRank calculations for the growing number of hosts in a fast and a scalable way. In the same way we intent to secure large scale networks that require fast and reliable computing to calculate the ranking of enormous graphs with thousands of vertices (states) and millions or arcs (links). In our proposed approach we focus on a parallel CostRank computational architecture on a cluster of PCs networked via Gigabit Ethernet LAN to evaluate the performance and scalability of our implementation. In particular, a partitioning of input data, graph files, and ranking vectors with load balancing technique can improve the runtime and scalability of large-scale parallel computations. An application case study of analogous Cost Rank computation is presented. Applying parallel environment models for one-dimensional sparse matrix partitioning on a modified research page, results in a significant reduction in communication overhead and in per-iteration runtime.We provide an analytical discussion of analogous algorithms performance in terms of I/O and synchronization cost, as well as of memory usage.

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Considering a piecewise linear oscillator with quasiperiodic excitation, we uncover the route of double grazing bifurcation of quasiperiodic torus to strange nonchaotic attractors (i.e., SNAs). The maximum displacement for double grazing bifurcation of the quasiperiodic torus can be obtained analytically. After double grazing of quasiperiodic orbits, the smooth quasiperiodic torus wrinkles increasingly with the continuous change of the parameter. Subsequently, the whole quasiperiodic torus loses the smoothness by becoming everywhere non-differentiable, which indicates the birth of SNAs. The Lyapunov exponent is adopted to verify the nonchaotic property of the SNA. The strange property of SNAs can be characterized by the phase sensitivity, the power spectrum, the singular continuous spectrum, and the fractal structure. Our detailed analysis shows that the SNAs induced by double grazing may exist in a short parameter interval between 1 T quasiperiodic orbit and 2 T quasiperiodic orbit or between 1 T quasiperiodic orbit and 4 T quasiperiodic orbit or between 1 T quasiperiodic orbit and chaotic motion. Noteworthy, SNAs may also exist in a large parameter interval after double grazing, which does not lead to any quasiperiodic or chaotic orbits.