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The research on the propagation of surface waves has received considerable attention in order to improve the efficiency and natural life of the surface acoustic wave devices, but the investigation on complex Rayleigh waves in functionally graded piezoelectric material (FGPM) is quite limited. In this paper, an improved Laguerre orthogonal function technique is presented to solve the problem of the complex Rayleigh waves in an FGPM half-space, which can obtain not only the solution of purely real values but also that of purely imaginary and complex values. The three-dimensional dispersion curves are generated in complex space to explore the influence of the gradient coefficients. The displacement amplitude distributions are plotted to investigate the conversion process from complex wave mode to propagating wave mode. Finally, the curves of phase velocity to the ratio of wave loss decrements are illustrated, which offers extra convenience for finding the high phase velocity points where the complex wave loss is near zero.

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We present RECFMM, a program representation and implementation of a recursive scheme for parallelizing the adaptive fast multipole method (FMM) on shared-memory computers. It achieves remarkable high performance while maintaining mathematical clarity and flexibility. The parallelization scheme signifies the recursion feature that is intrinsic to the FMM but was not well exploited. The program modules of RECFMM constitute a map between numerical computation components and advanced architecture mechanisms. The mathematical structure is preserved and exploited, not obscured nor compromised, by parallel rendition of the recursion scheme. Modern software system—CILK in particular, which provides graph-theoretic optimal scheduling in adaptation to the dynamics in parallel execution—is employed. RECFMM supports multiple algorithm variants that mark the major advances with low-frequency interaction kernels, and includes the asymmetrical version where the source particle ensemble is not necessarily the same as the target particle ensemble. We demonstrate parallel performance with Coulomb and screened Coulomb interactions.

The complex Rayleigh waves play an important role in the energy conversion efficiency of magneto-electro-elastic devices, so it is necessary to explore the wave propagation characteristics for the better applications in engineering. This paper modifies the Laguerre orthogonal polynomial to investigate the complex Rayleigh waves propagating in nonhomogeneous magneto-electro-elastic half-spaces. The improved method simplifies the calculation process by incorporating boundary conditions into the constitutive relations, shortens the solving time by transforming the solution of wave equation to an eigenvalue problem, and obtains all wave modes, including real and imaginary and complex wavenumbers. The three-dimensional curves of full frequency spectrum and phase velocities are presented for the better description of the conversion from complex Rayleigh wave modes to real wave ones; besides, the displacement distributions, electric and magnetic potential curves are obtained in thickness and propagation directions, respectively. Numerical results are analyzed and discussed elaborately in three cases: variation of nonhomogeneous coefficients, absence of magnetism, and absence of electricity. The results can be used to optimize and fabricate the acoustic surface wave devices of the nonhomogeneous magneto-electro-elastic materials.