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Palladium nanoparticles were prepared via solvothermal method using Na2PdCl4, AgNO3, and oleylamine. By choosing the optimal reaction time and reaction temperature, palladium nanoparticles with regular morphology and size within 10 nm were prepared. A vacuum freeze drying was applied to prevent aggregation of nanoparticles.

A new method based on Legendre orthogonal polynomials method (LOPM) is proposed to calculate the acoustic reflection and transmission coefficients at liquid/solid interfaces. The displacement solutions of each layer in a multilayer plate are fitted approximately by multiple groups of Legendre orthogonal polynomials. The stress components and the governing wave equations in the plate are derived. Reassembling the boundary conditions of liquid/solid interfaces, solid/solid interfaces and governing wave equations, the linear independent equations are established to calculate the reflection and transmission coefficients and the expansion coefficients of Legendre polynomials in displacement solutions. The transfer matrix method (TMM) is used to verify the accuracy of this proposed method. The relationship between reflection and transmission coefficients with oblique incident angles and frequencies is discussed. The influence of the polynomial cut-off order on the calculation results is analyzed, and the displacement and stress profiles in multilayer plate are reconstructed. The research extends the applicable range of LOPM, moreover, lays the foundation of non-destructive testing and inversion of mechanical properties and bonding quality of the multilayer structures by acoustic waves.

A numerical approach for analyzing the guided wave propagation behavior in a functionally graded material (FGM) hollow cylinder is presented. Based on the three-dimensional linear elasticity theory, the state matrix form of displacement and stress component is introduced into the differential governing equation. Combining with boundary conditions, the dispersion characteristic equations can be constructed. Meanwhile, taking the orthogonal completeness of Legendre series into consideration, the exhausting root findings of transcendental equation can be avoided, and it transforms the redundant integral operators into analytical expressions by means of its recursive properties. Moreover, it is not only unnecessary to stratify FGMs, and introducing the univariate nonlinear regression to modelling the arbitrary gradient distribution, and transforming the redundant integral operator into an analytical expression by means of recursive property. It should be pointed out that the coefficient matrix consists only zero and constant terms, which are related to the elastic constants and normalized wave number. And it provides a fast and effectively approach to extracting the dispersion curves, displacement distribution and stress profile, simultaneously. Meanwhile, the proposed method can get rid of the exhausted root-finding algorithm, and accordingly distinguish the different modes and the dispersion curves simultaneously. Finally, the typical non-stratified computation of dispersion curves of longitudinal guided wave for FGM hollow cylinder is realized. The accuracy of the proposed method is further demonstrated through comparison with the available data from Legendre polynomial method. Several numerical cases about iron based alumina FGM cylinder are studied, and the convergence of the present polynomial approach is discussed. Then the effect of the circumferential wave number, gradient variation, diameter-thickness ratio on the dispersion characteristics are illustrated. Moreover, the distribution characteristics of displacement and stress profiles of F(1,1) mode at a given frequency are discussed in detail. It is shown that the state-vector formalism and Legendre polynomial hybrid approach is a powerful tool in solving wave motions in FGM elastic cylinder, which lays a theoretical foundation for the non-destructive evaluation and quantitative characterization of the gradient structure characteristics of functionally graded hollow cylinder.

In this research, a numerical approach for analyzing the analytical solution of guided wave propagation characteristics in multilayer two-phased porous media is presented. Combing the global matrix method and Biot theory, and introducing the modulus-porosity-poisson’s ratio relation, the global dispersion equations of multi-layered porous materials are established. Also, the complex boundary conditions in multi-layered two-phase porous media are considered simultaneously. In order to confirm the feasibility and accuracy of the proposed method, the dispersion curves of the guided waves propagation in a single-layer porous graphite layer and a triple-layer graphite/copper/graphite model (porosity close to 0) were numerically calculated. The results were compared with the propagation characteristics of the guided waves in the corresponding stacked sequential laminates without porosity, which is done by the state-vector formalism and the Legendre polynomials method based on our previous work. Then, this approach is further applied to lithium ion battery. The change of porosity in graphite is used to simulate the state of charge of lithium-ion battery. The influence of porosity change on mode coupling effect of guided waves is analyzed, and the mapping relationship between porosity and corresponding dispersion curves is explored. Meanwhile, the research shows that, with the decrease of porosity (which means the state of charge increases), the phase velocity of the fundamental modes gradually increases, making the corresponding time of flight decrease gradually. Moreover, the theoretical model captures a meaningful relationship between state of charge and acoustic behavior. It gives theoretical support for nondestructive evaluation and quantitative estimation of the state characteristics of lithium-ion batteries.

In this paper, we determine the distribution of the length partition of a random multicurve of fixed topological type on a closed hyperbolic surface using the methods of Margulis' thesis and Mirzakhani's equidistribution theorem for horospheres. This distribution admits a polynomial density, whose coefficients can be expressed explicitly in terms of intersection numbers of psi-classes on the Deligne--Mumford compactification, and in particular it does not depend on the hyperbolic metric. This result generalizes prior work of M. Mirzakhani in the case of random pants decompositions. Results very close to ours are obtained independently and simultaneously by F. Arana-Herrera.