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A differential reproducing kernel (DRK) interpolation-based collocation method is developed for solving partial differential equations governing a certain physical problem. The novelty of this method is that we construct a set of differential reproducing conditions to determine the shape functions of derivatives of the DRK interpolation function, without directly differentiating the DRK interpolation function. In addition, the shape function of the DRK interpolation function at each sampling node is separated into a primitive function processing Kronecker delta properties and an enrichment function constituting reproducing conditions, so that the nodal interpolation properties are satisfied. A point collocation method based on the present DRK interpolation is developed for the analysis of one-dimensional bar problems, two-dimensional potential problems, and plane problems of elastic solids. It is shown that the present DRK interpolation-based collocation method is indeed a truly meshless approach, with excellent accuracy and fast convergence rate.

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A unified formulation of finite prism methods (FPMs) based on Reissner's mixed variational theorem is developed for the three-dimensional (3D) free vibration analysis of functionally graded (FG) carbon nanotube-reinforced composite (CNTRC) plates and laminated fiber-reinforced composite (FRC) plates, the edge conditions of which are considered such that one pair of opposite edges is simply supported and the other pair may be combinations of free, clamped or simply supported edges. The single-walled carbon nanotubes (CNTs) and polymer are regarded as the reinforcements and matrices, respectively, to produce the CNTRC plate. Four different distributions of CNTs varying in the thickness direction are considered (i.e. the uniformly distributed, and FG V-, rhombus-, and X-type variations), and the throughthickness distributions of effective material properties of the CNTRC plate are determined using the rule of mixtures. In the formulation, the CNTRC/FRC plate is divided into a number of finite prisms in the x À domain, in which the trigonometric functions and Lagrange polynomials are used to interpolate the y-direction and x À plane variations for the primary variables of each individual prism, respectively, and the related orders used for expansion of assorted primary variables in the thickness coordinate can be freely chosen. It is shown that the eight-and nine-node quadratic FPM solutions of frequency parameters of simply supported, CNTRC plates and laminated FRC plates are in excellent agreement with the exact 3D solutions available in the literature, and with the ones obtained using the ANSYS software for those plates with various boundary conditions.