Recently, the edge-based and node-based smoothed finite element method (ES-FEM and NS-FEM) has been proposed for Reissner–Mindlin plate problems. In this work, in order to utilize the numerical advantages of both ES-FEM and NS-FEM for static and vibration analysis, a hybrid smoothing technique based beta FEM ([Formula: see text]FEM) is presented for Reissner–Mindlin plate problems. A tunable parameter [Formula: see text] is introduced to tune the proportion of smoothing domains calculated by ES-FEM or NS-FEM, which controlled the accuracy of the results. Numerical illustrations in both static and free vibration analysis are conducted. The shear locking free property, converge property and dynamic stability are carefully examined via several well-known benchmark examples. Moreover, an experimental test is carefully designed and conducted for validations, in which the mode values and shape of a rectangular steel plate is tested. Numerical examples demonstrate the advantages of [Formula: see text]FEM, in comparison with the standard FEM, ES-FEM and NS-FEM using the same meshes. The numerical and experimental results are in good agreement with each other and the [Formula: see text]FEM achieves the best accuracy among all the methods for the static or free vibration analysis of plates.

The criticality type eigenvalues of the one-speed transport equation in a homogeneous slab with anisotropic scattering and Marshak boundary conditions have been studied. The scattering function is assumed to be a combination of linearly anisotropic and strongly forward-backward scattering. When the forward and backward scattering completely dominate over the 'ordinary' scattering, or the thickness of the slab approaches zero, the highly peaked angular flux at the central point of the slab was expressed by finite width delta functions. Using the finite width delta functions to analyse the high-order truncation error of the angular flux we could accurately obtain results with a low-order approximation. Numerical results for critical eigenvalues are obtained and tabulated for different scattering parameters including the extreme cases, while the standard spherical harmonics method gets a singularity.