Mixed Reissner variational layerwise plates and shallow shells with a hybrid FEM–PHS–RBF–FD discretization

A hybrid finite-element/polyharmonic-spline radial-basis-function-generated finite-difference discretization is developed for a layerwise laminated plate and shallow-shell model based on the Mixed Reissner Variational Theorem (RMVT). The three displacement components are continuous piecewise-linear functions through the thickness. The transverse stress vector [σz,τyz,τxz]T is introduced independently in each mid-surface element and each thickness element, interpolated by Q9 PHS–RBF cardinal functions in the plane and by linear functions through the thickness, and statically condensed. The in-plane stress components are then recovered from the partial compliance relation. The mid-surface weak form uses ordinary Q9 finite-element assembly, but the element basis is generated from an r5 polyharmonic spline with complete quadratic polynomial augmentation, so no RBF shape parameter is present. A consistent three-dimensional mass matrix supplies the constrained free-vibration eigenproblem. Constant shallow-shell curvature is introduced through the Donnell–Marguerre membrane strains without changing the mixed-stress condensation. Static deflection and fundamental frequency converge to matching layerwise Navier solutions; the first four flexural modes, transverse stress profiles, and the element stencil are displayed. Spherical panel results are also compared with the Reddy–Liu benchmarks.