Vibration analysis of arbitrarily shaped membranes using local radial basis function-based differential quadrature method

In this study, a recently developed local radial basis function-based differential quadrature (LRBFDQ) method is applied for the vibration analysis of arbitrarily shaped membranes. LRBFDQ method combines the good features of differential quadrature (DQ) approximation of derivatives and mesh-free nature of the radial basis functions (RBFs) in a local region. The derivative at a reference point is approximated as a linear weighted sum of functional values at a set of scattered points in the local supporting region of the reference point. The Helmholtz equation governing membrane vibration is directly discretized into algebraic equations, from which the wavenumbers (natural frequencies) and mode shapes of freely vibrating membranes are easily calculated. Owing to the properties of mesh-free and local approximation of the LRBFDQ method, the problems with arbitrarily shaped domains can be solved readily and accurately. In particular, for highly concave-shaped membranes and multi-connected membranes with a hole, very accurate numerical results can be easily obtained without the use of any domain decomposition technique. It is also shown that the LRBFDQ method can produce more accurate solutions than FEM when the two methods use nearly the same number of points in a domain.