Vibration analysis of polygonal plates using triangular energy element via global-local level mapping

A novel numerical integration strategy, based on global-local level mapping, is proposed in the Ritz method, for the vibration analysis of polygonal plates. At the global level, the polygonal plate is embedded in a rectangular domain, and then the rectangular domain is mapped to a unit square domain. The polygonal plate is then divided into multiple triangular domains. At the local level, each triangular domain is mapped to a right-angled triangular domain and then to a unit square domain, from which triangular energy elements are constructed using Gauss-Legendre quadrature via inverse mapping. By applying the global-local level mapping integration strategy to the vibration analysis of polygonal plates, the energy functionals in terms of global admissible functions can be established on the rectangular domain, and the integration of elements in the stiffness and mass matrices can be performed based on triangular energy elements. As a result, the energy functionals and computation procedures for arbitrarily shaped polygonal plates are standard, and will not be affected by variations in the polygonal plate's geometry. The vibrational behaviors of triangular, skew, trapezoidal, pentagonal, hexagonal, heptagonal, and octagonal plates are investigated, and compared with those reported in the literature. Results demonstrate the generalization, reliability, accuracy of the proposed method. This method is universal for solving 2D variational problems on complex geometric domain within Ritz formulation.