Development of square transverse vibration membrane formulation using an inverse approach

Common approaches in development of element formulations are based on minimization of the certain functions or residues using assumed shape functions. The choice of these shape functions strongly affects the accuracy of the solution and its convergence rate. Alternatively one may use an inverse approach and develop the element formulation by minimizing the element discretization error. This leads to the definition of the element formulation without the need to specify any shape functions. In this paper, a new mass matrix is developed for the 4 nodded square transverse vibration membrane elements by minimizing discretization error in element formulation. A parametric model is formed for the element and by assembling a set of these parametric elements, a model representing the governing equation of a transverse vibration membrane is obtained. The set of difference equations formed in the inner and boundary nodes of the model are compared with its corresponding analytical governing equations and the element parameters are obtained by minimizing the difference between finite element formulation and continuous model. Using this inverse approach, one obtains the element formulation with highest accuracy and convergence rate without the need to specify the element shape functions. The convergences rates of Eigen-values solution of new formulations are at least two times more than the corresponding consistent and lumped formulation.

کلیدواژه ها:

inverse method ، vibration membrane ، massmatrix ، Discritization Error