Eigenanalysis for membranes with stringers using the methods of fundamental solutions and domain decomposition

Abstract

We use a meshless numerical method to analyze the eigenanalysis of thin circular membranes with degenerate boundary conditions, composed by different orientations and structures of stringers. The membrane eigenproblem is studied by solving the two-dimensional Helmholtz equation utilizing the method of fundamental solutions and domain decomposition technique as well. The method of singular value decomposition is adopted to obtain eigenvalues and eigenvectors of the resulting system of global linear equation. The proposed novel numerical scheme was first validated by three circular membranes which are structured with a single edge stringer, two opposite edge stringers and an internal stringer. Present results for those three cases match very well with the solutions obtained by the analytical approach as well as by methods of dual boundary element, and finite element. The analysis is then extended to solve a completely new problem of a circular membrane with a cross stringer at the center of the membrane. We illustrate the proposed innovative numerical scheme which is simpler and more efficient to solve Helmholtz problems with degenerate boundary conditions. The good features of this scheme are not depending upon the treatments of mesh, singularity, hypersingularity, numerical integration and iterative procedure, which are generally required by other conventional mesh-dependent methods.