Eigenanalysis of membranes with stringers by method of fundamental solutions and domain decomposition technique

Abstract

We use a meshless numerical method to analyze the eigenanalysis of thin circular membranes with degenerate boundary conditions, composed by differentorientations and structures of stringers. The membraneeigenproblem is studied by solving the two-dimensionalHelmholtz equation utilizing the method of fundamental solutions and domain decomposition technique aswell. The method of singular value decomposition isadopted to obtain eigenvalues and eigenvectors of theresulting system of global linear equation. The proposed novel numerical scheme was first validated bythree circular membranes which are structured with a single edge stringer, two opposite edge stringers and an internal stringer. Present results for those three cases matchvery 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 solvea completely new problem of a circular membrane witha cross stringer at the center of the membrane. We illustrate the proposed innovative numerical scheme which issimpler and more efficient to solve Helmholtz problemswith degenerate boundary conditions. The good featuresof this scheme are not depending upon the treatmentsof mesh, singularity, hypersingularity, numerical integration and iterative procedure, which are generally requiredby other conventional mesh-dependent methods.