Application of Decoupled Scaled Boundary Finite Element Method to Solve Eigenvalue Helmholtz Problems

A novel element with arbitrary domain shape by using decoupled scaled boundary finite element (DSBFEM) is proposed for eigenvalue analysis of 2D vibrating rods with different boundary conditions. Within the proposed element scheme, the mode shapes of vibrating rods with variable boundary conditions are modelled and results are plotted. All possible conditions for the rods ends are incorporated in analysis. The considered element stiffness and mass matrix are developed and extracrted. This element is able to model any curved or sharp edges without any aproximation and also the element is able to model any arbitrary domain shape as a single element without any meshing. The coefficient matrices for the element such as mass and stiffness matrices are diagonal symmetric and all equations are decoupled by using Gauss-Lobatto-Legendre (G.L.L) quadrature. The element is used in order to calculate modal parameters by Finite element method for some benchmark examples and comparing the answers with Helmholtz equation solution. The most important achievment of this element is solving matrix equations instead of differential equations where cause faster calculations speed. The boundaries for this element are solved with matrix calculation and the whole interior domain with solving governing equations numerically wich leads us to an exact answer in whole domain. The introduced element is applied to calculate some benchmark example which have exact solution. The results shows accuracy and high speed of calculation for this method in comparison with other common methods.