Application of Decoupled Scaled Boundary Finite Element Method to Solve Eigenvalue Helmholtz Problems (Research Note)

Document Type : Original Article


Faculty of Civil Engineering, University of Semnan, Semnan, Iran


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.


1. Kontoni, D., Partridge, P. and Brebbia, C., "The dual reciprocity
boundary element method for the eigenvalue analysis of
helmholtz problems", Advances in Engineering Software and
Workstations,  Vol. 13, No. 1, (1991), 2-16. 
2. Zienkiewicz, O.C., Taylor, R.L., Nithiarasu, P. and Zhu, J., "The
finite element method, McGraw-hill London,  Vol. 3,  (1977). 
3. Press, W.H., Flannery, B.P., Teukolsky, S.A. and Vetterling,
W.T., "Numerical recipes, Cambridge University Press
Cambridge,  Vol. 3,  (1989). 
4. Hall, W.S., Boundary element method, in The boundary element
method. 1994, Springer.61-83. 
5. Bathe, K.-J., "Finite element procedures, Klaus-Jurgen Bathe, 
6. Davis, C., Kim, J.G., Oh, H.-S. and Cho, M.H., "Meshfree
particle methods in the framework of boundary element methods
for the helmholtz equation", Journal of Scientific Computing, 
Vol. 55, No. 1, (2013), 200-230. 
7. Miers, L. and Telles, J., "Meshless boundary integral equations
with equilibrium satisfaction", Engineering Analysis with
Boundary Elements,  Vol. 34, No. 3, (2010), 259-263. 
8. Roberts, J.E., "Mixed and hybrid methods", Handbook of
Numerical Analysis, Finite Element Methods,  Vol. 2, No.,
9. Cheng, Y. and Peng, M., "Boundary element-free method for
elastodynamics", Science in China Series G: Physics and
Astronomy,  Vol. 48, No. 6, (2005), 641-657. 
10. Brebbia, C.A. and Dominguez, J., "Boundary elements: An
introductory course, WIT press,  (1994). 
11. Song, C. and Wolf, J.P., "The scaled boundary finite-element
method—alias consistent infinitesimal finite-element cell
method—for elastodynamics", Computer Methods in Applied
Mechanics and Engineering,  Vol. 147, No. 3-4, (1997), 329355.
12. Khaji, N. and Khodakarami, M., "A new semi-analytical method
with diagonal coefficient matrices for potential problems",
Engineering Analysis with Boundary Elements,  Vol. 35, No.
6, (2011), 845-854. 
13. Khodakarami, M. and Khaji, N., "Analysis of elastostatic
problems using a semi-analytical method with diagonal
coefficient matrices", Engineering Analysis with Boundary
Elements,  Vol. 35, No. 12, (2011), 1288-1296. 
14. Khodakarami, M. and Fakharian, M., "A new modification in
decoupled scaled boundary method with diagonal coefficient
matrices for analysis of 2d elastostatic and transient
elastodynamic problems",  Asian Journal of Civil Engineering,
(2015), Vol. 16, No, 5, 709-732. 
15. Khodakarami, M., Khaji, N. and Ahmadi, M., "Modeling
transient elastodynamic problems using a novel semi-analytical
method yielding decoupled partial differential equations",
Computer Methods in Applied Mechanics and Engineering, 
Vol. 213, No., (2012), 183-195. 
16. Khalaj-Hedayati, H. and Khodakarami, M.I., "Solving elastostatic
bounded problems with a novel arbitrary-shaped element", Civil Engineering Journal, Vol.5,No. 9, (2019), 1941-1958.
17. Vaziri Astaneh, A., Mahmoudzadeh Kani, I. and Sadeghirad, A.,
"A collocation method with modified equilibrium on line
method for imposition of neumann and robin boundary
conditions in acoustics", International Journal of Engineering, 
Vol. 23, No. 1, (2010), 11-22. 
18. Khodadadi, M. and Damiri-Ganji, D., "Analytical solution of the
laminar boundary layer flow over semi-infinite flat plate:
Variable surface temperature", International Journal of
Engineering,  Vol. 23, No. 3, (2010), 215-222.