A Frobenius solution to the scaled boundary finite element equations in frequency domain for bounded media

• Published in: International journal for numerical methods in engineering Vol. 70; no. 12; pp. 1387 - 1408
• Main Authors: Yang, Z. J., Deeks, A. J., Hao, H.
• Format: Journal Article
• Language: English
• Published: Chichester, UK John Wiley & Sons, Ltd 18.06.2007; Wiley
• Subjects: Computational techniques; elastic dynamics; Exact sciences and technology; frequency domain; Fundamental areas of phenomenology (including applications); harmonic excitation; Mathematical methods in physics; Physics; scaled boundary finite element method; second-order non-homogeneous differential equations; Solid mechanics; Static elasticity (thermoelasticity...); Structural and continuum mechanics; Vibration, mechanical wave, dynamic stability (aeroelasticity, vibration control...); Vibration; Elasticity; harmonic excitation; Frequency domain method; Elastic wave; frequency domain; Modeling; Steady state; Power series; Confined space; Finite element method; Frobenius theorem; elastic dynamics; Series expansion; Rectangular plate; Frequency response; second-order non-homogeneous differential equations; Elastodynamics; Sinusoidal excitation; scaled boundary finite element method; Boundary element method
• ISSN: 0029-5981; 1097-0207
• DOI: 10.1002/nme.1926

The scaled boundary finite element method (FEM) is a recently developed semi‐analytical numerical approach combining advantages of the FEM and the boundary element method. Although for elastostatics, the governing homogeneous differential equations in the radial co‐ordinate can be solved analytically without much effort, an analytical solution to the non‐homogeneous differential equations in frequency domain for elastodynamics has so far only been obtained by a rather tedious series‐expansion procedure. This paper develops a much simpler procedure to obtain such an analytical solution by increasing the number of power series in the solution until the required accuracy is achieved. The procedure is applied to an extensive study of the steady‐state frequency response of a square plate subjected to harmonic excitation. Comparison of the results with those obtained using ABAQUS shows that the new method is as accurate as a detailed finite element model in calculating steady‐state responses for a wide range of frequencies using only a fraction of the degrees of freedom required in the latter. Copyright © 2006 John Wiley & Sons, Ltd.