A Frequency-Domain Approach for Transient Dynamic Analysis Using Scaled Boundary Finite Element Method (I): Approach and Validation

Solutions of the scaled boundary finite element method (SBFEM) are approximate in the circumferential direction in the finite element sense, and analytical in the radial direction. This semi-analytical nature makes the SBFEM particularly suitable for certain situations such as unbounded media and solids with stress singularities (e.g., cracks or corners). This method has been very successfully applied in elastostatics. Its application to elastodynamic problems, however, has lagged behind, primarily due to the lack of an effective solution procedure to the governing equilibrium equation system in the frequency domain, which consists of second-order nonhomogeneous differential equations with respect to the radial coordinate. The authors recently developed an easy-to-follow Frobenius solution procedure to this equation system and conducted initial validations against simple problems subjected to harmonic loadings [1]. This study further develops a frequency-domain approach for the general transient dynamic analysis, through combining the Frobenius solution procedure and the fast Fourier transform (FFT) technique. The complex frequency-response functions (CFRFs) are first computed using the Frobenius solution procedure. This is followed by a FFT of the transient load and a subsequent inverse FFT of the CFRFs to obtain the time history of responses. Two wave propagation problems, subjected to Heaviside step load and triangular blast load, are modeled with a small number of degrees of freedom using the new approach. The numerical results agree very well with the analytical solutions and those from FEM. Further applications of this approach to transient dynamic fracture problems are presented in the accompanying paper [2].