Time‐domain analysis of unbounded media using mixed‐variable formulations

Abstract

SUMMARYFormulation of a matrix-valued force-displacement relationship which can take radiation damping into account is of major importance when modelling unbounded domains. This can be done by means of fundamental solutions in space and time in connection with convolution integrals or by means of a frequency dependent boundary element representation, but for discrete frequencies only. In this paper a method for interpolating discrete values of dynamic sti ness matrices by a continuous matrix valued rational function is proposed. The coupling between interface degrees of freedom is fully preserved. Another crucial point in soil-structure interaction analysis is how to implement an approximation in the spectral domain into a time-domain analysis. Well-known approaches for the scalar case are based on the partial-fraction expansion of a scalar rational function. Here, a more general procedure, applicable to MDOF-systems, for the transformation of spectral rational approximations into the time-domain is introduced. Evaluation of the partial-fraction expansion is avoided by using the so-called mixed variables. Thus, unknowns in the time-domain are displacements as well as forces.

Time‐domain analysis of wave propagation in unbounded domains based on the scaled boundary finite element method

“…(2) is the scaled boundary finite element equation in displacement. It can be transformed into an equivalent formulation in dynamic stiffness (3).…”

“…In order to obtain a time-domain model of the far field, the low-frequency part S ∞ l of the dynamic stiffness S ∞ is approximated by a rational matrix-valued function [3,4].…”

“…The rational force-displacement relationship can be transformed into a system of linear equations algebraically using the 'mixed-variables technique' [3,4].…”

Wave propagation in the soil due to a falling chimney

“…al. [3] proposed the transformation of frequency-dependent dynamic stiffness relationships to the time-domain using the 'mixed variables technique'. Here, the force-displacement relationship is approximated by a rational function in terms of iΩ.…”

Time‐domain soil‐structure interaction analysis using the scaled boundary finite element method and rational stiffness approximations

“…The rational representation can be transformed into a system of linear equations in the frequency-domain using the mixed-variables technique [2]. This corresponds to a system of first-order differential equations in the time-domain (Eq.…”