Symmetric matrix-valued transmitting boundary formulation in the time-domain for soil–structure interaction problems

• Dynamic soil–structure interaction problems with an unbounded soil-domain require time domain formulations for both parts.
• Structures are described in the time-domain; soil by frequency-dependent stiffness: switching from frequency- to time-domain is required.
• A dynamic stiffness Padé-like rational representation has been extended: the new formulation preserves matrices symmetry in the time-domain.
• A soil–rotor–foundation interaction problem with a transient excitation has been solved.
• Results are presented and issues concerning sensitivity of linear algebraic solvers and their accuracy are discussed in detail.

Time-domain formulations for soil–structure interaction problems with an unbounded soil-domain (the so-called far-field) including wave propagation require such time-domain formulations for both parts, soil and structure. For the structure (the near-field), typically treated by a finite element approach, the time-domain is used from the very beginning of the procedure. However, for the unbounded soil a representation by means of a frequency-dependent dynamic stiffness is usually available and it becomes necessary to devise techniques for switching from the frequency- to the time-domain.For various special cases in solid mechanics (e.g. plane, cylindrical and spherical waves) one-dimensional formulations in space have been used to derive scalar dynamical stiffness, to establish corresponding rational functions in the frequency-domain and transfer them into the time-domain in order to couple the near- and the far-field.A complete three-dimensional analysis for pile-groups through a linear homogeneous unbounded soil-domain and the corresponding description in the time-domain have already been treated by Cazzani and Ruge (2012, 2013) by means of a fully matrix-valued rational representation of a set of dynamic stiffness matrices K(Ω)K(Ω), as a function of the angular frequency ΩΩ. However, the symmetry of the input stiffness K(Ω)K(Ω) has not been maintained for the corresponding representation in the time-domain.This paper presents a fully matrix-valued rational formulation which does transfer the symmetry of K(Ω)K(Ω) to the corresponding formulation in the time-domain. Thus, the numerical treatment of the whole soil–structure interaction problem, coupling the far-field and near-field systems, can take advantage of algorithms for symmetric algebraic problems.