Recursive evaluation of interaction forces of unbounded soil in the time domain

The interaction forces representing the contribution of the linear unbounded soil to the equations of motion of a nonlinear soil‐structure‐interaction analysis are specified in the form of convolution integrals. They can be evaluated recursively in the time domain. In this procedure, the forces at a specific time are computed from the displacements at the same time and from the most recent forces and most recent past displacements. It is, in principle, only approximate. When the dynamic‐stiffness coefficients can be expressed as the ratios of two polynomials in frequency, the appropriately chosen recursive equations are exact. Two possibilities of choosing a recursive equation are discussed.(i) The impulse‐invariant method, where the unknown recursive coefficients are calculated by solving a system of equations which are established by equating the rigorous and recursive formulations for a discretized unit impulse displacement. (ii) In the segment approach, the dynamic‐stiffness coefficients in the time domain are interpolated piecewise. Applying the z ‐transformation analytically then results in an explicit recursive equation without solving a system of equations.The recursive evaluation of the convolution integrals in the time domain leads to a dramatic reduction in the computational effort up to two and three orders of magnitude and in the storage requirement. This makes the time‐domain analysis using the substructure method computationally competitive with the corresponding direct (non‐recursive) frequency‐domain procedure of determining the complex response which is, however, applicable only to a linear (total) system.