Improvement of transient viscoelastic auxiliary states synthesis and their incorporation into the boundary element method

Abstract

The present research plan deals with the synthesis of a series of transient viscoelastic non-singular auxiliary states that may be incorporated into a Boundary Element procedure. The Boundary Element formulation based on the aforementioned non-singular viscoelastic solutions requires only the discretization of the boundary of the domain under analysis. The analysis may be performed in bounded or in unbounded domains. Initially the viscoelastic non-singular auxiliary solutions are synthesized in the frequency domain. In the sequence a numerical inverse Fourier transform yields the transient solutions. Viscoelastic constitutive relations are introduced by means of the elastic-viscoelastic correspondence principle. This formulation allows, in principle, the introduction of any viscoelastic model or measured viscoelastic properties. In the last years the group of the guest researcher has developed numerical strategies to synthesize frequency domain solutions at extremely high frequencies, which, in turn, can be used in conjunction with the Fast Fourier Transform (FFT) algorithm to obtain transient viscoelastic solutions at very small time steps. These transient viscoelastic solutions are able to capture in detail the wave propagation process in half-space surfaces, full-spaces or at the interfaces of dynamic soil-foundation interaction phenomena. These auxiliary states have been incorporated into a Boundary Element scheme resulting in a very stable time procedure to analyze transient viscoelastic phenomena, in particular soil-structure interaction problems. As will be described in a more complete manner along the text of this research plan, there is a need to improve the synthesis of these auxiliary states in two distinct ways. Both improvements are related to the causality of the response. The first topic to be addressed is the need to perform a mathematical treatment of the frequency signal (the auxiliary state) at very high frequencies. It is necessary to introduce a proper filter at the high frequency end of the signal to avoid abrupt discontinuities at the high end of the spectrum. This discontinuities cause disturbances at the very initial time instants of the transient solution. The second improvement is to generalize the viscoelastic models that can be analyzed by the auxiliary states. Up to the present time only the imaginary part of the complex viscoelastic parameters may be frequency dependent. This allow for the analysis of may viscoelastic models but it needs to be generalized in order to include a broader class of models or measured viscoelastic properties. This generalization will also make possible to address causal and non-causal viscoelastic models. These are the two main activities to be performed in the 10 weeks period, which is being foreseen for this postdoctoral stay at the Technische Universität Braunschweig. As will be described in this research plan the host, Prof. Heinz Antes, has been extremely active in the area of Boundary Element research for transient analysis. Together with his co-workers he has developed an alternative procedure to perform transient analysis of viscoelastic and poroelastic continua. The procedure that discussed in this plan is an alternative dot the Convolution Quadrature (CQM). It would be important to compare the potentialities of the CQM with the approach presented by the guest researcher. (AU)