New Fast Convolution Algorithm in Boundary-Element Methods for Two- and Three-Dimensional Linear Soil Consolidation Analysis

Over the last $25\text{years}$ , the time domain boundary element formulations for the linear consolidation theory of Biot involving fully coupled governing differential equations of flow through porous media and those of elastic deformation of porous skeleton have been fully developed and implemented for both single-region and multiregion two-dimensional plane strain, axisymmetric and three-dimensional problems. However, this storage-based convolution method used in those developments was not found to be suitable for solving large practical problems because of the substantial computer disk space requirements. In order to find a better alternative, an accurate integration-based scheme was developed by the present writers and co-workers, in which, the storage was eliminated by accurately recalculating the summation (involved in the time convolution) of the right-hand side at each step during the time marching process. Although this work was not published in any literature, by using this type of approach, solving large scale problems became possible in an accurate manner, but the computational cost was significantly high, and there was a further need to develop a more practical and efficient time stepping algorithm. In the present work, an efficient and simplified integration-based fast convolution algorithm for two- and three-dimensional soil consolidation analysis has been subsequently developed. In this new algorithm, all of the time convolution steps have been calculated by assuming an equivalent spatial and temporal averaged value of the variables over each element to represent the total effect. The number of Gauss points used has been calculated in an efficient manner based on time-embedded distance criteria to accurately capture the past effects. The efficiency and accuracy of this newly developed fast convolution algorithm are compared with the accurate integration-based convolution approach and also with the analytical and other available solutions. Examples of applications involving two- and three-dimensional practical soil consolidation problems are presented to demonstrate the usefulness of the developed algorithm.