Research Paper Parallel finite element analysis of seismic soil structure interaction using a PC cluster B. Zhao a,1 , Y. Liu b,⇑ , S.H. Goh b , F.H. Lee b a Department of Offshore Structure and Analysis, Keppel Offshore and Marine Technology Centre Pte Ltd., Shipyard Road 31, Singapore 628130, Singapore b Department of Civil and Environmental Engineering, National University of Singapore, Block E1A, #07-03, No. 1 Engineering Drive 2, Singapore 117576, Singapore article info Article history: Received 22 July 2015 Received in revised form 11 February 2016 Accepted 11 July 2016 Keywords: Parallel finite element analysis Pre-conditioned conjugate gradient algorithm Message passing interface Soil structure interaction Seismic excitation abstract This paper describes an implementation of a highly scalable parallel computational facility with high speedup efficiency using relatively low-cost hardware, which consists of a cluster of desktop personal computers (PCs) connected via a 10-Gigabit Ethernet. Two-levels of parallelization were implemented. Communication between different PCs was achieved using message passing interface (MPI) protocol. Domain decomposition was automated and based on element numbering. Domain continuity was assured largely by re-numbering the elements using a ‘‘front squasher” code prior to decomposition. Within each PC, the shared memory parallelization was implemented using either the open- multiprocessing (OpenMP) or the MPI protocol. Analysis of three different problems with number of degrees-of-freedom ranging from about 129,000 to about 2,260,000 shows a speedup efficiency generally above 70%. Super-linear speedup was achieved in several of the cases examined in this study, with the hybrid MPI-OpenMP approach generally performing better compared to the pure MPI method for paral- lelization. The results demonstrate the feasibility of acquiring a parallel computing facility with relatively modest outlay that is within the reach of consulting or engineering offices. Ó 2016 Elsevier Ltd. All rights reserved. 1. Introduction A common approach for analysing seismic effect on structures involves using the peak ground acceleration to generate equivalent static structural loadings which are then applied onto the structure (e.g. [38,55]. Such an approach does not consider seismic soil- structure interaction since the peak ground acceleration applies typically to ‘‘green field” condition, that is, without the structure in place. This approach is applicable to situations where soil- structure interaction effects are small compared to site-response effects (e.g. [50]. However, for more massive structures such as high-rise buildings, soil-structure interaction effects may signifi- cantly influence the base motion of the structure. An example of this is high-rise buildings constructed over soft soil strata, wherein the foundation often comprises dense grids of piles. In addition, if the building footprint is large and the soil is soft, the shear wave length of the soil may be of the same order as the footprint dimensions of the building. For instance, the shear wave velocity of soft Singapore marine clay is about 70 m/s to about 150 m/s [36]. In such situations, the wavelengths of shear and Rayleigh waves may be of the same order as the length of the building and significant phase differences may exist between different parts of the building during earthquake excitation, caus- ing building distortion. Such building distortional effects cannot be captured without analysing the building and its foundation. Most dynamic soil-structure interaction problems analysed to date are relatively small in scale, often involving individual foundations or small pile groups (e.g. [7,12]). For larger problems involving entire foundation systems, paral- lel computation may be required. Although early works on parallel computing were conducted mainly on vector and shared memory machines (e.g. [2,20,25]), these are not readily scalable and have been largely replaced by distributed memory machines which are more readily up-scaled. Distributed parallel computing has been used to analyse propagation of seismic waves through the Earth’s crust (e.g. [29]. However, these implementations usually involve explicit time-integration schemes, which are less often used in dynamic soil-structure interaction problems due to phase errors arising from mass lumping [23] and the difficulties in dealing with strain-rate-dependent damping. Instead, implicit http://dx.doi.org/10.1016/j.compgeo.2016.07.006 0266-352X/Ó 2016 Elsevier Ltd. All rights reserved. ⇑ Corresponding author at: Department of Civil & Environmental Engineering, National University of Singapore, 1 Engineering Drive 2, Singapore 117576, Singapore. E-mail addresses: ben.zhao@KOMtech.com.sg (B. Zhao), ceeliuy@gmail.com (Y. Liu), ceegsh@nus.edu.sg (S.H. Goh), ceeleefh@nus.edu.sg (F.H. Lee). 1 Former research fellow at National University of Singapore. Computers and Geotechnics 80 (2016) 167–177 Contents lists available at ScienceDirect Computers and Geotechnics journal homepage: www.elsevier.com/locate/compgeo