### A Scalability Study using Supercomputers for Huge Finite Element Variably Saturated Flow Simulations

#### Abstract

This paper describes the challenges and scalability results of running a large finite element model of variably saturated flow in a three-dimensional (3-D) levee on a large high performance, parallel computer using a mesh with more than a billion nodes and two billion elements. MPI (Message Passing Interface) was used for the parallelization. The original finite element model consisted of 3,017,367 nodes and 5,836,072 3-D prism elements. The model exhibited three characteristics which made the problem difficult to solve. First, the different soil layers had soil properties that differed by several orders of magnitude. Secondly, there existed a 5 ft x 6 ft x 6 ft region at the toe of the levee where the mesh was refined using 1 in x 1 in x 1 in 3-D prism elements having randomly generated soil properties. Thirdly, variably saturated flow in levees is governed by the highly nonlinear Richards equation. A utility program was written to increase the size of the original problem by an arbitrarily large factor by replicating the original mesh in the y direction. A factor of two, for instance, would exactly double the number of elements and double the number of nodes less the interface nodes connecting the two pieces. The original data set was run using 32, 64, 96, and 256 MPI processes (one core per process was used throughout this study) with time to solution taken for each of these process counts. The data set was then magnified by 2 and runs for 64, 128, 192, and 512 processes were made with time to solution again recorded. This procedure was repeated for different numbers of processes and magnification values. The largest data set was generated from a magnification of 350 yielding a mesh of 1,044,246,303 nodes and 2,042,625,200 3-D prism elements. The Cray XE6 and Cray XC30 computers were used in this study. A tabulation of results is presented and analyzed, as well as the significant challenges that occurred in scaling up the problem size. Weak and strong scalability results are also presented in this paper.

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