Performance Analysis of SIMD Vectorization of High-Order Finite-Element Kernels

Physics-based three-dimensional numerical simulations are becoming more predictive and are already essential for improving the understanding of natural phenomena, such as earthquakes, tsunami, flooding or climate change and global warming. Among the numerical methods available to support these simulations, Finite-Element formulations have been implemented in several major software packages. The efficiency of these algorithms remains a challenge due to the irregular memory access that prevents the squeezing out of the maximum level of performance out of current architectures. This is particularly true at the shared-memory level with several levels of parallelism and complex memory hierarchies. Despite significant efforts, automatic optimizations provided by compilers and high-level frameworks are often far from the performances obtained from hand-tuned implementations. In this paper, we have extracted a kernel from the EFISPEC software package developed at BRGM (the French Geological Survey). This application implements a high-order finite-element method to solve the elastodynamic equation. We characterize the performance of the extracted mini-app considering key parameters such as the order of the approximation, the memory access pattern or the vector length. Based on this study, we detail specific optimizations and we discuss the results measured as regards to the roofline performance model on Intel Broadwell and Skylake architectures.