In computational fluid dynamics, we often need to solve complex problems with high precision and efficiency. We propose a three-pronged approach to attain this goal. First, we use the discontinuous Galerkin spectral element method (DG-SEM) for its high accuracy. Second, we use graphics processing units (GPUs) to perform our computations to exploit available parallel computing power. Third, we implement a parallel adaptive mesh refinement (AMR) algorithm to efficiently use our computing power where it is most needed. We present a GPU DG-SEM solver with AMR and dynamic load balancing for the 2D wave equation. The DG-SEM is a higher-order method that splits a domain into elements and represents the solution within these elements as a truncated series of orthogonal polynomials. This approach combines the geometric flexibility of finite-element methods with the exponential convergence of spectral methods. GPUs provide a massively parallel architecture, achieving a higher throughput than traditional CPUs. They are relatively new as a platform in the scientific community, therefore most algorithms need to be adapted to that new architecture. We perform most of our computations in parallel on multiple GPUs. AMR selectively refines elements in the domain where the error is estimated to be higher than a prescribed tolerance, via two mechanisms: p-refinement increases the polynomial order within elements, and h-refinement splits elements into several smaller ones. This provides a higher accuracy in important flow regions and increases capabilities of modeling complex flows, while saving computing power in other parts of the domain. We use the mortar element method to retain the exponential convergence of high-order methods at the non-conforming interfaces created by AMR. We implement a parallel dynamic load balancing algorithm to even out the load imbalance caused by solving problems in parallel over multiple GPUs with AMR. We implement a space-filling curve-based repartitioning algorithm which ensures good locality and small interfaces. While the intense calculations of the high order approach suit the GPU architecture, programming of the highly dynamic adaptive algorithm on GPUs is the most challenging aspect of this work. The resulting solver is tested on up to 64 GPUs on HPC platforms, where it shows good strong and weak scaling characteristics. Several example problems of increasing complexity are performed, showing a reduction in computation time of up to 3× on GPUs vs CPUs, depending on the loading of the GPUs and other user-defined choices of parameters. AMR is shown to improve computation times by an order of magnitude or more.
| Identifer | oai:union.ndltd.org:uottawa.ca/oai:ruor.uottawa.ca:10393/44507 |
| Date | 13 January 2023 |
| Creators | Tousignant, Guillaume |
| Contributors | Mavriplis, Catherine |
| Publisher | Université d'Ottawa / University of Ottawa |
| Source Sets | Université d’Ottawa |
| Language | English |
| Detected Language | English |
| Type | Thesis |
| Format | application/pdf |
| Rights | Attribution 4.0 International, http://creativecommons.org/licenses/by/4.0/ |
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