Analysis and Implementation of Parallel Solvers for PDE Based Mesh Generation and Coupled Systems

基于偏微分方程的网格生成和耦合系统并行求解器的分析与实现

• 批准号: RGPIN-2018-04881
• 负责人: Haynes, Ronald
• 金额: $ 2.99万
• 依托单位: Memorial University of Newfoundland
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=713135
• 关键词: Analysis; Implementation; Parallel; Solvers; PDE

Many models of interest to engineers and scientists are written mathematically as partial differential equations (PDEs). Except for certain idealized situations, the resulting PDEs are not possible to solve analytically. Instead we rely on numerical approximations. Of particular interest is the development of efficient implementations and the analysis of adaptive algorithms for the solution of time-dependent PDEs in two or three spatial dimensions, possibly living on non-flat surfaces, whose solutions exhibit large solution variation, singularity formation or moving fronts. My students and I study methods which obtain an efficient solution by automatically concentrating computational effort in space and time regions in which the solution has this interesting, but difficult to track behaviour. The strategy works by using automatically generated moving spatial meshes to adapt to the desired features. This is achieved by formulating a (possibly nonlinear) PDE for the mesh, which depends on the unknown solution, and hence is coupled to the physical PDE. We wish to study algorithms specifically designed to take advantage of readily available compute clusters with hundreds or thousands of cores, hybrid CPU-GPU systems and even desktop machines with multiple cores. We propose mapping the adaptive solution of time dependent PDEs on surfaces to multicore environments by dividing the large problem into small pieces, computing on individual cores and then recombining to give a solution of the original problem using domain decomposition (DD) algorithms and preconditioners. We will provide implementations and analyze algorithms for the generation of the adaptive grids and the solution of the physical PDE in either an alternating or monolithic (one-shot) framework. To saturate very large numbers of cores, small scale parallelism in time will be added by computing simultaneous predictions and corrections. The resulting software will be applied to geophysical electromagnetic problems in scenarios relevant to natural resource exploration. Extending our solvers to PDEs defined on surfaces will allow for fast, scalable simulations on more realistic geometries of interest to computational scientists and engineers. Ultimately, we will provide a new, theoretically based, modular platform for the parallel adaptive solution of time dependent PDEs on general surfaces suitable for existing and emerging high performance computing hardware. This research program provides an impact chain for PDE based mesh generation methods, providing a software tool for computational scientists requiring the solution of complex problems. Theoretically, the research program will enhance our knowledge of the behaviour of DD algorithms for nonlinear coupled systems. Finally, it will provide HQP with mathematical expertise, computational capability, and key transferable skills highly sought by employers.

工程师和科学家感兴趣的许多模型在数学上被写成偏微分方程（PDE）。除了某些理想化的情况下，所得到的偏微分方程是不可能解析求解的。相反，我们依赖于数值近似。特别感兴趣的是开发有效的实现和分析的自适应算法的解决方案的时间依赖的偏微分方程在两个或三个空间维度，可能生活在非平坦的表面上，其解决方案表现出大的解决方案的变化，奇点形成或移动的前线。我的学生和我研究的方法，通过自动集中在空间和时间区域的计算工作，获得一个有效的解决方案，其中解决方案具有这种有趣的，但难以跟踪的行为。该策略的工作原理是使用自动生成的移动空间网格来适应所需的功能。 这是通过制定一个（可能是非线性的）PDE的网格，这取决于未知的解决方案，因此耦合到物理PDE。我们希望研究专门设计的算法，以利用具有数百或数千个核心的现成计算集群，混合CPU-GPU系统，甚至具有多个核心的台式机。我们建议映射的自适应解决方案的时间相关的偏微分方程表面上的多核环境划分成小块的大问题，计算个别核心，然后重新组合，以给出一个解决方案的原始问题，使用域分解（DD）算法和预条件。 我们将提供实现和分析算法的自适应网格的生成和解决方案的物理PDE在一个交替或单片（一杆）的框架。为了使非常大量的核饱和，将通过计算同时的预测和校正来添加时间上的小规模并行性。 由此产生的软件将应用于地球物理电磁问题的场景有关的自然资源勘探。将我们的求解器扩展到曲面上定义的偏微分方程，将允许对计算科学家和工程师感兴趣的更现实的几何形状进行快速，可扩展的模拟。 最终，我们将提供一个新的，基于理论的，模块化的平台，适用于现有的和新兴的高性能计算硬件的一般表面上的时间相关偏微分方程的并行自适应解决方案。该研究计划为基于偏微分方程的网格生成方法提供了一个影响链，为需要解决复杂问题的计算科学家提供了一个软件工具。理论上，研究计划将提高我们的知识的行为DD算法的非线性耦合系统。最后，它将为HQP提供数学专业知识，计算能力和雇主高度寻求的关键可转移技能。