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: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758990
• 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都是用偏微分方程 (PDE) 的数学形式编写的。除了某些理想化情况外，所得的偏微分方程无法通过分析求解。相反，我们依赖于数值近似。特别令人感兴趣的是高效实现的开发和自适应算法的分析，用于求解两个或三个空间维度中的时间相关偏微分方程，这些偏微分方程可能存在于非平坦表面上，其解表现出较大的解变化、奇点形成或移动前沿。我和我的学生研究的方法是通过自动将计算工作集中在空间和时间区域来获得有效的解决方案，其中解决方案具有这种有趣但难以跟踪的行为。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特征。 这是通过为网格制定一个（可能是非线性的）PDE 来实现的，该 PDE 取决于未知的解，因此与物理 PDE 耦合。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。我们建议将表面上的时间相关 PDE 的自适应解映射到多核环境，方法是将大问题分成小块，在各个核上进行计算，然后使用域分解（DD）算法和预处理器重新组合以给出原始问题的解。 我们将在交替或整体（一次性）框架中提供用于生成自适应网格和物理偏微分方程解的实现和分析算法。为了使大量核心饱和，将通过计算同时预测和校正来及时添加小规模并行性。 由此产生的软件将应用于与自然资源勘探相关的场景中的地球物理电磁问题。将我们的求解器扩展到表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真实的几何形状进行快速、可扩展的模拟。最终，我们将为时间相关偏微分方程的并行自适应解决方案提供一个新的、基于理论的模块化平台。适用于现有和新兴高性能计算硬件的通用表面。该研究项目为基于偏微分方程的网格生成方法提供了影响链，为需要解决复杂问题的计算科学家提供了软件工具。从理论上讲，该研究计划将增强我们对非线性耦合系统 DD 算法行为的了解。最后，它将为总部提供雇主高度寻求的数学专业知识、计算能力和关键可转移技能。