In this work, we propose an adaptive geometric multigrid method for the solution of large-scale finite cell flow problems. The finite cell method seeks to circumvent the need for a boundary-conforming mesh through the embedding of the physical domain in a regular background mesh. As a result of the intersection between the physical domain and the background computational mesh, the resultant systems of equations are typically numerically ill-conditioned, rendering the appropriate treatment of cutcells a crucial aspect of the solver. To this end, we propose a smoother operator with favorable parallel properties and discuss its memory footprint and parallelization aspects. We propose three cache policies that offer a balance between cached and on-the-fly computation and discuss the optimization opportunities offered by the smoother operator. It is shown that the smoother operator, on account of its additive nature, can be replicated in parallel exactly with little communication overhead, which offers a major advantage in parallel settings as the geometric multigrid solver is consequently independent of the number of processes. The convergence and scalability of the geometric multigrid method is studied using numerical examples. It is shown that the iteration count of the solver remains bounded independent of the problem size and depth of the grid hierarchy. The solver is shown to obtain excellent weak and strong scaling using numerical benchmarks with more than 665 million degrees of freedom. The presented geometric multigrid solver is, therefore, an attractive option for the solution of large-scale finite cell problems in massively parallel high-performance computing environments.

翻译：本文提出了一种自适应几何多重网格方法，用于求解大规模有限单元流问题。有限单元法通过将物理域嵌入到规则的背景网格中，旨在避免对边界适配网格的需求。由于物理域与背景计算网格之间的交叉，所得方程组通常数值病态，因此对切割单元进行适当处理成为求解器中的关键环节。为此，我们提出了一种具有良好并行特性的平滑算子，并讨论了其内存占用和并行化方面。我们提出了三种缓存策略，在缓存计算与实时计算之间取得平衡，并探讨了该平滑算子所提供的优化机会。研究表明，由于该平滑算子具有加法性质，可以在并行环境下以较小通信开销精确复制，这为并行设置带来显著优势，因为几何多重网格求解器因此不依赖于进程数量。通过数值算例研究了该几何多重网格方法的收敛性和可扩展性。结果表明，求解器的迭代次数保持有界，与问题规模和网格层级深度无关。该求解器在超过6.65亿自由度的数值基准测试中展现出了优异的弱扩展性和强扩展性。因此，所提出的几何多重网格求解器是大规模并行高性能计算环境下解决大规模有限单元问题的理想选择。