Structural problems play a critical role in many areas of science and engineering. Their efficient and accurate solution is essential for designing and optimising civil engineering, aerospace, and materials science applications, to name a few. When appropriately tuned, Algebraic Multigrid (AMG) methods exhibit a convergence that is independent of the problem size, making them the preferred option for solving structural problems. Nevertheless, AMG faces several computational challenges, including its remarkable memory footprint, costly setup, and the relatively low arithmetic intensity of the sparse linear algebra operations. This work presents AMGR, an enhanced variant of AMG that mitigates such limitations. Its name arises from the AMG reduction framework it introduces, and its flexibility allows for leveraging several features that are common in structural problems. Namely, periodicities, spatial symmetries, and localised non-linearities. For such cases, we show how to reduce the memory footprint and setup costs of the standard AMG, as well as increase its arithmetic intensity. Despite being lighter than the standard AMG, AMGR exhibits comparable scalability and convergence rates. Numerical experiments on several industrial applications prove AMGR’s effectiveness, resulting in up to 3.7x overall speed-ups compared to the standard AMG.

中文翻译：

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用于结构力学的基于 AMG 简化的高效预处理器


结构问题在科学和工程的许多领域中发挥着至关重要的作用。他们高效、准确的解决方案对于设计和优化土木工程、航空航天和材料科学应用等至关重要。经过适当调整后，代数多重网格 (AMG) 方法表现出与问题大小无关的收敛性，使其成为解决结构问题的首选。然而，AMG 面临着一些计算挑战，包括其显着的内存占用、昂贵的设置以及稀疏线性代数运算的相对较低的算术强度。这项工作提出了 AMGR，这是 AMG 的一种增强变体，可以缓解此类限制。它的名字来源于它引入的 AMG 简化框架，其灵活性允许利用结构问题中常见的几个功能。即周期性、空间对称性和局部非线性。对于这种情况，我们展示了如何减少标准 AMG 的内存占用和设置成本，并提高其算术强度。尽管比标准 AMG 更轻，AMGR 仍表现出可比的可扩展性和收敛速度。多个工业应用的数值实验证明了 AMGR 的有效性，与标准 AMG 相比，整体加速高达 3.7 倍。