面向复杂三维几何的谱/hp元方法的低阶细化预处理
Low-Order Refined Preconditioning for Spectral/hp Element Method for Complex, 3D Geometries
- King’s College London（伦敦国王学院）
- McLaren Racing Limited（迈凯伦赛车有限公司）
- Imperial College London（帝国理工学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文扩展低阶细化预处理至单纯形和混合单元，建立广义范德蒙德变换，在复杂三维几何中显著减少CG迭代次数并加速生产级模拟。
AI中文摘要：
低阶细化（LOR）预处理将高阶算子替换为在细化节点网格上的低阶离散。对于张量积单元，这两个算子在谱意义下等价，其界与多项式阶数$P$无关，但该构造不能直接推广到单纯形和混合单元离散。本文做出两项贡献：将LOR预处理推广到单纯形和混合单元离散，包括三角形、四面体和棱柱单元，并建立了一个广义的范德蒙德变换，将模态和节点LOR公式联系起来，表明由此产生的预处理谱和Krylov收敛性与高阶基无关。数值实验表明，在三角形网格上，尽管条件数增加，迭代次数仍受控增长；在四面体、棱柱和混合单元网格上，直到$P=6$，迭代次数均受控。每次外迭代使用一次代数多重网格V循环，在迭代次数和成本之间取得了最佳平衡。该方法应用于一个生产级不可压缩纳维-斯托克斯模拟，即赛车前翼和车轮构型，该模拟在由$2.87\ imes10^6$个混合棱柱和四面体单元组成的网格上离散，在$P=3$时产生$32.2\ imes10^6$个压力自由度。与Nektar++中默认的生产级静态凝聚对角预处理相比，LOR将平均压力共轭梯度（CG）迭代次数从$235.3$降至$5.5$，并将1000个时间步内的压力求解时间减少了16.1%。构建LOR预处理的一次性成本在生产模拟中得到摊销。
英文摘要：
Low-order refined (LOR) preconditioning replaces a high-order operator with a low-order discretisation on a refined nodal mesh. For tensor-product elements, the two operators are spectrally equivalent with bounds independent of the polynomial order $P$, but the construction does not extend directly to simplex and mixed-element discretisations. This work makes two contributions: it extends LOR preconditioning to simplex and mixed-element discretisations, including triangular, tetrahedral, and prismatic elements, and establishes a generalised Vandermonde transformation linking the modal and nodal LOR formulations, showing that the resulting preconditioned spectra and Krylov convergence are independent of the high-order basis. Numerical experiments show controlled iteration growth on triangular meshes despite increasing condition number, and controlled iteration counts up to $P=5$ on tetrahedral, prismatic, and mixed-element meshes. A single algebraic multigrid V-cycle per outer iteration gives the best balance of iteration count and cost. The method is applied to a production incompressible Navier-Stokes simulation of a race-car front-wing and wheel configuration, discretised on a mesh of $2.87\times10^6$ mixed prismatic and tetrahedral elements giving $32.2\times10^6$ pressure degrees of freedom at $P=3$. LOR reduces the mean pressure conjugate gradient (CG) iteration count from $235.3$ to $5.5$, and the pressure-solve time over 1000 timesteps by 16.1%, relative to the default production static-condensation diagonal preconditioner in Nektar++. The one-time cost of constructing the LOR preconditioner is amortised over the production simulation.
