独立覆盖流形法采取“分区级数解”的思想,将计算域划分为若干分区,在各个分区内部采用多项式级数等完备级数直接逼近真实物理场函数;各分区之间采用窄条带连接,通过单位分解函数将各分区级数连接成整体近似函数,在收敛意义上具有C₁连续性;计算网格具有任意形状、任意连接和任意加密的特性。基于计算网格的任意性,采用“一分为二”的网格分裂算法作为网格划分和细分的策略;基于收敛意义上的整体C₁连续性,采用物理场导数的连续程度作为误差控制的指标。以上组成了基于任意网格的自适应分析策略。通过数值试验验证了这一策略的可行性。在此基础上提出优化方案:采用绝对误差指标控制近似函数导数的连续性,以简化误差判断;采用网格预划分的方式优化凹角局部区域的网格分布。最后,通过方孔、重力坝等算例进行验证,表明优化方案可以大幅减少网格数量,从而节省算力。

[Objective] This study aims to optimize the two-dimensional adaptive analysis strategy of the independent cover-based manifold method, focusing on addressing its deficiencies in error control and mesh distribution, thereby significantly enhancing computational efficiency and engineering practicality. [Methods] Based on the arbitrarily shaped and connected cover meshes of the independent cover-based manifold method, a “split-one-into-two” mesh splitting algorithm was employed for arbitrary refinement, and the degree of continuity of physical field derivatives was adopted as the error control indicator, forming an adaptive analysis strategy. An optimization scheme was proposed. 1) Adopting an absolute error indicator to replace the relative error indicator: the original relative error indicator tended to cause over-refinement in regions of minor stress and was overly sensitive in concave corner singularity regions. Using the absolute error indicator not only simplified the error judgment logic but also permitted larger error thresholds to be set near singular points such as concave corners, thereby effectively avoiding over-refinement. 2) Introducing a local mesh pre-partitioning and short strip elimination strategy: to address the issue of excessively high mesh density and irregular distribution in concave corner regions, a local pre-partitioning strategy was proposed, which pre-set the initial mesh in these regions by inwardly offsetting and reversely extending the edges of the concave corner. Simultaneously, an adjacent point merging algorithm was introduced during the mesh splitting process, which avoided the generation of extremely short connection strips and improved the conditioning of the system equations. [Results] Verification through two typical hydraulic structure examples, the square-hole and the gravity-dam model, demonstrated that the optimized scheme achieved a breakthrough improvement in computational efficiency. For the square-hole example, the original adaptive strategy generated 310 covers, corresponding to 6 520 degrees of freedom (DOFs). Under the same accuracy objective, the optimized scheme required only 59 covers and 933 DOFs. This represented a reduction of approximately 81% in the number of meshes and approximately 86% in DOFs. For the gravity-dam example, the original strategy generated 228 covers and 4 810 DOFs, whereas the optimized scheme required only 106 covers and 2 354 DOFs, achieving significant results of over 53% reduction in the number of meshes and 51% reduction in DOFs. The most notable achievement of the optimized scheme was in the effective suppression of mesh over-refinement near concave corner singularity regions. The calculation results demonstrated that the new strategy could generate more reasonable meshes, while ensuring computational accuracy, it substantially reduced the computational scale, and greatly enhanced the computational efficiency. [Conclusion] The proposed optimization strategy significantly enhances the efficiency of adaptive analysis while maintaining high accuracy. Through absolute error control and local mesh pre-partitioning, it effectively solves the problems of mesh over-refinement and unreasonable distribution near concave corner singularities, laying a foundation for subsequent three-dimensional adaptive analysis and engineering applications. Future research includes: criteria for selecting error thresholds and the highest order of cover series; further automating the local mesh pre-partitioning process to enable it to handle more complex geometries, ultimately achieving the goal of efficient and fully automatic simulation analysis for hydraulic structures.