在球面上利用最优传输和一种 Monge-Ampère 类方程的数值解进行网格自适应
Mesh Adaptation on the Sphere using Optimal Transport and the Numerical Solution of a Monge-Ampère type Equation
AI总结：
本文首次在球面上用数值方法解 Monge-Ampère 类方程生成最优传输网格，该网格均匀分布于监控函数，保持原有网格连通性，适用于 r 自适应模拟。
摘要：
一种 Monge-Ampère 类方程首次在球面上被数值求解，以生成最优传输（OT）网格，这些网格相对于监控函数均匀分布。最优传输生成的网格保持原有网格的连通性，使其适用于 r 自适应模拟，在移动参考系中求解运动方程以避免在旧与新网格之间映射解及并行计算机上的负载平衡问题。半隐式求解 Monge-Ampère 类方程涉及对 Hessian 项的新线性化，利用指数映射将旧网格映射到新网格。Hessian 的行列式被计算为旧与新网格单元之间的体积变化，而非使用梯度的数值近似。OT 网格被生成以与球面上的质心 Voronoi 分割进行比较，发现 OT 网格在准确性、迭代次数与面扭曲度方面具有优势，但各向异性更高且非正交。研究表明，球面上的最优传输生成的网格不会扭曲，但梯度计算中的数值误差可能引入扭曲。最后，使用观测降水作为监控函数生成 OT 网格，以展示该技术的潜力。
英文摘要：
An equation of Monge-Ampère type has, for the first time, been solved numerically on the surface of the sphere in order to generate optimally transported (OT) meshes, equidistributed with respect to a monitor function. Optimal transport generates meshes that keep the same connectivity as the original mesh, making them suitable for r-adaptive simulations, in which the equations of motion can be solved in a moving frame of reference in order to avoid mapping the solution between old and new meshes and to avoid load balancing problems on parallel computers. The semi-implicit solution of the Monge-Ampère type equation involves a new linearisation of the Hessian term, and exponential maps are used to map from old to new meshes on the sphere. The determinant of the Hessian is evaluated as the change in volume between old and new mesh cells, rather than using numerical approximations to the gradients. OT meshes are generated to compare with centroidal Voronoi tesselations on the sphere and are found to have advantages and disadvantages; OT equidistribution is more accurate, the number of iterations to convergence is independent of the mesh size, face skewness is reduced and the connectivity does not change. However anisotropy is higher and the OT meshes are non-orthogonal. It is shown that optimal transport on the sphere leads to meshes that do not tangle. However, tangling can be introduced by numerical errors in calculating the gradient of the mesh potential. Methods for alleviating this problem are explored. Finally, OT meshes are generated using observed precipitation as a monitor function, in order to demonstrate the potential power of the technique.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
