Mesh adaptation on the sphere using optimal transport and the numerical solution of a Monge-Ampère type equation
Mesh adaptation on the sphere using optimal transport and the numerical solution of a Monge-Ampère type equation
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DOI：
10.1016/j.jcp.2015.12.018
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发表时间：
2015-12
期刊：
影响因子：
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通讯作者：
H. Weller;P. Browne;C. Budd;M. Cullen
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文献类型：
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作者：
H. Weller;P. Browne;C. Budd;M. Cullen
Monge-Ampère型方程，第一次，数值求解的表面上的球，以产生最佳运输（OT）网格，均匀分布的监控功能。最优传输生成的网格保持与原始网格相同的连通性，使它们适合于r-自适应模拟，其中运动方程可以在移动参考系中求解，以避免新旧网格之间的映射，并避免并行计算机上的负载平衡问题。Monge-Ampère型方程的半隐式解涉及Hessian项的新线性化，并且指数映射用于在球体上从旧网格映射到新网格。Hessian的行列式被评价为新旧网格单元之间的体积变化，而不是使用对梯度的数值近似。OT网格被生成以与球体上的质心Voronoi镶嵌进行比较，发现其优点和缺点; OT等分布更精确，收敛迭代次数与网格尺寸无关，减少了面偏斜度并且连通性不改变。然而，各向异性较高，OT网格是非正交的。结果表明，最佳的传输球导致网格不纠缠。然而，在计算网格势的梯度时，数值误差会引入缠结。最后，以实测降水量为监控函数，生成了OT网格，展示了该技术的潜力。
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 tessellations 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.