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Abstract
An equation of MongeAmpère 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 radaptive 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 semiimplicit solution of the MongeAmpère 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 nonorthogonal. 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.
Original language  English 

Pages (fromto)  102123 
Number of pages  22 
Journal  Journal of Computational Physics 
Volume  308 
Early online date  15 Dec 2015 
DOIs  
Publication status  Published  1 Mar 2016 
Keywords
 Adaptive
 Atmosphere
 Mesh
 Mesh generation
 Modelling
 MongeAmpére
 Optimal transport
 Refinement
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Dive into the research topics of 'Mesh adaptation on the sphere using optimal transport and the numerical solution of a MongeAmpère type equation'. Together they form a unique fingerprint.Projects
 1 Finished

Moving Meshes for Global Atmospheric Modelling
Natural Environment Research Council
1/09/15 → 31/08/18
Project: Research council
Profiles

Chris Budd
 Department of Mathematical Sciences  Professor
 EPSRC Centre for Doctoral Training in Statistical Applied Mathematics (SAMBa)
 Probability Laboratory at Bath
 Centre for Doctoral Training in Decarbonisation of the Built Environment (dCarb)
 Centre for Mathematical Biology
 Institute for Mathematical Innovation (IMI)
 Centre for Nonlinear Mechanics
 EPSRC Centre for Doctoral Training in Advanced Automotive Propulsion Systems (AAPS CDT)
Person: Research & Teaching, Affiliate staff