The geometry of r-adaptive meshes generated using optimal transport methods

Abstract

The principles of mesh equidistribution and alignment play a fundamental role in the design of adaptive methods [32], and a metric tensor M and mesh metric are useful theoretical tools for understanding a method's level of mesh alignment, or anisotropy. We consider a mesh redistribution method based on the Monge-Ampère equation [17],[9], [10], [8], [7], which combines equidistribution of a given scalar density function ρ with optimal transport. It does not involve explicit use of a metric tensor M, although such a tensor must exist for the method, and an interesting question to ask is whether or not the alignment produced by the metric gives an anisotropic mesh. For model problems with a linear feature and with a radially symmetric feature, we derive the exact form of the metric M, which involves expressions for its eigenvalues and eigenvectors. The eigenvectors are shown to be orthogonal and tangential to the feature, and the ratio of the eigenvalues (corresponding to the level of anisotropy) is shown to depend, both locally and globally, on the value of ρ = √ det M and the amount of curvature. We thereby demonstrate how the optimal transport method produces an anisotropic mesh along a given feature while equidistributing a suitably chosen scalar density function. Numerical results are given to verify these results and to demonstrate how the analysis is useful for problems involving more complex features, including for a non-trivial time dependant nonlinear PDE which evolves narrow and curved reaction fronts.

A high-fidelity multiresolution digital elevation model for Earth systems

“…But a mesh constructed from interpolated vertices does not necessarily comply with the terrain relief, and elevation errors are frequently reported as an input uncertainty (Bilskie and Hagen, 2013;Hunter et al, 2007;Nunalee et al, 2015;Wilson and Gallant, 2000). Although there are many situations where dynamic conditions are stressed as stronger impacts on predictions (Cea and Bladé, 2015;Budd et al, 2015), the underlying topography is still very important due to its increasingly improved fidelity to the Earth's surface (Bates, 2012;Tarolli, 2014), and a sophisticated topography transformation would be beneficial to reduce discrepancies arising from physical inconsistencies (Chen et al, 2015;Glover, 1999;Ringler et al, 2011).…”

Simulation of all-scale atmospheric dynamics on unstructured meshes

“…Because unstructured-meshes are essentially free of topological constraints, they offer flexibility unmatched by the established techniques operating on regular Cartesian grids. Admittedly, such grids enable computationally efficient static and dynamic mesh adaptivity via continuous mappings [109,171,66,13,14], yet their rigid connectivity imposes stringent constraints on the adapted grids. The unavailability of regular equidistant discretisation on a spherical surface is an apparent evidence of these constraints and a venerable force for advancement of flexible meshing in atmospheric models.…”

“…The Helmholtz problem (17) was discussed in [138,68]. In the NFTFV codes, we solve both (14) and (17) with a bespoke nonsymmetric preconditioned Generalised Conjugate Residual (GCR) approach, widely discussed in the literature; cf. [135] for a recent overview and a comprehensive list of references.…”

An unstructured-mesh atmospheric model for nonhydrostatic dynamics: Towards optimal mesh resolution

“…4 The consumed CPU time was 68 s. The corresponding solution on the adapted mesh containing 8830 nodes used 6000 time steps with δt = 3 s and four elliptic-solver iterations per time step, and took 464 s of CPU time.…”

“…Even though the latter enable computationally efficient static and dynamic mesh adaptivity via continuous mappings [31,56,21,3,4], their rigid connectivity imposes stringent constraints on adapted grids. Flexible unstructured meshes relax the constraints and offer alternative means for optimising variable resolution required for improved representation of complex physical processes in atmospheric flows.…”