Robust, multidimensional mesh-motion based on Monge–Kantorovich equidistribution

Abstract

a b s t r a c tMesh-motion (r-refinement) grid adaptivity schemes are attractive due to their potential to minimize the numerical error for a prescribed number of degrees of freedom. However, a key roadblock to a widespread deployment of this class of techniques has been the formulation of robust, reliable mesh-motion governing principles, which (1) guarantee a solution in multiple dimensions (2D and 3D), (2) avoid grid tangling (or folding of the mesh, whereby edges of a grid cell cross somewhere in the domain), and (3) can be solved effectively and efficiently. In this study, we formulate such a mesh-motion governing principle, based on volume equidistribution via Monge-Kantorovich optimization (MK). In earlier publications [1,2], the advantages of this approach with regard to these points have been demonstrated for the time-independent case. In this study, we demonstrate that MongeKantorovich equidistribution can in fact be used effectively in a time-stepping context, and delivers an elegant solution to the otherwise pervasive problem of grid tangling in mesh-motion approaches, without resorting to ad hoc time-dependent terms (as in moving-mesh PDEs, or MMPDEs [3,4]). We explore two distinct r-refinement implementations of MK: the direct method, where the current mesh relates to an initial, unchanging mesh, and the sequential method, where the current mesh is related to the previous one in time. We demonstrate that the direct approach is superior with regard to mesh distortion and robustness. The properties of the approach are illustrated with a hyperbolic PDE, the advection of a passive scalar, in 2D and 3D. Velocity flow fields with and without flow shear are considered. Three-dimensional grid, time-step, and nonlinear tolerance convergence studies are presented which demonstrate the optimality of the approach.Published by Elsevier Inc.

The scaling and skewness of optimally transported meshes on the sphere

“…An alternative approach for constructing an r-adaptive mesh is to explicitly prescribe the local scale of the mesh, via a (solution-dependent) mesh density monitor function, while imposing global regularity in the form of an optimal transport constraint. This has been implemented and analysed on the plane Williams, 2006, 2009;Chacón et al, 2011;Browne et al, 2014;Budd et al, 2015), and in recent papers, we have extended this to produce solutionadapted meshes on the sphere (Weller et al, 2016;McRae et al, 2018). These methods used an optimal transport approach linked to the solution of a (version of the) Monge-Ampère equation, posed on the tangent bundle to the sphere, to produce regular-looking meshes with the desired (spatially-varying) density of mesh points.…”

“…An alternative, and powerful, technique is to use the concept of optimal transport (Villani, 2003(Villani, , 2009; previous work using this approach includes Williams (2006, 2009);Chacón et al (2011);Browne et al (2014). We now seek a map F satisfying eq.…”

Three-dimensional CFD simulations with large displacement of the geometries using a connectivity-change moving mesh approach

“…Some studies try to impose a mesh motion that is directly adapted to the physical phenomena in question, using for instance either so-called Moving Mesh PDEs [32] or a Monge-Ampère equation [15]. However, interesting these approaches may be, they still seem to be time-consuming, especially in 3D, due to the solution of a non-linear equation, and it is unsure whether they can handle complex 3D geometries.…”

Numerical simulation of four-field extended magnetohydrodynamics in dynamically adaptive curvilinear coordinates via Newton–Krylov–Schwarz

“…According to this idea, an r-type variational grid generation technique based on a Monge-Kantorovich (MK) optimization process [23][24][25][26] is studied.…”

“…A single MongeAmpère equation [25][26][27] is obtained for the displacement potential x. It is a grid equation for the map xðnÞ, leading to a new grid equidistributed by the density function chosen by the user.…”