Mesh adaptation on the sphere using optimal transport and the numerical solution of a Monge-Ampère type equation

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

Optimal-transport-based mesh adaptivity on the plane and sphere using finite elements

In moving mesh methods, the underlying mesh is dynamically adapted without changing the connectivity of the mesh. We specifically consider the generation of meshes which are adapted to a scalar monitor function through equidistribution. Together with an optimal transport condition, this leads to a Monge-Amp\`ere equation for a scalar mesh potential. We adapt an existing finite element scheme for the standard Monge-Amp\`ere equation to this mesh generation problem; this is a mixed finite element scheme, in which an extra discrete variable is introduced to represent the Hessian matrix of second derivatives. The problem we consider has additional nonlinearities over the basic Monge-Amp\`ere equation due to the implicit dependence of the monitor function on the resulting mesh. We also derive the equivalent Monge-Amp\`ere-like equation for generating meshes on the sphere. The finite element scheme is extended to the sphere, and we provide numerical examples. All numerical experiments are performed using the open-source finite element framework Firedrake.Comment: Updated following reviews, 36 page

The scaling and skewness of optimally transported meshes on the sphere

In the context of numerical solution of PDEs, dynamic mesh redistribution methods (r-adaptive methods) are an important procedure for increasing the resolution in regions of interest, without modifying the connectivity of the mesh. Key to the success of these methods is that the mesh should be sufficiently refined (locally) and flexible in order to resolve evolving solution features, but at the same time not introduce errors through skewness and lack of regularity. Some state-of-the-art methods are bottom-up in that they attempt to prescribe both the local cell size and the alignment to features of the solution. However, the resulting problem is overdetermined, necessitating a compromise between these conflicting requirements. An alternative approach, described in this paper, is to prescribe only the local cell size and augment this an optimal transport condition to provide global regularity. This leads to a robust and flexible algorithm for generating meshes fitted to an evolving solution, with minimal need for tuning parameters. Of particular interest for geophysical modelling are meshes constructed on the surface of the sphere. The purpose of this paper is to demonstrate that meshes generated on the sphere using this optimal transport approach have good a-priori regularity and that the meshes produced are naturally aligned to various simple features. It is further shown that the sphere's intrinsic curvature leads to more regular meshes than the plane. In addition to these general results, we provide a wide range of examples relevant to practical applications, to showcase the behaviour of optimally transported meshes on the sphere. These range from axisymmetric cases that can be solved analytically to more general examples that are tackled numerically. Evaluation of the singular values and singular vectors of the mesh transformation provides a quantitative measure of the mesh aniso...Comment: Updated following reviewer comment

Fast three dimensional r-adaptive mesh redistribution

This paper describes a fast and reliable method for redistributing a computational mesh in three dimensions which can generate a complex three dimensional mesh without any problems due to mesh tangling. The method relies on a three dimensional implementation of the parabolic Monge–Ampère (PMA) technique, for finding an optimally transported mesh. The method for implementing PMA is described in detail and applied to both static and dynamic mesh redistribution problems, studying both the convergence and the computational cost of the algorithm. The algorithm is applied to a series of problems of increasing complexity. In particular very regular meshes are generated to resolve real meteorological features (derived from a weather forecasting model covering the UK area) in grids with over 2×107 degrees of freedom. The PMA method computes these grids in times commensurate with those required for operational weather forecasting

Conservation with moving meshes over orography

Adaptive meshes have the potential to improve the accuracy and efficiency of atmospheric modelling by increasing resolution where it is most needed. Mesh re-distribution, or r-adaptivity, adapts by moving the mesh without changing the connectivity. This avoids some of the challenges with h-adaptivity (adding and removing points): the solution does not need to be mapped between meshes, which can be expensive and introduces errors, and there are no load balancing problems on parallel computers. A long standing problem with both forms of adaptivity has been changes in volume of the domain as resolution changes at an uneven boundary. We propose a solution which achieves exact local conservation and maintains a uniform scalar field while the mesh changes volume as it moves over orography. This is achieved by introducing a volume correction parameter which tracks the cell volumes without using expensive conservative mapping. A finite volume solution of the advection equation over orography on moving meshes is described and results are presented demonstrating improved accuracy for cost using moving meshes. Exact local conservation and maintenance of uniform scalar fields is demonstrated and the correct mesh volume is preserved. We use optimal transport to generate meshes which are guaranteed not to tangle and are equidistributed with respect to a monitor function. This leads to a Monge-Ampère equation which is solved with a Newton solver. The superiority of the Newton solver over other techniques is demonstrated in the appendix. However the Newton solver is only efficient if it is applied to the left hand side of the Monge-Ampère equation with fixed point iterations for the right hand side

A level set based method for fixing overhangs in 3D printing

3D printers based on the Fused Decomposition Modeling create objects layer-by-layer dropping fused material. As a consequence, strong overhangs cannot be printed because the new-come material does not find a suitable support over the last deposed layer. In these cases, one can add some support structures (scaffolds) which make the object printable, to be removed at the end. In this paper we propose a level set method to create object-dependent support structures, specifically conceived to reduce both the amount of additional material and the printing time. We also review some open problems about 3D printing which can be of interests for the mathematical community

Emergence of a tracer source from air concentration measurements, a new strategy for linear assimilation

The measurement of atmospheric concentrations by a monitoring network is a promising tool for the identification of the widespread sources of trace species. The paper addresses the case of the species scattered linearly by a known meteorology. The question is classical: what can be said about the source from a set of measurements? Is it possible to guess from the values observed by the measurements that the source is spread close to the detectors, or that the tracer comes from a remote region? And, if the source was a point source, would it be possible to understand it by just considering these values? A part of the answers is a matter of practical sense: the resolution with which an emission can be retrieved will always be limited and probably lower for a remote region, even if the detectors and dispersion model are error free. The paper proposes a linear strategy of inference: to any set of values taken by the observed concentrations is associated linearly an estimate of the source. Doubled values lead to a doubled estimate. The method, based on adjoint techniques, is intended to optimise the resolution by quantifying, with the concept of illumination, which regions are well, poorly or not seen at all. The illumination tied to ordinary adjoint functions becomes excessive close to the detectors thus leading to inversion artefacts. This may be corrected by attributing each point of the space time domain a geometric and statistical weight. The adjoint functions are transformed. The choice of this renormalising function is constrained by an unambiguous entropic criterion preventing any overestimation of the available information that would lead to artefacts. It amounts to evenly distribute the information between the points organised with their weights as a 'known domain'. The theory is illustrated by calculations performed with the experimental source ETEX1

MICROPARTICLE SAMPLING AND SEAPARATIONENABLED BY DROPLET MICROFLUIDICS

This work reports design, device fabrication, modeling and experimental results on newsampling and separation principles in which liquid is transported in a droplet form on a plannerhydrophobic surface with no moving parts. The presented particle sampler and separatorconstitute core units for the handheld lab-on-a-chip-based airborne particle monitoring system.For the airborne particle sampling, a novel method is developed by which the particles onthe solid surface are swept and sampled by electrowetting-actuated moving droplets. Theoreticalanalysis and experimental works along with microfabricated testing devices are carried out toinvestigate the underlying physics and to optimize the sampling conditions. The samplingconcepts are examined and proved on a solid surface and perforated filter membrane showinghigh sampling efficiencies.For the particle separation, a new separation scheme is developed in which the mixedparticles are separated within a mother droplet by traveling-wave dielectrophoresis (tw-DEP).Using the subsequent operation of droplet splitting by way of electrowetting, the separatedparticles can be isolated into each split droplet according to the DEP properties of the particles.This in-droplet separation is examined with many combinations of particles in microfabricateddevices. By investigating the particle behavior as function of the frequency of the traveling waveDEP signal, the separation efficiencies are optimized.The above microfluidic units constitute key components for upstream particle sampling anddownstream sample processing in the lab on a chip system, providing the following advantages:extremely small amount use of samples/reagents (2) no external pressure source required forfluidic operations, (3) simple design and fabrication since no mechanical moving structure