Meshfree methods on manifolds for hydrodynamic flows on curved surfaces: A Generalized Moving Least-Squares (GMLS) approach

Gross, Ben; Trask, Nathaniel; Kuberry, Paul; Atzberger, Paul J.

doi:10.1016/j.jcp.2020.109340

Cited by 43 publications

(29 citation statements)

References 98 publications

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“…While these consequences of the Hodge decomposition should be known in the mathematics community, also other examples can be found, where the vorticity-stream function approach is carelessly used, see e.g. [4][5][6].…”

Section: Resultsmentioning

confidence: 99%

Vorticity‐stream function approaches are inappropriate to solve the surface Navier‐Stokes equation on a torus

Nitschke

Reuther

Voigt

2021

Proc Appl Math and Mech

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The growing interest in interfacial flows in engineering and biological applications demands to numerically solve the incompressible surface (Navier)-Stokes equation. A common approach, based on a vorticity-stream function formulation reformulates the equation into a system of scalar-valued surface partial differential equations, for which various numerical methods have been developed. However, on surfaces, which are not simply connected, this approach is not appropriate. We here explain the underlying situation and provide details and examples.

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Section: Resultsmentioning

confidence: 99%

Vorticity‐stream function approaches are inappropriate to solve the surface Navier‐Stokes equation on a torus

Nitschke

Reuther

Voigt

2021

Proc Appl Math and Mech

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“…We note that similar notions of using GFDMs for discretizing surface derivatives have been done in, for example, References 16 and 49, based on approximations of surface metric tensors. In contrast, the approach used here, based on Reference 32, does not rely on surface‐based metrics.…”

Section: Surface Derivativesmentioning

confidence: 94%

“…Visualization driven work has mostly considered only inviscid flow 13,14 . A lot of work has been done for steady surface flow, 15,16 and surface flow on surfaces of specific shapes, such as spheres 17 or other radial manifolds 15 . Atmospheric dynamics work to model flow on the surface of the Earth solve two‐dimensional simulations in the latitude‐longitude frame 18,19 .…”

Section: Introductionmentioning

confidence: 99%

A meshfree Lagrangian method for flow on manifolds

Suchde

2021

Numerical Methods in Fluids

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In this article, we present a novel meshfree framework for fluid flow simulations on arbitrarily curved surfaces. First, we introduce a new meshfree Lagrangian framework to model flow on surfaces. Meshfree points or particles, which are used to discretize the domain, move in a Lagrangian sense along the given surface. This is done without discretizing the bulk around the surface, without parametrizing the surface, and without a background mesh. A key novelty that is introduced is the handling of flow with evolving free boundaries on a curved surface. The use of this framework to model flow on moving and deforming surfaces is also introduced. Then, we present the application of this framework to solve fluid flow problems defined on surfaces numerically. In combination with a meshfree generalized finite difference method (GFDM), we introduce a strong form meshfree collocation scheme to solve the Navier–Stokes equations posed on manifolds. Benchmark examples are proposed to validate the Lagrangian framework and the surface Navier–Stokes equations with the presence of free boundaries.

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“…There are many interpolation function theories. Commonly used are the inverse distance weighted (IDW) interpolation method [12,13], the Kriging interpolation method [14][15][16], the moving least squares (MLS) interpolation method [17][18][19], and the radial basis functions interpolation method [20][21][22]. To compare the performance of interpolation functions, Franke [23] compared dozens of different interpolation algorithms in terms of parameters and modeling accuracy.…”

Section: Related Workmentioning

confidence: 99%

Repair of Geological Models Based on Multiple Material Marching Cubes

Zhong

Wang

2021

Mathematics

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In this paper, we present a multi-domain implicit surface reconstruction algorithm for geological modeling based on the labeling of voxel points. The improved algorithm sets a label for each voxel point to represent the type of its geological domain and then obtains all the voxel points in the void areas. After that, the improved algorithm modifies the labels of the voxel points in the void areas and finally reconstructs the geological models through the Multiple Material Marching Cubes (M3C) algorithm. The improved algorithm solves the problems of some unexpected overlaps and voids in geological modeling by setting and modifying the labels of the voxel points. Our key contribution is proposing a labeling processing method to repair the overlap and void defects generated in the geological modeling and realizing the improved M3C algorithm. The experimental results of some geological models show the performance of the improved method. Compared with the original method, the improved method can repair the overlap and void defects in geological modeling to ensure the raw structural adjacency relationships of the geological bodies.

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Meshfree methods on manifolds for hydrodynamic flows on curved surfaces: A Generalized Moving Least-Squares (GMLS) approach

Cited by 43 publications

References 98 publications

Vorticity‐stream function approaches are inappropriate to solve the surface Navier‐Stokes equation on a torus

Vorticity‐stream function approaches are inappropriate to solve the surface Navier‐Stokes equation on a torus

A meshfree Lagrangian method for flow on manifolds

Repair of Geological Models Based on Multiple Material Marching Cubes

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