A meshfree generalized finite difference method for surface PDEs

Suchde, Pratik; Kuhnert, Jörg

doi:10.1016/j.camwa.2019.04.030

Cited by 68 publications

(63 citation statements)

References 55 publications

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“…The parameters used here are adopted from conventions in volumetric meshfree flow simulations 28,30,31 . We set r min = 0.2 and r max = 0.45, as has been done for meshfree surfaces in References 26 and 32. This results in about 15 − 20 points in each neighborhood.…”

Section: Preliminariesmentioning

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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Section: Preliminariesmentioning

confidence: 99%

A meshfree Lagrangian method for flow on manifolds

Suchde

2021

Numerical Methods in Fluids

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“…Furthermore, the resulting preliminary velocity features a divergence which is very close to the targeted one. We note that in equations (28) and (29), the stress tensor S n+1 was determined according to equation (2).…”

Section: Step 1 Point Cloud Movementmentioning

confidence: 99%

“…For more details on the Eulerian procedure, we refer to [25], and for similar GFDM Eulerian formulations, we refer to [29]. For numerical validations of the velocity-pressure scheme used here, their implementations within a GFDM framework, and a comparison of GFDM results with other numerical methods on benchmark problems, we refer to our earlier work [4,10,13,21,24,25,30].…”

Section: Further Detailsmentioning

confidence: 99%

A meshfree generalized finite difference method for solution mining processes

et al. 2020

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Experimental and field investigations for solution mining processes have improved intensely in recent years. Due to today's computing capacities, three-dimensional simulations of potential salt solution caverns can further enhance the understanding of these processes. They serve as a "virtual prototype" of a projected site and support planning in reasonable time. In this contribution, we present a meshfree Generalized Finite Difference Method (GFDM) based on a cloud of numerical points that is able to simulate solution mining processes on microscopic as well as macroscopic scales, which differ significantly in both the spatial and temporal scale. Focusing on anticipated industrial requirements, Lagrangian and Eulerian formulations including an Arbitrary Lagrangian-Eulerian (ALE) approach are considered.

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“…MLS/RKPM/GFD approaches provide a compelling alternative by addressing accuracy issues through the explicit construction of approximations with polynomial reproduction properties and an accompanying rigorous approximation theory [82,99], but lack a stability theory. There have been several examples of successful discretization of scalar surface PDEs [89,92]. In Generalized Moving Least Squares (GMLS) this approach is extended to enable the recovery of arbitrary linear bounded target functionals from scattered data [66,99].…”

Section: Introductionmentioning

confidence: 99%