Point cloud movement for fully Lagrangian meshfree methods

Suchde, Pratik; Kuhnert, Jörg

doi:10.1016/j.cam.2018.02.020

Cited by 24 publications

(17 citation statements)

References 29 publications

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“…( 2), most authors tend to very small time steps. The same arguments of inaccuracy presented in [57] for volumetric flows carry over to the surface case here. Thus, we use the more accurate second order method for point cloud movement…”

Section: The Actual Movementsupporting

confidence: 63%

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A fully Lagrangian meshfree framework for PDEs on evolving surfaces

Suchde

Kuhnert

2019

Journal of Computational Physics

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We propose a novel framework to solve PDEs on moving manifolds, where the evolving surface is represented by a moving point cloud. This has the advantage of avoiding the need to discretize the bulk volume around the surface, while also avoiding the need to have a global mesh. Distortions in the point cloud as a result of the movement are fixed by local adaptation. We first establish a comprehensive Lagrangian framework for arbitrary movement of curves and surfaces given by point clouds. Collision detection algorithms between point cloud surfaces are introduced, which also allow the handling of evolving manifolds with topological changes. We then couple this Lagrangian framework with a meshfree Generalized Finite Difference Method (GFDM) to approximate surface differential operators, which together give a method to solve PDEs on evolving manifolds. The applicability of this method is illustrated with a range of numerical examples, which include advection-diffusion equations with large deformations of the surface, curvature dependent geometric motion, and wave equations on evolving surfaces.

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Section: The Actual Movementsupporting

confidence: 63%

“…In each case, the velocity is taken to be constant within each time step. In our earlier work for meshfree volumetric Lagrangian flows [57], we have shown the inaccuracies surrounding the first order movement similar to Eq. (2), which lead to large defects in volume conservation.…”

Section: The Actual Movementmentioning

confidence: 79%

A fully Lagrangian meshfree framework for PDEs on evolving surfaces

Suchde

Kuhnert

2019

Journal of Computational Physics

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“…Each step involves a second order in time movement. We note that most Lagrangian methods only use a first order movement, which has been shown to be extremely inaccurate in capturing rotational components of motion 33 . For time integration between time levels t n and t n + 1 with the velocity

\vec{v}

, we get

Δ {\vec{x}}_{1} = {\vec{v}}^{(n)} Δ t + \frac{1}{2} \frac{{\vec{v}}^{(n)} - {\vec{v}}^{(n - 1)}}{Δ t_{0}} {(Δ t)}^{2},

where bracketed superscripts indicate the time level,

Δ t = t^{n + 1} - t^{n}

is the current time step, and

Δ t_{0} = t^{n} - t^{n - 1}

is the previous time step.…”

Section: The Lagrangian Frameworkmentioning

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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“…The discretization procedure begins by moving the point cloud according to a second order method [27] by…”

Section: Step 1 Point Cloud Movementmentioning

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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Point cloud movement for fully Lagrangian meshfree methods

Cited by 24 publications

References 29 publications

A fully Lagrangian meshfree framework for PDEs on evolving surfaces

A fully Lagrangian meshfree framework for PDEs on evolving surfaces

A meshfree Lagrangian method for flow on manifolds

A meshfree generalized finite difference method for solution mining processes

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