Finite Pointset Method for the Simulation of a Vehicle Travelling Through a Body of Water

Numerical Meth Engineering

Kuhnert

Schröder

et al. 2017

Self Cite

SUMMARYLack of conservation has been the biggest drawback in meshfree generalized finite difference methods (GFDMs). In this paper, we present a novel modification of classical meshfree GFDMs to include local balances which produce an approximate conservation of numerical fluxes. This numerical flux conservation is done within the usual moving least squares framework. Unlike Finite Volume Methods, it is based on locally defined control cells, rather than a globally defined mesh. We present the application of this method to an advection diffusion equation and the incompressible Navier-Stokes equations. Our simulations show that the introduction of flux conservation significantly reduces the errors in conservation in meshfree GFDMs.

“…This is because the spatial derivatives used in such cases would be completely described by Eq. (13). Thus, the inclusion of Eq.…”

Section: Flux Conserving Differential Operatorsmentioning

confidence: 98%

“…(14) constitute an enhancement of the GFDM differential operators by the conservation condition Eq. (13). If the monomials m ∈ M are taken up to second order, Eq.…”

Section: Flux Conserving Differential Operatorsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A flux conserving meshfree method for conservation laws

Numerical Meth Engineering

Kuhnert

Schröder

et al. 2017

Self Cite

“…Meshfree GFDMs [14,18,26,38] are strong form methods. For volume-domains, they have been shown to be robust methods, with a wide variety of applications [9,24,42,45]. They have also been referred to under the name of Generalized Moving Least Squares (GMLS) [43].…”

Section: Differential Operatorsmentioning

confidence: 99%

A fully Lagrangian meshfree framework for PDEs on evolving surfaces

Journal of Computational Physics

Kuhnert

2019

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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.

A meshfree Lagrangian method for flow on manifolds

Numerical Methods in Fluids

2021

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.