Application of a generalized finite difference method to mould filling process

Computers & Mathematics with Applications

Kuhnert

2019

In this paper, we propose a novel meshfree Generalized Finite Difference Method (GFDM) approach to discretize PDEs defined on manifolds. Derivative approximations for the same are done directly on the tangent space, in a manner that mimics the procedure followed in volume-based meshfree GFDMs. As a result, the proposed method not only does not require a mesh, it also does not require an explicit reconstruction of the manifold. In contrast to some existing methods, it avoids the complexities of dealing with a manifold metric, while also avoiding the need to solve a PDE in the embedding space. A major advantage of this method is that all developments in usual volume-based numerical methods can be directly ported over to surfaces using this framework. We propose discretizations of the surface gradient operator, the surface Laplacian and surface Diffusion operators. Possibilities to deal with anisotropic and discontinous surface properties (with large jumps) are also introduced, and a few practical applications are presented.

Section: Numerical Results and Validationmentioning

confidence: 99%

Section: Basics and Nomenclaturementioning

confidence: 99%

A meshfree generalized finite difference method for surface PDEs

Computers & Mathematics with Applications

Kuhnert

2019

“…8). Alternatively, the normal field can be computed from the boundary discretization using surface normal computation methods [103,130,165,198].…”

Section: Meshfree Advancing Front Methodsmentioning

confidence: 99%

Point Cloud Generation for Meshfree Methods: An Overview

Arch Computat Methods Eng

Jacquemin

Davydov

2022

Meshfree methods are becoming an increasingly popular alternative to mesh-based methods of numerical simulation. The biggest stated advantage of meshfree methods is the avoidance of generating a mesh on the computational domain. However, even today a surprisingly large amount of meshfree literature ironically uses the nodes of a mesh as the point set that discretizes the domain. On the other hand, already existing efficient meshfree methods to generate point clouds are apparently not very well known among meshfree communities, which has led to recent work redeveloping existing algorithms. In this paper, we present a brief overview of point cloud generation methods for domains and surfaces and discuss their features and challenges, in particular in the context of applicability to industry-relevant complex geometries.

“…To use a more robust method to compute normals on point clouds, we use a minimization approach that maximizes the angles between the normal and the neighbouring points. This is based on the procedure used for computing normals at the free surface for volumetric flow by [52]. For an interior point i, the manifold normal n i is set to be the unit normal vector minimizing the functional…”

Section: Normal Computationmentioning

confidence: 99%

A fully Lagrangian meshfree framework for PDEs on evolving surfaces

Journal of Computational Physics

Kuhnert

2019

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.