16 research outputs found
A Meshfree Generalized Finite Difference Method for Surface PDEs
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 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
On Meshfree GFDM Solvers for the Incompressible Navier-Stokes Equations
Meshfree solution schemes for the incompressible Navier--Stokes equations are
usually based on algorithms commonly used in finite volume methods, such as
projection methods, SIMPLE and PISO algorithms. However, drawbacks of these
algorithms that are specific to meshfree methods have often been overlooked. In
this paper, we study the drawbacks of conventionally used meshfree Generalized
Finite Difference Method~(GFDM) schemes for Lagrangian incompressible
Navier-Stokes equations, both operator splitting schemes and monolithic
schemes. The major drawback of most of these schemes is inaccurate local
approximations to the mass conservation condition. Further, we propose a new
modification of a commonly used monolithic scheme that overcomes these problems
and shows a better approximation for the velocity divergence condition. We then
perform a numerical comparison which shows the new monolithic scheme to be more
accurate than existing schemes
A Meshfree Lagrangian Method for Flow on Manifolds
In this paper, 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
Higher-Order GFDM for Linear Elliptic Operators
We present a novel approach of discretizing diffusion operators of the form
in the context of meshfree generalized finite
difference methods. Our ansatz uses properties of derived operators and
combines the discrete Laplace operator with reconstruction functions
approximating the diffusion coefficient . Provided that the
reconstructions are of a sufficiently high order, we prove that the order of
accuracy of the discrete Laplace operator transfers to the derived diffusion
operator. We show that the new discrete diffusion operator inherits the
diagonal dominance property of the discrete Laplace operator and fulfills
enrichment properties. Our numerical results for elliptic and parabolic partial
differential equations show that even low-order reconstructions preserve the
order of the underlying discrete Laplace operator for sufficiently smooth
diffusion coefficients. In experiments, we demonstrate the applicability of the
new discrete diffusion operator to interface problems with point clouds not
aligning to the interface and numerically prove first-order convergence
A Meshfree Generalized Finite Difference Method for Solution Mining Processes
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