On meshfree GFDM solvers for the incompressible Navier–Stokes equations

Abstract

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.

Point cloud movement for fully Lagrangian meshfree methods

“…The boundary conditions used, including the ones at the free surface, are as detailed in [11,16]. While an inaccurate update of point locations represent one source of error in meshfree solution schemes to the incompressible Navier-Stokes equations, other sources of errors in the same have been discussed in our earlier work [27,28]. Inaccurate movement of the points on the free surface can result in a volume loss or gain, similar to that illustrated in the rotating disc example considered earlier.…”

A meshfree generalized finite difference method for surface PDEs

“…A significant amount of work has been done to modify GFDM differential operator for different ends, and they can be easily carried over to surface PDEs using the present work. These include developments such as higher order spatial discretizations [32], conservation [26,51], accuracy considerations [52], upwinding methods for advection [42,48], staggered methods [54], among others. A few examples of carrying over volumetric GFDMs developments to surface PDEs are shown in the coming sections.…”

Algebraic multigrid for the finite pointset method

“…We now re-organize (18) such that all terms referring to time step n + 1 move to the left hand side, while the terms depending on time step n are on the right hand side. At this point, in the segregated approach we need to replace p n+1 by an intermediate pressurep.…”

“…when the magnitude of the term Ψ η (v n+1 −v n+1 ) would require time steps that are too small to allow for an effective simulation, see e.g. [18] and the references therein for a detailed discussion. An option to overcome this problem is to solve for velocity and correction pressure in one single linear solve (coupled approach).…”

“…On the other hand, this experiment shows that in situations with small Reynolds numbers the segregated approach cannot be used and the coupled approach is the only option giving real- istic results. For a more detailed discussion on the advantages and disadvantages of the various approaches we refer to [18].…”