Meshfree methods

About: Meshfree methods is a research topic. Over the lifetime, 2216 publications have been published within this topic receiving 69596 citations.

The reproducing kernel particle Petrov–Galerkin method for solving two-dimensional nonstationary incompressible Boussinesq equations

Abstract: The meshless techniques are improved to simulate a wide range of the physical models. Two common forms of the meshless methods are strong and (local) weak forms. In the current paper, we employ the local weak form technique to simulate the two-dimensional non-stationary Boussinesq equations. In the present method the trial functions have been selected from the shape functions of the RKPM. Also, the test function is based upon a Ck function. At first, the time variable has been discretized via a finite difference scheme and then the space direction has been approximated by using the MLPG procedure. In this procedure, by employing the continuity equation, the two-dimensional non-stationary Boussinesq equations have been transformed to a pressure Poisson equation. After solving the obtained pressure Poisson equation, the velocity of fluid in the x- and y-directions and also the temperature can be updated, directly. Numerical examples confirm the ability of the developed technique.

Numerical Methods for the Modelling of Chip Formation

Abstract: The modeling of metal cutting has proved to be particularly complex due to the diversity of physical phenomena involved, including thermo-mechanical coupling, contact/friction and material failure. During the last few decades, there has been significant progress in the development of numerical methods for modeling machining operations. Furthermore, the most relevant techniques have been implemented in the relevant commercial codes creating tools for the engineers working in the design of processes and cutting devices. This paper presents a review on the numerical modeling methods and techniques used for the simulation of machining processes. The main purpose is to identify the strengths and weaknesses of each method and strategy developed up-to-now. Moreover the review covers the classical Finite Element Method covering mesh-less methods, particle-based methods and different possibilities of Eulerian and Lagrangian approaches.

A review on approaches to solving Poisson’s equation in projection-based meshless methods for modelling strongly nonlinear water waves

Abstract: Three meshless methods, including incompressible smooth particle hydrodynamic (ISPH), moving particle semi-implicit (MPS) and meshless local Petrov–Galerkin method based on Rankine source solution (MLPG_R) methods, are often employed to model nonlinear or violent water waves and their interaction with marine structures. They are all based on the projection procedure, in which solving Poisson’s equation about pressure at each time step is a major task. There are three different approaches to solving Poisson’s equation, i.e. (1) discretizing Laplacian directly by approximating the second-order derivatives, (2) transferring Poisson’s equation into a weak form containing only gradient of pressure and (3) transferring Poisson’s equation into a weak form that does not contain any derivatives of functions to be solved. The first approach is often adopted in ISPH and MPS, while the third one is implemented by the MLPG_R method. This paper attempts to review the most popular, though not all, approaches available in literature for solving the equation.

A Meshfree Method For Incompressible Fluid Flows with Incorporated Surface Tension

Abstract: A meshfree particle method is used to simulate free surface flows. This is a Lagrangian method. Flows are modeled by the incompressible Navier-Stokes equations. The particle projection method is used to solve the Navier-Stokes equations. The spatial derivatives are approximated by the weighted least squares method (WLS). The pressure Poisson equation is solved by a local iterative procedure with the help of WLS. Numerical experiments are presented for two dimensional cases. In the case of breaking dam problem the numerical result is compared with the experimental result. The surface tension effects are studied in different shapes of drops and Laplace's law is verified. Finally, the collisions of two drops are simulated.