Abstract

The paper describes some recent developments in finite element and particle methods for analysis of a wide range of bulk forming processes. The developments include new stabilized linear triangles and tetrahedra using finite calculus and a new procedure combining particle methods and finite element methods. Applications of the new numerical methods to casting, forging and other bulk metal forming problems and mixing processes are shown.

Abstract

In this work, the finite point method is applied to the solution of high‐Reynolds compressible viscous flows. The aim is to explore this important field of applications focusing on two main aspects: the easiness and automation of the meshless discretization of viscous layers and the construction of a robust numerical approximation in the highly stretched clouds of points resulting in such domain areas. The flow solution scheme adopts an upwind‐biased scheme to solve the averaged Navier–Stokes equations in conjunction with an algebraic turbulence model. The numerical applications presented involve different attached boundary layer flows and are intended to show the performance of the numerical technique. The results obtained are satisfactory and indicative of the possibilities to extend the present meshless technique to more complex flow problems.

Abstract

A method is presented for the solution of an incompressible viscous fluid flow with heat transfer using a fully Lagrangian description of the motion. Due to the severe element distortion, a frequent remeshing is performed in an efficient manner. An implicit time integration through a classical fractional step is presented. The non-linearities of the formulation are taken into account and solved with the fixed-point iteration method. The displacement and temperature solutions are coupled through the Boussinesq approximation. The Lagrangian formulation provides an elegant way of solving free-surface problems with thermal convection as the particles are followed during their motion. To illustrate the method, the Rayleigh–Bénard instability with and without free surface in two dimensions has been computed.

Abstract

This paper presents the Particle Finite Element Method (PFEM) and its application to multi-fluid flows. Key features of the method are the use of a Lagrangian description to model the motion of the fluid particles (nodes) and that all the information is associated to the particles. A mesh connects the nodes defining the discretized domain where the governing equations, expressed in an integral form, are solved as in the standard FEM.We have extended the method to problems involving several different fluids with the aim of exploiting the fact that Lagrangian methods are specially well suited for tracking any kind of interfaces.

Abstract

Abstract

Incompressible fluid analysis using the ISPH or MPS methods requires the solution of the pressure Poisson equation, which takes up most of the overall computation time. In addition, the iteration number for solving pressure Poisson equations may increase as the simulation model scale increases. This is a common problem in particle methods and the other implicit time integration solvers. In different methods, FEM, etc., good quality preconditioning, such as multigrid preconditioning, can significantly improve the convergence of iterative solution methods. There are two types of multigrid preconditioners, algebraic multigrid and geometric multigrid methods, but there are few examples of their application in particle methods. In this study, we attempted to develop a framework for a geometric multigrid preconditioner for solving the pressure Poisson equation in the ISPH. First, we focused on the geometric multigrid preconditioner using background cells, which are used for neighboring particle search, and implemented it on a GPU environment. Through a simple dam-break problem, we compared the computation time between the Conjugate gradient (CG) solver with a traditional preconditioner and the CG solver with a geometric multigrid preconditioner. We confirmed that the background cell-based geometric multigrid preconditioner is practical for the ISPH method.

Abstract

Particle methods such as the SPH and MPS methods have problems because it is difficult to treat curved bottom surfaces such as seabed surfaces accurately. In this study, regarding this problem, the curved bottom surfaces’ treatments have been improved using a coordinate transformation using the high-order second derivative model called SPH(2). Although the theory for the coordinate transformation was established in the MPS method, its accuracy did not give the desired accuracy because of the numerical errors of the second derivative models. Therefore, the numerical errors in these coordinate transformations were overcome by applying the second derivative model of SPH(2) to the coordinate transformation formulas. The superiority and validity of the proposed coordinate transformation using SPH(2) are demonstrated through validation examples such as the hydrostatic pressure and dam-break problems.