Abstract

This paper presents advances in recent work of the authors to derive a fractional step scheme based on the stabilized finite element method that allows overcoming the above mentioned problem, resulting in a efficient time accurate scheme.

The starting point is the modified governing differential equations for the incompressible turbulent viscous flow and the free surface condition incorporating the necessary stabilization terms via a finite calculus (FIC) procedure developed by the authors .  This technique is based on writting the different balance equations over a domain of finite size and retaining higher order terms. These terms incorporate the ingredients for the necessary stabilization of any transient and steady state numerical solution already at the differential equations level.

The resulting stabilized equations are integrated in space using the standard finite element method, and in time using an implicit and uncoupled second order fractional step method.

Abstract

Particle Methods are those in which the problem is represented by a discrete number of particles. Each particle moves accordingly with its own mass and the external/internal forces applied to it. Particle Methods may be used for both, discrete and continuous problems. In this paper, a Particle Method is used to solve the continuous fluid mechanics equations. To evaluate the external applied forces on each particle, the incompressible Navier–Stokes equations using a Lagrangian formulation are solved at each time step. The interpolation functions are those used in the Meshless Finite Element Method and the time integration is introduced by an implicit fractional‐step method. In this manner classical stabilization terms used in the momentum equations are unnecessary due to lack of convective terms in the Lagrangian formulation. Once the forces are evaluated, the particles move independently of the mesh. All the information is transmitted by the particles. Fluid–structure interaction problems including free‐fluid‐surfaces, breaking waves and fluid particle separation may be easily solved with this methodology. 

Abstract

In this paper, three different fractional step methods are designed for the three-field viscoelastic flow problem, whose variables are velocity, pressure and elastic stress. The starting point of our methods is the same as for classical pressure segregation algorithms used in the Newtonian incompressible Navier–Stokes problem. These methods can be understood as an inexact LU block factorization of the original system matrix of the fully discrete problem and are designed at the pure algebraic level. The final schemes allow one to solve the problem in a fully decoupled form, where each equation (for velocity, pressure and elastic stress) is solved separately. The first order scheme is obtained from a straightforward segregation of pressure and elastic stress in the momentum equation, whereas the key point for the second order scheme is a first order extrapolation of these variables. The third order fractional step method relies on Yosida's scheme. Referring to the spatial discretization, either the Galerkin method or a stabilized finite element formulation can be used. We describe the fractional step methods first assuming the former, and then we explain the modifications introduced by the stabilized formulation we employ and that has been proposed in a previous work. This discretization in space shows very good stability, permitting in particular the use of equal interpolation for all variables.