Generalized-alpha scheme in the PFEM for velocity-pressure and displacement-pressure formulations of the incompressible Navier-Stokes equations

TL;DR

This paper evaluates the performance of the Generalized-alpha time integration method within the Particle Finite Element Method (PFEM) for solving incompressible Navier-Stokes equations, comparing different implementations and formulations through various benchmark problems.

Contribution

It provides a detailed implementation analysis of the Generalized-alpha method in PFEM and compares its performance against other schemes for different formulations.

Findings

Generalized-alpha shows improved stability over Backward Euler and Newmark.

Displacement-pressure formulation offers advantages in fluid-structure interaction scenarios.

Performance varies with implementation approach and problem type.

Abstract

Despite the increasing use of the Particle Finite Element Method (PFEM) in fluid flow simulation and the outstanding success of the Generalized-alpha time integration method, very little discussion has been devoted to their combined performance. This work aims to contribute in this regard by addressing three main aspects. Firstly, it includes a detailed implementation analysis of the Generalized-alpha method in PFEM. The work recognizes and compares different implementation approaches from the literature, which differ mainly in the terms that are alpha-interpolated (state variables or forces of momentum equation) and the type of treatment for the pressure in the time integration scheme. Secondly, the work compares the performance of the Generalized-alpha method against the Backward Euler and Newmark schemes for the solution of the incompressible Navier-Stokes equations. Thirdly, the study is enriched by considering not only the classical velocity-pressure formulation but also the displacement-pressure formulation that is gaining interest in the fluid-structure interaction field. The work is carried out using various 2D and 3D benchmark problems such as the fluid sloshing, the solitary wave propagation, the flow around a cylinder, and the collapse of a cylindrical water column.