Evaluating the performance of the particle finite element method in parallel architectures

Abstract

This paper presents a high performance implementation for the particle-mesh based method called particle finite element method two (PFEM-2). It consists of a material derivative based formulation of the equations with a hybrid spatial discretization which uses an Eulerian mesh and Lagrangian particles. The main aim of PFEM-2 is to solve transport equations as fast as possible keeping some level of accuracy. The method was found to be competitive with classical Eulerian alternatives for these targets, even in their range of optimal application. To evaluate the goodness of the method with large simulations, it is imperative to use of parallel environments. Parallel strategies for Finite Element Method have been widely studied and many libraries can be used to solve Eulerian stages of PFEM-2. However, Lagrangian stages, such as streamline integration, must be developed considering the parallel strategy selected. The main drawback of PFEM-2 is the large amount of memory needed, which limits its application to large problems with only one computer. Therefore, a distributed- memory implementation is urgently needed. Unlike a sharedmemory approach, using domain decomposition the memory is automatically isolated, thus avoiding race conditions; however new issues appear due to data distribution over the processes. Thus, a domain decomposition strategy for both particle and mesh is adopted, which minimizes the communication between processes. Finally, performance analysis running over multicore and multinode architectures are presented. The Courant-Friedrichs-Lewy number used influences the efficiency of the parallelization and, in some cases, a weighted partitioning can be used to improve the speed-up. However the total cputime for cases presented is lower than that obtained when using classical Eulerian strategies.

Seakeeping with the semi-Lagrangian particle finite element method

“…The latter uses the Finite Volume Method (FVM) for discretisation and the Volume Of Fluid (VOF) method for capturing the multi-fluid interfaces. Gimenez et al did [10] similar performance studies for incompressible flow simulations using the SL-PFEM on parallel computers. They reported strong scalability of the SL-PFEM at par with OpenFOAM ® (using the solver pimpleFoam); however, a three-fold gain in speed (wall-time) was reported for the former for a chosen level of accuracy.…”

A pseudo-DNS method for the simulation of incompressible fluid flows with instabilities at different scales

“…As mentioned above, adopting a Lagrangian formulation allows for accurately and naturally convecting and spreading the instabilities, even with a coarse mesh. Good resolution may be obtained by using either the first [30,31] or the second generation of the Particle Finite Element Method (PFEM-2) [32][33][34][35] or its finite volume version, called Particle Finite Volume Method (PFVM) [36]. The readers are referred to these references for further details on these technologies.…”

Surface tension problems solved with the particle finite element method using large time-steps

“…In Herein, the efficient distributed-memory implementation presented in [21] and extended to the free-surface treatment (see Algorithm 1) is used to simulate each of next cases presented. Also, the numerical parameter θ p is set to 0 in every case so as to restart the pressure at each time step to allow larger time-steps and 3 iterations of steps 4 and 5 are done to improve the global first order [2].…”

“…This proposal not only tracks material propertiesas density, viscosity, etc., therefore eliminating the need of the non-linear convective term. Also, using an improved explicit integration named X-IVS (eXplicit Integration following the Velocity Streamlines) added to an implicit correction of diffusive terms, there is no limitation in the time step, being the required precision the only bound for the time-step [21]. The enhanced PFEM-2 version to solve multiphase problems, presented in [1] and validated in [2], preserves the large time-step goodnesses of the single-phase strategy, also includes enrichment strategies to capture discontinuities in the pressure gradient, i.e.…”

“…Also, θ µ can be 0 or 1 depending on the necessity or not of an accurate diffusion calculation when large Fourier numbers are employed, F = (ρg + F σ ) dΩ, and M and K are the standard mass and stiffness matrices of any FEM assembling. The computational implementation was done extending the high performing library presented in[21] and each test presented in this work was simulated with that house-made code.…”