ECCOMAS Thematic Conference on Computational Methods in Structural Dynamics and Earthquake Engineering M. Papadrakakis, D.C. Charmpis, N.D. Lagaros, Y. Tsompanakis (eds.) Rethymno, Crete, Greece, 13–16 June 2007
A FAST NEIGHBOUR-SEARCH ALGORITHM FOR FREE SURFACE FLOW SIMULATIONS USING SPH Giacomo Viccione1*, Vittorio Bovolin1, Eugenio Pugliese Caratelli2 1
Dipartimento di Ingegneria Civile, Università degli Studi di Salerno Via Ponte Don Melillo – 84084 – Fisciano (IT) e-mail: {gviccion, v.bovolin}@unisa.it 2
C.U.G.RI., Università degli Studi di Salerno P.zza V. Emanuele – 84080 – Penta di Fisciano (IT) e-mail:
[email protected]
Keywords: CFD, SPH, free surface flow, Verlet list, cell-linked list. Abstract. A fast neighbour search algorithm as a device used in SPH simulations, is presented to increase global execution speed. SPH, like other particle methods, is based on short-range interactions ruled by a cutoff distance rc, which defines a small influence region centered on each moving particle. Commonly, the number of neighbours inside such area is much lower than the total number of particles Ntot, hence, if no search procedures are employed, a great amount of unnecessary distance calculations should be estimated. Our search procedure is built by coupling the cell-linked list with the Verlet table. The available domain is therefore partitioned into cells with edge lengths larger than the interaction range of the kernel function plus the Verlet skin, that is “rc +Vskin”. All data structures are set up with 1D arrays with a resulting saving of physical memory. Indeed, the cell linked list is defined taking only those cells taken by at least one particle. The benefit is straightforward, especially for those problems characterized by moving domains much smaller than the simulation space size, in which a huge number of empty cells may be present. Concerning the Verlet list, the advantage arises mainly from the fact that the number of neighbours for each moving particle is in general not constant, although slightly variable. A list of cell neighbours for each cell is set up before a simulation starts. Such data structure avoids the neighbour cell identification, needed each time neighbouring particles have to be identified for a fluid particle which migrates to a new cell. An efficiency analysis is eventually carried out, varying the two spatial scales here introduced: “Dcell” and “Vskin”. As test problem, we consider a solitary wave run-up on a steep beach, simply generated with a vertical paddle. Results show as best CPU times are obtained using the cell-linked list and the Verlet table together.
Giacomo Viccione, Vittorio Bovolin and Eugenio Pugliese Carratelli
1
INTRODUCTION
Particle methods such as Smoothed Particle Hydrodynamics (SPH) have recently become quite popular in Computational Fluid Dynamics. Originally developed for astrophysical applications [1, 2] it has also been applied to other areas such as multi-phase flows [3], flows through porous media [4, 5], high explosive detonation and explosion [6, 7], hyper velocity impact and penetration [8, 9]. A comprehensive review of SPH theoretical aspects has recently been published [10], so they need not be discussed here. Here, we are concerned with one particular aspect of SPH, that is the handling of data and the neighbour search techniques which are needed to speed up free surface flow simulations. Code performance is measured in term of the computation time needed to carry out a numerical simulation. The total CPU time τtot may be considered as the sum of the integration time τint for the numerical integration in a strict sense (calculating particles interactions and fluid dynamical parameters) and the time τs for the identification of the local neighbourhood. Each computing particle, representing a small volume of moving domain, interacts with a number of neighbouring particles Nn (local neighbourhood), included within a short range rc; for large simulation problems, such a number is obviously much lower than the total number of particles Ntot considered in the problem. For two-dimensional problems Oger et al. [11], have shown that a cut off distance rc, encompassing at least 20 neighbouring particles provides acceptable results. A generic scalar or vector field fi at a given position ri taken by the fluid particle “i”, is given by a smooth interpolation from the values fk carried by those particles within the interaction sphere of radius rc =2h [12]: fi =
Nn
∑f
k
⋅ W ( ri − rk , h)
(1)
k =1
where W is the so called kernel or weighting function and h is known as the smoothing length. Here the cubic spline devised by Monaghan and Lattanzio [13] is adopted:
1 3 2 2 3 − q + 2 q 1 (2 − q )3 W (q, h ) = An (nd ) ⋅ 6 0
0 ≤ q
