numerical simulation of free surface ﬂows by lagrangian ... · numerical simulation of free...

Numerical simulation of free surface flowsby Lagrangian particle methods

L. Cueto-Felgueroso, I. Colominas, F. Navarrina & M. CasteleiroGroup of Numerical Methods in Engineering, Civil Engineering School,University of La Coruna, Spain

Abstract

This paper presents a Galerkin based SPH formulation with moving least-squaresmeshless approximation, applied to free surface flows. The Galerkin scheme pro-vides a clear framework to analyze several procedures widely used in the classicalSPH literature, suggesting that some of them should be reformulated in order todevelop consistent algorithms. The performance of the methodology proposed istested through various dynamic simulations, demonstrating the attractive ability ofparticle methods to handle severe distortions and complex phenomena.

1 Introduction

The endeavour to solve the continuum equations in a particle (as opposed tocell or element) framework, i.e. simply using the information stored at certainnodes or particles without reference to any underlying mesh, has given rise toa very active area of research: the class of so-called meshless, meshfree or particlemethods.

The origin of modern meshless methods could be dated back to the 1970s withthe pioneering works in generalized finite differences and vortex particle methods[1, 2]. However, the strongest influence upon the present trends is commonlyattributed to early Smoothed Particle Hydrodynamics (SPH) formulations [3, 4, 5],where a lagrangian particle tracking is used to describe the motion of a fluid.The extension to solid mechanics was introduced by Libersky, Petschek et al. [6]and Randles [7]. Johnson and Beissel proposed a Normalized Smoothing Func-tion (NSF) algorithm [8] and other corrected SPH methods have been developedby Bonet et al. [9, 10] and Chen et al. [11]. More recently, Dilts has introducedMoving Least Squares (MLS) shape functions into SPH computations [12].

Fluid Structure Interaction and Moving Boundary Problems 491

© 2005 WIT Press WIT Transactions on The Built Environment, Vol 84, www.witpress.com, ISSN 1743-3509 (on-line)

The ability of the Smoothed Particle Hydrodynamics method to handle severedistortions allows this technique to be successfully applied to simulate fluid flowsin complex CFD applications. Early SPH formulations included both a new approx-imation scheme and certain characteristic discrete equations (the so-called SPHequations), which may look quite strange for those researchers with someexperience in methods with a higher degree of formalism such as finite elements.The formulation described in this paper follows a different approach, and the dis-crete equations are obtained using a Galerkin weighted residuals scheme. Althoughthis new statement may result somewhat disconcerting for those accustomed to theclassical SPH equations, the Galerkin formulation provides a strong framework todevelop more consistent algorithms.

2 Meshfree approximation: Moving Least-Squares (MLS)

Let us consider a function u(xxxxxxxxxxxxxx) defined in a bounded, or unbounded, domain Ω.The basic idea of the MLS approach is to weighted u(xxxxxxxxxxxxxx), at a given point xxxxxxxxxxxxxx, througha polynomial least-squares fitting of u(xxxxxxxxxxxxxx) in a neighbourhood of xxxxxxxxxxxxxx as:

u(xxxxxxxxxxxxxx) ≈ u(xxxxxxxxxxxxxx) =m∑

i=1

pi(xxxxxxxxxxxxxx)αi(zzzzzzzzzzzzzz)|zzzzzzzzzzzzzz=xxxxxxxxxxxxxx = ppppppppppppppt(xxxxxxxxxxxxxx)αααααααααααααα(zzzzzzzzzzzzzz)|zzzzzzzzzzzzzz=xxxxxxxxxxxxxx (1)

where ppppppppppppppt(xxxxxxxxxxxxxx) is an m-dimensional polynomial basis and αααααααααααααα(zzzzzzzzzzzzzz)|zzzzzzzzzzzzzz=xxxxxxxxxxxxxx is a set of para-meters to be determined, such that they minimize the following error functional:

J(αααααααααααααα(zzzzzzzzzzzzzz)|zzzzzzzzzzzzzz=xxxxxxxxxxxxxx) =∫yyyyyyyyyyyyyy∈Ωxxxxxxxxxxxxxx

W (zzzzzzzzzzzzzz − yyyyyyyyyyyyyy, h)|zzzzzzzzzzzzzz=xxxxxxxxxxxxxx[u(yyyyyyyyyyyyyy) − ppppppppppppppt(yyyyyyyyyyyyyy)αααααααααααααα(zzzzzzzzzzzzzz)|zzzzzzzzzzzzzz=xxxxxxxxxxxxxx

]2dΩxxxxxxxxxxxxxx (2)

where W (zzzzzzzzzzzzzz − yyyyyyyyyyyyyy, h)|zzzzzzzzzzzzzz=xxxxxxxxxxxxxx is a symmetric kernel with compact support (denoted byΩxxxxxxxxxxxxxx) chosen among the kernels used in standard SPH, and the smoothing length hmeasures the size of Ωxxxxxxxxxxxxxx. The stationary conditions of J with respect to αααααααααααααα lead to

∫yyyyyyyyyyyyyy∈Ωxxxxxxxxxxxxxx

pppppppppppppp(yyyyyyyyyyyyyy)W (zzzzzzzzzzzzzz − yyyyyyyyyyyyyy, h)|zzzzzzzzzzzzzz=xxxxxxxxxxxxxx u(yyyyyyyyyyyyyy)dΩxxxxxxxxxxxxxx = MMMMMMMMMMMMMM(xxxxxxxxxxxxxx)αααααααααααααα(zzzzzzzzzzzzzz)|zzzzzzzzzzzzzz=xxxxxxxxxxxxxx (3)

where the moment matrix MMMMMMMMMMMMMM(xxxxxxxxxxxxxx) is

MMMMMMMMMMMMMM(xxxxxxxxxxxxxx) =∫yyyyyyyyyyyyyy∈Ωxxxxxxxxxxxxxx

pppppppppppppp(yyyyyyyyyyyyyy)W (zzzzzzzzzzzzzz − yyyyyyyyyyyyyy, h)|zzzzzzzzzzzzzz=xxxxxxxxxxxxxx ppppppppppppppt(yyyyyyyyyyyyyy)dΩxxxxxxxxxxxxxx (4)

In numerical computations, the global domain Ω is represented by a set of nnodes or particles, and the integrals in (3) and (4) can be computed by using those

492 Fluid Structure Interaction and Moving Boundary Problems


nodes inside Ωxxxxxxxxxxxxxx as quadrature points (nodal integration). Thus, rearranging (3) wecan write

αααααααααααααα(zzzzzzzzzzzzzz)|zzzzzzzzzzzzzz=xxxxxxxxxxxxxx = MMMMMMMMMMMMMM−1(xxxxxxxxxxxxxx)PPPPPPPPPPPPPPΩxxxxxxxxxxxxxxWWWWWWWWWWWWWWV (xxxxxxxxxxxxxx)uuuuuuuuuuuuuuΩxxxxxxxxxxxxxx (5)

where the vector uuuuuuuuuuuuuuΩxxxxxxxxxxxxxx contains certain nodal parameters of those nodes in Ωxxxxxxxxxxxxxx, andMMMMMMMMMMMMMM(xxxxxxxxxxxxxx) = PPPPPPPPPPPPPPΩxxxxxxxxxxxxxxWWWWWWWWWWWWWWV (xxxxxxxxxxxxxx)PPPPPPPPPPPPPP t

Ωxxxxxxxxxxxxxx , being

PPPPPPPPPPPPPPΩxxxxxxxxxxxxxx =(pppppppppppppp(xxxxxxxxxxxxxx1)pppppppppppppp(xxxxxxxxxxxxxx2) · · · pppppppppppppp(xxxxxxxxxxxxxxnxxxxxxxxxxxxxx)

); WWWWWWWWWWWWWWV (xxxxxxxxxxxxxx) = diag Wi(xxxxxxxxxxxxxx − xxxxxxxxxxxxxxi)Vi , i = 1, nxxxxxxxxxxxxxx

(6)The complete details can be found in [13]. In the above equations, nxxxxxxxxxxxxxx denotesthe total number of nodes within the neighbourhood of point xxxxxxxxxxxxxx and Vi and xxxxxxxxxxxxxxi

are, respectively, the tributary volume (used as quadrature weight) and the coordi-nates associated to node i. Note that the tributary volumes of neighbouring nodesare included in the matrix WWWWWWWWWWWWWWV , obtaining the MLS Reproducing Kernel ParticleMethod [13]. Otherwise, we can use WWWWWWWWWWWWWW instead of WWWWWWWWWWWWWWV being

WWWWWWWWWWWWWW (xxxxxxxxxxxxxx) = diag Wi(xxxxxxxxxxxxxx − xxxxxxxxxxxxxxi) , i = 1, . . . , nxxxxxxxxxxxxxx (7)

which corresponds to the classical MLS approximation (in the nodal integration ofthe functional (2), the same quadrature weight is associated to all nodes).Introducing (5) in (1) the interpolation structure can be identified as:

u(xxxxxxxxxxxxxx) = ppppppppppppppt(xxxxxxxxxxxxxx)M−1(xxxxxxxxxxxxxx)PPPPPPPPPPPPPPΩxxxxxxxxxxxxxxWWWWWWWWWWWWWWV (xxxxxxxxxxxxxx)uuuuuuuuuuuuuuΩxxxxxxxxxxxxxx = NNNNNNNNNNNNNN t(xxxxxxxxxxxxxx)uuuuuuuuuuuuuuΩxxxxxxxxxxxxxx (8)

And, therefore, the MLS shape functions can be written as:

NNNNNNNNNNNNNN t(xxxxxxxxxxxxxx) = ppppppppppppppt(xxxxxxxxxxxxxx)M−1(xxxxxxxxxxxxxx)PPPPPPPPPPPPPP ΩxxxxxxxxxxxxxxWWWWWWWWWWWWWWV (xxxxxxxxxxxxxx) (9)

It is most frequent to use a scaled and locally defined polynomial basis, instead ofthe globally defined pppppppppppppp(yyyyyyyyyyyyyy). Thus, if a function is to be evaluated at point xxxxxxxxxxxxxx, the basiswould be of the form pppppppppppppp(yyyyyyyyyyyyyy−xxxxxxxxxxxxxx

h ). The shape functions are, therefore, of the form

NNNNNNNNNNNNNN t(xxxxxxxxxxxxxx) = ppppppppppppppt(00000000000000)M−1(xxxxxxxxxxxxxx)PPPPPPPPPPPPPPΩxxxxxxxxxxxxxxWWWWWWWWWWWWWWV (xxxxxxxxxxxxxx) (10)

In the examples presented in this paper, a linear polynomial basis has been used,which provides linear completeness. Respect to the choice of kernel, a wide varietyof kernel functions have been proposed in the literature, most of them being splineor exponential functions. In this study, the following cubic spline was used

Wj(xxxxxxxxxxxxxx) = W (xxxxxxxxxxxxxx − xxxxxxxxxxxxxxj , ρ) =α

ρν

1 − 32s2 +

34s3 s ≤ 1

14(2 − s)3 1 < s ≤ 2

0 s > 2

(11)

where s = ‖xxxxxxxxxxxxxx − xxxxxxxxxxxxxxj‖ρ , ν is the number of dimensions and α takes the value 2

3 , 107π

or 1π in one, two or three dimensions, respectively. The coefficient α/ρν is a scale

factor necessary only if non-corrected SPH interpolation is being used, to assurethe normality property

∫WdV = 1. We do not use it in our MLS computations.



3 Discrete equations for fluid flow problems

3.1 Weighted residuals. Test and trial functions

We assume that the behaviour of a continuum can be analyzed by means of thefollowing governing equations: the continuity equation (conservation of mass) andthe momentum equation (conservation of linear momentum) [14, 15].

The meshless discrete equations can be derived using a weighted residuals for-mulation. Thus, the global weak (integral) form of the spatial momentum equationcan be written as:∫

Ω

ρdvvvvvvvvvvvvvv

dt· δvvvvvvvvvvvvvv dΩ = −

∫Ω

σσσσσσσσσσσσσσ : δllllllllllllll dΩ +∫

Ω

bbbbbbbbbbbbbb · δvvvvvvvvvvvvvv dΩ +∫

Γ

σσσσσσσσσσσσσσnnnnnnnnnnnnnn · δvvvvvvvvvvvvvv dΓ (12)

where Ω is the problem domain, Γ is its boundary, and certain functions δvvvvvvvvvvvvvv and vvvvvvvvvvvvvvare the approximations to the test and trial functions δvvvvvvvvvvvvvv and vvvvvvvvvvvvvv.

The spatially discretized equations are obtained after introducing meshless testand trial functions and their gradients in (12) as

δvvvvvvvvvvvvvv(xxxxxxxxxxxxxx) =n∑

i=1

δvvvvvvvvvvvvvviN∗i (xxxxxxxxxxxxxx), ∇∇∇∇∇∇∇∇∇∇∇∇∇∇δvvvvvvvvvvvvvv(xxxxxxxxxxxxxx) =

n∑i=1

δvvvvvvvvvvvvvvi ⊗∇xxxxxxxxxxxxxxN∗i (xxxxxxxxxxxxxx) (13)

vvvvvvvvvvvvvv(xxxxxxxxxxxxxx) =n∑

j=1

vvvvvvvvvvvvvvjNj(xxxxxxxxxxxxxx), ∇∇∇∇∇∇∇∇∇∇∇∇∇∇vvvvvvvvvvvvvv(xxxxxxxxxxxxxx) =n∑

j=1

vvvvvvvvvvvvvvj ⊗∇xxxxxxxxxxxxxxNj(xxxxxxxxxxxxxx). (14)

Thus, for each node (particle) i the following identity must hold:

n∑j=1

∫Ω

mijdvvvvvvvvvvvvvvj

dt= ffffffffffffff int

i + ffffffffffffffexti , i = 1, n (15)

where the mass term mij , the internal forces term ffffffffffffff inti and the external forces term

ffffffffffffffexti are:

mij =∫

Ω

ρN∗i (xxxxxxxxxxxxxx)Nj(xxxxxxxxxxxxxx)dΩ (16)

ffffffffffffff inti = −

∫Ω

σσσσσσσσσσσσσσ∇∇∇∇∇∇∇∇∇∇∇∇∇∇xxxxxxxxxxxxxxN∗i (xxxxxxxxxxxxxx)dΩ, ffffffffffffffext

i =∫

Ω

N∗i (xxxxxxxxxxxxxx)bbbbbbbbbbbbbb dΩ +

∫Γ

N∗i (xxxxxxxxxxxxxx)σσσσσσσσσσσσσσnnnnnnnnnnnnnn dΓ (17)

The MLS shape functions do not vanish on essential boundaries and, therefore,the boundary integral in (17) can be decomposed as:∫

Γ

N∗i (xxxxxxxxxxxxxx)σσσσσσσσσσσσσσnnnnnnnnnnnnnn dΓ =

∫Γu

N∗i (xxxxxxxxxxxxxx)σσσσσσσσσσσσσσnnnnnnnnnnnnnn dΓu +

∫Γn

N∗i (xxxxxxxxxxxxxx)σσσσσσσσσσσσσσnnnnnnnnnnnnnn dΓn (18)

being Γu and Γn, respectively, the parts of the boundary where essential and natu-ral boundary conditions are prescribed and Γ = Γu ∪ Γn.



Regarding the equation used for mass conservation, its Galerkin weak form isequivalent to a point collocation scheme and, thus, the continuity equation must beenforced at each particle i,

dρi

dt= −ρidiv(vvvvvvvvvvvvvv)i = −ρi

n∑j=1

vvvvvvvvvvvvvvj · ∇∇∇∇∇∇∇∇∇∇∇∇∇∇xxxxxxxxxxxxxxNj(xxxxxxxxxxxxxxi) (19)

where expression (14) for ∇∇∇∇∇∇∇∇∇∇∇∇∇∇vvvvvvvvvvvvvvi has been used.

3.2 Nodal integration

Nodal integration has been used, at least implicitly, in most SPH formulations, andlies in the basis of its early formulation. Obviously, this is the cheapest option andthe resulting scheme is truly meshless (no background mesh is needed).The particles are used as quadrature points and the corresponding integrationweights are their tributary volumes. Recalling the weak form derived in theprevious section, the discrete momentum equation can be written as:

MMMMMMMMMMMMMMaaaaaaaaaaaaaa = FFFFFFFFFFFFFF int + FFFFFFFFFFFFFF ext (20)

where

mij =n∑

k=1

ρkN∗i (xxxxxxxxxxxxxxk)Nj(xxxxxxxxxxxxxxk)Vk (21)

ffffffffffffff inti = −

n∑k=1

σσσσσσσσσσσσσσk∇∇∇∇∇∇∇∇∇∇∇∇∇∇xxxxxxxxxxxxxxN∗i (xxxxxxxxxxxxxxk)Vk (22)

ffffffffffffffexti =

n∑k=1

N∗i (xxxxxxxxxxxxxxk)bbbbbbbbbbbbbbkVk +

n∑k=1

N∗i (xxxxxxxxxxxxxxk)σσσσσσσσσσσσσσknnnnnnnnnnnnnnAk (23)

In the above, Vk represents the tributary volume associated to particle k. In themost frequent approach in SPH simulations, the particles are set up with certaininitial densities, volumes and, therefore, masses. These physical masses Mkremain constant during the simulation and densities are field variables updatedusing the continuity equation. Thus, particle volumes are obtained each time stepas Vk = Mk

ρk. Note that, in our formulation, the real or physical particle masses

Mk are different, in general, from the numerical masses mij given by (21), andderived in the Galerkin scheme.

Nodal integration is the origin of well known instabilities in meshless meth-ods. Beissel and Belytschko [16], and Bonet and Kulasegaram [10], have analyzedthe performance of this quadrature in the context of Element-Free Galerkin (EFG)and Corrected SPH (CSPH) methods, and proposed modified variational princi-ples based on least-squares stabilizations, requiring second derivatives of the shape



Figure 1: Horizontal gravity current: scheme of the initial configuration.

functions. Chen and coworkers developed a stabilized conforming nodal integra-tion [17], based on a strain smoothing, which requires the Voronoi diagram of thecloud of particles, at a high computational cost.

4 Examples and conclusions

A gravity current is the flow of a fluid into another fluid of different density [18].In the first example (Figures 1 and 2), we consider a tank with fluid of densityρ10 = 1000 kg/m3 (left) separated by a lock gate from another fluid of density

ρ20 = 1500 kg/m3 (right). The lock is rapidly (‘instantaneously’) removed and

fluid 2 flows under fluid 1. An experimental measurement of the velocity of thehead of the current was proposed by Rottman and Simpson (1983, cited by [18]):

vh ∼ 0.4(

gD∆ρ

ρ0

)1/2

(24)

where D is the depth of the tank. In this example, D = 0.06 m, g = 9.81 m/s2,ρ0 = 1000 kg/m3 and ∆ρ = 500 kg/m3. Thus, (24) predicts a head velocity ofvh = 0.217 m/s. In our numerical experiments we have obtained vh ∼ 0.2 m/s,which is in good agreement with the experimental value. The simulation involved5100 particles.

In the second example, a circular drop of water falls vertically at 2 m/s on amass of water initially at rest (Figures 3 and 4). The fluid density and viscosity areρ0 = 1000 kg/m3 and µ = 0.5 kg m−1 s−1, respectively. The total number ofparticles is 5539.

The results obtained demonstrate the ability of the proposed numerical approachto simulate complex unsteady fluid flow problems. The Galerkin approachconstitutes a clear framework to derive the discrete equations, while a moving



time=0.048 s time=0.108 s



Figure 2: Horizontal gravity current: simulation at various stages.

Figure 3: Fluid-Fluid Impact: scheme of the initial configuration.

least squares approximation significantly improves the standard smooth particlehydrodynamics kernel estimates. Within this general methodology it is possible toanalyze from a rigorous point of view some of the corrections that are generally



time=0.0018 s time=0.0099 s

time=0.0297 s time=0.0477 s

time=0.0657 s time=0.0837 s

time=0.1017 s time=0.1197 s

time=0.1377 s time=0.1557 s

Figure 4: Fluid-Fluid Impact: simulation at various stages.

introduced in the SPH method to obtain accurate solutions. Furthermore, it is nowpossible to derive new more consistent meshless numerical approaches.

This work has been partially supported by the SGPICT of the ‘Ministerio de Cien-cia y Tecnologıa’ of the Spanish Government (Grant DPI# 2001-0556), the ‘Xuntade Galicia’ (Grant # PGDIT01PXI11802PR) and the University of La Coruna.

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