The role of time integration in energy conservation in Smoothed Particle Hydrodynamics fluid dynamics simulations

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The role of time integration in energy conservation in

Smoothed Particle Hydrodynamics ﬂuid dynamics

simulations

Jose Luis Cercos-Pitaa,1,∗, Pablo Eleazar Merino-Alonsob,2, Javier

Calderon-Sanchezc,3, Daniel Duquec,4

aHedenstierna laboratory, Surgical Sciences Department,

Uppsala Universitet, 75185 Uppsala, Sweden

bM2ASAI Research Group,

ETS Ingenieros Navales, Universidad Polit´ecnica de Madrid,

28040 Madrid, Spain

cCEHINAV Research Group,

ETS Ingenieros Navales, Universidad Polit´ecnica de Madrid,

28040 Madrid, Spain

Abstract

The choice of a time integration scheme is a crucial aspect of any transient ﬂuid

simulation, and Smoothed-Particle Hydrodynamics (SPH) is no exception. The

inﬂuence of the time integration scheme on energy balance is here addressed.

To do so, explicit expressions allowing to compute the deviations from the en-

ergy balance, induced by the time integration scheme, are provided. These

expressions, computed a posteriori, are valid for diﬀerent integration methods.

Besides, a new formulation that improves energy conservation by enhancing sta-

bility, based on an implicit integration scheme, is proposed. Such formulation is

tested with the simulation of a two-dimensional non-viscous impact of two jets,

with no artiﬁcial dissipation terms. To the best of our knowledge, this is the

ﬁrst stable simulation of a non-dissipative system with a weakly-compressible

SPH method. A viscous case, the Taylor-Green vortex, has also been simulated.

∗Corresponding author

Email address: jl.cercos@upm.es (Jose Luis Cercos-Pita)

1ORCID: 0000-0002-3187-4048

2ORCID: 0000-0002-2630-3590

3ORCID: 0000-0003-0636-8853

4ORCID: 0000-0002-2248-5630

Preprint submitted to European Journal of Mechanics - B/Fluids December 18, 2024

arXiv:2210.12372v2 [physics.flu-dyn] 17 Dec 2024

Results show that an implicit time integration scheme also behaves better in a

viscous context.

Keywords: stability, time integration scheme, energy balance, SPH

1. Introduction

Smoothed-Particle Hydrodynamics (SPH) is a meshfree numerical method

in which continuum media are discretized as a set of particles, which move in

a Lagrangian manner [1]. There is no doubt that its meshless nature is the

feature which has drawn more attention to the model, that is indeed well suited

to problems dominated by complex geometries, such as simulations involving

free surface ﬂow, or ﬂows driven by large boundary displacements.

In addition to that, SPH is built starting from a relatively simple formulation

that can be applied to a wide variety of physical phenomena. Indeed, even

though the model was initially developed in astrophysics [2, 3], it quickly spread

to other disciplines, including free surface ﬂows [4], solid mechanics [5, 6] and

geomaterial mechanics [7].

An interesting feature of SPH which has been traditionally considered one of

its main beneﬁts is the conservation of both momentum and energy. The claim

that the method features exact energy conservation has been made several times

in the past [8, 9, 10], although literature may also be found (e.g. [11]) where

such conservation is shown to be linked to the accuracy of the time integration

algorithm. Therefore, a clear line of investigation to improve the stability of

the model comes from the analysis of the time integration scheme. Previous

research on this topic has already shown promising results [12, 13, 14, 15, 16].

Indeed SPH has been widely criticized for characteristic instabilities, partic-

ularly in its weakly-compressible SPH (WC-SPH) incarnation, which is by far

the most popular one. One of the most aggressive solutions to avoid stability

issues of WC-SPH is to implement it within a rigorous incompressible formula-

tion, leading to Incompressible-SPH (I-SPH) [17], a method that inherits some

good stability features of more conventional CFD methods [18], at the expense

of increased algorithm complexity.

Within the WC-SPH formulation, and probably motivated by the excel-

lent energy conservation properties, some authors attributed the instabilities to

spurious zero-energy modes [19, 20]. Lately, the focus has been set on tensile

instability [21, 22, 23]. The ﬁrst formulation designed to mitigate the pernicious

eﬀects of this instability was X-SPH, in which the velocity ﬁeld is smoothed at

each particle using information from its neighbors [21]. Along this line, some au-

thors [22] analyzed the convolution kernel, culminating in the work by Dehnen

and Aly [23], where it was demonstrated that Wendland kernels beneﬁt particle

packing.

Another methodology to deal with the tensile instability which is gaining

popularity is the Particle Shifting Technique (PST), in which the particles’

positions are slightly modiﬁed at the end of each time step in order to preserve

particle packing [18, 24, 25, 26, 27, 28]. Some authors who have dealt with

tensile instabilities are moving to the so-called Total Lagrangian formalism [29],

specially in solid dynamics [30, 31, 32].

A diﬀerent line of investigation to improve stability has been the application

of extra energy dissipation terms, the most straightforward one being artiﬁcial

viscosity [1]. However, research quickly targeted mass conservation as well,

by means of Shepard ﬁltering. Afterwards, the addition of dissipation terms

to the mass conservation equation has been investigated, resulting in the δ-

SPH [33], and Riemann solvers-based schemes [34]. The relation between both

formulations has been addressed in the past [35, 36].

In many of the research targeted at intrinsic instabilities just described, it is

not yet clear how novel algorithms and formulations may aﬀect the conservation

properties of the method. It is not unreasonable to suggest that these studies

have been circumvented due to the fact that the SPH community has tradi-

tionally regarded SPH as an exact energy conservation model, and the eﬀorts

have been accordingly directed towards solutions to known drawbacks of the

method. For instance, an energy analysis of the δ-SPH term was conducted in

Antuono et al. [37], demonstrating that it is intrinsically dissipating energy far

from the boundaries. However, such energy dissipation is presented as a perni-

cious side eﬀect of the model, a point which is not obvious, as discussed below.

Similarly, Green and Peir´o [38] examine energy conservation and partition be-

tween kinetic, potential, and compressible energies at the post-processing stage,

in order to assess diﬀerent models, for a long-duration simulation.

In Cercos-Pita et al. [39], violations of exact energy conservation were for-

mally demonstrated for the ﬁrst time. In such work, ﬂuid extensions are consid-

ered, and extra energy terms are shown to appear due to interactions with the

boundary. The investigation was also extended to other boundary formulations

[40].

Surprisingly, although both the inﬂuence of the time integration scheme in

the stability and the beneﬁts of eventual extra energy dissipation have been

already demonstrated, the role of the time integration scheme in the energy

conservation has not been addressed in the literature yet. This paper is therefore

devoted to this topic. For the sake of simplicity, we focus on the WC-SPH

formulation for non-viscous ﬂuids.

In order to analyze energy balance, spatial and time discretization are inde-

pendently considered: the former will be presented in Section 2 and the latter

in Section 3. They are combined afterwards in a total energy balance in Section

3.2. In Section 4 an implicit time integration scheme is proposed in order to

improve energy conservation. Then, numerical experiments are carried out in

Section 5, in order to support the theoretical ﬁndings. One of the simulations

presented in this Section would be the ﬁrst stable simulation of a non-dissipative

system with a weakly-compressible SPH method (to the best of our knowledge.)

The other is an application of this methodology to a viscous benchmark case.

Finally, conclusions are presented in Section 6.

2. Spatial discretization

This section deals chieﬂy with spatial discretization. The SPH governing

equations are introduced in Section 2.1. Power balance is discussed in Section

2.2, where contributions to energy variation are identiﬁed and separated.

2.1. SPH numerical model

Herein we focus on weakly compressible ﬂows, even though similar analy-

ses can be carried out for incompressible ﬂows, or even for diﬀerent physical

phenomena. Hence, the governing equation for the evolution of density is the

conservation equation:

dρ

dti

(t) = −ρi(t)⟨∇ · u⟩i(t) (1)

where ⟨·⟩ denotes SPH operators, and abusing the notation, any magnitude

resulting from the application of the SPH methodology. In the equation above

ρiis the density of an arbitrary i-th particle, and ui, its velocity.

The evolution of the velocity ﬁeld is a discrete version of the Navier-Stokes

momentum equation,

du

dti

(t) = −⟨∇p⟩i(t)

ρi(t)+µ

ρi(t)⟨∆u⟩i(t) + g−kpi⟨∇γ⟩i(t)

ρi(t),(2)

where piis the pressure, µthe viscosity coeﬃcient, gthe acceleration due to

external forces, and γthe Shepard renormalization factor (see, for instance [41]).

The extra term with the coeﬃcient k, which appears in this discrete version and

is absent in the continuum, is explained below.

In WC-SPH, these equations are closed by an equation of state (EOS) relat-

ing pressure and density:

pi(t) = p0+c2

0(ρi(t)−ρ0),(3)

where p0is the background pressure, ρ0is the reference density, and c0the

speed of sound in the ﬂuid. The latter is customarily set to a value high enough

that the ﬂuid behaves almost as if incompressible.

Incidentally, it may be highlighted that every single SPH related operator

can be split in 2 terms,

⟨·⟩ =⟨·⟩Ω+⟨·⟩∂Ω,(4)

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TheroleoftimeintegrationinenergyconservationinSmoothedParticleHydrodynamicsfluiddynamicssimulationsJoseLuisCercos-Pitaa,1,∗,PabloEleazarMerino-Alonsob,2,JavierCalderon-Sanchezc,3,DanielDuquec,4aHedenstiernalaboratory,SurgicalSciencesDepartment,UppsalaUniversitet,75185Uppsala,SwedenbM2ASAIResearchGro...

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The role of time integration in energy conservation in Smoothed Particle Hydrodynamics fluid dynamics simulations

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