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An explicit-implicit ﬁnite element model for the

numerical solution of incompressible Navier-Stokes

equations on moving grids

J. Martia,b, P.B. Ryzhakova

aCentre Internacional de M`etodes Num`erics en Enginyeria (CIMNE)

Gran Capit´an s/n, 08034 Barcelona, Spain

bDepartment of Civil and Environmental Engineering, Universitat Polit`ecnica de

Catalunya (UPC), 08034 Barcelona, Spain

Abstract

In this paper an eﬃcient mesh-moving Finite Element model for the simula-

tion of the incompressible ﬂow problems is proposed. The model is based on

a combination of the explicit multi-step scheme (Runge-Kutta) with an im-

plicit treatment of the pressure. The pressure is decoupled from the velocity

and is solved for only once per time step minimizing the computational cost of

the implicit step. Novel solution algorithm alleviating time step restrictions

faced by the majority of the former Lagrangian approaches is presented. The

method is examined with respect to its space and time accuracy as well as

the computational cost. Two numerical examples are solved: one involving a

problem on a domain with ﬁxed boundaries and the other one dealing with

a free surface ﬂow. It is shown that the method can be easily parallelized.

Keywords: incompressible Navier-Stokes, accuracy, Particle Finite

Element Method, Lagrangian, OpenMP, benchmark

1. INTRODUCTION

Lagrangian models for the simulation of ﬂow problems have been de-

veloped by several groups worldwide and successfully applied to a variety

of engineering problems over past three decades. They were found advan-

tageous for problems involving moving domain boundaries [1, 2, 3, 4, 5]

Email address: pryzhakov@cimne.upc.edu (P.B. Ryzhakov)

Preprint submitted to Computer Methods in Applied Mechanics and EngineeringMarch 8, 2019

and interfaces, particularly, free-surface ﬂows interacting with solid bodies

[6, 7, 8, 9, 10, 11]. They have been also applied to the simulation of diﬀerent

material forming processes [4, 12, 13, 14, 15]. Lagrangian methods are ad-

vantageous for such problems due to the intrinsic capability of tracking the

boundary evolution. Unlike their ﬁxed-grid counterparts, they do not suﬀer

numerical diﬀusion and preserve interfaces sharp. Nevertheless, Lagrangian

ﬂow models are generally characterized by higher computational costs due

to time step restrictions and necessity of mesh re-generation [16].

The main distinguishing feature of these approaches originates from the

fact that when written in the Lagrangian framework, the governing equa-

tions of the continuum (incompressible Navier-Stokes equations in our case)

become ”split” into a geometrical part (tracking the motion of the particles)

dx/dt = u(x, t) and a physical part (calculating how the ﬂow variables change

in time at each particle). Consequently, convective term disappears from the

momentum equation of the ﬂow, leading to several beneﬁts: no necessity of

stabilization for convection-dominated ﬂows and symmetry of the governing

system. However, the diﬃculty arises due to the necessity of ﬁnding an opti-

mal way of moving the particles and coupling the particle movement to the

solution of the momentum-continuity equations system. Thus, despite of the

fact that the convective term vanishes (source of non-linearity in classical

ﬁxed-grid approaches), the problem remains non-linear as the ﬂuid stresses

depend on both the velocity and the position of the particles, and this latter

one, in turn, depends on the velocity. This means that in order to solve the

problem numerically, an iterative method is needed. All iterative methods are

based on a linearization of the equations and also require an initial ”guess”

for starting the iterations. Depending on the way the particle movement is

modeled and coupled to the solution of Navier-Stokes equations, diﬀerent

versions of the Lagrangian models can be distinguished.

In this work we shall concentrate on the class of Lagrangian methods

known as Particle Finite Element Methods (PFEM), which are very similar

to the classical Lagrangian Finite Element method used in solids. It treats

the Lagrangian mesh nodes as immaterial particles, i.e. they follow the

convective velocity but do not have any mass associated to them. Other

popular Lagrangian methods such as Smooth Particle Hydrodynamics [17,

9, 6] and Material Points Methods (MPM) [18] (where particles are treated

as true material points with associated mass) lie outside the scope of the

present work.

Majority of the existing Lagrangian ﬁnite element-based ﬂuid models are

2

relying on fully implicit time integration schemes [19, 16, 8, 20, 11, 21, 22, 23].

According to these schemes, the domain where equations need to be solved

corresponds to the mesh deﬁned by the nodal (particle) positions which are

unknown. Typically, the particle positions are predicted using the last known

value of the velocity. The overall solution procedure consists in a non-linear

loop, where at each step the Navier-Stokes equations are solved and then the

nodal positions are updated according to newly obtained velocity value.

Unfortunately, in all the above-mentioned mentioned approaches, the pos-

sibility of element inversion at any non-linear iteration step introduces an

important limitation. Element inversion (negative Jacobians) leads to an

immediate failure of the implicit solver [24]. Thus, all the mentioned ap-

proaches require estimating a critical time step in order to ensure that no

element gets inverted. In practice, this may introduce a prohibitively strict

time step restrictions.

In order to alleviate the above-mentioned drawbacks of the iterative pro-

cedure involved in an implicit solution Idelsohn et al [25] proposed a scheme,

where the particles (nodes) were moved only once, prior to solving the Navier-

Stokes equations. Such mesh movement step was fully explicit. The obtained

conﬁguration was considered to be the end-of-step conﬁguration and was not

further updated. In order to ensure improved accuracy of this prediction,

the particles positions were obtained following the streamline corresponding

to velocity at the known time step. The advantage of this method was that

it allowed using large time steps without the danger of the element inversion

as the mesh was considered ﬁxed within an implicit step. Unfortunately, the

accuracy of the proposed scheme resulted to be of the ﬁrst order in time.

In the present work we strive to develop an alternative explicit-implicit

(semi-explicit) scheme, where domain conﬁguration is predicted in a more

accurate way, while the computational cost of the implicit step is further

reduced. This is achieved by using the fourth order Runge-Kutta scheme

for integrating both the velocity and the particle positions. Due to the im-

plicit nature of the pressure in incompressible ﬂows, pressure is integrated

implicitly. However, in order to reduce the computational cost of the over-

all method, we propose a methodology that requires solving the implicit

pressure equation only once per time step and not at each sub-step of the

Runge-Kutta scheme. Particular attention is given to the mesh update strat-

egy. A technique for ensuring that element inversion does not lead to time

steps limitation is proposed.

The paper is structured as follows: ﬁrst, the system of governing equa-

3

tions is presented. Then these are discretized in time and space. A combined

explicit-implicit time integration scheme is derived. Overall solution algo-

rithm is outlined, emphasizing the steps essential for minimizing the overall

computational cost. The paper concludes with two numerical examples: one

involving a problem on a domain with ﬁxed boundaries and the other one

involving a free surface ﬂow problem. The method is examined with respect

to space and time accuracy and computational cost. We note that there exist

nearly no study where temporal and spatial accuracy of a Lagrangian ﬁnite

element method for the ﬂuid problems have been analyzed.

2. EQUATIONS OF MOTION

2.1. Governing equations at continuum level

Let Ωtdenote a domain containing a viscous incompressible ﬂuid deﬁned

by the particles position xi=χ(Xi, t)|d

i=1 in the Cartesian spatial coordinates

at time t(dis the spatial dimension), where Xis the position vector in the

reference conﬁguration. The knowledge of the deformation map χfor all

the particles completely deﬁnes the motion and deformation of a continuum

body. The evolution of the velocity v=v(x, t), the pressure p=p(x, t)

and the position of a given material particle is governed by the following

equations:

Dx

Dt =v(1)

Dv

Dt =−1

ρ∇p+1

ρ∇ · 2µ∇v+∇Tv

2+b(2)

∇ · v= 0 (3)

where Dφ/Dt represents the total or material time derivative of a function

φ,µis the ﬂuid viscosity, ρis the density, and bis the body force.

At any time tEqs. (2) and (3) can be solved on the whole volume Ωt

occupied by the material body, specifying the surface forces ¯

tat certain sur-

faces Γσof the boundary Γ (Neumann boundary condition) of the body and

prescribing velocity ¯

vin the remaining parts Γvof the material’s boundary

(Dirichlet boundary condition):

−∇p·n+∇ · 2µ∇v+∇Tv

2·n=¯

ton Γσ(4)

4

v=¯

von Γv(5)

In the following section, the temporal and spatial discretization will be

presented.

2.2. Temporal integration

In order to integrate the governing equations of the incompressible ﬂow in

time, fully implicit schemes are the most commonly used. Thus, a non-linear

system given by the momentum and continuity equations is solved iteratively

at every time step providing the increments of the velocity and the pressure.

While some authors apply coupled velocity-pressure (monolithic solvers) [11],

most often, algorithms like fractional-step (pressure segregation)[26, 27] are

used to decouple the pressure from the velocity making the solution procedure

less computationally expensive. This method is used in several Lagrangian

ﬂuid models, such as e.g. [28, 19, 29, 30]. Alternatively, quasi-incompressible

approaches that introduce artiﬁcial compressibility into the ﬂuid are also

employed [21, 20, 31]. These approaches also allow decoupling the velocity

and the pressure solution steps.

Fully explicit methods for incompressible ﬂows cannot be eﬃciently used

due to the implicit nature of the pressure (a fully explicit method would

be governed by the acoustic pressure scale introducing prohibitively small

time step restrictions). However, recently explicit approaches with artiﬁ-

cial compressibility have been proposed leading to somewhat improved time

step estimates [32]. However, the stable time steps are directly dependent

on the artiﬁcial compressibility constant and thus for obtaining reasonably

large time step estimates one would need to strongly violate incompressibility

constraint.

Combined explicit-implicit (semi-explicit) methods that rely on integrat-

ing velocity explicitly while treating the pressure implicitly have been pro-

posed by several authors in the framework of ﬁxed grid (Eulerian) approaches.

These methods usually use multi-step Runge-Kutta schemes for the velocity

integration. In ﬁnite volumes context this was followed e.g. in [33]. In ﬁnite

diﬀerence framework this was done e.g. in [34] where a combination of 3rd-

order Runge-Kutta scheme for the convective term with a Crank-Nicholson

integration for the viscous term was proposed. In [35] a 4th-order Runge-

Kutta scheme equipped with solution of pressure Poisson’s equation at each

sub-step was developed. The above-mentioned semi-explicit ﬁnite volume

5

and ﬁnite diﬀerences approaches rely on computing the pressure at every

sub-step of the multi-step scheme, which is a computationally intensive op-

tion. An idea of reducing the number of implicit steps of an explicit-implicit

scheme has been proposed in [36] for a ﬁnite diﬀerence model. Similar idea

was followed in the for the ﬁxed-grid ﬁnite element models in [37] and [38].

In the present work, an explicit-implicit (semi-explicit) model based on

the combination of a 4th-order Runge-Kutta with an implicit pressure inte-

gration is derived in a Lagrangian framework for the ﬁrst time. In comparison

with similar Eulerian models the additional challenges arise due to the fact

that in the Lagrangian framework the nodes move. An additional dif-

ferential equation describing the movement of the nodes must be

added. In the present approach this equation is integrated nu-

merically using the same scheme as the one used for the velocity,

namely the Runge-Kutta. Nodal positions are updated at every

intermediate step of the Runge-Kutta scheme. As a consequence

of the nodal movement, the elemental domains (and corresponding

discrete operators) are updated at each intermediate step of the

Runge-Kutta scheme. Minimization of the computational cost of the

method is achieved by deriving a strategy that requires solving the implicit

problem (pressure) only once per time step. This is done via the application

of the fractional step method. The formulation details are presented below.

For the momentum Eq.(2), the Runge-Kutta scheme yields the following

time-discrete momentum equation residual [39]:

vn+1 −vn

∆t=1

6(r1+ 2r2+ 2r3+r4) (6)

where r1,r2,r3and r4are the intermediate residuals that are computed

according to the formulae (note that the superindex βidistinguishes the value

at the corresponding sub-step i):

First residual:

r1=1

ρ−∇pn+∇ · 2µ∇vn+∇Tvn

2+ρb(7)

vβ1=vn+∆t

2r1(8)

xβ1=Xn+∆t

2vn(9)

6

Second residual:

r2=1

ρ−∇pβ1+∇ · 2µ∇vβ1+∇Tvβ1

2+ρb(10)

vβ2=vn+∆t

2r2(11)

xβ2=Xn+∆t

2vβ1(12)

Third residual:

r3=1

ρ−∇pβ2+∇ · 2µ∇vβ2+∇Tvβ2

2+ρb(13)

vβ3=vn+ ∆tr3(14)

xβ3=Xn+ ∆tvβ2(15)

Fourth residual:

r4=1

ρ−∇pβ3+∇ · 2µ∇vβ3+∇Tvβ3

2+ρb(16)

Note that it is assumed that the body force bdoes not vary in time. In

case of a variable body force one must update it at every intermediate step.

However, this does not pose any additional diﬃculty as the body force is an

input data.

After evaluating each residual one has to update the velocity and the par-

ticle positions as well as the ﬂuid pressure. Eqs.(10,13,16) contain unknown

pressures pβ1,pβ2and pβ3. Computing these pressures would require solving

the Poisson’s equation at every intermediate step, which is computationally

expensive. Thus, we propose to evaluate the intermediate residuals r2and r3

using the historical pressure value (pβ1=pβ2=pn), while using pβ3=pn+1

in the last one (Eq.(16)). Consequently, the equation for the velocity at time

vn+1 can be written as

vn+1 =vn+∆t

6(r1+ 2r2+ 2r3+ˆ

r4)−∆t

6ρ∇pn+1 (17)

7

where ˆ

r4=1

ρh∇ · 2µ∇vβ3+∇Tvβ3

2+ρbi(note that ˆ

r4contains only the

eﬀect of viscous and body forces).

Following the idea of the fractional step approach [26], the momentum

equation is split into two parts by introducing the intermediate velocity ˜

v.

Adding and subtracting ˜

vto the above equation and performing a second

order split[40], Eq.(17) can be written as

˜

v=vn+∆t

6(r1+ 2r2+ 2r3+ˆ

r4)−∆t

6ρ∇pn(18)

vn+1 =˜

v+∆t

6ρ(−∇pn+1 +∇pn) (19)

Eq.(18) will be referred to as the ”fractional momentum” and Eq.(19)

will be called the ”end-of-step momentum” (or ”projection”) equation.

The Poisson’s equation for the pressure is obtained by applying the in-

compressibility condition(Eq.(3)) to Eq.(19), leading to

∇ · vn+1 =∇ · ˜

v+∇ · ∆t

6ρ(−∇pn+1 +∇pn)(20)

2.3. Spatial discretization.

For each mesh conﬁguration (that changes not only at every time step,

but also at every sub-step) we deﬁne the approximation for the components

of the velocity ﬁeld and the pressure using the FEM shape functions as

p(x) = NT(x)p(21)

vi(x) = NT(x) vi(22)

where virepresent the i component of the velocity ﬁeld at time t and

NTis the transpose vector of the shape functions at the same time step.

Multiplying every residual equation by a test function and integrating over

the domain deﬁned by the particles position at the tn,tn+1/2,tn+1/2and tn+1

and integrating by parts the diﬀusive term, we obtain the following discrete

equations:

First step (finding the values of vand xat t=tn+∆t/2using

the residual evaluated at t=tn):

8

r1=G(1

ρ)pn−K(µ

ρ)vn+F(23)

vβ1=vn+∆t

2M−1r1(24)

xβ1=Xn+∆t

2vn(25)

Second Step (finding the values of vand xat t=tn+ ∆t/2

using the residual evaluated at t=tn+ ∆t/2):

r2=G(1

ρ)pn−K(µ

ρ)vβ1+F(26)

vβ2=vn+∆t

2M−1r2(27)

xβ2=Xn+∆t

2vβ1(28)

Third Step (finding the values of vand xat t=tn+1 using

the residual evaluated at t=tn+ ∆t/2):

r3=G(1

ρ)pβ2−K(µ

ρ)vβ2+F(29)

vβ3=vn+ ∆tM−1r3(30)

xβ3=Xn+ ∆tvβ2(31)

Fourth Step (evaluating the velocity at t=tn+1 using the

residual evaluated at t=tn+1 ):

r4=G(1

ρ)pβ3−K(µ

ρ)vβ3+F(32)

Particle’s position xat time n+ 1 can be written as

xn+1 =Xn+∆t

6(vn+ 2vβ1+ 2vβ2+vβ3) (33)

9

Having discretized every residual, the discrete version of the Eqs. to solve

can be written as

˜

v=vn+∆t

6M−1(r1+ 2r2+ 2r3+ˆ

r4) + ∆t

6M−1G(1

ρ)pn(34)

∆t

6Lpn+1 =∆t

6Lpn−Dρ˜

v(35)

vn+1 =˜

v+∆t

6M−1G(1

ρ)(pn+1 −pn) (36)

with

K(µ

ρ)=X

elem ZΩ

µ

ρ∇N∇N+∇N∇TNdΩ (37)

M=X

elem ZΩ

NNdΩ (38)

F=X

elem ZΩ

NbdΩ (39)

G(1

ρ)=X

elem ZΩ

1

ρ∇NNdΩ (40)

Dρ=X

elem ZΩ

ρN∇NdΩ (41)

L(1

ρ)=X

elem ZΩ

1

ρ∇N∇NdΩ (42)

Eq.(35) must be stabilized to avoid numerical oscillation due equal order

of approximation for the velocity and pressure. Many of the existing tech-

niques can be used (e.g., Galerkin/least-squares, various variational multi-

scale methods(ASGS,OSS) [41, 42] or ﬁnite calculus [43]). In this present

work, the ASGS method was implemented. This leads to the following sta-

bilized form of Eq. (35):

∆t

6L+Lτpn+1 =∆t

6Lpn−Dρ˜

v−Sτ

˜

v−vn

∆t+Fτ(43)

10

The operators corresponding to the stabilization terms are:

Fτ=X

elem

τZΩ

ρ∇NbdΩ (44)

Sτ=X

elem

τZΩ

ρ∇NNdΩ (45)

Lτ=X

elem

τZΩ

∇N∇NdΩ (46)

where the stabilization coeﬃcient τis deﬁned as τ=4µ

ρh2+1

∆t−1

,

where his the element size (see e.g. [39]).

We note that since the nodal positions (and thus elemental do-

mains) are updated at every sub-step of the scheme, the elemental

matrices and vectors deﬁned by Eqs. (37)-(42) must be recomputed

at each sub-step correspondingly.

3. OVERALL SOLUTION STRATEGY

Re-meshing. In Lagrangian Finite Element models the mesh follows the ﬂow

motion. When applied to ﬂuid ﬂow problems, mesh deformations are large

and therefore re-meshing may be necessary. The main challenge consists not

only in the computational cost of the re-meshing procedure, but in the fact

that when the mesh distortions are large it may happen that the time step

size must be reduced in the run-time in order to avoid element inversions. In

practice this often leads to prohibitively small time steps. Element inversion

in the context of a fully implicit method leads to an immediate failure of the

solver. On the other hand, in an explicit method the error is localized. This

means that even if an element is inverted, one may continue the simulation

provided that an error is rectiﬁed posteriorly.

Therefore to ensure that the present explicit-implicit method does not

suﬀer from the restrictions faced by the former fully implicit ones, we propose

to perform the re-meshing not at the end of the time step, but at the end

of the explicit step. This ensures that when the implicit step (solution of

the Poisson’s equation) is performed no inverted elements are present in the

mesh.

11

During the explicit step the following remedy has been implemented: in

case an element was found to undergo an inversion at one of the sub-steps of

the Runge-Kutta scheme, prior to re-meshing the velocity and the pressure

values are recomputed at all its nodes using the values of the neighboring

elements. This strategy allows working with constant time steps governed

exclusively by the stability criteria (given by Courant-Friedrich-Levy (CFL)

number) and not by the local mesh deformations.

The procedure implemented is explained in detail using an ex-

ample shown in Fig. 1. Let us consider an element surrounded by

a large red circle in Fig. 1(a). Let us consider that a large nodal

velocity (designated by an error) leads it to inversion, and results

in a conﬁguration shown in Fig. 1(b). For the sake of simplicity of

the explanation we shall consider that the other nodes undergo a

much smaller motion.

The velocity and the pressure values at the nodes of the in-

verted element (see red circles in Fig. 1(c)) are erroneous. We

propose to interpolate them from the corresponding patches. For

instance, for the node Iwe identify the patch (large blue circle in

Fig. 1(d) indicates the connectivities) and subsequently compute

the approximated value as an average of the values of the patch

nodes belonging to non-inverted elements (these nodes are distin-

guished by green circles). For example, we compute the pressure

at the node Ias follows: pI=1

6P6

i=1 pi(Arabic numbers stand for

the nodes of ”good” elements, while Roman numbers indicate the

nodes of the inverted element). The same procedure is performed

for each node of the inverted element and for each primary variable

(velocity and pressure).

Once the explicit step is over, the re-meshing is performed. Re-

meshing and boundary identiﬁcation used in the present work follow standard

procedure implemented in all former versions of the PFEM. Therefore, these

are not discussed here and the reader is referred to e.g. [20, 19, 8] for the

corresponding details. The re-meshed conﬁguration corresponding to

the above-described case is shown in Fig. 1(e).

Solution strategy. The problem to be solved can be formulated as: ”given

the nodal positions, the velocity and the pressure at time step tn, ﬁnd these

variables at tn+1. The overall solution strategy according to the method

proposed in the present work can be summarized as follows:

12

(a) Initial conﬁg. (b) Snap-through (c) Element erasing

(d) Interpolation (e) After re-meshing

Figure 1: Velocity ﬁelds at t=0.5 s

4. EXAMPLES

The present model was implemented in Kratos Multi-Physics code, an

academic Open Source software [44]. For solution of pressure Poisson’s equa-

tion conjugate gradient (CG) solver equipped with a diagonal pre-conditioner

was used. The convergence tolerance was set to 10−6.

4.1. Example with an analytic solution

This example proposed in [40] was used in several references for assess-

ing the accuracy and computational eﬃciency of incompressible ﬂow FEM

solvers. Navier-Stokes equations are solved here in a unit square domain

equipped with homogeneous boundary conditions for the velocity at all the

boundaries. The force vector corresponding to the following analytic solution

is applied:

vx(x,y, t) = f(x)f0(y)g(t) (47)

vy(x,y, t) = −f0(x)f(y)g(t)

where

f(x) = 100x2(1 −x)2(48)

g(t) = cos(4πt)e−t

13

1. Knowing the velocity v, pressure pand nodal position Xcorresponding

at time tnperform the explicit step:

•Evaluate r1using Eq.(23). Find vβ1and xβ1(Eq.(24), Eq.(25)).

•Move particles to the new position xβ1.

•Update elemental matrices and vectors according to the

new nodal positions (Eqs. 37-42).

•Evaluate r2using Eq.(26). Find vβ2and xβ2(Eq.(27),Eq.(28)).

•Move particles to the new position xβ2.

•Update elemental matrices and vectors (Eqs. 37-42).

•Evaluate r3using Eq.(29). Find vβ3and xβ3(Eq.(30), Eq.(31)).

•Move particles to the new position xβ3.

•Update elemental matrices and vectors (Eqs. 37-42).

•Find ˜

vand xn+1 solving Eq.(34) and Eq.(33), respectively.

•Move particles to the new position xn+1.

2. Check if any element was inverted at the explicit step.

•If yes: interpolate the values at the nodes of the inverted elements

from the neighbors.

•If not: go to step 3.

3. Re-mesh the ﬂuid domain (if necessary).

4. Perform the implicit step: solve the Poisson’s equation for the pressure

(Eq.(43)). Result: pn+1.

5. Correct the velocity to obtain a divergence-free solution. Result: vn+1

(Eq.(36)).

6. Go to the next time step.

Table 1: Explicit-implicit solution algorithm for the incompressible ﬂow problems simula-

tion on a moving grid.

One of the pressure solutions satisfying the problem is

p= 100x2(49)

14

(a) vxat t=0.5 s(b) vyat t=0.5 s

Figure 2: Velocity ﬁelds at t=0.5 s

We adopt it as an initial condition for the pressure. The force vector to be

prescribed at the nodes at every step is obtained by plugging Eqs. (48) and

(49) inside the momentum equation (Eq.(2)). Note that in contrast to the

typical situation of force vector being the gravity (and thus constant), the

body force vector in the present example depends both on time and space and

therefore must be recomputed at each sub-step of the Runge-Kutta scheme.

The simulation time was set to 0.5 s, the kinematic viscosity νof the

ﬂuid was 0.001 m2/s (ν=µ/ρ)and the density ρ= 1 kg/m3. The example

was solved using diﬀerent mesh resolutions and time step sizes in order to

assess its spatial and temporal accuracy. The meshes used were structured

and contained 20x20, 40x40, 100x100 and 200x200 triangular elements (cor-

responding element sizes were 0.05, 0.025, 0.01 and 0.005 m, respectively).

The time step sizes were varied from 0.0005 sto 0.0025 susing the ﬁnest

mesh (0.005 m) in order to minimize the error in space.

Spatial accuracy. Figs. 2(a) and 2(b) display the horizontal and the vertical

velocity ﬁelds at the end of the simulation (t=0.5 s).

Fig. 3(a) shows temporal evolution of the horizontal velocity vxat the

ﬁxed spatial point (x,y)=(0.75, 0.25). The solution corresponds to the mesh

of 200 ×200 and time step of 0.0005 s. Numerical and analytic solutions are

superimposed and one can see that these are coincident. Fig. 3(b) provides

an insight of the accuracy in terms of convection of the particles: it shows

the trajectory of the particle located initially at (0.75, 0.25) obtained using

15

(a) Velocity evolution at ﬁxed spatial point

(0.75, 0.25)

(b) Trajectories of the particle located at

(0.75, 0.25) at t=0

Figure 3: Velocity ﬁeld and velocity evolution in the veriﬁcation example.

diﬀerent mesh sizes. One can see that for mesh size h≤0.01 the obtained

trajectory is practically indistinguishable from the analytic solution.

In order to quantify the spatial accuracy of the method the ﬁnal position

of the particle located initially at the point (x,y)=(0.75, 0.25) was recorded

using diﬀerent mesh resolutions (time step was set to 0.0005 sand the error

was measured. The ﬁnal location of this particle is shown in Fig. 4(a). In

order to assess the eﬀect of the re-meshing (nodal reconnection) the test was

executed with and without re-meshing.

One can see that as the mesh size decreases the ﬁnal location obtained

using the numerical simulation is approaching the analytic result. A simi-

lar tendency (in terms of absolute value of the error) is observed with and

without re-meshing. Important observation is that the re-meshing aﬀects the

result but does not aﬀect the rate of convergence signiﬁcantly and tends to

the same result as the mesh size diminishes. This can be also seen in Fig.

4(b), where the relative error versus mesh size is displayed in logarithmic

scale. Error was computed as

Err = p(xnum −xr)2+ (ynum −yr)2)

px2

r+ y2

r

(50)

where (xr,yr) is the location of the particle according to the reference so-

lution (analytic one, in this case) and (xnum,ynum) are the corresponding

simulation results. In both cases (with and without re-meshing) the spatial

16

(a) Final position (b) Relative error

Figure 4: Final position (t=0.5 s) of the particle initially located at (0.25, 0.75) obtained

using diﬀerent mesh resolutions.

error decreases quadratically with the mesh size h.

Time accuracy. Next the time accuracy is assessed. Fig. 5(a) shows the

ﬁnal position of the particle initially located at (0.75, 0.25) obtained on the

mesh of 200 ×200 elements using 4 diﬀerent time steps ranging from 0.0025

to 0.0005 s. Since the spatial error remains constant for a given mesh and

dominates for small time step sizes, in order to assess the time convergence

the numerical solution obtained using an very small time step (dt=0.0001)

was chosen as the reference solution. Thus, one avoids the dependence of

the result on the spatial error. Fig. 5(b) shows the relative error in the ﬁnal

position versus time step size in a logarithmic scale. One can see that the con-

vergence rate is quadratic. Fig. 6(a) shows the error in velocity computed at

the considered particle. Once again, the error rate is quadratic. In the same

ﬁgure we display the results obtained using a fully implicit second order back-

ward diﬀerentiation method (BDF2) [38, 45]. One can see that the present

approach shows nearly identical convergence in both the value and the slope.

The error convergence rates obtained indicate that the proposed explicit-

implicit scheme provides attractive convergence characteristics in spite of

the fact that the pressure is solved only once per time step. Therefore, for

the situations where time step estimates according to Courant-Friedrichs-

Levy (CFL) criterion are favorable, the proposed integration method can be

very advantageous from the point of view of computational eﬃciency in com-

parison with the commonly used fully implicit schemes such as e.g. BDF2 or

17

(a) Final position (b) Relative error

Figure 5: Final position (t=0.5 s) of the particle initially located at (0.25, 0.75) obtained

using diﬀerent time step sizes.

Crank-Nicholson method.

Comparison of the results obtained using the present method

with the ones reported in [46]. In the reference the particle posi-

tions are obtained using explicit streamline integration using the

historical velocity vnaccording to the method originally proposed

in [47]. One can see that the rate of convergence of the present

method is similar to that of the reference. However, the error

itself is considerably smaller (nearly one order of magnitude) in

case of using the present methodology. This suggests that, as ex-

pected, the domain conﬁguration prediction obtained by the ex-

plicit step using Runge-Kutta scheme is more accurate than that

of the streamline integration, which is done using only the histori-

cal velocity values.

Computational eﬃciency. Computational cost of the time integration of the

present method is governed by the cost of the implicit step, which involves

the solution of the Poisson’s equation (if no re-meshing is performed). This

is due to the fact that the explicit step is ”cheaper” than the implicit step per

se and, moreover, can be easily parallelized. Not pretending to present an

eﬃciency study we provide some indications of the computational cost of the

diﬀerent steps of the method and the speed-up obtained when parallelizing

it.

The 4 explicit steps of Runge-Kutta scheme involve assembly of global

18

(a) Versus ﬁxed-grid approach (b) Versus streamline integration

Figure 6: Relative error in the horizontal velocity at 0.5 s. Comparison of the present

approach with the former methods.

vectors (residual of the momentum equation), which involve loops over all the

elements of the model. These loops were parallelized (note that the elemental

contributions to the global vectors are independent among themselves) using

the OpenMP library.

In order to assess the computational eﬃciency a cluster composed of

homogeneous machines and Slurm resource scheduler were used. The used

cluster node was of the class Bull bullx B510 with NUMA architecture and

two processors Intel Xeon E5-2670, each one with 8 cores, 4x8 GB of RAM

per CPU and four memory channels. As ﬂoating-point operations are used

intensively, hyper-threading was disabled. All saving power options were also

disabled. At every execution, only one processor has been used in order to

avoid the use of Intels QPI, the exploitation of L3 cache and the use of the

proper NUMA node memory. To do this, all executions have been launched

using NUMActl to select sequentially the cores of the processor.

Table 2 shows the average computational times of the explicit and the im-

plicit steps obtained on two meshes (one of 6000 elements and one of 300000

elements approximately). The tests were carried out using a sequential im-

plementation and a parallel implementation. One can see that in a sequential

implementation the explicit step is approximately 2.5-3 times ”cheaper” than

the implicit one. Parallel implementation leads to a considerable speed-up,

which is, however, lower than the ideal. One can see the comparison between

the ideal and the obtained speed-up of the explicit step on Fig. 7. For ex-

ample, when 8 cores are used the observed speed-up is about the factor of

6, which is satisfactory considering that the ideal speed-up would have been

19

Simulation Expl. step Impl. step

Mesh1, sequential 0.015 s0.039 s

Mesh1, 8 cores 0.0028 s0.0076 s

Mesh2, sequential 0.75 s2.3s

Mesh2, 8 cores 0.14 s0.35 s

Table 2: Computational cost of diﬀerent solution steps of the proposed strategy.

around 8.

Figure 7: Speed-up obtained via OpenMP parallelization of the explicit step

4.2. Free surface sloshing

In this example we analyze the application of the present method to the

simulation of a free-surface ﬂuid sloshing [2]. Initial domain conﬁguration

is shown in Fig. 8(a). Homogeneous boundary condition is prescribed at

all the walls of the domain. The dimensions are: H=0.1 mand h=0.02 m.

The free surface shape is a sinusoid. The properties are: density ρ= 1000

kg/m3, gravity g=−9.8m/s2. The viscosity is ν= 0.0001 m2/s (ν=µ/ρ)

if not mentioned otherwise. A total time of 1 sis simulated. The domain

is discretized with an unstructured triangular mesh of size 0.002 m(6000

elements approximately).

Fig. 8(b) shows the evolution of the free surface at the left wall of the

container. The result obtained using present methodology is compared with

the results obtained by using an implicit solver proposed in [48]. The implicit

solver results obtained using a time step dt=0.001 sare taken as a reference

20

(a) Initial domain geometry (b) Wave height at the left wall

Figure 8: Free-surface ﬂuid sloshing example.

solution. The simulation using the present model was carried out using

dt=0.005 s. One can see a good agreement between the curves.

Fig. 9 shows the domain conﬁguration and the corresponding velocity and

pressure distribution at 3 time instances. One can see a nearly hydrostatic

pressure distribution. Maximum velocity of around 0.18 m/s is observed.

Time step size. In order to see the advantage of the strategy proposed in the

present paper the example was simulated using the present approach and

an implicit one [48], where mesh is moved at every non-linear iteration of

the solution of the momentum equation. We note that a strategy similar

to [48] is used in the majority of the Lagrangian FEM ﬂuid models such as

[19, 49, 21]. Thus, in we shall refer to this strategy as ”standard”.

Fig. 10(a) shows the time step sizes used throughout the simulation

carried out using viscosity of 0.0001 m2/s. In this case, due to relatively

high viscosity, the mesh deformations are moderate. For CFL=0.5 both the

present and the standard strategies maintained the maximum time step size.

For CFL=1 the standard implicit strategy required to diminish the time step

due to element deformation at several instances, while the method proposed

here allowed to maintain the constant time step.

Fig. 10(b) shows the time step sizes used throughout the simulation

carried out using viscosity of 0.00001 m2/s. Here the mesh deformation

becomes signiﬁcantly larger due to lower viscosity. Thus, even at CFL=0.5

the standard approach requires diminishing time steps on many occasions. At

CFL=1 the number of diminished time steps further increases. On the other

21

(a) vat t=0.055 s(b) vat t=0.22 s(c) vat t=0.305 s

(d) pat t=0.055 s(e) pat t=0.22 s(f) pat t=0.305 s

Figure 9: Free-surface sloshing: velocity and pressure distributions at diﬀerent time in-

stances.

22

(a) Viscosity ν=0.0001 m2/s (b) Viscosity ν=0.00001 m2/s

Figure 10: Used time step size in the simulation of a free-surface sloshing.

hand, the method proposed here maintains constant time step at both values

of CFL since the implicit step is always performed on a newly generated mesh,

while the mesh movement is done prior to this step explicitly.

5. SUMMARY AND CONCLUSIONS

In this paper we presented an explicit-implicit Lagrangian Finite Ele-

ment model for the simulation of incompressible ﬂow problems. The main

idea consisted in combining an explicit multi-step scheme (Runge-Kutta) for

velocity integration with an implicit integration of the pressure. We have

shown that applying the fractional step approach to the governing equa-

tions one can obtain an integration scheme that requires solving the pressure

Poisson’s equation only once per time step. The overall solution algorithm

presented here allowed working with constant time steps without the danger

of inverting an element at the implicit step (which was the main problem of

the majority of formerly proposed Lagrangian ﬁnite element ﬂuid solvers).

The numerical tests carried out revealed that the proposed method is

characterized by quadratic convergence in time and space. It was discovered

that its time accuracy is identical with that of the commonly used implicit

methods. On the other hand, the computational cost of the simulations using

our model is very attractive being governed by the cost of the solution of the

linear Poisson’s equation for the pressure, while the formerly proposed fully

implicit fractional step-based schemes additionally required costly iterative

implicit solution for the velocity. It was shown that the explicit step of the

method can be very easily parallelized using the OpenMP library.

23

Even though the method proposed here deﬁnes a very attractive version

of the PFEM for the problems where stable time step estimates are favor-

able, one must keep in mind that for practical purposes it still relies on re-

constructing of the entire mesh. This step is generally non-parallelizable and

can deﬁne a bottleneck of the model. Developing a method that optimizes

the mesh re-construction (for example, by considering local re-meshing or re-

connection, rather than re-constructing the entire mesh) deﬁnes the next step

that must be done for establishing a new generation of eﬃcient Lagrangian

ﬂuid solvers.

Acknowledgments. The authors also express their gratitude to Mr. A. Burgos

for the help with the parallelization of the explicit solver.

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