Engineering Analysis with Boundary Elements 37 (2013) 187–196

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Engineering Analysis with Boundary Elements

0955-79

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journal homepage: www.elsevier.com/locate/enganabound

A meshless solution for two dimensional density-driven groundwater flow

Karel Kovarık n, Juraj Muzık

University of Zilina, Faculty of Civil Engineering, Department of Geotechnics, Univerzitna 8215/1, 010 26 Zilina, Slovakia

a r t i c l e i n f o

Article history:

Received 3 May 2012

Accepted 18 October 2012Available online 30 November 2012

Keywords:

Solute transfer

Density-driven flow

Meshless method

Radial basis functions

97/$ - see front matter & 2012 Elsevier Ltd. A

x.doi.org/10.1016/j.enganabound.2012.10.005

esponding author. Tel.: þ421 905 659763.

ail address: kovarik@fstav.uniza.sk (K. Kovarı

a b s t r a c t

Density-driven groundwater flow is a complicated nonlinear problem in groundwater hydraulics. The local

boundary integral method is a promising meshless scheme that is used for solving several difficult

problems in different areas. This method applies the boundary integral equations to the local domain

around every node. The nodes can be randomly distributed in the domain and on the global boundary.

Therefore, this method is characterised as meshless. The unknown potentials and concentrations in all of

the nodes are approximated by interpolation to obtain a system of linear equations. Solving this system of

equations leads to the numerical solution for the main problem. In this paper, a combination of the radial

basis function interpolation and the local boundary element method is used to solve groundwater flow

problem combined with the transport of pollution, which also influences the density of groundwater.

& 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Density-driven groundwater flow appears mainly in saltwaterintrusion and geothermal processes. Some environmental problems,such as leakage from landfills, can also be influenced by changes inthe density and viscosity of groundwater. Modelling density-drivenflow problems requires a coupled groundwater flow and transportnumerical model. The coupling is realised using the state equationsthat link density and viscosity variations to pollution concentration ortemperature. This coupled problem is nonlinear; therefore, thesimulation usually requires large meshes and extensive computa-tional time, even for simulations of testing examples. Because of thehigh computational costs, most authors have focused on vertical 2Dnumerical models, although the problems are generally three-dimensional. The typical numerical methods used to solve theseproblems are based on different formulations of the finite elementmethod [1,2] or the discontinuous Galerkin method [3,4]. In thispaper, we present a meshless numerical method based on the localboundary integral element method (LBIEM) to reduce the largecomputational requirement.

The local boundary integral method (LBIEM) was introduced byZhu et al. [5]. This method is characterised as meshless becausedistributed nodal points that cover the domain are employed. Thesenodal points can be randomly distributed over the domain. Everynode is surrounded by a circular mesh of boundary elementscentred at this point. The unknown variable at this point is thenexpressed using a boundary integral equation on this local mesh. Allof these unknown variables are approximated by interpolation toobtain a system of linear equations. Solving this system of

ll rights reserved.

k).

equations leads to the numerical solution. Zhu et al. [5] used themoving least squares (MLSs) method for the interpolation, but theradial basis functions (RBFs) interpolation has also been used [6,7].

Here, the solution of the coupled groundwater flow-masstransfer problem, based on the LBIEM–RBF is presented.

2. Governing equations and local integral formulation

Density-driven groundwater flow can be written in terms of anequivalent fresh water potential (see also [3])

rS@h

@tþE @r

@tþr � ðrqÞ ¼ 0, ð1Þ

where h is the equivalent fresh water potential, E is the porosity ofthe porous medium, and r is the density of the solution. q is theDarcy velocity defined as

q¼�K rhþr�r0

r0

r z!

� �, ð2Þ

where K is the matrix of hydraulic conductivities and r0 is theinitial density of fresh water.

To simplify the groundwater flow equation (1), we used theBoussinesq approximation, i.e. density variations are neglectedand only the buoyancy term of the Darcy equation depends on thedensity [1]. The differential equation of 2D groundwater flowwith variable density is now expressed as

@

@xKx@h

@x

� �þ@

@yKy

@h

@yþr�r0

r0

� �� �¼ S

@h

@t, ð3Þ

where we denote the hydraulic conductivities ½LT�1� in x and y

directions as Kx and Ky, respectively. The solute mass conservation

Internal point

K. Kovarık, J. Muzık / Engineering Analysis with Boundary Elements 37 (2013) 187–196188

can be written in terms of the solute concentrations as

r � ðrDrC�qrCÞ ¼@ðrECÞ@t

, ð4Þ

where C is the solute concentration and D is the dispersion tensor.The flow and transport equations are coupled by a state equationlinking the density to the solute concentration. For the density,we use a linear model

r¼ r0þðrc�r0ÞC , ð5Þ

where rc is the density of injected fluid. C is the relativeconcentration defined as

C ¼C

Cmax, ð6Þ

where Cmax is the maximum mass concentration. Eq. (3) can betransformed to the following shape

@2h

@x2þ

Ky

Kx

@2h

@y2¼

S

Kx

@h

@t�

Ky

Kxr0

@r@y: ð7Þ

To transform (7) to the Poisson equation, we use the followingtransformation of coordinates

~x ¼ x ~y ¼ y

ffiffiffiffiffiffiKx

Ky

s, ð8Þ

and we obtain

@2h

@ ~x2þ@2h

@ ~y2¼

S

Kx

@h

@t�

1

r0

ffiffiffiffiffiffiKy

Kx

s@r@ ~y: ð9Þ

To solve (9) in a two-dimensional domain O with a boundary G,we apply the weighting residual principle with the Green inte-gration formula (see [8]) and we obtain the following integralform for the source point k

ckhkþ

ZG

@un

kj

@nhj dG�

ZG

un

kj

@hj

@ndG�

ZO

un

kjbj dO¼ 0 k¼ 1 . . .N: ð10Þ

Here, bj is the value of the right side of (7) at point j, and un

kj is thefundamental solution of the Laplace equation. The constant ck hasvalues from 0 to 1, where 0.5 corresponds to the point k being onthe smooth boundary.

Regular network

Irregular network

Boundary pointCorner point

Internal point

Boundarypoint

Segment

Fig. 1. Supporting points.

3. Radial basis functions interpolation

Radial basis functions (RBFs) are a powerful tool for approx-imating functions from scattered data. Due to their mesh-freenature, RBFs have received increasing attention for solving dif-ferent types of partial differential equations. The first attempt wasreported by Kansa [9]. The full implementation of the RBF methodis limited because its coefficients matrix becomes ill conditionedas the number of nodes increases. To overcome this difficulty, Shuet al. [10] developed a local RBF method, in which the approx-imation is based only on several local supporting points. This localRBF method was used for interpolation here. The unknownfunction U is approximated in a sub-domain, which forms theneighbourhood or support of a reference point i by the weightedsum of the radial basis functions and polynomials

Uðxi,yiÞ ¼Xn

j ¼ 1

ljWðrijÞþXm

j ¼ 1

wjpjðxi,yiÞ, ð11Þ

where lj and wj are the weights, WðrijÞ are the radial basisfunctions, and pj is a basis for polynomial space with degreem�1. m is the order of W, and n is the number of field nodes inthe neighbourhood of a reference point. Some of the most popularRBFs can be found in [11]. In our paper, we used multiquadricfunctions (MQs) because they are continuously differentiable and

integrable. MQ approximation is also attractive due to its expo-nential convergence rate (see [12]). These functions are defined as

WðrijÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2

ijþR2q

, ð12Þ

where rij is a distance between points i and j. R is the so-calledshape factor of the multiquadric function. Franke [13] comparedvarious RBFs and found that the multiquadric functions generallyperform better for the interpolation of 2D scattered data but thatthe optimal value of the shape factor should be used. We appliedthe formula according to Hardy [14] slightly modified to the localRBFs (see [6]) and we calculated the optimal value of the shapefactor at point i as

Ri ¼0:815

n

Xn

j ¼ 1

dj, ð13Þ

where dj is the distance between j-th local supporting point andits closest neighbour point. The optimal value of the shapeparameter according (13) is variable, and it changes from pointto point.

There are several different strategies to find local supportingpoints (see [15]). Here we used the following three strategies:

�

Prescribed number of supporting points. A support for point i

is created by the given number of closest points. This strategyworks well for points inside the area, but the boundary pointsrequire corrections (see [15]).

� Prescribed radius. The user can define the maximum distance

from point i, and a support is created by the points inside thiscircle. There could be some difficulties if the radius is toosmall; therefore, the number of supporting points is insuffi-cient for interpolation.

K. Kovarık, J. Muzık / Engineering Analysis with Boundary Elements 37 (2013) 187–196 189

�

Fig. 3. The local network around an internal point.

Segmentation. The neighbourhood of the point i is divided intoa given number of segments, and a support is created by theclosest point in every segment. This is a useful strategy,particularly for randomly placed points (Fig. 1).

The nine supporting points for point i inside of domain O(including the point i itself) were used for all regular networksand 12 segments (i.e. 13 supporting points including the point i

itself) were used for irregular networks using randomly distrib-uted points in all numerical examples in our paper (Fig. 1).

Coefficients lj and wj in Eq. (11) can be determined byrequiring (11) to be satisfied at the Ni nodes surrounding thepoint of interest. This algorithm is described in detail in [6]and [7]. Using this technique, we obtain a set of RBF shapefunctions Fj, and we can write

Uðxi,yiÞ ¼Xn

j ¼ 1

FjUj: ð14Þ

This formula is used to develop the meshless local formulationin the next section.

2.0 m

1.0

m

C=1C=0

q

C

C

z

z

=0

=0

Fig. 4. The geometry and boundary conditions of the Henry problem.

Table 1The parameters of the Henry problem.

4. Meshless local integral formulation

The area of interest O is covered by nodes on the globalboundary G and inside the area (Fig. 2). Consider a local circularsub-domain Ok with boundary Lk centred at every point k. Thissub-domain is regular around all of the internal points, but in thepoints on the global boundary, this local boundary is the inter-section of the global boundary and the local sub-domain Gk (seeFig. 3). The boundary Lk [ Gk is then divided into severalboundary elements (see Fig. 3). The value of the potential in nodek can be calculated from the potential values in the local elementsusing (10) as

ckhkþ

ZLk[Gk

@un

kj

@nhjdG�

ZLk[Gk

un

kj

@hj

@ndG�

ZOk

un

kjbj dO¼ 0: ð15Þ

To avoid the term @hj=@n, defined on the local boundary, weused a companion solution (see [5]). This solution satisfies theLaplace equation and is equal to the fundamental solution on thelocal boundary. For the 2D Laplace equation the companion

Fig. 2. Points in the global area.

Symbol Quantity Value Unit

E Porosity 0.35 –

K Hydraulic conductivity 1.00102�10�2m s�1

Dm Molecular diffusion (Henry solution) 6.6�10�6 m2 s�1

Dm Molecular diffusion (Pinder solution) 2.31�10�6 m2 s�1

q Discharge of fresh water 6.6�10�5 m2 s�1

r0 Fresh water density 1000 kg m�3

rc Solute density 1025 kg m�3

Ss Specific storage 0.0 m�1

solution is defined as

uc ¼1

2p log1

r0

� �, ð16Þ

where r0 is the radius of the local domain Ok. Eq. (15) changes to

ckhkþ

ZLk[Gk

@un

kj

@nhj dG�

ZGk

ðun

kj�uckjÞ@hj

@ndG�

ZOk

ðun

kj�uckjÞbj dO¼ 0:

ð17Þ

The potential in the local boundary element j can be nowapproximated from the potential values for the n neighbourhood

K. Kovarık, J. Muzık / Engineering Analysis with Boundary Elements 37 (2013) 187–196190

points using the RBF interpolation according to (14), and we obtain

ckhkþXn

m ¼ 1

ZLk[Gk

@un

kj

@nFjm dGhm�

Xn

m ¼ 1

ZGk

ðun

kj�uckjÞ@Fjm

@ndGhm

�Xn

m ¼ 1

ZOk

ðun

kj�uckjÞFjm dObm ¼ 0, ð18Þ

where Fjm are RBF shape functions between the local boundaryelement j and the neighbourhood point m. Eq. (18) can be formulated

1.00.50.0

1.0

0.5

Fig. 5. The velocity vectors for t

Fig. 6. The two networks used for th

0.15.00.0

1.0

0.5

Fig. 7. The solute mass concentration

in matrix form as

ckhkþXn

m ¼ 1

Hkmhm�Xn

m ¼ 1

Gkmhm�Xn

m ¼ 1

Skmbm ¼ 0, ð19Þ

where the hm denotes the values of the potential in the neighbour-hood points of the reference point k. The matrices Hkm, Gkm, and Skm

can be written as

Hkm ¼Xne

j ¼ 1

ZLkj[Gkj

@un

kj

@nFjm dG,

2.01.5

he original Henry problem.

e solution of the Henry problem.

0.25.1

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

s for the original Henry problem.

K. Kovarık, J. Muzık / Engineering Analysis with Boundary Elements 37 (2013) 187–196 191

Gkm ¼Xne

j ¼ 1

ZGkj

ðun

kj�uckjÞ@Fjm

@ndG,

Skm ¼Xne

j ¼ 1

ZOkj

ðun

kj�uckjÞFjm dO, ð20Þ

where Lkj is the j-th internal local boundary element, Gkj is thej-th local boundary element situated on the intersection of the globalboundary, and Okj is the triangular area element (Fig. 3).

A backward difference was used to approximate the timederivatives of the potential on the right side of (4). We obtain arecurrent system of equations to solve the potential at everynode. The mass transfer equations are solved using the same LBIEmethod and algorithms similar to those used for the potentialflow equation. The equations of flow and mass transfer are thencoupled by the equation of state, which makes the fluid density afunction of the mass solute fraction. The coupling scheme wasrealised by a sequential-iterative approach using the modifiedPickard algorithm according to [16]:

�

Step 1: Solve the transfer equations. � Step 2: Update the fluid flow properties r, m, Kx, and Ky. � Step 3: Solve the potential flow. � Step 4: Compute the velocities of flow. � Step 5: Test the convergence of the process.

0.15.00.0

1.0

0.5

0.1

Fig. 8. The solute mass concentrations for the

0.15.00.0

Original Henry solution

Lee, Cheng

Gotovac et al.

Present analysis

Voss and Souza

Soto Meca et al.

Fig. 9. A comparison of the isochlor C¼0.5 for the

This modified scheme converges faster than the classical Pickardalgorithm (see [16]).

5. Numerical example

Unfortunately, we cannot use the usual verification procedurebased on exact analytical solutions in this case due to the non-linear nature of the density-driven problems and we must relyonly on comparison with other numerical solutions. Therefore,the LBIEM model has been compared with two standard numer-ical test cases, the Henry saltwater intrusion problem and theElder salt convection problem.

All numerical examples were solved on a workstationequipped with two Intel Xeon E5620 CPUs and 8 GB memory.The generalised minimal residual (GMRES) method with simpleJacobi preconditioning was used to solve both systems of equa-tions (for potentials and concentrations) in every time step anditeration.

5.1. Simulation of the Henry problem

The Henry problem is one of the most popular tests used fordensity driven flow models. This is a 2D problem describingsaltwater intrusion into a confined rectangular aquifer t‘hat wasinitially saturated with freshwater. The geometry of the problem

0.25.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Pinder modification of the Henry problem.

0.25.1

original Henry problem from different authors.

Table 2The parameters of the Elder problem.

Symbol Quantity Value Unit

E Porosity 0.1 –

k Intrinsic permeability 4.845�10�13 m2

Dm Molecular diffusion 3.565�10�6 m2 s�1

m Dynamic viscosity 1�10�3 Pa s

r0 Freshwater density 1000 kg m�3

rc Solute density 1200 kg m�3

bC Coefficient of expansivity 1.20 –

Ss Specific storage 0.0 m�1

Fig. 11. The geometry and boundary conditions of the Elder problem.

0.25.10.15.00.0

Pinder and Cooper

Segol et al.

Gotovac et al.

Present analysis

Fig. 10. A comparison of the isochlor C¼0.5 for the Pinder modification from different authors.

K. Kovarık, J. Muzık / Engineering Analysis with Boundary Elements 37 (2013) 187–196192

is shown in Fig. 4. The boundary conditions for flow consist of twoimpermeable parts in the top and bottom of the aquifer. The rightvertical part of the boundary is considered the seaside boundary,and a hydrostatic pressure is defined along it. A constant inflow offreshwater into the solved area is assumed along the oppositevertical part of the boundary. Therefore, the boundary conditionsfor solute transport are quite simple. The maximum concentra-tion Cmax ¼ 1 is assigned to the seaside right vertical part of theboundary, and the freshwater condition (C¼0) is defined alongthe opposite left vertical part. The zero-flux conditions are usedon both horizontal parts of the boundary (Fig. 4). The properties ofthis problem are listed in Table 1.

In the right bottom part of the rectangular domain (where thedensity is the highest), the gradient of the hydraulic head isoriented vertically upward, and the gravitational force pointsvertically downward. These two forces generate a nearly hor-izontal flow of seawater into the aquifer. The solute densitydecreases along the bottom part of the boundary as a result ofthe influence of the freshwater flow from the left-hand side.Finally, the velocity directions are redirected back to the upperright side (Fig. 5).

The simulation was performed on 861 regularly distributednodes (41 horizontally and 21 vertically) and on 1147 randomlydistributed nodes (Fig. 6). The initial condition of the problem wasan aquifer filled by freshwater. Two different coefficients ofmolecular diffusion were used for the simulation (Table 1). Thefirst one corresponds to the Henry solution [17], and the secondone corresponds to the Pinder solution [18]. A uniform time stepof 60 s was used throughout the simulation. The simulations wererun for 400 min. The resulting isochlors are presented inFigs. 7 and 8. The iterative scheme used the modified Pickardalgorithm, and the subsequent iterations were employed until the

maximum L2 error in the concentration value for every time stepwas less than 1�10�5.

For the isochlor C¼0.5, Fig. 9 compares the simulation resultsof the original Henry problem with those of other reports[17,19–21]. The results of Voss and Souza’s simulation [22], whichused slightly different boundary condition and did not employ theBoussinesq approximation, are also included in this figure. Simi-larly, Fig. 10 compares the isochlor C¼0.5 for the Pinder mod-ification with those of different authors [18,23,20].

5.2. Simulation of the Elder problem

The original Elder problem (see [24]) represents a freegeothermal convection problem. This problem can be trans-formed to a modified one with parameters adapted to ground-water flow through porous media, where the density dependenceis caused by the concentration field (see [25]). The 2D domain is avertically oriented rectangular area filled with a homogeneousisotropic porous medium. The groundwater flow is driven only byfluid density differences. The saltwater concentration on the topof the domain increases the solute density and creates anunstable situation which leads to the down-welling of solutefrom higher concentrations and to the formation of solute finger-ing. In homogeneous isotropic media, we can determine the

010 0035

150

75

Soto Meca et al.

Present analysis, 861 nodes

Ackerer,Younes

Fig. 12. A comparison of the isochlors 0.2, 0.4 and 0.6 for the Elder problem after 20 years.

Table 3Characteristics of networks for the Elder problem.

No. of nodes Dimensions Level Type of flow

231 21�11 3.32 Down-welling

861 41�21 4.32 Up-welling

3321 81�41 5.32 Up-welling

5151 101�51 5.64 Down-welling

Table 4The computing times for the Elder problem.

No. of

nodes

CPU time

(s)

Condition

no.

Mean number of

iterations

Mean number of

GMRES iterations

231 0.848 7.0793 5.45 97.04

861 15.756 5.1459 9.19 172.09

3321 750.113 22.1037 9.45 392.00

5151 2837.054 79.5578 11.15 507.17

0

10

20

30

40

50

60

70

80

90

0

500

1000

1500

2000

2500

3000

0Number of nodes

Tim

e [s

]

Con

ditio

n nu

mbe

r

Time

Condition No.

1000 2000 3000 4000 5000 6000

Fig. 13. The CPU time and condition numbers.

K. Kovarık, J. Muzık / Engineering Analysis with Boundary Elements 37 (2013) 187–196 193

influence of free convection by the value of the Rayleigh numberRa. This number is the ratio of the buoyancy and conductiveforces. The dimensionless Rayleigh number can be defined as(see [1])

Ra¼KHbCDC

EDm, ð21Þ

where K is the hydraulic conductivity, H is the height of the modeldomain, Dm is the coefficient of molecular diffusion, DC is thesolute mass concentration difference between the top and bottomof the domain and bC is the coefficient of expansivity [1], definedas

bC ¼1

r0

@r@C¼

rc�r0

r0Cmax: ð22Þ

The minimum value of the critical Rayleigh number in rectangularareas is Racr ¼ 4p2. It is valid only if the aspect ratio of the domaina¼L/H is an integer (see [26]). In theory, simulations withRaoRacr are conductive, whereas systems with Ra4Racr exhibitconvective and unstable flow. The solved problem has the aspectratio a¼L/H¼4; therefore, the critical Rayleigh number isRacr ¼ 4p2 ¼ 39:478. The coefficients used in the solution of theproblem are presented in Table 2. The Rayleigh number calculatedfor this problem is Ra¼ 4004Racr . The problem is convective andunstable flow should occur.

Owing to symmetry, only half of the original aquifer is usuallysolved. The dimensions of the domain and the boundary condi-tions are presented in Fig. 11.

There is a problem concerning the potential boundary condi-tions because only no-flux conditions are given; therefore, asteady potential solution does not exist. Several authors (see[2]) overcame this problem by assigning a fictitious zero-headcondition in one node located in the upper left part of the aquifer.However, this artificial condition could influence the velocityfield. Another possibility for solving this problem is a transforma-tion of (3) to a stream-function form [21]. The stream function ccan be defined as

qx ¼@c@y

qy ¼�@c@x: ð23Þ

Eq. (3) for steady flow through an isotropic area is now trans-formed to [27]

@qx

@y�@qy

@x¼

K

r0

@r@x: ð24Þ

Combining Eqs. (23), (24), and (5) we obtain

@2c@x2þ@2c@y2¼ KbC

@C

@x: ð25Þ

K. Kovarık, J. Muzık / Engineering Analysis with Boundary Elements 37 (2013) 187–196194

Here, K is the hydraulic conductivity, and bC is the coefficient ofexpansivity defined in (22). The boundary conditions can now bewritten as c¼ 0 for the entire boundary, and no problems withthe steady flow solution occur.

Similar to the previous subsection, there are several publishedprofiles available for the Elder salt convection problem. Previouslypublished profiles by Soto Meca et al. [21] and Ackerer andYounes [3] are compared to the present analysis in Fig. 12.

The solutions of the Elder problem available in the literaturediffer depending on the level of numerical discretisation and ondifferent time integration schemes. Numerous published results(see e.g. [1,21,2]) discussed the existence of an up-wellingor down-welling flow in the central part of the original domain(i.e. near the axis of symmetry) after 20 years simulation time.

4 y 01

Fig. 14. The solute mass concentrations for the Elder problem, isochlors 0.2, 0.4

4 y 01

Fig. 15. The streamline patterns for the Elder problem, (a) 231

Down-welling flow has usually been found for coarse and for verydense meshes (see [28,29]). According to [28] the grid can becharacterised by its level L, defined as N¼ 22Lþ1, where N is thenumber of square elements in the grid. This level is applied to asolution that used the finite element or finite volume methods.Diersch and Kolditz [29] reported that down-welling flowoccurred when Lo4, and at increasing L an up-welling flowappeared. When L46, a central down-welling developed again.Frolkovic and De Schepper [28] reported that the central down-welling reappeared again for LZ7. To test this phenomenon, fourdifferent examples have been chosen with 231, 861, 3321, and5151 regularly distributed nodes. The regular networks enabledbetter comparisons of the influence of a density of nodes on theresulting mass concentrations.

y 02y

, and 0.6, (a) 231 nodes, (b) 861 nodes, (c) 3321 nodes, and (d) 5151 nodes.

y02y

nodes, (b) 861 nodes, (c) 3321 nodes, and (d) 5151 nodes.

010 0035

150

75

Diersch, Kolditz

Present analysis, 5151 nodes

Frolkovic, De Schepper

Fig. 16. A comparison of isochlors 0.2, 0.4 and 0.6 for the Elder problem, dense networks, after 20 years.

010 0035

150

75

0.1

0.2

0.3

0.4 0.50.

6

0.7

0.9

Fig. 17. A random network for the Elder problem, solute mass concentrations after 20 years.

K. Kovarık, J. Muzık / Engineering Analysis with Boundary Elements 37 (2013) 187–196 195

Fig. 14 shows the interpolated solute concentrations computedafter 4, 10 and 20 years for all four networks of nodes, respec-tively. The streamlines patterns for all four networks computedafter 4, 10, and 20 years are presented in Fig. 15. The computedsalinities for the dense network after 20 years are compared tothe similar results by [29,28] in Fig. 16.

The dimensions of networks, its level and the flow characteristicare presented in Table 3. The total CPU time, condition numbers forthe system of equations in the last time step, the mean number ofiterations in one time step, and the mean number of GMRESiterations in one solution are presented in Table 4. We used 1000regular time steps, and the CPU times were measured as a sum forthe entire computation. The condition numbers were calculated asL2 norm, and they change during the computation. Table 4 showsthat these numbers increase with the number of nodes and stronglyinfluence the convergence of the GMRES method, thus influencingthe resulting CPU time (Fig. 13).

The distribution of salinities confirms the previous finding, andthe CPU times show the effectiveness of this meshless method. Aswe noted above, we used a regular network of nodes to enable acomparison with previous published results. The LBIEM-RBFmethod is a truly meshless and we can use also randomlydistributed points. Fig. 17 shows a randomly distributed networkof 3097 points and the interpolated salinity concentrations after20 years.

6. Conclusions

This paper presents a possible use of the LBIEM-RBF meshlessmethod to model density-driven flow. This method appears to beeffective and useful for modelling the density-driven flow. Thisresearch is at its initial stages, and a follow-up study should focuson the modification of existing algorithms to enable distributedprocessing. Choosing suitable tools that allow parallel solving of verylarge network systems, which usually exist in practical solutions, areneeded.

Acknowledgments

This contribution is the result of the project implementation‘‘Support of the Research and Development for Centre of Excellencein Transport Engineering’’ (ITMS: 26220120031) supported by theResearch and Development Operational Programme funded by theERDF and the Scientific Grant Agency of Slovak Republic (VEGA) No.1-0789-12.

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