method of fundamental solution (mfs) coupled with particle...

Scientia Iranica A (2013) 20(5), 1327{1336

Sharif University of TechnologyScientia Iranica

Transactions A: Civil Engineeringwww.scientiairanica.com

Method of Fundamental Solution (MFS) coupled withParticle Swarm Optimization (PSO) to determineoptimal phreatic line in uncon�ned seepage problem

Sh. Shahrokhabadia;1;� and A.R. Ahmadib

a. Department of Civil Engineering, Vali-e-Asr University of Rafsanjan, Rafsanjan, P.O. Box 518, Iran.b. Kerman Graduate University of Technology, Kerman, P.O. Box 76315-117, Iran.

Received 5 April 2012; received in revised form 14 December 2012; accepted 19 February 2013

KEYWORDSMesh-free;Particle SwarmOptimization (PSO);Method ofFundamental Solution(MFS);Radial Basis Function(RBF).

Abstract. In this research, Method of Fundamental Solution (MFS) is coupled withParticle Swarm Optimization (PSO) technique to determine the optimal phreatic linein uncon�ned seepage problems. To model the uncon�ned boundary (phreatic line), aformulation with oating geometry is derived. Regarding the use of fundamental solutionof the Laplace equation, expressed in the Radial Basis Functions (RBF), a boundary typeof the mesh-free method can be established. In this research, an objective function, basedon principle of minimum potential energy, is formed to control the position of uncon�nedboundary. MFS and PSO are utilized simultaneously to �t the phreatic line, using 4thdegree polynomials, satisfying the ow continuity and energy principle. E�ciency andaccuracy of the proposed method are veri�ed through examples. The obtained results arein a good agreement with other numerical and experimental models.c 2013 Sharif University of Technology. All rights reserved.

1. Introduction

Seepage is one of the most important phenomena ingeotechnical engineering which can be divided intocon�ned and uncon�ned problems. The con�nedproblems refer to problems with known boundaries andthe uncon�ned ones refer to problems with unknownboundaries. Determination of the phreatic line is a fun-damental step in solving uncon�ned seepage problems.Broad studies, mostly numerically-based, have beenconducted in an e�ort to simulate this phenomenonin earth dams [1-4]. Advancements in computationaltechnologies and data analysis have rendered numerical

1. Present address: Department of Civil and EnvironmentalEngineering, Mississippi State University, P.O. Box 9546,USA.

*. Corresponding author. Tel: +98 913 291 5015;Fax: +98 3913202233E-mail addresses: [email protected] (Sh.Shahrokhabadi), [email protected] (A.R. Ahmadi)

methods as valuable tools in almost all engineering�elds. Finite Element Method (FEM) and BoundaryElement Method (BEM) have been used extensively inmodeling heat and mass transfer problems, especiallyin situations with moving boundaries (e.g. phreatic linein uncon�ned seepage problem) [5-7]. In this kind ofproblems, which is a complicated and time-consumingprocess, the major di�culty is the mesh generationand/or regeneration (as strategy may require). Anexcessively re�ned mesh is usually needed to simulatesuch problems, leading to large matrices and highercomputational costs. Thus, special strategies for meshgeneration must be adopted in FEM [7,8]. On theother hand, BEM is based on solving integral equationsalong the boundaries. Singularities on the boundariesare possible which render this approach as an in exiblemethod. Furthermore, BEM produces large matricesthat makes this approach time-consuming in iterativesolution procedures.

Over the last decade, a new group of numer-

1328 Sh. Shahrokhabadi and A.R. Ahmadi/Scientia Iranica, Transactions A: Civil Engineering 20 (2013) 1327{1336

ical techniques, known as mesh-free methods, havebeen developed in computational mechanics. Mostimportantly, these methods o�er a convenient approachin solving problems with variable geometry [9-12].The basic idea in mesh-free methods is based onthe Radial Basis Function (RBF), which introducesthe relationship between two distinct nodes, useful inscattered data interpolation [13,14]. RBF is de�ned asa function that de�nes the Euclidean distance betweentwo nodes [15,16].

The most popular RBF functions are:

r2m�2 log(r); (1a)

(r2 + c2)m2 ; (1b)

e��r; (1c)

where r = kx� xjk is the Euclidean distance betweenany point, and node j in the problem domain m,c, � are constant values. Eq. (1a) is introduced asthe generalized Thin Plate Spline (TPS). Eq. (1b) isde�ned as the generalized multiquadric and Eq. (1c)is the Gaussian function. In this research, Eq. (1a)is selected as the RBF; it has been shown that bet-ter results are to be expected when value of m isone [17].

Usually, mesh-free methods are divided into twomain categories: domain type and boundary type.Since in domain type formulation any single RBF can-not satisfy the governing equations, obtaining a viablesolution would require a large number of collocationpoints for both domain and boundary of the problem.On the other hand, boundary type formulation requirescollocation nodes on the boundary of the problemonly. In boundary type problems, fundamental solutionof the governing linear di�erential equation can beselected as the RBF solution space [18]. This ap-proach automatically satis�es the governing equation.Moreover, source nodes are set outside the problemdomain, which contains no singularities, and only a fewcollocation points are needed on the boundary to solvethe problem [19].

Method of Fundamental Solution (MFS) is usedin solving uncon�ned seepage problem. However,the ow continuity over the downstream slope is notconsidered [20].

In this research, the governing equation andboundary conditions of the problem are explained,followed by a brief description on MFS and RBF. Thenthe Particle Swarm Optimization (PSO) algorithm isbrie y explained, and it is applied using a 4th degreepolynomial as a phreatic line. At the end, numericalresults are shown and compared to other numericalmethods and indoor testing.

2. Governing equation and boundaryconditions

Uncon�ned seepage problems can be considered aslaminar steady state ow in a homogenous x�y-plane.These assumption leads to Laplace equation as theproblem's mathematical model [21].

@2�@x2 +

@2�@y2 = 0; (2)

where �(x; y) is the total energy.Based on energy principle, the total energy on

both domain and boundary of the problem is:

� = EH + PH; (3)

where EH is the elevation head, and PH is the pressurehead at every node of the problem's boundary.

Boundary conditions, regarding satisfaction ofthe ow continuity, in the computational domain arede�ned as [22] (see Figure 1):

1. Impermeable boundaries: Flow velocity is zeroalong these segments (� = 0) or (@�@n = 0), theseboundaries de�ne streamlines ( 1).

2. Equipotential lines: Hydrostatic pressure is appliedto these boundaries, and the total head (�) isconstant along these segments (�01, �02).

3. Phreatic line: This boundary is the upper stream-line in the problem domain and pore pressure is zeroon this surface (PH=0). In this paper, the phreaticline is modeled using a 4th degree polynomialfunction ( 2).

4. Seepage face: On this face, water seeps out of thedam and the total energy equals each node eleva-tion, and pressure head equals zero. This boundaryis neither an equipotential nor a streamline (�).

Figure 1. Flow through a 2-D homogenous dam.

Sh. Shahrokhabadi and A.R. Ahmadi/Scientia Iranica, Transactions A: Civil Engineering 20 (2013) 1327{1336 1329

Since equipotential lines and streamlines are perpen-dicular at upstream, the phreatic line is perpendicularto the upstream slope and the phreatic line must betangent to downstream slope to satisfy ow continuity.

Generally, head loss leads to ow through adam; consequently, the mathematical de�nition of headloss de�nes a path that descends along seepage ow.Eventually, total energy at (x = x1, y = �1) is constantand equal to upstream total head (� = �01).

In this research, the equation of the phreatic lineis assumed as a fourth degree polynomial, which isintroduced by Eq. (4):

P4(x) = c0 + c1x+ c2x2 + c3x3 + c4x4: (4)

Constraints depicted in Figure 1, are as follows:

P4(x) = �01jx=x1 ; (5a)

rP4(x) = � tan�1(�)jx=x1 ; (5b)

rP4(x) = tan(�)jx=x4 ; (5c)

rP4(x) < 0jx=xi (i = 1; 2; � � �m); (5d)

where r is the gradient operator ( ddx ) and m is the

number of nodes on phreatic line. Eq. (5) should beconsidered in optimization procedure as constraints ofobjective function.

The node at x5 is dependent on the node at x4to guarantee that the phreatic line is tangent to thedownstream slope:

x5 = x4 � �x;�5 = �4 + �x: tan(�); (6)

where �x is introduced as a small distance between x4and x5.

Since phreatic line is a 4th degree polynomial, itis necessary to evaluate this function at least at �venodes. Nodal coordinates are de�ned as follows:

P4(x) = �01jx1=0; (7a)

P4(x) = �2jx2= 13 (x4�x1); (7b)

P4(x) = �3jx3= 23 (x4�x1); (7c)

P4(x) = �4jx4=C��4: cot(�); (7d)

P4(x) = �5jx5=�5 : (7e)

In Eqs. (7b) to (7d), �i(i = 2 � 4) is determinediteratively. �0i is a constant value and �5 depends on�4 in solution steps. C is a geometry value (toe ofthe dam) which is de�ned in Figure 1. Based on the

third boundary condition (phreatic line), an objectivefunction is de�ned as:

f =mXi=1

Z(P4(xi)� �(xi; yi))2; (8)

where m is the number of nodes on phreatic line.The main purpose of this research is to introduce

a function to satisfy the imposed constraints (Eq. (5))while minimizing the goal function given by Eq. (8).

3. Method of Fundamental Solution (MFS)

In mesh-free methods, the total energy during the nthiteration step is a linear combination of N Radial BasisFunctions (RBFs) [23].

�(n)(xi; yi) =nXi=1

�(n)i qi(x; y); (9)

where qi(x; y) is an RBF with its source node locatedat (xi; yi), and �(n)

i is the RBF weight.Fundamental solution of Laplace operator is se-

lected as RBF, that is:

qi(x; y) = ln(di); (10)

where di is Euclidean distance from a boundary nodeon the computational boundary to ith RBF center(source node) [19]. Source nodes are set outside ofthe computational domain. Thus, the solution formsatis�es the governing equation automatically even atRBF centers. The main process is to �nd �(n)

i , which isaccomplished by solving a system of equations formedfrom the boundary conditions.

Partial derivatives of the total energy de�ne the ow velocity, i.e.:�

@�@x

�n=

NXi=1

�(n)i@qi@x

;

�@�@y

�n=

NXi=1

�(n)i@qi@y

: (11)

There are N unknowns in each iteration (�(n)i , i =

1; 2; � � �N). The constraint conditions yield the follow-ing system of simultaneous equations to solve:264�1;1 � � � �1;N

... �i;j...

�N;1 � � � �N;N

37524�1:�N

35 =

24 b1:bN

35 ; (12)

where �i;j = qj(xi; yi) and bi = �(xi; yi). Solution ofEq. (12) yields RBF weights that can be used alongwith Eq. (9) to compute the total energy at everyboundary point.


4. Particle Swarm Optimization (PSO)

Considering the non-linear objective function, whichis in the form of multi criterion, will lead to solvenonlinear simultaneous equations. On the other hand,the objective function must be solved through freederivative optimization methods, because in the cal-culation procedure, only the numerical value of theobjective function can be calculated [8]. Therefore,application of an optimization procedure which is notdependent on objective function derivative is useful forthese kinds of problems. The proposed algorithm foroptimization procedure is Particle Swarm Optimization(PSO).

There are several evolutionary methods that canbe applied to solve the optimization problems. Amongthe available solution techniques, PSO is proved to berobust, e�ective and easy to apply [24]. This methodis based on a very simple framework and can be usedeasily with primitive mathematical operators. Thismethod does not need much computer memory, whilethe speed of computation is relatively fast [25].

Swarm behavior of bird societies has inspireda simple and highly e�ective optimization algorithmby Eberhart and Kennedy [26]. PSO includes a setof individuals for which their knowledge is improvediteratively in the search space. The position andvelocity of the individuals de�ne what is known as par-ticles. PSO has no crossover or transformation betweenparticles. Moreover, particles are never substitutedby other individuals during the run. Conceptually,in the search domain, PSO tries to attract particleswith high probability for �tness [27]. Each particlehas a memory function which allows it to rememberthe best position it has experienced thus far. Also,the best global position is visited by the entire swarm.The information obtained thus far is divided into twoparts; the �rst part includes the particle's memory ofits past state and the second part entails memory ofthe society. PSO has a �tness evaluation measure thattakes each particle's position, and returns a �tnessvalue for it. Global Best is de�ned as the maximum�tness value that swarm has met. In addition, LocalBest is de�ned as the maximum �tness value thateach particle has individually experienced; hence, eachparticle remembers both the Global and the Local best.

The canonical PSO has been widely used inengineering and science [28-30]. Now, position (xi)and velocity (vi) for each particle form the set of allparticles with population size of M (i = 1; 2; � � � ;M)are de�ned as:vi(t+ 1) =wvi(t) + c1r1(t)(pbesti(t)� xi(t))

+ c2r2(t)(gbest(t)� xi(t)); (13)

xi(t+ 1) = xi(t) + vi(t+ 1); (14)

where c1 and c2 are acceleration factors (constantvalues); t shows the tth iteration; r1 and r2 are weightsgenerated randomly with values between 0 and 1; w isan inertia weight that controls the impact of velocityfrom previous iteration on newly computed velocity;gbest is the Global Best and pbest is the Local Best.

Generally, PSO algorithm includes the following�ve main steps:

1. Randomly generation of an initial position vector xand the related velocity vector v for all particles inthe population set.

2. Evaluating �tness value of each particle.

3. Comparing the �tness value of each particle to itslocal best.

If the current value is better than the previousone, then it is replaced with the new value. Whenindividual particle's �tness measure is compared tothat of population's best, i.e. Global best, if thecurrent value is better than the previous one, it isreplaced as Global Best.

4. Changing velocity and position, using Eqs. (13) and(14).

In conventional methods, usually a �xed num-ber of trials are carried out with the minimum valuefrom all the trails taken as the global best. In thecurrent study, the limitation of conventional PSOis solved based on termination proposal, introducedby Cheng et al. [25].

If gbest gets stable after N2 iterations, the al-gorithm will terminate by de�nition of the followingcriteria:

jfg � fsf j � "; (15)

where xsf and fsf mean the best solution in thecurrent state and its related objective functionvalue. " is the tolerance of termination. PSO isnot sensitive to optimization parameters in mostproblems. It is the brilliant specialty which ranksPSO as a high recommended method in Geotechni-cal problems [31].

5. Repeating steps 2 to 4 until the described termina-tion criterion, based on Eq. (15).

In optimization method design variables, vari-able bounds, constraints and penalty function aresummarized in Table 1.

In this study, the external penalty functionmethod as one of the most common forms of thepenalty function in the structural optimization is em-ployed to transform the constrained problem into theunconstrained one, as follows [24,32,33]:

~f(x) = f(x)(1 + rppf); (16)


Table 1. Design variables, variable bounds, constraints and penalty function in this research.

Case DesignVariables

Variablebounds (m)

Penaltyfunction

Goalfunction

Constraints

Example 1 Eqs. (7b) to (7d) 0 to 24 Eq. (16) Eq. (8) Eq. (5)Example 2 Eqs. (7b) to (7d) 0 to 3.2 Eq. (16) Eq. (8) Eq. (5)Example 3 Eqs. (7b) to (7d) 0 to 0.46 Eq. (16) Eq. (8) Eq. (5)

where ~f(x) and rp are modi�ed function (�tness func-tion) and an adjusting coe�cient, in a row, and pf isthe penalization factor, which is de�ned as the sum ofall active constraints violations.

After determination of variable designs in eachiteration a 4th degree polynomial is formed, all theproduced nodes on the curve should be considered ifthey satisfy the goal function (Eq. (8)) and the relatedconstraints which are determined based on Eq. (5).

5. Numerical studies

Here, some examples are solved using the presentedmethodology in order to verify the proposed method.First, a rectangular dam is analyzed and compared toavailable results [20,34,35]. Then, a trapezoidal damis considered which is available in literature [7]. Next,the results are veri�ed by a laboratory test [36].

Solution procedure starts with generation of ran-dom initial values for �1, �2, �3 (Eq. (7)); thementioned items should satisfy geometrical conditions.Therefore, the variable bonds vary from 0 to 24, 3.2,and 0.46 m in Examples 1, 2, and 3, respectively.Thereby, forming individual set in PSO algorithm.Every individual line is solved in the seepage problem,using mesh-free concept. The obtained results foreach node on the proposed phreatic line ((�i (i =1; 2; � � � ; Ns)), where Ns is the number of nodes onphreatic line, is supplied to PSO to check the objectivefunction and the related constraints. This procedurecontinues until the stop criterion is reached. The owchart of the present method is illustrated in Fig-ure 2.

Particle size, maximum velocity, maximum par-ticle movement amplitude and maximum number ofiteration are considered as important factors in PSO,that a�ect convergence rate and hence the cost ofcomputation. In this research, the best values ofmentioned parameters are achieved through severalruns (Table 2).

MFS is based on the concept of RBF; hence the

Table 2. PSO parameters used in solution procedure.

Particlesize(n)

Max.velocity

Max.particlemove

N1 N2

30 0.5 0.009 500 200

Figure 2. Flow chart for the particle swarm optimizationand method of fundamental solution.

Table 3. MFS parameters used in this solution procedure.

ExampleSource

point-boundarypoint distance (cm)

Nodedistribution

distance (cm)1 20 502 30 50

distance between source nodes and boundary nodesplays an important role in progression of the solution.Table 3 lists the parameters used in mesh-free analysis.

Geometry of the rectangular dam is as follows:

1. Upstream water level (total energy-upstream) is24 m. Thus �01 = �1 = 24 m.

2. The length of the dam is 16 m.3. Downstream water level (total energy-downstream)

is 4 m. Thus �02 = 4 m.


Figure 3. Instances of individual in PSO algorithm tosolve Example 1. a) Individual 1; b) Individual 2; and c)Individual 3.

Some of PSO individuals, selected randomly, are shownin Figure 3.

The phreatic line, obtained from presentedmethod is illustrated and compared with the previousstudies (see Figure 4).

The node where the boundary condition number(3) and (4) meet is called separation node (see Fig-ure 1). Di�erent separation nodes which are the resultsof various methods are listed in Table 4.

Geometry of the trapezoidal dam is as follows:

1. Upstream water level (total energy-upstream) is3.2 m. Thus �01 = �1 = 3:2 m.

Figure 4. Comparison of phreatic line from previousstudies and present method.

Table 4. Separation point achieved in di�erent methods.

Method Height (m)

MFS 12.88FDM 12.79FEM Not presentedBEM 12.68MFS+PSO (presented method) 12.86

Table 5. Separation point coordinates achieved indi�erent methods.

Method X (m) Y (m)

Ouria & Tou�gh (2009) 3.248 1.64MFS+PSO (presented method) 3.382 1.537

2. The length of the dam is 9.6 m.

3. Downstream water level (total energy-downstream)is zero. Thus �02 = 0.

Some of the PSO individuals, selected randomly, areshown in Figure 5.

The Phreatic line obtained from the presentmethod is illustrated and compared to the resultsreported by other researchers (see Figure 6) [7]. Table 5lists the separation node.

The outcomes of research are compared with theresults of seepage ow through earth dam which is con-ducted in laboratory. The layout of the experimentalfacility is re-drawn in Figure 7. The dam is compactedwith nature sedimentation with natural density of sand.The porosity is estimated to be 0.2, and the saturatedhydraulic conductivity (K) is determined in the steady-state ow case, with K = 3:5e�4 m

s . The pressure inthe dam is measured by piezometric tubes [36]. Theresults are illustrated and compared with experimentaldata in Figure 8.

Note that the results obtained from the present


Figure 5. Instances of individual in PSO algorithm to solve Example 2. a) Individual 1; b) Individual 2; c) Individual 3;and d) Individual 4.

Figure 6. Resulted phreatic line from present methodand �nite element method.

Figure 7. Redrew layout of experimental facility [36].

method are exible and can accurately model theuncon�ned boundary, which renders a larger range ofselection for individuals (phreatic line) as comparedto previous studies; furthermore, the computed resultssatisfy ow continuity.

In this research, only boundary of the problem is

Figure 8. Resulted phreatic line from present methodand indoor testing.

modeled, and hence the number of collocation nodesand source nodes are much less than similar solutionprocedures that are based on domain type formula-tions, whether they are mesh-free or �nite elementmethods. Totally, all the mentioned e�orts translate totime saving in both modeling and computing. It shouldalso be noted that the present method is sensitive to thedistance between source nodes and collocation nodes;hence suitable results are achieved through trial anderror in node positioning.

6. Discussion

The advantage of current method is the usage of a


fully mesh-free method, dealing only with the oatingboundary of the problem. In optimization procedure,each individual proposes a di�erent curve which shouldbe analyzed. Ouria and Tou�gh used an optimizationprocedure which was based on Finite Element Method(FEM) analysis, but the procedure needs a specialstrategy to establish a convergent method, because theregularity of generated meshes has profound e�ectson convergence of the results [7]. A study doneby Shahrokhabadi and Tou�gh introduced the useof Natural Element Method (NEM) instead of FEM.Despite much less sensitivity on mesh generation,the mentioned procedure should consider the problemdomain to be tessellated by Delaunay triangles. Ac-cordingly, the problem domain should be analyzed aswell, which is not the aim in �nding optimum phreaticline. Moreover, Delaunay tessellation is a kind of meshgeneration in this approach. Subsequently, it is not afully mesh-free method [37].

The recent study proposes a method which usesonly the boundaries of the problem, and by using asimple algorithm, results in the same achievements inmuch less time-consuming process. The method alsoconsiders the ow continuity which was not consideredin the previous studies. Specially, Figure 4 shows thatthe previous studies did not consider ow continuity,which means that phreatic line should be tangent ondownstream slope. It could not easily be obtained be-cause the slope of rectangular dam is �

2 at downstream(in the case of rectangular dam). However, introducingsuitable constraints in optimization procedure of thepresented method ful�lls this goal.

Eventually, the proposed method bene�ts fromMethod of Fundamental Solution (MFS). This leadsto the analysis of all proposed individuals (curves)by PSO. In the similar way, if FEM was used, somerandomly individuals which led to inappropriate meshgeneration would cause diverge. Therefore, this kindof individuals should be omitted prior to analysis,but in the current method, all individuals can beanalyzed.

Regarding the 15 times of running the algorithm,the stability of the proposed method is observed inTable 6. It is noteworthy that the stability of thepresented method is acceptable based on the shownstandard deviation.

Table 6. Stability of presented method with respect toworst, mean, best, and standard deviation.

Case Worst Mean Best Standarddeviation

Example 1 1.65 0.7984 0.039 0.5389

Example 2 1.2 0.1447 0.0029 0.3275

Example 3 0.1892 0.0269 0.0003 0.0704

7. Conclusion

Here, the Method of Fundamental Solution (MFS) iscoupled with the particle swarm algorithm to simulateuncon�ned seepage problem. Fundamental solution ofLaplace operator satis�es the governing equation. Inaddition, the entire source nodes are placed outsidethe computational domain, and thus singularities donot appear. The computational procedure is theboundary type of Mesh-free methods. The meshgeneration is not an issue here; in addition, only fewcollocation points are needed along the boundary withassociated source nodes to model and solve the seepageproblem successfully without numerical integration, allof which translates into time saving in modeling andcomputations, as well as low requirement on computingresources.

Particle Swarm Optimization (PSO) is adapted to�nd the best possible 4th degree polynomial, therebyminimizing the error function and satisfying ow con-tinuity constraints.

PSO is an optimization procedure that does notrequire the computation of error function's derivative.Simplicity of PSO, as well as MFS, leads to the theintroduction of a new robust procedure that is not lim-ited by the mesh generation or the usual optimizationissues.

Accuracy of the present method is con�rmed viasolving a standard rectangular homogeneous dam thathas been studied by other researchers. A trapezoidaldam is also investigated and accurate results have beenobtained with satisfaction of ow continuity constrains,and the accuracy of the present method is also evalu-ated through experimental data, which is available inother researches.

References

1. Harr, M.E., Groundwater and Seepage, Dover Publica-tions, Dover, US, pp. 64-73 (1962).

2. Mishra, G. and Singh, A. \Seepage through a Levee",Int. J. of Geomech., 74(5), pp. 74-79 (2005).

3. Numerov, S. \Solution of problem of seepage withoutsurface of seepage and without evaporation or in�ltra-tion of water from free surface", NUG, 25 (1942).

4. Pavlovsky, N. \Seepage through earth dams", Instit.Gidrotekhniki i Melioratsii, Leningrad, Translated byUS Corps of Engineers (1931).

5. Blyth, M. and Pozrikidis, C. \A comparative studyof the boundary and �nite element methods for theHelmholtz equation in two dimensions", Eng. Anal.Bound. Elem., 31(1), pp. 35-49 (2007).

6. Long, S.Y., Kuai, C., Chen, J. and Brebbia, C.\Boundary element analysis for axisymmetric heatconduction and thermal stress in steady state", Eng.Anal. Bound. Elem., 12(4), pp. 293-303 (1993).


7. Ouria, A. and Tou�gh, M.M. \Application of Nelder-Mead simplex method for uncon�ned seepage prob-lems", Appl. Math. Model., 33(9), pp. 3589-3598(2009).

8. Wu, N.J., Tsay, T.K. and Young, D. \Meshless numer-ical simulation for fully nonlinear water waves", Int. J.Numer. Meth. Fluids., 50(2), pp. 219-234 (2006).

9. Chen, J.S., Wang, D. and Dong, S.B. \An extendedmeshfree method for boundary value problems", Com-put. Method. Appl. Mech. Engrg., 193, pp. 1085-1103(2004).

10. Gu, Y.T. and Zhang, L. \Coupling of the meshfree and�nite element methods for determination of the cracktip �elds", Eng. Fract. Mech. 2008, 75(5), pp. 986-1004(2004).

11. Kim, Y., Kim, D.W., Jun, S. and Lee, J.H. \Mesh-free point collocation method for the stream-vorticityformulation of 2D incompressible Navier-Stokes equa-tions", Comput. Methods. Appl. Mech. Engrg., 196(33-34), pp. 3095-3109 (2007).

12. Liu, G.R. \Mesh free methods: moving beyond the�nite element method", CRC (2003).

13. Franke, R. \Scattered data interpolation: Tests ofsome methods", Math. Comput., 38(157), pp. 181-200,American Mathematical Society, US (1982).

14. Hardy, R.L. \Multiquadric equations of topographyand other irregular surfaces", J. Geophys. Res., 76(8),pp. 1905-1915 (1971).

15. Chen, W. and Tanaka, M. \A meshless, integration-free, and boundary-only RBF technique* 1", Comput.Math. Appl., 43(3-5), pp. 379-391 (2001).

16. Ling, L. and Kansa, E.J. \Preconditioning for radialbasis functions with domain decomposition methods",Math. Comput. Model, 40(13), pp. 1413-1427 (2004).

17. Power, V. \A comparison analysis between unsym-metric and symmetric radial basis function collocationmethods for the numerical solution of partial di�er-ential equations", Comput. Math. Appl., 43(3-5), pp.551-583 (2002).

18. Golberg, M., The Method of Fundamental Solutions forPotential, Helmholtz and Di�usion Problems, Comput.Mech., Boston, pp. 103-176 (1999).

19. Wu, N.J. and Tsay, T.K. \Applicability of the methodof fundamental solutions to 3-D wave-body interactionwith fully nonlinear free surface", J. Eng. Math.,63(1), pp. 61-78 (2009).

20. Chaiyo, K., Chantasiriwan, S. and Rattanadecho, P.\The method of fundamental solutions for solving freeboundary saturated seepage problem", Int. Commun.Heat. Mass., 38(12), pp. 249-254 (2010).

21. Das, B.M., Advanced Soil Mechanics, Taylor & Francise-Library, pp. 210-212, Taylor & Francis, New York,USA (2008).

22. Connor, J.J. and Brebbia, C.A., Finite ElementTechniques for Fluid Flow, pp. 215-216, Newnes-Butterworths, London, UK (1976).

23. Golberg, M.A. \The method of fundamental solutionsfor Poisson's equation", Eng. Anal. Bound. Elem.,16(3), pp. 205-213 (1995).

24. Salajegheh, E., Salajegheh, J., Seyedpoor, S. andKhatibinia, M. \Optimal design of geometrically non-linear space trusses using an adaptive neuro-fuzzyinference system", Scientia Iranica, Transactions A:Civil Engineering, 16(5), pp. 403-14 (2009).

25. Cheng, Y., Li, L., Chi, S. and Wei, W. \Particle swarmoptimization algorithm for the location of the criticalnon-circular failure surface in two-dimensional slopestability analysis", Comput. Geotech., 34(2), pp. 92-103 (2007).

26. Kennedy, J. and Eberhart, R. \Particle swarm opti-mization", Neural Networks Conf., IEEE Int., Wash-ington, DC, USA, 4, pp. 1942-1948 (1995).

27. Assareh, E., Behrang, M., Assari, M. and Ghan-barzadeh, A. \Application of PSO (particle swarmoptimization) and GA (genetic algorithm) techniqueson demand estimation of oil in Iran", Energy, 35(12),pp. 5223-5229 (2010).

28. Brits, R. Engelbrecht, A. and Van den Bergh, F.\Locating multiple optima using particle swarm op-timization", Appl. Math. Comput., 189(2), pp. 1859-1883 (2007).

29. Pan, H., Wang, L., and Liu, B. \Particle swarmoptimization for function optimization in noisy envi-ronment", Appl. Math. Comput., 181(2), pp. 908-919(2006).

30. Yang, I. \Performing complex project crashing analysiswith aid of particle swarm optimization algorithm",Int. J. Proj. Manag., 25(6), pp. 637-646 (2007).

31. Cheng, Y., Li, L. and Chi, S. \Performance studieson six heuristic global optimization methods in thelocation of critical slip surface", Comput. Geotech.,34(6), pp. 462-484 (2007).

32. Rajeev, S. and Krishnamoorthy, C. \Discrete opti-mization of structures using genetic algorithms", J.Struct. Eng-ASCE., 118(5), pp. 1233-1250 (1992).

33. Khatibinia, M., Salajegheh, E., Salajegheh, J. andFadaee, M. \Reliability-based design optimization ofreinforced concrete structures including soil-structureinteraction using a discrete gravitational search algo-rithm and a proposed metamodel", Eng. Optimiz., pp.1147-65 (2013).

34. Chen, J.T., Hsiao, C.C., Chiu, Y.P. and Lee, Y.T.\Study of free surface seepage problems using hy-persingular equations"', Commun. Numer. Meth. En.,23(8), pp. 755-769 (2007).

35. Westbrook, D. \Analysis of inequality and residual ow procedures and an iterative scheme for free surfaceseepage", Int. J. Numer. Method. Engrg., 21(10), pp.1791-1802 (1985).


36. Jun, F.U. and JIN, S. \A study on unsteady seepage ow through dam", J. Hydrodyn., 21(4), pp. 499-504(2009).

37. Shahrokhabadi, S. and Tou�gh, M. \The solutionof uncon�ned seepage problem using natural elementmethod (NEM) coupled with genetic algorithm (GA)",Appl. Math. Model., 37(5), pp. 2775-2786 (2013).

Biographies

Shahriar Shahrokhabadi received his MSc fromShahid Bahonar University of Kerman, Iran, in 2009.He was a faculty member in Civil Engineering De-partment of Vali-e-Asr University of Rafsanjan. He

is currently doing his PhD program in MississippiState University, Mississippi, USA. He has over 7years experience in both computational mechanics. Hismajor research includes mesh-free methods, especiallyNatural Element Method (NEM).

Ali Reza Ahmadi is a Research Associate Professorat International Center of Science, High Technology& environmental Science. He received his PhD inMechanical Engineering from University of Kansas in2003. He received his D.E. in Civil Engineering fromUniversity of Kansas in 1988. His areas of research in-cludes computational mechanics, computational math-ematics and Finite Element Method (FEM).

method of fundamental solution (mfs) coupled with particle...

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