A 3D BEM model for liquid sloshing in baffled tanks

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng 2008; 76:1419–1433Published online 19 June 2008 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nme.2363

A 3D BEMmodel for liquid sloshing in baffled tanks

R. D. Firouz-Abadi1, H. Haddadpour1,∗,†, M. A. Noorian1 and M. Ghasemi2

1Department of Aerospace Engineering, Sharif University of Technology, Tehran 1458889694, Iran2School of Engineering, Tarbiat Modarres University, Tehran, Iran

SUMMARY

The present work aims at developing a boundary element method to determine the natural frequenciesand mode shapes of liquid sloshing in 3D baffled tanks with arbitrary geometries. Green’s theorem isused with the governing equation of potential flow and the walls and free surface boundary conditionsare applied. A zoning method is introduced to model arbitrary arrangements of baffles. By discretizingthe flow boundaries to quadrilateral elements, the boundary integral equation is formulated into a generalmatrix eigenvalue problem. The governing equations are then reduced to a more efficient form that ismerely represented in terms of the potential values of the free surface nodes, which reduces the size ofthe computational matrices considerably. The results obtained using the proposed model are verified incomparison with the literature and very good agreement is achieved. Finally, a number of example tankshaving common configurations are used to investigate the effect of baffle on sloshing frequencies andsome conclusions are outlined. Copyright q 2008 John Wiley & Sons, Ltd.

Received 10 February 2008; Accepted 8 March 2008

KEY WORDS: liquid sloshing; boundary element method; natural frequencies; baffled tank

1. INTRODUCTION

Liquid sloshing phenomenon in a tank is a field of interest and appeals to a great number ofresearchers in the field of fluid dynamics. Liquid tanks are considered as important parts ofmunicipal facilities systems, power plants, oil industry, and naval and aerospace systems. Hence,a good understanding of the physical behavior of liquid sloshing and the factors affecting thesloshing behavior of liquids is of significant importance in the process of design of such tanks.

Hydrodynamic forces exerted on the walls of the tank as a result of sloshing of the liquid insidemay damage the whole system. The damage due to such a failure can accompany rather oppressiveconsequences and much bigger costs than the exclusive value of the tank itself and what is therein. Damage to water or fuel tanks in the state of emergency following an earthquake, the effects

∗Correspondence to: H. Haddadpour, Department of Aerospace Engineering, Sharif University of Technology, Tehran1458889694, Iran.

†E-mail: Haddadpour@sharif.edu

Copyright q 2008 John Wiley & Sons, Ltd.

1420 R. D. FIROUZ-ABADI ET AL.

of liquid sloshing on satellite flight dynamics, maneuvering of a supertanker in a hurricane andmany other examples confirm the importance of the field of liquid sloshing.

A number of expansive and valuable series of research projects have been carried out in the fieldsof sloshing phenomenon modeling, evaluation of the natural frequencies and corresponding modeshapes of liquid vibrations in a tank, study on the linear and non-linear characters of the liquid flow,sloshing analysis in the case of small gravity, coupled-fields solution of the structural vibrationand the liquid sloshing, in addition to the optimization and control of sloshing characteristics.

Abramson and Silverman [1] collected what had been published concerning liquid sloshing tothe 1966 mainly dealing with aerospace applications. Furthermore, the study carried out by Ibrahim[2] may be referred to as one of the most distinguished projects in the recent years through whichhe has brought together a large deal of past research dedicated to liquid sloshing in the form of aninvaluable book. He has covered almost all of the research contents in the field at avail until itspublication date by a comprehensive review and leaving references to more than 2600 instances.

Numerical studies on the sloshing problem have a growing up interest to many researchersand industries along with the development of microprocessors and computational techniques inthe recent decades. The application of such techniques as the finite difference, finite volume,finite element, boundary element and pseudo-spectral methods has become very common in thisfield. Among all, the boundary element method (BEM) has turned out to be well popular andhas had thriving applications in the linear and non-linear simulations of liquid sloshing problems.Discretizing the boundaries instead of discretizing the entire flow field and subsequently cuttingdown on the costs of memory and time is one of the most essential reasons for preferring the useof BEM over other methods.

Nakayama and Washizu [3] used BEM for the analysis of the 2D non-linear sloshing problemand Iseki et al. [4] employed a cubic spline BEM in the study of sloshing in 3D tanks. Later, apanel method based on source distribution along the tank boundaries was implemented by Hwanget al. [5] to investigate the dynamic behavior of sloshing in a partially filled spherical tank underexternal excitation. Sen [6] presented a cubic spline boundary integral method as a solution to theliquid sloshing problem in which the inclusion of both conditions of the continuity of velocityand the continuity of potential were possible. Abe [7] used the BEM along with the weightedresiduals formulation of the pressure equation into the solution of 2D unsteady non-linear sloshingvia utilizing an adaptive meshing based on r -method. Christensen and Brunty [8] developed acomputer code for the hydrodynamic analysis of sloshing by taking the advantage of BEM. Abeand Sakuraba [9] introduced and applied an hr-adaptive meshing method for the 2D non-linearanalysis of water free surface motion. They made use of the BEM as well as the weighted residualsmethod to estimate the velocity potential and boundary conditions at the free surface. Dutta andLaha [10] employed a low-order BEM for the evaluation of free and forced sloshing of liquidin 3D tanks. Having reformulated the linearized Navier–Stokes equation into the Laplace andHelmholtz equations, Zang et al. [11] used BEM to investigate the liquid sloshing in a 2D movingtank while the viscosity effect is taken into account. Zhu and Saito [12] used a multiple domainBEM for sloshing analysis in a 2D tank with an internal structure as well as the vibration of theinternal structure in the tank. Donescu and Virgin [13] developed a solution for the simulationof large-amplitude motion of floating bodies in liquid with free surface using an implicit BEM.Gedikli and Erguven [14] presented a variational BEM for evaluating the sloshing problem usingthe Hamiltonian method and assessed the effect of baffle dimensions and position on the naturalfrequencies of a rigid cylindrical tank. Kita et al. [15] employed the Trefftz-type BEM for thesimulation of the sloshing phenomenon of the fluid in a rectangular vessel. Ning and Teng [16]

Copyright q 2008 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2008; 76:1419–1433DOI: 10.1002/nme

A 3D BEM MODEL FOR LIQUID SLOSHING IN BAFFLED TANKS 1421

used a 3D higher-order BEM in the time domain to simulate a non-linear irregular wave tank. Theyapplied a new Green function in the whole fluid field so that only the incident surface and freesurface are discretized for the integral equation. Assuming a laminar boundary layer, Jamali [17]introduced a solution to the liquid surface motion via a combinatory use of BEM and the finitedifference method and then calculated the damping rate. Chen et al. [18] employed a 3D BEMto study the sloshing behavior of cylindrical and cubic liquid tanks under harmonic and seismicexcitations.

In this paper BEM is used to develop a computational tool for the calculation of the naturalfrequencies and mode shapes of liquid sloshing in 3D baffled tanks with arbitrary geometries. Theliquid is considered to be inviscid and incompressible and the amplitude of oscillation is assumedto be small. Having followed the procedure that is described in this paper, a numerical model isobtained that is represented only in terms of the potential values of liquid free surface nodes, whichis computationally more efficient than a full boundary element model. The introduced model isverified in comparison with the literature and used to carried out an extensive investigation of theeffect of baffle on sloshing frequencies of some example liquid tanks.

2. GOVERNING EQUATIONS

In general, liquid sloshing problem is described through the use of Navier–Stokes equations,kinematic and dynamic conditions of liquid free surface, and the non-slipping condition on thewalls. However, in spite of completion, the above-described method typically faces difficultiessuch as mathematical and computational complexities. Thus, the use of more simple simulationpatterns is much appealing to most of the engineers. Based on the potential flow theory and withthe help of some simplifying assumptions, the problem of sloshing can be modeled in terms ofthe velocity potential. Assuming that the flow is inviscid and irrotational, the governing equationof sloshing of an incompressible fluid in a three dimensional tank is described by the Laplaceequation:

∇2�=0 (1)

where � is the velocity potential. The non-penetration condition results in the following boundarycondition on the tank walls:

��

�n=0 (2)

where �/�n denotes the outward normal derivative operator. The kinematic and dynamic boundaryconditions on the free surface can be expressed as follows:

��

�t+ ��

�x��

�x+ ��

�y��

�y− ��

�z= 0 (3)

��

�t+ 1

2|∇�|2+g� = const (4)

where � is the free surface elevation. Assuming small values of � and therefore small disturbancepotentials, the combination of Equations (3) and (4) can be linearized to obtain the following

equation governing the motion of liquid free surface:

��

�n=−1

�2��t2

3. DEVELOPING THE BEM FORMULATION

The velocity potential field generated by a unit source is known as the fundamental solution ofthe Laplace equation, which is expressed as

�∗ = 1

4�r(6)

where r is the distance from the source point. Applying Green’s second identity to the flow regionbounded externally by surface S and excluding the sphere of radius ε about point p (see Figure 1),one obtains∫

V−Vε

(�∇2�∗−�∗∇2�)dV =∫S(�q∗−�∗q)dS+

∫Sε

(�q∗−�∗q)dS=0 (7)

where q=��/�n and q∗ =��∗/�n. Assuming that � and q are well-behaved functions, after somemathematical simplifications the following integral equation is derived from Equation (7):

cp�p+∫S(�q∗−�∗q)dS = 0 (8)

cp = �

4�(9)

where � represents the internal spatial angle at the source point p. If the source point is located on asmooth and flat boundary, cp is equal to 0.5 and it is equal to 1.0 for the points in the flow domain.

Figure 1. Schematic of flow field and boundaries.

The boundary integral equation (8) can be solved numerically by discretizing the flow boundaryinto small quadrilateral elements. The integration is then performed over all boundary elementswhich their contributions are added together to complete the boundary integral in Equation (8),namely

ci�i −NIO∑j=1

∫S jq�∗dS j +

NIO+NW∑j=1

∫S j

�q∗dS j =0 (10)

where NIO and NW are the numbers of inflow/outflow and wall elements, respectively. The potentialand flux density at any point within each element may be approximated by using appropriateinterpolation functions among their nodal values

� =Nu (11)

q =Nq (12)

where N is a row matrix containing the element shape functions, and u and q are the vectors ofpotential and flux density values at the nodes of the element, respectively. Introducing the vectorforms given in Equations (11) and (12) into Equation (10), the resulting equation is transformedinto the local coordinate system of the boundary elements (�,�) to obtain

ci�i −NIO∑j=1

Gi jq j +NIO+NW∑

j=1Hi ju j =0 (13)

in which

Gi j =∫ 1

−1�∗N|J|d�d� (14)

Hi j =∫ 1

−1q∗N |J| d�d� (15)

where |J| is the determinant of the Jacobian matrix of the transformation from the global Cartesiansystem to the local coordinate system of the element. Using the point collocation method by theevaluation of Equation (13) at all of the boundary nodes, the following matrix equation form isobtained:

AU−BQ=0 (16)

where A and B are called the influence matrices, and U and Q are the two vectors containing thepotential and flux density values at all of the boundary element model nodes, respectively.

Let us consider (�=1,q|S =0) as an evident solution to the Laplace equation. Substituting thissolution into Equation (13) and using a simple matrix algebra show that the sum of the terms ineach rows of the A matrix must be zero. Using this interesting result, the off-diagonal terms ofthe A matrix can be calculated using conventional numerical integration schemes and each of thediagonal terms, which includes ci multipliers, can be found by summing the off-diagonal terms inthe same row and reversing the sign.

Since the flux density on the wall elements is zero, one may reform Equation (16) as follows:

A11UIO+A12UW−B11QIO = 0 (17)

A21UIO+A22UW−B21QIO = 0 (18)

where UIO and UW are two vectors involving the potential values at the inflow/outflow nodes andwall nodes, respectively, and the vector QIO includes the normal flux values at the inflow/outflownodes. Equation (18) can be solved to find UW in terms of the potential and flux density of theinflow/outflow nodes as

UW=A−122 (B21QIO−A21UIO) (19)

Introducing the above expression of UW into Equation (17) yields the following reduced forms:

AIOUIO−BIOQIO = 0 (20)

UW =KWUIO (21)

AIO =A11−A12A−122 A21 (22)

BIO =B11−A12A−122 B21 (23)

KW =A−122 B21(B11−A12A

−122 B21)

−1(A11−A12A−122 A21)−A−1

22 A21 (24)

Equation (20) provides a reduced-form boundary element model that involves only the potentialand flux density values of the inflow/outflow nodes. Furthermore, Equation (21) signifies the factthat the velocity potential on the wall boundaries statically depends on the potential values on theinflow/outflow boundaries.

3.1. Zoning method for baffled tanks

Consider a typical baffled tank as shown in Figure 2(a). Generally speaking, baffles are thin platesthat are devised in a liquid tank to increase the damping of liquid sloshing and usually cause tochange sloshing frequencies. For such baffled tanks, a boundary element analysis similar to thatpresented in Equation (16) loses its efficiency and in fact becomes inaccurate as the boundaryelements become very close together. This problem arises when we are integrating the fundamentalsolution to find the influence of the nodes of one element on the nodes of a very close element. Asthe baffle becomes thinner, the distance r becomes smaller and the fundamental solution q∗ risessharply. Consequently, the numerical integration becomes less efficient and inaccurate to computethe boundary integrals over the surface of the baffles.

Zoning a tank, such as done in Figure 2(b), resolves the problem of modeling the baffles. Thetank is divided into some individual zones in a way that there is no baffle in a zone. Each zone isa complete boundary element model that has common boundary elements with the other zones ininterface surfaces. The problem of efficiency and accuracy of numerical integration is completelyremoved when the elements facing each other on the baffles are in different zones. The influencematrices are calculated for each zone independently and then combined together to obtain theinfluence matrices of the whole system.

Figure 2. Schematic of a baffled tank divided into three non-baffled zones.

A zoning method like this includes some other valuable advantages as well. Cutting down theCPU time because of computing smaller influence matrices for individual zones instead of largeinfluence matrices for the whole model, introducing a degree of sparsity and increasing the bandinglevel of the whole system matrices are some of these advantages.

3.2. Standard eigenvalue form

For a baffled liquid tank, Equation (20) gives the boundary element model of each zone individually.However, it must be noted that for any two zones that are common in some interface surfaces, theflux density across the interfaces is negative of each other. Evaluating Equation (20) for all zonesand bringing together the reduced form boundary element models of them yield the boundaryelement model of the entire flow field in the tank. Finally, the flux density of the free surfacenodes is replaced with the expression given in Equation (5) to obtain the following system ofequations:

[A1 A2

]=0 (25)

where x is a vector whose elements are the potential values of the free surface nodes and y is avector including the potential and flux density values at all of the interface nodes. Equation (25)includes no inertia term associated with the y vector, and thus it has a number of zero eigenvaluesequal to the number of y elements. In fact, it reveals that the potential and flux density values atthe interface nodes can be directly expressed in terms of the potential values at the free surfacenodes. Using a simple matrix algebra similar to that accomplished before in Equations (16)–(24),Equation (25) can be revised as follows:

gMx+Kx= 0 (26)

y=K∗x (27)

where M and K are the sloshing mass and stiffness matrices, respectively, and are defined as

M= B1−A2A−14 B1 (28)

K= A1−A2A−14 A3 (29)

and K∗ is obtained as

K∗ = A−14 B1(B1−A2A

−14 B1)

−1(A1−A2A−14 A3)−A−1

4 A3 (30)

Equation (26) provides a general numerical model for sloshing analysis in baffled liquid tanks witharbitrary geometries. It is stated only in terms of the potential values of the free surface nodes,and thus the associated matrices will be reduced in size and the computational costs decrease.Assuming x= xei�t , Equation (26) can be simply transformed into a standard eigenvalue problem todetermine the sloshing natural frequencies � and corresponding mode shapes of velocity potentialx on the free surface. For each potential mode, Equation (27) may be used to find the velocitypotential and flux density on the interface elements, and subsequently using Equation (21) foreach zone, the velocity potential on the walls is determined. Then using the unsteady Bernoulli’sequation, the hydrodynamic pressure exerted on the walls of each zone is calculated as

p=−��UW (31)

where � is the liquid density. Furthermore, the free surface elevation at each mode is obtained asfollows:

−→� =−�

gx (32)

4. NUMERICAL RESULTS

This section provides some examples to verify the present BEM for the determination of the sloshingnatural frequencies. Subsequently, a number of liquid tanks with commonly used configurationsare used to investigate the effect of baffle on the sloshing frequencies.

4.1. Verification examples

Case 1: Upright cubic tank. The following analytical expression gives the sloshing frequencies ofan upright cubic liquid tank [1, 2]:

�2nm= g�

√m2+n2 tanh

�H√m2+n2

Figure 3. Variation of dimensionless sloshing frequencies in a cubic tank vs liquid depth: solidline, analytical; points, present BEM.

Figure 3 illustrates the variation of the lowest three sloshing frequencies against liquid depth andgives a comparison between the present BEM and the exact results.

Case 2:Cylindrical tank. For a cylindrical tank, the natural frequencies of sloshing are determinedby [1, 2]

�2mn = gmn

in which the values of mn are calculated as

dxJm(x)

∣∣∣∣x=mn

=0 (35)

Figure 4. Variation of dimensionless sloshing frequencies in a cylindrical tank vs liquid depth: solid line,analytical; points, present BEM.

Figure 5. Variation of dimensionless sloshing frequencies in a baffled cylindrical tank vs liquid depth:solid line, Reference [14], h/H =0.9; dashed line, Reference [14], h/H =0.8; and points, present BEM.

Figure 6. The geometric dimensions and boundary element mesh of the fuel tank.

Figure 7. The lowest five sloshing modes of the fuel tank.

where Jm is the mth Bessel function of the first kind. The effect of baffle dimension and positionon the natural frequencies of a rigid cylindrical tank is investigated by Gedikli and Erguven [14].Figures 4 and 5 show that how the sloshing frequencies in a cylindrical tank vary with the liquiddepth and baffle parameters. The graphs show the lowest three frequencies corresponding to thefirst circumferential harmonic 1n and make a clear comparison between the introduced model andthe literature. The good agreement between the results verifies the introduced model well.

4.2. Sloshing frequencies and modes in a fuel tank

In order to highlight the capability of the present model of sloshing in liquid tanks with arbitrarygeometries, a fuel tank is considered with the geometry drawn in Figure 6. The tank is partially filledand the fluid is 1.25m deep. Figure 6 also illustrates the boundary element mesh used to determinethe sloshing frequencies and modes. Figure 7 represents the sloshing modes concerned to thelowest five sloshing frequencies. The figure involves the free surface elevation and hydrodynamicpressure on the tank wall for the considered modes.

4.3. Supplementary examples and more results

Cubic and cylindrical liquid tanks are known as commonly used configurations in many engineeringsystems. We are interested in finding the effect of a riming baffle on the sloshing frequenciesof a cubic tank as well as a cylindrical tank. For both types of tank, the lowest three sloshingfrequencies are calculated as functions of baffle position and depicted in Figures 8 and 9. Thegraphs show a decrease in frequencies as the baffle nears the free surface, which can be explainedas the effect of the shallow liquid region that is formed on the baffle.

Figure 10 shows a typical type of baffle setup in rectangular tanks. The tank has a length-to-widthratio of b/a=2 and the liquid depth parameter is h/a=1. The variation of sloshing frequenciesagainst the baffle width e/h when d/b=0.5 is given in Figure 11. Figure 11 also shows how

0.6 0.7 0.8 0.9

Figure 8. Variation of dimensionless sloshing frequencies in the baffled cubic tank vs thebaffle position (h/a=1,e/a=0.15).

0.6 0.7 0.8 0.9

Figure 9. Variation of dimensionless sloshing frequencies in the baffled cylindrical tank vs thebaffle position (h/r =2,e/r =0.2).

Figure 10. A rectangular liquid tank with a typical type of baffle setup.

Figure 11. Variation of dimensionless sloshing frequencies in the baffled rectangular tankvs the baffle width and position.

Figure 12. Variation of dimensionless sloshing frequencies in the spherical tank vs thebaffle position (e/r =0.25,h/r =1.5).

the frequencies change vs the baffle position d/b when e/h=0.4. It is clearly realized from thegraphs that the lowest sloshing frequency has a considerable decrease as the baffle becomes wider;however, it varies a little when the position of the baffle changes. Furthermore, the results revealthat there is no remarkable changes in the higher frequencies when the baffle parameters change.

The spherical tanks are known as useful configurations having extensive applications in manyindustries. Having a thorough understanding of the baffle effect on the sloshing characteristics ofthe spherical tanks is of great interest and importance. Figure 12 illustrates the variation of lowestthree sloshing frequencies for a baffled spherical tank for various baffle positions.

The last example is a combined conical–cylindrical tank having a riming baffle in the cylin-drical section. This type of liquid tank is widely used in civil facilities as liquid reservoirs anddetermination of sloshing characteristics has a primary importance in seismic design and analysisof such tanks. Figure 13 shows that how the sloshing frequencies change when the baffle nearsthe free surface.

5. CONCLUSIONS

A numerical model has been proposed for the determination of sloshing frequencies in tanks havingarbitrary geometries. The BEM was used along the walls and free surface boundary conditions toobtain the equations governing the liquid motion. A zoning method was introduced for modelingof baffled tanks and the governing equations was reduced to a particular form that is only in termsof potential values of the free surface nodes. The major advantage of the proposed particular formis due to the reduction in the size of computational matrices, which reduces the computationalcosts. The present model was verified in comparison with the literature and its capability as ageneral tool for sloshing analysis has been demonstrated. In addition, some tanks having typicalconfigurations were used to examine the effect of baffle position on the sloshing frequencies. The

Figure 13. Variation of dimensionless sloshing frequencies in the conical–cylindrical tank vs the baffleposition (a/r =0.4,h/r =1.0,=45◦).

performed investigations confirm that the use of baffles in liquid tanks may change some sloshingfrequencies while it does not affect some other frequencies.

REFERENCES

1. Abramson HN, Silverman S. The dynamic behavior of liquids in moving containers. NASA SP-106, NASA,Washington, DC, U.S.A., 1966.

2. Ibrahim RA. Liquid Sloshing Dynamics. Cambridge University Press: Cambridge, 2005.3. Nakayama T, Washizu K. The boundary element method applied to the analysis of two-dimensional nonlinear

sloshing problems. International Journal for Numerical Methods in Engineering 1981; 17(11):1631–1646.4. Iseki T, Shinkai A, Nakatake K. Boundary element analysis of 3-dimensional sloshing problem by using cubic

spline element. Naval Architect (Japan) 1989; 166:355–362.5. Hwang JH, Kim IS, Seol YS, Lee SC, Chon YK. Numerical simulation of liquid sloshing in three dimensional

tanks. Computers and Structures 1992; 5(44):339–342.6. Sen D. A cubic-spline boundary integral method for two-dimensional free-surface flow problems. International

Journal for Numerical Methods in Engineering 1995; 38(11):1809–1830.7. Abe K. R-adaptive boundary element method for unsteady free-surface flow analysis. International Journal for

Numerical Methods in Engineering 1996; 39(16):2769–2787.8. Christensen ER. Brunty J. Launch vehicle slosh and hydroelastic loads analysis using the boundary element

method. Thirty-eighth AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference,Kissimmee, FL, 7–10 April 1997.

9. Abe K, Sakuraba S. An hr-adaptive boundary element for water free-surface problems. Engineering Analysiswith Boundary Elements 1999; 23:223–232.

10. Dutta S, Laha MK. Analysis of the small amplitude sloshing of a liquid in a rigid container of arbitrary shapeusing a low-order boundary element method. International Journal for Numerical Methods in Engineering 2000;47(44):33–48.

11. Zang Y, Xue S, Kurita S. A boundary element method and spectral analysis model for small-amplitude viscousfluid sloshing in couple with structural vibrations. International Journal for Numerical Methods in Fluids 1998;23(1):69–83.

12. Zhu R, Saito K. Multiple domain boundary element method applied to fluid motions in a tank with internalstructure. Journal of the Society of Naval Architects of Japan 2000; 188:135–141.

13. Donescu P, Virgin LN. An implicit boundary element solution with consistent linearization for free surface flowsand non-linear fluid–structure interaction of floating bodies. International Journal for Numerical Methods inEngineering 2001; 51(4):379–412.

14. Gedikli A, Erguven ME. Evaluation of sloshing problem by variational boundary element method. EngineeringAnalysis with Boundary Elements 2003; 27(9):935–943.

15. Kita E, Katsuragawab J, Kamiya N. Application of Trefftz-type boundary element method to simulation oftwo-dimensional sloshing phenomenon. Engineering Analysis with Boundary Elements 2004; 28(6):677–683.

16. Ning DZ, Teng B. Numerical simulation of fully nonlinear irregular wave tank in three dimension. InternationalJournal for Numerical Methods in Engineering 2006; 53(12):1847–1862.

17. Jamali M. BEM modeling of surface water wave motion with laminar boundary layers. Engineering Analysiswith Boundary Elements 2006; 30(1):14–21.

18. Chen YH, Hwang WS, Ko CH. Sloshing behaviors of rectangular and cylindrical liquid tanks subjected to harmonicand seismic excitations. Journal of Earthquake Engineering and Structural Dynamics 2007; 36(12):1701–1717.

A 3D BEM model for liquid sloshing in baffled tanks

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