Evaluation of sloshing problem by variational boundary element method

Engineering Analysis with Boundary Elements 27 (2003) 935–943 www.elsevier.com/locate/enganabound

Evaluation of sloshing problem by variational boundary element method A. Gedikli*, M.E. Ergu¨ven Civil Engineering Faculty, Technical University of Istanbul, 34469 Maslak, Istanbul, Turkey Received 22 February 2002; revised 26 February 2003; accepted 26 February 2003

Abstract In this study, a Variational Boundary Element Method (VBEM) based on the Hamilton’s principle that produces symmetric matrices is presented. The effect of a rigid baffle on the natural frequencies of the liquid in a cylindrical rigid container is then investigated by the use of VBEM. Boundary integral equations are regularized by moving the singular points outside the domain. Fluid motion is assumed to be irrotational, incompressible and inviscid. Linearized free surface conditions are taken into consideration. q 2003 Elsevier Ltd. All rights reserved. Keywords: Variational method; Boundary element method; Hamilton’s principle; Cylindrical container; Baffle; Vibration; Sloshing

1. Introduction Phenomenon of earthquake damages on nuclear power plants, petroleum tanks, etc. creates an important research area in seismic analysis of liquid storage tanks [1,2]. Behaviour of liquid in rigid tanks was examined by some authors during past so many decades [3,4]. After that, studies on Fluid-Structure Interaction (FSI) problems were presented assuming liquid container as an elastic shell [5 – 7]. There are some papers, in which the different seismic analysis methods are compared, on partially liquid filled or empty containers [8 – 10]. The Finite Element Methods (FEM) [11], the Boundary Element Methods (BEM) [12] and the representation of fluid behaviour by series of generalized functions [13] can be presented as some of the methods in handling wet structures. Some authors have made use of mass-spring model of the liquid, [14 –18] while others tend to solve the potential problems [19 – 21]. The method of perturbation expansions is also used for calculation of the water waves [22]. A semianalytical method using Fourier series and FEM was studied by Ref. [23]. Numerical inversion of transformation methods [24], Panel Method based on the Boundary Integral Technique [25], FEM [26] and Galerkin’s approach [10] were used to solve the liquid sloshing in the cylindrical tanks, that are elastic or rigid, with the linear or non-linear * Corresponding author. 0955-7997/03/$ - see front matter q 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0955-7997(03)00046-8

free surface conditions. Compressibility of the liquid is considered for the analysis of dams [27,28]. The BEM can be used to evaluate the natural frequencies and the natural modes of the liquid. The technique of superposition of the modes was then used for the seismic analyses [4,19]. Passive systems can be configured with the additional structural elements, such as mass-damper-spring systems or baffles in the liquid storage tanks [29,30]. The problem of natural oscillations of an ideal fluid in cylindrical containers with annular ribs as additional structural elements was solved analytically by Trotsenko [31]. But results were given for only one annular rib. The same problem with several ribs was also solved using direct BEM by Ergu¨ven and Gedikli [32 – 36]. Unfortunately, Dutta and Laha [37] have published the problem solved in Ref. [34] by using the same solution method. A variational principle for the coupled dynamic system of a solid and a liquid field with a free surface was derived with the aid of Hamilton and Toupin’s dual principles [38]. A variationally coupled BEM – FEM was developed to analyse dynamic response, including free-surface sloshing motion of liquid in cylindrical storage tanks with/without baffles subjected to horizontal ground motion [35]. FSI analysis of 3D rectangular tanks by a variationally coupled BEM – FEM was solved by Koh et al. [39]. Non-singular symmetric boundary element formulation for elastodynamics, which is based on a generalized variational principle, was given by Gaul and Moser [40].

A. Gedikli, M.E. Ergu¨ven / Engineering Analysis with Boundary Elements 27 (2003) 935–943

936

In the presented study, the functional that is based on the Hamilton’s principle including the Lagrange multiplier derived from Ref. [41] is presented for Variational Boundary Element Method (VBEM) that produces symmetric matrices [35]. Independent variables of the functional are the hydrodynamic pressure and displacement potential field. The Lagrange multiplier is used to substitute the conservation law of momentum into the functional. Displacement potential field is defined by the use of global interpolation functions evaluated from the fundamental solution of the Laplace equation. In the presence of a baffle, the behaviour matrices of VBEM for each of the regions is obtained by separating the volume of liquid into two subregions. Later, two systems of equations are assembled by imposing the continuity conditions for the displacement potential and displacements of the neighbor points on the interface of subregions. An ideal fluid is taken into consideration, i.e. fluid is incompressible, inviscid and the motion of fluid is irrotational. A linear free surface condition in terms of kinematic and dynamic conditions is used. Foundation, container and baffle are assumed as rigid structures. A discretization for the liquid in a container being a shell of revolution is presented as an application to VBEM. If desired, it is also possible for any shape of container. The effect of a rigid baffle on the natural frequencies of the liquid in a cylindrical container is illustrated by the use of presented method. Regularized boundary integral equations is used to solve the problem.

2. Governing equations The governing equations of motion of liquid are based on the hydrodynamic pressure, p and the displacement potential, f: It is clear that the selected pair of ðp; fÞ does not allow the solutions for the hydrostatic problems. Other selections are possible for the hydrostatic/hydrodynamic problems. The selections of ðp; p0 ; fÞ; ðh; fÞ and ðp0 ; f_ Þ are some of them, where h and p0 are the elevation of free surface and hydrostatic pressure, respectively [42 – 44]. A superimposed dot represents the first derivative with respect to time. A harmonic boundary value problem can be expressed in terms of the displacement potential and hydrodynamic pressure as follows Governing equation of motion 72 f ¼ 0 in the region of the liquid; Rf

ð1Þ

where rf is the mass density of liquid and Sf is the whole of the surface of the liquid. Linear free surface condition can be given in terms of kinematic and dynamic conditions such as

f€ þ gf;z ¼ 0 on Sfs

where g is gravitational acceleration. It should be noticed that Eq. (5) is not valid for large displacements and velocities.

3. Introduction to variational boundary element method A functional based on the meaning of energy is given for the VBEM, such as for FEM. Unlike FEM, it requires global interpolation functions that are the combinations of fundamental solutions of the governing equations of motion for the liquid. A functional including independent variables of hydrodynamic pressure, p and displacement potential, f and the Lagrange multiplier, l can be given as follows [41] ( ðt2 r ð 1 ð 2 f pl ðp; f; lÞ ¼ f_ ;i f_ ;i dR 2 p dS 2grf Sfs 2 Rf t1 ) ð € þ lðp þ rf fÞdS dt ð6Þ Sfs

where a subscript after a comma corresponds to the derivation in the direction, i; and there is a summation rule on the repeated indices. The terms of the functional correspond to the kinetic energy, potential energy and the conservation of momentum, respectively. The Hamilton’s principle requires that the first variation has stationarity condition leading to Euler – Lagrange equations and vanishes, dpl ¼ 0: The first variation of the functional can be obtained by integration by parts of the related terms with respect to time and applying divergence theorem to the volume integrals in the following form " ð t2 ð dpl ðp; f; lÞ ¼ rf df{f€;ii }dR t1

f;z ¼ h on the free surface ; Sfs

ð

Sr

df{ 2 f€;n }dS

df{ 2 f€;n þ l€}dS ( ) ð p þ dp 2 þ l dS grf Sfs # ð € þ dl{p þ rf f}dS dt þrf

Sfs

Sfs

ð2Þ

"

þ 2 rf

ð3Þ þrf

Dynamic condition p þ rf f€ ¼ 0 on Sf þ Rf

Rf

ð

þrf

Kinematic conditions

f;n ¼ 0 on the rigid surface; Sr

ð5Þ

ð4Þ

¼0

ð Sr

ð Rf

df{f_ ;ii }dR

df{f_ ;n }dS þ rf

ð

#t2   _ _ df{f;n 2 l}dS   Sfs t1

ð7Þ

A. Gedikli, M.E. Ergu¨ven / Engineering Analysis with Boundary Elements 27 (2003) 935–943

where n is outward normal to the surface. The Hamilton’s principle says that df_ ðt1 Þ and df_ ðt2 Þ are equal to zero. Volume integrals of Eq. (7) include only f; while surface integrals include the terms of p; f and l: Euler– Lagrange equations can be obtained by Eq. (7) df ! f€;ii ¼ 0 in Rf

ð8Þ

f€;n ¼ 0 on Sr

ð9Þ

l€ 2 f€;n ¼ 0 on Sfs

ð10Þ

p ¼ 0 on Sfs grf

ð11Þ

dl ! p þ rf f€ ¼ 0 on Sfs

ð12Þ

dp ! l 2

The Lagrange multiplier can be chosen from Eqs. (11) and (12) as either l ¼ f;n or l ¼ p=grf : Here, the second one is more suitable than the first one which causes non-symmetric matrices in VBEM. After this point, VBEM gets away from a FEM formulation based on Eq. (7). In order to represent a new VBEM formulation, the following equation can be given by applying divergence theory to the volume integral of Eq. (6) and taking into account that the displacement potential satisfies Eq. (8), ðt2 ð t1

f_ ;i f_ ;i dR dt

Rf

¼2

ðt2 ð t1

Sf

ff€;n dS dt þ

ð Sf

t2   _ ff;n dS t

ð13Þ

1

The functional related to VBEM is obtained by substituting Eqs. (11) and (13) into pl ; [35], ( ! ) 2 ðt2 ð rf ð 1 p pðp; fÞ ¼ 2 ff€;n dS þ þ pf€ dS dt g Sfs 2rf 2 Sf t1 ( )t2  rf ð  þ ff_ ;n dS  ð14Þ  2 Sf t1

937

4. Fundamental solution and VBEM discretization The formulation given above has no restriction on the shape of the container, i.e. it may have any shape. A cylindrical container with/without a baffle is taken into consideration as an application to the presented formulation in the rest of this study, (Fig. 1). Hydrodynamic pressure field p can be discretized over the constant elements. Unlike this, displacement potential field f should be expressed in terms of fundamental functions. The fundamental solution of the 3D potential problem is well-known expression fp ðs; QÞ ¼ 1=RsQ ; where RsQ is the distance between receiver, s and source, Q points. As the receiver coincides into source, a singularity of 1=RsQ takes place and fp ðs; QÞ is infinite. Otherwise, it has a finite value. The singularity is removed by taking the source points to the exterior of the liquid domain. However, the following question immediately arrives; where should these points be located? The answer is that there is, in fact, no restriction, if they are in different locations. The source points will be located on the line normal to the boundary (Fig. 2). The suitable distance, ds ¼ lq0i qi l; giving the most accurate result can be determined by comparing the wellknown analytical solutions with the numerical results, where q0i is the mid point of ith segment on r – z plane. The source ring over which the source is distributed is depicted to gain the efficiency of the calculations instead of a source point (Fig. 2). For the asymmetric problem, the distribution of intensity of the source throughout the ring can be defined as cðQÞ ¼ cos w; where Q is a source point on the ring. Here qi is called as ith head point. Hydrodynamic pressure field, p; can be discretized over the constant elements, because p is only required on the free surface, Sfs ; as it is seen from the variational form given by Eqs. (15) and (16). So, p can be defined as pðs; u; tÞ ¼ NT p cos u; on Sfs

ð17Þ

where N and p are the vectors of piece-wise step functions and nodal values of p defined as N ¼ {N1 ðsÞ; …; Nnfs ðsÞ} and p ¼ {p1 ðtÞ; …; pnfs ðtÞ}; respectively. s and nfs are

This functional also satisfies Euler– Lagrange equations, if f satisfies Eq. (8). The first variation, dp; can be obtained by applying the similar procedure to p as applied to pl : The following expressions can be written form dp; dropping the integrals over the time, 1 ð 1ð dpp dS þ dpf€ dS ¼ 0 grf Sfs g Sfs 2 rf

ð Sf

dff€;n dS þ

1ð dfp€ dS ¼ 0 g Sfs

ð15Þ

ð16Þ

Since Eqs. (15) and (16) include only surface integrals, they are suitable forms to be used for VBEM. This forms can be discretized over the boundary elements.

Fig. 1. Cross-section of cylindrical container. (a) Without baffle, (b) with baffle.

A. Gedikli, M.E. Ergu¨ven / Engineering Analysis with Boundary Elements 27 (2003) 935–943

938

as follows: ð2p

fp;n ðs; QÞcðQÞdw

  4 2 ¼ rq 2 2 1 2 rs nr þ ðzq 2 zs Þnz rqs k 

1 2 2  2 1 EðkÞ 2 2 KðkÞ 1 2 k2 k2 k 1  2 Gðqi ; sÞnr ð21Þ 2

G;n ðqi ; sÞ ¼

0

Substituting Eqs. (17) and (19) into Eqs. (20) and (21), and then into Eqs. (15) and (16), equations of motion can be obtained in matrix form as " #( ) " p #( ) p p€ K 0 0 Mpf ð22Þ ¼ pf T f s s€ 0 0 M 2M Fig. 2. The source ring and the head point.

the receiver point on r – z plane and the number of elements on free surface, respectively. u is the circumferential angle defined in Fig. 2. The piece-wise step functions is given as follows: ( 1; s [ Gi Ni ðsÞ ¼ ð18Þ 0; s  Gi The field of displacement potential, f; can be expressed in terms of fp ðs; QÞ; because f is required all over the domain Rf as it is seen from the variational form given by Eqs. (15) and (16). The displacement potential for the asymmetric motion of liquid can be defined as

fðs; u; tÞ ¼ GT s cos u; on Sf þ Rf

ð19Þ

where G and s are the vectors of global interpolation functions and intensity factors of source rings where and s ¼ {s1 ðtÞ; …; sn ðtÞ}; G ¼ {Gðq1 ; sÞ; …; Gðqnf ; sÞ} respectively. Here nf is the number of the rings. The global interpolation functions for the asymmetric motion can be given as the potential of the source ring on the receiver point, s Gðqi ; sÞ ¼

ð2p

fp ðs; QÞcðQÞdw

Elements of sub-matrices in Eq. (22) can be defined by notifying that dS ¼ rs ds du and eliminating cos2 u term from integrals, as follows: The mass matrix for domain ð f Mi;j ¼ rf f Gðqi ; sÞG;n ðqj ; sÞrs ds ð23Þ G

The mass matrix for free surface pf Mi;j ¼

1ð Gðqj ; sÞrs ds g Gfsi

ð24Þ

The stiffness matrix for free surface 8 1 ð > : i–j 0

ð25Þ

In Eqs. (22) – (25), Gf ; the contour of whole liquid surface, Sf ; and Gifs ; ith piece of contour of free surface, Sfs ; are both defined on the r – z plane as shown in Fig. 1,

Gf ¼ Gr þ Gfs and Gfs ¼

nfs X

Gfsi

ð26Þ

i¼1

Eliminating parameter, s; in Eq. (22), a standard form of eigensystem can be obtained as follows Mp€ ¼ Kp p

ð27Þ

0

4 ¼ rqs





2 2 2 1 KðkÞ 2 2 EðkÞ k2 k

ð20Þ

2 2 where RsQ ¼ lrs rQ l; k ¼ 4rq rs =rqs ; rqs ¼ ðrq þ rs Þ2 þ ðzq 2 2 zs Þ and ðnr ; nz Þ are direction cosines of the outward normal. KðkÞ and EðkÞ are the first and the second kind complete elliptic integrals, respectively [45,46]. There is a symmetrical property that Gðqi ; sÞ ¼ Gðs; qi Þ: Derivation of Gðqi ; sÞ with respect to outward normal can be written

where M is represented as M ¼ Mpf ðMf Þ21 ðMpf ÞT

ð28Þ

5. Baffle and subregions In the existence of a baffle, it separates liquid domain into two subregions (Fig. 3). Then Eq. (22) can be written for

A. Gedikli, M.E. Ergu¨ven / Engineering Analysis with Boundary Elements 27 (2003) 935–943

939

Fig. 3. Boundary elements, (a) without baffle; (b) with baffle.

upper subregion 2

38 9 > p€ > < > = 6 7> 6 cf 7 6 0 0 M1 7 c€ 1 4 5> > > : > ; pf T cf T f € s 1 M1 M1 2M 2 p 38 9 0 0 > K >p> > 6 7< = c 7 c ¼6 0 K 0 1 4 5> 1 > > : > ; s1 0 0 0 0

0

Mpf 1

and for the lower subregion as follows 2 3( ) " #( ) 0 Mcf Kc2 0 c2 c€ 1 4 5 2 ¼ T 0 0 s2 s€2 Mcf 2Mf2 2

ð29Þ

ð30Þ

where subscripts represent subregion numbers. Here c1 and c2 are the vectors of nodal values of hydrodynamic pressure on the contour, G c of interface of subregions. Because of ds1 ¼ 2ds2 on the interface, the following equation can be obtained from Eq. (25) Kc1 þ Kc2 ¼ 0

ð31Þ

which implies the continuity condition for the displacement. On the other hand, the field of hydrodynamic pressure is continuous at any point on the contour of interface. So, the continuity condition for p can be expressed as follows: c1 2 c2 ¼ 0

ð32Þ

Equations of motion of liquid can be obtained in matrix form by assembling the matrices given in Eqs. (29) and (30)

by using Eqs. (31) and (32) as follows 38 9 2 0 0 0 Mpf > p€ > 1 > 7> 6 > > > 7> 6 > > cf cf < = 7 6 0 € p M 0 M c 1 2 7 6 7 6 7> s€ > 6 fp fc f 0 7> 6 M1 M1 2M1 1 > > > > 5> 4 > > : ; fc f € s 2 0 2M2 0 M2 38 9 2 p K 0 0 0 > >p> > > 7> 6 > > > 6 0 0 0 0 7> < 7 pc = 6 7 6 ¼6 7 6 0 0 0 0 7> > s > > 5> 4 > 1> > > : > ; 0 0 0 0 s2

ð33Þ

where pc ¼ c1 ¼ c2 : Eliminating parameters of pc ; s1 and s2 in Eq. (33), a standard form of eigensystem can be reobtained as in Eq. (27), but the mass matrix in Eq. (27) should be evaluated in the following forms M ¼ Mpfp 2 Msub

ð34Þ

and f 21 pf T Mpfp ¼ Mpf 1 ðM1 Þ ðM1 Þ

ð35Þ

Msub ¼ Mpfc Mcfc ðMpfc ÞT

ð36Þ

f 21 cf T Mpfc ¼ Mpf 1 ðM1 Þ ðM1 Þ

Mcfc ¼

f 21 cf T ½Mcf 1 ðM1 Þ ðM1 Þ sub

ð37Þ þ

f 21 cf T 21 Mcf 2 ðM2 Þ ðM2 Þ 

ð38Þ

It is clear that M vanishes as the inner radius of baffle is vanishing. The natural frequency of the liquid should decrease for the smaller inner radius since the stiffness matrix remains unchanged. This is a well-known situation from the literature [29 – 31, 36,47].

A. Gedikli, M.E. Ergu¨ven / Engineering Analysis with Boundary Elements 27 (2003) 935–943

940

equation:

Table 1 Convergence test by number of elements n

J 01 ðln Þ ¼ 0

Frequencies (Hz) f1

f2

f3

2 4 6 8 10 20 40 60 80

0.583 0.579 0.577 0.577 0.577 0.576 0.576 0.576 0.576

1.348 1.204 1.172 1.161 1.155 1.148 1.146 1.146 1.145

– 1.666 1.555 1.512 1.492 1.465 1.458 1.457 1.456

Exact

0.576

1.145

1.456

In particular, l1 ¼ 1.8412, l2 ¼ 5.3314, l3 ¼ 8.5363, l4 ¼ 11.7060, etc. 6.1. Convergence tests Three factors should be tested to assure that the presented method tends to the exact solution. These are the boundary element mesh, the numerical integration quadrature and the distance of the head point to the boundary. In the examples for convergence test, the height of the liquid is taken as H ¼ 1/2 m. In the first example, it is investigated how the approximation depends on the boundary element mesh. The number of elements per unit length is taken as n: Frequencies corresponding to the first three modes are given in Table 1 as a convergency test for meshing. In the second example, it is investigated how the approximation depends on the numerical integration quadrature. The number of elements per unit length is taken as n ¼ 10. Frequencies corresponding to all of the available modes are given in Table 2 as a convergency test for nG : In the last example of convergency tests, it is investigated how the approximation depends on the distance of the head point to the boundary. The number of elements per unit length is taken as n ¼ 10. Frequencies corresponding to all of the available modes are given in Table 3 as a convergency test for the source distance. It is seen from the first two tables given above that the number of segments are more dominant than nG : A better result converging to the exact solution can only be obtained by using the improved mesh of boundary element, but it takes much more time for calculation. When a large value of ds is used, the systems of equations given by Eqs. (28), (35), (37) and (38) are ill-conditioned and a precision more than double is required for numerical processes. On the contrary, a small value of ds yields mistakes in numerical integrations. It is seen from the last table that the distance of the head point from the boundary

H ¼ 1/2 m, R ¼ 1 m, n ¼ number of elements per unit length.

Proceeding in a similar assembling process as given above, the solution can be obtained for a number of baffles.

6. Numerical examples In the following examples, the radius and the height of cylindrical container are taken as R ¼ H ¼ 1 m, unless they are not given. The distance between the head point and the segment is equal to the length of the segment, ds ¼ ls and all segments have the same length. Gauss – Legendre quadrature with seven points is used for numerical integrations, nG ¼ 7, [45]. Gravitational acceleration is g ¼ 9.807 m/s2. An analytical expression for the nth sloshing frequency of liquid in cylindrical containers is given in the literature, Aslam [20], as

v2n ¼



g H ln tanh ln R R

ð40Þ

ð39Þ

in which the values of ln ; corresponding to the first circumferential harmonic, are obtained from the following

Table 2 Convergence test by number of points of Gauss–Legendre integration nG

Frequencies (Hz) f1

f2

f3

f4

f5

f6

f7

f8

f9

f10

1 3 5 7

0.592 0.577 0.577 0.577

1.155 1.155 1.155 1.155

1.465 1.493 1.492 1.492

1.715 1.790 1.790 1.790

1.933 2.081 2.080 2.080

2.125 2.376 2.374 2.374

2.301 2.675 2.670 2.670

2.476 2.946 2.935 2.935

2.614 3.115 3.091 3.091

2.725 3.234 3.207 3.207

Exact

0.576

1.1545

1.456

1.705

1.922

2.116

2.293

2.458

2.612

2.757

H ¼ 1/2 m, R ¼ 1 m, number of elements per unit length ¼ 10.

A. Gedikli, M.E. Ergu¨ven / Engineering Analysis with Boundary Elements 27 (2003) 935–943

941

Table 3 Convergency test by distance of source to the boundary ds ¼ als

Frequencies (Hz)

a

f1

f2

f3

f4

f5

f6

f7

f8

f9

f10

0.050 0.250 0.500 1.000 1.500 2.000

0.939 0.582 0.578 0.577 0.577 0.577

1.626 1.165 1.157 1.155 1.155 1.155

2.162 1.509 1.493 1.492 1.492 1.492

2.671 1.812 1.788 1.790 1.790 1.791

3.173 2.104 2.069 2.080 2.084 2.085

3.662 2.391 2.346 2.374 2.390 2.395

4.121 2.671 2.615 2.670 2.713 2.734

4.514 2.926 2.857 2.935 3.025 3.083

4.796 3.123 3.042 3.091 3.156 3.243

5.247 3.333 3.205 3.207 3.326 3.476

Exact

0.576

1.145

1.456

1.705

1.922

2.116

2.293

2.458

2.612

2.757

H ¼ 1/2 m, R ¼ 1 m, nfs ¼ nbottom ¼ 10, nwall ¼ 5.

may be selected as, ds < ls ; by which the best results are obtained.

The natural frequencies corresponding to the first three modes are plotted for the various deepness and inner radius

of the baffle, Fig. 5. The number of elements per unit length are taken as n ¼ 10. The results of VBEM are, for the first mode, near to that of Direct BEM evaluated by Mu¨ller [47]. VBEM differs less than 0.025 Hz from Ref. [47] and 0.001 Hz from the exact solutions for the first mode. The exact solutions in the literature are available only for r ¼ 0 and r ¼ R: As in the previous example, the accuracy declines, while the order of vibration mode increases. The second example for the baffle is given to compare the natural frequencies corresponding to the first mode, for the various inner radius of the baffle and the deepness h ¼ 1/2 m, obtained by VBEM and Trotsenko [31], Fig. 6. The number of elements per unit length are taken as n ¼ 10. Additionally, the displacements of the liquid particles on the r – z plane, for the first mode, are shown in Fig. 7 using 20 elements per unit length. In the study given by Trotsenko [31], the natural frequencies havebeenanalyticallyobtainedfromthefirstapproximation,so they differ from VBEM less than 0.015 Hz. As a result, from the examples given above, a baffle in a shallow position declines the natural frequencies more than the deeper one does.

Fig. 4. The natural frequencies of the liquid in the cylindrical container. R ¼ 1 m.

Fig. 5. The natural frequencies of the liquid in the cylindrical container with the baffle, for deepness of baffle h ¼ 0.1 m and 4h ¼ 0.2 m. H ¼ R ¼ 1 m.

6.2. Free vibration in cylindrical container Natural frequencies of the liquid is plotted for the various ratios H=R in Fig. 4. Ten elements per unit length are taken for the bottom of the container and the free surface. It is taken as n ¼ Ceilingð10H=RÞ for the wall, where Ceiling is a function which returns the least integer greater than or equal to its argument. Frequencies obtained by VBEM are so close to the exact solution that the greatest difference between them is 0.001 Hz, for the first mode. VBEM differs less than 0.002 Hz from Ref. [36] and 0.001 Hz from the exact solutions for the first mode. The accuracy of the given method declines, while the order of vibration mode increases. 6.3. Free vibration in cylindrical container with a baffle

942

A. Gedikli, M.E. Ergu¨ven / Engineering Analysis with Boundary Elements 27 (2003) 935–943

As well known, the smaller natural frequency of a structure causes the smaller design forces in seismic analysis. By the way, a baffle can be used for the purpose to decrease the natural frequencies of the liquid. The applicability of the presented method to the liquid sloshing problems is validated by numerical examples. Final remarks are in order concerning some extensions including the elastic containers or large amplitude sloshing and also certain applications to the seismic design or time history analysis which will be followed by the next study. Fig. 6. The first natural frequency of the liquid in the cylindrical container with the baffle. H ¼ R ¼ 1 m, h ¼ r ¼ 1/2 m.

Fig. 7. Displacement field of the first mode. H ¼ R ¼ 1 m, h ¼ r ¼ 1/2 m. w1 ¼ 0.605 Hz.

7. Conclusion The singularities of boundary integrals are vanished by taking the source points outside the domain. It is shown from the numerical examinations that the most suitable distance between a source point and the boundary should be, for the accurate results, approximately equal to the length of the boundary segment. The most important advantage of the presented VBEM is that it produces symmetric matrices, so it can be easily applied over any method producing symmetric system of equations, such as FEM. The disadvantage is that it does not produce sparse or band matrices. Like any approximated method, VBEM gives also more accurate results for a lower order mode than it does for the higher one.

References [1] Hanson RD. Behavior of liquid storage tanks in the great Alaska earthquake of 1964. Washington DC: National Academy of Sciences; 1973. p. 331– 9. [2] Epstein HJ. Seismic design of liquid-storage tanks. J Struct Div, ASCE 1976;102:1659–73. [3] Jacobsen LS. Impulsive hydrodynamics of fluid inside a cylindrical tank and of fluid surrounding a cylindrical pier. Bull Seismol Soc Am 1949;39:189–204. [4] Graham EW, Rodriguez AM. The characteristics of fuel motion which affect airplane dynamics. J Appl Mech 1952;19:381–8. [5] Veletsos AS. Seismic effects in flexible liquid storage tanks. Proc Int Assoc Earthquake Engng Mech, Rome 1974;. [6] Veletsos AS, Tang Y. Rocking response of liquid storage tanks. J Engng Mech Div, ASCE 1987;113:774–1792. [7] Zhaolin W, Bin Q, Xuduo C. Viscoelastic plate fluid interactive vibration and liquid sloshing suppression. Acta Mech Sinica 1990;8: 273 –80. [8] Davidovici V, Haddadi A. Calcul pratique de re´servoirs en zone sismique. Annales de I’institut Technique du Batiment et des Travaux Publics 1982;409. [9] Tedesco JW, Kostem CN, Kalnins A. Free vibration analysis of cylindrical liquid storage tanks. Comput Struct 1987;26:957– 64. [10] Utsumi M, Kimura K, Sakata M. The non-stationary random vibration of an elastic circular cylindrical liquid storage tank in simulated earthquake excitation. Jpn Soc Mech Engrs 1987;30:467– 75. [11] Pinson LD, Brown CG. A finite element method for non-axisymmetric vibrations of pressurized shells of revolution partially filled with liquid. AIAA 1973;73–339. [12] Komatsu K. Hydroelastic analaysis using axisymmetric fluid finite elements, Technical Report of National Auro Space Laboratory, Japan, TR-478; 1976 [13] Housner JM, Herr RW. Vibrations of incompressible liquid–elastic tank combinations using a series representation of the liquid. Comput Meth Fluid-Struct Interact Prob, ASME 1978;AMD 26:49– 64. [14] Kana DD. Status and research needs for prediction of seismic response liquid containers. Nucl Engng Des 1982;69:205–21. [15] Housner GW. Dynamic pressures on the accelerated liquid containers. Bull Seismol Soc Am 1957;47:15–35. [16] Housner GW. The dynamic behavior of water tanks. Bull Seismol Soc Am 1963;53:381– 7. [17] Haroun MA, Housner GW. Complications in the free vibration analysis of tanks. J Engng Mech Div, ASCE EM5 1982;108:801 –18. [18] Haroun MA, Housner GW. Dynamic characteristics of liquid storage tanks. J Engng Mech Div, ASCE EM5 1982;108:783– 800. [19] Hunt B. Seismic-generated water waves in axisymmetric tanks. Part J Engng Mech, ASCE 1987;113(5):653–70. paper no. 21447. [20] Aslam M, Godden WG, Scalise DT. Earthquake sloshing in annular and cylindrical tanks. J Engng Mech Div, ASCE EM3 1979;105: 371 –89.

A. Gedikli, M.E. Ergu¨ven / Engineering Analysis with Boundary Elements 27 (2003) 935–943 [21] Hunt B, Priestley N. Seismic water waves in a storage tank. Bull Seismol Soc Am 1978;68:487–99. [22] Stoker JJ. Water waves. New York: Interscience; 1957. Sec. 2.1. [23] Wunderlich W, Springer H, Goebel W. In: Mang HA, Kuhn G, editors. Discretization and solution techniques for liquid filled shells of revolution under dynamic loading. IUTAM/IACM Symposium, Heidelberg: Springer; 1989. [24] Ujihashi S, Matsumoto H, Nakahara I, Shigeta M. Axisymmetrical impulsive response of an infinite circular cylindrical shell filled with liquid. Jpn Soc Mech Engrs 1987;30:262. see also p. 568–73. [25] Hwang JH, Kim IS, Seol YS, Lee SC, Chon YK. Numerical simulation of liquid sloshing in three dimensional tanks. Comput Struct 1992;44:339–42. [26] Aslam M. Finite element analysis of earthquake induced sloshing in axisymmetric tanks. Int J Numer Meth Engng 1981;17: 59–170. [27] Crawford C. Earthquake design loading for thin arch dams. Proc World Conf Civ Engrs, NZ 1965;. [28] Westergaard HM. Water pressures on dams during earthquakes. Trans, ASCE 1933;98:418– 33. [29] Schwind RG, Scotti RS, Jo¨rgen S. The effect of the baffles on tank sloshing. Report No. LMSC-A642961 (Contract No. NASI-4065), Lockheed Missiles and Space Co.;1964 [30] Silveria MA, Stephens DG, Leonard HW. An experimental investigation of the damping of liquid oscillations in cylindrical tanks with various baffles. NASA TN-715; 1961 [31] Trotsenko VA. On the free vibrations of liquid in a cylinder with annular ribs. Sov Appl Mech, Prikladnaya Mekhanika 1967;3(12): 73–7. [32] Ergu¨ven ME, Gedikli A. The natural oscillations of a fluid in hollow containers with additional structural elements. Proc Ninth Eur Conf Earthquake Engng, Moscow 1990;7-C:53– 60. [33] Ergu¨ven ME, Gedikli A. The natural oscillations of a fluid in cylindrical containers with additional structural elements. Proc Fifth Int Conf Bound Elem Technol, Comput Engng Bound Elem, Delaware 1990;1:23–33.

943

[34] Ergu¨ven ME, Gedikli A. The effect on the natural frequencies of the axisymmetric container-liquid system with additional structural elements. Bound Elem XIV, Seville 1992;2:467–74. [35] Gedikli A. Fluid-structure interaction using variational BE –FE methods in cylindrical tanks. PhD Thesis, Istanbul Technical University, Istanbul, Turkey; 1996 [36] Gedikli A, Ergu¨ven ME. Seismic analysis of a liquid storage tank with a baffle. Int J Sound Vib 1999;223:141– 56. [37] Dutta S, Laha MK. Analysis of the small amplitude sloshing of a liquid in a rigid container of arbitrary shape using a low-order boundary element method. Int J Numer Meth Engng 2000;47: 1633– 48. [38] Yamamato Y. A variational principle for a solid–water interaction system. Int J Engng Sci 1981;19(12):1757–63. [39] Koh HM, Kim JK, Park JH. Fluid-structure interaction analysis of 3-D rectangular tanks by a variationally coupled BEM – FEM and comparison with test results. Earthquake Engng Struct Dyn 1998; 27:109–24. [40] Gaul L, Moser F. Non-singular symmetric boundary element formulation for elastodynamics. Engng Anal Bound Elem 2001;25: 843–9. [41] Liu WK, Uras RA. Variational approach to fluid structure interaction with sloshing. Nucl Engng Des 1988;106:69 –85. [42] Olson LG, Bathe KJ. Analysis of fluid structure interactions. A direct symmetric coupled formulation based on the fluid velocity potential. Comput Struct 1985;21(1/2):21 –32. [43] Everstine GC. A symmetric potential formulation for fluid-structure interaction. Letter to the Editor. J Sound Vib 1981;79:157–60. [44] Bathe KJ, Hahn W. On transient analysis of fluid-structures systems. Comput Struct 1979;10:383 –91. [45] Abramowitz M, Stegun IA. Dover: handbook of mathematical functions. ; 1972. [46] Liggett JA, Liu PL-F. The boundary integral equation method for porous media flow. London: George Allen & Unwin; 1983. [47] Mu¨ller M. Dreidimensionale Elastodynamische Analayse von Tanks mit Fluidbenetzten Einbauten. Braunschweig: Ser Mech 1993;8.